Donaldson type invariants for algebraic surfaces: Transition of moduli stacks

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1972 Takuro Mochizu...

Author: Takuro Mochizuki

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1972

Takuro Mochizuki

Donaldson Type Invariants for Algebraic Surfaces Transition of Moduli Stacks

ABC

Takuro Mochizuki Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 Japan [email protected]

ISBN: 978-3-540-93912-2 e-ISBN: 978-3-540-93913-9 DOI: 10.1007/978-3-540-93913-9 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008943979 Mathematics Subject Classification (2000): 14D20, 14J60, 14J80 c 2009 Springer-Verlag Berlin Heidelberg ° This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper 987654321 springer.com

To my friends Yoshinobu Akahori and Shu Kawaguchi for memory of our informal student seminar

Preface

In this monograph, we define and investigate an algebro-geometric analogue of Donaldson invariants by using moduli spaces of semistable sheaves with arbitrary ranks on a polarized projective surface. We may expect the existence of interesting “universal relations among invariants”, which would be a natural generalization of the “wall-crossing formula” and the “Witten conjecture” for classical Donaldson invariants. Our goal is to obtain a weaker version of such relations, in other brief words, to describe a relation as the sum of integrals over the products of moduli spaces of objects with lower ranks. Fortunately, according to a recent excellent work of L. G¨ottsche, H. Nakajima and K. Yoshioka, [53], a wall-crossing formula for Donaldson invariants of projective surfaces can be deduced from such a weaker result in the rank two case. We hope that our work in this monograph would, at least tentatively, provides a part of foundation for the further study on such universal relations. In the rest of this preface, we would like to explain our motivation and some of important ingredients of this study. See Introduction for our actual problems and results.

Donaldson Invariants Let us briefly recall Donaldson invariants. We refer to [22] for more details and precise. We also refer to [37], [39], [51] and [53]. Let X be a compact simply connected oriented real 4-dimensional C ∞ -manifold with a Riemannian metric g. Let P be a principal SO(3)-bundle on X. A connection ∇ of P is called an anti-selfdual (ASD) connection if the curvature F (∇) satisfies ∗g F (∇) = −F (∇), where ∗g denotes the Hodge star operator associated to g. For simplicity, let us restrict ourselves to the case that P comes from a principal SU (2)-bundle. Note that P is determined by its second Chern class c2 := c2 (P ). Let b+ (X) be the number of positive eigenvalues of the intersection form on H 2 (X). Let M (c2 , g) denote the moduli space of ASD connections of P . If g is sufficiently general, itis known that M (c2 , g) is naturally a C ∞ -manifold with dimR M (c2 , g) = 8c2 − 3 1 + b+ (X) . vii

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Let d := dimR M (c2 , g)/2 and write d = l + 2m. We consider integrals X,g m Φd α1 · · · · · αl · p := μ(α1 ) ∪ · · · ∪ μ(αl ) · μ(P )m

(0.1)

M (c2 ,g)

for αi ∈ H2 (X, Q) (i = 1, . . . , l) and p ∈ H0 (X, Q). The map μ : H∗ (X, Q) −→ H ∗ M (c2 , g), Q is formally given by μ(α) = c2 (P)/α, where P is the universal principal bundle on X × M (c2 , g). Although M (c2 , g) is not compact in general, μ(αi ) are naturally extended to the cohomology classes on the Uhlenbeck compactification M (c2 , g). Moreover, let A∗ (X) be the symmetric algebra generated by H2 (X) ⊕ H0 (X), gives which is graded by giving degree 2 − i/2 to elements of Hi (X). Then, ΦX,g d a linear map Ad (X) −→ Q. Thus, we obtain a map X,g ΦX,g := Φd : A∗ (X) −→ Q. (0.2) d≥0

It is called a Donaldson invariant of X. If b+ (X) > 1, (0.1) are shown to be independent of the choice of general g. They were successfully applied in the study of low-dimensional differential topology, until the appearance of Seiberg-Witten invariants which are defined more easily and believed to contain equivalent information in most cases. Nowadays, the attention to Donaldson invariants has been rather limited. So the author should explain why he would like to study a generalization of their algebro-geometric analogue in this comparatively long monograph. Although they might be less interesting in terms of topological application, there would exist attractive problems of “universal relations among invariants” which are natural generalizations of the “wallcrossing formula” (“Kotschick-Morgan conjecture”) and the “Witten conjecture”. We remark that P. Kronheimer [76] studied such a generalization of Donaldson invariants for real 4-dimensional manifolds by using the moduli of higher rank objects from a viewpoint of differential geometry. It was also investigated in mathematical physics. (See [84], for example.) Kotschick-Morgan Conjecture and Witten Conjecture Let us recall the conjectures for an explanation of the motivation of our study, although they are beyond the scope of this monograph. We should remark that they have been studied intensively from mathematical viewpoints, by V. Y. Pidstrigach– A. N. Tyurin, B. Chen, and in particular P. Feehan–T. Leness. In the case b+ = 1, the integrals (0.1) depend on the choice of the metric g. Let us recall the result of D. Kotschick–J. Morgan in [74], following [51]. Let H 2 (X, R)+ denote the set of α ∈ H 2 (X, R) such that α2 > 0. Then, H 2 (X, R)+ /R+ has two connected components. Let us fix one component Ω + . The period point ω(g) is the point in Ω + defined by the closed two form which is harmonic with respect to g and satisfies ∗g ω(g) = ω(g). An element ξ ∈ H 2 (X, Z) is called of type d, if

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(d + 3)/4 + ξ 2 ∈ Z≥0 . For such ξ, let W ξ := L ∈ Ω + L · ξ = 0 be called a wall of type d. The chambers of type d are the connected components in the complement depends only of the walls of type d in Ω + . Kotschick and Morgan proved that ΦX,g d on the chamber in which ω(g) is contained. It is very interesting to ask what happens when ω(g) crosses a wall of type d, that is called a wall-crossing phenomenon. 1 2 X X : Ad (X) −→ Q given by ΦX,g − ΦX,g = δξ,d , We obtain the linear map δξ,d d d ξ where the chambers of ω(g1 ) and ω(g2 ) are divided with W . They conjectured X (αd ) (α ∈ H2 (X)) is the polynomial in the intersection numbers α2 and that δξ,d α · ξ whose coefficients depend only on ξ 2 , d and the homotopy type of X. Very X can be expressed interestingly, assuming the conjecture, G¨ottsche showed that δξ,d by using modular forms, which is called a wall-crossing formula. ([50]. See also [51].) Since the appearance of Seiberg-Witten invariants, it has been expected that Seiberg-Witten invariants and Donaldson invariants give equivalent information in most cases. Let χ and τ denote the Euler number and the signature of X, respectively. Let m(X) = 2 + 7χ + τ /4. Let Λ denote the set of isomorphism classes of line bundles L on X such that c1 (L)2 = 2χ + 3τ . For L ∈ Λ, let nL denote the Seiberg-Witten invariant, i.e., the number of the solutions of the Seiberg-Witten equation associated to a Spinc -structure whose determinant bundle is L. Then, Donaldson invariants and Seiberg-Witten invariants are expected to be related by the following impressive formula: nL · ec1 (L)·α ΦX (α) = 2m(X) eα·α/2 L∈Λ

(See [130] and [21] for more precise and details.) Both conjectures claim the existence of universal relations which are expressed in impressive ways. It might be desirable to understand them as a part of larger universal relations by extending the framework to the higher rank cases, and moreover we might dream to understand some geometry behind them.

Donaldson Invariants in Algebraic Geometry We recall how Donaldson invariants were discussed in algebraic geometry. Let X be a smooth projective surface with an ample line bundle OX (1). Let ω be a Kahler metric which represents the first Chern class of OX (1). Let E be a holomorphic vector bundle on X such that det(E) OX . A hermitian metric h of E is called Hermitian-Einstein with respect to ω, if Λω F (h) = 0 is satisfied, where F (h) is the curvature of the canonical unitary connection ∇h of (E, h), and Λω denotes the adjoint of the multiplication of ω. Let P be the principal SU (2)-bundle associated to E with a hermitian metric h. Then, the induced connection ∇h on P is ASD, if and only if h is Hermitian-Einstein. Therefore, considering ASD-connections is equivalent to considering holomorphic vector bundles with Hermitian-Einstein metrics.

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Recall a very deep correspondence between objects in differential geometry and algebraic geometry, so called the Kobayashi-Hitchin correspondence. (See [18], [19], [128].) Let E be a holomorphic vector bundle on X such that det(E) OX . Then, E is called μ-stable if and only if degω (F ) := X c1 (F ) · ω < 0 for any coherent subsheaf F of E. According to the Kobayashi-Hitchin correspondence, E is μ-stable if and only if E is indecomposable and has a Hermitian-Einstein metric. Moreover, such a metric is unique up to obvious ambiguity. Thus, we obtain the bijective correspondence between μ-stable bundles and holomorphic indecomposable vector bundles with Hermitian-Einstein metrics, more strongly the homeomorphism of the moduli spaces. Stable bundles and more general stable sheaves have been studied in algebraic geometry. A moduli space M(c2 ) of semistable sheaves was constructed as the projective variety by D. Gieseker and M. Maruyama in the framework of geometric invariant theory, which gives a compactification of moduli of stable vector bundles, called the Gieseker-Maruyama compactification. It was known that there exists the projective morphism of the Gieseker-Maruyama compactification to the Uhlenbeck compactification. It is natural to expect that the integral (0.1) can be defined as the evaluation over M(c2 ). It doesn’t work in general, since a Kahler metric is not necessarily generic and the moduli space M(c2 ) does not have the expected dimension. But, if c2 is sufficiently large, it is known that M(c2 ) has the expected dimension, and the evaluation over M(c2 ) gives the Donaldson invariants according to J. Li and Morgan. (See [81] and [94].) By using the blow up formulas due to R. Fintushel and R. J. Stern [34], which relate the Donaldson invariants of X and the

of X in a point, the full Donaldson invariants can be defined in a purely blow up X algebro-geometric way. (See [53] for more details.) Note that we will later give a different construction of invariants in an algebro-geometric way. (They are the same, if the moduli has the expected dimension.) Each method has its own advantage.

Reduction to Sum of Integrals over the Products of Hilbert Schemes G. Ellingsrud-L. G¨ottsche and R. Friedman-Z. Qin studied the wall crossing phenomena of Donaldson invariants in algebraic geometry. (See [26] and [36].) Although we omit the details here, their results briefly say that if X is a smooth X can be described as the sum of integrals of naturally induced projective surface, δξ,d cohomology classes over the products of Hilbert schemes of points on X, under the assumption that the wall W ξ is good. Such a formula is called a weak wall-crossing formula in this monograph, and actually one of our main goals is to show it without any assumption on the walls by using intrinsic smoothness of the moduli spaces.

Hilbert Schemes of Points on a Surface The author thinks that it is a proper goal to describe some relations among invariants as the sum of integrals over the products of Hilbert schemes, although not final.

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For an explanation, we recall a brief history of the study of Hilbert schemes X [n] of n-points on a connected projective surface X. We refer to [49] and [98] for more details. The first important result is the irreducibility and smoothness of X [n] . Among several important and pioneering works, one of the most impressive and famous results is a formula of G¨ottsche. It is irresistible to reproduce it here. Let bi (X) denote the i-th Betti numbers of X. Let PX [n] (z) denote the Poincar´e polynomial of X [n] in a variable z. Then, G¨ottsche’s formula is ∞ n≥0

PX [n] (z)tn =

(1 + z 2m−1 tm )b1 (X) (1 + z 2m+1 tm )b3 (X) . (1 − z 2m−2 tm )(1 − z 2m tm )b2 (X) (1 − z 2m+2 tm ) m≥1

Many mathematicians were attracted by the formula for deeper understanding. For example, it has been shown to hold in the level of Grothendieck group of varieties ([11], [52], [16]). In the other direction, a deep observation was given by Nakajima and I. Grojnowski ([56] and [97]), who constructed geometrically a representation of Heisenberg algebra on n H ∗ (X [n] ). Since then, a considerable number of studies have been conducted on Hilbert schemes of points, and hence they have been much more familiar than moduli of sheaves with higher ranks. It could be one reason why we try to express some quantity as the sum of integrals over the products of Hilbert schemes. But moreover, in principle, we may expect that some information of Hilbert schemes X [n] are comprehensible from that of the original surface X with universal and lucid procedures which are independent of X. The formula of G¨ottsche and the NakajimaGrojnowski construction are typical. The principle was enforced by the results on cobordism classes, some integrals and others. (For example, see [27] and [80].) That is the main reason why the author believes it worthwhile to express something in terms of Hilbert schemes, like a weak wall-crossing formula in the rank two case. Fortunately for the author, after the solution of Nekrasov’s conjecture ([99], [100], [101], [102] [106]), G¨ottsche, Nakajima and Yoshioka successfully showed that the wall-crossing formula for Donaldson invariants of projective surface can be deduced from the weak wall-crossing formula! (See [53].) We refer to the lecture notes and the survey by Nakajima and Yoshioka for this story. ([99] and [102]).

Virtual Fundamental Classes We should emphasize that our moderate goal, such as a weak wall-crossing formula, is already related with important, interesting and rather recent development in mathematics, so called derived algebraic geometry. In their study of the wallcrossing phenomena, Ellingsrud-G¨ottsche and Friedman-Qin used a sequence of flips to relate the moduli spaces corresponding to two polarizations. Their argument can work if flips are neat, which is related with the assumption that the wall is good. But, singularity may appear in general. Singularity of moduli spaces causes difficult but attractive problems in many aspects of mathematics. Since 1990’s, several mathematicians, such as P. Deligne,

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V. Drinfeld and M. Kontsevich, have proposed significance of intrinsic smoothness of moduli spaces. (See [71].) This revolutionary idea was employed most efficiently in the Gromov-Witten theory. In constructing Gromov-Witten invariants, we need to evaluate some cohomology classes over fundamental classes of moduli stacks of stable maps. However, the moduli stacks may not be smooth and of expected dimension, and hence they do not have fundamental classes in the naive sense. As a solution of this difficulty, L. Behrend-B. Fantechi, K. Fukaya-K. Ono and J. Li-G. Tian gave constructions of virtual fundamental classes by using intrinsic smoothness called perfect obstruction theory or Kuranishi structure ([6], [40] and [82]). Rather recently, virtual fundamental classes have also been used in the study of Donaldson-Thomas invariants [121] for three dimensional Calabi-Yau varieties. As for our problems, we will define our algebro-geometric analogue of Donaldson invariants by using virtual fundamental classes, which allows us to utilize intrinsic smoothness effectively in the study of transitions of moduli stacks. More specifically, we can use a technique of localization with respect to torus actions to obtain weak wall crossing formulas and similar formulas, instead of flips.

Master Space Another important ingredient is the excellent and beautiful idea of master space due to M. Thaddeus [120]. Let G be a linear reductive group. Let U be a projective variety with a G-action. Let Li (i = 1, 2) be G-polarizations of U . Then, we have the open subset U ss (Li ) of semistable points of U with respect to Li . It is interesting and significant to compare the quotient stacks Mi := U ss (Li )/G (i = 1, 2). (Since the moduli stacks of semistable sheaves have such descriptions, it is clearly related with our problem.) For that purpose, Thaddeus introduced the idea of master space. Let us con−1 sider the G-variety TH := P(L−1 1 ⊕ L2 ) on U , the projectivization of the vector bundle L1 ⊕ L2 . We have the canonical polarization OP (1) on TH. We have the canonically defined G-action on TH, and OP (1) gives the G-polarization. The set of the semistable points is denoted by THss . Then we obtain the quotient stack M := THss /G. Let π : THss −→ M denote the projection. Let Gm denote a one dimensional torus Spec C[t, t−1 ]. We have the Gm -action on TH given by ρ(t)·[x : y] = [t·x : y], where [x : y] denotes the homogeneous coordinate of TH along the fiber direction. Since ρ commutes with the action of G, it induces a Gm -action ρ on M . We have the natural inclusions THi = P(L−1 i ) −→ TH. Because OP (1)| THi = ss = U (L ) and inclusions Mi −→ M . Since THi are compoLi , we have THss i i nents of the fixed point set of the action ρ, the stacks Mi are the components of the fixed point set of the action ρ. The action ρ may have fixed points not contained in M1 ∪ M2 . Let x ∈ THss . In general, π(x) is fixed by the action ρ if and only if Gm · x ⊂ G · x. So we may have the components of the fixed points π(x) such that x is not fixed by ρ. Such a component is called exceptional. In a sense, the information on the difference of

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the quotient stacks Mi (i = 1, 2) is contained in the exceptional fixed point sets. As already mentioned, we will extract this information by using a Gm -localization technique in our situation. We have one more technical remark. In the study of the transition of moduli stacks of objects with higher ranks, one of the main difficulties for the author is the appearance of objects which are decomposable into more than three components. In our argument, it causes that the master space is not Deligne-Mumford. To avoid such difficulty, we introduce enhanced master spaces, which requires a detailed and interesting investigation in controlling fixed point sets.

Further Study This monograph is tentative and experimental, partially because of rapid and intensive development of the theory of stacks. For example, J. Lurie, B. To¨en and their collaborators are rewriting the foundation of algebraic geometry. (See [124] for overview.) Their theory, so called derived algebraic geometry or homotopical algebraic geometry, seems to provide us with a powerful tool in constructing perfect obstruction theories, and hence virtual fundamental classes. But, unfortunately, it is beyond the author’s ability to use their theory at this moment. He believes that it should be one of the themes in the study of stacks to make it easy to deal with the objects and the formalisms in our work. He hopes to study these problems from the viewpoint of derived algebraic geometry once more in future. (However, he also believes that we will obtain the same invariants even if we adopt another method in constructing perfect obstruction theory.) One of the important results missing in our theory is the blow-up formula, i.e., a comparison of invariants for X and a blow up of X. The author originally intended to develop a theory “without blow up”. But it seems to contain interesting problems for such a comparison. For example, see the recent attractive work of Nakajima and Yoshioka, [103] and [104]. From a perfect obstruction theories, we obtain not only virtual fundamental classes but also virtual structure sheaves, which seem very attractive subjects. For example, it would be very interesting to find a formula to express the Euler numbers of the tensor products of line bundles and the virtual structure sheaves. (See [28], [54] and [101] for related topics.) As mentioned in the beginning, we hope that our weak relations would lead to more impressive formulas. Moreover, it would be desirable to understand such formulas in a geometric way, as in the case of G¨ottsche’s formula and the NakajimaGrojnowski construction. The study of wall-crossing phenomena for Donaldson-Thomas invariants has been developing intensively. (See [4], [73], [114], [123], for example.) The author expects that we will have wall crossing phenomena in many other situations. It would be very interesting if some of the techniques used in this monograph could be applied.

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Acknowledgements I started this project in 1998, my last year at the graduate school in Kyoto University. I continued the study in Osaka City University and the Institute for Advanced Study. I resumed it in 2006 at the Max-Planck Institute for Mathematics, and continued the ´ study at the Institut des Hautes Etudes Scientifiques. Then, the final version was written at Kyoto University, Department of Mathematics, and Research Institute for Mathematical Sciences. During the course of the study, I visited the Tata Institute of Fundamental Research and the International Centre for Theoretical Physics. I am heartily grateful to the institutions and the colleagues for the excellent mathematical environment, and I would like to express my deep gratitude to those who kindly supported me. The final manuscript is completely different from the first version after several major revisions, and it took very long time for publication partially because of my poor ability in writing, which I apologize to those who kindly showed their interest in this study. Anyway, I feel relief that the manuscript is published at last. My mathematical interest has been clearly and deeply influenced by Kenji Fukaya. I learned many things through his talks and lectures. I was also very encouraged by his interest in my study. It is my great pleasure to express my gratitude to him. I thank Hiraku Nakajima for his stimulating questions, which made me resume this study. And, I am grateful to Lothar G¨ottsche, H. Nakajima and K¯ota Yoshioka for their amazing work and their interest in this study. I also thank Barbara Fantechi for her valuable comments and suggestions. I express my gratitude to K. Fukaya, Yoshihiro Fukumoto, Mikio Furuta, Ryushi Goto, Yoshitake Hashimoto, Yukio Kametani, Hiroshi Konno, Norihiko Minami, Hiroshi Ohta, Kaoru Ono, Tatsuru Takakura, who kindly listened to a series of my bad talks on this study in Gero. It is impossible to express how I was glad and encouraged and how I appreciate them. I thank Yoshifumi Tsuchimoto and Akira Ishii for their constant encouragement and advice since my undergraduate student days. All what I do in mathematics is based on what I learned from them. Special thanks go to Mark de Cataldo, Pierre Deligne, William Fulton, David Gieseker, Masaki Kashiwara, Akira Kono, Mikiya Masuda, Claude Sabbah, Carlos Simpson, Tomohide Terasoma, Michael Thaddeus and Kari Vilonen for their help. I am willing to express my gratitude to Takeshi Abe, K¨ursat Aker, Andreas Bender, Ryan Budney, Ron Donagi, Simon Donaldson, Michihiko Fujii, Koji Fujiwara, Rajendra Vasant Gurjar, Claus Hertling, Michiaki Inaba, Masaharu Kaneda, Atushi Kasue, Tsuyoshi Kato, Hiroshige Kajiura, Nariya Kawazumi, Yoshikata Kida, Maxim Kontsevich, Kong Liang, Kefeng Liu, Pierre Matsumi, Atsushi Matsuo, Vikram B. Mehta, Luca Migliorini, Hiroaki Narita, Shin-ichi Ohta, So Okada, Tony Pantev, Ramadas T. Ramakrishnan, Utkir Rozikov, Masa-Hiko Saito, Christian Sevenheck, Szilard Szabo, Alexander Usnich, Kimiko Yamada, Satoshi Yamaguchi for their kindness and some discussion. I still remember vigorous and powerful

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lectures by Masaki Maruyama on semistable sheaves. I thank Fumiharu Kato for some valuable information on stacks and his kindness. I would like to express my deep thanks to the anonymous referee for his careful and patient reading. I also appreciate his useful and practical advice in writing. I thank the partial supports by the Ministry of Education, Culture, Sports, Science and Technology, the Japan Society for the Promotion of Science, the Sumitomo Foundation, and the Sasagawa Foundation. I wish to acknowledge the National Scientific Foundation for a grant DMS 9729992, although any opinions, findings and conclusions or recommendations expressed in this material do not necessarily reflect the views of the National Scientific Foundation. Kyoto, June 2008

Takuro Mochizuki

Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Construction of Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Orientation and Reduced L-Bradlow Pairs . . . . . . . . . . . . . . . 1.2.2 Virtual Fundamental Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Transition Formulas in a Simple Case . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Rank Two Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Dependence on Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Reduction to Integrals over Hilbert Schemes . . . . . . . . . . . . . 1.5 Higher Rank Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 pg > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Transition Formula in the Case pg = 0 . . . . . . . . . . . . . . . . . . 1.5.3 Weak Intersection Rounding Formula . . . . . . . . . . . . . . . . . . . 1.6 Master Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Master Space in the Case that a 2-Stability Condition is Satisfied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Gm -Localization Method (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Enhanced Master Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4 Gm -Localization Method (II) . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 3 5 6 9 9 11 12 12 13 15 16 17 19 20 22

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Some Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Product and Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Coherent Sheaves on a Product . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Quotient Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Signature in Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Filtrations and Complexes on a Curve . . . . . . . . . . . . . . . . . . . 2.1.7 Virtual Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.8 Compatible Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 25 26 26 26 26 27 28 29

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2.3

2.4

2.5

2.6

2.7

3

Geometric Invariant Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 GIT Quotient and Algebraic Stacks . . . . . . . . . . . . . . . . . . . . . 2.2.2 Mumford-Hilbert Criterion and Some Elementary Examples Cotangent Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Quotient Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Some More Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Obstruction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Definition and Fundamental Theorems . . . . . . . . . . . . . . . . . . 2.4.2 Easy Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Locally Free Subsheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Filtrations of a Vector Bundle on a Curve . . . . . . . . . . . . . . . . Equivariant Complexes on Deligne-Mumford Stacks with GIT Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Locally Free Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Equivariant Representative . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary Remarks on Some Extremal Sets . . . . . . . . . . . . . . . . . . . 2.6.1 Preparation for a Proof of Proposition 4.3.3 . . . . . . . . . . . . . . 2.6.2 Preparation for a Proof of Proposition 4.4.4 . . . . . . . . . . . . . . Twist of Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Weight of the Induced Action . . . . . . . . . . . . . . . . . . . . . . . . . .

Parabolic L-Bradlow Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Sheaves with Some Structure and their Moduli Stacks . . . . . . . . . . . . 3.1.1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Parabolic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 L-Bradlow Pairs and Reduced L-Bradlow Pairs . . . . . . . . . . . 3.1.4 Type and Moduli Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Tautological Line Bundle and Relations Among Some Moduli Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Hilbert Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Hilbert Polynomials of Coherent Sheaves . . . . . . . . . . . . . . . . 3.2.2 Hilbert Polynomials of Parabolic Sheaves . . . . . . . . . . . . . . . . 3.2.3 Hilbert Polynomial for Parabolic L-Bradlow Pairs . . . . . . . . . 3.2.4 Hilbert Polynomials Associated to a Type . . . . . . . . . . . . . . . . 3.3 Semistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Semistability Conditions and the Associated Moduli Stacks . 3.3.2 Harder-Narasimhan Filtration and Partial Jordan-H¨older Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 (δ, )-Semistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Some Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Fundamental Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Boundedness of Semistable L-Bradlow Pairs . . . . . . . . . . . . . 3.4.3 Boundedness of Yokogawa Family . . . . . . . . . . . . . . . . . . . . . .

29 29 30 32 32 34 39 44 44 46 48 51 55 55 56 57 57 59 61 61 61 63 64 64 65 66 68 69 71 71 71 72 72 73 73 76 77 78 78 79 80

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3.6

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1-Stability Condition and 2-Stability Condition . . . . . . . . . . . . . . . . . 3.5.1 Parabolic Sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Parabolic L-Bradlow Pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Parabolic L-Bradlow Pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quot Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Quotient Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Quotient Quasi-Parabolic Sheaves and MaruyamaYokogawa Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.4 Quotient L-Bradlow Pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.5 Quotient Reduced L-Bradlow Pair . . . . . . . . . . . . . . . . . . . . . . 3.6.6 Quotient Reduced L-Bradlow Pair . . . . . . . . . . . . . . . . . . . . . . 3.6.7 Oriented Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.8 Quotient Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 84 85 88 89 89 89 90 91 93 94 94 95

Geometric Invariant Theory and Enhanced Master Space . . . . . . . . . . 97 4.1 Semistability Condition and Mumford-Hilbert Criterion . . . . . . . . . . 98 4.1.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.1.2 Mumford-Hilbert Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1.3 A Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.1.4 Proof of the Claim 1 in Proposition 4.1.2 . . . . . . . . . . . . . . . . 103 4.1.5 Proof of the Claim 2 of Proposition 4.1.2 . . . . . . . . . . . . . . . . 104 4.1.6 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 Perturbation of Semistability Condition . . . . . . . . . . . . . . . . . . . . . . . . 106 4.2.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.2.2 δ+ -Semistability and δ− -Semistability . . . . . . . . . . . . . . . . . . 107 4.2.3 (δ, )-Semistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3 Enhanced Master Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.3.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.3.2 Proof of Lemma 4.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.3.3 Proof of Lemma 4.3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.3.4 Proof of Lemma 4.3.6, Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.3.5 Proof of Lemma 4.3.6, Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.3.6 Proof of Lemma 4.3.6, Step 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.4 Fixed Point Set of Torus Action on Enhanced Master Space . . . . . . . 120 4.4.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.4.2 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.4.3 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.4.4 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.4.5 Step 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4.6 Step 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.4.7 End of the Proof of Proposition 4.4.4 . . . . . . . . . . . . . . . . . . . . 127 4.5 Enhanced Master Space in Oriented Case . . . . . . . . . . . . . . . . . . . . . . 128 4.5.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.5.2 Obvious Fixed Point Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

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4.5.3 Fixed Point Sets Associated to Decomposition Types . . . . . . 130 4.5.4 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.5.5 Ambient Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.5.6 Fixed Point Set of the Ambient Space . . . . . . . . . . . . . . . . . . . 131 4.5.7 Proof of Proposition 4.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Decomposition of Exceptional Fixed Point Sets . . . . . . . . . . . . . . . . . 134 4.6.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.6.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.6.3 Construction of the Stack S . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.6.4 Universal Sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Simpler Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.7.1 Case in Which a 2-Stability Condition is Satisfied . . . . . . . . . 138 4.7.2 Oriented Reduced L-Bradlow Pairs . . . . . . . . . . . . . . . . . . . . . 140

Obstruction Theories of Moduli Stacks and Master Spaces . . . . . . . . . 145 5.1 Deformation of Torsion-Free Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.1.1 Construction of a Basic Complex . . . . . . . . . . . . . . . . . . . . . . . 146 5.1.2 The Trace-Free Part and the Diagonal Part . . . . . . . . . . . . . . . 148 5.1.3 Preparation for Master space . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.1.4 Basic Complex on the Moduli Stack M(m, y) . . . . . . . . . . . . 151 5.1.5 Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.2 Relative Obstruction Theory for Orientations . . . . . . . . . . . . . . . . . . . 154 5.2.1 Construction of a Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.2.2 Relative Obstruction Property . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.3 Relative Obstruction Theory for L-Sections . . . . . . . . . . . . . . . . . . . . . 156 5.3.1 Construction of a Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.3.2 Relative Obstruction Property . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.3.3 Preparation for Obstruction Theory of Master Space . . . . . . . 160 5.3.4 Preparation for Proposition 6.2.1 . . . . . . . . . . . . . . . . . . . . . . . 161 5.3.5 Another Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.4 Relative Obstruction Theory for Reduced L-Sections . . . . . . . . . . . . . 164 5.4.1 Construction of a Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.4.2 Relative Obstruction Property . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.4.3 Preparation for Obstruction Theory of Master Space . . . . . . . 168 5.4.4 Preparation for Proposition 6.2.1 . . . . . . . . . . . . . . . . . . . . . . . 170 5.5 Relative Obstruction Theory for Parabolic Structures . . . . . . . . . . . . . 170 5.5.1 Construction of a Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.5.2 Relative Obstruction Property . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.5.3 Decomposition into the Trace-Free Part and the Diagonal Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.6 Obstruction Theory for Moduli Stacks of Stable Objects . . . . . . . . . . 174 5.6.1 Relative Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.6.2 Construction of Complexes and Morphisms . . . . . . . . . . . . . . 175 5.6.3 Obstruction Theories of Quot Schemes and Moduli Stacks . . 176 5.6.4 Obstruction Theories of Moduli Stacks of Stable Objects . . . 177

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Obstruction Theory for Enhanced Master Spaces and Related Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.7.1 Enhanced Master Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.7.2 Substack M 5.7.3 Moduli Stack M(m, y , [L]) . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.7.4 Moduli Stack M(m, y, L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 s (

5.7.5 Moduli Stack M y , α∗ , +) . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.7.6 Case in Which a 2-Stability Condition is Satisfied . . . . . . . . . 185 5.7.7 Oriented Reduced L-Bradlow Pairs . . . . . . . . . . . . . . . . . . . . . 186 5.8 Moduli Theoretic Obstruction Theory of Fixed Point Set . . . . . . . . . . 187 5.8.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.8.2 Moduli Stack of Split Objects with Orientations . . . . . . . . . . 188 5.8.3 Embedding into Moduli Stack of Non-Split Objects . . . . . . . 190 5.8.4 Some Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.8.5 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.8.6 Proof of Proposition 5.8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 5.8.7 Case in Which a 2-Stability Condition is Satisfied . . . . . . . . . 199 5.8.8 Oriented Reduced L-Bradlow Pairs . . . . . . . . . . . . . . . . . . . . . 200 5.9 Equivariant Obstruction Theory of Master Space . . . . . . . . . . . . . . . . 201 5.9.1 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

) and ob(M

) . . . . . . . . . . . . . . 204 5.9.2 Gm -Equivariant Lift of Ob(M 5.9.3 Comparison of Gm -Equivariant Structures of τ≥−1 LM

. . . . 205

∗ ) . . . . . . . . . . . . . . . . . . . . 207 5.9.4 Gm -Equivariant structure of Ob(M 5.9.5 Proof of Proposition 5.9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5.9.6 Proof of Proposition 5.9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5.9.7 Case in Which a 2-Stability Condition is Satisfied . . . . . . . . . 209 5.9.8 Oriented Reduced L-Bradlow Pairs . . . . . . . . . . . . . . . . . . . . . 210 6

Virtual Fundamental Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.1 Perfectness of Obstruction Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.1.1 Moduli Stacks of Semistable Objects . . . . . . . . . . . . . . . . . . . . 213 6.1.2 Master Spaces and Some Related Stacks . . . . . . . . . . . . . . . . . 214 6.1.3 Vanishing of Some Cohomology Groups . . . . . . . . . . . . . . . . . 216 6.1.4 Proof of the Propositions in Subsection 6.1.1 . . . . . . . . . . . . . 219 6.1.5 Expected Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 6.2 Comparison of Oriented Reduced Case and Unoriented Unreduced Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.2.1 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.2.2 Proof of Proposition 6.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.3 Rank One Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 6.3.1 Moduli of L-Abelian Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 6.3.2 Involutivity and Relation with Seiberg-Witten Invariants . . . 227 6.3.3 Parabolic Hilbert Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 6.3.4 Splitting into Moduli of Abelian Pairs and Parabolic Hilbert Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

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6.5 6.6

7

6.3.5 Morphism to Moduli of Abelian Pairs . . . . . . . . . . . . . . . . . . . 232 6.3.6 Morphism to Parabolic Hilbert Scheme . . . . . . . . . . . . . . . . . . 237 6.3.7 Mixed Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 6.3.8 Proof of Proposition 6.3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 Bradlow Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 6.4.1 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 6.4.2 Construction of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.4.3 Compatibility of the Obstruction Theories . . . . . . . . . . . . . . . 244 6.4.4 Ambient Smooth Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Comparison with Full Flag Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Parabolic Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.6.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.6.2 Construction of a Stack B with an Obstruction Theory . . . . . 254 6.6.3 Compatibility of the Obstruction Theories . . . . . . . . . . . . . . . 257 6.6.4 Smooth Ambient Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 6.6.5 Proof of Proposition 6.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.1.1 Ring Rl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 7.1.2 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 7.1.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 7.1.4 Equivariant Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 7.1.5 Ring RCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 7.1.6 Equivariant Euler Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 7.1.7 Twist by Line Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.2 Transition Formulas in Simpler Cases . . . . . . . . . . . . . . . . . . . . . . . . . 276 7.2.1 Basic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 7.2.2 Twist with the Euler Class of the Relative Tangent Bundle . . 279 7.2.3 Oriented Reduced L-Bradlow Pairs . . . . . . . . . . . . . . . . . . . . . 282 7.3 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 7.3.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 7.3.2 Easy Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 ss (

y , α∗ , +) . . . . . . . . . . . . . . . . . . . . . . . . . . 290 7.3.3 Integrals over M 7.3.4 Another Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 7.3.5 Deformation Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 7.4 Rank Two Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 7.4.1 Reduction to the Sum of Integrals over the Products of Hilbert Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 7.4.2 Dependence on Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . 296 7.4.3 Proof of Theorem 7.4.3 (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 7.4.4 Proof of Theorem 7.4.3 (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 7.5 Higher Rank Case (pg > 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 7.5.1 Transition Formula in the Case pg > 0 . . . . . . . . . . . . . . . . . . 303

Contents

xxiii

7.5.2

Reduction to the Sum of Integrals over the Products of Hilbert Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 7.5.3 Independence from Polarizations in the Case pg > 0 . . . . . . . 309 7.6 Transition Formula (pg = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 7.6.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 7.6.2 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 7.6.3 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 7.6.4 Step 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 7.7 Weak Wall Crossing Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 7.7.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 7.7.2 Proof of Theorem 7.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 7.7.3 Weak Wall Crossing Formula in the Rank 3 Case . . . . . . . . . . 323 7.7.4 Weak Intersection Rounding Formula in the Rank 3 Case . . . 325 7.7.5 Transition for a Critical Parabolic Weight . . . . . . . . . . . . . . . . 330 7.8 Weak Intersection Rounding Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 331 7.8.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 7.8.2 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 7.8.3 Preparation from Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . 335 7.8.4 Proof of Theorem 7.8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

Chapter 1

Introduction

Since we have explained the background and the motivation of the study in Preface, we will explain our problems and results, which are actually investigated in this monograph. In Section 1.1, we explain the problems. In Section 1.2, we discuss the main issues for construction of invariants. In Section 1.3, transition formulas are stated under an assumption which makes the problems much simpler. They are enough for the study of invariants in the rank 2 case, and the results are explained in Section 1.4. Generalization to the higher rank case is discussed in Section 1.5. We explain how to use master spaces for our problems in Section 1.6.

1.1 Problems Let X be a smooth projective surface with an ample line bundle OX (1) over the complex number field C. For simplicity, we assume in this introduction that X is simply connected. We set pg := dim H 2 (X, OX ). Let T ype denote the set of cohomology classes of X obtained as the Chern character of some torsion-free sheaves on X. We say that a torsion-free sheaf E is of type y, if ch(E) = y. For any y ∈ T ype, let Mss (y) denote a moduli stack of semistable (resp. stable) torsionfree sheaves of type y. One of our main problems is the following. Problem 1.1. Construct Donaldson type invariants by using Mss (y). Let E u denote the universal sheaf over Mss (y)×X. Let A∗ (X) be the Chow group of X. Let P (E u ) be a polynomial in the slant products chi (E u )/b for elements b ∈ A∗ (X) and i ∈ Z≥ 0 . Naively speaking, we would like to define the number: P (E u ), or equivalently, deg P (E u ) ∩ [Mss (y)] (1.1) Φ(y) := Mss (y)

Namely, we would like to obtain a 0-cycle by taking the cap product of P (E u ) and the fundamental class of Mss (y), and we would like to obtain the number by taking T. Mochizuki, Donaldson Type Invariants for Algebraic Surfaces: Transition of Moduli Stacks, Lecture Notes in Mathematics 1972, DOI: 10.1007/978-3-540-93913-9 1, c Springer-Verlag Berlin Heidelberg 2009

1

2

1 Introduction

its degree. As mentioned in Preface, Donaldson invariants for smooth projective surfaces can be defined in terms of moduli spaces of semistable sheaves of rank 2. It is natural to ask what invariants would be obtained by considering integrals as in (1.1) for moduli stacks of semistable sheaves of higher ranks. But, there are two main issues in making (1.1) well-defined. (A) The moduli stack Mss (y) is not Deligne-Mumford. For Artin stacks, there is no well-established theory on the degree of 0-cycles, or in other words, the push forward of cycles. (B) The moduli stack Mss (y) and the related moduli stacks may not be pure dimensional. Hence, they may not have fundamental classes in the naive sense. We will give a brief account for the above issues in Section 1.2. Remark 1.1.1 P. Kronheimer [76] studied such a generalization of Donaldson invariants for real 4-dimensional manifolds by using the moduli of higher rank objects from a viewpoint of differential geometry. (See also [135].) It was also investigated in mathematical physics. (See [84], for example.) Remark 1.1.2 Related issues were studied for moduli stack of sheaves on CalabiYau threefolds by R. P. Thomas [121] in the construction of Donaldson-Thomas invariants. See also the recent work of R. Pandharipande and R. P. Thomas. ([112]– [114].) After constructing the invariants, we will be interested in universal relations among invariants, and moreover geometry behind them. (See Preface for the related conjectures on Donaldson invariants.) In particular, we study the following problems in this monograph. Problem 1.2. Clarify the dependence of Φ(y) on the polarization OX (1).

Problem 1.3. Express Φ(y) as the sum of integrals over the products of moduli spaces of objects with rank one. As for Problem 1.2, we will show that Φ(y) is independent of the choice of OX (1) in the case pg > 0, and we will obtain a “weak wall-crossing formula” in the case pg = 0. As for Problem 1.3, we will give such a formula in the case pg > 0. See Section 1.4 for more details in the rank 2 case, and Section 1.5 in the higher rank case. As mentioned in Preface, the related problems for Donaldson invariants (the rank 2 case) have been intensively studied. For example, they were investigated by V. Y. Pidstrigach–A. N. Tyurin, [115], B. Chen [12], and in particular P. Feehan–T. Leness ([29]–[33] and others), by using differential geometric techniques. From algebrogeometric viewpoints, Problem 1.2 was studied by G. Ellingsrud-L. G¨ottsche [26] and R. Friedman-Z. Qin [36], and Problem 1.3 was discussed by C. Okonek, A. Schmitt and A. Teleman [110]. Rather recently, D. Joyce developed an attractive and huge theory to obtain invariants from abelian categories provided with semistability conditions. (For example, see [66].) His theory can be applied to the category of coherent sheaves

1.2 Construction of Invariants

3

on a smooth projective surface whose anti-canonical sheaf is nef. (For example, see [122].) He also studied wall-crossing phenomena of the invariants for variation of semistability conditions. But, his invariant is “motivic”, and it seems different from ours. Remark 1.1.3 As explained in the next section, we will consider moduli stacks y ) of oriented semistable sheaves of type y, instead of Mss (y). So we preMss (

fer to using the symbol Φ(

y ) instead of Φ(y).

1.2 Construction of Invariants 1.2.1 Orientation and Reduced L-Bradlow Pairs Let us consider the issue (A). We will replace Mss (y) with another proper DeligneMumford stack which is a nice approximation of Mss (y). Recall that stable sheaves have non-trivial automorphism groups given by constant multiplications. To eliminate such automorphisms, we first consider orientations of torsion-free sheaves. Let y be as in Section 1.1. Let a be the first Chern class determined by y, i.e., the H 2 (X)-component. We take an algebraic line bundle La on X such that c1 (La ) = a. An oriented torsion-free sheaf (E, ρ) on X of type y is a pair of a coherent torsion-free sheaf E of type y and an isomorphism ρ : det(E) La . Let y ) (resp. Ms (

y )) denote a moduli stack of semistable (resp. stable) oriented Mss (

torsion-free sheaves of type y. A constant multiplication t · idE gives an automory ) is Deligne-Mumford. phism of (E, ρ) if and only if trank E = 1. Hence, Ms (

y ) = Mss (

y ), the issue (A) can be avoided with this easy modification. If Ms (

u denote the universal sheaf over Ms (

u ) be a Namely, let E y ) × X, and let P (E u

)/b for elements b ∈ A∗ (X) and i ∈ Z≥ 0 . polynomial in the slant products chi (E We can use the following integral instead of (1.1), modulo the issue (B):

u ) P (E (1.2) Φ(

y) = Ms (

y)

y ) instead of Mss (y), i.e., Remark 1.2.1 There is another reason to consider Mss (

an obstruction theory of oriented sheaves is more suitable. We also recall the notion of L-Bradlow pair. Let L be a line bundle on X. A morphism φ : L −→ E is called an L-section, and a pair (E, φ) is called an L-Bradlow pair. For simplicity, we always assume φ = 0 for an L-Bradlow pair in this introduction. Let P br denote the set of polynomials δ with rational coefficients such that (i) deg(δ) ≤ 1, (ii) δ(t) > 0 for any sufficiently large t. Recall that δ-semistability and δ-stability conditions are defined for L-Bradlow pairs, when we are given δ ∈ P br . (See Subsection 3.3.1.) We use the symbol Mss (y, L, δ) (resp. Ms (y, L, δ)) to denote a moduli stack of semistable (resp. stable) L-Bradlow pairs of type y. We say that δ is critical, if Mss (y, L, δ) = Ms (y, L, δ). It can be shown that there are only

4

1 Introduction

finitely many critical parameters. For a non-critical parameter δ, the moduli stack Mss (y, L, δ) is proper and Deligne-Mumford. If δ is sufficiently small, we have the morphism π : Ms (y, L, δ) −→ Mss (y). If moreover L = OX (−m) for a sufficiently large m and Mss (y) = Ms (y), then π is a (CN − {0})-bundles for some N . Hence, Ms (y, OX (−m), δ) for small δ can be regarded as an approximation of Mss (y). We may consider integrals over Ms (y, OX (−m), δ) instead of Mss (y). However, as explained later, we will use virtual fundamental classes by constructing perfect obstruction theories of our moduli stacks. And, if pg > 0, we obtain the vanishing of the virtual fundamental class of Mss y, OX (−m), δ induced by a natural obstruction theory. We need some more modification. y , L, δ) of oriented L-Bradlow pairs. It is a Let us consider a moduli stack Mss (

Gm -torsor over Mss (y, L, δ), where Gm denotes a one dimensional torus. Hence, it is Deligne-Mumford except for a finite number of δ ∈ P br . However, it is not proper. So we consider the Gm -action on Mss (y, L, δ) given by multiplication on L-sections t · (E, ρ, φ) := (E, ρ, t·φ), and take the quotient stack which is denoted by Mss (y, [L], δ). It is Deligne-Mumford and proper except for finite number of δ ∈ P br . It can be regarded as a moduli stack of oriented reduced L-Bradlow pairs. We refer to Subsection 3.1.3 for such a moduli theoretic meaning. y , L, δ). Remark 1.2.2 There is an e´ tale proper morphism of Mss (y, [L], δ)toMss (

In other words, they are isomorphic up to an e´ tale proper morphism. However, natural obstruction theories are different. Let us consider the case L = O(−m). If δ is sufficiently small, it is non-critical, y , [O(−m)], δ) −→ Mss (

y ). and we have the naturally defined morphism π : Mss (

be the Hilbert polynoIf m is sufficiently large, the morphism π is smooth. Let H y mial associated to y, i.e., Hy (m) := X Td(X) · y · ch O(m) for any integer m. In the case Mss (

y ) = Ms (

y ), the morphism π is a PHy (m)−1 -bundle. Hence, we ss y , [O(−m)], δ) as a nice approximation of Mss (

y ). regard M (

Let Θrel denote the relative tangent bundle of the smooth morphism π, and let y ) by the Eu(Θrel ) denote its Euler class. We would like to define the number Φ(

following formula, modulo the issue (B):

u ) · Eu(Θrel ) (1.3) P (E Φ(

y ) := Hy (m) Mss (

y ,[O(−m)],δ) Needless to say, we have to ask the following: (C)

Is (1.3) independent of the choice of m?

If Mss (

y ) = Ms (

y ), (1.3) is clearly independent of the choice of m, and it is compatible with (1.2). We will obtain an affirmative answer of (C) in Subsection 7.3.1. Remark 1.2.3 Although we do not discuss the parabolic structure in this introduction, we will consider invariants obtained from moduli stacks of oriented parabolic

1.2 Construction of Invariants

5

torsion-free sheaves and oriented parabolic reduced L-Bradlow pairs. And, in general, it is not clear whether (1.3) is independent of the choice of m. Instead, we can show the existence of the following limit, for a line bundle L such that L−1 is ample:

u ) · Eu(Θrel ) (1.4) P (E lim m→∞ Mss (

NLm (y) m y ,[L ],α∗ ,δ) Here, y denotes a type of parabolic sheaves, and α∗ denotes a system of weight. Moreover, the limit is independent of the choice of L. Hence, we may adopt (1.4) as the definition of Φ(

y , α∗ ).

1.2.2 Virtual Fundamental Classes There are several well-established methods to deal with a problem similar to (B), which were originally prepared for construction of Gromov-Witten invariants ([71], [6], [82], [40]). In this paper, we follow the method of K. Behrend and B. Fantechi. Namely, we will show that the moduli stacks mentioned in Subsection 1.2.1 are naturally provided with perfect obstruction theories in the sense of [6]. (We will review obstruction theory in Subsection 2.4.1.) This will allow us to construct virtual fundamental classes. Such an obstruction theory may perhaps be well known. For example, it is standard that for any vector bundle E on X (i) the first cohomology group H 1 (X, End(E)) gives the space of infinitesimal deformations, (ii) the second cohomology group H 2 (X, End(E)) gives the space of obstructions. However, the author does not know an appropriate literature to deal with obstruction theories in the sense of [6] for moduli stacks of reduced oriented L-Bradlow pairs and master spaces (Section 1.6), which are suitable for our arguments using the localization technique. Therefore, we give a detailed argument in Chapter 5. As mentioned in Preface, there is another way to define invariants in a purely algebro-geometric way, and to avoid difficulty of non-smoothness of the moduli spaces by using “blow up formula”. (See [81], [94] and Section 1.1 of [53].) However, there is an advantage in defining invariants by using virtual fundamental classes. We know a nice localization theory for virtual fundamental classes due to T. Graber and R. Pandharipande [55]. As explained later, we can use it to study transition of invariants such as Problems 1.2 and 1.3. Even if we have generic smoothness of our moduli stacks, transition of moduli stacks may be singular. Hence, it is desirable to have perfect obstruction theories and to exploit intrinsic smoothness of moduli. Remark 1.2.4 R. P. Thomas gave an obstruction theory of moduli of coherent sheaves with fixed determinant in a different manner [121]. Remark 1.2.5 Recently, the theory of “derived stacks” has been developed, which seems to provide us with a general and powerful tool to construct obstruction theories for some kind of stacks. (See [124] for an overview of the theory.) For example,

6

1 Introduction

the results in [125] imply the construction of an obstruction theory of moduli stacks of semistable sheaves. It is not clear to the author, at the present moment, whether we can directly apply their results and methods to moduli stacks of semistable oriented reduced L-Bradlow pairs and master spaces, although it seems quite promising. The author would like to come back to this problem in future. However, the author also expects that the same “invariants” would be obtained, even if we adopt a different way in constructing obstruction theories, at least if H 1 (X, OX ) = 0. We should have transition formulas as argued in this monograph, thanks to which it can be reduced to a comparison of the invariants obtained from moduli stacks of objects of rank one. We should have a splitting as in Proposition 6.3.8, and hence it can be reduced to a comparison of the invariants obtained from abelian pairs. (See Subsection 7.5.2 for such a reduction in the case H 2 (X, O) = 0. We may also obtain such a reduction in the case H 2 (X, O) = 0, although the formula would be more complicated.) In the case H 1 (X, O) = 0, moduli spaces of abelian pairs are smooth, and obstruction theories are given by obstruction bundles, which should be as in Subsection 6.3.1. We expect that a comparison of the invariants could be done in this way. Remark 1.2.6 Although we use virtual fundamental classes for defining integrals,

u ), if there are we prefer to using a description like (1.2) instead of [Ms ( y)]vir P (E no risk of confusion.

1.3 Transition Formulas in a Simple Case Let us explain typical results on transition of invariants. We remark that we will use the symbol pX to denote the projection U × X −→ U for any stack U. y , [L], δ) Integrals over Ms (

There is the relative tautological line bundle Orel (1) on Mss (

y , [L], δ). (See Subsection 3.1.5 for the definition.) Let ω denote the first Chern class of Orel (1). If δ is y , [L], δ) = Ms (

y , [L], δ), we set a non-critical parameter, i.e. Mss (

u ) · ωk Φ(

y , [L], δ) := P (E (1.5) Ms (

y ,[L],δ)

We should remark that the integral (1.5) is defined by using the virtual fundamental y , [L], δ)]vir as mentioned in Subsection 1.2.2. But, we prefer to using a class [Ms (

description like (1.5) if there are no risk of confusion. Let δ be critical. We take parameters δ− < δ < δ+ such that |δκ − δ| are sufficiently small for κ = ±. We would like to describe the difference y , [L], δ− ) Φ(

y , [L], δ+ ) − Φ(

1.3 Transition Formulas in a Simple Case

7

as the sum of integrals over the products of moduli stacks of objects with lower ranks. Such a description is called a transition formula. In this section, we explain the results when the following condition is satisfied: (2-stability condition) We say that that the 2-stability condition holds for (y, L, δ), if the automorphism group of any (E, φ) ∈ Mss (y, L, δ) is {1} or Gm .

Preparation To state transition formulas, we prepare some notation. For any y ∈ T ype, the H 0 (X)-component is denoted by rank(y). The reduced Hilbert polynomial Hy / rank(y) is denoted by Py . When a parameter δ is given, we put Pyδ := (Hy + δ)/ rank(y). We set S(y, δ) := (y1 , y2 ) ∈ T ype2 y1 + y2 = y, Pyδ = Pyδ1 = Py2 . Transition Formula I For a given (y1 , y2 ) ∈ T ype2 , we put ri := rank yi . We also set y2 ). M(y1 , y 2 , L, δ) := Mss (y1 , L, δ) × Mss (

On M(y1 , y 2 , L, δ) × X, we have the sheaf E1u which is induced by the universal

u sheaf on Mss (y1 , L, δ) × X via the natural projection. We also have the sheaf E 2 ss which is induced by the universal sheaf on M (

y2 ) × X via the natural projection. Let us describe some cohomology classes on M(y1 , y 2 , L, δ). For that purpose, it is convenient to use equivariant cohomology classes. Let Gm denote a one dimensional torus. Let A∗ denote the bivariant theory. Recall that the Gm -equivariant bivariant group of a point R(Gm ) is defined to be the limit of A∗ (Pm ) with respect to inclusions. Let t be the first Chern class of the tautological line bundle. Then, R(Gm ) Q[t]. It is convenient to use the symbol ew·t to denote the trivial line bundle with a Gm action of weight w ∈ Q. Note that “an action of rational weight” makes sense by considering a covering of Gm . An element of the K-theory of Gm -equivariant perfect complexes is called a virtual Gm -equivariant vector bundle in this monograph. We have the following virtual Gm -equivariant vector bundle on M(y1 , y 2 , L, δ):

u ·er1 (t−ω1 )/r2 N0 (y1 , y2 ) := −RpX ∗ RHom E1u ·e−t , E 2 u r1 (t−ω1 )/r2

2 ·e −RpX ∗ RHom E , E1u ·e−t

u ·er1 (t−ω1 )/r2 +RpX ∗ Hom L·e−t , E 2

8

1 Introduction

Here, we put ω1 := c1 (Or(E1u ))/r1 , and ew·ω1 denotes Or(E1u )w/r1 formally. (See Section 2.7 for a rational power of line bundles.) We obtain its Gm -equivariant Euler class Eu(N0 (y1 , y2 )) ∈ A∗ M(y1 , y 2 , L, δ) ⊗ R(Gm )[t−1 ]. It is invertible in the ring A∗ M(y1 , y 2 , L, δ) ⊗ R(Gm )[t−1 ].

u · er1 (t−ω1 )/r2 , we also obtain By replacing E with E1u · e−t ⊕ E 2

u · er1 (t−ω1 )/r2 ∈ A∗ M(y1 , y 2 , L, δ) ⊗ R(Gm ) P E1u · e−t ⊕ E 2 Hence, we obtain

u · er1 (t−ω1 )/r2 · tk P E1u · e−t ⊕ E 2 ∈ A∗ M(y1 , y 2 , L, δ) Υ (y1 , y2 ) := Res t=0 Eu(N0 (y1 , y2 )) Here “Rest=0 ” means taking the coefficient of t−1 . Theorem 1.3.1 (A special case of Theorem 7.2.1) Assume that the 2-stability condition holds for (y, L, δ). Then, we have the following equality: y , [L], δ− ) = Υ (y1 , y2 ) (1.6) Φ(

y , [L], δ+ ) − Φ(

(y1 ,y2 )∈S(y,δ)

M(y1 ,

y2 ,L,δ)

The contributions of (y1 , y2 ) are 0, if pg > 0 and rank(y1 ) > 0.

Transition Formula II The following is called the i-vanishing condition for (y, L, δ): (i-vanishing condition) We have H j (X, L−1 ⊗ E) = 0 for any j ≥ i and for any (E, φ) ∈ Mss (y, L, δ). When a 1-vanishing condition is satisfied, we will be interested in integrals of another kind of cohomology classes. Let M(

y ) denote a moduli stack of oriented torsion-free sheaves of type y which are not necessarily semistable. If the 1-vanishing condition holds for (y, L, δ), the y , [L], δ) −→ M(

y ) is smooth. The relative tangent bundle is demorphism Mss (

noted by Θrel . We put NL (y) := Td(X) · y · ch(L−1 ) = rank Θrel + 1. X

If δ is non-critical, we consider the following integral:

u ) · Eu(Θrel ) y , [L], δ) := P (E Φ1 (

NL (y) ss M (

y ,[L],δ) We will be interested in the transition of Φ1 (

y , [L], δ) for variation of δ.

(1.7)

1.4 Rank Two Case

9

Let δ be critical. For (y1 , y2 ) ∈ S(y, δ), we put y1 , [L], δ) × Mss (

y2 ). M(

y1 , y 2 , [L], δ) := Mss (

u which is the pull back of the On M(

y1 , y 2 , [L], δ) × X, we have the sheaf E 1 ss y1 , [L], δ) × X via the natural projection. Similarly, we universal sheaf on M (

u which is the pull back of the universal sheaf on Mss (

have the sheaf E y2 ) × X 2 via the natural projection. Let R(Gm ) = Q[s] be the Gm -equivariant bivariant group of a point. Let ew·s denote the trivial line bundle with the Gm -action of weight w. We have the following y1 , y 2 , [L], δ): virtual Gm -equivariant vector bundle on M(

u −s/r1

1 ·e

2u ·es/r2 ,E − RpX ∗ RHom E u s/r2

·e

u ·e−s/r1 − RpX ∗ RHom E ,E 2 1

2 · es/r2 denote its equivariant Euler class.

1 · e−s/r1 , E Let Q E Theorem 1.3.2 (A special case of Theorem 7.2.4) Assume that the 2-stability condition and the 1-vanishing condition hold for (y, L, δ). y , [L], δ+ ) = Φ1 (

y , [L], δ− ). • If pg > 0, we have Φ1 (

• If pg = 0, we have the following equality: y , [L], δ+ ) − Φ1 (

y , [L], δ− ) = Φ1 (

NL (y1 ) NL (y)

(y1 ,y2 )∈S(y,δ)

Ψ (y1 , y2 )

M(

y1 ,

y2 ,[L],δ)

The cohomology classes Ψ (y1 , y2 ) are given as follows:

u · e−s/r1 ⊕ E

u · es/r2 P E Eu(Θ1,rel ) 1 2 Ψ (y1 , y2 ) = Res · u

· e−s/r1 , E

u · es/r2 s=0 NL (y1 ) Q E 1 2

(1.8)

Here, Θ1,rel denotes the relative tangent bundle of the naturally defined smooth y1 , [L], δ) −→ M(

y1 ). morphism Mss (

1.4 Rank Two Case 1.4.1 Dependence on Polarizations Since a 2-stability condition is always satisfied in the rank two case, we can study Problems 1.2 and 1.3 by using Theorem 1.3.1. We explain our result for Problem 1.2 in this subsection. Let a and n denote the first and second Chern classes of y.

10

1 Introduction

In the following, we identify H 4 (X, Z) and Z via the natural orientation of X. Let N S(X) denote the subgroup of H 2 (X, Z) generated by algebraic 1-cycles on X. The intersection pairing of α, β ∈ N S(X) is denoted by α · β. Let N S(X)R := N S(X) ⊗ R. The ample cone C is the cone in N S(X)R generated by the first Chern classes of ample line bundles. y ) to To distinguish the dependence on polarizations H, we use the symbol MH (

denote a moduli stack of torsion-free sheaves of type y which are semistable with respect to H.For ξ ∈ N S(X) such that a2 − 4n ≤ ξ 2 < 0, we set W ξ := c ∈ C c · ξ = 0 , which is called the wall determined by ξ ∈ N S(X). Connected components of C − W ξ are called chambers. It is well known that the moduli y ) depends only on the chamber to which H belongs. We define MH (

u ). ΦH (

y ) := P (E MH (

y)

We would like to study how ΦH (

y ) changes when polarizations cross a wall W ξ . We set S0 (y, ξ) := (a0 , a1 ) ∈ N S(X)2 a0 + a1 = a, a0 − a1 = m · ξ (m > 0) . For a non-negative integer l, let X [l] denote a Hilbert scheme of l-points in X. For any (a0 , a1 ) ∈ S0 (y, ξ), we put X(a0 , a1 ) := X [n0 ] × X [n1 ] . n0 +n1 =n−a0 ·a1

On X [n0 ] × X [n1 ] × X, we have the sheaf Iiu , which is the pull back of the universal ideal sheaf over X [ni ] × X via the natural projection. Let eai denote the holomorphic line bundle on X whose first Chern class is ai . It is uniquely determined up to isomorphisms, that X is simply connected in this intro since we have assumed duction. Let Q I0u ea0 −s , I1u ea1 +s be the equivariant Euler class of the following virtual Gm -equivariant vector bundle X [n0 ] × X [n1 ] : − RpX ∗ RHom I0u · ea0 −s , I1u · ea1 +s − RpX ∗ RHom I1u · ea1 +s , I0u · ea0 −s Theorem 1.4.1 Let C+ and C− be chambers which are divided by a wall W ξ . Let H+ and H− be ample line bundles contained in C+ and C− , respectively. We assume (H− , ξ) < 0 < (H+ , ξ). y ) = ΦH− (

y ). Namely, the invariants do not • In the case pg > 0, we have ΦH+ (

depend on the choice of generic polarizations. • In the case pg = 0, we have the following equality:

1.4 Rank Two Case

11

ΦH+ (

y ) − ΦH− (

y) =

(a0 ,a1 )∈S0 (y,ξ)

P I0u · ea0 −s ⊕ I1u · ea1 +s Res Q I0u · ea0 −s , I1u · ea1 +s X(a0 ,a1 ) s=0

It is called a weak wall crossing formula. Remark 1.4.2 Under the assumption that the wall W ξ is “good”, the weak wall crossing formula was proved for Donaldson invariants in [26] and [36], which was refined in [53]. Remark 1.4.3 In their amazing work [53], L. G¨ottsche-H. Nakajima-K. Yoshioka established the way to derive the wall crossing formula from the weak wall crossing formula! Remark 1.4.4 K. Yamada proved the independence of invariants from polarizations in some cases.

1.4.2 Reduction to Integrals over Hilbert Schemes Let us argue Problem 1.3 in the case rank(y) = 2. Let a and n denote the first and second Chern classes of y. We also assume pg > 0. For any a1 ∈ N S(X), we put a2 := a − a1 . Let eai denote a holomorphic line bundle whose first Chern class is ai . Let Iiu denote the universal ideal sheaves over X [ni] × X. Let Z i denote the universal 0-schemes over X [ni ] × X. We set Ξi := pX ∗ OZi ⊗ eai . We use the same symbols to denote the pull back of them via appropriate morphisms. Then, we obtain the following cohomology class on X [n1 ] × X [n2 ] : P I1u ea1 −s ⊕ I2u ea2 +s Eu(Ξ1 ) · Eu(Ξ2 e2s ) Ψ (a1 , a2 , n1 , n2 ) := Res s=0 (2s)n1 +n2 −pg Q I1u ea1 −s , I2u ea2 +s In the case c1 (O(1)) · a1 < c1 (O(1)) · a2 , we put A(a1 , y) := n1 +n2 =n−a1 ·a2

X [n1 ] ×X [n2 ]

In the case c1 (O(1)) · a1 = c1 (O(1)) · a2 , we set A(a1 , y) := n1 +n2 =n−a1 ·a2 n1 >n2

X [n1 ] ×X [n2 ]

Ψ (a1 , a2 , n1 , n2 )

Ψ (a1 , a2 , n1 , n2 )

Recall that an abelian pair is defined to be a pair of a holomorphic line bundle L and a section φ of L. For c ∈ N S(X), let M (c) denote a moduli space of abelian pairs (L, φ) such that c1 (L) = c. We can show the following proposition. (See Subsections 6.3.1–6.3.2.)

12

1 Introduction

Proposition 1.4.5 Assume that X is simply connected, and pg > 0. Moreover, we assume that the virtual fundamental class of M (c) is not 0. Then, the expected dimension of M (c) is 0. Hence, the virtual fundamental class of M (c) can be regarded as a number, denoted by SW(c). bethe SeibergLet a ∈ N S(X) be determined by 2c = a + c1 (KX ). Let SW(a) Witten invariant associated to a Spinc -structure ξ such that c1 det(ξ) = a. Then, SW(c) = SW(a). We put SW(a1 ) = 0 a1 ∈ N S(X) a1 · c1 (OX (1)) ≤ a · c1 (OX (1)) /2

SW(X, y) :=

Theorem 1.4.6 (Theorem 7.4.1) Assume pg > 0 and H 1 (X, O) = 0. Assume Py > PK and χ(y) − 1 ≥ 0, where K denotes the canonical line bundle of X, and we put χ(y) := Td(X) · y for the Todd class Td(X). Then, we have the following equality:

u ) + P (E Mss (

y)

SW(a1 ) · 21−χ(y) · A(a1 , y) = 0

a1 ∈SW(X,y)

Remark 1.4.7 The author learned the idea to use Bradlow pairs for reduction of Donaldson invariants to Seiberg-Witten invariants, in [21] and [42]. Theorem 1.4.6 gives some relation among the invariants. However, the above result is not completely satisfactory, because the author has no idea to calculate the coefficients 21−χ(y) · A(a1 , y).

u ), the integrals over Mss (

For an appropriate choice of P (E y ) and Mss (

y ·eω ) are the same. It is not clear how the coefficients depend on such twisting.

1.5 Higher Rank Case 1.5.1 pg > 0 Let us consider the higher rank case without assuming a 2-stability condition. If pg > 0 is satisfied, transition formulas (Theorems 1.3.1 and 1.3.2) can be generalized rather easily. Actually, we have the formally same result. Theorem 1.5.1 (Theorem 7.5.1) Assume pg > 0. Let Φ(

y , [L], δ) be as in (1.5). Then, the following equality holds:

1.5 Higher Rank Case

13

Φ(

y , [L], δ+ ) − Φ(

y , [L], δ− ) =

u · er1 (t−ω1 )/r2 · tk P E1u · e−t ⊕ E 2 Res Eu N0 (y1 , y2 ) M(y1 ,

y2 ,L) t=0

(1.9)

(y1 ,y2 )∈S1 (y,δ)

Here, we put S1 (y, δ) := (y1 , y2 ) ∈ S(y, δ) rank(y1 ) = 1 . We have used the symbol M(y1 , y 2 , L) instead of M(y1 , y 2 , L, δ), because a δ-semistability condition is trivial in the case rank(y1 ) = 1. y , [L], δ) Assume that the 1-vanishing condition holds for (y, L, δ), and let Φ1 (

be as in (1.7). Then, we have Φ1 (y, [L], δ+ ) = Φ1 (y, [L], δ− ). We can solve Problem 1.2. Let ΦH (

y ) be as in Subsection 1.4.1. Theorem 1.5.2 (Theorem 7.5.3) ΦH (

y ) are independent of the choice of generic polarizations in the case pg > 0. As for Problem 1.3, we obtain an immediate generalization of Theorem 1.4.6 by using the formula (1.9). We do not reproduce it here. See Subsection 7.5.2 for more details.

1.5.2 Transition Formula in the Case pg = 0 Our transition formula is comparatively complicated if pg = 0. We restrict ourselves to the case in which the 1-vanishing condition holds for (y, L, δ), and we consider y , [L], δ) as in (1.7). integrals Φ1 (

Preparation For each positive integer k, we put Sk (y, δ) := Y = (y1 , . . . , yk ) ∈ T ypek Pyi = Pyδ . For each element Y = (y1 , . . . , yk ) ∈ Sk (y, δ), we put |Y | = put k rank(yi ) . W (Y ) := 1≤j≤i rank(yj ) i=1

k i=1

yi . We also

We set S(y, δ) :=

(y0 , Y ) ∈ T ype × Sk (y, δ) y0 + |Y | = y .

k

For any (y0 , Y ) ∈ S(y, δ), we put

14

1 Introduction k M(

y0 , Y , [L]) := Mss y 0 , [L], δ− × Mss (

yi ). i=1

u denote the sheaf over M(

y0 , Y , [L]) × X which is obtained as the pull back Let E 0 of the universal sheaf over M(

y0 , [L], δ− ) × X via the natural projection. We use

u in similar meanings. the symbols E i The Induced Cohomology Classes on M(

y0 , Y , [L]) Let (y0 , Y ) ∈ S(y, δ), where Y = (y1 , . . . , yk ). Let G = Gkm be a k-dimensional torus. The G-equivariant bivariant group of a point R(G) is the limit of A∗ (Pm )k with respect to inclusions. It is naturally identified with Q[t1 , . . . , tk ], where ti corresponds to the first Chern class of the tautological line bundle of the i-th := R(tk , . . . , t1 ). (See Subsection 7.1.6 for Pm . It is contained in the ring R(G) R(tk , . . . , t1 ).) For variables t1 , . . . , tk , we set T0 := −

j>0

tj , 0≤h<j rank(yh )

Ti := −

ti tj + . rank(y ) rank(y h i) 0≤h<j

j>i

Let ew·ti denote the trivial line bundle provided with the G-action which is induced by the action of the i-th Gm of weight w. We have the following virtual G-equivariant vector bundle over M(

y0 , Y , [L]):

i · eTi , E

j · eTj − RpX ∗ RHom E

j · eTj , E

i · eTi −RpX ∗ RHom E We have their equivariant Euler classes

j · eTj ∈ A∗ M(

i · eTi , E y0 , Y , [L]) ⊗ R(G). Q E y0 , Y , [L]) ⊗ R(G). They are invertible in the ring A∗ M(

We set

1 · eT1 , . . . , E

k · eTk :=

i · eTi , E

j · eTj .

0 · eT0 , E Q E Q E i<j

Let Θ0 rel denote the vector bundle over M(

y0 , Y , [L]) obtained from the relative y0 , [L]). Then, we obtain tangent bundle of the smooth map M(

y0 , [L], δ) −→ M(

y0 , Y , [L]) : the following element of A∗ M(

k

u Ti P Eu(Θ0,rel ) i=0 Ei · e Ψ (y0 , Y ) := Res · · · Res ·

0 · eT0 , . . . , E

k · eTk t1 =0 tk =0 NL (y0 ) Q E

1.5 Higher Rank Case

15

Statement In a sense, much part of this paper is devoted to a proof of the following theorem. Theorem 1.5.3 We have the following formula: Φ1 (

y , [L], δ+ ) − Φ1 (

y , [L], δ− ) = NL (y0 ) (y0 ,Y )∈S(y,δ)

NL (y)

· W (Y ) ·

,[L]) M(

y0 ,Y

Ψ (y0 , Y ) (1.10)

1.5.3 Weak Intersection Rounding Formula We can study Problem 1.2 by using (1.10), and we obtain a generalization of a weak wall crossing formula. (See Theorem 7.7.1.) We do not reproduce it here. The formula itself is not so easy to treat, partially because it contains integrals over moduli stacks of semistable sheaves with higher ranks. To extract a more accessible quantity from our invariants, we consider “intersection rounding phenomena”, or in other words, “wall crossing of wall crossing”. The general case will be discussed in Section 7.8. In this subsection, we explain the result in the rank 3 case. See Subsection 7.7.4 for more details. W ξ2 H−+ q H−−

q

qH++ q

W ξ1

H+−

We take an element ξ = (ξ1 , ξ2 ) ∈ N S(X)2 such that ξ1 and ξ2 are linearly independent, and we put W ξ := W ξ1 ∩ W ξ2 . A connected component T of W ξ \ W =W ξi W is called a tile. For each tile T , there exist four chambers C++ , C+− , C−− and C−+ with the following properties: • The closure of Cκ1 ,κ2 contains T . • Take an ample line bundle Hκ1 ,κ2 ∈ Cκ1 ,κ2 . Then, the signature of the intersection pairing c1 (Hκ1 ,κ2 ) · ξi is κi We define y ) := ΦH++ (

y ) − ΦH+− (

y ) − ΦH−+ (

y ) + ΦH−− (

y ). DξT Φ(

We would like to express DξT Φ(

y ) as the sum of integrals over the products of Hilbert schemes.

16

1 Introduction

Let S(2, 1) be the set of a = (a0 , a1 , a2 ) ∈ N S(X)3 with the following property: • a0 + a1 − 2a2 = A1 · ξ1 and a0 − a1 = A2 · ξ2 for some Ai > 0. Let S(1, 2) be the set of a = (a0 , a1 , a2 ) ∈ N S(X)3 with the following property: • 2a0 − (a1 + a2 ) = A1 · ξ1 and a1 − a2 = A2 · ξ2 for some Ai > 0. For each a, we set N (y, a) = n + X(y, a) :=

a20 + a21 + a22 − a2 , 2

2

X [ni ] .

n0 +n1 +n2 =N (y,a) i=0

Here, a and n denote the first Chern class and the second Chern class of y, respectively. Proposition 1.5.4 y ) is independent of the choice of tiles T . Hence, we can omit to denote T . • DξT Φ(

• The following equality holds: y) = Dξ Φ(

a∈S(1,2)

+

P I0u ea0 −t1 ⊕ I1u ea1 +t1 /2−t2 ⊕ I2u ea2 +t1 /2+t2 Res Res t1 Q I0u ea0 −t1 , I1u ea1 +t1 /2−t2 , I2u ea2 +t1 /2+t2 X(y,a) t2

P I0u ea0 −t1 /2−t2 ⊕ I1u ea1 −t1 /2+t2 ⊕ I2u ea2 +t1 Res Res t1 Q I0u ea0 −t1 /2−t2 , I1u ea1 −t1 /2+t2 , I2u ea2 +t1 X(y,a) t2

a∈S(2,1)

It is called a weak intersection rounding formula in the rank 3 case. See Theorem 7.8.2 for the general case. Remark 1.5.5 The author expects that Dξ Φ(

y ) can be described in a more beautiful way like the wall crossing formula in the rank 2 case ([50], [53]). That is the reason why “weak” is added.

1.6 Master Space As mentioned in Preface, we use master spaces due to M. Thaddeus as one of the most important ingredient in this study. We explain how to utilize it in our situation.

1.6 Master Space

17

1.6.1 Master Space in the Case that a 2-Stability Condition is Satisfied We give an outline of the construction of a master space for our moduli stacks, when a 2-stability condition is satisfied (see Section 1.3). To begin with, we give a remark. It is known that coarse moduli schemes of semistable torsion-free sheaves are obtained as the categorical quotient of the sets of the semistable points of some projective varieties provided with actions of reductive groups. But, we will discuss moduli stacks of semistable parabolic sheaves or semistable parabolic L-Bradlow pairs, although we omit to mention parabolic structure in this introduction. The author does not know whether such a description is available for moduli of such parabolic objects. Hence, we need a modification of the construction for master spaces, which we explain in the paragraph “Construction”. But, it is rather technical. So the author recommends the reader to skip to the paragraph “Fixed point set”.

Construction Let y ∈ T ype. We put y(m) := y · ch O(m) . Let Hy be the Hilbert polynomial associated to y. Let Vm be an Hy (m)-dimensional vector space. We put Vm,X := Vm ⊗ OX . The projectivization of Vm is denoted by Pm . Let Q(m, y) denote a quot scheme, that is a moduli scheme of quotient sheaves of Vm,X whose Chern character is y(m). By using Q(m, y), it is easy to construct a moduli scheme Q(m, y ) of quotient sheaves of Vm,X with orientations whose Chern character is y(m). We fix an inclusion ι : O(−m) −→ L. Let δ ∈ P br . Then, we obtain the subscheme Qss (m, y, [L], δ) of Q(m, y) × Pm , which is the moduli scheme of the quotient sheaves (q : Vm,X → E) with non-trivial reduced L-sections [φ] : L −→ E(−m) (see Subsection 3.1.3 for reduced L-sections), satisfying the following condition: • The Chern character of E is y(m), and the pair (E(−m), [φ]) is δ-semistable. We put Qss (m, y , [L], δ) := Qss (m, y, [L], δ) ×Q(m,y) Q(m, y ). 2 Let det(y(m)) denote the H (X)-component ofi y(m). We take a line bundle L such that c1 (L) = det y(m) . We may assume H (X, L) = 0 (i > 0). We denote by Zm the projectivization of H 0 (X, L), which is called the Gieseker space. We put Am := Zm × Pm For a δ ∈ P br , the ample Q-line bundle L is given on Am as follows: L := OZm Pyδ (m) ⊗ OPm δ(m) We have a naturally defined SL(Vm )-action on Am , and L gives an SL(Vm )polarization. Let Ass m (L) denote the open subset of the semistable points with respect to L. It is standard to show that we have an SL(Vm )-equivariant closed immersion Qss (m, y, [L], δ) −→ Ass m (L). We take a large integer k such that L⊗ k is actually a line bundle. We take a nonzero rational number γ whose absolute value is sufficiently small. We put Lγ := L⊗ k ⊗ OPm (γ). Let Ass m (Lγ ) denote the open subset of the semistable points with

18

1 Introduction

respect to Lγ . We take δ− < δ < δ+ sufficiently closely. Let sign(γ) denote the signature of γ. It can be shown that we have a naturally induced closed immersion Qss (m, y, [L], δsign(γ) ) −→ Ass m (Lγ ). We take rational numbers γ2 < 0 < γ1 such that |γi | are sufficiently small. We take a large number k such thatk · (γ1 − γ2 ) = 1. Let π : Am −→ Pm denote the projection. We put B := P π ∗ OPm (0) ⊕ π ∗ OPm (1) , which is a P1 -bundle over Am . The tautological line bundle is denoted by OP (1). We put OB (1) := k ss denote the open OP (1) ⊗ L⊗ γ1 . It gives an SL(Vm )-polarization of B. Let B subset of the semistable points of B with respect to OB (1). We have natural inclusions B1 := P π ∗ OPm (0) ⊂ B, B2 := P π ∗ OPm (1) ⊂ B.

ss We remark OB (1)|Bi = L⊗k γi . Let Bi denote the semistable points of Bi with respect to OB (1). We set

TH

ss

:= Qss (m, y , [L], δ) ×A B ss ,

ss

i := Qss (m, y , [L], δ) ×A Biss . TH

ss

:= TH / GL(Vm ), which is the master space in this case. If the We put M

is Deligne2-stability condition is satisfied for (y, L, δ), it is easy to show that M Mumford and proper. ss

i := TH

1 Ms (

GL(Vm ). We can observe M We put M y , [L], δ+ ) and i s

y , [L], δ− ). M2 M (

Fixed Point Set

. The fixed point set is as follows: We have a naturally defined Gm -action on M

2

Gm (y1 , y2 )

1 M M M (y1 ,y2 )∈S(y,δ)

Gm (y1 , y2 ) are isomorphic to moduli stacks of (E1 , φ, E2 ; ρ) with the folHere, M lowing properties: • (E1 , φ) is a δ-stable L-Bradlow pair with ch(E1 ) = y1 . • E2 is a semistable torsion-free sheaf with ch(E2 ) = y2 . • ρ is an orientation of E1 ⊕ E2 .

Gm is isomorphic to Mss (y1 , L, δ) × Mss (

y2 ) up to It is easy to observe that M e´ tale proper morphisms. Namely, we have the following diagram of proper DeligneMumford stacks: a b

Gm ←−− M −− S −−−−→ Mss (y1 , L, δ) × Mss (

y2 )

Here, a and b are e´ tale proper morphisms.

1.6 Master Space

19

1.6.2 Gm -Localization Method (I)

to obtain the transition formula (Theorem We explain how to use the master space M 1.3.1), when the 2-stability condition is satisfied for (y, L, δ). We use the notation in Section 1.3. Let M(

y , [L]) denote a moduli stack of oriented reduced L-Bradlow

−→ M(

pairs of type y. Let ϕ : M y , [L]) be the naturally defined morphism. Let T (1) denote a trivial line bundle on M(

y , [L]) with the Gm -action of weight 1. We

u , ϕ∗ Orel (1) and ϕ∗ T (1). Therefore, have naturally induced Gm -actions on ϕ∗ E

: we obtain the following Gm -equivariant cohomology classes on M

u · c1 ϕ∗ Orel (1) k , Φt := P ϕ∗ E

Φt := Φt · c1 ϕ∗ T (1)

The master space is naturally provided with a perfect obstruction theory, and hence we class. Therefore, we can consider the polynomial have a virtual fundamental ∗ Φ ∈ Q[t]. Since ϕ T (1) is the trivial line bundle, the specialization at t = 0 is t

M trivial. We use the localization theory of virtual fundamental classes due to GraberPandharipande [55]. We have the following equality in Q[t, t−1 ]:

M

Φt =

i=1,2

i M

t · Φt

i )) Eu(N(M

+

(y1 ,y2 )∈S(y,δ)

Gm (y1 ,y2 ) M

t · Φt

Gm (y1 , y2 )) Eu N(M

Gm (y1 , y2 )) denote the virtual normal bundles. Therefore,

i ) and N(M Here, N(M we obtain the following equality:

i=1,2

Φt

i ) Eu N(M

Res

i t=0 M

+

(y1 ,y2 )∈S(y,δ)

Res

Gm (y1 ,y2 ) t=0 M

Φt

Gm (y1 , y2 )) Eu N(M

= 0.

(1.11)

It is easy to observe that the first term of the left hand side of (1.11) can be rewritten as follows: u k

u ) · ω k = −Φ(

P (E ) · ω + P (E y , [L], δ+ ) + Φ(

y , [L], δ− ) −

1 M

2 M

Gm (y1 , y2 ) is isomorphic to M(y1 , y 2 , L, δ) up to e´ tale proper morNote that M phisms. By a formal calculation, it can be shown that the second term in (1.11) equals the right hand side of (1.6). Thus, we obtain the transition formula in the case that the 2-stability condition is satisfied for (y, L, δ).

20

1 Introduction

1.6.3 Enhanced Master Space

in Subsection 1.6.1 is not If the 2-stability condition is not satisfied for (y, L, δ), M Deligne-Mumford. So we introduce enhanced master space which we explain in this subsection. Although this subsection is quite technical, it is one of the cores in this monograph. so we include it in this introduction. We use the notation in Subsection 1.6.1.

Construction Take a sufficiently large m. We set N := Hy (m) = dim Vm . Let Flag(Vm , N ) denote the full flag variety of Vm : Flag(Vm , N ) = F1 ⊂ F2 ⊂ · · · ⊂ FN = Vm dim Fi = i Then, we set Bi := Bi × Flag(Vm , N ).

B := B × Flag(Vm , N ),

Let Gl (Vm ) denote the Grassmannian variety of l-dimensional subspaces of Vm , which has a canonical polarization OGl (1). We have the morphism ρl : Flag(Vm , N ) −→ Gl (Vm ) given by ρl (F∗ ) = Fl . Take small positive numbers ni (i = 1, 2, . . . , N ). We obtain the following SL(Vm )-polarization of B: OB (1) := OB (1) ⊗

N

ρ∗l OGl (Vm ) (ni )

l=1

Let Bss and Biss denote the set of semistable points with respect to OB (1). In this case, we set TH

ss

ss

i := Qss (m, y , [L], δ) ×A Biss . TH

:= Qss (m, y , [L], δ) ×A Bss ,

And, we define

ss

:= TH / GL(Vm ). M

It is called an enhanced master space. Proposition 1.6.1 (Proposition 4.3.3 and Proposition 4.5.1) The enhanced mas is Deligne-Mumford and proper. ter space M ss

i := TH / GL(Vm ). It can be shown that M

1 is the full flag bundle We put M i s

u (m). Similarly, M

2 is over M (

y , [L], δ+ ) associated to the vector bundle pX ∗ E s y , [L], δ− ). the full flag bundle over M (

1.6 Master Space

21

Fixed Point Set

. To describe the exceptional fixed We have a naturally defined Gm -action on M point sets, we need some preparation. Definition 1.6.2 Let y ∈ T ype, δ ∈ P br and m ∈ Z>0 . A decomposition type for (m, y, δ) is defined to be a datum I := (y1 , y2 , I1 , I2 ) as follows: • y = y1 + y2 in T ype such that Pyδ1 = Pyδ . • N = I1 I2 such that |Ii | = Hyi (m), where Hyi denote the Hilbert polynomials associated to yi . The set of decomposition types for (m, y, δ) is denoted by Dec(m, y, δ). We put k(I) := min(I2 ) − 1 for I = (y1 , y2 , I1 , I2 ) ∈ Dec(m, y, δ). We introduce the notion of (δ, )-semistability. Definition 1.6.3 Let (E, φ) be an L-Bradlow pair on X, and let F be a full flag of H 0 X, E(m) . Let be any positive integer. We say that (E, φ, F) is (δ, )semistable, if the following conditions are satisfied: • (E, φ) is δ-semistable. • Take any Jordan-H¨older filtration of (E, φ) with respect to δ-semistability: E (1) ⊂ E (2) ⊂ · · · ⊂ E (i−1) ⊂ (E (i) , φ) ⊂ · · · ⊂ (E (k) , φ) Then, we have F ∩ H 0 X, E (i−1) (m) = {0} and F ⊂ H 0 X, E (j) (m) for j < k. ss y, L, (δ, ) denote a moduli stack of such tuples (E, φ, F). In the Let M ss y , L, (δ, ) as usual. We obtain a stack oriented case, we use the symbol M ss y , L, (δ, ) as in the case of Mss (

ss y , [L], (δ, ) from M y , [L], δ). M Then, the fixed point set is as follows:

1 M

2 M

Gm (I) M

I∈Dec(m,y,δ)

Gm (I) is isomorphic to a moduli stack of objects (E1 , φ, E2 , ρ, F (1) , F (2) ) Here, M with the following properties: (1) 0 • (E1 , φ) is a δ-semistable L-Bradlow pair, and F is a full flag of H (X, E1 (m)) (1) such that (E1 , φ, F ) is δ, k(I) -semistable. • E2 is a semistable torsion-free sheaf, and F (2) is a full flag of H 0 (X, E2 (m)) (2) such that E2 , Fmin(I2 ) is -semistable reduced O(−m)-Bradlow pair, where denotes any sufficiently small positive number. • ρ is an orientation of E1 ⊕ E2 .

Gm (I) is isomorphic to M ss y1 , L, (δ, k(I)) × M ss (

Therefore, M y2 , +) up to ss e´ tale proper morphisms, where M (

y2 , +) denotes a moduli stack of oriented semistable sheaves (E2 , ρ2 ) with full flags F (2) as in the above condition.

22

1 Introduction

1.6.4 Gm -Localization Method (II)

to obtain the transition forWe explain how to use the enhanced master space M mulas. The argument is partially the same as that in Subsection 1.6.2. We have the

−→ M(

naturally defined morphism ϕ : M y , [L]). Let T (1) denote the trivial line bundle on M(

y , [L]) with the Gm -action of weight 1. We have the naturally defined

u , ϕ∗ Orel (1), ϕ∗ Θ rel and ϕ∗ T (1). We consider the following Gm -action on ϕ∗ E

cohomology classes on M :

u ) · c1 ϕ∗ Orel (1) k · Eu(Θrel ) , t := P (ϕ∗ E Φ Hy (m)!

t · c1 ϕ∗ T (1) Φt := Φ

By the argument in Subsection 1.6.2, we obtain the following equality: i=1,2

t Φ

i ) Eu N(M

Res

i t=0 M

+

I∈Dec(m,y,δ)

Res

Gm (I) t=0 M

t Φ

Gm (I))) Eu(N(M

= 0.

(1.12)

i ) and N(M

Gm (I)) denote the virtual normal bundles. Since M

i are the Here, N(M ss y , [L], δκ ) (κ = ±), it is not difficult to observe that the full flag bundles over M (

y , [L], δ− ). first term of the left hand side of (1.12) is equal to −Φ(

y , [L], δ+ ) + Φ(

In the case pg > 0, the second term of (1.12) is significantly simplified, because we

Gm can show vanishing of the virtual fundamental class of M (I) if rank(y1 ) > 1, where I = (y1 , y2 , I1 , I2 ). If rank(y1 ) = 1, the δ, k(I) -semistability condition ss y1 , L, (δ, k(I)) is the full flag bundle over M(y1 , L) is trivial. Therefore, M ss y2 , +) is the associated to the vector bundle pX ∗ E1u (m). On the other hand, M (

ss flag variety bundle over M y 2 , [O(−m)], for any sufficiently small positive number . (See Subsection 4.6.1 for more details.) Hence, we can easily observe that the second term of the left hand side of (1.12) is equal to the right hand side of (1.9). Thus, we obtain the transition formula in the case pg > 0. In the case pg = 0, we need more additional argument. In this case, we can iden ss y 1 , [L], (δ, ) . ss y1 , L, (δ, ) and M tify the virtual fundamental classes of M y , [L], δ− ) Thanks to the equality (1.12), we obtain the expression Φ(

y , [L], δ+ )−Φ(

as the sum of terms of the following form: Ψ (y1 , y2 ) (1.13) ss (

y1 ,[L],(δ, ))×Mss (

y2 ) M

ss (

y1 , [L], (δ, )) as in the case of We do not have a simple description of M ss M (

y , [L], δ± ). Hence, (1.13) are not directly related with integrals over the product of moduli stacks of reduced L-Bradlow pairs and semistable sheaves.

1.6 Master Space

23

1 connecting M ss y 1 , [L], (δ, ) and So we take the enhanced master space M ss y 1 , [L], δ− . Such a space can be constructed by the method in Subsection M 1.6.3. We have only to choose appropriate numbers ni (i = 1, . . . , N ). Let S( ) de(2) (2) (2) (2) note the set of decomposition types I(2) = (y1 , y2 , I1 , I2 ) with the following property: (2)

(2)

• y1 + y2 = y1 . (2) (2) (2) • I1 I2 = 1, . . . , Hy1 (m) and |Ii | = Hy(2) (m). i

(2)

• {1, . . . , } ⊂ I1 . Then, applying the localization method again, we obtain the following:

Ψ (y1 , y2 ) =

ss (

y1 ,[L],(δ, ))×Mss (

y2 ) M

+

I(2) ∈S( )

Ψ (y1 , y2 )

ss (

y1 ,[L],δ− )×Mss (

y2 ) M (2)

(2)

Ψ (2) (y1 , y2 , y2 )

Gm (I(2) )×Mss (

y2 ) M 1

(2)

(1.14)

We can apply this procedure inductively. Note rank(y1 ) < rank(y1 ) < rank(y). Hence, the inductive process will stop. In principle, we can obtain the general transition formula. However, the general formula would be comparatively complicated, and it is less interesting for the author at this moment. We restrict ourselves to the transition formula in the special case, as in Theorem 1.5.3.

Chapter 2

Preliminaries

In Section 2.1, we prepare some convention. In Section 2.2, we review basic results from the geometric invariant theory. In particular, we recall a sufficient condition for a quotient stack to be Deligne-Mumford and proper. We also recall Mumford-Hilbert criterion, and look at some easy examples. The results will be used in Chapter 4. In Section 2.3, we review some basic facts on cotangent complexes. Then, we recall how to express cotangent complexes of quotient stacks in Subsection 2.3.2, which will be used in Chapter 5 frequently. We also study some more examples in Subsection 2.3.3, which will be used in Sections 6.3, 6.4 and 6.6. In Section 2.4, we review obstruction theory in the sense of K. Behrend-B. Fantechi [6]. We explain a naive strategy to construct obstruction theories of moduli stacks in Subsection 2.4.2. We recall an obstruction theory of locally free subsheaves in Subsection 2.4.3. It gives obstruction theories of moduli spaces of torsion-free quotient sheaves over a smooth projective surface. The result will be used in Section 5.6. We also obtain the smoothness of moduli spaces of quotient torsion-sheaves over a smooth projective curve, although we will not use it later. In Subsection 2.4.4, we recall an obstruction theory of filtrations of a vector bundle on a smooth projective curve. It will be used to construct a relative obstruction theory for quasi-parabolic structures. In Section 2.5, we recall some standard results for equivariant complexes on Deligne-Mumford stacks with GIT construction, which will be used in Section 5.9. In Section 2.6, we give some elementary remarks on extremal sets, which are used in Sections 4.3–4.4. In Section 2.7, we give remarks on the twist of line bundles.

2.1 Some Convention 2.1.1 Product and Projection Let S be a scheme. Let Y be an algebraic stack over S. Let g : T −→ U be a morphism of algebraic stacks over S. The naturally induced morphism T ×S Y −→ U ×S Y is denoted by gY or simply by g. T. Mochizuki, Donaldson Type Invariants for Algebraic Surfaces: Transition of Moduli Stacks, Lecture Notes in Mathematics 1972, DOI: 10.1007/978-3-540-93913-9 2, c Springer-Verlag Berlin Heidelberg 2009

25

26

2 Preliminaries

Let X and U be algebraic stacks over S. We use the symbol pX to denote the projection forgetting the X-component: pX : U ×S X −→ U,

pX (u, x) = u

Similarly, pU denotes the projection U ×S X −→ X.

2.1.2 Vector Bundles Let V be a vector bundle on an algebraic stack Y . The sheaf of local sections of V is also denoted by the same symbol V , if there are no risk of confusion. But, we use some particular notation in the following case: For vector bundles Vi (i = 1, 2), let Hom(V1 , V2 ) denote the sheaf of homomorphisms from V1 to V2 . The corresponding vector bundle is denoted by N (V1 , V2 ). Let F be a vector bundle on Y . The complement of the image of the 0-section in F is denoted by F ∗ , i.e., F ∗ := F − Y , and the dual bundle of F is denoted by F ∨ . The projectivization of F is denoted by P(F ∨ ) or PF .

2.1.3 Coherent Sheaves on a Product Let X be a flat scheme over S, and let U be an algebraic stack over S. A coherent sheaf E over U ×S X is called a U -coherent sheaf, if it is flat over U . A U -coherent sheaf E is called a U -torsion free sheaf, if E|{u}×S X is torsion-free for each u ∈ U . We will often omit to denote “U -”, if there are no risk of confusion. When we are given a line bundle OX (1) on X which is relatively ample over S, we use the symbol E(m) to denote E ⊗ p∗U OX (m) for any coherent sheaf E on U ×S X.

2.1.4 Quotient Stacks Let Z be an algebraic stack over S provided with an action of a group scheme G over S. Then, we use the symbols ZG or Z/G to denote the quotient stack.

2.1.5 Signature in Complexes We follow the signature convention in [68]. We recall some of them for later use in our situation. Let X be an algebraic stack over S. For two bounded OX -complexes C • and D• , let Hom(C • , D• ) denote the complex whose i-th terms are

2.1 Some Convention

27

Hom C j , Dk ,

k−j=i

and whose differentials are given as follows: Hom C j , Dk −→ Hom C j , Dk+1 ⊕ Hom C j−1 , Dk a −→ dD ◦ a, (−1)k−j+1 a ◦ dC Let us look at some examples. For a complex C • , we denote the dual complex Hom(C • , OX ) by C • ∨ . The differentials are as follows: Hom(C n , OX ) −→ Hom(C n−1 , OX ),

a −→ (−1)n+1 · a ◦ dX

−1 0 • −1 For two term complexes C • = → D0 ), the → C ) and D = (D (C • • differentials of the complex Hom C , D are given as follows:

Hom C 0 , D−1 −→ Hom C 0 , D0 ⊕ Hom C −1 , D−1 a −→ dD ◦ a, a ◦ dC Hom C 0 , D0 ⊕ Hom C −1 , D−1 −→ Hom C −1 , D0 (b1 , b2 ) −→ −b1 ◦ dC + dD ◦ b2 We will often use the dual Hom(C • , D• )∨ whose differentials are given as follows: Hom D0 , C −1 −→ Hom(D0 , C 0 ) ⊕ Hom(D−1 , C −1 ) a −→ (−dC ◦ a, a ◦ dD )

Hom(D , C ) ⊕ Hom(D−1 , C −1 ) −→ Hom D−1 , C 0 0

0

(b1 , b2 ) −→ −b1 ◦ dD − dC ◦ b2

2.1.6 Filtrations and Complexes on a Curve Let D be a smooth projective curve over S. Let Ea (a = 1, 2) be coherent OD modules which are flat over S. Assume that we are given a decreasing filtration F (Ea ) = (Fi (Ea ) | i = 1, . . . , l) of Ea such that Coki (Ea ) := Ea /Fi+1 (Ea ) are flat over S. Let Va,• = (Va,−1 → Va,0 ) be locally free resolutions of Ea (a = 1, 2). We set Va(1) := Va,0 ,

Va(l+1) = Va,−1 ,

Va(i) := Ker Va,0 −→ Coki (Ea ) ,

(i = 2, . . . , l).

28

2 Preliminaries (i+1)

(i)

(i)

(1)

(l+1)

(i)

Let fi : VD −→ VD , ti : VD −→ VD and si : VD −→ VD denote the inclusions. Let us consider the complex C1 (V1∗ , V2∗ ) given as follows: l+1 l (1) (l+1) d−1 (i) (i) d0 (i+1) (i) Hom V1 , V2 −→ Hom V1 , V2 −→ Hom V1 , V2 i=1

i=1

Here, the first term stands in the degree −1. The differentials di are given as follows: (2.1) d−1 (a) = si ◦ a ◦ ti i = 1, . . . , l + 1 d0 (b1 , . . . , bl ) = −f1 ◦b1 +b2 ◦f1 , −f2 ◦b2 +b3 ◦f2 , . . . , −fl ◦bl +bl+1 ◦fl (2.2) We have the naturally defined morphism: ϕ = (ϕi ) : C1 (V1∗ , V2∗ ) −→ Hom V1,• , V2,•

(2.3)

(1) More precisely, ϕ0 is the projection induced and by the identifications V0 = V (l+1) , ϕ1 is given by ϕ1 (ai ) = si+1 ◦ ai · ti , and ϕ2 is the identity. We V−1 = V can directly check that ϕ is the morphism of complexes. We put

C2 (V1∗ , V2∗ ) := Cone(ϕ)[−1]. The following lemma is easy to check. Lemma 2.1.1 The complexes Ci (V1∗ , V2∗ ) and the morphism ϕ : C1 (V1∗ , V2∗ ) −→ Hom(V1 • , V2 • ) depend only on (E1 , F ) and (E2 , F ) in the derived category D(D). Notation 2.1.2 We denote Ci (V1∗ , V2∗ ) by RHom i (E1∗ , E2∗ ).

If Ea and Ea /Fj (Ea ) are locally free sheaves for a = 1, 2 and j = 1, . . . , l, then we have vanishings Hi Hom 1 (E1 , E2 ) = 0 (i = 0), and H0 Hom 1 (E1 , E2 ) is isomorphic to the sheaf of homomorphisms of E1 to E2 which preserve the filtrations.

2.1.7 Virtual Vector Bundle Let G be a group scheme over S. Let Y be an algebraic stack over S provided with a (possibly trivial) G-action. Let KG (Y ) denote the K-group of G-equivariant perfect complexes. Elements of KG (Y ) are called virtual G-equivariant vector bundles in this monograph. We often omit to distinguish G.

2.2 Geometric Invariant Theory

29

2.1.8 Compatible Diagrams Let Ai,j (i = 1, 2) (j = 1, 2, 3, 4) be objects in some category. Assume that we are given morphisms ϕj : A1,j −→ A2,j . We also assume that we are given commutative diagrams (CD)i : ai → Ai,2 Ai,1 −−−− ⏐ ⏐ ⏐ ⏐ ci ! bi ! d

Ai,3 −−−i−→ Ai,4 We say that (CD)1 and (CD)2 are compatible with respect to the morphisms ϕj (j = 1, 2, 3, 4), if every face of the naturally obtained cube is commutative. It is equivalent to the commutativity of the following diagrams: A1,1 −−−−→ A1,2 ⏐ ⏐ ⏐ ⏐ ! !

A1,1 −−−−→ A1,3 ⏐ ⏐ ⏐ ⏐ ! !

A2,1 −−−−→ A2,2

A2,1 −−−−→ A2,3

A1,2 −−−−→ A1,4 ⏐ ⏐ ⏐ ⏐ ! !

A1,3 −−−−→ A1,4 ⏐ ⏐ ⏐ ⏐ ! !

A2,2 −−−−→ A2,4

A2,3 −−−−→ A2,4

2.2 Geometric Invariant Theory 2.2.1 GIT Quotient and Algebraic Stacks Let k be an algebraically closed field with characteristic 0. Let G be a linear reductive group over k. Let Y be a projective variety over k, provided with a G-action ρ. Let L be an ample line bundle on Y with a G-action which is a lift of ρ. The lift is also denoted by ρ. We recall some basic definitions. A point y ∈ Y is semistable with respect to L, if there exists a G-invariant section s of L⊗ n for some n > 0 such that s(y) = 0. A point y ∈ Y is defined to be stable with respect to L, if there exists a G-invariant section s of L⊗ n for some n > 0 such that (i) s(y) = 0, (ii) any G-orbits contained in Y −s−1 (0) are closed. Let Y s (L) (resp. Y ss (L)) denote the set of the stable (resp. semistable) points with respect to L. The fundamental theorem of D. Mumford is the following. Proposition 2.2.1 ([96]) There exists a uniform categorical quotient π : Y −→ Y ss //G. Moreover, the following holds:

30

2 Preliminaries

• The map π is affine and universally submersive. • Y ss //G is a projective variety. • There exists the open subset Y s //G of Y ss //G, such that (i) π −1 (Y s //G) = Y s , (ii) π : Y s −→ Y s //G is a universal geometric quotient of Y s . Proof See Proposition 1.9, Theorem 1.10 and Page 40 in [96].

We combine it with some results of A. Vistoli in [129]. Let Y sf denote the set of the stable points of Y whose stabilizers are finite. In this situation, we obtain the quotient stack Y sf /G, which is Deligne-Mumford. See Sections 2 and 7 of [129] for more details on such quotient stacks. We recall one of his results. Proposition 2.2.2 ([129]) The naturally induced morphism Y sf /G −→ Y sf //G is proper. Proof The map Y sf −→ Y sf //G is a universal geometric quotient. In particular, it is universally submersive, and the geometric fibers are precisely the orbits of geometric points of X. Therefore, Y sf //G is a quotient of Y sf by G in the sense of Vistoli. (See Page 630 of [129].) Applying Proposition 2.11 of [129], we can conclude that the map Y sf /G −→ Y sf //G is proper. Corollary 2.2.3 Let Z be a variety over k with a G-action. Let Φ : Z −→ Y be a G-equivariant immersion with the following property: • The stabilizer groups of any points of Z are finite. • The image Φ(Z) is contained in Y s (L). • Φ : Z −→ Y ss (L) is proper. Then, Z/G is Deligne-Mumford and proper. Proof We can regard Z/G as a substack of Y sf /G. We can also regard Z//G as a closed subscheme of Y sf //G ⊂ Y ss //G. Since Y ss //G is projective, Z//G is also projective. According to the previous lemma, the morphism Z/G −→ Z//G is proper. Therefore, Z/G is proper.

2.2.2 Mumford-Hilbert Criterion and Some Elementary Examples Let Y , L and G be as above. Let λ : Gm −→ G be a one-parameter subgroup. We put P (λ) := limt→0 λ(t)·P . Then, λ acts on the fiber L|P (λ) . The weight is denoted by μλ (P, L). Proposition 2.2.4 (Mumford-Hilbert criterion, [96]) The point P is semistable (resp. stable) with respect to L, if and only if μλ (P, L) ≥ 0 (resp. μλ (P, L) > 0) for any one-parameter subgroup λ. Remark 2.2.5 We use the convention to identify a vector bundle and the sheaf of its sections. Hence, the above definition of μλ is the same as that given in [96].

2.2 Geometric Invariant Theory

31

For later use, we recall some elementary examples. Let V be a vector space over an algebraically closed field k of characteristic 0 with a base u1 , . . . , uN . Take wi = 0 and wi ≤ wi+1 . Let λ be the one-parameter w1 , . . . , wN ∈ Z such that subgroup of SL(V ) given by λ(t) · ui = twi · ui . Let V (i) denote the subspace Vw denote the weight decomposition of λ, generated by u1 , . . . , ui . Let V = i.e., λ preserves the decomposition, and the action on Vw is the multiplication of tw . We put Gj := w≤j Vw . We denote a point of P(V ∨ ) by [v] by using a representative v ∈ V − {0}. Let us consider the right SL(V )-action on P(V ∨ ) given by g · [v] := [g −1 (v)], which can be lifted to the action on OP(V ∨ ) (1). Lemma 2.2.6 ([96]) μλ [v], OP(V ∨ ) (1) = min i vi ∈ Gi . In other words, μλ [v], O(1) = wi · dim V (i) ∩ v − dim V (i−1) ∩ v i

=

j · dim Gj ∩ v − dim Gj−1 ∩ v .

(2.4)

j

Here v denotes the subspace generated by v. Proof According to the weight decomposition V = Vi , we have the decompo sition v = vi . In P(V ∨ ), we have % " # $ −i λ(t)[v] = λ(t)−1 v = t · vi . We put i0 := max i | vi = 0 = min i v ∈ Gi . It is easy to see lim λ(t)[v] = [vi0 ].

t→0

The weight of λ on OP(V ∨ ) (1)|[vi0 ] is i0 . Thus, the first claim is obtained. The second claim follows from the first one. Let Gl (V ) denote the Grassmann variety of l-dimensional subspaces of V : Gl (V ) := ι : W ⊂ V dim W = l . &l ∨ &l given by W −→ We have the Pl¨ucker embedding Gl (V ) −→ P V W ⊂ &l V . It induces a polarization OGl (V ) (1) of Gl (V ). The group SL(V ) has the right action on Gl (V ) given by ι −→ g −1 ◦ ι, which can be lifted to that on OGl (V ) (1). Lemma 2.2.7 For any point W of Gl (V ), we have the following equality: N rank W ∩ V (i) − rank W ∩ V (i−1) · wi μλ W, OGl (V ) (1) = i=1

=

j∈Z

j · dim

W ∩ Gj . W ∩ Gj−1

(2.5)

32

2 Preliminaries

Proof For any J = (j1 < j2 < · · · < jl ), we put uJ := uj1 ∧ · · · ∧ ujl and l &l denote the such uJ gives a base of V . Let λ wJ := i=1 wji . Collection of &l wJ V induced by λ. We have λ(t)(u one-parameter subgroup of SL J ) = t · uJ . Let us take a base v1 , . . . , vl of W of the form v h = uih + j wI . We have μλ W, OGl (V ) (1) = μλ z, OP(& l V ∨ ) (1) = wI according to Lemma 2.2.6. Then, it is easy to derive the claim of the lemma. We also have the Grassmann variety G l of l-dimensional quotients: G l (V ) := q : V −→ Q dim Q = l &l V given by the correspondence We have the Plucker embedding G l (V ) −→ P &l &l &l q −→ q: V −→ Q . It induces a polarization OGl (V ) (1). Lemma 2.2.8 ([96], [87]) Let q : V −→ Q be a point of G l (V ). We put W := Ker(q). Then, we have the following equality: N wi · dim V (i) ∩ W − dim V (i−1) ∩ W − 1 μλ q, OGl (V ) (1) = i=1

( ' W ∩ Gj Gj . j · dim − dim = W ∩ Gj−1 Gj−1 j=1 N

(2.6)

Proof We put W (i) := Gi ∩ W Gi−1 ∩ W . By using the natural isomorphism Vi . It is easy to see that the limit Gi /Gi−1 Vi , we regard W (i) as the subspaces of q

: V −→ Vi /W (i) . The weight of λ on limt→0 λ(t) · q is given by the quotient (i) . Then, it is easy to deduce the claim. OGl (V ) (1)| q is −i · dim Vi /W Remark 2.2.9 We have the obvious isomorphism Gl (V ) G N −l (V ). However, it does not preserve the semistability conditions on the varieties induced by the Pl¨ucker embeddings.

2.3 Cotangent Complex 2.3.1 Basic Facts Recall some fundamental property of cotangent complexes from [64], [79] and [111]. Let X and Y be Deligne-Mumford stacks with e´ tale site. For any morphism f : X −→ Y of Delinge-Mumford stacks, the cotangent complex was introduced by L. Illusie [64] as a complex of OX -modules. It is denoted by LX /Y or Lf . Recall that the cotangent complex controls deformations of f in the following sense

2.3 Cotangent Complex

33

(Section 3 [64]). Let T be a scheme over Y, and let h : T −→ X be a Y-morphism. Let T be a Y-scheme such that T is a closedY-subscheme of T and the corresponding ideal J is square-zero, i.e., J 2 = {f · g f, g ∈ J} = 0. Proposition 2.3.1 (Illusie, [64]) We have the obstruction class o(h) ∈ Ext1 (h∗ LX /Y , J) with the following property: • The morphism h can be extended over T , if and only if o(h) vanishes. In the case o(h) = 0, the set of the extension classes is a torsor over the group Ext0 h∗ LX /Y , J . Cotangent complexes have a nice functorial property. For example, we have the distinguished triangle for a morphism Y −→ Z, f ∗ LY/Z −→ LX /Z −→ LX /Y −→ f ∗ LY/Z [1] in the derived category D(X ). As for general Artin stacks with lisse-´etale site, cotangent complexes with some good functorial property have been introduced by G. Laumon, L. Moret-Bailly and M. Olsson (Section 17 of [79] and Section 8 of [111]). For any Artin stack X , Olsson

(X ) of the projective systems introduced the category Dqcoh K = (· · · → K≥−n−1 → K≥−n → · · · → K≥ 0 ) in D+ (X ) such that K≥ −n −→ τ≥−n K≥−n and τ≥ −n K≥ −n−1 −→ τ≥ −n K≥ −n are isomorphisms. Here τ≥ −n denotes the canonical n-th truncation functor. See

(X ). Let f : X −→ Y be a quasi-compact [111] for the functorial property of Dqcoh and quasi-separated morphism of Artin stacks. Then, we can associate −n−1 −n ≥0

→ L≥ LX /Y = Lf = · · · → L≥ X /Y X /Y → · · · → LX /Y ∈ Dqcoh (X ) to each f with the following property (Theorem 8.1 [111]): −n • If X and Y are algebraic spaces, L≥ X /Y are isomorphic to τ≥ −n LX /Y in + Dqcoh (X ), where the latter LX /Y denotes the usual cotangent complex defined by Illusie. • When we are given a 2-commutative diagram of Artin stacks f

X −−−−→ ⏐ ⏐ g! h

X ⏐ ⏐ !

Y −−−−→ Y,

34

2 Preliminaries

we have the functorial morphism Lf ∗ LX /Y −→ LX /Y . If the diagram is 2-Cartesian, and if one of g or h is flat, then the morphism Lf ∗ LX /Y −→ LX /Y is an isomorphism. • Let f : X −→ Y be a morphism of Artin stacks. Let g : Y −→ Z be another morphism. Then, we have the distinguished triangle Lf ∗ LY/Z −→ LX /Z −→ LX /Y −→ Lf ∗ LY/Z [1]

(X ). in Dqcoh

The following properties can be derived directly from the construction. (See Section 8 of [111] for the construction of LX /Y .) −n • Each L≥ X /Y is an object in Dqcoh (X ). • If f is smooth and representable, then LX /Y is quasi-isomorphic to its 0-th cohomology sheaf, which is isomorphic to the locally free sheaves of Kahler dif−n ferentials ΩX /Y . In general, if f is smooth, any L≥ X /Y is of perfect amplitude [−n,1]

0 contained in [0, 1]. In particular, they are isomorphic to L≥ X /Y .

Remark 2.3.2 M. Aoki generalized the deformation theory of Illusie, and showed a natural generalization of Proposition 2.3.1 for Artin stacks [2].

2.3.2 Quotient Stacks Let S be a variety. Let G be a group scheme smooth over S. Let Y be a smooth S-scheme with a G-action. The quotient stack is denoted by YG . Let Z be an Artin stack over S with a morphism F : Z −→ YG . We have the corresponding G-torsor P (F ) over Z and the G-equivariant map F : P (F ) −→ Y : F

P (F ) −−−−→ Y ⏐ ⏐ ⏐ ⏐ π! ! Z

F

−−−−→ YG

Let us describe F ∗ LYG /S on Z. We have a map α : F ∗ ΩY /S −→ ΩP (F )/Z on P (F ), which is the composite of the differential F∗ ΩY /S −→ ΩP (F )/S and the natural projection ΩP (F )/S −→ ΩP (F )/Z . Proposition 2.3.3 F ∗ LYG /S is represented by the descent of Cone(−α)[−1] with respect to the natural G-action. Proof We recall the construction of LYG /S in this case. We set m+1

Y

(m)

) *+ , := Y ×YG · · · ×YG Y .

2.3 Cotangent Complex

35

(m) We have the YG −→ S. We obtain complexes natural morphisms Y −→ (m) (m) := ΩY (m) /S −→ ΩY (m) /YG on Y , where ΩY (m) /S stands in the C degree 0. We have the strictly simplicial structure given by the naturally defined quasi-isomorphisms πi∗ C (m−1) −→ C (m) (i = 0, 1, . . . , m), where πi denote By definition, the projections Y (m) −→ Y (m−1) forgetting the i-th components.

(YG ) is represented by C (m) m = 0, 1, . . . . LYG /S ∈ Dqcoh m+1

) *+ , We put P (F ) := P (F ) ×Z · · · ×Z P (F ). We have the naturally defined mor

(Z) is represented phisms F (m) : P (F )(m) −→ Y (m) . Then, F ∗ LYG /S ∈ Dqcoh (m)∗ (m) m = 0, 1, . . . . We have the following commutative diagram: by F C (m)

F (m) ∗ ΩY (m) /S −−−−→ F (m) ∗ ΩY (m) /YG ⏐ ⏐ ⏐ ⏐ =! ! b

F (m) ∗ ΩY (m) /S −−−−→

ΩP (F )(m) /Z

Here, b is the composite of the differential F (m) ∗ ΩY (m) /S −→ ΩP (F )(m) /S and the natural projection ΩP (F )(m) /S −→ ΩP (F )(m) /Z . Let qi : Y ×S Gm −→ Y (i = 0, 1, . . . , m) be the morphism given by qi (y, g1 , . . . , gm ) = y · g1 · · · · · gi . They induce an isomorphism Y ×S Gm −→ Y (m) . Under the identification, qi is the projection onto the i-th component. Similarly, we have the identification P (F ) ×S Gm P (F )(m) , under which F (m) is given by F (m) (y, g1 , . . . , gm ) = F(y), g1 , . . . , gm . Let ρm denote the projection of P (F ) ×S Gm onto Gm . We have the subcomplex ∗ id ρm ΩGm −→ ρ∗m ΩGm of F ∗ C (m) . It is compatible with the simplicial structure. (m)

(m) , and C m = 0, 1, . . . also repreThe quotient complexes are denoted by C

(Z). Then, it follows that F ∗ LYG /S is given as the descent sents F ∗ LYG /S in Dqcoh ∗ α (0)

= F ΩY /S −→ ΩP (F )/Z with respect to the natural G-action. of C Example 2.1. Let k be a field. Let GL(R) denote the R-th general linear group over k. Let kGL(R) denote the quotient stack of Spec(k) with the trivial GL(R)action. Let E be a vector bundle on a k-variety X of rank R, and let f : X −→ kGL(R) be the classifying map. Then, we have f ∗ LkGL(R) /k End(E)[−1]. Let H denote the composite of F and the canonical map YG −→ SG . Let P (H) denote the G-torsor over Z corresponding to H. Since we have the natural isomor : P (F ) −→ S be the lift phism P (H) P (F ), we do not distinguish them. Let H of H. Let π denote the projection P (F ) −→ Z. We have the canonical isomorphism ∗ ΩS/S ΩP (F )/Z . We also have the canonical isomorphism π ∗ H ∗ LSG /S [1] H G π ∗ F ∗ LYG /SG F ∗ ΩY /S . We obtain the following corollary.

36

2 Preliminaries

Corollary 2.3.4 The morphism F ∗ LYG /SG −→ H ∗ LSG /S [1] on Z is obtained as the descent of α : F∗ ΩY /S −→ ΩP (F )/Z . Proof We have the distinguished triangle H ∗ LSG /S −→ F ∗ LYG /S −→ F ∗ LYG /SG −→ H ∗ LSG /S [1]. By Proposition 2.3.3, we understand the morphism H ∗ LSG /S −→ F ∗ LYG /S . Then, we understand the morphism F ∗ LYG /SG −→ H ∗ LSG /S [1]. Let us argue the naturality of the expression in Proposition 2.3.3. Let Gi (i = 1, 2) be smooth S-group schemes with a homomorphism a : G1 −→ G2 . Let Yi (i = 1, 2) be S-schemes provided with G-actions. For simplicity, we assume that Yi are smooth. Let g : Y1 −→ Y2 be an equivariant morphism through the morphism a. Let [g] : Y1 G1 −→ Y2 G2 denote the induced morphism. Let h1 : Z −→ Y1 G1 be a morphism. The composite [g] ◦ h1 is denoted by h2 . We would like to obtain an expression of the morphism h∗2 LY2,G2 /S −→ h∗1 LY1,G1 /S . hi : We have corresponding Gi -torsors Pi over Z with G-equivariant morphisms P −→ Yi . We can identify P2 = (P1 ×S G2 )/G1 , where the G1 -action on P1 ×S G2 is given by g1 (y, g2 ) = (yg1−1 , g1 g2 ). Let ι : P1 −→ P2 denote the natural inclusion. We have the following commutative diagram: h

P1 −−−1−→ Y1 ⏐ ⏐ ⏐ ⏐ g! ι! h

P2 −−−2−→ Y2 It induces the following commutative diagram of G1 -equivariant sheaves on P1 : ι∗ α

ι∗ h∗2 ΩY2 /S −−−−2→ ι∗ ΩP2 /Z ⏐ ⏐ ⏐ ⏐ ! ! α1 → ΩP1 /Z h∗1 ΩY1 /S −−−−

Note that the descent of Cone(−ι∗ α2 ) with respect to the G1 -action is naturally isomorphic to the descent of Cone(−α2 ) with respect to the G2 -action. Lemma 2.3.5 The morphism c : h∗2 LY2 G2 /S −→ h∗1 LY1 G1 /S is the descent of the induced morphism Cone(−ι∗ α2 )[−1] −→ Cone(−α1 )[−1]. Proof According to the functorial construction of Proposition 2.3.3, we have only to consider the case Y1 = Y2 =: Y . Let us consider the case Y = S. Since h∗i LSGi /S are isomorphic to the H 1 -cohomology sheaves H1 h∗i LSGi /S , we have only to identify the pull back of H1 (c) via the pull back P1 −→ Z, which can be done easily. Let us consider the general case. Let ki : Z −→ SGi denote the naturally defined morphism. We have the distinguished triangles ki∗ LSGi /S −→

2.3 Cotangent Complex

37

h∗i LYi,Gi /S −→ h∗i LYi,Gi /SGi . By the above argument, we know the induced morphism k2∗ LSG2 /S −→ k1∗ LSG1 /S . The isomorphism h∗2 LY2,G2 /SG2 h∗1 LY1,G1 /SG1 is easy to understand. Hence, we can identify h∗2 LY2,G2 /S h∗1 LY1,G1 /S . Remark 2.3.6 Let G1 be a smooth group scheme over S. Assume that Y is provided with a G1 -action, which commutes with the G-action. It induces a G1 -action on YG . Moreover, assume that Z is also provided with a G1 -action such that F is G1 -equivariant. Then, we have the naturally induced G1 -action on the complex Cone(−α)[−1], which commutes with the G-action. It induces a G1 -action on the descent of Cone(−α)[−1] on Z. In particular, we obtain a G1 -equivariant repre sentative of F ∗ LYG /S . Let π : Y −→ YG denote the canonical projection. By Proposition 2.3.3, LYG /S α on YG is the descent of ΩY /S −→ ΩY /YG given on Y with respect to the natural G-action, where ΩY /S stands in the degree 0. Lemma 2.3.7 Let g denote the tangent space of G at the unit, or equivalently the vector space of the right invariant vector fields, and let g∨ denote the dual. Then, ΩY /YG g∨ ⊗ OY . Proof Let p1 , p2 : Y ×S G −→ Y be given by the natural projection and the G-action. Let r1 , r2 , r3 : Y ×S G2 −→ Y ×S G be given by r1 (y, g, h) = (y, g), r2 (y, g, h) = (y, gh) and r3 (y, g, h) = (yg, h). We have the following commutative diagram: r1 ,r2 p1 Y ×S G2 −−−−→ Y ×S G −−−−→ Y ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ p2 ! r3 ! π! p1 ,p2

Y ×S G −−−−→

Y

−−−−→ YG

Then, Ωπ is obtained as the descent of Ωp2 by the identification r1∗ Ωp2 r2∗ Ωp2 . Hence, the claim of the lemma follows. Let ΘY /S denote the relative tangent bundle of Y /S. The G-action on Y induces the map A : g ⊗ OY −→ ΘY /S . The dual of A is denoted by A∨ . Lemma 2.3.8 The map α : ΩY /S −→ g∨ ⊗OY is given by the dual of −A. Namely, we have π ∗ LYG /S Cone(A∨ )[−1]. Proof Let pi : Y ×YG Y −→ Y denote the projection onto the i-th component. We have the following factorization of p∗1 α: p∗1 ΩY /S −→ ΩY ×YG Y /S −→ Ωp2 p∗1 ΩY /YG Each morphism is induced by the natural differential. Let us take the identification Y ×YG Y Y ×S G, for which p1 and p2 correspond to the natural projection onto Y and the G-action, respectively. Let y be any closed point of Y , and let e be the unit of G. We have p1 (y, e) = p2 (y, e) = y. We denote the differential of pi at (y, e) by T(y,e) pi . Let us consider

38

2 Preliminaries

the specialization of the dual of p∗1 α at (y, e). Then, it is the composite of the inclusion Ker(T(y,e) p2 ) ⊂ T(y,e) and the natural projection T(y,e) (Y ×S G) −→ (Y ×S G) Ty Y . Since we have Ker T(y,e) p2 (−Av, v) v ∈ g g, the map is −A. Since α can be recovered from p∗1 α, the claim of the lemma is proved. Remark 2.3.9 Since F ∗ LYG /S is obtained as the descent of F ∗ Cone(A∨ )[−1] for a morphism F : Z −→ YG , Lemma 2.3.8 is useful for calculation. Example 2.2. Let k be a field. Let Wi (i = −1, 0) be Ri -dimensional vector spaces over k. Let N (W−1 , W0 ) denote the vector space of linear maps from W−1 to W0 . We have the right GL(W−1 ) × GL(W0 )-action on N (W−1 , W0 ) given by (g−1 , g0 ) · f = g0−1 ◦ f ◦ g−1 . Hence, we obtain the quotient stack Y (W• ) := N (W−1 , W0 )GL(W−1 )×GL(W0 ) . Let X and U be algebraic stacks over k. Let Vi (i = −1, 0) be vector bundles on U × X whose ranks are Ri . Let f : V−1 −→ V0 be a morphism of OU ×X -modules. Then, we obtain a morphism Φf : U × X −→ Y (W• ). We claim that Φ∗f LY (W• )/k is represented by the following complex: α Hom V0 , V−1 −→ Hom V0 , V0 ⊕ Hom(V−1 , V−1 ). Here Hom V0 , V−1 stands in degree 0, and the map α is given by α(a) = f ◦ a, −a ◦ f . ∨ We remark that it is isomorphic to Hom V• , V• ≤0 [−1]. (See Subsection 2.1.5.) To show the claim, we have only to be careful on signatures. We can argue it formally. Let f be an element of N (W−1 , W0 ). The differential of the action of GL(W−1 ) × GL(W0 ) gives the map: End(W−1 ) ⊕ End(W0 ) −→ Tf N (W−1 , W0 ) = N (W−1 , W0 ),

(2.7)

(a−1 , a0 ) −→ −a0 ◦ f + f ◦ a−1 f

If we regard W−1 −→ W0 as a complex, (2.7) can be regarded as Hom(W• ,W• )≥ 0 . ∨ Then, the cotangent complex is represented by Hom(W• , W• )≥ 0 [−1], according to Lemma 2.3.8. Example 2.3. Let X be an algebraic stack. Let Ea (a = −1, 0) be vector bundles on X with a morphism f : E−1 −→ E0 . We regard E−1 as a group scheme over X, which acts on E0 through f . The quotient stack is denoted by Q(E0 , E−1 ). For simplicity, we assume that f is an injection as a morphism of OX -modules. Let E0 /E−1 denote the quotient OX -module. A section of g of E0 /E−1 corresponds to a morphism Φ(g) : X −→ Q(E0 , E−1 ). The correspondence is given as π follows: From a section g, we obtain an extension 0 −→ E−1 −→ G −→ OX −→ 0.

2.3 Cotangent Complex

39

We obtain a E−1 -torsor P = π −1 (1) with an equivariant morphism P −→ E0 , i.e., a morphism Φ(g) : X −→ Q(E0 , E−1 ). The pull back of the cotangent complex Φ(g)∗ LQ(E0 ,E−1 )/X is denoted by (E−1 −→ E0 )∨ . Let us consider the following diagram: ψ

Y −−−−→ S ⏐ ⏐ ⏐ ⏐ π1 ! π! ψ

YG −−−−→ SG We have the natural isomorphisms: π ∗ LYG /SG LY /S ,

π ∗ ψ ∗ LSG /S [1] ψ∗ LS/SG LY /YG .

Lemma 2.3.10 Under the isomorphisms above, π ∗ LYG /SG −→ π ∗ ψ ∗ LSG /S [1] is the same as the natural morphism LY /S −→ LY /YG . Proof We have the natural isomorphisms: π ∗ LYG /S Cone LY /S −→ LY /YG [−1],

π ∗ ψ ∗ LSG /S ψ∗ LS/SG [−1]

The natural morphism π ∗ ψ ∗ LSG /S −→ π ∗ LYG /S is induced by ψ∗ LS/SG −→ LY /YG . The distinguished triangle π ∗ ψ ∗ LSG /S −→ π ∗ LYG /S −→ π ∗ LYG /SG −→ π ∗ ψ ∗ LSG /S [1] is identified with the following: ψ∗ LS/SG [−1] −→ Cone LY /S → LY /YG [−1] −→ LY /S −→ ψ∗ LS/SG Then, the claim of the lemma follows.

2.3.3 Some More Examples The technical results in this subsection will be used in Sections 6.3–6.6. The author recommends the reader to skip here. Let X be a smooth connected projective surface, and let U1 be a quasi-compact algebraic stack. Let U0 be a substack of U1 . Let F be a U1 -coherent sheaf with a section ϕ over U1 × X. We assume the following: (A):

pX ∗ F is locally free, and U0 is contained in the induced section ϕ of pX ∗ F.

Assume we are given a data as follows: • A commutative diagram on U1 × X:

40

2 Preliminaries

V0,−1 −−−−→ V1,−1 ⏐ ⏐ ⏐ ⏐ ! ! V0,0 −−−−→ V1,0 ⏐ ⏐ ⏐ ⏐ ! ! E0

−−−−→

E1

−−−−→ F

Here, Ei are U1 -coherent sheaves, Va,b are locally free sheaves, and the sequences 0 −→ Va,−1 −→ Va,0 −→ Ea −→ 0 and 0 −→ E0 −→ E1 −→ F −→ 0 are exact. • A section φ of E1 such that the composite O −→ E1 −→ F is ϕ. In this subsection, such a data is called a resolution of (F, ϕ). Note that the restriction of φ to U0 × X induces a section φ0 of E0 . Note that there always exists such a resolution. For example, we have the following construction. Take a sufficiently large integer m0 , and we put E1 := p∗X pX∗ F(m0 ) ⊗ p∗U1 OX (−m0 ), E1 := E1 ⊕ OU1 ×X . The natural morphism E1 −→ F and ϕ induce a morphism π1 : E1 −→ F. We set E0 := Ker π1 . We take a sufficiently large m1 , and we put Va,0 := p∗X pX∗ Ea (m1 ) ⊗ p∗U1 OX (−m1 ), Va,−1 := Ker Va,0 −→ Ea . Then, we obtain a diagram with the desired property. Let us return to the general situation. Let Za (a = 0, 1) be the quotient stacks of N (OU1 ×X , Va,0 ) via the natural actions of N (OU1 ×X , Va,−1 ). We obtain the following commutative diagram: Φ(φ)

U1 × X −−−−→ ⏐ j1 X ⏐

Z1 ⏐ ⏐

Φ(φ0 )

U0 × X −−−−→ Z0 By using an argument in Subsection 2.3.2, Φ(φ)∗ LZ0 /Z1 is represented by the following: ∨ k(E• , V•,• , φ) := j1∗ X Cone Hom(OU1 ×X , V1• )∨ −→ Hom OU1 ×X , V0• Here, j1X : U0 × X −→ U1 × X denotes the inclusion. We have the induced morphism r(E• , V•,• , φ) : k(E• , V•,• , φ) −→ LU0 ×X/U1 ×X . We set Ob(E• , V•,• , φ) := RpX∗ k(E• , V•,• , φ) ⊗ ωX .

2.3 Cotangent Complex

41

Then, we have the induced morphism: ob(E• , V•,• , φ) : Ob(E• , V•,• , φ) −→ LU0 /U1 Lemma 2.3.11 k(E• , V•,• , φ) and r(E• , V•,• , φ) depend only on (F, ϕ) in the derived category D(U0 × X). In particular, Ob(E• , V•,• , φ) and ob(E• , V•,• , φ) depend only on (F, ϕ) in the derived category D(U0 ). Hence, we denote them by k(F, ϕ), r(F, ϕ), Ob(F, ϕ) and ob(F, ϕ), respectively. Proof It is standard that k(E• , V•,• , φ) and Ob(E• , V•,• , φ) are independent of the choice of resolutions. We would like to show the independence of r(E• , V•,• , φ) and

, φ ). We set ob(E• , V•,• , φ). Assume we are given another (E• , V•,• E1

= E1 ⊕ E1 ,

:= V1,0 ⊕ V1,0 , V1,0

E0

:= Ker E1

−→ F ,

V1,−1 := V1,−1 ⊕ V1,−1 := Ker V1,0 −→ E1

.

Let π : V1,0 −→ E1

denote the natural morphism. We can take a locally free sheaf A with a surjection A −→ π −1 (E0

). We set

:= V0,0 ⊕ V0,0 ⊕ A, V0,−1 := Ker V0,0 −→ E0

. V0,0

Then, (E•

, V•,• , φ

) with a naturally defined diagram gives a resolution of F. More

. over, we have the natural inclusions Ea ⊂ Ea

and Va,b ⊂ Va,b

Let Za (a = 0, 1) be the quotient stacks of N (OX , Va,0 ) via the naturally in

). We obtain the following diagram: duced actions of N (OX , Va,−1 Φ(φ)

U1 × X −−−−→ ⏐ ⏐ Φ(φ0 )

c

Z1 −−−1−→ ⏐ ⏐

Z1

⏐ ⏐

(2.8)

c

U0 × X −−−−→ Z0 −−−0−→ Z0

The morphisms ci are naturally induced ones. The composites c1 ◦ Φ(φ) and c0 ◦ Φ(φ0 ) are the morphisms induced by φ

and φ

0 , respectively. Hence, we obtain the following:

, φ

) LU0 ×X/U1 ×X ←−−−− k(E• , V•,• , φ) ←−−−− k(E•

, V•,•

, φ

): Hence, we obtain the following factorization of ob(E•

, V•,•

LU0 /U1 ←−−−− Ob(E• , V•,• , φ) ←−−−− Ob(E•

, V•,• , φ

)

, φ

). Thus, we are Similarly, we can compare Ob(E• , V• , φ ) and Ob(E•

, V•,• done.

42

2 Preliminaries

Lemma 2.3.12 Let (Fi , ϕi ) (i = 1, 2) satisfy Condition (A). If we are given a morphism g : F1 −→ F2 such that ϕ2 = g ◦ ϕ1 , we have the factorization of ob(F2 , ϕ2 ): Ob(F2 , ϕ2 ) −−−−→ Ob(F1 , ϕ1 ) −−−−→ LU0 /U1 Proof We take sufficiently large integers mj , and apply the above construction of resolutions to Fi . Then, the claim is clear. Now, we assume Ri pX ∗ F = 0 for i > 0. We put V := pX ∗ F. A section ϕ of V is induced by ϕ. We have the following commutative diagram: j2

U0 −−−−→ ⏐ ⏐ j1 !

U1 ⏐ ⏐ i!

(2.9)

ϕ

U1 −−−−→ V Here i is the 0-section, and ja are the natural inclusions. Proposition 2.3.13 Ob(F, ϕ) is isomorphic to j2∗ LU1 /V V∨ [1], and ob(F, ϕ) is the same as the morphism κ : j2∗ LU1 /V −→ LU0 /U1 induced from the diagram (2.9). Proof We have the naturally defined morphism a1 : p∗X V −→ F, for which we have ϕ = a1 ◦ p∗X ϕ. By Lemma 2.3.12, we have the following factorization of ob(F, ϕ): ob(p∗ V,p∗ ϕ)

b

X −→ LU0 /U1 Ob(F, ϕ) −−−0−→ Ob(p∗X V, p∗X ϕ) −−−−X−−−−

Let us look at ob(p∗X V, p∗X ϕ) more closely. In the construction for (p∗X V, p∗X ϕ), we may choose E1 = V1,0 = p∗X V,

E0 = V1,−1 = V0,0 = V0,−1 = 0.

Then, Z0 = U1 × X and Z1 = p∗X V. The diagram (2.8) is given as follows: j1

U0 × X −−−−→ ⏐ ⏐ j2 ! p∗ Xϕ

Z0 ⏐ ⏐ i!

U1 × X −−−−→ Z1

U1 × X ⏐ ⏐ !

(2.10)

p∗X V

Here i denotes the 0-section. We have k = j1∗ LZ0 /Z1 p∗X V∨ |U0 ×X [1], and the morphism r : k −→ LU0 ×X/U1 ×X is the same as the pull back of κ. In particular, we have the following factorization of ob(p∗X V, p∗X ϕ): b κ Ob(p∗X V, p∗X ϕ) = V∨ [1] ⊗ RpX ∗ p∗U0 ωX −−−1−→ V∨ [1] −−−−→ LU0 /U1

2.3 Cotangent Complex

43

It is easy to see that the composite b1 ◦b0 is an isomorphism, under the assumption Ri pX ∗ F = 0 (i > 0). Thus, the proof of Proposition 2.3.13 is finished. We have a similar result for a smooth projective curve. Since the argument is similar and simpler, we explain only the statement. Let D be a smooth projective curve. Let F be a U1 -coherent sheaf on U1 × D with a section ϕ. Assume the following: (A’) pD ∗ F is locally free, and U1 is contained in the 0-set of the induced section ϕ of pD ∗ F. Let (E0 → E1 ) be a locally free resolution of F on U1 × D with a section of φ of E1 such that the composite O −→ E1 −→ F is ϕ. It is called a resolution of (F, ϕ). A section φ0 of V0 | U0 ×D is induced. We put k(E• , ϕ) := Hom(OU0 ×D , V•|U0 ×D )∨ . Let us construct a morphism r(E• , ϕ) : k(E• , ϕ) −→ LU0 ×D/U1 ×D . We put Za := N (O, Ea ) for a = 0, 1. Then, we have the naturally defined morphism Z0 −→ Z1 . The sections φ and φ0 induce the following commutative diagram: j

U0 × D −−−−→ ⏐ ⏐ !

Z0 ⏐ ⏐ !

U1 × D −−−−→ Z1 It induces a morphism r(E• , ϕ) : k(E• , ϕ) j ∗ LZ0 /Z1 −→ LU0 ×D/U1 ×D . It can be shown that r(E• , ϕ) and k(E• , ϕ) depend only on (F, ϕ), as in Lemma 2.3.11. Therefore, we use the symbols r(F, ϕ) and k(F, ϕ) to denote them. We set Ob(F, ϕ) := RpD ∗ k(F, ϕ) ⊗ ωD . We have the induced morphism ob(F, ϕ) : Ob(F, ϕ) −→ LU0 /U1 . It is functorial as in Lemma 2.3.12. Now, we assume Ri pD ∗ F = 0 for i > 0. We put V := pX ∗ F. We have the induced section ϕ. We obtain the diagram (2.10). It induces a morphism κ : V∨ [1] −→ LU0 /U1 . Proposition 2.3.14 Under the assumption Ri pD ∗ F = 0 for i > 0, we have the following commutative diagram: ob(F ,ϕ)

Ob(F, ϕ) −−−−−→ LU0 /U1 ⏐ ⏐ ⏐ ⏐ =! ! V∨ [1]

κ

−−−−→ LU0 /U1

Proof It can be shown by an argument used in the proof of Proposition 2.3.13.

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2.4 Obstruction Theory 2.4.1 Definition and Fundamental Theorems In the study of Gromov-Witten theory, M. Kontsevich, J. Li-G. Tian, K. BehrendB. Fantechi and K. Fukaya-K. Ono introduced the notion of virtual fundamental classes of moduli stacks with some good structure. (See [71], [82], [6] and [40]. See also the recent work of I. Ciocan-Fontanine and M. Kapranov [15].) In this paper, we follow the framework of Behrend-Fantechi. See [6] for more details and precise. The paper [55] of T. Graber-R. Pandharipande is also very useful, in which they studied localization of virtual fundamental classes. Definition 2.4.1 Let S be an algebraic stack, and let X be an algebraic stack over S. Let E • be an object in D(X ) such that Hi (E • ) are coherent (i = −1, 0, 1). A homomorphism φ : E • −→ LX /S is called an obstruction theory for X /S, if Hi (φ) (i ≥ 0) are isomorphic and H−1 (φ) is surjective. In that case, E • is also called an obstruction theory for X /S. Because Hi (LX ) = 0 for i > 1, the condition implies Hi (E • ) = 0 for i > 1. If X is Deligne-Mumford, we also have H1 (E • ) = H1 (LX ) = 0. We will often use the following theorem of Behrend-Fantechi. Proposition 2.4.2 (Theorem 4.5, [6]) Let X be a Deligne-Mumford stack over S. Let φ : E • −→ LX /S be a morphism in D(X ). The following conditions are equivalent. • φ is an obstruction theory. • Let T and T be S-schemes such that T is a closed subscheme of T whose ideal sheaf J is square-zero. Let g : T −→ X be a morphism over S. (A1) g can g : T −→ X over S, if and only if be extended to a morphism φ∗ o(g) = 0 in Ext1 g ∗ E • , J , where o(g) is the obstruction class of g. (See Proposition 2.3.1.) (A2) If φ∗ o(g) = 0, the set of the extension classes of g is a torsor over the group Ext0 g ∗ E • , J . We recall the notion of perfect obstruction theory in the sense of BehrendFantechi with a minor generalization. Definition 2.4.3 Let φ : E • −→ LX /S be an obstruction theory of an algebraic stack X over S. It is called perfect, if it is quasi-isomorphic to a complex of locally free sheaves F −1 → F 0 → F 1 in the derived category D(X ). In that case, the number − rank F 1 + rank F 0 − rank F −1 is well defined on each connected component of X . It is called the expected dimension of X over S with respect to φ.

2.4 Obstruction Theory

45

If X is Deligne-Mumford, we have H1 (E • ) = H1 (LX ) = 0 for any obstruction theory E • . Hence, a perfect obstruction theory is quasi-isomorphic to a two-term locally free complex F −1 → F 0 . The important and foundational theorem of Behrend and Fantechi is the following. (See also [82].) Proposition 2.4.4 (Section 5 [6]) Let X be a Deligne-Mumford stack over a smooth scheme S. Let A∗ (X ) denote the Chow group of X with rational coefficient. A perfect obstruction theory φ : E • −→ LX /S induces an element [X , φ] ∈ Ad (X ) called virtual fundamental class, where d is the expected dimension with respect to φ. If X is smooth, [X , φ] is the Euler class of the vector bundle H1 E • ∨ . We often use the notation [X ] instead of [X , φ], if there are no risk of confusion. Remark 2.4.5 In Definition 2.4.3, E • is assumed to be quasi-isomorphic to a complex F • of locally free sheaves. According to A. Kresch [75], the existence of such a global complex is not necessary for construction of virtual fundamental class. Namely, Proposition 2.4.4 holds, if E • is locally quasi-isomorphic to a two-term locally free complex. Let S, X , φ : E • −→ LX /S be as in Proposition 2.4.4. Let S be a smooth scheme, and let ι : S −→ S be a morphism. We set X := X ×S S with the natural morphism ι : X −→ X . Proposition 2.4.6 (Proposition 7.2, [6]) The induced morphism ι∗ φ : ι∗ E • −→ LX /S is also a perfect obstruction theory. Let [X , ι∗ φ] denote the associated virtual fundamental class. Assume that ι is either (i) a closed regular immersion, or (ii) flat. Then, we have the relation ι∗1 [X , φ] = [X , ι∗ φ]. Let Xi (i = 1, 2) be algebraic stacks over S provided with obstruction theories φi : Ei• −→ LXi /S . Assume we have the following commutative diagram: f

X1 −−−−→ ⏐ ⏐ g!

X2 ⏐ ⏐ !

h

Y1 −−−−→ Y2 −−−−→ S Recall the following definition in [6]. Definition 2.4.7 We say that φi are compatible over h, if we have the following morphism of distinguished triangles on X1 : f ∗ E2• ⏐ ⏐ !

−−−−→

E1• ⏐ ⏐ !

−−−−→ g ∗ LY1 /Y2 −−−−→ ⏐ ⏐ !

g ∗ LX2 /S −−−−→ LX1 /S −−−−→ LX1 /X2

f ∗ E2• [1] ⏐ ⏐ !

−−−−→ g ∗ LX2 /S [1]

We will use the following theorem for comparison of virtual fundamental classes.

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Proposition 2.4.8 (Proposition 7.5, [6]) Assume Xi are Deligne-Mumford, and the obstruction theories φi are perfect. If φi are compatible over h, then we have the equality h! [X2 , φ2 ] = [X1 , φ1 ] at least if h is smooth or Yi are smooth over S.

2.4.2 Easy Example Let X be a smooth variety over k. We would like to construct an obstruction theory of the moduli spaces M of some objects on X. A naive strategy is summarized as follows (See [6], for example): 1. Take a classifying stack Y of such objects over X. It means that such objects over U × X bijectively correspond to morphisms Φ : U × X −→ Y over X. For example, recall that vector bundles of rank R over U × X correspond to morphisms U × X −→ XGL(R) over X. 2. Any classifying maps Φ : U × X −→ Y induce morphisms Φ∗ LY /X −→ LU ×X/X on U ×X. Let ωX denote the dualizing complex on X, i.e., it is the canonical line bundle shifted by the dimension of X. Then, we obtain the following morphisms on U : ObU := RpX∗ Φ∗ LY /X ⊗ ωX −→ RpX∗ p∗X LU/k ⊗ ωX −→ LU/k . In particular, a morphism ObM −→ LM on M is induced by a universal object. 3. We hope that the morphism ObM −→ LM is an obstruction theory, in some cases. Note that the property is local, once the morphism is given globally. Thus we have only to check the claim for sufficiently small e´ tale open subsets of M. Proposition 2.4.2 provides us with a useful tool to check it. Remark 2.4.9 In general, we need some modification in construction of ObM to obtain a good obstruction theory. Let us look at the easiest example. Let F and V be vector bundles on X. Let U be any scheme over k, and let f : p∗U (F ) −→ p∗U (V ) be a morphism of OU ×X modules over U × X. It is easy to see that such a morphism f corresponds to a morphism Φf : U × X −→ N (F, V ) over X. We obtain a complex g(f ) := Φ∗f LN (F,V )/X and a morphism g(f ) −→ LU ×X/X in the derived category D(U × X).

2.4 Obstruction Theory

47

Lemma 2.4.10 g(f ) is represented by p∗U Hom(V, F ). Proof Let π : N (F, V ) −→ X denote the natural projection. Since the morphism N (F, V ) −→ X is smooth, the cotangent complex LN (F,V )/X is quasi-isomorphic to ΩN (F,V )/X π ∗ Hom(V, F ), and thus Φ∗f LN (F,V )/X p∗U Hom(V, F ). We set Ob(f ) := RpX∗ g(f ) ⊗ ωX . Then, we obtain morphisms Ob(f ) −→ RpX∗ LU ×X/X ⊗ ωX −→ LU in the derived category D(U ). The composite is denoted by ob(f ). Now, let M (F, V ) denote a moduli scheme of morphisms F −→ V , i.e., maps U −→ M (F, V ) bijectively correspond to f : p∗U (F ) −→ p∗U (V) on U × X. It is easy to see that M (F, V ) is isomorphic to the vector space H 0 X, Hom(F, V ) . We have the universal morphism f u : p∗M (F,V ) (F ) −→ p∗M (F,V ) (V ) over M (F, V ) × X. It induces a morphism ob(f u ) : Ob(f u ) −→ LM (F,V ) . Lemma 2.4.11 ob(f u ) gives an obstruction theory of M (F, V ). Proof It is almost obvious from the universal properties of N (F, V ) and M (F, V ). However, we give a rather detailed argument as an explanation. We have only to check the conditions (A1) and (A2) in Proposition 2.4.2. Since the claim is local, we can check the claim for any sufficiently small open subset U of M (F, V ). Let T be an affine scheme over k. A morphism g : T −→ U induces morphisms gX : T × X −→ U × X and gX = Φf u ◦ gX : T × X −→ N (F, V ) over X. Let T denote a scheme such that T is embedded in T whose ideal J is square-zero. Deformations of ∗the morphisms g and gX is controlled by the groups LN (F,V )/X , JX , respectively. We have the followExti g ∗ LU/k , J and Exti gX ing commutative diagram: h −→ Exti g ∗ Ob(f u ), J Exti g ∗ LU/k , J ∗ ↓ ∗↑ = ∗ Exti gX LU ×X/X , JX −→ Exti gX (g), JX −→ Exti gX LN (F,V )/X , JX We have the obstruction classes o(g) ∈ Ext1 g ∗ LU/k , J ,

∗ o( gX ) ∈ Ext1 gX g, JX

of the morphisms g and gX respectively. By the functoriality of the cotangent complex, the obstruction class o(g) is mapped to the obstruction class o( gX ) in the above diagram.

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If the image h o(g) is 0, the class o( gX ) is 0. Hence, gX can be extended to a morphism T × X −→ N (F, V ), which induces a morphism of p∗T (F ) −→ p∗T (V ) on T × X. By the universal property of M (F, V ), we obtain a morphism T −→ M (F, V ) which is an extension of g. Therefore, the condition ∗ (A1) is satisfied. LN (F,V )/X , J is Similarly, we can show that Ext0 g ∗ LU/k , J −→ Ext0 gX an isomorphism by the universality of M (F, V ) and N (F, V ). Hence, the condition (A2) is also satisfied. Thus we are done.

2.4.3 Locally Free Subsheaves Let X be a smooth projective variety over k with an ample line bundle OX (1). Let V be a locally free sheaf on X. Let W denote an R-dimensional k-vector space. We denote W ⊗ OX by WX . We have the natural right GL(W )-action on N (WX , V ). The quotient stack is denoted by Yquo (W• ). We consider deformations of locally free subsheaves of V with rank R. Let U be any k-scheme. Any locally free subsheaf f : F −→ p∗U V on U × X with rank F = R induces a morphism Φ(F, f ) : U × X −→ Yquo (W• ) over X. We put g(F, f ) := Φ(F, f )∗ LYquo (W• )/X Ob(F, f ) := RpX∗ g(F, f ) ⊗ ωX Then, we obtain morphisms g(F, f ) −→ LU ×X/X on U × X, and ob(F, f ) : Ob(F, f ) −→ LU on U . The following lemma can be shown by using the argument explained in Subsection 2.3.2. Lemma 2.4.12 g(F, f ) is represented by Cone(α)[−1] of the morphism α : Hom(p∗U V, F ) −→ Hom(F, F ) given by α(a) = a ◦ f .

Remark 2.4.13 We put V−1 := F and V0 := p∗U V , and we regard V• = (V−1 → V0 ) as a complex, where V0 stands in the degree 0. Then, Cone(α) is naturally isomorphic to Hom(V−1 [1], V• )∨ [−1]. Let H be a polynomial. We have a moduli scheme of quotients (q : V −→ Q) of V such that the Hilbert polynomials of Q are H. Let M (V, H) denote the open subscheme which consists of the points (q : V −→ Q) such that Ker(q) are locally free. Then, we have the universal family f u : F u −→ p∗M (V,H) (V ) defined over M (V, H) × X. A morphism ob(F u , f u ) : Ob(F u , f u ) −→ LM (V,H) is induced.

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49

Proposition 2.4.14 ob(F u , f u ) is an obstruction theory of M (V, H). Proof Let N be a sufficiently the condition ON for the large number satisfying family F u , i.e., we have H i X, F u (N )|{q}×X = 0 for any q ∈ M (V, H) and u i > 0, and F|{q}×X (N ) are globally generating for any q ∈ M (V, H). We put u u ∗ F := pX pX∗ F (N ) ⊗ O(−N ). We have the natural surjection g : F u −→ F u . We put F = O(−N )⊕ d , where d = rank Fu . We have the Grassmaniann bundle π : Gr(F , R) −→ X associated to the vector bundle F , i.e., the fiber of π over a point x ∈ X is the Grassmann variety of R-dimensional quotient spaces of the vector space F |x . We denote the universal quotient bundle over Gr(F , R) by Q. Then, we have the vector bundle Yquo := N (Q, π ∗ V ) over Gr(F , R), which is a variety smooth over X. We have the natural morphism π1 : Yquo −→ Yquo (W• ). We would like to check the conditions (A1) and (A2) in Proposition 2.4.2. Let U be any sufficiently small open set of M (V, H), on which we can assume that there exists an isomorphism Fu p∗U F . Thus, the surjection γ : p∗U F −→ F u is given on U × X. From γ and f u , we obtain a morphism Φ(γ, F u , f u ) : U × X −→ Yquo over X. By the argument in Subsection 2.3.2, we can show that the complex Φ(γ, F u , f u )∗ LYquo /X is represented by the cone Cone(β)[−1] for the morphism β : Hom(p∗U V, F u ) ⊕ Hom(F u , p∗U F ) −→ Hom(F u , F u ) β(b1 , b2 ) = b1 ◦ f u − f u ◦ b2 . We can also show that the natural morphisms Cone(α)[−1] −→ Cone(β)[−1] −→ LU ×X/X is the same as the factorization: Φ(F u , f u )∗ LYquo /X −→ Φ(γ, F u , f u )∗ LYquo /X −→ LU ×X/X associated to U × X −→ Yquo −→ Yquo (W• ). We set g(γ, F u , f u ) := Φ(γ, F u , f u )∗ LYquo /X Ob(γ, F u , f u ) := RpX∗ g(γ, F u , f u ) ⊗ ωX . Let T be an affine scheme. Let g : T −→ U be a morphism, which induces gX : T × X −→ U × X. We put gX := Φ(F u , f u ) ◦ gX and g X := Φ(γ, F u , f u ) ◦ gX . For any coherent sheaf J on T , we have the following commutative diagram:

50

2 Preliminaries ∗ Exti (g ∗ LU/k , J) −−−−→ Exti (gX LU ×X/X , JX ) ⏐ ⏐ ⏐ ⏐ hi1 ! ! ∗ g(γ, F u , f u ), JX Exti g ∗ Ob(γ, F u , f u ), J −−−−→ Exti gX ⏐ ⏐ ⏐ ⏐ hi2 ! ! i ∗ i ∗ Ext (g Ob(F u , f u ), J) −−−−→ Ext gX g(F u , f u ), JX

(2.11)

Let T be an affine scheme into which T is embedded closedly such that the corresponding ideal J is square-zero. According to the deformation theory of Illusie, we have the obstruction g and g X in the groups ∗ classesu of uthe morphisms g(γ, F , f ), J respectively. The classes are deExt1 (g ∗ LU/k , J) and Ext1 gX gX ) in the diagram noted by o(g) and o(

gX ). By functoriality, o(g) is mapped to o(

(2.11). If h11 (o(g)) is 0, then the morphism g X can be extended. Note that the cohomology sheaves Ri pX∗ Hom(F u , p∗U F )⊗ωX vanish unless i = 0, because of our choice of N . Thus, we have the isomorphisms Exti g ∗ Ob(γ, F u , f u ), J Exti g ∗ Ob(F u , f u ), J forany i> 0 and for any coherent sheaf J on T . Hence, h12 ◦ h11 (o(g)) = 0 implies h11 o(g) = 0. Then, the morphism g X can be extended over T × X, and hence gX can also be extended over T × X. Therefore, we obtain a locally free subsheaf F

∗ F u . By the universal property of of p∗T (V ) on T × X, which is an extension of gX M (V, H), the morphism g can be extended over T . Therefore, the condition (A1) is satisfied. Let us check the condition (A2). We set ∗ Hom(F u , p∗U F ) ⊗ ωX , J H0 := Ext0 pX∗ gX = H 0 T, g ∗ End pX ∗ F u (m) ⊗ J ∗ LYquo /X , JX H1 := Ext0 g ∗ Ob(g, F u , f u ), J = Ext0 g X H2 := Ext0 g ∗ Ob(F u , f u ), J We obtain an exact sequence 0 −→ H0 −→ H1 −→ H2 −→ 0. According to the

: T × X −→ Yquo theory of Illusie, H1 parameterizes the set of the extensions g X of g X . The natural action of H0 on H1 determines the equivalence relation on H1 ,

∼ g X if and only if π1 ◦ g X = π1 ◦ g X , because H0 and it is easy to see that g X u parameterizes the deformations of the morphism F −→ F . Thus the set of the extensions of the morphism T × X −→ Yquo (W• ) over T × X is a torsor over the group H2 . By the universal property of M (V, H) and Yquo (W• ), the set of the extensions of g over T is also a torsor over H2 . Namely the condition (A2) is satisfied.

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51

Usually, we consider deformations of quotients of V . Let H be a polynomial, and let Quot(V, H) denote a quot scheme which parameterizes the quotient sheaves of V whose Hilbert polynomials are H. We have the universal quotient q u : p∗Quot(V,H) (V ) −→ Qu on Quot(V, H) × X. We denote the kernel of q u by F u , and the inclusion F u −→ p∗Quot(V,H) (V ) is denoted by f u . Let us consider the case dim X = 1. Let HV denote the Hilbert polynomial of V . Then, Quot(V, H) parameterizes the locally free subsheaves of V whose Hilbert polynomials are HV −H. Therefore, we have obtained an obstruction theory ob(F u , f u ) : Ob(F u , f u ) −→ LQuot(V,H) . Proposition 2.4.15 In the case dim(X) = 1, the obstruction theory ob(f u ) is perfect. The scheme Quot(V, H) is smooth, if H is a constant, i.e., H is a Hilbert polynomial of sheaves of finite length. Proof To show the perfectness of Ob(F u , f u ), we have only to show that RpX ∗ g(F u , f u )∨ is perfect of amplitude contained in [0, 1]. Let q be any point of Quot(V, H). We u and Q := V /F . The complex g(F u , f u )∨ put F := F|{q}×X |{q}×X is Cone(γ)[−1] for the natural morphism γ : Hom(F, F ) −→ Hom(F, V ), which is quasi= 0 for any point isomorphic to Hom(F, Q). Hence, H i X, g(F u , f u )∨ |{q}×X q ∈ Quot(V, H) unless i = 0, 1. Then, the desired perfectness easily follows. Let us show the second claim. If H is a constant, we have H 1 X, g(F u , f u )∨ |{q}×X = 0 for any q ∈ Quot(V, H). Let T be any affine scheme over k, and let g : T −→ Quot(V, H) be a morphism. Then, Ext1 g ∗ Ob(F u , f u ), J = 0 for any coherent OT -module J, and hence any obstruction classes vanish. Thus we obtain the smoothness. Remark 2.4.16 Let us consider the case dim X = 2. Let Qtf (V, H) denote the open subset of Q(F, H) corresponding to torsion-free quotient sheaves. It gives an u are open subset of a moduli stack of locally free subsheaves of V . Then, F|{q}×X tf locally free for any q ∈ Q (V, H). Therefore, we obtain an obstruction theory ob(F u , f u ) : Ob(F u , f u ) −→ LQtf (V,H)/k from Proposition 2.4.14.

2.4.4 Filtrations of a Vector Bundle on a Curve Let S be a scheme over k, and let D be a smooth projective curve over S provided with an ample line bundle O(1). The projection D −→ S is denoted by p. Let V and F be locally free sheaves on D provided with an injective morphism f : F −→ V . Assume that the quotient is S-flat.

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Let Hi be polynomials. Let g : T −→ S be an S-scheme. We have the induced curve DT := D ×S T over T . The induced morphism DT −→ D is denoted by g. We obtain a morphism g∗ F −→ g∗ V on DT . Let F (T, g) denote the set of the filtrations V ∗ of g∗ V on DT , g∗ V = V (1) ⊃ V (2) ⊃ · · · ⊃ V (l) ⊃ V (l+1) = g∗ F, with the following property: • The quotients Coki := V (1) /V (i+1) are T -flat. • The Hilbert polynomials of Coki| Dt are Hi for any i = 1, . . . , l and t ∈ T . The functor F is representable by an S-scheme, which can be shown by the standard technique using quot schemes. Let gM (H∗ ) : M (H∗ ) −→ S denote a moduli S-scheme. We have the universal filtration on M (H∗ ) ×S D: ∗ (1) ∗ ⊃ V (2) ⊃ · · · ⊃ V (l) ⊃ V (l+1) = gM gM (H∗ ) V = V (H∗ ) F.

To construct an obstruction theory of M (H∗ ), we introduce some stacks. Take vector spaces Wi (i = 2, . . . , l) over k such that rank Wi = rank V (i) =: ri . We put W (i) := Wi ⊗ OD (i = 2, . . . , l). We set W (1) := V and W (l+1) := F . .l We define Y0 := N (W (l+1) , W (1) ) and R1 := i=1 N (W (i+1) , W (i) ). We put .l G(W∗ ) := i=2 GL(Wi ). We have the natural right G(W∗ )-action on R1 . Let Y1 denote the quotient stack of R1 by the G(W∗ )-action. By composition of the maps, we obtain a morphism φ : R1 −→ Y0 , which induces Y1 −→ Y0 . We put Y2 := D. Then, the morphism F −→ V induces a morphism Y2 −→ Y0 . We put Y := Y1 ×Y0 Y2 . Let V ∗ denote a filtered vector bundle on DT as above. We obtain morphisms Φi (V ∗ ) : DT −→ Yi . We have the naturally defined morphism: G(V ∗ ) : Φ0 (V ∗ )∗ LY0 /D −→ Φi (V ∗ )∗ LYi /D i=1,2

We use the notation in Subsection 2.1.6. We put g(V ∗ ) := C2 (V ∗ , V ∗ )∨ [−1]. By the argument in Subsection 2.3.2, the cone of G(V ∗ ) is represented by the complex g(V ∗ ). Thus, wehave the naturally defined morphism g(V ∗ ) −→ LDT /D . We put Ob(V ∗ ) := Rp∗ g(V ∗ ) ⊗ ωDT /T . Then, we obtain a morphism ob(V ∗ ) : Ob(V ∗ ) −→ LT /S . Applying the construction to the universal filtered bundle V ∗ on M (H∗ ) ×S D, we obtain ob(V ∗ ) : Ob(V ∗ ) −→ LM (H∗ )/S . Lemma 2.4.17 The morphism ob(V ∗ ) gives an obstruction theory of M (H∗ ). Proof In the following, we will shrink S without mention. Take locally free sheaves J (i) (i = 2, . . . , l) on D such that (i) there exist surjections J (i) −→ V (i) ,

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53

) denote (ii) R1 p∗ Hom(J (i) , V (i) ) = 0. For any S-scheme g : T −→ S, let F(T ∗ the set of (V , ϕ∗ ) on DT as follows: • V ∗ denotes a filtration of g∗ V as above. • ϕ∗ denotes a tuple of surjections of g∗ J (i) onto V (i) . (H∗ ). We have The functor F is representable by a scheme which is denoted by M ∗ (i) (i) on M (H J , V the locally free sheaves Ni := Hom gM ∗ ) ×S D. Let p : (H∗ ) M (H∗ )×S D −→ M (H∗ ) denote the projection. Then, M (H∗ ) is isomorphic to an (H∗ ) ×S D, we have the universal open subset of the vector bundle p∗ Ni . On M filtration V ∗ with the tuple of surjective morphisms ϕu∗ . Let Gr(J (i) , ri ) be the Grassmannian bundles of ri -dimensional quotient spaces associated to the vector bundles J (i) . We have the universal quotient bundle Qi . .l (i) We put Z := , ri ), where the fiber product is taken over D. The i=2 Gr(J (i) pull back of Qi via the projection Z −→ Gr(J (i) , ri ) are denoted by W (i = 2, . . . , l). The pull back of V and F via the projection Z −→ D are de (l+1) (1) and W (l+1) respectively. Then, we put Y0 := N W (1) , noted by W ,W . l (i+1) , W (i) ) and Y2 := Z. We have the natural morphisms Y1 := i=1 N (W Yi −→ Y0 (i = 1, 2) as above. The fiber product Y1 ×Y0 Y2 is denoted by Y . The inclusions Y −→ Yi are denoted by ji . On Y , we have the natural morphism j0∗ LY0 /D −→ i=1,2 ji∗ LYi /D , whose cone is denoted by Ob(Y ). Then, we have the naturally defined morphism ob(Y ) : Ob(Y ) −→ L , and it gives an obstrucY /D

tion theory for Y over D. (Basic example in [6]). Let g : T −→ S be an S-scheme. From (V ∗ , ϕ∗ ) on DT , we obtain morphisms Φi (V ∗ , ϕ∗ ) : DT −→ Yi . Therefore, we obtain Φ(V ∗ , ϕ∗ )∗ Ob(Y ) −→ LDT /T . ∗ , ϕ∗ ) := Rp∗ Ob(Y ) ⊗ ωD/S , and then we obtain a morphism We put Ob(V ∗ , ϕ∗ ) : Ob(V ∗ , ϕ∗ ) −→ LT /S . ob(V ∗ , ϕ∗ ). We have the morphisms Let us describe the complex Ob(V Hom V (i) , J (i) −→ Hom V (i) , V (i) , ai −→ ϕi ◦ ai . It induces a morphism of the complexes α:

l

Hom V (i) , J (i) [−1] −→ g(V ∗ )

i=2

We put g(V ∗ ) := Cone(α). By using the argument in Subsection 2.3.2, we can show that g(V ∗ ) represents Φ(V ∗ , ϕ∗ )∗ Ob(Y ).

54

2 Preliminaries

Applying the above construction to (V ∗ , ϕu∗ ), we obtain a morphism ∗ , ϕu ) −→ L ∗ , ϕu ) : Ob(V . ob(V ∗ ∗ M (H∗ )/S ∗ , ϕu ) gives an obstruction theory of M (H∗ ) Lemma 2.4.18 The morphism ob(V ∗ over S. (H∗ ) be a morphism, and let J be a coherent sheaf on T . Proof Let h : T −→ M The pull back of J via DT −→ T is denoted by JD . We have the induced morphism (H∗ ) ×S D. We set

hD := Φ(V ∗ , ϕu∗ ) ◦ hD . We have the following h : DT −→ M commutative diagram: ψ J Ext1 h∗ LM −−−−→ Ext1 h∗ Ob, (H∗ )/S , J ⏐ ⏐ ⏐ ⏐ ! ∗ 1 g(V ∗ ), JD Ext1 h∗D LM (H∗ )×S D/D , JD −−−−→ Ext hD Let T be an S-scheme such that T is embedded as a closed subscheme and that the corresponding ideal J is square-zero. We have the obstruction classes o(h) and 1 ∗ o(

hD ) in Ext1 h∗ LM /S , JX . It is easy to see that (H∗ )/S , J and Ext hD LY J is the same as the image of o(

ψ o(h) ∈ Ext1 h∗ Ob, hD ) via the composite of the following morphisms: ∗ b b J hD LY /S , JD −−−1−→ Ext1 h∗D g(V ∗ ), JD −−−2−→ Ext1 h∗ Ob, Ext1

gives an obhD ) = 0. Since Ob Hence, the vanishing of ψ o(h) implies b1 o(

struction theory for Y , it implies that

h can be extended to a morphism T ×S (H∗ ) by the D −→ Y . Then, we obtain an extension of h to a morphism T −→ M universal property of M (H∗ ). Therefore, the condition (A1) of Proposition 2.4.2 is checked. The condition (A2) can also be checked easily, and the proof of Lemma 2.4.18 is finished. (H∗ ) −→ M (H∗ ), which is smooth. We have the Let π denote the projection M following commutative diagram: ∗

u

Φi (V ,ϕ∗ ) (H∗ ) ×S D −− M −−−−−→ ⏐ ⏐ πD !

M (H∗ ) ×S D

Φi (V ∗ )

−−−−→

Yi ⏐ ⏐ ! Yi

2.5 Equivariant Complexes on Deligne-Mumford Stacks with GIT Construction

55

(H∗ ) ×S D: We obtain the following morphism of the distinguished triangles on M ∗ πD g(V ∗ ) ⏐ ⏐ !

−−−−→ Φ(V ∗ , ϕu∗ )∗ Ob(Y ) −−−−→ ⏐ ⏐ !

∗ πD LM (H∗ )×S D/D −−−−→

l i=2

LM (H∗ )×S D/D

−−−−→

Hom(V (i) , J (i) ) −−−−→ ⏐ ⏐ !

∗ πD g(V ∗ )[1] ⏐ ⏐ !

(2.12)

∗ LM (H∗ )×S D/M (H∗ )×S D −−−−→ πD LM (H∗ )×S D/D [1]

Hence, we obtain the following morphism of the distinguished triangles: ∗ , ϕ∗ ) −−−−→ π ∗ Ob(V ∗ ) −−−−→ Ob(V ⏐ ⏐ ⏐ ⏐ ! ! π ∗ LM (H∗ ) −−−−→ LM (H∗ )/S −−−−→ i

∨ p∗ Hom(J (i) , V (i) ) −−−−→ π ∗ Ob(V ∗ )[1] ⏐ ⏐ ⏐ ⏐ ϕ! ! LM (H∗ )/M (H∗ )

(2.13)

−−−−→ π ∗ LM (H∗ )/S [1]

∨ (i) , V (i) ) are isomorphic It is easy to see that both LM (H∗ /S) and i p∗ Hom(J to their 0-th cohomology sheaves, and that ϕ is an isomorphism. Then, the claim of Lemma 2.4.17 immediately follows from Lemma 2.4.18.

2.5 Equivariant Complexes on Deligne-Mumford Stacks with GIT Construction The results in this section will be used when we consider equivariant obstruction theory of master spaces in Section 5.8.

2.5.1 Locally Free Resolution Let Gi (i = 1, 2) be linear reductive groups over k. Let U be a quasi-projective variety over k provided with an action of G1 × G2 . We assume that there exists a

56

2 Preliminaries

G1 × G2 -equivariant embedding into a projective space PN . The closure of U in PN is denoted by U . The G1 × G2 -equivariant polarization is denoted by O(1). We assume that U is contained in the open subset of the stable points of U with respect to the polarization O(1) and the G2 -action. We assume that M = U/G2 is a separated Deligne-Mumford stack. The projection U −→ M is denoted by π. Lemma 2.5.1 Let F be a G1 -equivariant coherent sheaf on M. Then, there exists a G1 -equivariant locally free sheaf V on M with a G1 -equivariant surjection φ : V −→ F. Proof There exists a coherent sheaf G on U such that G|U = π ∗ F. There exists a large number N such that G(N ) is globally generating. Then, π ∗ F(N ) is also globally generating. We may take a G1 × G2 -equivariant subspace W of H 0 U, π ∗ F(N ) such that W ⊗ O(−N ) −→ π ∗ F is surjective. We have only to take descent of W ⊗ O(−N ) and the morphism. Corollary 2.5.2 Let F• be a bounded G1 -equivariant complex of coherent sheaves on M. Assume that there exist integers M1 and M2 such that the following holds: • For any point of M, there exists a neighbourhood U such that F•|U is isomorphic to a coherent locally free complex G•U in D(U) where GiU = 0 unless M1 ≤ i ≤ M2 . Then, there exists a global G1 -equivariant coherent locally free complex G• F in D(M), where Gi = 0 unless M1 ≤ i ≤ M2 .

2.5.2 Equivariant Representative We recall that the morphism of M to the coarse scheme is finite (Proposition 2.2.2). Lemma 2.5.3 Let Ci • (i = 1, 2) be G1 -equivariant bounded complexes of coherent sheaves on M. We assume that C1 • is perfect. Let ϕ be an element of the G1 -invariant part of Ext0 (C1 • , C2 • ). Then, we can take a G1 -equivariant perfect 1 • with a G1 -equivariant morphism ψ : C 1 • −→ C2 , such that (i) C 1 • complex C is G1 -equivariantly quasi-isomorphic to C1 • , (ii) ψ represents ϕ. Proof We give only an indication. We may assume that C2,i = 0 unless |i| < N . We take a sufficiently large number N1 , and we replace C1 • with a G1 -equivariant 1,i , C2,j = 0 for any 1 • with the property Extk C quasi-isomorphic complex C 0 1,• , C2,• ) is k > 0 and i > −N1 , and for any j. Then, Ext (C1,• , C2,• ) Ext0 (C isomorphic to the first cohomology of the following: 1,i , C2,j ) −→ 1,i , C2,j ) −→ 1,i , C2,j ) Ext0 (C Ext0 (C Ext0 (C −i+j=−1

−i+j=0

Since G1 is assumed to be reductive, the claim is clear.

−i+j=1

2.6 Elementary Remarks on Some Extremal Sets

57

Let B (i) (i = 1, 2) be G1 -equivariant bounded complexes on M. We assume that B is perfect. Let φ be an element of the G1 -invariant part of Ext0 B (1) , B (2) . (1) with G1 -equivariant morphisms We take a G1 -equivariant perfect complex B (1) −→ B (i) such that (i) a1 is a quasi-isomorphism, (ii) the diagram ai : B (1)

a1 (1) a2 B −→ B (2) B (1) ←−

represents φ. We have the natural G1 -equivariant structure on the cone Cone(a2 ).

(1) with G1 -equivariant morAssume we have another G1 -equivariant complex B (1) (i)

phisms

ai : B −→ B such that the diagram

a1 (1)

a2 B −→ B (2) B (1) ←− (1)

represents φ. Then, there exists a G1 -equivariant complex B with G1 -equivariant (1) (1) and g : B (1) −→ B

(1) such that the morphisms with morphisms f : B −→ B following diagrams are commutative up to homotopy for i = 1, 2: (1)

B ⏐ ⏐ g!

f (1) −−−−→ B ⏐ ⏐ ai !

ai

(1) −−− −→ B (i) B

By an argument used in the proof of Lemma 2.5.3, we may assume that the homotopy is also G1 -equivariant. Then, we have the G1 -equivariant quasi-isomorphisms: a2 ◦ g) Cone(a2 ◦ f ) −→ Cone(a2 ) Cone(

a2 ) ←− Cone(

In this sense, the G1 -equivariant complex Cone(a2 ) is uniquely determined up to G1 -equivariant quasi-isomorphisms. We denote it by Cone(ϕ).

2.6 Elementary Remarks on Some Extremal Sets The results in this section will be used when we study geometric invariant theory for enhanced master spaces in Sections 4.3–4.4.

2.6.1 Preparation for a Proof of Proposition 4.3.3 Let us consider a vector space U =

N

Q · ei . We put ei + j · ei . fj := (j − N ) i=1

i≤j

i>j

58

2 Preliminaries

The following lemma is well known and easy to prove. N N Lemma 2.6.1 Take any element ρ = i=1 ai · ei ∈ U satisfying j=1 aj = 0 and a2 ≤ · · · ≤ aN . Then, there exist non-negative rational numbers bj such that a1 ≤ ρ = bj · fj . s Let r1 , . . . , rs be positive integers such that j=1 rj = N . We put Rj := i≤j ri . We set vj := ei (j = 1, . . . , s). Rj−1
For an integer j ∈ {1, . . . , s}, we put y(j) := −(N − Rj )

vh + R j

h≤j

vh .

h>j

For a pair of integers (i1 , i2 ) such that 1 ≤ i1 < i2 ≤ s, we define x(i1 , i2 ) := −(N − Ri2 ) vh + Ri1 vh . h≤i1

i2
For an integer i0 ∈ {1, . . . , s}, we set S(i0 ) := (i1 , i2 ) ∈ Z2 1 ≤ i1 < i0 < i2 ≤ s . Lemma 2.6.2 Let v =

s j=1

aj · vj be an element of U satisfying the following:

a1 ≤ a2 ≤ · · · ≤ as ,

s

rj · aj = 0.

(2.14)

j=1

Take an integer i0 such that 1 ≤ i0 ≤ s. • Assume ai0 > 0. Then, there exist the non-negative rational numbers b(i1 , i2 ) for (i1 , i2 ) ∈ S(i0 ) and the non-negative rational numbers cj (1 ≤ j < i0 ) such that the following equality holds: v=

b(i1 , i2 ) · x(i1 , i2 ) +

(i1 ,i2 )∈S(i0 )

i 0 −1

cj · y(j).

(2.15)

j=1

• Assume ai0 = 0. Then, there exist the non-negative rational numbers b(i1 , i2 ) for (i1 , i2 ) ∈ S(i0 ) such that the following holds: v= b(i1 , i2 ) · x(i1 , i2 ). (i1 ,i2 )∈S(i0 )

• Assume ai0 < 0. Then, there exist the non-negative rational numbers b(i1 , i2 ) for (i1 , i2 ) ∈ S(i0 ) and the non-negative rational numbers cj (i0 < j ≤ N ) such that the following holds:

2.6 Elementary Remarks on Some Extremal Sets

v=

59

b(i1 , i2 ) · x(i1 , i2 ) +

N

cj · y(j).

j=i0 +1

(i1 ,i2 )∈S(i0 )

Proof We use an induction on the number d(v) := # i ai = ai+1 . In the case d(v) = 0, the claim is obvious. Let v be as in the lemma such that d(v) = m + 1. Take the integers h1 and h2 satisfying the following: a1 = a2 = · · · = ah1 < ah1 +1 ,

as = as−1 = · · · = ah2 +1 > ah2 .

We remark the following: • In the case ai0 > 0, we have h1 < i0 . • In the case ai0 = 0, we have h1 < i0 < h2 . • In the case ai0 < 0, we have i0 < h2 . Let us argue the case ai0 > 0. If we have i0 ≤ h2 , we put as follows: ah1 +1 − ah1 ah2 +1 − ah2 a i · vi , f := min , v := v − f · x(h1 , h2 ) = N − Rh2 Rh1 If we have i0 ≥ h2 + 1, we put as follows: v := v − g · y(h2 ),

g :=

ai0 − ah2 2Rh2

It is easy to see that the numbers a i satisfy the condition (2.14), and that we have d(v ) ≤ d(v) − 1. By the hypothesis of the induction, we have the expression for v

as in (2.15) with the non-negative coefficients. Then, we obtain the desired expression for v. The cases ai0 = 0 or ai0 < 0 can be argued similarly.

2.6.2 Preparation for a Proof of Proposition 4.4.4 Let N (α) (α = 1, 2) be positive integers. Let us consider a vector space as follows: U =U

(1)

⊕U

(2)

,

U

(α)

=

(α) N

(α)

Q · ei .

i=1

s(α) (α) (α) (α) Let r1 , . . . , rs(α) (α = 1, 2) be positive integers such that j=1 rj = N (α) . (α) (α) (α) We put Rj = h≤j rh . We set Ω (α) = i ei . We also put (α)

vj

:= (α)

(α)

ei (α)

Rj−1

j = 1, . . . , s(α) .

60

2 Preliminaries

For each integer j ∈ {1, . . . , s(2)}, we set (2) (2) (2) (2) y (2) (j) := −(N − Rj ) · vh + Rj · vh . h≤j

h>j

For each integer j ∈ {1, . . . , s(1)}, we put (1) (1) x1 (j) := −N (2) · vh + Rj · Ω (2) , h≤j

x2 (j) := N (2) ·

(1)

(1)

vh + (Rj−1 − N (1) ) · Ω (2) .

h≥j

Lemma 2.6.3 Let v = following conditions: (α)

a1

(α)

≤ a2

α=1,2

(α)

j

aj

(α)

≤ · · · ≤ as(α) ,

(α)

· vj

be any element of U satisfying the

α=1,2

(α)

rj

(α)

· aj

= 0.

(2.16)

j

Take an integer i0 such that 1 ≤ i0 ≤ s(1). Then, there exist non-negative rational numbers c(j) ≥ 0 (j = 1, . . . , s(2)), d1 (i) ≥ 0 (i = 1, . . . , i0 ), d2 (i) ≥ 0 (i = i0 + 1, . . . , s(1)) and a rational number A such that the following holds:

v=

s(2) j=1

c(j) · y (2) (j) +

d1 (i) · x1 (i) +

i

d2 (i) · x2 (i)

i>i0

+ A · N (2) Ω (1) − N (1) · Ω (2) . (2.17) (2)

(2)

= · · · = as(2) from the beginning. (1) (1) We use an induction on the number d(v) = # i | ai = ai+1 . If d(v) = 0, we have v = A · N (2) · Ω (1) − N (1) · Ω (1) for some A, and hence the claim is clear. Let v be an element as in the lemma such that d(v) = m + 1 > 0. Let us take the (1) (1) (1) integer h1 satisfying a1 = ah1 < ah1 +1 . In the case i0 > h1 , we put as follows: Proof By Lemma 2.6.1, we may assume a1

(1)

v := v −

(1)

ah1 +1 − ah1 x1 (h1 ) N (2)

In the case i0 ≤ h1 , we put as follows: (1)

v = v −

(1)

ah1 +1 − ah1 · x2 (h1 ). N (2)

Then, v satisfies the conditions (2.16) and d(v ) < d(v). By the hypothesis of the induction, we have the expression for v as in (2.17). Hence, we obtain the desired expression for v.

2.7 Twist of Line Bundles

61

2.7 Twist of Line Bundles This subsection is a preparation for Section 4.6.

2.7.1 Construction Let Y be an algebraic stack over a field k. Let Gm denote the one dimensional algebraic torus Spec k[t, t−1 ]. Let I denote the trivial line bundle on Y . A point of I is denoted by (y, u) where y ∈ Y and u ∈ I|y . For each integer n, T (n) denote the line bundle I with the Gm -action by t · (y, u) := (y, tn ·u). Let L be any line bundle on Y . Let L∗ denote the complement of the image of the 0-section, i.e., L∗ := L − Y . Let π : L∗ −→ Y denote the naturally defined projection. A point of L∗ is also denoted by (y, v), where y ∈ Y and v ∈ π −1 (y). Let us fix an integer r. We consider the Gm -action on L∗ given by t · (y, v) := (y, tr v). We have the naturally defined Gm -action on π ∗ T (n). It induces a line bundle In on the algebraic stack L∗ /Gm . Let ϕ : L∗ /Gm −→ Y denote the naturally induced morphism. Lemma 2.7.1 We have canonical isomorphisms In ⊗ Im In+m and I−n In−1 and I0 OL∗ /Gm . We also have a canonical isomorphism I−r ϕ∗ L. Proof The first claim is obvious. Let us show the second claim. Let us denote a point of π ∗ L by (y, v, u ), where y ∈ Y , v ∈ π −1 (y) and u ∈ L|y . The trivial Gm on L induces the Gm -action on π ∗ L over L∗ , which is given by t·(y, v, u ) = action r

y, t · v, u . On the other hand, let us denote a point of π ∗ T (−r) by (y, v, u)where y ∈ Y, v ∈ π −1 (y) and u ∈ T (−r)|y . The action is denoted by t·(y, v, u) = y, tr v, t−r u . We have the naturally defined isomorphism π ∗ T (−r) −→ π ∗ L given by (y, v, u) −→ (y, v, u · v), which is Gm -equivariant. Therefore, we obtain the iso morphism I−r ϕ∗ L.

2.7.2 Weight of the Induced Action (i)

We use the notation in the previous subsection. Let Gm (i = 1, 2) denote one(1) (2) dimensional tori Spec k[ti , t−1 i ]. Let us consider the action of Gm × Gm on L given by (t1 , t2 ) · (y, v) := y, t1 ·t2 ·v . (1) (2) Let T (n1 , n2 ) denote the trivial line bundle I with the Gm × Gm -action given n1 n2 (1) (2) by (t1 , t2 ) · (y, u) = y, t1 · t2 · u . Then, we have the Gm × Gm -line bundle (2) π ∗ T (n1 , n2 ) on L∗ . We obtain the line bundle In2 on L∗ /Gm , and we have the (1) induced Gm -action on In2 .

62

2 Preliminaries (1)

Lemma 2.7.2 The weight of the Gm -action on In2 is n1 − n2 . (i) (1) (2) Proof We put G m := Spec k[si ]. Let us consider the morphism Gm × Gm −→ (1) (2) (1) (2) Gm × Gm given by (s1 , s2 ) −→ s1 , s−1 1 · s2 . The induced Gm × Gm -action ∗ r on L and T (n1 , n2 ) is given by (s1 , s2 ) · (y, v) = y, s2 · v and (s1 , s2 ) · (y, u) = n1 −n2 n2 (1) y, s1 · s2 · u . Therefore, the weight of the Gm -action on In2 is given by n1 − n2 .

Chapter 3

Parabolic L-Bradlow Pairs

In this chapter, we recall some basic definitions. All of them are more or less standard. Our purpose is to fix the meanings in this monograph. In the following of this chapter, X will denote a smooth connected projective variety over an algebraically closed field k of characteristic 0. Let PicX denote the Picard variety of X. We fix a base point x0 ∈ X, and hence we have a Poincar´e bundle PoinX on PicX ×X. In Section 3.1, we review the basic notion. In Subsections 3.1.1–3.1.3, we recall the definition of some structure on torsion-free sheaves such as orientation, parabolic structure, L-section, and reduced L-section. In Subsection 3.1.4, we prepare the symbols to describe some moduli stacks. In Subsection 3.1.5, we introduce relative tautological line bundles of moduli stacks of oriented reduced LBradlow pairs. We also see the relation among moduli stacks of oriented reduced L-Bradlow pairs and unoriented unreduced L-Bradlow pairs. In Section 3.2, we recall the definition of Hilbert polynomials for torsion-free sheaves with some additional structures. They lead the naturally defined semistability conditions, which are discussed in Section 3.3. We recall the concepts of HarderNarasimhan filtration and partial Jordan-H¨older filtration in Subsection 3.3.2. Then, we introduce the notion of (δ, )-semistability condition in Subsection 3.3.3, which is useful to control the transitions of moduli stacks of δ-semistable L-Bradlow pairs for variation of δ. In Section 3.4, we review the boundedness of some families. In Subsection 3.4.1, we recall foundational theorems. Then, in Subsection 3.4.2, we recall the boundedness of δ-semistable L-Bradlow pairs when δ is varied. The important observation is due to M. Thaddeus. In Subsection 3.4.3, we show the boundedness of Yokogawa family, which will be used to show the properness of some morphisms in Chapter 4. In Section 3.5, we recall 1-stability and 2-stability conditions. In Section 3.6, we recall some moduli schemes of quotient sheaves with some additional structures.

T. Mochizuki, Donaldson Type Invariants for Algebraic Surfaces: Transition of Moduli Stacks, Lecture Notes in Mathematics 1972, DOI: 10.1007/978-3-540-93913-9 3, c Springer-Verlag Berlin Heidelberg 2009

63

3 Parabolic L-Bradlow Pairs

64

3.1 Sheaves with Some Structure and their Moduli Stacks 3.1.1 Orientation Let E be a U -coherent sheaf on U × X. We have the morphism detE : U −→ PicX induced by the determinant line bundle det(E) of E, which satisfies the condition det(E)|{u}×X PoinX | {detE (u)}×X . The morphism will be simply denoted by det, if there are no risk of confusion. The line bundles det∗E PoinX and det(E) are not necessarily isomorphic. Example 3.1. Assume k = C. Let c be an element of the second cohomology group H 2 (X, Z) of (1, 1)-type, and let PicX (c) denote the Picard variety of line bundles on X whose first Chern classes are c. Assume H i (X, L) = 0 (i > 0) for any line bundle onPicX (c). We bundle L ∈ PicX (c). Then, pX∗ (PoinX ) gives the vector obtain the associated projective space bundle Pc = P pX∗ (PoinX )∨ on PicX (c). Let π denote the natural projection Pc × X −→ PicX ×X. We have the line bundle L(a) = π ∗ PoinX ⊗ p∗X OPc (a) for each a ∈ Z. Here OPc (1) denotes the tautological bundle of the projective space Pc −→ PicX (c), and OPc (a) = bundle OPc (1)⊗ a . The determinant bundle det L(a) is obviously L(a) itself. On the other hand, detL(a) is given by the projection π. Thus, L(a) and det∗L(a) PoinX are not isomorphic, if a = 0. We recall the notion of orientation. (For example [110].) Definition 3.1.1 (Orientation) An orientation of a U -coherent sheaf E on U × X is defined to be an isomorphism ρ : det(E) −→ det∗E PoinX on U . Such a tuple (E, ρ) is called an oriented U -coherent sheaf. An isomorphism of two oriented sheaves (E, ρ) and (E , ρ ) is defined to be an isomorphism χ : E −→ E satisfying ρ = χ∗ (ρ) := ρ ◦ χ. ∗ The restrictions detE PoinX |{u}×X and det(E)|{u}×X are isomorphic for any point u ∈ U by definition of detE , so that the push-forward pX∗ Hom det(E), det ∗E PoinX

(3.1)

is a line bundle on U . Definition 3.1.2 (Orientation bundle) The line bundle (3.1) is called the orientation bundle of E. It is denoted by Or(E). If E is oriented, then the orientation bundle Or(E) is naturally isomorphic to the trivial line bundle OU , i.e., an orientation is equivalent to a trivialization of Or(E). Example 3.2. Let L(a) be as in Example 3.1. The orientation bundle Or(L(a), Pc ) induced by L(a) is isomorphic to

3.1 Sheaves with Some Structure and their Moduli Stacks

65

pX∗ Hom L(a), det ∗L(a) Poin pX∗ ◦ p∗X OPc (−a) OPc (−a) Orientation bundles are additive in the following sense. Lemma 3.1.3 Let Ei (i = 1, 2) be U -coherent sheaves on U × X. Then, we have a natural isomorphism Or(E1 ⊕ E2 ) Or(E1 ) ⊗ Or(E2 ). Proof The natural isomorphism det(E1 ⊕ E2 ) det(E1 ) ⊗ det(E2 ) is given, and hence det∗E1 PoinX ⊗ det∗E2 PoinX det∗E1 ⊕E2 PoinX . It induces the following isomorphism: Hom det(E1 ⊕ E2 ), det ∗E1 ⊕E2 PoinX Hom det(E1 ), det ∗E1 PoinX ⊗ Hom det(E2 ), det ∗E2 PoinX .

(3.2)

Therefore, we obtain the isomorphism Or(E1 ) ⊗ Or(E2 ) −→ Or(E1 ⊕ E2 ).

3.1.2 Parabolic Structure See [87] and [134] for details on the notion of parabolic sheaf. Our terminology slightly differs from theirs. We remark that it is also different from that in the author’s other papers ([93], for example.) Let D be a Cartier divisor of X. A U -parabolic sheaf, or simply parabolic sheaf, on U × (X, D) is defined to be a tuple E, F∗ (E), α∗ : • E is a U -coherent sheaf on U × X. • F∗ (E) denotes a filtration of E: E = F1 (E) ⊃ F2 (E) ⊃ · · · ⊃ Fl (E) ⊃ Fl+1 (E) = E(−D). Here E(−D) denotes E ⊗ p∗U O(−D). We assume that Coki (E) := E/Fi+1 (E) are flat on U . • α∗ = (α1 , . . . , αl ) is a tuple of numbers 0 < α1 < α2 < · · · < αl ≤ 1. It is called a system of weights, or simply a weight. Such a tuple(E, F∗ , α∗ ) will be often denoted just by E∗ . The underlying filtration F∗ is called a quasi-parabolic structure. Isomorphisms of parabolic sheaves are defined naturally. The number l is called the depth. For any parabolic sheaf E∗ , we put Gri (E) := Fi (E)/Fi+1 (E), which are U -coherent sheaves on U × D. Remark 3.1.4 We will often use the word “parabolic” even if a system of weights is not given. (Subobject and quotient object) Let E∗ be a parabolic torsion-free sheaf defined over U × (X, D). For any subsheaf E ⊂ E and any quotient sheaf E −→ E

, we have the induced parabolic structures on E and E

. Namely, we put Fi (E ) := Fi (E) ∩ E , and Fi (E

) :=

66

3 Parabolic L-Bradlow Pairs

Im(Fi (E) −→ E

). The parabolic structures are called the induced parabolic structures. We always consider the induced parabolic structures on subsheaves and quotient sheaves. (Condition Om ) Let (E, F∗ ) be a U -quasi-parabolic sheaf on U ×(X, D). Let m be a positive integer. We say that (E, F∗ ) satisfies the condition Om , if the following holds: • Fi (E)(m)|{u}×X and Coki (E)(m)|{u}×X are generated by its global sections, for all i = 1, . . . , l + 1 and for all u ∈ U . • The higher cohomology groups of Fi (E)(m)|{u}×X and Coki (E)(m)|{u}×X vanish, for all i = 1, . . . , l + 1 and for all u ∈ U . When we are given a U -quasi-parabolic sheaf (E, F∗ ) on U × (X, D), the open subset U is determined by the condition Om . (Twist for quasi-parabolic sheaves) Let m be an integer, and let E be a U -coherent sheaf defined over U × X. Recall that E(m) denotes the coherent sheaf E ⊗ p∗U OX (m). If E has a quasi-parabolic E(m) of induced parabolic structure F structure F∗ (E), we have the naturally ∗ E(m). The tuple E(m), F∗ (E(m)) is denoted by E∗ (m).

3.1.3 L-Bradlow Pairs and Reduced L-Bradlow Pairs Let L be a line bundle over X. Definition 3.1.5 (L-section) Let E be a U -coherent sheaf on U × X. A morphism φ : p∗U L −→ E is called an L-section. An OX -section is an ordinary section. Definition 3.1.6 Let (E, φ) be a pair of a U -coherent sheaf on X and an L-section. We say that φ is non-trivial everywhere, if φ|{u}×X = 0 for every point u ∈ U . Definition 3.1.7 (L-Bradlow pair) A parabolic L-Bradlow pair (E∗ , φ) on U × (X, D) is a pair of a U -torsion-free parabolic sheaf E∗ and an L-section φ : p∗U L −→ E. An isomorphism between two such pairs (E∗ , φ) and (E∗ , φ ) is defined to be an isomorphism χ : E∗ −→ E∗ satisfying φ = χ∗ (φ) := χ ◦ φ. (Oriented L-Bradlow pair) An oriented parabolic L-Bradlow pair (E∗ , φ, ρ) on U × (X, D) is a pair of a parabolic L-Bradlow pair (E∗ , φ) and an orientation of E. An isomorphism of two such pairs is defined naturally. Remark 3.1.8 We are mainly interested in parabolic L-Bradlow pairs (E∗ , φ) such that φ is non-trivial everywhere. We will sometimes assume it implicitly. Definition 3.1.9 Let (E∗ , φ) and (E∗ , φ ) be parabolic L-Bradlow pairs on X. We say that (E∗ , φ ) is a subobject of (E∗ , φ) if the following conditions hold:

3.1 Sheaves with Some Structure and their Moduli Stacks

• E is a subsheaf of E, and the parabolic structure is the induced one. • If the image of φ is contained in E, we have φ = φ. Otherwise, φ = 0.

67

We also introduce the notion of reduced L-section. Definition 3.1.10 (Reduced L-section) Let L be a line bundle over X. Let E be a U -coherent sheaf on U × X. A reduced L-section of E is defined to be a pair (M, [φ]) of a line bundle M on U and a morphism [φ] : p∗X (M ) ⊗ p∗U (L) −→ E. A reduced L-section is often denoted just by [φ] instead of (M, [φ]), if there are no risk of confusion. Definition 3.1.11 Let (E, [φ]) be a pair of a U -coherent sheaf on U × X and a reduced L-section. We say that [φ] is non-trivial everywhere, if [φ]|{u}×X = 0 for each u ∈ U . Let (M, [φ]) be a reduced L-section of E which is non-trivial everywhere. Then, (M, [φ]) induces a morphism [φ] : M −→ pX∗ Hom(L, E). Assuming = 0 (i > 0) for each u ∈ U , we obtain a section U −→ Hi X, Hom(L, E)|{u}×X P pX ∗ (Hom(L, E))∨ . Conversely, such a section U −→ P pX ∗ (Hom(L, E))∨ induces a reduced L-section which is non-trivial everywhere. Definition 3.1.12 (Parabolic reduced L-Bradlow pair) • A parabolic reduced L-Bradlow pair (E∗ , M, [φ]) on U × (X, D) is defined to be a pair of a torsion-free parabolic sheaf E∗ on U × (X, D) and a reduced L-section (M, [φ]) which is non-trivial everywhere. It is often denoted by (E∗ , [φ]) instead of (E∗ , M, [φ]). An isomorphism of two reduced L-Bradlow pairs (Ei ∗ , Mi , [φi ]) (i = 1, 2) is defined to be a pair (χ, η) of an isomorphism χ : E1 ∗ −→ E2 ∗ and η : M1 M2 such that the following diagram is commutative: φ1

p∗X M1 ⊗ p∗U L −−−−→ E1 ⏐ ⏐ ⏐ ⏐ χ! η⊗idL ! φ2

p∗X M2 ⊗ p∗U L −−−−→ E2 • An oriented parabolic reduced L-Bradlow pair is defined to be a tuple of a parabolic reduced L-Bradlow pair (E∗ , [φ]) with an orientation ρ of E. An isomorphism between two oriented reduced L-Bradlow pair is naturally defined. Remark 3.1.13 In the definition, we assume that [φ] is non-trivial everywhere, contrast to the case of non-reduced L-Bradlow pair. Remark 3.1.14 If U = Spec(k), a parabolic reduced L-Bradlow pair is just a parabolic L-Bradlow pair whose L-section is non-trivial, up to isomorphisms.

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68

Remark 3.1.15 We will often use the words “L-Bradlow pair” and “reduced L-Bradlow pair” instead of “parabolic L-Bradlow pair” and “parabolic reduced L-Bradlow pair”, if there are no risk of confusion. We will also use the words “quasi-parabolic L-Bradlow pair” and “quasiparabolic reduced L-Bradlow pair”, if a system of weights is not given. We will use the following auxiliary notion. Definition 3.1.16 Let L = (L1 , L2 ) be a tuple of line bundles on X. • A parabolic L-Bradlow pair on U × (X, D) is defined to be a tuple (E∗ , φ) of a U -parabolic torsion-free sheaf E∗ on U × (X, D) and a pair φ = (φ1 , φ2 ) of Li -sections φi (i = 1, 2). • An oriented parabolic reduced L-Bradlow pair on U × (X, D) is defined to be a tuple (E∗ , [φ], ρ) of a U -parabolic torsion-free sheaf E∗ on U × (X, D), a pair [φ] of reduced Li -sections [φi ] which are non-trivial everywhere, and an orientation ρ. Isomorphisms are defined naturally.

3.1.4 Type and Moduli Stacks Let H ∗ denote e´ tale cohomology theory with Q -coefficient for some prime , or singular cohomology theory with Q-coefficient if the ground field k is the com dim X plex number field. We put H ev (X) := i=0 H 2i (X). If D is a smooth divisor, ∞ let T ype denote the set of (y, y1 , y2 , . . . . . .) ∈ H ev (X) ⊕ i=1 H ev (Y ) satis fying the condition i≥1 yi = y|D . In the general case, T ype denotes the set of ∞ (y, y1 , y2 , . . . . . .) ∈ H ev (X) ⊕ i=1 H ev (X) satisfying yi = y · 1 − ch O(−D) . i≥1

In the following, y · 1 − ch O(−D) is denoted by y|D for simplicity of the notation, even if D is not necessarily smooth. For any quasi-parabolic sheaf (E, F∗ ) on X of depth l, we obtain the following element of T ype: type E, F∗ := ch(E), ch Gr1 (E) , . . . , ch Grl−1 (E) , ch Grl (E) , 0, . . . Let T ype denote the subset of T ype which consists of type E, F∗ for some quasiparabolic sheaves (E, F∗ ). Let y = (y, y1 , y2 , . . . . . .) be any element of T ype. The number depth(y) := max{i | yi = 0} is called the depth of y. The element y is called the H ev (X)component of y, and (y1 , y2 , . . . , ) is called the parabolic part of y. The H 0 (X)component of y is called the rank of y, and it is denoted by rank(y) or rank y.

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69

The H 2i (X)-components of y are called the i-th Chern character of y. We put χ(y) := Td(X) · y. We set T yper := y ∈ T ype rank(y) = r . We denote by T ype◦ the set of types whose parabolic components are trivial, i.e., y1 = y|D and yi = 0 for i > 1. We often identify y and y in that case, and we regard T ype◦ as a subset of H ∗ (X). We put T ype◦r := T ype◦ ∩ T yper . We have the sum y (1) + y (2) of two elements y (i) ∈ T ype (i = 1, 2) by taking component-wise summation. For any quasi-parabolic sheaf (E, F∗ ) on X, we obtain the element type E, F∗ of T ype. In general, let (E, F∗ ) be a U-quasi-parabolic sheaf on U × X. If we have an element y ∈ T ype such that type (E, F∗| )|{u}×X = y for any closed point u ∈ U , then (E, F∗ ) is called of type y. If U is connected, such an element always exists. Definition 3.1.17 The type of an (oriented) parabolic L-Bradlow pairs are defined to be the type of the underlying quasi-parabolic sheaves. We introduce the following notation. Notation 3.1.18 In each line, the left hand side denotes a moduli stack of objects in the right hand side: M(y): Quasi-parabolic sheaves of type y. M(

y ): Oriented quasi-parabolic sheaves of type y. M y, L : Quasi-parabolic L-Bradlow pairs of type y whose L-sections are nontrivial everywhere.

, L : Oriented quasi-parabolic L-Bradlow pairs of type y whose L-sections M y are everywhere. non-trivial My, [L]: Quasi-parabolic reduced L-Bradlow pairs of type y.

, [L] : Oriented quasi-parabolic reduced L-Bradlow pairs of type y. M y

, [L] : Oriented quasi-parabolic reduced L-Bradlow pairs of type y. M y The condition Om determines the open substacks of the moduli stacks. They

), M(m, y, L), M(m, y

, L), M(m, y, [L]), are denoted by M(m, y), M(m, y

, [L]), and M(m, y

, [L]). M(m, y When the parabolic part of y is trivial, we often use the symbols M(y), M(

y ), M(m, y), M(m, y ), etc..

3.1.5 Tautological Line Bundle and Relations Among Some Moduli Stacks Let y ∈ T ype, and let L be a line bundle on X. Let E u (L), E u (L) and E u [L] denote the universal sheaves over M(

y , L) × X, M(y, L) × X and M(

y , [L]) × X respectively. The universal L-sections of E u (L) and E u (L) are denoted by φ u and φu , respectively. The universal reduced L-section of E u [L] is denoted by [φ u ].

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70

We have the Gm -action ρ1 on M(

y , L) given by ρ1 (t) · (E, F∗ , φ, ρ) := E, F∗ , t·φ, ρ . It is easy to observe that the quotient stack is isomorphic to M(

y , [L]). Therefore, y , [L]). The associated line bundle we can regard M(

y , L) as a Gm -torsor over M(

is denoted by Orel (−1). We put Orel (1) := Orel (−1)∨ , and we obtain Orel (n) in the obvious manner. Definition 3.1.19 The line bundle Orel (1) is called the relative tautological line bundle of M(

y , [L]). It is also called the tautological line bundle. We can obtain the same line bundles with the method explained in Subsection 2.7.1. Namely, let T (n) denote the trivial line bundle on M(

y , [L]) provided with y , L) −→ M(

y , [L]) denote the natural the Gm -action of the weight n. Let π1 : M(

projection. Then, we have the Gm -line bundle π1∗ T (n). By taking descent, we obtain a line bundle In . We have the natural isomorphism In Orel (n). We remark that Orel (−1) appears in the domain of the universal reduced Lsection [φ u ]. Namely, [φ u ] is the morphism: p∗M( y,[L]) L ⊗ p∗X Orel (−1) −→ E u [L]. To see that, we have only to observe that ∗ ∗ E u [L] T (−1) −→ π1,X π ∗ [φ u ] : p∗M( y,L) L ⊗ π1,X

is equivariant with respect to ρ1 , and they give the universal objects over M(

y , L) × X.

, [O(−m)]) −→ M(m, y

) is a projective space Remark 3.1.20 Note that M(m, y

, [O(−m)]) is the relative tautological bundle. The restriction of Orel (1) to M(m, y line bundle. y , L) given by On the other hand, we have the Gm -action ρ2 on M(

ρ2 (t) · (E, F∗ , φ, ρ) := (E, F∗ , φ, t·ρ). It is easy to observe that the quotient stack is isomorphic to M(y, L). Thus, we can regard M(

y , L) as a Gm -torsor on M(y, associated line bundle is clearly L). The isomorphic to the orientation bundle Or E u (L) . Let r be the rank of y. The obvious multiplication of E gives the isomorphism E, t−1 ·φ, tr ρ (E, φ, ρ) To see it, note the following. Let f : E1 −→ E2 be an isomorphism. Then, the orientation ρ of E2 induces the following orientation of E1 : det(f )

det(E1 ) −−−−→ det(E2 ) −−−−→ det∗ Poin. ρ

3.2 Hilbert Polynomials

71

The L-morphism φ of E2 induces the following L-morphism of E1 : f −1

φ

L −−−−→ E2 −−−−→ E1 . Therefore, we have ρ1 (t) = ρ2 (tr ). Hence we have the naturally defined morphism κ : M(

y , [L]) −→ M(y, L), which is e´ tale and finite of degree r−1 . However, the morphism κ does not preserve the universal object. We have the relation κ∗X E u (L) E u [L] ⊗ Orel (1), and hence κ∗ Or E u (L) Orel (r).

3.2 Hilbert Polynomials 3.2.1 Hilbert Polynomials of Coherent Sheaves Let OX (1) be an ample line bundle on a projective variety X. Let E be a coherent sheaf on X. In this paper, the non-reduced Hilbert polynomial of E is denoted by such that HE (m) is equal HE , i.e., HE is the unique polynomialof Q-coefficients to the Euler number (−1)i dim H i X, E(m) . In the case rank(E) > 0, the reduced Hilbert polynomial of E is denoted by PE , i.e., PE := HE / rank(E). We also use the symbol h0 (E) to denote dim H 0 (X, E).

3.2.2 Hilbert Polynomials of Parabolic Sheaves We recall the notions of parabolic Hilbert polynomials and parabolic degree. They were introduced by Maruyama and Yokogawa in [87] to which we refer for more details and precise. Let E∗ := (E, F∗ , α∗ ) be a parabolic sheaves of depth l. We put i := αi+1 − αi (i = 1, . . . , l). Recall that the non-reduced parabolic Hilbert polynomial HE∗ is defined to be HE∗ (t) := HE(−D) (t) +

l

αi · HGri (E) = HE (t) −

i=1

l

i · HCoki (E) .

i=1

The reduced parabolic Hilbert polynomial PE∗ (t) is defined to be PE∗ (t) := HE∗ (t) rank(E) Because HE = HE(−D) +

l i=1

HGri (E) , we obtain the following lemma.

Lemma 3.2.1 We have the inequality HE∗ (t) ≤ HE (t) and PE∗ (t) ≤ PE (t) for any sufficiently large t.

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72

The parabolic degree is defined to be par-deg(E∗ ) = deg(E)+ l (dim X − 1)! · the coefficient of tdim X−1 in i=1 αi · HGri (t)

(3.3)

Note that we have the inequality deg(E) ≤ par-deg(E∗ ). The parabolic slope μ(E∗ ) is defined to be par-deg(E∗ )/ rank(E). Then, we have the inequality μ(E) ≤ μ(E∗ ) ≤ μ(E) + deg(D) for the usual slope μ(E) of a torsion-free sheaf E. We also put h0 (E∗ ) := α1 h0 E(−D) + i h0 Fi+1 (E) .

3.2.3 Hilbert Polynomial for Parabolic L-Bradlow Pairs We recall Hilbert polynomials for Bradlow pairs, following [119]. Notation 3.2.2 Let P br denote the set of polynomials δ of R-coefficients such that (i) deg(δ) ≤ dim X − 1, (ii) δ(t) > 0 for any sufficiently large t. For any δ ∈ P br , the coefficient of td−1 in δ is denoted by δtop , which may be 0. The total order ≤ on the set P br is defined as follows: Let δ and δ be elements of P br . Then, δ ≤ δ if and only if δ(t) ≤ δ (t) for any sufficiently large t. Let (E∗ , φ) be a parabolic L-Bradlow pair on (X, D). For any δ ∈ P br , the δ of (E∗ , φ) is defined as follows: non-reduced δ-Hilbert polynomial H(E ∗ ,φ) δ H(E ∗ ,φ)

:= HE∗ + (E∗ , φ) · δ,

(E∗ , φ) :=

1 (φ = 0) 0 (φ = 0)

(3.4)

δ δ := H(E / rank E. The reduced δ-Hilbert polynomial is defined to be P(E ∗ ,φ) ∗ ,φ) δ Similarly, the slope μ(E∗ ,φ) is defined by

μδ (E∗ , φ) := μ(E∗ ) + (dim X − 1)! · (φ) · δtop / rank(E). Let L = (L1 , L2 ) be a pair of line bundles on X. We take δ := (δ1 , δ2 ) ∈ br 2 P . Let (E∗ , φ) on (X, D) be a parabolic L-Bradlow pair. The δ-Hilbert polyδ := H + nomial is defined by H(E E ∗ i=1,2 (E∗ , φi ) · δi , where (E∗ , φi ) are ∗ ,φ) δ δ given as in (3.4). We also put P(E = H (E∗ ,φ) / rank E. ∗ ,φ)

3.2.4 Hilbert Polynomials Associated to a Type Let z be any element of i≥1 H i (X). Because z k = 0 for a large integer k, we ∞ have the polynomial exp(t·z) = k=0 (k!)−1 (t · z)k . In the case c = c1 OX (1) ,

3.3 Semistability

73

it is denoted by ch OX (t) . When we substitute t = m for some integer m, it is the same as the ordinary meaning. Let y = (y, y1 , y2 , . . . , ) be an element of T ype. We put Hy (t) := Td(X) · ch OX (t) · y, X

Hy,i (t) :=

Td(X) · ch OX (t) · yj .

X

j≤i

When D is smooth regard yi as an element of H ∗ (D), we put Hy,i (t) := andwe Td(D) · ch OX (t) · j≤i yj . When the parabolic part of y is trivial, we D use the symbol Hy (t) instead of Hy (t). If we are given a system of weights α∗ , we put i := αi+1 − αi . And we set Hyα∗ := Hy −

i · Hy,i ,

Pyα∗ :=

Hyα∗ . rank y

Let d := dim X. We also put d−1 d−1 y · c1 OX (1) − i · yj · c1 OX (1) deg(y, α∗ ) := X

X j≤i

μ(y, α∗ ) :=

deg(y, α∗ ) rank y

If the parabolic part of y is trivial, then they are denoted by deg(y) and μ(y), respectively. If we are given an element δ ∈ P br , we put Hyα∗ δ := H α∗ + δ,

Pyα∗ δ :=

μ(y, α∗ , δ) := μ(y, α∗ ) +

Hyα∗ δ , rank y

(d − 1)! · δtop rank(y)

3.3 Semistability 3.3.1 Semistability Conditions and the Associated Moduli Stacks Let (E∗ , φ) be a parabolic L-Bradlow pair on (X, D), and let δ ∈ P br . Recall that (E∗ , φ) is called δ-semistable, if the following inequality holds for each sub-object (E∗ , φ ) of (E∗ , φ): δ δ P(E (t) ≤ P(E ,φ) (t) ∗ ∗ ,φ )

(t is sufficiently large)

(3.5)

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74

If the strict inequality holds in (3.5) for each non-trivial subobject, (E∗ , φ) is called δ-stable. Since parabolic reduced L-Bradlow pairs on X are just parabolic L-Bradlow pairs whose L-sections are non-trivial, the notion of δ-(semi)stability of parabolic reduced L-Bradlow pairs is also given. The δ-(semi)stability of oriented parabolic (reduced) L-Bradlow pairs on X is defined by the δ-(semi)stability of the underlying parabolic (reduced) L-Bradlow pairs. Remark 3.3.1 Let E be a torsion-free sheaf on X. We can regard it as a parabolic sheaf E∗ canonically. Namely, the quasi-parabolic structure is given by F2 = E(−D) ⊂ F1 = E, and the weight is given by α1 = 1. Then, we have HE∗ = HE , and hence the semistability of (E∗ , φ) is equivalent to the semistability of (E, φ). If the weight is given by α1 = 0, then we have HE∗ = HE(−D) , and hence the semistability of (E∗ , φ) and (E, φ) are not equivalent, as remarked in Remark 1.1.1 in [87]. We also have the δ-μ-semistability in the standard way. Namely, a parabolic LBradlow pair (E∗ , φ) on (X, D) is called δ-μ-semistable, if the inequality μδ (E∗ , φ ) ≤ μδ (E∗ , φ) holds for any sub-objects (E∗ , φ ) ⊂ (E∗ , φ). If the strict inequality holds for any subjects such that 0 < rank E < rank E, (E∗ , φ) is called δ-μ-stable. It is easy to check the following implication: δ-μ-stable =⇒ δ-stable =⇒ δ-semistable =⇒ δ-μ-semistable Similarly, we have the notion of δ-μ-semistability and δ-μ-stability of parabolic reduced L-Bradlow pairs. In the family case, a U -parabolic L-Bradlow pair (E∗ , φ) on U × X is called δ-(semi)stable, if (E∗ , φ)|{u}×X is δ-(semi)stable for each u ∈ U . We obtain δ(semi)stability conditions in the oriented case and in the reduced case similarly. Remark 3.3.2 As usual, we have only to consider sub-objects (E∗ , φ ) ⊂ (E∗ , φ) such that E are saturated in order to check δ-(semi)stability of (E∗ , φ). Let Li (i = 1, 2) be line bundles on X, and let ι : L1 −→ L2 be a non-trivial morphism. Let (E∗ , φ) be a parabolic L2 -Bradlow pair on (X, D). We obtain the parabolic L1 -Bradlow pair (E∗ , φ ◦ ι). When we consider the semistable condition for parabolic L-Bradlow pairs, the choice of L is not essential in the following sense. Lemma 3.3.3 A pair (E∗ , φ) is δ-semistable, if and only if (E∗ , φ◦ι) is δ-semistable. Proof Let F be a saturated subsheaf of E. The image φ ◦ ι(L1 ) is contained in F if and only if the image φ(L2 ) is contained in F . Hence, the claim is clear. Lemma 3.3.4 Let (Ei ∗ , φi ) (i = 1, 2) be δ-semistable parabolic L-Bradlow pairs δ δ (t) = P(E (t). Let f : (E1 ∗ , φ1 ) −→ (E2 ∗ , φ2 ) be a non-trivial with P(E 1 ∗ ,φ1 ) 2 ∗ ,φ2 )

3.3 Semistability

75

morphism. We have the induced L-Bradlow pairs (Ker(f )∗ , φ ), (Im(f )∗ , φ

) and (Cok(f )∗ , φ

). Then, they are also δ-semistable. A similar claim holds for δ-μ-semistability. Proof We put (E3 ∗ , φ3 ) := Im(f )∗ , φ

. From the δ-semistability of (Ei ∗ , φi ) (i = 1, 2), we obtain the following inequality for sufficiently large t: δ δ δ (t) ≤ P(E (t) ≤ P(E (t). P(E 1 ∗ ,φ1 ) 3 ∗ ,φ3 ) 2 ∗ ,φ2 )

(3.6)

δ δ Thanks to the assumption P(E (t) = P(E (t), the equalities hold in (3.6). 1 ∗ ,φ1 ) 2 ∗ ,φ2 ) Then, it is easy to derive the claims of the lemma by definition of semistability.

Corollary 3.3.5 Any automorphisms of stable objects are multiplication of constants. We introduce the following notation. In each line, the left hand side denotes a moduli stack of objects in the right hand side: Ms (y, α∗ ): Stable parabolic sheaves of type y with weight α∗ . y , α∗ ): Stable oriented parabolic sheaves of type y with weight α∗ . Ms (

Ms (y, L, α∗ , δ): δ-Stable L-Bradlow pairs of type y with weight α∗ , whose L-sections are non-trivial everywhere. y , L, α∗ , δ): δ-Stable oriented L-Bradlow pairs of type y with weight α∗ , Ms (

whose L-sections are non-trivial everywhere. Ms (y, [L], α∗ , δ): δ-Stable reduced L-Bradlow pairs of type y with weight α∗ . y , [L], α∗ , δ): δ-Stable oriented reduced L-Bradlow pairs of type y with Ms (

weight α∗ . For moduli stacks of semistable objects, we use the symbols Mss (y, α∗ ), Mss (

y , α∗ ), Mss (y, L, α∗ , δ), Mss (

y , [L], α∗ , δ), etc.. If the parabolic part of y is trivial, we often use the symbols Ms (

y ), Ms (

y , [L], δ), s M (y, L, δ), etc.. Similarly, we have the notion of δ-(semi)stability for (oriented, reduced) y , [L], α∗ , δ) L-Bradlow pairs. We also have δ-μ-(semi)stability. We denote by Ms (

a moduli stack of δ-stable oriented reduced L-Bradlow pairs of type y with weight α∗ , for example. sm (

), we denote by M y , α∗ ) ⊂ M(m, y y , α∗ ) the full flag bundle assoIf Ms (

u u

ciated to the vector bundle pX ∗ E (m), where E denotes the universal bundle over sm y sm y

, [L], α∗ , δ and M

, [L], α∗ , δ , Ms (

y , α∗ ) × X. We use the symbols M etc., in similar ways. If there are no risk of confusion, we will often omit to denote m.

3 Parabolic L-Bradlow Pairs

76

3.3.2 Harder-Narasimhan Filtration and Partial Jordan-H¨older Filtration Let (E∗ , φ) be a δ-semistable parabolic L-Bradlow pair. In this monograph, a filtration (E1 ∗ , φ1 ) ⊂ (E2 ∗ , φ2 ) ⊂ · · · ⊂ (Ek ∗ , φk ) = (E∗ , φ) is called a partial Jordan-H¨older filtration with respect to δ-semistability, if each δ δ = P(E . Each Gri (E) := Ei /Ei−1 (Ei ∗ , φi ) is δ-semistable such that P(E i ∗ ,φi ) ∗ ,φ) has the induced parabolic structure and the L-section Gri (φ), and the parabolic δ δ = P(E . L-Bradlow pair Gri ∗ (E), Gri (φ) is δ-semistable with P(Gr i ∗ (E),Gri (φ)) ∗ ,φ) If each Gri ∗ (E), Gri (φ) is δ-stable, the filtration is called a Jordan-H¨older filtration with respect to δ-semistability. It can be shown Jordan that the length of H¨older filtration and the collection of graded objects Gri ∗ (E), Gri (φ) are independent of the choice of Jordan-H¨older filtrations. Similarly, we have the notions of partial Jordan-H¨older filtration and JordanH¨older filtration with respect to δ-μ-semistability. pair. There exists the unique Lemma 3.3.6 Let (E∗ , φ) be a parabolic L-Bradlow increasing filtration F = Fi (E) | i = 1, 2, . . . of E with the following property: F • The induced objects GrF i (E), Gri (φ) are δ-semistable. δ δ (t) hold for any • The inequalities P(Gr F (E),GrF (φ )) (t) > P(GrF (E),GrF (φ i i+1 )) i i i+1 i+1 sufficiently large t. The filtration is called the Harder-Narasimhan filtration with respect to the δsemistability. We also have the unique Harder-Narasimhan filtration with respect to the δ-μsemistability. Proof We give only an outline. We use an induction on rank(E). In the case rank(E) = 1, the claim is trivial. Take a sufficiently negative number C. We know that the family of saturated subsheaves E of E such that deg(E ) ≥ C is bounded. (See Proposition 3.4.1 below.) Therefore, the family of saturated subsheaves E of δ δ

E such that P(E ≥ P(E ,φ) is bounded, where (E∗ , φ ) denotes the subobject of ∗ ∗ ,φ )

(E∗ , φ) associated to E . Hence we obtain the finiteness of the set S of polynomials P with the following property: • P (t) ≥ P(E∗ ,φ) (t) for any sufficiently large t. δ • There exist a saturated subsheaf E of E such that P(E ,φ ) = P . ∗

We say P ≤ Q if P (t) ≤ Q(t) for any sufficiently large t. It gives the total order of S. Let P0 be the maximum. Let T (P0 ) denote the family of saturated subsheaves δ = P0 . Then, (E∗ , φ) is δ-semistable for any E ∈ T (P0 ). E of E such that P(E ∗ ,φ) It is also easy to see that E1 + E2 ∈ T (P0 ) for Ei ∈ T (P0 ). Therefore, we have the maximum E1 in T (P0 ) with respect to inclusion relation.

3.3 Semistability

77

:= E/E1 , and let π : E −→ E

denote the natural projection. We put E

δ δ

⊂ E

be any saturated subsheaf. If P Let E

) ≥ P(E1 ,φ1 ) , then we have

,φ (E δ δ P(π −1 (E

),φ ) ≥ P(E1 ,φ1 ) , which contradicts our choice of E1 . Therefore, we have δ P(δE ,φ ) < P(E . By applying the hypothesis of the induction, we have the 1 ,φ1 )

with respect to δ-semistability. Together

∗ , φ) Harder-Narasimhan filtration of (E with the above remark, we obtain the Harder-Narasimhan filtration of (E∗ , φ) with respect to δ-semistability.

3.3.3 (δ, )-Semistability Let P be a polynomial, and let r be a positive integer. Let m be a sufficiently large integer satisfying the following condition: δ • Let (E∗ , φ) be a δ-parabolic L-Bradlow pair with P(E = P and rank E ≤ r. ∗ ,φ) Then, E∗ satisfies the condition Om .

Definition 3.3.7 Let (E∗ , [φ]) be a parabolic reduced L-Bradlow pair on X with δ 0 X, E(m) . Let be any positive integer. = P , and let F be a full flag of H P(E ∗ ,φ) We say that (E∗ , [φ], F) is (δ, )-semistable, if the following conditions are satisfied: A(δ, ): (E∗ , [φ]) is δ-semistable. B(δ, ): Take any partial Jordan-H¨older filtration of (E∗ , [φ]) with respect to δsemistability: (1)

(2)

⊂ E∗

(i−1)

(i)

(p)

⊂ · · · ⊂ E∗ ⊂ (E∗ , φ) ⊂ · · · ⊂ (E∗ , φ) Then, we have F ∩ H 0 X, E (i−1) (m) = {0} and F ⊂ H 0 X, E (j) (m) for j < p. ss Let M m y, [L], α∗ , (δ, ) denote a moduli stack of such tuples (E∗ , [φ], F). In ss

, [L], α∗ , (δ, ) as usual. the oriented case, we use the symbol M m y Similarly, we have the notion of (δ, )-semistability for a tuple (E∗ , φ, F) of an L-Bradlow pair (E∗ , φ) with φ = 0 and a full flag F of H 0 (X, E(m)). A moduli ss y, L, α∗ , (δ, ) . If there are no risk of confusion, we omit stack is denoted by M m to denote m. E∗

Remark 3.3.8 Assume that is sufficiently large. The second condition in B(δ, ) is trivial. The first condition in B(δ, ) is equivalent to i = 1 for any J-H filtration. Therefore, (q, E∗ , φ, F) is (δ, )-semistable if and only if (E∗ , φ) is δ− -semistable for a parameter δ− < δ such that |δ − δ− | is sufficiently small. Lemma 3.3.9 Let (E∗ , [φ], F) be a reduced L-Bradlow pair with a full flag F of H 0 X, E(m) . We assume that it is (δ, )-semistable. Then, the automorphism group of (E∗ , [φ], F) is Gm .

3 Parabolic L-Bradlow Pairs

78

Proof Let f be an endomorphism of (E∗ , [φ], F). have the generalized eigen We p decomposition (E∗ , [φ], F) = (E1 ∗ , [φ1 ], F (1) )⊕ i=2 (Ei ∗ , F (i) ). But, the (δ, )semistability condition is not satisfied if p ≥ 2. Therefore, f has the unique eigenvalue. Let N be the nilpotent part of f . Let h be the integer such that N h = 0 and N h+1 = 0. Assume h ≥ 1, and we will derive a contradiction. We put E1 := Im N h and E2 := Ker N h . We have the naturally induced parabolic structures and the L-sections φi of Ei . Because N (φ) = 0, we have φ1 = 0. Then, we obtain a partial Jordan-H¨older filtration E1 ∗ ⊂ (E2 ∗ , φ2 ) ⊂ (E∗ , φ) with respect to δ-semistability, according to Lemma 3.3.4. Note that we have the induced filtrations F (1) on H 0 X, E1 (m) and F (3) on H 0 X, E/E2 (m) . The induced isomorphism H 0 X, E/E2 (m) −→ H 0 X, E1 (m) (1)

(3)

has to preserve the filtrations F (i) above. However, we have F = 0 and F = 0 thanks to the (δ, )-semistability. We arrive at the contradiction, and hence we have h = 0.

3.4 Some Boundedness 3.4.1 Fundamental Theorems Let OX (1) be a very ample line bundle. Let D denote a Cartier divisor of X. We recall several fundamental theorems, by following D. Huybrechts-M. Lehn ([62]). Proposition 3.4.1 (A. Grothendieck, Lemma 2.5, [57]) Let D be a bounded family of coherent sheaf F on X. Let C be any positive number. Then, we have the boundedness of the family of torsion-free coherent sheaves F

with the following property: • deg(F

) ≤ C. • There exists a member F ∈ D such that F

is a quotient sheaf of F .

For any torsion-free coherent sheaf E on X, we have the Harder-Narasimhan filtration with respect to the standard semistability. We denote the slope of the first (resp. last) term of the Harder-Narasimhan filtration by μmax (E) (resp. μmin (E)). Proposition 3.4.2 (M. Maruyama [85]) Let H be a polynomial, and let C be a constant. We have the boundedness of the family of torsion-free coherent sheaves F on X satisfying μmax (F ) ≤ C and HF = H. We use the notation [x]+ = max{x, 0} for any real number x. Proposition 3.4.3 (C. Simpson [117]) Let r be a positive integer. Then, there is a positive constant c such that the following inequality holds for every μ-semistable sheaf F satisfying rank(F ) < r and μ(F ) < μ:

3.4 Some Boundedness

79

1 h0 (F ) ≤ d−1 ([μ + c]d+ ). rank(F ) g d! d Here d := dim X and g := c1 OX (1) ∩ [X].

3.4.2 Boundedness of Semistable L-Bradlow Pairs Let y ∈ T ype, and let α∗ be a system of weights. Let L be a line bundle over X, and let δ (0) ∈ P br . Let SS(y, L, α∗ , δ (0) ) denote the family of parabolic L-Bradlow pairs (E∗ , φ) such that φ = 0 with the following property: • The type of E∗ is y, and the weight of the parabolic structure is given by α∗ . • (E∗ , φ) is δ-μ-semistable for some δ ≤ δ (0) in P br . Lemma 3.4.4 The family SS(y, L, α∗ , δ (0) ) is bounded. Proof Let (E∗ , φ) be a member of SS(y, L, α∗ , δ (0) ). It is δ-μ-semistable for some δ ∈ P br with δ ≤ δ (0) . Let E be the first member of the Harder-Narasimhan filtration of E with respect to the standard semistability. Let d := dim X. Then, we have the following inequalities: μmax (E) = μ(E ) ≤ μ(E∗ ) +

(E , φ) · (d − 1)! · δtop rank E

(0)

≤ μ(E∗ ) +

(d − 1)! · δtop (E, φ) · (d − 1)! · δtop ≤ μ(E∗ ) + . rank E rank E

(3.7)

The last term depends only on (y, δ (0) ). Thus, we obtain the boundedness from Proposition 3.4.2. Recall the following important observation due to M. Thaddeus [119]. Proposition 3.4.5 Take an element y ∈ T ype and a line bundle L on X. Let d := dim X. Assume r = rank(y) > 1. Let δ be an element of P br satisfying the following condition: (d − 1)! · δtop >

r μ(y, α∗ ) − deg(L) . r−1

(3.8)

Then, there does not exist a δ-semistable parabolic L-Bradlow pair (E∗ , φ) of type y such that φ = 0. Proof Let (E∗ , φ) be a δ-semistable L-Bradlow pair such that φ = 0. The L-section φ generates the subsheaf E of E with rank E = 1. Thanks to the δ-semistability, we have the following inequality: μ(E∗ ) + (d − 1)! · δtop ≤ μ(E∗ ) +

(d − 1)! · δtop . r

3 Parabolic L-Bradlow Pairs

80

We also have μ(E∗ ) ≥ μ(E ) = deg(E ) ≥ deg(L). Therefore, we obtain the following: ( ' 1 · (d − 1)! · δtop ≤ μ(E∗ ) − deg(L). 1− r Thus we are done.

Corollary 3.4.6 The family SS y, L, α∗ := δ∈P br SS y, L, α∗ , δ is bounded. Proof It follows from Proposition 3.4.5 and Lemma 3.4.4.

(0)

(0)

Let L = (L1 , L2 ) be a pair of line bundles on X. Let δ (0) = (δ1 , δ2 ) ∈ (P br )2 . Let SS(y, L, α∗ , δ (0) ) denote the family of parabolic L-Bradlow pairs (E∗ , φ) of type y with weight α∗ such that φi = 0, which is (δ1 , δ2 )-μ-semistable (0) for some δi ≤ δi . By an argument employed in the proof of Lemma 3.4.4, we can show the following: Lemma 3.4.7 The family SS(L, α∗ , δ (0) ) is bounded.

By an argument used in the proof of Proposition 3.4.5, we can show the following lemma. (0)

Lemma 3.4.8 Fix δ1 ∈ P br . There exists δ2 such that the following holds for any (0) δ2 ≥ δ2 : • There does not exist any (δ1 , δ2 )-semistable L-Bradlow pair (E∗ , φ1 , φ2 ) of type y such that φ2 = 0. In particular, the family δ2 SS L, α∗ , (δ1 , δ2 ) is bounded.

3.4.3 Boundedness of Yokogawa Family Let y ∈ T ype, and let L be a line bundle on X. Let us fix a system of weights α∗ , and we put i := αi+1 − αi . For each positive integer m, let us take an Hy (m)dimensional vector space Vm . Let Pm denote the projectivization of Vm , i.e., Pm := P(Vm∨ ). We also fix an inclusion ι : O(−m) −→ L. as Definition 3.4.9 A Yokogawa datum of type (y, m) is a tuple q, E∗ , φ, W∗ , [φ] follows: • (E∗ , φ) is a parabolic L-Bradlow pair over X, such that type(E∗ ) = y. • q is a generically surjective morphism Vm,X −→ E(m), where Vm,X := Vm ⊗ OX . • W∗ = (W1 , . . . , Wl ) is a tuple of subspaces of Vm such that dim Wi = Hy (m)− Hy,i (m). We assume that H 0 (q)(Wi ) is contained in H 0 X, Fi+1 (E)(m) . is a point of Pm , and there exists a non-zero scalar c such that H 0 (q)(φ) = • [φ] c · ι(φ), where ι(φ) denotes the section of E(m) induced by ι and φ.

3.4 Some Boundedness

81

Definition 3.4.10 Let δ ∈ P br , and let K be any non-negative number. We use the symbol YOK(m, K, y, L, δ) to denote the set of Yokogawa data q, E∗ , φ, W∗ , [φ] such that the following inequality holds for any subspace W ⊂ Vm : Pyδ,α∗ (m) rank EW

−(W, [φ])δ(m)−

l

i dim(Wi ∩W )−α1 dim(W )+K ≥ 0.

i=1

(3.9) is Here, EW denotes the subsheaf of E(m) generated by W and q, and (W, [φ]) the number 1 ([φ] ∈ PW ) or 0 ([φ] ∈ PW ). We remark (W, [φ]) = (EW (−m), φ), where (EW (−m), φ) is given as in (3.4). For each positive integer N , we set YOK(N, K, y, L, δ) :=

/

YOK(m, K, y, L, δ).

m≥N

Following Yokogawa [134], we consider the family YOK(N, K, y, L, δ) of quasiparabolic L-Bradlow pairs (E, F∗ , φ) of type y such that there exists a lift ∈ YOK(N, q, E, F∗ , φ, W∗ , [φ] K, y, L, δ). It is called the Yokogawa family. Proposition 3.4.11 There exist a small positive number K0 = K0 (y, L, δ) and a large integer N0 = N0 (y, L, δ) such that the following holds: • The family YOK N0 , K0 , y, L, δ is bounded, and it satisfies the condition Om for any m ≥ N0 . ∈ YOK N0 , K0 , y, L, δ , the morphism q is surjec• For any (q, E∗ , φ, W∗ , [φ]) tive. In particular, H 0 (q) : Vm −→ H 0 (X, E(m)) is isomorphic. • All members of YOK(N0 , K0 , y, L, δ) are δ-semistable. Proof We follow the argument of Yokogawa [134]. We begin with the following lemma. Lemma 3.4.12 Let E∗ , φ be a parabolic L-Bradlow pair contained in the family . Let E denote a quotient sheaf YOK(N, K, y, L, δ) with a lift q, E∗ , φ, W∗ , [φ] of E. Then, the following inequality holds for any m ≥ N : Pyδ,α∗ (m) ≤

h0 E∗ (m) + (E , φ) · δ(m) + K . rank(E )

(3.10)

Proof Let W denote the kernel of the composite of the following morphisms: H 0 (q) Vm −−−−→ H 0 X, E(m) −−−−→ H 0 X, E (m)

3 Parabolic L-Bradlow Pairs

82

Let EW denote the subsheaf of E(m) generated by the image of W via q. By (3.9), we obtain the following inequality: i dim(Wi ∩W )−α1 dim(W )+K ≥ 0. Pyδ,α∗ (m) rank(EW )+(W, φ)δ(m)− i

We have the inequalities dim(Wi ∩ W ) ≥ dim(Wi ) − h0 Fi+1 (E )(m) for i = + (E , [φ ]) = 1, where φ is the 1, . . . , l. We also have the equalities (W, [φ]) induced L-section of E by φ. Since q is generically surjective, we have rank(E ) + rank(EW ) = rank(E). Thus, we obtain the following inequality: 0 ≤ Pyδ,α∗ (m) · rank E − δ(m) − − α1 dim(W ) −

i · dim(Wi ) +

i δ,α∗ Py (m)

i · h0 Fi+1 (E )(m)

i

· rank E + (E , φ ) · δ(m) + K.

We have the following equality: Pyδ,α∗ (m) · rank E − δ(m) − = H(m) −

i · Hi (m) −

i · dim(Wi ) − α1 dim(W )

i

i · dim(Wi ) − α1 dim(V ) + α1 dim(V /W ).

i

Because dim(Wi ) = H(m) − Hi (m) and dim(V ) = H(m), the right hand side equals α1 dim(V /W ). We have the inequality dim(V /W ) ≤ dim H 0 X, E (m) . Hence, we obtain the following inequality: 0 ≤ − rank(E ) · Pyδ,α∗ (m) + h0 E∗ (m) + (E , φ) · δ(m) + K. Then, (3.10) immediately follows.

Lemma 3.4.13 The family YOK(N1 , K, y, L, δ) is bounded for some large N1 . d Proof We put d := dim X and g := c1 OX (1) ∩[X]. Take a sufficiently negative number C satisfying the following inequality for any sufficiently large t: 1 g d−1 d!

d C + tg + c + δ(t) + K < P δ,α∗ (t).

Take a large integer N1 such that the following inequalities hold for any m > N1 : d C + mg + c + δ(m) + K < P α∗ ,δ (m). C + mg + c > 0, δ(m) > 0, g d−1 · d! We will show that YOK(N1 , K, y, L, δ) is bounded. Let (E∗ , φ) ∈ YOK(N1 , K, y, L, δ). Let E denote the last member of the Harder-Narasimhan filtration of E with respect to the standard semistability. Assume μ(E ) < C, and we will derive a contradiction. By Proposition 3.4.3, we have the following inequality:

3.4 Some Boundedness

83

" #d μ(E ) + mg + c + h0 E∗ (m) h0 (E (m)) ≤ ≤ . rank(E ) rank(E ) g d−1 d! By the assumption μ(E ) < C, we obtain the following inequality: h0 E∗ (m) + (E , φ) · δ(m) + K (C + mg + c)d ≤ + δ(m) + K < P δ,α∗ (m). rank(E ) g d−1 d! (3.11) However, this contradicts with (3.10). Thus we obtain μ(E ) > C. It implies μmax (E) < C for some constant C , and thus we obtain the boundedness of YOK(N1 , K, y, δ) according to Proposition 3.4.2. Then, there exists an integer N2 such that the family YOK(N2 , K, y, L, δ) satisfies the condition Om for any m ≥ N2 . Lemma 3.4.14 Assume K1 is strictly smaller than α1 . Then, the map H 0 (q) : Vm −→ H 0 X, E(m) ∈ YOK(N 2 , K1 , y, L, δ). In particuis isomorphic for any q, E, F∗ , φ, W∗ , [φ] lar, q is surjective. Proof We have only to check that H 0 (q) is injective. Let W denote the kernel of H 0 (q). Then, we obtain the following inequality from (3.9): − i · dim(Wi ∩ W ) − α1 · dim(W ) + K1 ≥ 0. Because K1 < α1 , we obtain dim(W ) = 0, i.e., H 0 (q) is injective.

Let us finish the proof of Proposition 3.4.11. Let us consider the family S1 of parabolic L-Bradlow pairs (E∗ , φ ) with the following property: • There exists a member (E∗ , φ) ∈ YOK(N2 , K1 , y, L, δ) such that (E∗ , φ ) is isomorphic to the last member of Harder-Narasimhan filtration of (E∗ , φ) with respect to δ-semistability. Since S1 is bounded, there are only a finite number of polynomials which are the Hilbert polynomials of members in S1 . Therefore, there exists a small positive number K0 < K1 and a large integer N0 ≥ N2 such that the following holds for any m ≥ N0 . • Let (E∗ , φ) be a member of YOK(N0 , K0 , y, L, δ), which is not δ-semistable. Let (E∗ , φ ) denote the last member of the Harder-Narasimhan filtration of δ (E∗ , φ) with respect to δ-semistability. Then, the inequality P(E (t) + K0 < ∗ ,φ ) δ,α∗ Py (t) holds for any t ≥ m. • The family S1 satisfies the condition Om . be its Let (E∗ , φ) be a member of YOK(N0 , K0 , y, L, δ), and let (q, E∗ , φ, W∗ , [φ]) 0 , K0 , y, L, δ). Assume (E∗ , φ) is not δ-semistable, and let (E∗ , φ ) lift in YOK(N

84

3 Parabolic L-Bradlow Pairs

be the last member of the Harder-Narasimhan filtration of (E∗ , φ) with respect to the δ-semistability. Then, we obtain the following inequality from (3.10) and the second condition: h0 E∗ (m) + (E , φ) · δ(m) + K0 δ,α∗ δ ≤ P(E Py (m) ≤ (m) + K0 ∗ ,φ ) rank(E ) This contradicts with the first condition. Thus we are done.

3.5 1-Stability Condition and 2-Stability Condition 3.5.1 Parabolic Sheaf Let y = (y, y1 , . . . , yl ) ∈ T ype, and let α∗ = (α1 , . . . , αl ) be a system of weights. Definition 3.5.1 • We say that the 1-stability condition holds for (y, α∗ ), if we have Ms (y, α∗ ) = Mss (y, α∗ ). • We say that the 2-stability condition holds for (y, α∗ ), if the automorphism group of E∗ ∈ Mss (y, α∗ ) is Gm or G2m . Lemma 3.5.2 The 2-stability condition for (y, α∗ ) is equivalent to the following condition: • Let E∗ be a parabolic sheaf of type y with weight α∗ , which is polystable but not stable. Then, we have the unique decomposition E∗ = E1 ∗ ⊕ E2 ∗ , where Ei ∗ are stable parabolic sheaves with weight α∗ such that E1 ∗ E2 ∗ . Proof Assume that the 2-stability condition holds. Let E∗ be a polystable parabolic sheaf of type y with weight α∗ . We have a decomposition E∗ = E1 ∗ ⊕ E2 ∗ . If one of Ei ∗ is not stable, then by taking graduation of a Jordan-H¨older filtration, we obtain a polystable parabolic sheaf E∗ = E1 ∗ ⊕ E2 ∗ ⊕ E3 ∗ of type y with weight α∗ . However, the automorphism of E∗ is G3m , which contradicts with the 2-stability condition. Assume that the condition above holds. Let E∗ be a semistable parabolic sheaf of f : E∗ −→ E∗ be an endomorphism. The eigenvalues of type y with weight α∗ . Let N f are constant. Let E∗ = i=1 Ei,∗ be the generalized eigen decomposition of f . N We have the decomposition f = i=1 fi . If N ≥ 3, the length of a Jordan-H¨older filtration is longer than 3, and hence we have a polystable object which has more than three stable components. Hence N ≤ 2. In the case N = 2, it can be shown that Ei ∗ are stable by the same argument. Hence the automorphism group is G2m . Let us consider the case N = 1. If E∗ is stable, the automorphism group is Gm . In the case E∗ is not stable, the length of the Harder-Narasimhan filtration

3.5 1-Stability Condition and 2-Stability Condition

85

is 2. Moreover, the graded components are not mutually isomorphic. Hence the automorphism group is Gm . In the above argument, the following corollary is proved. Corollary 3.5.3 Assume that the 2-stability condition holds for (y, α∗ ). Let E∗ be a semistable parabolic sheaf of type y with weight α∗ . Then, one of the following holds: • E∗ is stable. • E∗ is uniquely decomposed into E1 ∗ ⊕ E2 ∗ , where Ei ∗ are stable such that E1 ∗ E2 ∗ . • We have the non-split exact sequence 0 −→ E1 ∗ −→ E∗ −→ E2 ∗ −→ 0, where Ei ∗ are stable such that E1 ∗ E2 ∗ .

3.5.2 Parabolic L-Bradlow Pair Let L be a line bundle on X, and let δ be an element of P br . Definition 3.5.4 • We say that the 1-stability condition holds for (y, α∗ , L, δ), if we have Ms (y, [L], α∗ , δ) = Mss (y, [L], α∗ , δ). • We say that the 2-stability condition holds for (y, α∗ , [L], δ), if the automorphism group of (E∗ , [φ]) ∈ Mss (y, [L], α∗ , δ) is Gm or G2m . Definition 3.5.5 Fix a type y ∈ T ype, a system of weights α∗ and a line bundle L. A parameter δ ∈ P br is called critical for (y, α∗ , L), if the 1-stability condition does not hold for (y, α∗ , L, δ). The set of such critical parameters is denoted by Cr(y, α∗ , L). Lemma 3.5.6 The set Cr(y, α∗ , L) is finite. Proof We may assume rank(y) > 1 and μ(y, α∗ ) ≥ 0. Recall Proposition 3.4.5. Let d := dim X. We can take a sufficiently negative number C such that there does not exist δ-semistable L-Bradlow pairs for any δ with (d − 1)!δtop ≥ −C. Let S1 denote the family of L-Bradlow pairs (E∗ , φ ) with the following property: • There exists a member (E∗ , φ) of SS(y, L, α∗ ) such that (E∗ , φ ) is a saturated subobject of (E∗ , φ). • deg(E∗ ) ≥ C. Since S1 is bounded (Proposition 3.4.1), we obtain the finiteness of the set T1 of polynomials H such that HE∗ = H for some (E∗ , φ ) ∈ S1 .

3 Parabolic L-Bradlow Pairs

86

Let δ be an element of Cr(y, L, α∗ ). Then, there exists a δ-semistable (E∗ , φ) such that φ = 0, which has a non-trivial partial Jordan-H¨older filtration (E∗ , φ ) ⊂ (E∗ , φ). We have the following equality: deg(E∗ ) + (d − 1)!δtop deg(E∗ ) + (E , φ ) · (d − 1)!δtop =

rank E rank E Because (d − 1)!δtop ≤ −C, we obtain the following inequality: deg(E∗ ) ≥ rank(E ) · μ(E∗ ) − (d − 1)!δtop ≥ C δ δ Hence, (E∗ , φ ) is a member of S1 . Because P(E = P(E ,φ) , there exist H ∈ T1 , ∗ ∗ ,φ ) r1 ∈ {1, . . . , r − 1} and ∈ {0, 1} satisfying the following: ' ( 1 H HE∗ − − , ·δ = r r1 r r1

Then, finiteness of Cr(y, L, α∗ ) follows from finiteness of T1 .

Corollary 3.5.7 If δ = δ is sufficiently close to some element δ ∈ P br , the 1-stability condition holds for (y, L, α∗ , δ ). If δ is sufficiently close to 0, the 1-stability condition for (y, L, α∗ , δ ). Lemma 3.5.8 Let δ0 be an element of P br . If δ1 ∈ P br is sufficiently close to δ0 , any δ1 -semistable L-Bradlow pair is also δ0 -semistable. Proof Let S, S1 and T1 be as in the proof of Lemma 3.5.6. Let r1 ∈ {1, . .. , r − 1} and ∈ {0, 1}. We put P (H, r1 , , δ) := r1−1 H + · δ − r−1 Hyα∗ + δ for any δ ∈ P br and H ∈ T1 . If δ1 is sufficiently close to δ0 , the following holds: (A) P (H, r1 , , δ1 )(t) > 0 implies P (H, r1 , , δ0 )(t) ≥ 0 for any sufficiently large t. Let δ1 be as above. We may assume that the 1-stability condition holds for (y, L, α∗ , δ1 ). Let (E∗ , φ) ∈ Ms (y, L, α∗ , δ1 ), which is not δ0 -semistable. Then, there exists a saturated subobject (E∗ , φ ) of (E∗ , φ) with the following property: δ0 δ0 • P(E ,φ ) (t) > P(E ,φ) (t) for any sufficiently large t. ∗ ∗

δ1 δ1 • P(E ,φ ) (t) < P(E ,φ) (t) for any sufficiently large t. ∗ ∗

Then, (E∗ , φ ) is a member of S1 thanks to the first inequality. Therefore, the two inequalities contradict with the condition (A) above. Thus we are done. By a similar argument, we can show the following. Lemma 3.5.9 Let y be an element of T ype, and let α∗ be a system of weights. Assume δ is sufficiently small. Then, E∗ is semistable if (E∗ , φ) is a δ-stable L-Bradlow pair.

3.5 1-Stability Condition and 2-Stability Condition

87

Proof Let S be a family of μ-semistable parabolic torsion-free sheaves of type y with weight α∗ . Let S denote the family of parabolic torsion-free sheaves E∗ with the following property: • There exists E∗ ∈ S such that E∗ is a saturated subobject of E∗ . Moreover, μ(E∗ ) = μ(y, α∗ ). Let T denote the set of the polynomials P such that P = Pyα∗ and P = PE∗ for some E∗ ∈ S. Since the families S and S are bounded, the set T is finite. We take a positive number δ1 satisfying the following: 0 < δ1 <

1 min |P − Pyα∗ | P ∈ T 10 · rank(y)

We regard δ1 as a polynomial of degree 0. Let (E∗ , φ) be a δ1 -semistable L-Bradlow pair of type y with weight α∗ . It is easy to observe that E∗ is μ-semistable. Let E∗ be a subobject of E∗ . In the case μ(E∗ ) < μ(E∗ ), we obviously have PE∗ < PE∗ . Assume μ(E∗ ) = μ(E∗ ) and PE∗ = PE∗ . Then, E∗ is a member of S, and PE∗ is a member of T . By δ1 -semistability of (E∗ , φ), we have the following: PE∗ +

(E∗ , φ) · δ1 δ1 ≤ PE∗ +

rank E rank E

It implies PE∗ < PE∗ by our choice of δ1 .

By an argument used in the proof of Lemma 3.5.2, we obtain the following lemma. Lemma 3.5.10 The 2-stability condition for (y, α∗ , L, δ) is equivalent to the following: • Let (E∗ , φ) be a parabolic L-Bradlow pair of type y with weight α∗ such that φ = 0, which is δ-polystable but not δ-stable. Then, we have the unique decomposition (E∗ , φ) = (E1 ∗ , φ1 ) ⊕ E2 ∗ , where (E1 ∗ , φ1 ) is δ-stable and E2 ∗ is stable. Moreover, when the 2-stability condition holds for (y, α∗ , L, δ), one of the following holds for any δ-semistable parabolic L-Bradlow pair (E∗ , φ) of type y with weight α∗ such that φ = 0. • (E∗ , φ) is δ-stable. • (E∗ , φ) has the unique decomposition (E∗ , φ) = (E1 ∗ , φ1 ) ⊕ E2 ∗ , where (E1 ∗ , φ1 ) is δ-stable and E2 ∗ is stable. • We have the non-split exact sequence 0 −→ (E1 ∗ , φ1 ) −→ (E∗ , φ) −→ E2 ∗ −→ 0 or 0 −→ E2 ∗ −→ (E∗ , φ) −→ (E1 ∗ , φ1 ) −→ 0, where (E1 ∗ , φ) is δ-stable and E2 ∗ is stable.

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88

3.5.3 Parabolic L-Bradlow Pair Let L = (L1 , L2 ) be a pair of line bundles on X, and let δ = (δ1 , δ2 ) be elements 2 of P br . Similarly, we have 1-stability and 2-stability conditions. Definition 3.5.11 • We say that the 1-stability condition holds for (y, α∗ , L, δ), if we have Ms (y, [L], α∗ , δ) = Mss (y, [L], α∗ , δ). • We say that the 2-stability condition holds for (y, α∗ , L, δ), if the automorphism group of any (E∗ , [φ]) ∈ Mss (y, [L], α∗ , δ) is Gm or G2m . Definition 3.5.12 • Fix a type y ∈ T ype, a system of weights α∗ and a pair of line bundles L. A parameter δ ∈ (P br )2 is called critical for (y, α∗ , L), if the 1-stability condition does not hold for (y, α∗ , L, δ). The set of such critical parameters is denoted by Cr(y, α∗ , L). • We also put as follows: Cr(y, α∗ , L, δ1 ) := δ2 ∈ P br (δ1 , δ2 ) ∈ Cr(y, α∗ , L) Any element δ2 ∈ Cr(y, α∗ , L, δ1 ) is called critical for (y, α∗ , L, δ1 ).

We can show the following lemma by using Lemma 3.4.8 and an argument used in the proof of Lemma 3.5.6. Lemma 3.5.13 The set Cr(y, L, α∗ , δ1 ) is finite.

By an argument used in the proof of Lemma 3.5.8, we can show the following lemma. Lemma 3.5.14 If δ2 are sufficiently close to δ2 , any (δ1 , δ2 )-semistable L-Bradlow pair is also (δ1 , δ2 )-semistable. If δ2 is sufficiently close to 0, the L1 -Bradlow pair (E∗ , φ1 ) is δ1 -semistable for any (δ1 , δ2 )-semistable L-Bradlow pair (E∗ , φ1 , φ2 ). Lemma 3.5.15 Assume that δ1 + δ2 is sufficiently small as in Lemma 3.5.9. Then, if (E∗ , φ1 , φ2 ) is a (δ1 , δ2 )-semistable L-Bradlow pair, E∗ is semistable. Lemma 3.5.16 If both δi are sufficiently small, the following claims hold: • If the 1-stability condition holds for (y, α∗ ), then the 1-stability condition holds also for (y, α∗ , L, δ). • Even if the 1-stability condition does not hold for (y, α∗ ), the 2-stability condition holds for (y, α∗ , L, δ). • If the 1-stability condition does not hold for δ = (δ1 , δ2 ), the equality δ1 /r1 = δ2 /r2 holds for some decomposition r1 + r2 = rank(y).

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Proof We take δi as in the proof of Lemma 3.5.9. The first claim is clear. Let us show the second claim. Let (E∗ , φ1 , φ2 ) be a (δ1 , δ2 )-polystable L-Bradlow pair of type y with weight α∗ such that φi = 0. We remark that E∗ is semistable. Since δi are sufficiently small, the number of stable components are at most 2. If it is decomposed, it is of the form (E1 ∗ , φ1 , 0) ⊕ (E2 ∗ , 0, φ2 ), where (Ei ∗ , φi ) are δi -stable with φi = 0. The components are not isomorphic. Hence the 2-stability condition holds. The third claim also immediately follows.

3.6 Quot Schemes 3.6.1 Preliminary Let y ∈ T ype◦ . The H 2 (X)-component of y is denoted by det(y). Let y(m) denote y · ch O(m) for any integer m. Let Hy be the Hilbert polynomial associated to y (Subsection 3.2.4). Take a large integer m such that Hy (m) > 0. We also assume that any line bundles M with c1 (M ) = det y(m) satisfy H i (X, M ) = 0 (i > 0). We take an Hy (m)dimensional vector space Vm over k. We denote Vm ⊗ OX by Vm,X .

3.6.2 Quotient Sheaves A pair of U -coherent sheaf E on U × X and a surjection q : p∗U Vm,X −→ E is called a U -quotient sheaf of Vm,X , which is denoted by (q, E) or simply by q. A U -quotient sheaf (q, E) with an orientation ρ is called an oriented U -quotient sheaf of Vm,X . The type of (q, E) or (q, E, ρ) is defined to be the type of E(−m). (See Subsection 3.1.4 for the type.) Recall that the moduli functor of quotient sheaves of Vm,X with type y is representable by a projective scheme ([57]). We denote it by Q(m, y). Let (q u , E u ) denote the universal quotient sheaf of Vm,X on Q(m, y) × X. A point of Q(m, y) is denoted by the corresponding quotient q, E . We have the right action of GL(Vm ) on Q(m, y) given by g · (q, E) := (q ◦ g, E). Let (q, E) be a U -quotient sheaf of Vm,X with type y defined over U × X. We say that q, E satisfies (TFV)-condition, if the following holds for any u ∈ U : 0 X, E is the induced map V −→ H (TFV): E|{u}×X is torsion-free, m |{u}×X i isomorphic, and H X, E|{u}×X vanish for any i > 0. In general, it determines the maximal open subset of U on which (TFV)-condition holds. In particular, we obtain the open subset of Q(m, y) determined by (TFV)condition, which is denoted by Q◦ (m, y). Let Or(E u ) denote the orientation bundle, which is the line bundle on Q(m, y). The moduli functor of oriented quotient sheaves of Vm,X with type y is representable

90

3 Parabolic L-Bradlow Pairs

by Q(m, y ) := Or(E u )∗ . We have the induced right action of GL(Vm ) on Q(m, y ). We put Q◦ (m, y ) := Q◦ (m, y) ×Q(m,y) Q(m, y ).

3.6.3 Quotient Quasi-Parabolic Sheaves and Maruyama-Yokogawa Construction Let D be a Cartier divisor of X. Let y and Vm,X be given as above. A U -quasiparabolic quotient sheaf (q, E, F∗ ) of Vm,X on U × (X, D) is defined to be a U -quotient sheaf (q, E) of Vm,X with a U -quasi-parabolic structure F∗ of E at D. The type of U -quasi-parabolic quotient sheaf (q, E,F∗ ) of Vm,X is defined to be the type of the underlying U -quasi-parabolic sheaf E(−m), F∗ . (See Subsection 3.1.4 for the type.) Let y be an element of T ype, whose H ∗ (X)-component is y. Let q, E, F∗ be a U -quotient quasi-parabolic sheaf of Vm,X with type y on U × (X, D). We say that q, E, F∗ satisfies (TFV)-condition, if (q, E) satisfies (TFV)-condition and if the following holds for any u ∈ U and for any i: (TFV): The restrictions Fi (E)|{u}×X and Coki (E)|{u}×X are generated by global sections, and the higher cohomology groups of Fi (E)|{u}×X and Coki (E)|{u}×X vanish. In general, it determines the maximal open subset of U on which (TFV)-condition holds. We set Hi := Hy,i . Let Qm,i denote a scheme representing the moduli functor of the quotient sheaves of Vm,X whose Hilbert polynomials are Hi . We have the natural right GL(Vm )-action on Q m,i . We have the open subset Um,i of Qm,i given by the conditions (i) H j X, Ei = 0 for any j > 0, (ii) Vm −→ H 0 X, Ei is surjective. Let Gm,i denote the Grassmannian variety, which is a moduli of Hi (m)-dimensional quotient space of the vector space Vm . We have the GL(Vm )equivariant morphism of Um,i to Gm,i by the correspondence: qi , Ei −→ H 0 (qi ) : Vm → H 0 X, Ei (m) Let Qtf (m, y) denote the open subset of Q(m, y) consisting of the points corresponding to the torsion-free quotients. For their construction of a moduli of semistable parabolic sheaves, Maruyama and Yokogawa constructed a subscheme .l Γ of Qtf (m, y) × i Um,i . (See section 3 of [87].) The scheme Γ is a moduli of quotient quasi-parabolic sheaves (q, E, F∗ ) of Vm,X with type y satisfying the following conditions, (i) E is torsion-free, (ii) the higher cohomology groups of Coki (E) vanish, (iii) Vm −→ H 0 X, Coki (E) are surjective for any i. Moreover, (TFV)-condition determines the open subset of Γ , which is denoted by Q◦ (m, y). The scheme Q◦ (m, y) represents the moduli functor of quotient quasiparabolic sheaves of Vm,X with type y satisfying (TFV)-conditions. We have the universal objects on Q◦ (m, y) × X, which is denoted by q u , E u , F∗u . We have the right GL(Vm )-action on Q◦ (m, y) given by g · (q, E, F∗ ) := (q ◦ g, E, F∗ ).

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91

Let PicX c denote the component of the Picard variety of X such that any c satisfy c e line bundle M ∈ Pic (M ) = c. Let Poin 1 X c denote the Poincar´ X bundle on PicX c × X. Then, we obtain the following locally free sheaf: rank 0y

Zm := pX ∗ Hom Vm,X , PoinX det y(m) .

(3.12)

The projectivization Zm is called the Gieseker space. We have the natural right action of GL(Vm ) on Zm . & r q gives the GL(Vm )It is known that the correspondence (q, E) −→ H 0 is known to be equivariant morphism of Q◦ (m, y) to Zm , which . .an immersion. × i Gm,i . Taking Therefore, we obtain a morphism Q◦ (m, y) × i Um,i −→ Zm. composition with the inclusion Q◦ (m, y) −→ Q◦ (m, y).× i Um,i , we obtain a GL(Vm )-equivariant morphism Q◦ (m, y) −→ Zm × i Gm,i . The following lemma was shown in [87]. . Lemma 3.6.1 ([87], Proposition 3.2) The morphism Q◦ (m, y) −→ Zm × i Gm,i is an immersion. Proof We give only some remarks. Since the morphism Q◦ (m, y) −→ Zm is an immersion, we have only to consider the morphism of Q◦ (m, y) to Q◦ (m, y) × . Gm,i . Recall the precise result of Maruyama and Yokogawa: Let α∗ be a system of weights. Let Γ ss denote the open subscheme of Γ such that the corresponding parabolic sheaves (E, F∗ , α∗ ) are semistable. We may assume that (TFV)-condition holds for each member Γ ss . . They showed that the morphism of Γ ss to Q◦ (m, y) × i Gm,i is an immersion. Their argument can be summarized as follows: . . (i) Construct a subscheme Z of.Q(m, y) × i Qm,i × i Gm,i such that the projection of Z to Q(m, y) × i Gm,i is immersive. (Z is denoted by Δ01 ×Q Δ02 ×Q · · · ×Q Δ0l in [87].) . −→ i Gm,i gives the graph G, which is a subset of (ii) The morphism Γ ss . . can be shown that G is contained in Z. Then, Q(m, y) × i Qm,i × i Gm,i . It . the projection of G to . Q(m, y) × i Gm,i is immersive. Hence the morphism Γ ss −→ Qtf (m, y) × i Gm,i is an immersion. In their argument, we only need the fact that each member of Γ ss satisfies (TFV)condition. (It is stated as the . conditions (3.0.1) and (3.0.2) in [87].) Hence the mor phism Q◦ (m, y) −→ Zm × i Gm,i is immersive. See [87] for more details.

3.6.4 Quotient L-Bradlow Pair Let L be a line bundle on X. Fix a non-trivial morphism ι : OX (−m) −→ L. Let q : p∗U Vm,X −→ E be a U -quotient sheaf with type y defined over U × X satisfying

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92

(TFV)-condition. We have the morphism E(−m) ⊗ L−1 −→ E induced by ι. Let φ be an L-section of E(−m). Then, ι and φ induce an OX -section ι(φ) of E. Definition 3.6.2 (Quotient L-section) Let (q, E) be a quotient sheaf of Vm,X de∗ fined over U × X. An OX -section φ of pU Vm,X is called a quotient L-section of q, E with respect to ι, if there exists an L-section of E(−m) such that ι(φ) = q ◦φ. that the element q ◦ φ ∈ Γ U × X, E is contained in The condition means Γ U × X, E(−m) ⊗ L−1 , where the inclusion E(−m) ⊗ L−1 −→ E is given by ι. Definition 3.6.3 A quotient U -quasi-parabolic L-Bradlow pair of Vm,X with type y on U ×(X, D) is defined to be a pair q, E, F∗ , φ of a quotient U -quasi-parabolic sheaf (q, E, F∗ ) of Vm,X with type y over U × (X, D) and a quotient L-section φ with respect to ι. Let us construct a moduli scheme Q◦ (m, y, L) of quotient quasi-parabolic L-Bradlow pairs of Vm,X with type y satisfying (TFV)-condition, whose L-section ◦ ◦ ∗ m, y, O m, y (−m) := Q is non-trivial everywhere. We put Q X × Vm . Let ◦ ◦ π denote the projection of Q m, y, O(−m) × X onto Q m, y × X. On Q◦ m, y, O(−m) ×X, we have the quotient quasi-parabolic sheaf π ∗ (q u , E u , F∗u ) u of Vm,X with type y. We also have the OX -section φ of p∗Q◦ (m,y,O(−m)) Vm,X , which is induced by the identity of Vm . ◦ Let Λ denote the composite of the following morphisms on Q m, y, O(−m) × X, where the last quotient sheaf is induced by ι: u

φ

π∗ qu

OQ◦ (m,y,O(−m)) −→ p∗Q◦ (m,y,O(−m)) Vm,X −→ π ∗ E u −→

π∗ E u . π ∗ E u (−m) ⊗ L−1

Recall the following result (Lemma 4.3 of [133]) due to Yokogawa. Lemma 3.6.4 Let f : X −→ S be a proper morphism of Noetherian schemes and φ : I −→ F be an OX -morphism of coherent sheaves. Assume that F is flat over S. Then, there exists a unique closed subscheme Z of S such that for all morphism g : T −→ S, g ∗ φ = 0 if and only if g factors through Z. Remark the following easy lemma. Lemma 3.6.5 Let E be a U -coherent sheaf on X ×U such that E|X×{u} is torsionfree for any u ∈ U . Let D be a Cartier divisor of X. Then, E ⊗ OD is flat over U . Proof Let f be any local section of OX , and let F be any OU -coherent sheaf. We have only to show the injectivity of Gf of E⊗OU F induced by f . the endomorphism By considering the support of pX Ker(Gf ) , it is reduced to the case U = Spec(K) and F = OU for some field K. In the case, the claim is trivial. According to the above two lemmas, the vanishing condition of Λ gives the closed subscheme of Q◦ m, y, O(−m) , which is Q◦ (m, y, L). By the construction, it is easy to see that Q◦ (m, y, L) has the desired property. We denote the

3.6 Quot Schemes

93

u universal pair on Q◦ (m, y, L) × X by q u , E u , F∗u , φ . We remark that we have u the unique L-section φu of E u (−m)such that ι(φu ) = q∗u (φ ). ◦ )-action on Q m, y, O(−m) is given by g · (q, E, F∗ , φ) = m The right GL(V −1 q ◦ g, E,F∗ , g ◦ φ . It can be naturally lifted to an action on the universal object. Note Q◦ m, y, L is a closed GL(Vm )-stable subscheme of Q◦ m, y, O(−m) .

3.6.5 Quotient Reduced L-Bradlow Pair Let ι : OX −→ L(m) be the fixed non-trivial morphism. Let q : p∗U Vm,X −→ E be a U -quotient sheaf of Vm,X defined over U × X. A reduced OX -section [φ] of p∗U Vm,X and q induce a reduced OX -section q∗ ([φ]) of E. A reduced L-section [φ] of E(−m) and ι induce a reduced OX -section ι([φ]) of E. Definition 3.6.6 • A reduced OX -section [φ] of p∗U Vm,X is called a quotient reduced L-section if there exists a reduced L-section of E(−m) such that ι([φ]) = q∗ ([φ]). • A quotient U -quasi-parabolicreduced L-Bradlow pair of Vm,X with type y on U ×X is defined to be a tuple q, E, F∗ , [φ] of quotient U -quasi-parabolic sheaf (q, E, F∗ ) of Vm,X with type y and a reduced L-section [φ] of E(−m). We also assume that [φ] is non-trivial everywhere. Let us construct a scheme Q◦ (m, y, [L]) representing the moduli functor of quotient quasi-parabolic reduced L-Bradlow pair of Vm,X with type y satisfying (TFV)◦ condition. We have the free Gm -action on the scheme Q (m, y, L) defined by t·(q, E, F∗ , φ) = q, E, F∗ , t·φ . Then, we set Q◦ (m, y, [L]) := Q◦ (m, y, L)/Gm . It is a closed subscheme of Q◦ (m, y) × Pm . The naturally defined morphism π : Q◦ (m, y, [L]) × X −→ Q◦ (m, y) × X induces the quotient quasi-parabolic u sheaf q u , E u , F ∗u := (π ∗ q u , π ∗ E u , π ∗ F∗u ). The morphism φ naturally induces u a reduced OX -section [φ ] : p∗Q◦ (m,y,[L]) OX ⊗ OPm (−1) −→ p∗Q◦ (m,y,[L]) Vm,X . We also have the induced reduced L-section [φu ] : p∗Q◦ (m,y,[L]) L ⊗ OPm (−1) −→

E(−m). p∗ ◦ Q (m,y,[L])

Lemma 3.6.7 q u , E u , F ∗u , [φu ] is the universal object, i.e., Q◦ (m, y, [L]) has the desired universal property. Proof We give only an outline. By Lemma 3.6.4, we may assume (i) L = O(−m), (ii) ι is the identity. Let (q, E, F∗ , [φ]) denote a U -quotient quasi-parabolic reduced L-Bradlow pair, satisfying (TFV)-condition. We have the map F : U −→ Q◦ (m, y) corresponding to (q, E, F∗ ). We have the locally free sheaf pX ∗ E and pX ∗ E u on U and Q◦ (m, y) respectively. Let P1 and P2 denote the projectivization of them. We have Ψ ∗ P2 P1 , and we have the natural morphism Ψ : P1 −→ P2 . We remark that P2 is naturally

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isomorphic to Q◦ m, y, [O(−m)] = Q◦ m, y × Pm . The pull back of the natuu rally defined reduced L-section [φ ] on P2 × X is the same as the naturally defined reduced L-section on P1 × X. The reduced L-morphism [φ] induces a section f : U −→ P1 . It is easy to see that [φ] is the pull back of the naturally defined reduced L-section on P1 × X by fX . Thus we are done. Since the above Gm -action and the GL(Vm )-action on Q◦ (m, y, L) are commutative, we obtain the induced right GL(Vm )-action on Q◦ (m, y, [L]) and a universal object. We also obtain a GL(Vm )-equivariant immersion of Q◦ (m, y, [L]) to . Zm × i Gm,i × P(Vm∨ ).

3.6.6 Quotient Reduced L-Bradlow Pair Let L = (L1 , L2 ) be a pair of line bundles on X. Quotient quasi-parabolic reduced L-Bradlow pairs of Vm,X with type y can be given as in Definition 3.6.6. It is easy to construct a scheme Q◦ (m, y, [L]) representing the moduli functor of quotient quasi-parabolic reduced L-Bradlow pairs of Vm,X with type y satisfying (TFV)conditions. In fact, we have only to take the fiber product of Q◦ (m, y, [Li ]) (i = (m, y, [L]) 1, 2) over Q◦ (m, y). We have the natural right GL(Vm )-action on Q◦. and the GL(Vm )-equivariant immersion Q◦ (m, y, [L]) −→ Zm × i Gm,i × Pm × Pm .

3.6.7 Oriented Objects

, [L]) := Q◦ (m, y, [L]) ×Q◦ (m,y) Q◦ (m, y ), which represents the We put Q◦ (m, y moduli functor of quotient oriented quasi-parabolic reduced L-Bradlow pairs of Vm,X with type y satisfying (TFV)-condition. We naturally have the universal ob ), Q◦ (m, y

, L), Q◦ (m, y

, [L]), etc., are given, ject. Similarly, the schemes Q◦ (m, y which represent appropriate functors respectively. Let Z m be as in (3.12). We have the following naturally induced Cartesian diagram:

m Q◦ (m, y ) −−−−→ Z ⏐ ⏐ ⏐ ⏐ ! ! Q◦ (m, y) −−−−→ Zm The morphisms are GL(Vm )-equivariant. Hence, we obtain a GL(Vm )-equivariant morphism:

) −→ Z m × Gm,i Q◦ (m, y i

3.6 Quot Schemes

95

3.6.8 Quotient Stacks We have the universal quotient quasi-parabolic sheaf q u , E u , F∗ of p∗Q◦ (m,y) Vm,X ◦ ◦ over u Qu (m,y) × X. The GL(Vm )-action uon Q (m, y) is naturally lifted to that on q , E , F∗ . By taking the descent of E (−m), F ∗ , we obtain a quasi-parabolic sheaf (E u , F∗ ) on Q◦ (m, y)/ GL(Vm ) × (X, D). The following lemma is well known. Lemma 3.6.8 Let y ∈ T ype. The quotient stack Q◦ (m, y)/ GL(Vm ) is isomorphic to M(m, y), and the quasi-parabolic sheaf (E u , F∗ ) gives a universal object. Proof Let g : T −→ M(m, y) be a morphism. We have the corresponding T -quasi-parabolic sheaf (E, F∗ ) on T × X of type y, satisfying the condition Om . We obtain the vector bundle V := pX ∗ E(m) on T . Let P denote the associated principal GL(Vm )-bundle, and let π : P −→ T denote the projection. On P , we have the equivariant trivialization π ∗ V Vm ⊗ OP . Let πX : P × X −→ T × X ∗ E(m) induced by π. We have the equivariant morphism q : p∗P Vm,X −→ πX ∗ on P × X. We obtain the quotient quasi-parabolic sheaf q, πX E(m), F∗ on P × X which is naturally GL(Vm )-equivariant. It also satisfies (TFV)-condition. Therefore, we obtain the GL(Vm )-equivariant morphism P −→ Q◦ (m, y), in other words, the morphism T −→ Q◦ (m, y)/ GL(Vm ). In particular, we obtain M(m, y) −→ Q◦ (m, y)/ GL(Vm ). Conversely, let g : T −→ Q◦ (m, y)/ GL(Vm ). We have the corresponding GL(Vm )-torsor P (g) over T with the GL(Vm )-equivariant morphism g : P (g) −→ ∗ ∗ u (q u ) : p∗P (g) Vm,X −→ gX E Q◦ (m, y). On P (g) × X, we have the quotient gX ∗ with the quasi-parabolic structure gX F , which is GL(Vm )-equivariant. By taking with respect to the action, we obtain the T -quasi-parabolic sheaf u the descent E (−m), F∗ on T × X. It satisfies the condition Om . Therefore, we obtain the morphism of T to M(m, y). In particular, we obtain Q◦ (m, y)/ GL(Vm ) −→ M(m, y). It is easy to check that they are mutually inverse.

), By the same argument, we obtain the description of moduli stacks M(m, y

, [L]), M(m, y

, [L]), etc., as the quotient M(m, y, L), M(m, y, [L]), M(m, y

), Q◦ (m, y, L), Q◦ (m, y, [L]), Q◦ (m, y

, [L]), Q◦ (m, y

, [L]), stacks of Q◦ (m, y etc. Universal objects are also constructed by the same procedure.

Chapter 4

Geometric Invariant Theory and Enhanced Master Space

We recall how to construct moduli stacks by using the geometric invariant theory. Then, we construct enhanced master spaces in our situation, and we describe the fixed point set with respect to a natural torus action. In Section 4.1, we review a basic result on the geometric invariant theory for our construction of moduli stacks of δ-semistable parabolic L-Bradlow pairs. In Section 4.2, we consider a perturbation of a δ-semistability condition. Namely, we multiply a full flag bundle to quot schemes, and we argue what is obtained for small perturbation of semistability conditions. The results in Sections 4.3–4.4 are one of the cores of this paper, which are useful in showing our transition formula. In Section 4.3, we construct our enhanced master space, and we show that it is Deligne-Mumford and proper. In Section 4.4, we study the fixed point set with respect to a natural torus action. In Section 4.5, we construct an enhanced master space in the oriented case, and we give a description of the stack theoretic fixed point set with respect to the natural torus action. They are essentially just a reformulation of the results in the previous sections. We give a more convenient description of the fixed point set in Section 4.6, i.e., we observe that they are isomorphic to the product of moduli stacks of objects with lower ranks, up to e´ tale proper morphisms. In some simpler cases, we do not have to consider enhanced master spaces. The statements for such cases are given in Section 4.7. In this chapter, let X be a smooth connected d-dimensional projective variety over an algebraically closed field k of characteristic 0, and let OX (1) be a very ample line bundle. Put g := X c1 (OX (1))d . Let D denote a Cartier divisor of X. We do not have to assume smoothness of D in this chapter. We sometimes use the symbol k to denote some numbers. We hope that there are no confusion.

T. Mochizuki, Donaldson Type Invariants for Algebraic Surfaces: Transition of Moduli Stacks, Lecture Notes in Mathematics 1972, DOI: 10.1007/978-3-540-93913-9 4, c Springer-Verlag Berlin Heidelberg 2009

97

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4 Geometric Invariant Theory and Enhanced Master Space

4.1 Semistability Condition and Mumford-Hilbert Criterion 4.1.1 Statement We use the notation in Section 3.6. Let y = (y, y1 , y2 , . . . , yl ) ∈ T ype. (See Subsection 3.1.4.) We have the associated polynomials Hy , Hy,1 , Hy,2 , . . . , Hy,l . (See Subsection 3.2.4.) In the following, m denotes a sufficiently large integer. Let Vm be an Hy (m)-dimensional vector space over k, and let Pm denote the projectivization. Let Vm,X := Vm ⊗ OX . Let Zm denote the Gieseker space over Pic det(y(m)) , and let Gm,i denote the Grassmann variety of Hy,i (m)dimensional quotients of Vm . We set Am (y) := Zm × Gm,i , Am (y, [L]) := Am (y) × Pm , i (2) Am (y, [L]) := Am (y) × P(1) m × Pm .

Here, L (resp. L = (L1 , L2 )) denotes a line bundle (a pair of line bundles) on (i) X, and Pm := Pm . Recall that we have obtained the SL(Vm )-equivariant immersions Ψm of Q◦ (m, y) to Am (y). (See Subsection 3.6.3). By fixing an in◦ clusion ι : O(−m) −→◦ L, we have a closed immersion Q (m, y, [L]) −→ ◦ Q m, y, [O(−m)] = Q (m, y) × Pm . Therefore, we obtain an equivariant immersion of Q◦ (m, y, [L]) to Am (y, [L]). Similarly, we obtain an equivariant immersion of Q◦ (m, y, [L]) to Am (y, [L]) by fixing inclusions ιi : O(−m) −→ Li . The immersions are denoted by Ψm . Since Zm is a projective space bundle on PicX det(y(m)) , we have the relative tautological bundle OZm (1). We also have the canonical polarizations OGm,i (1) and OPm (1) of Gm,i and Pm respectively. They are SL(Vm )-equivariant line bundles. For a positive number A and a tuple of positive numbers B∗ = (Bi | i = 1, . . . , l), we formally consider the line bundles on Am (y) given as follows, although it is not precisely a line bundle when the numbers are not integers: Ly (A, B∗ ) := OZm (A) ⊗

l

OGm,i (Bi ).

i=1

Similarly, for positive numbers C and Cj (j = 1, 2), we formally consider Ly,L (A, B∗ , C) := Ly (A, B∗ ) ⊗ OPm (C)

on Am (y, L),

Ly,L (A, B∗ , C1 , C2 ) := Ly (A, B∗ ) ⊗ OP(1) (C1 ) ⊗ OP(2) (C2 ) m

m

on Am (y, L).

When the numbers are positive rational numbers, Ly (A, B∗ ) gives a SL(Vm )equivariant Q-polarization of Am (y). Even if the numbers are not rational, the Hilbert-Mumford criterion formally provides the semistability condition with respect to Ly (A, B∗ ). So, let Ass m (y, A, B∗ ) denote the set of semistable points of

4.1 Semistability Condition and Mumford-Hilbert Criterion

99

the SL(Vm )-action on Am (y) with respect to Ly (A, B∗ ). Similarly, we obtain the ss schemes Ass m (y, L, A, B∗ , C) and Am (y, L, A, B∗ , C1 , C2 ). The semistability (resp. stability) condition with respect to a system of weights α∗ determines the open subset Qss (m, y, α∗ ) (resp. Qs (m, y, α∗ )) of Q◦ (m, y). Namely, it is the maximal open subset of Q◦ (m, y) which consists of the points (q, E, F∗ ) such that the parabolic sheaf (E(−m), F∗ , α∗ ) is semistable. Similarly, we obtain the open subset Qss (m, y, [L], α∗ , δ) (resp. Qs (m, y, [L], α∗ , δ)) of Q◦ (m, y, [L]) determined by the semistability (stability) condition with respect to a system of weights α∗ and a parameter δ ∈ P br . We also obtain the open subsets Qss (m, y, [L], α∗ , δ) and Qs (m, y, [L], α∗ , δ) of Q◦ (m, y, [L]) in similar ways. The first claim of the following proposition was proved by Maruyama-Yokogawa [87] and the second claim was proved by Yokogawa in [134]. Proposition 4.1.1 There exists an integer N (y, α∗ ) such that the following holds for any m ≥ N (y, α∗ ): 1. The image of Qss (m, y, α∗ ) via Ψm is contained in Ass m, y, Pyα∗ (m), ∗ . Thus, we obtain a morphism Ψ m : Qss (m, y, α∗ ) −→ Ass m, y, Pyα∗ (m), ∗ . 2. The morphism Ψ m above is proper. In particular, it is a closed immersion. By the same argument, we can show the following proposition. Proposition 4.1.2 There exists an integer N1 (y, L, α∗ , δ) such that the following claims hold for any m ≥ N1 (y, L, α∗ , δ): α ,δ ∗ (m), ∗ , δ(m) . Hence, we obtain a 1. Ψm (Qss (m, y, [L], α∗ , δ)) ⊂ Ass m Py morphism: α∗ ,δ (m), ∗ , δ(m) Ψ m : Qss (m, y, [L], α∗ , δ) −→ Ass m Py 2. The morphism Ψ m above is proper. In particular, it is a closed immersion. Similarly, there exists a large integer N1 (y, L, α∗ , δ) such that there exists a SL(Vm )-equivariant closed immersion for each m ≥ N1 (y, L, α∗ , δ): α∗ ,δ Ψ m : Qss (m, y, [L], α∗ , δ) −→ Ass (m), ∗ , δ(m) . m y, [L], Py Here, we put δ(m) := (δ1 (m), δ2 (m)). Although we need only a minor modification, we will later give a rather detailed proof of the claims for (y, L, α∗ , δ) in Proposition 4.1.2, for the convenience of the reader. We closely follow the arguments of [87] and [134]. We also use the argument in [62]. Since the claim for (y, L, α∗ , δ) can be shown similarly, we omit to give its proof. We also obtain the following proposition. Proposition 4.1.3 Ψm Qs (m, y, [L], α∗ , δ) ⊂ Asm (y, [L], α∗ , δ). Similar claims hold for Qs (m, y, α∗ ) and Qs (m, y, [L], α∗ , δ).

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4 Geometric Invariant Theory and Enhanced Master Space

The claim for Qs (m, y) was proved by Maruyama-Yokogawa in [87]. Since the argument is similar to that used in the proof of the first claim of Proposition 4.1.2, we give just some remarks in the proof of Proposition 4.1.2. Remark 4.1.4 If αi and the coefficients of δ and δi (i = 1, 2) are rational, it is standard to obtain projective coarse moduli schemes of δ-semistable parabolic L-Bradlow pairs or δ-semistable parabolic L-Bradlow pairs from Proposition 4.1.2. Even if the numbers are not rational, we can obtain coarse moduli schemes of δ-stable parabolic reduced L-Bradlow pairs, if the 1-stability condition holds for (y, L, α∗ , δ). Take a sufficiently large N > 0, and take rational numbers A, B∗ = (Bi | i = 1, . . . , l) and C which are close to N · Pyδ,α∗ (m), N · ∗ and N · δ(m) respectively. If a point of A(m, y, [L]) is stable with respect to Ly,L Pyδ,α∗ (m), ∗ , δ(m) , then it is stable with respect to Ly,L (A, B∗ , C). Thus, we can obtain a coarse scheme by using the geometric invariant theory (Proposition 2.2.1). Therefore, if the 1-stability condition holds, we obtain a projective coarse moduli scheme of semistable ones. Before going into a proof of Proposition 4.1.2, we give some consequences about the property of the moduli stacks. Corollary 4.1.5 If the 1-stability condition holds for (y, α∗ ), the moduli stack y , α∗ ) is Deligne-Mumford and proper. Ms (

Proof Under the assumption, we obtain Qss (y, α∗ ) = Qs (y, α∗ ). It is easy to show finiteness of the stabilizer of any point z ∈ Qs (y, α∗ ) with respect to the SL(Vm )-action from Corollary 3.3.5. Then, the quotient stack Qs (y, α∗ )/ SL(Vm ) is Deligne-Mumford and proper, according to Proposition 4.1.1, Proposition 4.1.3 and Proposition 2.2.2. Note we have the e´ tale proper morphisms: Qs (y, α∗ )/ SL(Vm ) −→ Qs (y, α∗ )/ PGL(Vm ) M(

y , α∗ ) −→ Qs (y, α∗ )/ PGL(Vm ) Hence, it is easy to deduce the claimed property of M(

y , α∗ ).

Similarly, we obtain the following. Proposition 4.1.6 • If the 1-stability condition holds for (y, L, α∗ , δ), then Ms (

y , [L], α∗ , δ) and Ms (y, L, α∗ , δ) are Deligne-Mumford and proper. y , [L], α∗ , δ) is • If the 1-stability condition holds for (y, L, α∗ , δ), then Ms (

Deligne-Mumford and proper. Remark 4.1.7 We can show Corollary 4.1.5 and Proposition 4.1.6 directly. For example, since the automorphism groups of any stable oriented parabolic sheaves are y , α∗ ) is Deligne-Mumford. The properness can be shown by Langton’s finite, Ms (

trick [78]. However, since we would like to use master spaces, we argue the GITconstruction.

4.1 Semistability Condition and Mumford-Hilbert Criterion

101

4.1.2 Mumford-Hilbert Criterion be a Yokogawa Let us start the proof of Proposition 4.1.2. Let (q, E∗ , φ, W∗ , [φ]) , we obtain the tuple of quotients datum of type (y, m) (Definition 3.4.9). From W ∗ . Vm /Wi | i = 1, . . . , l , which gives the point of i Gm,i . It is denoted by Vm /W∗ . Then, we obtain the following element of Am (y, [L]): r 0 0 Ψ q, E∗ , φ, W∗ , [φ] := H q , Vm /W∗ , [φ] ∈ Am (y, [L]). (4.1) is given as in Definition Let W be a subspace of Vm . The number (W, [φ]) 3.4.10. Let EW denote the subsheaf of E(m) generated by the image of W via q. is contained in Ass P α∗ ,δ (m), ∗ , δ(m) , if Lemma 4.1.8 Ψ q, E∗ , φ, W∗ , [φ] m y and only if the following inequalities hold for any non-trivial subspace W ⊂ Vm : Pyα∗ ,δ (m)·rank(EW )−(W, [φ])·δ(m)−

i ·dim(Wi ∩W )−α1 ·dim(W ) ≥ 0. (4.2)

is a semistable point, if and only if In other words, Ψ (q, E∗ , φ, W∗ , [φ]) ∈ YOK(m, 0, y, L, δ). (q, E∗ , φ, W∗ , [φ]) is contained in As P α∗ ,δ (m), ∗ , δ(m) , if and The point Ψ q, E∗ , φ, W∗ , [φ] m y only if the strict inequalities hold in (4.2) for any subspace W ⊂ Vm . 1l Proof We give only an indication. We set L1 := OZ Pyα∗ (m) ⊗ i=1 OGm i (i ) −1 α ,δ and L2 := OZ r · δ(m) ⊗ OPm (δ(m)). We have L Py ∗ (m), ∗ , δ(m) = L1 ⊗ L2 . We put N := dim Vm = Hy (m). Let u1 , . . . , uN be a basis of Vm satisfying the following: wi = 0. (4.3) λ(t) · ui = twi · ui , w1 ≤ w2 ≤ · · · ≤ wN , We remark that we can regard (4.3) as the condition for λ, when we fix a basis u1 , . . . , uN . The number μλ (P, L1 ) is calculated in [87] (see also [7]): μλ (P, L1 ) =

−Pyα∗ (m)

N

rank E (i) − rank E (i−1) wi

i=1

−

l j=1

j

N

rank Wj ∩ V (i−1) − rank Wj ∩ V (i) + 1 · wi .

i=1

(4.4)

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4 Geometric Invariant Theory and Enhanced Master Space

Here, E (i) denote the subsheaves of E(m) generated by u1 , . . . , ui via q, and V (i) denote the subspaces of Vm generated by u1 , . . . , ui . The number μλ (P, L2 ) equals δ(m) ×

1 rank E (i) − rank E (i−1) · wi − r i=1 N

+

N

∩V dimφ

(i)

∩V − dimφ

(i−1)

· wi

(4.5)

i=1

denotes the subspace of Vm generated by φ. The number μλ (P, L) can Here, φ be obtained as the sum of (4.4) and (4.5). In particular, it is linear with respect to (w1 , . . . , wN ). Therefore, μλ (P, L) ≥ 0 holds for any λ satisfying (4.3), if and only if μfk (P, L) ≥ 0 for any k = 1, . . . , N − 1, where fk are given as follows: k

) *+ , fk = (k − N, . . . , k − N , k . . . , k). According to the calculation of Maruyama and Yokogawa, we have the following: μfk (P, L1 ) = Hy (m) ·

Pyα∗ (m) · rank E (k) −

l

i · dim Wi ∩ V (k) − α1 · dim V (k) .

i=1

(4.6) By a direct calculation, we have the following: ' ( (k) rank E (k) − V , [φ] . μfk (P, L2 ) = Hy (m) · δ(m) · r

(4.7)

Hence, μfk (P, L) depends only on V (k) . Therefore, we obtain the function F(P,L) on the sets of the non-trivial subspaces {0 = W V }, and P is semistable with respect to L, if and only if F(P,L) (W ) ≥ 0 for any subspace W . Since F(P,L) (W ) is the left hand side of (4.2) multiplied with Hy (m) > 0, we are done.

4.1.3 A Lemma Following Huybrechts-Lehn [62], we show the following lemma. Lemma 4.1.9 Let y be an element of T ype. There exists an integer N4 with the following property: • Let (E∗ , φ) be a δ-semistable parabolic L-Bradlow pair of type y such that φ = 0. Then, the following inequality holds for any integer m ≥ N4 and any subsheaf F ⊂ E:

4.1 Semistability Condition and Mumford-Hilbert Criterion

h0 F∗ (m) + (F, φ ) · δ(m) δ ≤ P(E (m). ∗ ,φ) rank(F )

103

(4.8)

Here, φ denotes the L-section of F induced by φ. δ δ • If the equality holds in (4.8), (F∗ , φ ) is δ-semistable with P(F = P(E ,φ) . ∗ ,φ ) ∗ • If (E∗ , φ) is δ-stable, the strict inequality holds in (4.8). Proof We have the inequality μ(F ) ≤ μδ (F∗ ) ≤ μδ (E∗ ) for any subsheaf F ⊂ E, and hence μmax (F ) ≤ μ(E∗ ) + (d − 1)!δtop =: C0 . We also have the inequality μmin (F ) ≤ μ(F ) by definition. Hence, we obtain the following inequality by using Proposition 3.4.3 (see [62]): 1 h0 (F (m)) ≤ d−1 rank(F ) g d!

'

1−

" #d 1 C0 + mg + c + rank(F ) ( " #d 1 μ(F ) + mg + c + . + rank(F )

We can take a sufficiently negative number C with the following property: • For any positive integer r < rank(E) and for any sufficiently large t > 0, the following inequalities hold: ' ( 1 δ(t) 1 1 d d 1 − (C0 + tg + c) + (C + tg + c) + ≤ Pyα∗ (t). d−1 g d! r r r (4.9) Note that the coefficient of td in both sides are the same by construction, and thus we can take such C. Let N5 be a large number such that C + mg + c > 0 for any m > N5 . Let S be the family of the sheaves F with the following property: • There exists a δ-semistable parabolic reduced L-Bradlow pair (E∗ , φ) of type y with weight α∗ such that F is a saturated subsheaf of E. We divide S into two families by the conditions (i) μ(F ) < C, or (ii) μ(F ) ≥ C. If F is contained in the family (i), then the desired inequality holds for any m ≥ N5 because of our choices of C and N5 . Since the family (ii) is bounded by Proposition 3.4.1, we can take a large number N6 such that the family (ii) satisfies the condition Om for any m ≥ N6 . Then, the desired inequalities hold for any m ≥ N6 , because of δ-semistability of (E∗ , φ). We have only to put N4 := max{N5 , N6 }.

4.1.4 Proof of the Claim 1 in Proposition 4.1.2 Let N1 (y, L, α∗ , δ) be an integer larger than N4 in Lemma 4.1.9 and N0 (y, L, δ) in be a point of Qss (m, y, [L], α∗ , δ). The Proposition 3.4.11. Let q, E(m), F∗ , [φ]

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4 Geometric Invariant Theory and Enhanced Master Space

0 reduced L-section [φ] ofE is induced by [φ] and ι. By definition, H (q) gives the 0 isomorphism Vm H X, E(m) . Let W be a subspace of Vm , which generates the subsheaf EW of E(m) via q. By Lemma 4.1.9, we have the following inequality: h0 (EW,∗ ) + EW (−m), φ · δ(m) ≤ Pyδ,α∗ (m). (4.10) rank(EW )

We also have ≤ (EW (−m), φ), dim(W ) ≤ h0 (EW ), dim(W ∩Wi ) ≤ h0 (Fi (EW )). (W, [φ]) Thus, we obtain the following inequality: α1 dim(W ) +

i dim(Wi ∩ W ) ≤ α1 · h0 (EW ) + i · h0 (Fi (EW )) = h0 EW ∗ . (4.11)

is By substituting (4.11) (4.2). Hence, Ψm (q, E(m), F∗ , [φ]) α ,δto (4.10), we obtain ss ∗ contained in Am Py (m), ∗ , δ(m) . Remark 4.1.10 We obtain Proposition 4.1.3 by using the same argument and the second claims in Lemma 4.1.8 and Lemma 4.1.9.

4.1.5 Proof of the Claim 2 of Proposition 4.1.2 Take a discrete valuation ring R over k. We denote the quotient field by K. Assume that we have the following diagram: f

Spec(K) −−−−→ ⏐ ⏐ !

Qss (m, y, [L], α∗ , δ) ⏐ ⏐

! Φ α ,δ g ∗ Spec(R) −−−−→ Ass (m), ∗ , δ(m) m Py

(4.12)

We have only to show the existence of a lift Spec(R) −→ Qss (m, y, [L], α∗ , δ). Let XK and XR denote X × Spec K and X × Spec R, respectively. The morphism f gives the tuple (qK , EK∗ (m), [φK ]) of a quotient parabolic sheaf qK , EK∗ (m) and a quotient reduced L-section [φK ] defined over XK . The tuple (qK , EK∗ (m)) satisfies (TFV)-condition. An L-section [φK ] of EK is induced by [φK ] and ι, and the parabolic L-Bradlow pairs (EK ∗ , φK ) with weight α∗ is δ-semistable. As in [134] p. 502–503, EK∗ (m) can be extended to a parabolic torsion-free sheaf ER∗ (m) over XR . The morphism qK can be extended to the morphism qR : Vm ⊗ OXR −→ ER (m) such that the restriction of qR to the closed fiber

4.1 Semistability Condition and Mumford-Hilbert Criterion

105

is generically surjective. Since Pm is proper, we can extend [φK ] to [φR ] over R. 0 We put WK,i :=.H X ⊗ K, Fi+1 (E(m)) (i = 1, . . . , l) which give subspaces of Vm ⊗ K. Since i Gm,i are proper, we obtain the subbundles WR,i of Vm ⊗ R over Spec R. We put WR,∗ = (WR,i | i = 1, . . . , l). The family qR , ER ∗ , WR ∗ , [φR ] induces a morphism Spec(R) −→ Am (y, [L]) as in (4.1). Since Am (y, [L]) is separated, it coincides with g in the diagram (4.12). Let q0 , E0∗ , W0∗ , [φ0 ] denote the specialization of qR , ER∗ , WR∗ , [φR ] to the φ0 of E0 . The tuple closed point of Spec R. We also have the induced L-section q0 , E0∗ , φ0 , W0∗ , [φ0 ] is a Yokogawa datum such that Ψ q , E 0 0 ∗ , φ0 , W0 ∗ , [φ0 ] ss α∗ ,δ is contained in Am Py (m), ∗ , δ(m) . By Lemma 4.1.8,

q0 , E0∗ , φ0 , W0∗ , [φ0 ] ∈ YOK(m, 0, y, L, δ).

Recall m ≥ N1 (y, L, α∗ , δ) ≥ N0 (y, L, δ), where N0 (y, L, δ) is as in Proposition 3.4.11. Hence, we know (i) q0 is surjective, (ii) E0,∗ (m) satisfies (T F V )-condition, (iii) the parabolic L-Bradlow pair (E0,∗ , φ0 ) is δ-semistable. , E , W , [φ ] gives a map fR : Spec(R) −→ Qss m, y, [L], Therefore, q R R ∗ R∗ R α∗ , δ , whose restriction to Spec(K) is f . It is clear that fR gives the lift of g. Thus the claim 2 of Proposition 4.1.2 is proved.

4.1.6 Complement We give a consequence of the proof. We put V := Vm , Q := Qss (m, y, [L], α∗ , δ) and L := Ly,L (Pyδ,α∗ (m), ∗ , δ(m)). Lemma 4.1.11 Let z = (q, E, F∗ , [φ]) be a point of Q. Let V = V ⊕ V

be a decomposition, and let λ be the one-parameter subgroup of SL(V ) given by

t− rank V · idV ⊕trank V · idV Let E (m) denote the subsheaf generated by V . We have the induced parabolic structure and the L-Bradlow pair φ of E . Then, (E∗ , φ ) is δ-semistable with δ α∗ ,δ , if and only if μλ (z, L) = 0 holds. P(E = Py ∗ ,φ ) Proof Assume μλ (z, L) = 0. We put W := H 0 X, E (m) . We remark V ⊂ W E (m) = EW . and hence E (m) := EV ⊂ EW ⊂E (m). Therefore, 0 0 Let Wi denote the kernel of H X, E(m) −→ H X, Coki+1 (m) . From the calculation in the proof of Lemma 4.1.8, we have the following: 0 = μλ (z, L) = Hy (m) × Pyα∗ ,δ (m) · rank E − (E , φ ) · δ(m) −

i · dim(Wi ∩ W ) − α1 · dim(W ) . (4.13)

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4 Geometric Invariant Theory and Enhanced Master Space

Therefore, we obtain the following inequality: Pyα∗ ,δ (m) =

α1 · dim(W ) +

≤

i · dim(Wi ∩ W ) + (E , φ ) · δ(m) rank E

h0 (E∗ (m)) + (E , φ) · δ(m) ≤ Pyα∗ ,δ (m) rank E

(4.14)

Here, the first inequality is obtained in (4.11), and the second inequality follows from Lemma 4.1.9. We can conclude that the equality holds in (4.14). Then, (E∗ , φ ) is δ-semistable according to the second claim of Lemma 4.1.9.

Assume (E∗ , φ ) is δ-semistable. condition

Om holds for (E∗ , φ ), Note that the

because it holds for (E∗ , φ ) ⊕ (E/E )∗ , φ , where φ denotes the induced L-section on the quotient E/E . Hence, we have h0 (E∗ ) = α1 · dim V + i · dim Wi ∩ V . Then, μλ (z, L) = 0 follows from the calculation of μλ (z, L) in the proof of Lemma 4.1.8. See (4.6) and (4.7). We remark λ = fk and V = V (k) in this case. Corollary 4.1.12 Let z = (q, E, F∗ , [φ]) ∈ Q. Let λ be a one parameter subgroup of SL(V ). Let V = Vi be the weight decomposition of λ, i.e., λ preserves the decomposition, and the weight on Vi is i. Let E (i) be the subsheaf of E(m) generated structure by Vj (j ≤ i) via q. We have the induced L-section φi and the parabolic δ of E (i) (−m). Then, all E (i) (−m)∗ , φi are δ-semistable with P(E (i) (−m) ,φ ) = ∗ i δ , if and only if μ (z, L) = 0 holds. P(E λ ,φ) ∗

Proof We put Ui = j≤i Vj and Ui := j>i Vi . Let λi be the one-parameter − rank Ui · id ⊕ trank Ui · idUi . It is easy to see that subgroup of SL(V ) given by t Ui . ai λ can be expressed as λi with ai ∈ Q>0 . The condition μλ (z, L) = 0 implies μλi (z, L) = 0. Therefore, the claim immediately follows from Lemma 4.1.11.

4.2 Perturbation of Semistability Condition 4.2.1 Preliminary We continue to use the notation in Subsection 4.1.1. We put V := Vm , Q := Qss (m, y, [L], α∗ , δ) and L := Ly,L (Pyδ,α∗ (m), ∗ , δ(m)). For simplicity, we assume that αi and the coefficients of δ are rational. Take a sufficiently large number k such that L⊗ k is a line bundle on A := Am (y, [L]). For a rational number γ, we put Lγ := L⊗ k ⊗ OPm (γ). Let Ass (Lγ ) (resp. As (Lγ )) denote the set of the semistable (resp. stable) points of A with respect to Lγ .

4.2 Perturbation of Semistability Condition

107

We put N := dim Vm = Hy (m). Let Flag(V, N ) denote the full flag variety: F∗ = 0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ FN = V dim Fi /Fi−1 = 1 . (4.15) Let Gl (V ) denote the Grassmann variety of l-dimensional subspaces of V . We have the natural morphism ρl : Flag(V, N ) −→ Gl (V ). Let OGl (V ) (1) denote the canonical polarization of Gl (V ). For a tuple of positive rational numbers n∗ = (n1 , n2 , . . . , nN ), we put as follows: OFlag (n∗ ) :=

N

ρ∗i OGi (V ) (ni ).

i=1

:= Q × Flag(V, N ) and A := A × Flag(V, N ). We have the induced We put Q −→ A. For a tuple n∗ and a rational number γ, let us consider the map Ψm : Q following Q-line bundle: n∗ ) := Lγ ⊗ OFlag (n∗ ) = L⊗ k ⊗ OP (γ) ⊗ OFlag (n∗ ) L(γ, m n∗ ). We Let Ass (γ, n∗ ) denote the set of the semistable points with respect to L(γ, −1 ss will be interested in the open subset Ψm A (γ, n∗ ) .

4.2.2 δ+ -Semistability and δ− -Semistability Let δ+ and δ− denote elements of P br such that (i) δ− < δ < δ+ , (ii) δ+ and δ− are sufficiently close to δ. The following lemma is clear from Lemma 3.5.8. Lemma 4.2.1 When δ+ (resp. δ− ) is sufficiently close to δ, a parabolic L-Bradlow pair (E∗ , φ) is δ+ -semistable (δ− -semistable) if and only if the following condition holds: • Let us take any partial Jordan-H¨older filtration of (E∗ , φ) with respect to δ-semistability: (p) (1) (1) (2) (2) ⊂ E∗ , φ ⊂ · · · ⊂ E∗ , φ(p) = E∗ , φ . E∗ , φ Then, we have φ(i) = 0 for i < p (resp. φ(1) = 0). Moreover, any δ+ -semistable (resp. δ− -semistable) L-Bradlow pair is also δ+ stable (resp. δ− -stable). ss For κ = ±, we set Qκ := Qss m, y, [L], α∗ , δκ . They are independent of choices of δκ when δκ are sufficiently close to δ according to the previous lemma. We denote the signature of γ by sign(γ). The absolute value of γ is denoted by |γ|.

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4 Geometric Invariant Theory and Enhanced Master Space

Proposition 4.2.2 Assume that |γ| is sufficiently small, and that ni are sufficiently −1 ss A (γ, n∗ ) = Qss smaller than |γ|. Then, Ψm sign(γ) × Flag(V, N ). In particular, ss × Flag(V, N ) −→ Ass (γ, n∗ ). Moreover, we obtain the closed immersion Q sign(γ)

the image is contained in As (γ, n∗ ).

Proof Let us begin with the following lemma. Lemma 4.2.3 Assume that the absolute value of γ = 0 is sufficiently small. ss −1 A (Lγ ) = Qss • Ψm sign(γ) . ss • The induced morphism Ψm : Qss sign(γ) −→ A (Lγ ) is a closed immersion. s Moreover, the image is contained in A (Lγ ). Proof Let us show the first claim. Let z := (q, E∗ , φ) ∈ Q. As a preparation, we consider the following two cases: (A) There exists a partial Jordan-H¨older filtration E∗ ⊂ (E∗ , φ) with respect to δ-semistability. (B) There exists a partial Jordan-H¨older filtration (E∗

, φ) ⊂ (E∗ , φ) with respect to δ-semistability. In the case (A), we put V := H 0 X, E (m) , and take a complement V

of V

in V . Let λ(A) be the one parameter subgroup given by

t− rank V idV ⊕ trank V idV . Because μλ(A) (z, L) = 0, we have μλ(A) z, Lγ = γ · μλ z, OPm (1) = γ · rank V . In the case (B), we put V

:= H 0 X, E

(m) , and take a complement V of V

in V . Let us consider the one parameter subgroup λ(B) given by

t− rank V idV ⊕ trank V idV Then, we have

μλ(B) (z, Lγ ) = −γ · rank V

−1 From the above consideration, we easily obtain Ψm (Ass (Lγ )) ⊂ Qss sign(γ) . Let us show the reverse implication. We use the argument in the proof of Lemma 4.1.8. Let u1 , . . . , uN be any base of V . Let (w1 , . . . , wN ) ∈ ZN such that

wi ≤ wi+1 ,

N

wi = 0.

i=1

Let λ be the one parameter subgroup of SL(V ) given by λ(t) · ui = twi · ui . We have μλ (z, Lγ ) = k · μλ (z, L) + γ · μλ z, OPm (1) . As seen in the proof of Lemma 4.1.8, μλ (z, L) is linear with respect to (w1 , . . . , wN ). It is also well

4.2 Perturbation of Semistability Condition

109

known that μλ z, OPm (1) is linear with respect to (w1, . . . , wN ). (See Lemma 2.2.6, for example.) Therefore, we have only to show μfh z, Lγ > 0 for any h = 1, . . . , N − 1. Let us consider the case μfh z, L > 0. We have k ·μfh z, L ≥ 1 by our choice of k. The absolute value of μfh z,OPm (1) is dominated by dim V . Therefore, if γ is sufficiently small, we have μfh z, Lγ > 0. Let us consider the case μfh (z, L) = 0. Let W denote the subspace of V generated by u1 , . . . , uh . Let EW denote the subsheaf of E(m) generated by W and q. We put φ := φ if the image of φ is contained in EW , and φ := 0 otherwise. Then, (E (−m), φ ) is δ-semistable (Lemma 4.1.11). By the considerations (A) and (B), if (E∗ , φ) is δsign(γ) -semistable, we obtain μfh (z, Lγ ) > 0. Thus, the first claim is proved. In the above argument, we have shown μλ (Ψm (z), Lγ ) > 0 for any point z of ss s ss Qss sign(γ) , and thus Ψm (Qsign(γ) ) ⊂ A (Lγ ). Since Ψm : Q −→ A (L0 ) is proper ss ss (Proposition 4.1.2), the properness of Qsign(γ) −→ A (Lγ ) follows from the first claim. Thus, the proof of Lemma 4.2.3 is finished. Let us return to the proof of Proposition 4.2.2. Let z ∈ Q × Flag(V, N ). Let u1 , . . . , uN be a base of V . Let (w 1 , . . . , wN ) and λ be as in the proof z, OFlag (n∗ ) is linear with respect to of Lemma 4.2.3. By Lemma 2.2.7, μ λ z, L(γ, n μ ) is also linear. Therefore, we have only to show w1 , . . . wN . Hence, λ ∗ μfh z, L(γ, n∗ ) > 0 for any h = 1, . . . , N − 1. We have the decomposition: nh · μfh z, ρ∗j OGj (V ) (1) . μfh z, L(γ, n∗ ) = μfh z, Lγ + j

If we take a large integer k such that L⊗k is a line bundle, then we have γ k ⊗ k μfh z, L⊗ ≥ 1, because μ z, L is a positive integer. On the other hand, fh γ ∗γ 2 μfh z, ρi OGi (V ) (1) is dominated by 2(dim if ni are sufficiently V ) . Therefore, small, the contribution of OFlag (n∗ ) to μfh z, L(γ, n∗ ) is sufficiently small. Thus we are done.

4.2.3 (δ, )-Semistability ss (δ, ) denote the maximal subset of Q, which Let be a positive integer. Let Q consists of the points q, E∗ , [φ], F such that (E∗ , [φ], F) is (δ, )-semistable. (See Subsection 3.3.3 for (δ, )-semistability.) Proposition 4.2.4 There exist negative rational number γ and a tuple of positive rational numbers n∗ , for which the following holds: −1 ss ss (δ, ) = Ψm A (γ, n∗ ) . • Q

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4 Geometric Invariant Theory and Enhanced Master Space

ss (δ, ) −→ Ass (γ, n∗ ) is a closed immersion. The • The induced morphism Q s image is contained in A (γ, n∗ ). Before going into a proof of Proposition 4.2.4, we give a consequence. ss y, [L], α∗ , (δ, ) of (δ, )-semistable objects Corollary 4.2.5 A moduli stack M is Deligne-Mumford and proper. Proof From Lemma 3.3.9, Proposition 4.2.4 and Proposition 2.2.2, the quotient ss (δ, )/ SL(Vm ) is Deligne-Mumford and proper. Since we have the e´ tale stack Q proper morphisms ss (δ, )/ PGL(Vm ) and ss (δ, )/ SL(Vm ) −→ Q Q ss y ss (δ, )/ PGL(Vm ),

, [L], α∗ , (δ, ) −→ Q M y

, [L], α∗ , (δ, ) . we easily obtain the desired properties for M Let us start the proof of Proposition 4.2.4. When is larger than N , the (δ, )semistability is equivalent to the δ− -semistability (Remark 3.3.8). Hence, the claim follows from Proposition 4.2.2. Therefore, we will assume < N in the following argument. We take γ and n∗ satisfying the following condition: Condition 4.2.6 Let K0 (y, L, δ) be the number as in Proposition 3.4.11. Take a small rational number satisfying the following: 0<<

K0 (y, L, δ) 100 · N 100

For simplicity, we assume −1 is a prime integer. Take an irrational number a > 0 satisfying the following: ( ' 1 · a < < · a.

−

Take prime numbers p1 , . . . , p such that p1 > 100·N 100 and pi > 100·N 100 ·pi−1 . We also assume the following: i < min · a − , − ( − −1 ) · a pi Take integers q1 , . . . , q such that the following holds: qi − a < 1 , (i = 1, . . . , ). pi i pi We put γ := − and ni := qi /pi (i = 1, . . . , ). We remark n1 > n2 > · · · > n

and the following inequalities:

4.2 Perturbation of Semistability Condition

γ+

i · ni > 0,

111

γ+

i=1

i · ni + ni0 (i0 − 1) < 0

1≤i≤

i=i0

We also take prime numbers pi (i = + 1, . . . , N ) satisfying pi > 100N 100 pi−1 (i = + 1, . . . , N ) and the following: ⎧ ⎫ ⎪ ⎪ ⎪ N

⎨ ⎪ ⎬ i 100 100N < min i · ni − , i · ni + ni0 (i0 − 1) − ⎪ ⎪ pi ⎪ ⎪ i= +1 1≤i≤

⎩ i=1 ⎭ i=i0

We put ni := p−1 i (i = + 1, . . . , N ).

We remark the following elementary fact. Let qi = 0 be an Lemma 4.2.7 Let p1 , . . . , pN be mutually distinct prime numbers. N integer which is coprime to pi , for each i. Then, the sum i=1 qi /pi cannot be an integer. N Proof If i=1 qi /pi = a is an integer, we have the relation: ⎛ ⎞ q1 ·

j>1

N N ⎟ ⎜ pj + p1 · ⎜ a · p + qj · pi ⎟ i ⎝ ⎠ = 0. i=2

j=2

i≥j, i=j

This contradicts with the assumption that q1 and p1 are coprime.

We give Let us start the proof of Proposition 4.2.4. Let z = (q, E∗ , φ, F) ∈ Q. preliminary considerations. (A) Let us consider the case that there exists a partial Jordan-H¨ older filtration E∗ ⊂ (E∗ , φ) with respect to δ-semistability. We put V := H 0 X, E (m) , and we take a complement V

of V in V . Let λ denote the one parameter subgroup of SL(V ) given by t− rank V idV ⊕ trank V idV . Then, we have the following equality: n∗ ) = k · μλ (z, L) + γ · μλ z, OP (1) + μλ z, OFlag (n∗ ) μλ z, L(γ, ni − dim Fi ∩ V rank V

+dim Fi Fi ∩ V rank V

= γ rank V +

rank V − ni dim Fi /Fi ∩ V ni dim Fi ∩ V rank V

= γ+ i=1

+

N i= +1

i=1

ni − dim(Fi ∩ V ) rank V

+ dim Fi /Fi ∩ V rank V .

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4 Geometric Invariant Theory and Enhanced Master Space

By our choice of and n∗ , the right hand side is larger than 0, if and only if F ∩V1 = {0}. Moreover, if it is larger than 0, it is strictly larger than 0 according to Lemma 4.2.7. (B) Let us consider the case that there exists a partial Jordan-H¨older filtration (E∗

, φ) ⊂ (E∗ , φ) with respect to δ-semistability. We put V

:= H 0 X, E

(m) and take a complement V of V

in V . Consider the one parameter subgroup λ given by t− rank V idV ⊕trank V idV . Then, we have the following equality: n∗ ) = μλ z, L(γ, − γ rank V +

ni − dim Fi ∩ V

rank V +dim Fi /Fi ∩ V

rank V

i=1

=

N

−γ −

rank V + ni dim Fi ∩ V ni dim Fi /Fi ∩ V

rank V

i=1

+

N

i=1

ni − dim(Fi ∩ V

) rank V + dim Fi /Fi ∩ V

rank V

.

i= +1

By our choice of γ and n∗ , it is strictly smaller than 0, if and only if F ⊂ V

. Namely, it is larger than 0 if and only if F ⊂ V

. Moreover, if it is larger than 0, it is strictly larger than 4.2.7. 0 according to Lemma ss (δ, ) from the above preliminary considWe obtain Ψ−1 Ass (δ, n∗ ) ⊂ Q eration. Let us show the reverse implication. We use the standard argument as in ss (δ, ). Let the proof of Lemma 4.1.8. Let z = (q, E∗ , φ, F) be a point of Q u1 , . . . , uN be abase of V , and let (w1 , . . . , wN ) be an element of ZN such that subgroup of SL(V ) given by wi ≤ wi+1 and wi = 0. Letλ be the one-parameter n ) is linear with respect to (w1 , . . . , wN ). λ(t) · ui = twi · ui . Then, μλ z, L(γ, ∗ n ) > 0 for any h. Hence, we have only to show μfh z, L(γ, ∗ ⊗k ≥ 1. First, let us consider the case μfh z, L > 0. We have μfh z, L −1 −100 z, O Since |γ| and n are smaller than 100 · (dim V ) , we have μ (γ) ⊗ i fh P −1 we obtain μfh z, L(γ, n∗ ) > 0. Next, let us conOFlag (n∗ ) < 10 . Hence, sider the case μfh z, L = 0. Let E denote the subsheaf of E(m) generated by

u1 , . . . , uh . We put image of φ is contained in E (−m), and φ := 0 φ := φ if the

Jordan-H¨older filtration as otherwise. Then, E (−m)∗ , φ ⊂ (E ∗ , φ) is a partial in (A) or (B). Therefore, we have μfh z, L(γ, n∗ ) > 0 in this case, too. Hence, the first claim of Proposition 4.2.4 is obtained. Since we have shown that μfh z, L(γ, n∗ ) > 0 for any h, we obtain that the ss (δ, )) is contained in As (γ, n∗ ). Let us show the properness of Ψ : image Ψ(Q ss Q (δ, ) −→ Ass (γ, n∗ ). We use the argument in Subsection 4.1.5. Assume that we have the following diagram:

4.3 Enhanced Master Space

113 f ss (δ, ) Spec K −−−−→ Q ⏐ ⏐ ⏐ ⏐ ! ! g Spec R −−−−→ Ass (γ, n∗ )

Let XK and XR denote X × Spec K and X × Spec R respectively. We denote the projection A −→ A by π. Let qK , EK ∗ , [φK ], FK,∗ be the objects on XK corresponding to f . As in Subsection 4.1.5, we obtain the objects (qR , ER ∗ , φR , WR ∗ , [φR ]) on XR . It induces the morphism f1 : Spec R −→ A, which is the same as π ◦ g. We also obtain the Yokogawa datum (q0 , E0 ∗ , φ0 , W0,∗ , [φ0 ]). Since |γ| and ni are sufficiently small, the tuple q0 , E0∗ , φ0 , W0 ∗ , [φ0 ] is contained in the Yokogawa family YOK(N0 , K0 , y, L, δ) for N0 = N0 (y, L, δ) and

gives a point of Q=Qss (m, y, [L], α∗ , δ), K0 = K0 (y, L, δ). Hence, q0 , E0∗ , [φ] according to Proposition 3.4.11. It implies the image of f1 = π ◦ g is contained in Ass (L0 ). Then, the desired properness immediately follows from the first claim.

4.3 Enhanced Master Space 4.3.1 Construction We use the notation in Section 4.2. Take a rational number γ2 < 0, whose absolute value is sufficiently small. We will consider the following two situations: γ1 is a sufficiently small positive rational number, and ni are sufficiently smaller than γ1 . (II) γ1 and n∗ are as in Condition 4.2.6.

(I)

In both cases, |γ1 | is assumed to be sufficiently smaller than |γ2 |. We also assume the following: i · ni + |γ2 | ≤ K0 (y, L, δ). (4.16) i

Here K0 (y, L, δ) denotes the constant in Proposition 3.4.11. Let k be a number such that k · (γ1 − γ2 ) = 1. We consider the following Q-line bundles on Q: 1 , n∗ )⊗ k , L1 := L(γ

2 , n∗ )⊗ k . L2 := L(γ

Then, we have L2 = L1 ⊗ OPm (−1). Let π1 : A −→ Pm denote the projection. We set B := P π1∗ OPm (0) ⊕ π1∗ OPm (1)

114

4 Geometric Invariant Theory and Enhanced Master Space

We put O (1) := OP (1)⊗ L1 , where OP (1) denotes the tautological bundle over A. B ∗ Let Bss (resp. Bs ) denote the set of the of P π1 OPm (0) ⊕ π1∗ OPm (1) over Q. semistable (resp. stable) points with respect to OB (1). We set B1 := P π1∗ OPm (0) and B2 := P π1∗ OPm (1) . We naturally regard Bi The following lemma is clear by construction. as closed subschemes of B. Lemma 4.3.1 The restriction of OB (1) to Bi is Li . The Q-line bundle OB (1) gives a GL(Vm )-equivariant polarization. THi := Bi × Q and TH∗ := TH − TH1 ∪ TH2 . We put TH := B ×A Q, A We remark that TH∗ is isomorphic to Qss (m, y, L, α∗ , δ) × Flag(V, N ). We also The following lemma is obvious from Proposition 4.2.2, put THss := Bss ×A Q. Proposition 4.2.4 and our choice of the constants. Lemma 4.3.2 • In the case (I), THss ×TH TH1 = Qss + × Flag(V, N ). ss • In the case (II), TH ×TH TH1 = Qss (δ, ). • In both cases, THss ×TH TH2 = Qss − × Flag(V, N ). The quotient stack THss / SL(V ) is called the enhanced master space. In the rest of this section, we will show the following proposition. Proposition 4.3.3 The stack THss / SL(V ) is Deligne-Mumford and proper. According to Corollary 2.2.3, the proposition is obtained from the following three lemmas. Lemma 4.3.4 Let us consider the SL(V )-action on THss . The stabilizer of any point z ∈ THss is finite and reduced. As a result, THss / SL(V ) is DeligneMumford. Lemma 4.3.5 If m is sufficiently large, then the morphism THss −→ Bss is proper. Lemma 4.3.6 The image of THss −→ Bss is contained in Bs .

4.3.2 Proof of Lemma 4.3.4 ss The claim is obvious for any 1 ∪ TH2 . So we discuss the point z ∈ TH ∩ TH stabilizer of a point z = q, E∗ , [φ], F∗ , u ∈ THss ∩ TH∗ . Let g ∈ SL(V ) be (i) denote the generalized eigen any element such that g · z = z. Let V = V decomposition of g. Correspondingly, we have the decomposition: l (i) (i) (i) (1) q, E∗ , [φ], F = q (1) , E∗ , [φ], F (1) ⊕ q , E∗ , F . i=2

Lemma 4.3.7 l ≤ 2.

4.3 Enhanced Master Space

115

Proof Assume l > 2, and we will derive a contradiction. Let us consider the one (3) (2) parameter subgroup λ given by t− rank V idV (2) ⊕ trank V idV (3) . It is easy to see that λ fixes the point z. Since we have μλ (z, L) = μλ z, OPm (1) = 0, we have the following equality: N (2) (3) ni · − rank V (3) ·rank Fi +rank V (2) ·rank Fi . (4.17) μλ z, OB (1) = i=1

By our choice of n∗ , the right hand side of (4.17) can be 0, if and only if the following equality holds for any i: (2)

− rank V (3) · rank Fi

(3)

+ rank V (2) · rank Fi (2)

= 0. (2)

However, there exists a number i0 such that rank Fi0 +1 = rank Fi0 + 1 and (3) (3) rank Fi0 +1 = rank Fi0 . Thus μλ z, OB (1) = 0. Let λ−1 denote the one −1 −1 z, O (t) = λ(t) . Then, one of μ (1) or parameter subgroup given by λ λ B μλ−1 z, OB (1) is negative. Hence, z cannot be semistable. Lemma 4.3.8 Let N be a nilpotent endomorphism of (E∗ , [φ], F). Then N = 0. Proof Assume N = 0, and we will derive a contradiction. There exists the positive integer j such that N j = 0 and N j+1 = 0. It is easy to obtain N (φ) = 0. We obtain the subsheaves Im N j and Ker(N j ) of E. Then, we obtain the naturally induced parabolic L-Bradlow pairs (Im N∗j , φ ) ⊂ (Ker N∗j , φ

) ⊂ (E∗ , φ), which gives the partial Jordan-H¨older filtration with respect to δ-semistability, according to Lemma 3.3.4. We take subspaces Ki (i = 1, 2, 3) of V satisfying the following condition: K1 = H 0 X, Im N j , K1 ⊕ K2 = H 0 X, Ker N j , K1 ⊕ K2 ⊕ K3 = V. We remark that N j induces the isomorphism K3 −→ K1 . From the inclusion K3 K1 ⊂ V , we have the induced filtration FK1 . From the isomorphism V (K1 ⊕ K2 ), the filtration FK3 is induced. We remark N FK3 h ⊂ FK1 h . Let −1 us consider the one-parameter subgroup λ given by K1 ⊕ t idK3 , for which t id wehave μλ (z, L) = μλ z, OPm (1) = 0. Note N FK3 h ⊂ FK1 h for any h, and N FK3 h0 FK1 h0 for some h0 . Hence, we obtain μλ z, OFlag (n∗ ) < 0. Thus, the point z cannot be semistable. From Lemmas 4.3.7 and 4.3.8, we obtain the following lemma. such that z = (q, E∗ , [φ], F, u) ∈ Lemma 4.3.9 Let (q, E∗ , [φ], F) be a point of Q THss ∩ TH∗ . Then, either one of the following holds: 1. The automorphism group of (E∗ , [φ], F) is Gm . 2. There exists the unique decomposition

(1) (2) q, E∗ , [φ], F = q (1) , E∗ , [φ(1) ], F (1) ⊕ q (2) , E∗ , F (2) ,

and the automorphism group of (q, E∗ , [φ], F) is G2m .

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4 Geometric Invariant Theory and Enhanced Master Space

Let us finish the proof of Lemma 4.3.4. In the first case in Lemma 4.3.9, the stabilizer of (q, E∗ , [φ], F, u) with respect to the SL(V )-action is trivial. In the second case, we put V (i) := H 0 X, E (i) (m) ⊂ V . Then, the intersection G2m ∩ SL(V ) consists of the elements ρ(t) = ta · idV (1) ⊕tb · idV (2) satisfying a · rank V (1) + b · rank V (2) = 0. By considering the action along the direction of which is given by ρ(t)u = ta · u, we obtain that the stabilizer is the fiber TH /Q, finite.

4.3.3 Proof of Lemma 4.3.5 be a Yokogawa datum. Recall that we obtain the point Let (q, E∗ , φ, W∗ , [φ]) ∈ A := Am (y, [L]). Ψ (q, E∗ , φ, W∗ , [φ]) Lemma 4.3.10 Assume that m is larger than the constant N (y, L, δ) in Proposition , where π 3.4.11. Let z be a point of Bss such that π(z) = Ψ q, E∗ , [φ], W∗ , [φ] denotes the naturally defined projection B −→ A. Then, (E∗ , [φ]) is δ-semistable, q is onto, and the condition Om holds for (E∗ , [φ]). Proof Let W be any subspace of V . Take a complement W of W in V . Consider the one-parameter subgroup of SL(V ) given by t− rank W idW ⊕ trank W idW . We have the following: nj · μλ z, OGl (V ) (1) μλ z, OB (1) = k · k · μλ z, L + k

+ k max γ1 · μλ z, OPm (1) , γ2 · μλ z, OPm (1) . (4.18) The first term in the right hand side is i dim(Wi ∩ W ) k · k · Hy (m) · P α∗ ,δ (m) · rank EW −

− α1 dim(W ) − (W, [φ]) · δ(m) .

The absolute of the second term in the right hand side of (4.18) can be domivalue n nated by k i=1 ni ·i·dim V . The absolute value of the third term can be dominated by k · |γ2 | · dim V . Recall dim V = Hy (m). Since we have assumed that |γi | and nj are sufficiently small as in (4.16), we obtain the following inequality: Pyα∗ ,δ (m) · rank EW −

i · dim(Wi ∩ W ) − α1 dim(W ) − (W, [φ]) · δ(m) + K ≥ 0.

Here K = K0 (y, L, δ) denotes the constant in Proposition 3.4.11. Namely, ∈ YOK(m, K, y, L, δ). (q, E∗ , φ, W∗ , [φ]) Therefore, the claim of the lemma follows from Proposition 3.4.11.

(4.19)

4.3 Enhanced Master Space

117

Now, we use the argument in the last part of the proof of Proposition 4.2.4. Assume that we have a diagram: f

Spec K −−−−→ THss ⏐ ⏐ ⏐ ⏐ ! ! g π Spec R −−−−→ Bss −−−−→ A

We can show the composite π ◦ g is contained in Ass (L0 ), by using Lemma 4.3.10. Then, the desired properness follows from the definition of THss .

4.3.4 Proof of Lemma 4.3.6, Step 1 Let us show that the image of THss is contained in Bs . Let z = (q, E∗ , [φ], F, u) be a point of THss . We have only to consider the case u = 0. a base of V , and let (w1 , . . . , wN ) be an element of ZN such Let u1 , . . . , uN be wi = 0. Let λ be the one-parameter subgroup of SL(V ) that wi ≤ wi+1 and given by λ(t)·ui = twi ·ui . We will not distinguish the elements w = (w1 , . . . , wN ) and λ. We have the following decomposition: μλ z, OB (1) = k · k · μλ z, L + k · μλ z, OFlag (n∗ ) + k max γi · μλ z, OPm (1) . i=1,2

Recall that we have the expression λ =

aj · fj for aj ≥ 0, where fj =

j

*+ , ) j − N, . . . , j − N , j . . . , j . Lemma 4.3.11 If μλ (z, L) = 0, then μfh (z, L) = 0 for any h such that ah = 0. Proof Since (E∗ , [φ]) is δ-semistable, the claim immediately follows. We put S1 := j μfj (z, L) = 0 and S2 := j μfj (z, L) > 0 . Lemma 4.3.12 For any element 0 = ρ = j∈S2 aj · fj with aj ≥ 0, the following holds: k · μρ z, L + μρ z, OFlag (n∗ ) + min γi · μρ z, OPm (1) > 0. i=1,2

Proof We put Fi (ρ) := k · μρ (z, L) + μρ (z, OFlag (n∗ )) + γi · μρ (z, OPm (1)) for i = 1, 2. We have only to show Fi (ρ) > 0. Since Fi are linear with respect to ρ, we have only to show Fi (fj ) > 0 for any j ∈ S2 . We remark k · μfj (z, L) ≥ 1. The number μfj (z, OFlag (n∗ )) is dominated by dim(V ) and ni (i = 1, . . . , N ). The number γi · μfj (z, OPm (1)) is dominated by dim(V ) and γi . Since nj and γi are sufficiently small, the claim is clear.

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4 Geometric Invariant Theory and Enhanced Master Space

Lemma 4.3.13 To show μλ z, O (1) > 0 for any λ, we have only to show the B following inequality for 0 = ρ = j∈S1 aj · fj : k · μρ (z, L) + μρ z, OFlag (n∗ ) + max γi · μρ (z, OPm (1)) > 0. i=1,2

Proof We have the decomposition λ = λ(1) + λ(2) , where λ(l) = Assume (4.20) holds for λ(1) . Then, we obtain the following:

j∈Sl

(4.20) aj · fj .

1 z, O k · μ μ (1) = (l) (z, L) + μρ(l) (z, OFlag (n∗ )) ρ ρ B k

l=1,2

+ max γj · μρ(1) (z, OPm (1)) + γj · μρ(2) (z, OPm (1)) j=1,2

≥ k · μρ(1) (z, L) + μρ(1) (z, OFlag (n∗ )) + max γj · μρ(1) (z, OPm (1)) + k · μρ(2)

j=1,2

z, L + μρ(2) z, OFlag (n∗ ) + min γj · μρ(2) z, OPm (1) > 0. j=1,2

Thus we are done.

4.3.5 Proof of Lemma 4.3.6, Step 2 To show (4.20), we give some preliminary consideration. (A) If there exists a partial Jordan-H¨older filtration E∗ ⊂ (E∗ , φ) with respect to δ-semistability, take a decomposition V = V ⊕ V

such that V = H 0 X, E (m) and V = V ⊕ V

. Consider the one parameter subgroup λ given by t− rank V · idV ⊕ trank V ·idV . Then, we have μλ (z, L) = 0, μλ z, OPm (1) = rank V > 0, and the following equality: μλ (z, OFlag (n∗ )) =

nj · − rank V

· dim Fj ∩ V + rank V · dim

Fj . Fj ∩ V

From the semistability of z, we obtain the following: γ1 · rank V +

nj · − rank V

· dim(Fj ∩ V ) + rank V · dim

Fj ≥ 0. Fj ∩ V

(4.21)

We remark that the strict inequality holds in (4.21). In the case (I), it is obvious. In thecase (II), it follows from our choice of n∗ and Lemma 4.2.7. Hence, we obtain μλ z, OB (1) > 0 for λ as above.

4.3 Enhanced Master Space

119

(B) If there exists a partial Jordan-H¨older filtration (E∗

, φ

) ⊂ (E∗ , φ) with re

spect to δ-semistability such

that φ = 0, let us take a decomposition V = V ⊕ V

0 such that V = H X, E (m) . Consider the one-parameter subgroup λ given by t− rank V idV ⊕ trank V idV . We have μλ (z, L) = 0, μλ (z, OPm (1)) =

− rank V and the following: μλ (z, OFlag (n∗ )) = ' nj · − rank V · rank Fj ∩ V

+ rank V

· rank

Fj Fj ∩ V

( .

From the semistability of z, we obtain the following: − γ2 · dim V

' + nj · − rank V · rank Fj ∩ V

+ rank V

· rank

Fj Fj ∩ V

( ≥0 (4.22)

In both cases (I) and (II), the strict inequality holds in (4.22). Hence, we have μλ z, OB (1) > 0 for λ as above. (C) Let us consider the case that there exists a partial Jordan-H¨older filtration E∗ ⊂ (E∗

, φ) ⊂ (E∗ , φ) with respect to δ-semistability. We take a decomposition V = V ⊕V

⊕V

such that V = H 0 (X, E (m)) and V ⊕V

= H 0 (X, E

(m)). Consider the one-parameter subgroup λ given by t− rank V ·idV ⊕ trank V ·idV . Then, we have μλ (z, L) = μλ (z, OPm (1)) = 0. Hence, we obtain the following inequality from semistability of the point z:

' ' nj − rank V

· rank(Fj ∩ V ) + rank V · rank

Fj Fj ∩ (V ⊕ V

) ≥0

((

(4.23)

According to our choice of n∗ and Lemma 4.2.7, the strict inequality holds in (4.23). Therefore, we have μλ (z, OB (1)) > 0 for λ as above.

4.3.6 Proof of Lemma 4.3.6, Step 3 For ρ = fj ∈S1 aj · fj , let V = Vi be the weight decomposition. We have the number i0 such that φ ∈ Gi0 − Gi0 −1 . We put ri := dim Vi . of U = ρ as an element nWe use the notation in Subsection 2.6.1. We regard

Q · e . Then, we have the expression ρ = a · v such that a ≤ a 2 ≤ i j 1 j i=1

· · · ≤ as and ri · ai = 0.

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4 Geometric Invariant Theory and Enhanced Master Space

Assume a i0 > 0. Then, we have the following expression by Lemma 2.6.2:

ρ=

b(i1 , i2 ) · x(i1 , i2 ) +

i 0 −1

cj · y(j).

j=1

(i1 ,i2 )∈S(i0 )

Here the coefficients b(i1 , i2 ) and cj are non-negative, and one of b(i1 , i2 ) or cj is positive. We remark that we have the following linearity: i 0 −1 cj · μy(j) (z, OPm (1)). max γi · μρ (z, OPm (1)) = γ1 ·

i=1,2

j=1

Hence, the following holds: 0 −1 i μρ z, OB (1) = b(i1 , i2 ) · μx(i1 ,i2 ) z, OB (1) + cj · μy(j) z, OB (1) .

j=1

We have the positivity μx(i1 ,i2 ) z, OB (1) > 0 and μy(j) z,OB (1) > 0 from (C) and (A), respectively. Therefore, we obtain the positivity μρ z, OB (1) > 0. By similar arguments, we can show the desired positivity in the cases a i0 = 0 and a i0 < 0. Thus, the image of THss is contained in Bs . Therefore, we obtain Lemma 4.3.6 and hence Proposition 4.3.3.

4.4 Fixed Point Set of Torus Action on Enhanced Master Space 4.4.1 Preliminary We continue to use the notation in Section 4.3. Let Gm be a one dimensional torus. We have the Gm -action ρ on B = P OPm (0) ⊕ OPm (1) given by ρ(t) · [u1 : u2 ] := [t · u1 : u2 ]. It induces the Gm -action on TH, which is also denoted by ρ. Since it commutes with the SL(V )-action, we obtain the induced action ρ on THss / SL(V ). We would like to argue the fixed point set of the enhanced master space. The stack theoretic fixed point set (see [55]) is given in our case, as follows: We have the SL(V ) × Gm -equivariant closed immersion THss −→ Bs . Therefore, we have an open neighbourhood U of THss / SL(V ) in Bs / SL(V ), which is Gm invariant, Deligne-Mumford and smooth. The embedding THss / SL(V ) −→ U is Gm -equivariant. The fixed point set U Gm of U is defined to be the 0-set of the vector field induced by the Gm -action. Then, the stack theoretic fixed point set (THss / SL(V ))Gm of THss / SL(V ) is defined to be the intersection (THss / SL (V )) ∩ U Gm .

4.4 Fixed Point Set of Torus Action on Enhanced Master Space

121

However, we restrict ourselves to the set theoretic fixed point set in this section. In other words, we will consider only the closed points of the fixed point set, although it is not difficult to look at the stack theoretic fixed point set. We will later argue the stack theoretic fixed point set of the enhanced master space in the oriented case. ss ss Let π be the projection TH −→ TH / SL(V ). We will use the symbol q, E∗ , [φ], F, u to denote a point of TH, where (q, E∗ , [φ], F) denotes a point and u denotes a point of the fiber of TH −→ Q over (q, E∗ , [φ], F). of Q, Lemma 4.4.1 Let z = q, E∗ , [φ], F, u be a point of THss . The point π(z) is contained in the fixed point set, if and only if one of the following holds: 1. z ∈ TH1 ∪ TH2 . 2. We have the unique decomposition: (1)

(2)

(q, E∗ , [φ], F) = (q (1) , E∗ , [φ(1) ], F (1) ) ⊕ (q (2) , E∗ , F (2) ) ss ∩ TH∗ . Assume that the conProof We have only to consider the z ∈ TH case(i) (i) 0 := H X, E (m) , and we consider the one paramdition 2 holds. We put V (2) (1) eter subgroup λ of SL(V ) given by t− rank V · idV (1) ⊕trank V · idV (2) . It fixes q, E∗ , [φ], F , and it acts non-trivially along the direction of the fiber TH −→ Q, − rank V (2) u. Therefore, the action ρ fixes π(z). as λ(t)u = t Conversely, if π(z) is a fixed point with respect to ρ, then we obtain the oneparameter subgroup of SL(V ) which fixes (q, E∗ , [φ], F) by Lemma 4.3.9. Hence, it has the decomposition. The uniqueness follows from Lemma 4.3.4.

Let z = (q, E∗ , [φ], F) be a point of THss ∩ TH∗ such that π(z) is contained in (THss / SL(V ))Gm . We have the decomposition as in Lemma 4.4.1. Then, we (1) (2) obtain the types y 1 = type(E∗ ) and y 2 = type(E∗ ). We also obtain the decomposition I1 I2 = N := {1, . . . , N }: (j) (j) Ij := i ∈ N Fi /Fi−1 = 0 The datum (y 1 , y 2 , I1 , I2 ) is called the decomposition type of z. In general, we prepare the following definition. Definition 4.4.2 A decomposition type for (m, y, α∗ , δ) is defined to be a datum I := (y 1 , y 2 , I1 , I2 ) as follows: • y = y 1 + y 2 in T ype such that Pyα1∗ ,δ = Pyα∗ ,δ . • N = I1 I2 such that |Ii | = Hyi (m). The set of such decomposition types is denoted by Dec(m, y, α∗ , δ).

We remark that the condition Om -holds for Mss (y 1 , L, α∗ , δ) and Mss (y 2 , α∗ ) for a decomposition type (y 1 , y 2 , I1 , I2 ), if Mss (y 1 , L, α∗ , δ)×Mss (y 2 , α∗ ) = ∅.

122

4 Geometric Invariant Theory and Enhanced Master Space

4.4.2 Statements Let I := (y 1 , y 2 , I1 , I2 ) ∈ Dec(m, y, α∗ , δ). We take a decomposition V = V (1) ⊕ V (2) such that dim V (i) = Hyi (m). We put P(i) := P(V (i)∨ ). Then, we put Q(1) := Qss m, y 1 , [L], α∗ , δ and Q(2) := Qss (m, y 2 , α∗ ). For i = 1, 2, we set @ ? (i) filtration indexed by N , (i) (i) . Flag(V , Ii ) := F∗ dim GrF = 1 (j ∈ Ii ), or = 0 (j ∈ Ii ) j It is also denoted by Flag(i) for short. Clearly, Flag(i) are isomorphic to the full flag varieties of V (i) . We put (i) := Q(1) × Flag(i) , Q

split (I) := Q (1) × Q (2) . Q

We set split (I) −→ Q. Then, we have the naturally defined morphism Q split (I), THsplit (I) := TH ×Q Q

split (I) TH∗split (I) := TH∗ ×Q Q

split (I). THi,split (I) := THi ×Q Q We have the naturally defined closed immersion ι : THsplit (I) −→ TH. (1) (2) Let z = (q1 , E∗ , [φ], F (1) ), (q2 , E∗ , F (2) ), u be a point of THsplit (I). Let min(I2 ) denote the minimum of I2 . We will prove the following lemma in Subsection 4.4.5. Lemma 4.4.3 In the case (II) (see Subsection 4.3.1), if ι(z) is contained in THss , then min(I2 ) > holds. (2)

(2)

(2)

(2)

We set Fmin := Fmin(I2 ) . We remark Fmin −1 = 0 and dim Fmin = 1. We also (2) (2) remark that the pair E∗ , Fmin can be regarded as a reduced OX (−m)-Bradlow pair on X. We will prove the following proposition. Proposition 4.4.4 ι(z) is contained in THss , if and only if the following conditions hold: • z ∈ TH∗split (I). (2) (2) • E∗ , Fmin is an -semistable reduced O(−m)-Bradlow pair for any sufficiently small > 0. (1) • (E∗ , [φ], F (1) ) is δ, min(I2 ) − 1 -semistable.

4.4.3 Step 1 Let G1 denote the subgroup of GL(V (1) ) × GL(V (2) ) determined by det(g) = 1, i.e., G1 := {(g1 , g2 ) ∈ GL(V (1) ) × GL(V (2) ) | det(g1 ) · det(g2 ) = 1}.

4.4 Fixed Point Set of Torus Action on Enhanced Master Space

123

Lemma 4.4.5 The following two conditions are equivalent: • μλ ι(z), OB (1) ≥ 0 for any one parameter subgroup λ of SL(V ). • μλ ι(z), OB (1) ≥ 0 for any one parameter subgroup λ of G1 . Proof The first condition clearly implies the second condition. Let us show the reverse implication. Assume the second condition holds. Let λ : Gm −→ SL(V ) be a one-parameter subgroup.We have the decomposition λ = λ(1) + λ(2) such that μλ(1) z, L = 0 and μλ(2) z, L > 0. By the same argument as that employed in the proof of Lemma 4.3.13, we have only to show the following inequalities: k · μλ(1) (z, L) + max γj · μλ(1) (z, OP (1)) + μλ(1) (z, OFlag (n∗ )) > 0 (4.24) j=1,2

k · μλ(2) (z, L) + min γj · μλ(2) (z, OP (1)) + μλ(2) (z, OFlag (n∗ )) > 0 (4.25) j=1,2

Since ni and γj are sufficiently small, we can show that the inequality (4.25) always holds by the same argument as the proof of Lemma 4.3.12. So we may and will assume μλ (z, L) = 0. Let V = i Vi denote the weight decomposition of λ. We put Gj := i≤j Vi . We have the number i0 determined by φ ∈ Gi0 − Gi0 −1 . We have only to show the following: Fj ∩ Gi nj i · dim ≥ 0. max γj · i0 + j=1,2 F j ∩ Gi−1 j i We put H0 := V (1) and H1 := V , which give the filtration of V . We have the natural identification GrH (V ) V . Since F is compatible with the decomposition V = V (1) ⊕ V (2) , the induced filtration by F is the same as F. Let G denote the induced filtration by G on GrH (V ) V : Gi ∩ V (1) := Gi ∩ V (1) ,

Gi ∩ V (2) :=

Gi , Gi ∩ V (1)

Gi := Gi ∩ V (1) ⊕ Gi ∩ V (2) .

Because φ ∈ V (1) , we have φ ∈ Gi 0 − Gi 0 −1 . (a) Let us take any decomposition V (a) = Ki (a = 1, 2) such that Gj ∩ V (a) = (a) (1) (2) i≤j Ki . We put Ki := Ki ⊕Ki . Then, we obtain the one parameter subgroup Ki . λ of G1 whose weight decomposition is Lemma 4.4.6 We have μλ z, L = 0. Proof Let Ei (m) denote the subsheaf of E(m) generated by Gi . Each Ei has the induced parabolic structure and the reduced L-section [φi ]. Then, the filtration · · · ⊂ (Ei ∗ , [φi ]) ⊂ (Ei+1 ∗ , [φi+1 ]) ⊂ · · · is a partial Jordan-H¨older filtration by Corollary 4.1.12. Let us consider the filtration H of E given by H0 := E (1) and H1 := E. and the reduced L-section On GrH (Ei ), we have the induced parabolic structure [GrH (φi )]. Then, the tuple GrH (Ei )∗ , GrH (φi ) is δ-semistable.

124

4 Geometric Invariant Theory and Enhanced Master Space

We have the canonical isomorphism GrH (E) E. We regard GrH (Ei )(m) as the subsheaf of E(m), and then Gi generates GrH (Ei )(m). Thus, we obtain μλ (z, L) = 0 from Corollary 4.1.12. Let us return to the proof of Lemma 4.4.5. Since we have assumed the second condition in the lemma, μλ z, OB (1) ≥ 0 holds. Then, we obtain the following inequality from Lemma 4.4.6: Fj ∩ Gi

max γj · i0 + nj i · dim ≥ 0.

j=1,2 Fj ∩ Gi−1 j i Hence, we have only to show the following inequality:

i · dim

i

Fj ∩ Gi

Fj ∩ Gi ≤ i · dim .

Fj ∩ Gi−1 F j ∩ Gi−1 i

We put M := max{i | V (i) = 0}, and we put H := Fj . Then, we have the following equalities:

i · dim

H ∩ Gi H ∩ Gi = i · dim H ∩ Gi−1 H ∩ Gi−1 i≤M = i · dim(H ∩ Gi ) − (i + 1) dim H ∩ Gi i≤M

i≤M −1

= dim H · M −

dim H ∩ Gi .

i≤M −1

Hence, we have only to show dim H ∩ Gi ≤ dim H ∩ Gi for each i. But we have the equality H ∩ Gi ∩ V (1) = H ∩ Gi ∩ V (1) and the following inclusion: H ∩ Gi ⊂ H ∩ Gi ∩ V (2) . H ∩ Gi ∩ V (1) Thus, we obtain the desired inequality, and Lemma 4.4.5 is proved.

4.4.4 Step 2 We give some preliminary consideration. (2) (2) (O1) Assume that there exists a partial Jordan-H¨older filtration E1 ∗ ⊂ E2 ∗ . We (2) (2) (2) (2) take a decomposition V (2) = V1 ⊕ V2 such that V1 = H 0 X, E1 (m) . Let (2)

(2)

λ be the one parameter subgroup given by t− rank V2 idV (2) ⊕ trank V1 idV (2) . In 1 2 the case, we have the following:

4.4 Fixed Point Set of Torus Action on Enhanced Master Space

125

μλ z, OB (1) = N (2) (2) (2) (2) nj − rank V2 · dim Fj ∩ V1 + rank V1 · dim j=min(I2 )

(2)

Fj (2) Fj

∩

(2) V1

(4.26) Since nmin(I2 ) is sufficiently larger than nj (j > min(I2 )), (4.26) is larger than 0, (2) (2) if and only if Fmin(I2 ) ∩ V1 = {0}. We also remark that (4.26) cannot be 0. In particular, we obtain the following: Lemma 4.4.7 If ι(z) is contained in Bss , then the reduced O(−m)-Bradlow pair (2)

(E∗ , Fmin ) is -stable for any sufficiently small number > 0.

4.4.5 Step 3 We give some preliminary consideration. (O2) Let λ be the one-parameter subgroup of G1 given by trank V

(2)

idV (1) ⊕ t− rank V

(1)

idV (2) .

Let πi denote the projection of THsplit (I) onto THsplit (I) ∩ THi . Let us consider the points zi := π i (z) which are fixed with respect to λ. We have μλ (zi , L) = 0 for i = 1, 2, and μλ z2 , OPm (γ2 ) = γ2 · rank V (2) . Since nj are sufficiently smaller z μ , O (1) < 0. than |γ2 |, we always have the inequality λ 2 B Let us consider the condition μλ z1 , OB (1) ≥ 0. It is equivalent to the following inequality: (4.27) μλ z1 , OPm (γ1 ) + μλ z1 , OFlag (n∗ ) ≥ 0. In the case (I), we have γ1 > 0, and ni are sufficiently smaller than γ1 . Hence, the inequality (4.27) always holds. More strongly, the strict inequality holds. In the case (II), the inequality (4.27) can be rewritten as follows: γ1 rank V (2) + ' ' ni dim(Fi ∩ V (1) ) rank V (2) − dim i

=

γ1 +

(dim Fi ∩ V (1) )ni i=1

− +

i≥ +1

'

ni dim

i=1

ni dim(Fi ∩ V

(1)

'

Fi Fi ∩ V (1)

) rank V

(2)

Fi Fi ∩ V (1)

(

( rank V (1)

rank V (2)

(

− dim

rank V (1) '

Fi Fi ∩ V (1)

(

( rank V

(1)

≥0 (4.28)

126

4 Geometric Invariant Theory and Enhanced Master Space

The inequality (4.28) is equivalent to the condition F ⊂ V (1) , by our choice of γ1 and n∗ . Moreover, if the inequality holds, the strict inequality holds by our choice of n∗ . in Bss, we have Now, we give a proof of Lemma 4.4.3. If ι(z) is contained μλ z, OB (1) ≥ 0 for λ as above. Therefore, we obtain μλ z1 , OB (1) ≥ 0, and hence F ⊂ V (1) . It means min(I2 ) > . Thus, Lemma 4.4.3 is proved.

4.4.6 Step 4 We put k := min(I2 ) in the following argument. We give some more preliminary considerations. (1) (1) (O3) Assume there exists a partial Jordan-H¨older filtration E1 ∗ ⊂ (E∗ , φ) with (1) (1) respect to δ-semistability. We take a decomposition V (1) = V1 ⊕ V2 such that (1) (1) V1 = H 0 X, E1 (m) , and we consider the one-parameter subgroup λ given (1) (2) by t− rank V idV (1) ⊕ trank V1 idV (2) . We have μλ (z, L) = μλ z, OPm (1) = 0 1 for such one parameter subgroups. Therefore, the condition μλ z, OB (1) ≥ 0 is equivalent to the following inequality: 0 ≤ μλ z, OFlag (n∗ ) (1) (1) (1) (2) + rank V1 · dim Fi ni · − rank V (2) · dim Fi ∩ V1 = i

=−

k−1

(1) (1) ni · rank V (2) · dim Fi ∩ V1

i=1

(1) (1) (1) (2) + rank V1 · dim Fk + nk · − rank V (2) · dim Fk ∩ V1 (1) (1) (1) (2) + rank V1 · dim Fi . (4.29) ni · − rank V (2) · dim Fi ∩ V1 +

i>k

Recall that ni are sufficiently smaller than ni−1 . Hence, the inequality (4.29) holds, (1) (1) (1) (1) if and only if Fk−1 ∩ V1 = Fk ∩ V1 = {0}. (O4) Assume that there exists a partial Jordan-H¨older filtration (1)

(1)

(E2 ∗ , φ) ⊂ (E∗ , φ) (1)

(1)

with respect to δ-semistability. We take a decomposition V (1) = V2 ⊕ V1 such (1) (1) that V2 = H 0 X, E2 (m) , and we take the one-parameter subgroup λ given by (1) (2) trank V idV (1) ⊕ t− rank V1 · idV (2) . Because μλ (z, L) = μλ z, OPm (1) = 0, 1 the condition μλ z, OB (1) ≥ 0 is equivalent to the following:

4.4 Fixed Point Set of Torus Action on Enhanced Master Space

0≤

rank V

ni

· rank

(2)

=

(1)

+

i>k

rank V

(2)

· rank

ni

(1)

∩ V2

rank V (2) · rank

(1)

Fk (1)

(1)

(1)

−

(1) rank V1

(1)

−

·

(2) rank Fi

(1)

Fi

Fi

F ∩ V2 k (1) Fi Fi

ni · rank V (2) rank

i=1

+ nk

k−1

(1)

Fi Fi

i

127

(1)

∩ V2

(1) rank V1

·

(2) rank Fk

(1)

∩ V2

−

(1) rank V1

·

(2) rank Fi

(4.30) Since ni is sufficiently smaller than ni−1 , the inequality (4.30) holds if and only if (1) (1) Fk−1 ⊂ V2 . We obtain the following claim from the above preliminary considerations. Lemma 4.4.8 If ι(z) is contained in Bss , the parabolic reduced O(−m)-Bradlow (1) pair (E∗ , φ, F (1) ) is δ, min(I2 ) − 1 -semistable.

4.4.7 End of the Proof of Proposition 4.4.4 When ι(z) is contained in Bss , it is easy to see z ∈ TH∗split (I). We have already shown that the other two conditions are satisfied (Lemma 4.4.7 and Lemma 4.4.8). Let z ∈ THsplit (I) be a point which satisfies the conditions in Proposition 4.4.4. Let u1 , . . . , uN (1) be a base of V (1) , and let uN (1) +1 , . . . , uN bea base of V (2) . Let wi = 0. Let λ be (w1 , . . . , wN ) be an element of ZN such that wi ≤ wi+1 and the one-parameter subgroup of G, given by λ(t)·ui = twi ·ui . We do not distinguish λ and (w1 , . . . , wN ). We have the following decomposition: μλ z, OB (1) = k · k · μλ z, L + k μλ z, OFlag (n∗ ) + k max γi · μλ (z, OPm (1)) . i=1,2

We put S1 := h μfh (z, L) = 0 and S2 := h μfh (z, L) > 0 , where fh = h

*+ , ) h − N, . . . , h − N , h, . . . , h . Since nj and γi are sufficiently small, we can show that the following inequality holds for any h ∈ S2 by the argument used in the proof of Lemma 4.3.12: k · μfh (z, L) + μfh (z, OFlag (n∗ )) + min γi · μfh (z, OPm (1)) > 0. (4.31) i=1,2

128

4 Geometric Invariant Theory and Enhanced Master Space

By the argument in the proof of Lemma 4.3.13, we have only to show the following inequalities for any 0 = ρ = j∈S1 aj · fj with aj ≥ 0: F (ρ) := k · μρ (z, L) + μρ (z, OFlag (n∗ )) + max γi · μρ (z, OPm (1)) > 0 (4.32) i=1,2

Let us show (4.32). We have the weight decomposition V (α) = put N

(α)

:= dim V

(α)

s(2)

j

(α)

Vj

of ρ. We

(α) . We put rj

(α) := dim Vj . We use the notation in Subsection (α) (α) as a · vj satisfying (2.16). Let i0 be the j

2.6.2. Then, ρ can be expressed number determined by φ ∈ i≤i0 Vi − expression:

ρ=

s(α)

c(j) · y (2) (j) +

j=1

i≤i0 −1

d1 (i) · x1 (i) +

i
Vi . By Lemma 2.6.3, we have the

d2 (i) · x2 (i)

i>i0

+ A · N (2) Ω (1) − N (1) · Ω (2) .

Here c(j), d1 (i) and d2 (i) are non-negative numbers, and A is a rational number. One of c(j), d1 (i), d2 (i) or A is not zero. Because μκ (z, OPm (1)) = 0 for κ = y (2) (j), x1 (i) or x2 (i), we have the following linearity:

F (ρ) =

s(2)

c(j) · F y (2) (j) + d1 (i) · F x1 (i) + d2 (i) · F x2 (i)

j=1

i
i>i0

+ F A · N (2) Ω (1) − N (1) · Ω (2) .

We obtain F y (2) (j) > 0, F x1 (i) > 0 and F x2 (i) > 0 from the preliminary considerations (O1), (O3) and (O4) respectively. We have F A · N (2) Ω (1) − N (1) · Ω (2) > 0 in the case A = 0, from the preliminary consideration (O2). Therefore, we obtain the desired positivity, and the proof of Proposition 4.4.4 is finished.

4.5 Enhanced Master Space in Oriented Case 4.5.1 Construction

:= Qss (

We use the notation in Subsections 4.3.1 and 4.2.1. We put Q y , [L], α∗ , δ) ∗

:= and Q := Q × Flag(V, N ). Then, we set TH := TH × Q and TH ∗

Q

We remark TH Qss (m, y

, L, α∗ , δ) × Flag(V, N ). We also put TH∗ ×TH TH.

4.5 Enhanced Master Space in Oriented Case ss

129 ss

= THss ×TH TH. We have the natural GL(V )-action on TH . The quotient TH stack is called the enhanced master space in the oriented case, and it is denoted by

. It is also called the master space for abbreviation. M

is Deligne-Mumford and proper. Proposition 4.5.1 M Proof From Proposition 4.3.3, THss / PGL(V ) is Deligne-Mumford and proper.

−→ THss / PGL(V ), which is e´ tale We have the naturally defined morphism M

is also Deligne-Mumford and proper. and proper. Hence, M u u

× X. It inWe have the universal quotient object q , E , F∗ , [φu ], ρu on Q ∗ × X, E u (−m), F∗ , [φu ], ρu on TH duces the oriented reduced L-Bradlow pair πX × X −→ Q

× X denotes the natural projection. By takwhere πX : TH ing descent with respect to the GL(V )-action, we obtain the oriented reduced

M , F∗M , [φM ], ρM ). We also have the induced full flag F M of L-Bradlow pair (E

M

(m). pX ∗ E which is also denoted by ρ. It inThe Gm -action ρ on TH induces that on TH,

duces the Gm -action on M , which is denoted by ρ. Let us look at the stack theoretic

in the following subsections. fixed point set of M

4.5.2 Obvious Fixed Point Sets ss

ss

ss

:= THi ×TH TH

i := TH / GL(V ) . We have the substacks M We put TH i i

2 gives the (i = 1, 2). The stack M2 can be easily described. By construction, M s (

moduli stack M y , [L], α∗ , δ− ) of oriented δ− -stable reduced L-Bradlow pairs (E∗ , [φ], ρ) of type y with full flags F of H 0 X, E(m) . It is easily related with a moduli stack Ms (

y , [L], α∗ , δ− ) of δ− -semistable oriented reduced L-Bradlow

u over Ms (

pairs. We have the universal sheaf E y , [L], α∗ , δ− ) × X. We obtain u s

the locally free sheaf pX ∗ E (m) on M (

y , [L], α∗ , δ− ). The associated full flag

2 . bundle is isomorphic to M

1 is isomorphic to a moduli stack of δ+ 1 . In the case (I), M Let us look at M semistable oriented reduced L-Bradlow pairs (E∗ , [φ], ρ) of type y with full flags

, [L], α∗ , δ+ , as in F of H 0 (X, E(m)). It is related with a moduli stack Ms y

1 is a moduli stack of (δ, )-semistable tuples

2 . In the case (II), M the case of M (E∗ , [φ], ρ, F).

M

i × X is

M , the restriction of (E , F∗M , [φM ], ρM ) to M By construction of E naturally isomorphic to the universal object, which is given by the moduli theoretic

M

i . It is also easy to observe that the weight of ρ on E meaning of M

×X is 0. |M i

130

4 Geometric Invariant Theory and Enhanced Master Space

4.5.3 Fixed Point Sets Associated to Decomposition Types

∗ := M

− Let us describe the components of the fixed point set contained in M

(M1 ∪ M2 ). We use the notation in Subsection 4.4.2. Let I = (y 1 , y 2 , I1 , I2 ) be a decomposition type. In the case (II) (see Subsection 4.3.1), we assume ⊂ I1 . (1) (δ, k) denote the maximal open subset of We put k := min(I2 ) − 1. Let Q (1) Q determined by the (δ, k)-semistability. (It is open by Proposition 4.2.4.) Let (2) determined by the condition that the (2) denote the maximal open subset of Q Q + (2) reduced O(−m)-Bradlow pair E∗ , Fk+1 is -stable for any sufficiently small (1) (δ, k) × Q (2) −→ Q. Let > 0. We have the naturally defined morphism Q +

(1) (δ, k) × Q (2) and Q We also put over Q. Q split (I) denote the fiber product of Q + ss ∗

(1) TH )×GL(V (2) )-action split (I) := TH × Qsplit (I). We have the natural GL(V Q

ss

Gm (I). According to Proposition on TH split (I). The quotient stack is denoted by M

Gm (I) −→ M

. 4.4.4, we have the naturally defined morphism ϕI : M Lemma 4.5.2

Gm (I) is a moduli stack of tuples (E∗(1) , φ, F (1) ), (E∗(2) , F (2) ), ρ as follows: 1. M (1)

• (E∗ , φ) is a δ-semistable parabolic L-Bradlow pair of type y 1 , and F (1) is (1) a full flag of pX ∗ E (1) (m) such that (E∗ , φ, F (1) ) is (δ, k)-semistable. (2) • E∗ is a semistable parabolic sheaf of type y 2 , and F (2) is a full flag of (2) pX ∗ E (2) (m) such that (E (2) , Fmin ) is a reduced O(−m)-Bradlow pair. • ρ is an orientation of E (1) ⊕ E (2) .

∗M 2. We have decompositions ϕ∗I E = E1M ⊕ E2M and ϕ∗I F M = F1M ⊕ F2M . The

M pull back of the reduced L-section [φM ] of E naturally gives an L-section φM 1

of E1M . M

∗ M M M gives a universal object Then, the object (E1M∗ , φM 1 , F1 ), (E2 ∗ , F2 ), ϕI ρ Gm

× X in the moduli theoretic meaning above. over M

is OP (−1)∗ . Then, the claim is clear Proof We recall the fibration TH∗ −→ Q m by construction.

Gm (I) and the restriction Later, we will give a more convenient description of M M

G

m (I).

, F∗ , [φ], ρ to M of E

4.5.4 Statement We put as follows:

4.5 Enhanced Master Space in Oriented Case

S(m, y) :=

131

⎧ ⎨ Dec(m, y, α∗ , δ) ⎩

(case (I))

I = (y 1 , y 2 , I1 , I2 ) ∈ Dec(m, y, α∗ , δ) ⊂ I1 (case (II))

Gm denote the stack theoretic fixed point set of M

with Proposition 4.5.3 Let M respect to the action ρ. Then, we have the following isomorphism:

1 M

2

Gm (I).

Gm M M M I∈S(m,y)

We will prove the proposition in the following subsections.

4.5.5 Ambient Stack We have the Poincar´e bundle Poin on Pic det(y(m)) × X. Then, we obtain the &r

m := pX ∗ Hom vector bundle Z Vm,X , Poin on Pic det(y(m)) . Recall

m . We have the natural prothat the Gieseker space Zm is the projectivization of Z ss

m ×Z Bss . We have the naturally jection B −→ Zm . Thus, we put B := Z m ss

defined GL(V )-action on B . The quotient stack is denoted by B . The Gm -action

ss ρ on Bss naturally induces an action on B . Since it commutes with the action of GL(V ), we obtain a Gm -action ρ on B . ss

ss We have the naturally defined closed immersion TH −→ B , which is GL(Vm ) × Gm -equivariant. Therefore, we obtain the closed Gm -equivariant im −→ B . Since M

is Deligne-Mumford and separated, we can take a mersion M

in B , which is Deligne-Mumford Gm -equivariant open neighbourhood B of M and separated. We may assume that B is also smooth. Let BGm denote the fixed point set of B with respect to the Gm -action ρ, i.e., the 0-set of the vector field

Gm is induced by the Gm -action. Recall that the stack theoretic fixed point set M

. defined to be BGm ×B M

4.5.6 Fixed Point Set of the Ambient Space

ss

ss

ss We set Bi := Bi ×B B . The quotient stack Bi / GL(V ) is denoted by Bi . It

i = Bi ×B M

. gives an open subset of BGm . It is easy to see M (m). Let N = (N , N ), r = (r , r ), k Recall N := H y 1 2 1 2 ∗ := k1,j , k2,j j = 1, . . . , l , I := (I1 , I2 ) be as follows: • Ni are positive integers such that N1 + N2 = N . • ri are positive integers such that r1 + r2 = r.

132

4 Geometric Invariant Theory and Enhanced Master Space

• ki,j are positive integers such that k1,j + k2,j = Hy,j (m). • I1 I2 = N . We remark Such a tuple Q := (N , r, k∗ , I) is called a decomposition type for A. that a decomposition type I = (y 1 , y 2 , I1 , I2 ) induces the decomposition type Q(I) as follows: for A, Ni = |Ii |,

ri = rank(y i ),

ki,j = Hyi ,j (m).

the

m . For a decomposition type Q = (N , r, k∗ , I) for A, Z Z We put A := A× m

locally closed regular subvariety C1 (Q) of A is the set of the points (f, K∗ , [φ], F) satisfying the following conditions: • There exists the unique decomposition V = V (1) ⊕ V (2) .

m is contained in H 0 X, Hom &r1 V (1) ⊗ &r2 V (2) , L for some • 0 = f ∈ Z X X (i) (i) line bundle L ∈ Pic det(y(m)) , where V := V ⊗ O . X X . • K∗ = Ki i = 1, . . . , l ∈ i Qm,i is compatible with the decomposition (1) (2) V = V (1) ⊕ V (2) , i.e., we have the decomposition Kj = Kj ⊕ Kj such that (i)

(i)

Kj are quotients of V (i) . We also assume dim Kj = ki,j . • [φ] is contained in the projectivization of V (1) . • We have the decomposition F = F (1) ⊕F (2) compatible with the decomposition (i) (i) V = V (1) ⊕ V (2) . Moreover, Fj /Fj−1 = 0 if and only if j ∈ I (i) . We put B∗ := B − Bi and C2 (Q) := B∗ ×A C1 (Q). We have the natural GL(V )-action on C2 (Q), and the quotient stack is denoted by C 3 (Q), which is the closed substack of B . We put C3 (Q) = C 3 (Q) ∩ B. The following lemma can be checked easily. Lemma 4.5.4 C3 (Q) are open subsets of BGm .

4.5.7 Proof of Proposition 4.5.3 We put as follows: m := BG 0

C3 Q(I)

I∈S(m,y)

Gm (I) −→ B factors through C3 Q(I) . Therefore, we have the The morphism M

Gm (I) −→ BGm . We obtain the following morphism: morphism M 0

×B BGm

Gm (I) −→ M M ψ1 : 0 I∈S(m,y)

4.5 Enhanced Master Space in Oriented Case

133

Lemma 4.5.5 ψ1 is isomorphic.

×B BGm be a morphism. Then, we obtain a GL(V )-torsor Proof Let g : T −→ M 0 P (g) over T with the following data: • The GL(V )-equivariant decomposition V1 ⊕V2 of V := p∗P (g) VX over P (g)×X. &r • The non-trivial morphism f : V −→ det∗ Poin, which is contained in Hom

r1 0

V1 ⊗

r2 0

V2 , det ∗ Poin .

is contained in M

∗ , and hence in

×B BGm −→ M The composite T −→ M 0

, L, α∗ , δ)/ GL(Vm ). Therefore, we obtain The GL(V )-equivariant oriQss (m, y ented quotient parabolic L-Bradlow pair q, E∗ (m), ρ, φ on P (g) × X, where q : p∗P (g) Vm,X −→ E(m) satisfies (TFV)-condition. From the equality ρ◦

r 0

q = f,

we obtain the decomposition E = E (1) ⊕ E (2) = q V (1) ⊕ q V (2) . The claim is clear on the open subset where E is locally free. Since E is torsion-free and q is surjective, the decomposition is obtained on the whole space. By taking the descent with respect to the GL(V )-action, we obtain (1) (2) (q1 , E∗ , φ1 , F (1) ), (q2 , E∗ , F (2) ), ρ on T × X. We remark that the decomposition data is determined on each connected component of T . The conditions in Proposition 4.4.4 is satisfied for the specializa(1) (2) tion of (q1 , E∗ , φ1 , F (1) ) and (q2 , E∗ , F (2) ) to any closed fibers {u}×X. Hence, A Gm we obtain the morphism T −→ M (I). In particular, we obtain the morphism

×B BGm −→ A M

Gm (I). It is easy to see that ψ1 and ψ2 are mutually ψ2 : M 0 inverse. Then, we obtain the following:

2

1 M M

Gm ×B BGm

Gm (I) = M

Gm ×B B1 ∪ B2 ∪ BGm = M M 0

A Gm

. Since B(Q) M (I) is the closed substack of M Gm Gm

and Bi are open in B , it is easy to see that Mi and M (I) are unions of connected components of the fixed point set. Thus, the proof of Proposition 4.5.3 is finished.

1 M

2 In particular, M

134

4 Geometric Invariant Theory and Enhanced Master Space

4.6 Decomposition of Exceptional Fixed Point Sets 4.6.1 Statement Let I = (y 1 , y 2 , I1 , I2 ) be a decomposition type. We would like to decompose

Gm (I) into the product of two moduli stacks up to e´ tale proper morphisms. (See M Proposition 4.6.1.) We put k := min(I2 ) − 1. We use the notation in Subsection 4.5.3. We introduce some moremoduli stacks. (1) ss y

1 , [L], α∗ , (δ, k) denote a moduli stack of (E∗ , [φ(1) ], ρ(1) , F (1) ) Let M as follows: (1)

• (E∗ , [φ(1) ], ρ(1) ) is a δ-semistable oriented reduced L-Bradlow pair of type (1) y 1 , and F (1) is a full flag of pX ∗ E (1) (m) such that (E∗ , [φ(1) ], F) is (δ, k)semistable. u u u u ss y

, φ , ρ , F denote a universal object over M

1 , [L], α∗ , (δ, k) ×X. Let E 1∗ 1 1 1 ss y 1 , L, α∗ , (δ, k) denote a moduli stack of (E∗(1) , φ(1) , F (1) ) as folLet M lows: (1)

• (E∗ , φ(1) ) is a δ-semistable L-Bradlow pair of type y 1 such that φ(1) is nontrivial everywhere, and F (1) is a full flag of pX ∗ E (1) (m) such that the tuple (1) (E∗ , φ(1) , F (1) ) is (δ, k)-semistable. ss y 1 , L, α∗ , (δ, k) × X is denoted by (E u , φu , F u ). A universal object over M 1∗ 1 1 (2) ss y

2 , α∗ , + denote a moduli stack of tuples (E∗ , ρ(2) , F (2) ) as folLet M lows: (2)

• (E∗ , ρ(2) ) is semistable oriented parabolic sheaf of type y 2 , and F (2) is a full flag of pX ∗ E (2) (m). (2) (2) • (E∗ , Fmin ) is an -semistable reduced O(−m)-Bradlow pair for any sufficiently small > 0.

u , ρu , F u ). ss (

y 2 , α∗ , +) × X is denoted by (E A universal object over M 2 2 2 Proposition 4.6.1 We put ri := rank y i . There exists an algebraic stack S with the following properties: • There exists the following diagram: G

Gm (I) ←−F−−− S −−− ss y ss y

1 , [L], α∗ , (δ, k) × M

2 , α∗ , + M −→ M (4.33) The morphisms F and G are e´ tale proper of degree (r1 · r2 )−1 and r2−1 , respec

Gm (I) and M ss y ss y

1 , [L], α∗ , (δ, k) × M

2 , α∗ , + tively. In this sense, M are isomorphic up to e´ tale proper morphisms. ss y

1 , [L], α∗ , (δ, k) . • Let O1,rel (1) denote the tautological line bundle of M There exists the line bundle O1,rel (1/r2 ) on S such that O1,rel (1/r2 )r2 = G∗ O1,rel (1), and we have the following relations:

4.6 Decomposition of Exceptional Fixed Point Sets

135

u ⊗ O1,rel (1), F ∗ E1M G∗ E 1

u ⊗ O1,rel (−r1 /r2 ) F ∗ E2M G∗ E 2

(4.34)

The weight of the Gm -action ρ on E1M and E2M are −1 and r1 /r2 , respectively. Corollary 4.6.2 We have the following diagram: G ss y 1 , L, α∗ , (δ, k) × M ss y

Gm (I) ←−F−−− S −−−

2 , α∗ , + (4.35) −→ M M Here G is e´ tale proper of degree (r1 · r2 )−1 . We have the following relations:

F ∗ E1M G ∗ E1u ,

u ⊗ Or(E u )−1/r2 F ∗ E2M G ∗ E 2 1

Here, we put Or(E1u )−1/r2 := O1,rel (−r1 /r2 ).

Before going into a proof, we give a remark on the inductive process to investigate the transition of moduli stacks, heuristically. Let δ be critical. Let δ− and δ+ be sufficiently close to δ such that δ− < δ < δ+ . It is interesting to investigate differy , [L], α∗ , δ− ) and Mss (

y , [L], α∗ , δ+ ). We make the enhanced ence between Mss (

master space as above. Then, Mi (i = 1, 2) are isomorphic to the full flag variety y , [L], α∗ , δκ ) for κ = ±. So we can derive some information on bundles over Mss (

Gm (I) by using the the difference of the moduli stacks from the fixed point sets M

Gm (I) are isomorphic to localization technique. According to Corollary 4.6.2, M ss y ss (

1 , [L], α∗ , (δ, ) × M M y 2 , α∗ , +) up to e´ tale proper morphisms, where = min(I2 ) − 1. ss (

ss (

The stack M y 2 , α∗ , +) is related with M2 := M y 2 , [O(−m)], α∗ , ),

u over where is a sufficiently small number. We have a universal oriented sheaf E 2 M2 × X with the universal reduced section [φu2 ]. We obtain the vector bundle u u

(m) with the line subbundle Q ⊂ pX ∗ E

(m) induced by [φu ]. We pX ∗ E 2 2 u

(m) /Q. Then, the full flag bundle associated to C is isomorphic put C := pX ∗ E 2 ss y 2 , α∗ , +). to M (

ss y

1 , [L], α∗ , (δ, ) is not so easy to describe. However, we The structure of M

(1) again, so that M

(1) and M

(1) are isomorcan make the enhanced master space M 1 2 ss ss y

1 , [L], α∗ , (δ, ) and the full flag bundle over M y

1 , [L], α∗ , δ− , phic to M respectively. Thus, we can proceed inductively. Because rank(y 1 ) < rank(y), the process will stop, and we will obtain a dey , [L], α∗ , δ+ ) and Mss (

y , [L], α∗ , δ− ) in scription of difference between Mss (

terms of the products of moduli stacks of semistable objects with lower ranks. We use such an argument in Section 7.6.

136

4 Geometric Invariant Theory and Enhanced Master Space

4.6.2 Preliminary

(1) We use the notation in Section 4.4. Let Q denote the maximal open subset of

(2) (1) ◦

1 , L) × Flag determined by the (δ, k)-semistability. Let Q denote Q (m, y (2) ◦

the maximal open subset of Q (m, y 2 ) × Flag , which consists of the points (2) (q2 , E2 ∗ , ρ2 , F (2) ) such that (E2 ∗ , Fmin ) is -semistable for any small > 0. −1 Let T1 := Spec k[t1 , t1 ] be a one dimensional torus. We have the T1 -action on

(1) (2) × Q given as follows: Q t1 ·

E1 ∗ , φ, ρ1 , F (1) , E2 ∗ , ρ2 , F (2) := (2) E1 ∗ , φ, t1 ·ρ1 , F (1) , E2 ∗ , t−1 ·ρ , F 2 1

(1) ss

(2) ×Q /T1 . We By construction, we have the isomorphism TH split (I) Q (i) (i)

have the naturally defined actions GL(Vi ) on Q . We put Mi := Q / GL(Vi ). Then, we obtain the following description:

Gm

M

(1) (2) M1 × M2 ×Q Q (I) GL(V1 ) × GL(V2 ) × T1 T1

Let us look at the right hand side more closely. The stack M1 is isomorphic to a (1) moduli stack of tuples (E∗ , φ(1) , ρ(1) , F (1) ) as follows: (1)

• (E∗ , φ(1) , ρ(1) ) is a δ-semistable oriented L-Bradlow pair of type y 1 , and F (1) (1) is a full flag of pX ∗ E (1) (m) such that (E∗ , φ(1) , F (1) ) is (δ, k)-semistable. ss y

2 , α∗ , + . The inThe quotient stack M2 is isomorphic to a moduli stack M duced T1 -actions on Mi (i = 1, 2) are given as follows: (1) (1) t1 · E∗ , φ(1) , ρ(1) , F (1) := E∗ , φ(1) , t1 ·ρ(1) , F (1) (2) (2) (2) , F (2) . t1 · E∗ , ρ(2) , F (2) := E∗ , t−1 1 ·ρ

4.6.3 Construction of the Stack S r1 r2 We put T1 := Spec k[s1 , s−1 , 1 ]. Let T1 −→ T1 be the morphism given by t1 = s1 where ri = rank y i . We have the naturally induced T1 -action on M1 × M2 . We set

S :=

M1 × M2 . T1

4.6 Decomposition of Exceptional Fixed Point Sets

137

Then, we have the following morphism: F

Gm (I) S −−−−→ (M1 × M2 )/T1 M

Here F is e´ tale proper of degree (r1 · r2 )−1 . Let us look at the T1 -action on Mi (i = 1, 2). The induced T1 -action on M2 is trivial, i.e., (2) (2) (2) 1 ·r2 ·ρ(2) , F (2) E∗ , ρ(2) , F (2) . s1 · E∗ , ρ(2) , F (2) = E∗ , s−r 1 The isomorphism is given by the following diagram: E (2) ⏐ −r ⏐ s1 1 ! E (2)

s

−r1 r2 (2)

·ρ

det(E (2) ) −−1−−−−−→ det∗ Poin ⏐ ⏐ ⏐ −r r ⏐ id! s1 1 2 ! det(E (2) )

ρ(2)

(4.36)

det∗ Poin

−−−−→

The induced T1 -action on M1 is given as follows: (1) (1) s1 · E∗ , φ(1) , ρ(1) , F (1) = E∗ , φ(1) , sr11 r2 ·ρ(1) , F (1) (1) E∗ , sr12 ·φ(1) , ρ(1) , F (1) . The isomorphism is given by the following diagram: L −−−−→ E (1) ⏐ ⏐ ⏐ r ⏐ s12 ! id! s

r2

φ

1 −→ E (1) L −−−

s

r1 r2

ρ(1)

det(E (1) ) −−1−−−−→ det∗ Poin ⏐ ⏐ ⏐ r r ⏐ s11 2 ! id!

(4.37)

ρ(1)

det(E (1) ) −−−−→ det∗ Poin

Let T1 := Spec k[u1 , u−1 1 ], which acts on M1 by u1 · E (1) , φ(1) , ρ(1) , F (1) := E (1) , u1 ·φ(1) , ρ(1) , F (1) . We have the homomorphism T1 −→ T1 given by u1 = sr12 , which induces the morphism G: G S = (M1 /T1 ) × M2 −→ (M1 /T1 ) × M2 ss y ss (y 2 , α∗ , +)

1 , [L], α∗ , (δ, k) × M M

The morphism G is e´ tale proper of degree 1/r2 .

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4 Geometric Invariant Theory and Enhanced Master Space

4.6.4 Universal Sheaf ss y

1 , [L], α∗ , (δ, k) . We remark that π : M1 × M2 −→ We put M1 := M M1 × M2 is isomorphic to O1,rel (−1)∗ −→ M1 × M2 , and the T1 -action on O1,rel (−1)∗ is given by s · v = sr2 v on each fiber. We use the argument in Section 2.7. Let T (n) denote the trivial line bundle on M1 × M2 with the T1 -action of weight n. It induces the T1 -line bundle π ∗ T (n) over M1 ×M2 . Taking the descent, we obtain the line bundle In over M1 /T1 × M2 . We put O1,rel (1/r2 ) := I1 . According to Lemma 2.7.1, it satisfies O1,rel (1/r2 )r2 G∗ O1,rel (1).

u (i = 1, 2) denote the pull back of E

u via the morphism M1 × M2 × Let E i i X −→ M1 × M2 × X. By the construction, we have the T1 -equivariant sheaves ss

TH ⊕ E

TH on M1 × M2 × X, which is induced by the sheaf on TH as in E 1 2 Subsection 4.5.1. Taking account of the Gm -action ρ, we have

1 u ⊗ π ∗ T (r2 ),

1 TH E E

2 TH E

2 u ⊗ π ∗ T (−r1 ) E

by the diagrams (4.36) and (4.37). Therefore, we obtain (4.34). We put T2 := Spec k[s2 , s−1 2 ], and let T2 −→ Gm be a morphism given by (1) (2)

t = sr22 . On Q ×Q , the induced T2 -action is given as follows: s2 · (E1 ∗ , φ, ρ1 , F1 ), (E2 ∗ , ρ2 , F2 ) = (E1 ∗ , sr22 φ, ρ1 , F1 ), (E2 ∗ , ρ2 , F2 ) . Hence, the actions of T1 and T2 on M1 × M2 are the same. Taking account of the

u and E

u . Therefore, T2 -action, the induced bundles over M1 × M2 × X are E 1 2 ∗ u

⊗Ir is −r2 , and that on ϕ∗ E

u ⊗I−r the weight of the induced T2 -action on ϕ E 1 2 2 1 is r1 , thanks to Lemma 2.7.2. Thus, the proof of Proposition 4.6.1 is finished.

4.7 Simpler Cases 4.7.1 Case in Which a 2-Stability Condition is Satisfied We study the case that the 2-stability condition is satisfied for (y, L, α∗ , δ). We give only the statements without proof. In this case, we do not have to consider enhanced master spaces and (δ, )-semistability. We use the notation in Subsections 4.1.1 and 4.2.1. We take a positive rational number γ1 and a negative rational number γ2 such that |γi | are sufficiently small. We take a large rational number k such that k · (γ1 − γ2 ) = 1. We put Li := L⊗k γi . We have L2 = L1 ⊗ OPm (−1). Let us consider B := P OPm (0) ⊕ OPm (1) over A. We put B1 = P OPm (0) and B2 = P OPm (1) , which are naturally regarded as closed subschemes of B. We have the tautological line bundle OP (1), and we put OB (1) := OP (1) ⊗ L1 . We

4.7 Simpler Cases

139

have the natural SL(V )-action on B, and OB (1) gives the equivariant polarization. Let B ss denote the set of the semistable points of B with respect to OB (1). We put THss := Q ×A B ss . We have the natural SL(V )-action on THss . We can show the following proposition by the argument in the proof of Proposition 4.3.3. In fact, it is much easier. Proposition 4.7.1 The quotient stack THss / SL(V ) is Deligne-Mumford. We put TH

ss

where Q

:= Qss (

:= THss ×Q Q, y , [L], α∗ , δ). We define ss

:= TH / GL(V ), M which is called the master space in the oriented case. We have the Gm -action ρ as in the enhanced case. From Proposition 4.7.1, we obtain the following.

is Deligne-Mumford and proper. Proposition 4.7.2 M ss TH i

ss

ss / GL(V TH i ss

i := and M := Bi ×A TH ) for i = 1, 2. We put

1 and M

2 are isomorphic to M (

Thanks to Lemma 4.2.3, M y , [L], α∗ , δ+ ) and

with Mss (

y , [L], α∗ , δ− ), respectively. They give the obvious fixed point sets of M respect to ρ.

−(M

1 ∪ M

2 ). It is easy to observe that M

∗ is an open substack

∗ := M We put M

, L). Let us look at the components of the fixed point set contained in of M(m, y

∗ . A decomposition type is defined to be I := (y 1 , y 2 ) ∈ T ype2 satisfying M y 1 + y 2 = y and Pyα∗ ,δ = Pyα1∗ ,δ = Pyα2∗ . For such a I := (y 1 , y 2 ), we consider a

Gm (I) of (E∗(1) , φ), E∗(2) , ρ as follows: moduli stack M (1)

• (E∗ , φ) is δ-stable L-Bradlow pair of type y 1 . (2) • E∗ is stable parabolic sheaf of type y 2 . • ρ is an orientation of E (1) ⊕ E (2) . Note that the 2-stability condition for (y, L, α∗ , δ) implies the 1-stability conditions for (y 1 , L, α∗ , δ) and (y 2 , α∗ ) if Mss (y 1 , L, α∗ , δ) × Mss (y 2 , α∗ ) = ∅. We

Gm (I) −→ M

as in Subsection 4.5.3. have the naturally defined morphism M Let S(y, α∗ , δ) denote the set of decomposition types. Then, we can show the following proposition by the argument in the proof of Proposition 4.5.3.

2 A

1 M

Gm (I) is the stack theoretic fixed Proposition 4.7.3 M I∈S(y,α∗ ,δ) M

with respect to ρ. point set of M

∗M , [φM ], ρM ) on We naturally have the oriented reduced L-Bradlow pair (E

M M

i × X

M × X, as in Subsection 4.5.1. The restriction of (E∗ , [φ ], ρM ) to M has the universal property with respect to the moduli theoretic meaning above.

−→ M(m, y

, [L]) denote the naturally defined morphism. The reLet ϕ : M ∗

∗ is canonically trivialized. Therefore, the restriction striction of ϕ Orel (1) to M

∗ M

gives the L-section, which is denoted by φM . Then, the restricof [φ ] to M

Gm M M M

tion of (E , φ , ρ ) to M (I) has the universal property with respect to

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4 Geometric Invariant Theory and Enhanced Master Space

the moduli theoretic meaning above. Correspondingly, we have the decomposition

M E = EM ⊕ EM .

Gm (I) |M

1

2

Gm (I) into the product of two moduli stacks up It is convenient to decompose M to e´ tale proper morphisms. By the argument in the proof of Proposition 4.6.1, we

Gm (I) up to e´ tale proper morphisms. obtain the following description of M Proposition 4.7.4 We put ri := rank y i . There exists an algebraic stack S with the following properties: • There exists the following diagram: G

Gm (I) ←−F−−− S −−−

1 , [L], α∗ , δ × Mss y

2 , α∗ −→ Mss y M The morphisms F and G are e´ tale proper of degree (r1 · r2 )−1 and r2−1 , respectively. We also have the following diagram: G

Gm (I) ←−F−−− S −−−

2 , α∗ −→ Mss y 1 , L, α∗ , δ × Mss y M

(4.38)

Here, G is e´ tale proper of degree (r1 r2 )−1 . y 1 , [L], α∗ , δ). We • Let O1,rel (1) denote the tautological line bundle of Mss (

use the same symbol to denote the pull back via an appropriate morphism. Then, there exists the line bundle O1,rel (1/r2 ) on S such that O1,rel (1/r2 )r2 = G∗ O1,rel (1), and we have the following relation:

1u ⊗ O1,rel (1), F ∗ E1M G∗ E

2u ⊗ O1,rel (−r1 /r2 ) F ∗ E2M G∗ E

(4.39)

u and E

u are induced by the universal sheaves over Mss (

Here E y 1 , [L], α∗ , δ)× 1 2 ss y 2 , α∗ ) × X, respectively. We also have the following relation: X and M (

F ∗ E1M G ∗ E1u ,

u ⊗ Or(E u )−1/r2 F ∗ E2M G ∗ E 2 1

Here, E1u is the pull back of the universal sheaf over Mss (y 1 , L, α∗ , δ)×X, and Or(E1u )−1/r2 denotes O1,rel (−r1 /r2 ).

• We have the induced Gm -action ρ on E1M and E2M . The weights are −1 and r1 /r2 , respectively.

4.7.2 Oriented Reduced L-Bradlow Pairs 2 Let L = (L1 , L2 ) be a pair of line bundles on X. Let δ = (δ1 , δ2 ) ∈ P br . We can argue the transition of moduli stacks Mss (y, L, α∗ , δ) for variation of δ1 . For simplicity, we restrict ourselves to the case that both δi are sufficiently small. Recall the results in Subsection 3.5.3. Then, a 2-stability condition is always satisfied, and the problem is simple as in Subsection 4.7.1. We give only the statements without proof.

4.7 Simpler Cases

141

If the 1-stability condition does not hold for (y, L, α∗ , δ), we have the positive integers ri (i = 1, 2) satisfying the following: r1 + r2 = r,

δ1 δ2 = r1 r2

We take δ1,− and δ1,+ such that δ1,− < δ1 < δ1,+ , which are sufficiently close to δ1 . We set δ κ := (δ1,κ , δ2 ) for κ = ±. We would like to compare the moduli stacks Mss (y, L, α∗ , δ − ) and Mss (y, L, α∗ , δ + ). We use the notation in Subsection 4.1.1. We put A := Am (y, [L]). We set L := Ly,L Pyα∗ ,δ (m), ∗ , δ(m) which gives a GL(V )-polarization on A. We put Lγ := L ⊗ OP(1) (γ) for a rational m number γ. Let Ass (Lγ ) denote the set of the semistable points of A with respect to Lγ . We set Q := Qss (m, y, [L], α∗ , δ). We have the GL(V )-actions on A and Q. We also have the equivariant immersion Ψm : Q −→ A. The δ κ -semistability condition determines the open subset Qss κ . The following lemma can be shown by the argument in the proof of Lemma 4.2.3. Lemma 4.7.5 Assume that the absolute value of γ = 0 is sufficiently small. ss −1 A (Lγ ) = Qss • Ψm sign(γ) . ss • The induced morphism Ψm : Qss sign(γ) −→ A (Lγ ) is a closed immersion. Moreover, the image is contained in As (Lγ ). We take a positive rational number γ1 and a negative rational number γ2 such that |γi | are sufficiently small. We take a large integer k such that k (γ1 − γ2 ) = 1. (−1). We put Li := L⊗k γi . We have L2 = L1 ⊗ OP(1) m Let us consider B := P OP(1) (0) ⊕ OP(1) (1) over A. We put B1 = P OP(1) (0) m m m and B2 = P OP(1) (1) , which are naturally regarded as the closed subschemes of B. m We have the tautological line bundle OP (1), and we put OB (1) := OP (1) ⊗ L1 . We have the natural SL(V )-action on B, and OB (1) gives the equivariant polarization. Let B ss denote the set of the semistable points of B with respect to OB (1). We put THss := Q ×A B ss . We have the natural SL(V )-action on THss . As in the case of Proposition 4.7.1, the following proposition can be shown easily. Proposition 4.7.6 The quotient stack THss / SL(V ) is Deligne-Mumford.

ss

where we put Q

:= Q(

:= THss ×Q Q, y , [L], α∗ , δ). We put We put TH ss

:= TH / GL(V ), which is called the master space. We have the Gm -action ρ M as usual. From Proposition 4.7.6, we obtain the following.

is Deligne-Mumford and proper. Proposition 4.7.7 M ss TH i

ss

ss i / GL(V TH

i :=

∗ := , and we put M We put := Bi ×A TH ). We put M ∗

M − (M1 ∪ M2 ). It is easy to observe that M is an open substack of a moduli stack

, L1 , [L2 ]) of tuples (E∗ , ρ, φ1 , [φ2 ]) as follow: M(m, y

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4 Geometric Invariant Theory and Enhanced Master Space

• E∗ is a parabolic sheaf with an orientation ρ satisfying the condition Om . • φ1 is an L1 -section of E∗ such that φ1 = 0. • [φ2 ] is a reduced L2 -section of E∗ such that [φ2 ] = 0.

1 and M

2 are isomorphic to Mss (

From Lemma 4.7.5, M y , [L], α∗ , δ + ) and

with M (

y , [L], α∗ , δ − ), respectively. They give the obvious fixed point sets of M respect to ρ. Let us look at the other components of the fixed point set. A decomposition type is defined to be a datum I := (y 1 , y 2 ) ∈ T ype2 satisfying the following: ss

y 1 + y 2 = y,

Pyα∗ = Pyα1∗ = Pyα2∗ ,

ri = rank(y i )

Gm (I) be a moduli stack of the For a decomposition type I := (y 1 , y 2 ), let M (1) (2) objects E∗ , φ1 , E∗ , [φ2 ], ρ as follows: (1)

• (E∗ , φ1 ) is δ1 -stable L1 -Bradlow pair of type y 1 . (2) • (E∗ , [φ2 ]) is δ2 -stable reduced L2 -Bradlow pair of type y 2 . • ρ is an orientation of E (1) ⊕ E (2) . Note that δi are sufficiently small, and hence the 1-stability conditions hold for (y 1 , L1 , α∗ , δ1 ) and (y 2 , L2 , α∗ , δ2 ). We have the naturally defined morphism

Gm (I) −→ M

as in Subsection 4.5.3. M Let S(y, α∗ , δ) denote the set of decomposition types. Then, we can show the following proposition by the argument in the proof of Proposition 4.5.3.

2 A

Gm (I) is the stack theoretic fixed

1 M Proposition 4.7.8 M I∈S(y,α∗ ,δ) M

with respect to ρ. point set of M

∗M , [φM ], [φM ], ρ) on We naturally have the oriented reduced L-Bradlow pair (E 1 2

M M

i × X

], ρ) to M M × X, as in Subsection 4.5.1. The restriction of (E∗ , [φ1 ], [φM 2

i . has the universal property with respect to the moduli theoretic meaning of M (i)

, [L]) which are the pull back Let Orel (1) denote the line bundles on M(m, y

, [Li ]) via the natural morof the relatively tautological line bundles on M(m, y

−→ M(m, y

, [L]) −→ M(m, y

, [Li ]). Let ϕ : M

, [L]) denote phism M(m, y ∗ (1) ∗

the naturally defined morphism. The restriction of ϕ Orel (1) to M is canonically

M trivialized. Thus, the restriction [φM

∗ induces the L1 -section of E , which is 1 ]|M

(2) denoted by φM := ϕ∗ Orel (−1). Then, [φM 1 . We put I 2 ] gives the morphism

(2) M M M M

Gm (I) has the I ⊗ L2 −→ E . The restriction of (E , φ1 , [φ2 ], ρ) to M universal property with respect to the moduli theoretic meaning above. Correspond

M M

M ingly, we have the decomposition E

Gm (I) = E1 ⊕ E2 . |M (2)

Gm (I) into the product of two moduli stacks up It is convenient to decompose M to e´ tale proper morphisms. By the argument in the proof of Proposition 4.6.1, we

Gm (I) up to e´ tale proper morphisms. obtain the following description of M

4.7 Simpler Cases

143

Proposition 4.7.9 We put ri := rank y i . There exists an algebraic stack S with the following property: • There exists the following diagram: G

Gm (I) ←−F−−− S −−−

1 , [L1 ], α∗ , δ1 × Mss y

2 , [L2 ], α∗ , δ2 −→ Mss y M (4.40) The morphisms F and G are e´ tale proper of degree (r1 · r2 )−1 and r2−1 , respectively. We also have the following diagram: G

Gm (I) ←−F−−− S −−−

2 , [L2 ], α∗ , δ2 −→ Mss y 1 , L1 , α∗ , δ1 × Mss y M (4.41) Here G is e´ tale proper of degree (r1 r2 )−1 . • Let Oi,rel (1) denote the pull back of the relative tautological line bundle of Mss (y i , [Li ], α∗ , δ). There exists the line bundle O1,rel (1/r2 ) on S such that O1,rel (1/r2 )r2 G∗ O1,rel (1), and we have the following relation:

1u ⊗ O1,rel (1), F ∗ E1M G∗ E

2u ⊗ O1,rel (−r1 /r2 ), F ∗ E2M G∗ E

F ∗ I (2) G∗ O2,rel (−1) ⊗ O1,rel (−r1 /r2 ).

u are the pull back of the universal sheaves over Mss (

Here, E y i ,[Li ],α∗ ,δi )×X. i We also have the following:

F ∗ E1M G ∗ E1u ,

2u ⊗ Or(E1u )−1/r2 F ∗ E2M G ∗ E

Here, E1u is the pull back of the universal sheaf over Mss (y 1 ,L1 ,α∗ ,δ1 ) × X, and Or(E1u )−1/r2 denotes O1,rel (−r1 /r2 ).

(2) are −1, r1 /r2 and r1 /r2 , respec• The weights of ρ on E1M , E2M and I Gm (I) |M tively.

Chapter 5

Obstruction Theories of Moduli Stacks and Master Spaces

Let X be a smooth connected projective surface with a base point over an algebraically closed field k of characteristic 0. Let D be a smooth hypersurface of X. In this chapter, we study obstruction theories of moduli stacks of some kinds of stable objects on X with parabolic structure along D. The naive strategy for construction was explained in Subsection 2.4.2. We will also discuss obstruction theories for master spaces. In Section 5.1, we study an obstruction theory associated to a torsion-free sheaf E on U × X, where X is a smooth projective surface. We put g(V• ) := Hom(V• , V• )∨ [−1], Ob(V• ) := RpX ∗ g(V• ) ⊗ ωX for a resolution V• of E. In Subsection 5.1.1, we explain how to obtain the morphisms g(V• ) −→ LU ×X/X and Ob(V• ) −→ LU . In Subsection 5.1.2, we observe that g(V• ) is decomposed into the trace-free part and the diagonal part, and that the diagonal part is related to the determinant bundle. In Subsection 5.1.3, we give some factorization which will be useful for construction of obstruction theories of master spaces. In Subsection 5.1.4, we give an obstruction theory of the open subset of a moduli stack of torsion-free sheaves determined by the condition Om , by directly applying the construction in Subsection 5.1.1. As a special case, we look at an obstruction theory of the moduli of line bundles in Subsection 5.1.5, which will be used in the construction of a relative obstruction theory for orientations in Section 5.2. In Section 5.3, we study a relative obstruction theory for L-sections. In Subsections 5.3.1–5.3.2, we construct a complex with a morphism, and show its relative obstruction property. See also Subsection 5.3.5 for another construction, which might be useful for simplification. In Subsection 5.3.3, we give a factorization which will be useful to construct obstruction theories of master spaces. In Section 5.4, we argue a relative obstruction theory for reduced L-sections. We need some modification to the construction in Section 5.3. In Section 5.5, we study a relative obstruction theory for parabolic structures. By pulling them together, we can construct obstruction theories of moduli stacks considered in this monograph, which is explained in Section 5.6. T. Mochizuki, Donaldson Type Invariants for Algebraic Surfaces: Transition 145 of Moduli Stacks, Lecture Notes in Mathematics 1972, DOI: 10.1007/978-3-540-93913-9 5, c Springer-Verlag Berlin Heidelberg 2009

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5 Obstruction Theories of Moduli Stacks and Master Spaces

Then, we construct obstruction theories of master spaces in Section 5.7. Once we have factorizations as in Subsections 5.1.3, 5.3.3 and 5.4.3, the construction is easy. We also obtain obstruction theories for some related stacks. In Subsections 5.7.6–5.7.7, we give only the statements in some easier cases for explanation. In Section 5.8, we investigate obstruction theories of the fixed point sets. In Section 5.9, we study equivariant obstruction theories of master spaces and the induced obstruction theory of the fixed point sets.

5.1 Deformation of Torsion-Free Sheaves 5.1.1 Construction of a Basic Complex Let U be any algebraic stack over k, and let E be a torsion-free U -coherent sheaf defined over U × X. Assume that we are given a locally free resolution V• = (V−1 → V0 ) of E on U × X, i.e., Vi are locally free sheaves of finite ranks, and we have a surjection V0 −→ E whose kernel is V−1 . The inclusion V−1 ⊂ V0 is denoted by f . We put as follows (see Subsection 2.1.5): g(V• ) := Hom(V• , V• )∨ [−1] Let W−1 and W0 be vector spaces over k such that rank Wi = rank Vi . We denote Wi ⊗ OX by Wi X . We put GL(W• ) := GL(W−1 ) × GL(W0 ). We have the natural right GL(W• )-action on the vector bundle N (W−1 X , W0 X ) (see Subsection 2.1.2) given by (g−1 , g0 ) · a = g0−1 ◦ a ◦ g−1 . Here gi ∈ GL(Wi ) and a ∈ N (W−1 X , W0 X ). The quotient stack is denoted by Y (W• ). Then, we obtain the classifying map Φ(V• ) : U × X −→ Y (W• ) over X associated to V• , which induces the morphism Φ(V• )∗ LY (W• )/X −→ LU ×X/X . As explained in Example 2.2, Φ(V• )∗ LY (W• )/X is represented by the complex g(V• )≤1 . We have the naturally defined morphism g(V• ) −→ g(V• )≤1 . Hence, we obtain a morphism g(V• ) −→ LU ×X/X . Let ωX denote the dualizing complex of X, and we set Ob(V• ) := RpX ∗ g(V• ) ⊗ ωX . Then, we have the naturally defined morphism ob(V• ) : Ob(V• ) −→ LU .

5.1 Deformation of Torsion-Free Sheaves

147

Lemma 5.1.1 The object Ob(V• ) and the morphism ob(V• ) depend only on E in the derived category D(U ), in the sense that they are independent of the choice of a resolution V• . Proof It is standard that g(V• ) depends only on E in the derived category D(U×X). (1) Hence, Ob(V• ) depends only on E in D(U ). Let V• be another resolution. We (1) would like to compare the two morphisms ob(V• ) and ob(V• ) in the derived category D(U ). We set (2)

V0

(1)

(2)

:= V0 ⊕ V0 ,

(2)

V−1 := ker(V0 (2)

−→ E).

(1)

(2)

We have the natural morphisms V• −→ V• and V• −→ V• . Hence, we have (2) only to compare the morphisms ob(V• ) and ob(V• ). Note that Vi is a subbundle (2) (2) (2) (2) (i, j = 0, −1) of Vi , i.e., we have the filtration V• ⊂ V• . Let Hom Vi , Vj (2)

denote the sheaf of OX -morphisms Vi

(2)

−→ Vj

naturally form a complex of sheaves Hom

preserving the filtrations. They

(2) (2) (V• , V• ).

We set

g(V• , V• ) := Hom (V• , V• )∨ [−1]. (2)

(2)

(2)

We have the following naturally defined quasi-isomorphisms: Hom(V• , V• ) ←−−−− Hom (V• , V• ) −−−−→ Hom(V• , V• ) (2)

(2)

(2)

(2)

They induce the following quasi-isomorphisms: γ1

γ2

(2)

(2)

g(V• ) −−−−→ g(V• , V• ) ←−−−− g(V• )

(5.1)

In D(U × X), the quasi isomorphism (5.1) is the same as the standard one. (2) (2) (2) Let Wi be vector spaces such that rank Wi = rank Vi (i = 1, 2). We (2) (2) (2) fix inclusions Wi ⊂ Wi . We denote Wi ⊗ OX by Wi X . We have the fil(2) (2) (2)

trations Wi X ⊂ Wi X . Let Hom (Wi X , Wj X ) be the sheaf of OX -morphisms (2)

(2)

Wi X −→ Wj X preserving the filtrations. The corresponding vector bundle is denoted by N (Wi X , Wj X ). We have the natural morphisms: (2)

(2)

(2) (2) (2) (2) N (Wi X , Wj X ) ←−−−− N Wi X , Wj X −−−−→ N Wi X , Wj X Let GL (Wi ) be the subgroup of GL(Wi ), which consists of the elements (2) (2) of GL(Wi ) preserving the filtrations. We have the natural right GL (W1 ) × (2) (2) (2) (2) GL (W2 )-action on N (Wj , Wl ). The quotient stack is denoted by Y (W• ). (2)

(2)

We have the naturally defined homomorphisms GL (Wi ) −→ GL(Wi ) and (2) GL (Wi ) −→ GL(Wi ). Hence, we have the following morphisms: (2)

(2)

Y (W• ) ←−−−− Y (W• ) −−−−→ Y (W• ) (2)

(2)

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5 Obstruction Theories of Moduli Stacks and Master Spaces

(2) (2) : U × X −→ From the tuple V• , V• , we obtain a morphism Φ V• , V• (2)

Y (W• ). Then, it can be shown by the argument in Subsection 2.3.2, that the (2) ∗ (2) LY (W (2) )/X is represented by g(V• , V• )≤1 . We also have complex Φ V• , V• • the following commutative diagram: Φ(V• )∗ LY (W• )/X −→ Φ(V• , V• )∗ LY (W (2) )/X ←− Φ(V• )∗ LY (W (2) )/X • • ↑ ↑ ↑ γ1 γ2 (2) (2) −→ g(V• , V• )≤1 ←− g(V• )≤1 g(V• )≤1 (2)

(2)

Then, we obtain the following commutative diagram: =

=

LU ×X/X −→ LU ×X/X ←− LU ×X/X ↑ ↑ ↑ (2) ∗ (2) ∗ ∗ Φ(V• ) LY (W• )/X −→ Φ(V• , V• ) LY (W (2) )/X ←− Φ(V• ) LY (W (2) )/X • • ↑ ↑ ↑ (2) (2) −→ g(V• , V• ) ←− g(V• ). g(V• ) (2) (2) We put Ob V• , V• := RpX ∗ g(V• , V• ) ⊗ ωX . We obtain the following diagram in D(U ): =

−−−−→

LU ⏐ ⏐

LU ⏐ ⏐

=

←−−−−

LU ⏐ ⏐

(2) (2) Ob(V• ) −−−−→ Ob V• , V• ←−−−− Ob(V• ) Thus we are done.

If E is a vector bundle of rank R, we may take V0 = E and V−1 = 0. In this case, the construction can be reworded as follows: We have the classifying map Φ(E) : U × X −→ XGL(R) associated to E. It induces an isomorphism: Hom(E, E) Φ(E)∗ LXGL(R)/X −→ LU ×X/X We set Ob(E) := RpX ∗ Hom(E, E) ⊗ ωX . Then, we obtain a morphism ob(E) : Ob(E) −→ LU/k .

5.1.2 The Trace-Free Part and the Diagonal Part We have the homomorphism GL(W• ) −→ Gm given by (f−1 , f0 ) −→ det(f−1 )−1 · det(f0 ).

5.1 Deformation of Torsion-Free Sheaves

149

It induces a morphism w : Y (W• ) −→ XGm . The composite w ◦ Φ(V• ) : U × X −→ XGm is equal to the classifying map of the determinant bundle det(E) det(V0 ) ⊗ det(V−1 )−1 , which is denoted by Φ(det(E)). Then, we obtain the following commutative diagram: ∗ Φ det(E) LXGm /X −−−−→ Φ(V• )∗ LY (W• )/X −−−−→ LU ×X/X ⏐ ⏐ ⏐ ⏐ O[−1]

i

−−−−→

g(V• )

Here, the map i : O[−1] −→ g(V• ) is given as follows: O −→ Hom(V0 , V0 ) ⊕ Hom(V−1 , V−1 ),

f −→ (f · idV0 , −f · idV−1 )

We have the trace map tr : g(V• ) −→ O[−1]: Hom(V0 , V0 ) ⊕ Hom(V−1 , V−1 ) −→ O,

(f0 , f−1 ) −→ tr(f0 ) + tr(f−1 )

We put Ker(tr) := g◦ (V• ), and gd (V• ) := Im(i). We have the decomposition g(V• ) = g◦ (V• ) ⊕ gd (V• ), and hence Ob(V• ) = Ob◦ (V• ) ⊕ Obd (V• ). The complexes g◦ (V• ) and Ob◦ (V• ) (resp. gd (V• ) and Obd (V• )) are called the trace-free part (resp. the diagonal part). E : U −→ M(1). We also The determinant bundle induces the morphism det have the following commutative diagram: U ×X ⏐ ⏐ E,X ! det

Φ(V• )

−−−−→ Y (W• ) ⏐ ⏐ w!

(5.2)

M(1) × X −−−−→ XGm Because w ◦ Φ(V• ) = Φ(det(E)), we obtain the following commutative diagram: LU ×X/X ⏐ ⏐

←−−−−

Φ(V• )∗ LY (W• )/X ⏐ ⏐

∗

∗ det E,X LM(1)×X/X ←−−−− Φ(det(E)) LXGm /X

←−−−− g(V• ) ⏐ ⏐ gd (V• )

Therefore, we obtain the following commutative diagram: LU ⏐ ⏐

←−−−− Ob(V• ) ⏐ ⏐

∗ LM(1) det E

←−−−− Ob (V• ) d

(5.3)

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5.1.3 Preparation for Master space We put A(W• ) := XGL(W0 ) . We have the naturally defined morphism Γ : Y (W• ) −→ A(W• ). We put Ψ (V• ) := Γ ◦ Φ(V• ), and then Ψ (V• )∗ LA(W• )/X is represented by the complex Hom(V0 , V0 )∨ [−1]. For these representatives, the natural morphism Ψ (V• )∗ LA(W• )/X −→ Φ(V• )∗ LY (W• )/X is represented by the inclusion of Hom(V0 , V0 ) to Hom(V0 , V0)⊕Hom(V−1 , V−1 ). We put h(V• ) := Hom(V0 , V• )∨ [−1] and ObG (V• ) := RpX ∗ h(V• )⊗ωX . Then, we obtain the following diagram: h(V• ) ⏐ ⏐ !

−−−−→

g(V• ) ⏐ ⏐ !

Ψ (V• )∗ LA(W• )/X −−−−→ Φ(V• )∗ LY (W• )/X −−−−→ LU ×X/X Therefore, we obtain the morphisms ObG (V• ) −→ Ob(V• ) −→ LU/k . Now, we assume the following condition (C) for E and V• : (C1) The higher cohomology groups H i X, E|{u}×X (i > 0) vanish for any point u ∈ U . (C2) We put V := pX ∗ (E). Then, we have V0 = p∗X V , and the morphism V0 −→ E is the same as the naturally induced one. We put B(W• ) := Spec(k)GL(W0 ) . We remark A(W• ) = X × B(W• ). Because V0 = p∗X V , we have rank W0 = rank V , and hence we have the classifying map Ψ (V ) : U −→ B(W• ) of V . We have the following commutative diagram: Φ(V• )

−−−−→ Y (W• ) −−−−→ ⏐ ⏐ Γ!

U ×X ⏐ ⏐ Ψ (V )X !

X ⏐ ⏐ !

(5.4)

=

B(W• ) × X −−−−→ A(W• ) −−−−→ X Thus, we obtain the following diagram on U × X: LU ×X/X ⏐ ⏐

←−−−− Φ(V• )∗ LY (W• )/X ←−−−− g(V• ) ⏐ ⏐ ⏐ ⏐

Ψ (V )∗X LB(W• )×X/X ←−−−− Ψ (V• )∗ LA(W• )/X ←−−−− h(V• )

(5.5)

5.1 Deformation of Torsion-Free Sheaves

151

Hence, we obtain the following diagram on U : LU ⏐ ⏐

←−−−− Ob(V• ) ⏐ ⏐

(5.6)

τ

1 −− ObG (V• ). Ψ (V )∗ LB(W• ) ←−−

It is easy to see that both ObG (V• ) and Ψ (V )∗ LB(W• ) are isomorphic to Hom(V , V )[−1] under the condition (C). Lemma 5.1.2 The morphism τ1 in (5.6) is an isomorphism. Proof The composite of the following naturally defined morphisms is an isomorphism: RpX ∗ Hom(V0 , V0 ) → Hom(V−1 , V0 ) ⊗ ωX −→ RpX ∗ Hom(V0 , V0 ) ⊗ ωX −→ Hom(V , V ) (5.7) Then, the claim of the lemma immediately follows.

5.1.4 Basic Complex on the Moduli Stack M(m, y) Let y ∈ H ∗ (X) be a Chern character of a coherent sheaf on X. Let H := Hy denote the polynomial associated to y. We take an H(m)-dimensional vector space Vm . We have the scheme Q◦ (m, y). (See Subsection 3.6.2.) We consider the universal quotient q u : p∗Q◦ (m,y) Vm,X −→ E u (m) defined over Q◦ (m, y) × X. We set V0u := p∗Q◦ (m,y) Vm,X ,

u V−1 := ker V0u −→ E u (m) .

u −→ V0u is denoted by f u . We put The inclusion V−1

V := Vm ⊗ OQ◦ (m,y) = pX ∗ V0u . We have the morphism π : Q◦ (m, y) −→ M(m, y) := Q◦ (m, y)/GL(Vm ). The latter is an open subset of a moduli stack of torsion-free sheaves of type y, determined by the condition Om . The descent of E u , V• and V with respect to the GL(Vm )-action are denoted by E u , V• and V . The sheaf E u has the universal property. We put W0 := Vm , and we take a vector space W−1 such that dim W−1 = H(m) − rank(y). Applying the result in Subsection 5.1.1, we obtain a complex Ob(V• ) with a morphism ob(V• ) : Ob(V• ) −→ LM(m,y)/k . We obviously have π ∗ Ob(V• ) = Ob(V•u ).

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Lemma 5.1.3 We have the following morphism of the distinguished triangles on Q◦ (m, y): u , f u ) −→ Hom(V , V ) −→ π ∗ Ob(V• )[1] π ∗ Ob(V• ) −→ Ob(V−1 ↓ ↓ ↓ ↓ ∗ ∗ π LM(m,y) −→ LQ◦ (m,y) −→ LQ◦ (m,y)/M(m,y) −→ π LM(m,y) [1].

Proof We use the notation in Subsection 2.4.3. We have the following commutative diagram: u Φ(V−1 ,f u )

Q◦ (m, y) −−−−−−−→ Y (W• )quo −−−−→ ⏐ ⏐ ⏐ ⏐ π0 ! ! M(m, y)

−−−−→

Y (W• )

X ⏐ ⏐ !

−−−−→ X

u The composite π0 ◦ Φ(V−1 , f u ) is Φ(V•u ). Thus, we obtain the following morphism of distinguished triangles on Q◦ (m, y) × X: a

u , f u )∗ LYquo (W• )/X −→ Φ(V•u )∗ LY (W• )/X −→ Φ(V−1 ↓ ↓

π ∗ LM(m,y)×X/X −→

LQ◦ (m,y)×X/X

−→

u , f u )∗ LYquo (W• )/Y (W• ) −→ Φ(V•u )∗ LY (W• )/X [1] Φ(V−1 ↓ ↓ ∗ −→ π LM(m,y)×X/X [1] LQ◦ (m,y)×X/M(m,y)×X

(5.8)

u Recall that Φ(V•u )∗ LY (W• )/X and Φ(V−1 , f u )∗ LYquo (W• )/X are represented by u u u g(V• )≤1 and g(V−1 , f )≤1 , respectively. It is easy to see that the morphism a is u , f u )≤1 . We set represented by the naturally defined morphism g(V•u )≤1 −→ g(V−1 u , f u) . k(V•u ) := Cone g(V•u ) −→ g(V−1

We obtain the following morphism of distinguished triangles on Q◦ (m, y) × X: u −→ g(V−1 , f u ) −→ k(V•u ) g(V•u ) ↓ ↓ ↓ π ∗ LM×X/X −→ LQ◦ (m,y)×X/X −→ LQ◦ (m,y)×X/M(m,y)×X

−→

g(V•u )[1] ↓

−→ π ∗ LM(m,y)×X/X [1]

(5.9)

5.1 Deformation of Torsion-Free Sheaves

153

Hence, we obtain the following morphism of distinguished triangles on Q◦ (m, y): Ob(V• ) −→ Ob(V−1 , f ) −→ RpX∗ k(V• ) ⊗ ωX −→ Ob(V• )[1] ↓ ↓ ↓ϕ ↓ π ∗ LM −→ LQ◦ (m,y) −→ LQ◦ (m,y)/M(m,y) −→ π ∗ LM(m,y) [1]. ∨ ∨ u u , f u ) = Hom V−1 [1], V•u [−1]. Recall g(V•u ) = Hom V•u , V•u [−1] and g(V−1 It is easy to observe that k(V•u ) is represented by the complex u , V0u ), Hom(V0u , V0u ) → Hom(V−1

where the first term stands at the degree 0. Under the identification, the morphism ϕ is induced by the identity of Hom(V0u , V0u ). Then, it is easy to check that RpX∗ (k(V• )) and LQ◦ (m,y)/M(m,y) are quasi-isomorphic to their 0-th cohomology sheaves Hom(V , V ), and that the morphism ϕ in the diagram is an isomorphism as in Lemma 5.1.2. Corollary 5.1.4 ob(V• ) gives an obstruction theory for M(m, y).

We also use the symbols Ob(m, y) and ob(m, y) to denote Ob(V• ) and ob(V• ).

5.1.5 Line Bundles Let Poin denote the Poincar´e bundle on Pic ×X. We have the classifying map Φ(Poin) : Pic ×X −→ XGm associated to Poin. We put g(Poin) := Φ(Poin)∗ LXGm /X , Ob(Poin) := RpX ∗ g(Poin) ⊗ ωX . We obtain a morphism ob(Poin) : Ob(Poin) −→ LPic on Pic. Because g(Poin) O[−1], we have an isomorphism in the derived category D(Pic): Ob(Poin) H 0 (X, O)∨ ⊗ OPic [−1] ⊕ H 1 (X, O)∨ ⊗ OPic [0] ⊕ H 2 (X, O)∨ ⊗ OPic [1]

(5.10)

We should remark that the decomposition is not canonical, but the morphism Ob(Poin) −→ LPic −→ H 1 (X, O)∨ ⊗ OPic [0] H1 Ob(Poin) induces the canonical decomposition: τ≥0 Ob(Poin) H 1 (X, O)∨ ⊗ OPic [0] ⊕ H 0 (X, O)∨ ⊗ OPic [−1] We also remark that the composite τ≤−1 Ob(Poin) −→ LPic is trivial.

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5.2 Relative Obstruction Theory for Orientations 5.2.1 Construction of a Complex We use the notation in Subsections 5.1.1–5.1.2. Let F1 : U1 −→ U be an Artin ∗ (E) over stack over U . Assume that we are given an orientation ρ of the sheaf F1X U1 × X. We have the morphism detE : U1 −→ Pic induced by det(E). We denote detE × idX : U1 × X −→ Pic ×X by detE,X , and then ρ : det∗E,X Poin det(E). Let M(1) denote a moduli stack of line bundles, i.e., M(1) = PicGm . Let π denote the projection Pic −→ M(1). We have the following commutative diagrams induced by det(E): det

E → U1 −−−− ⏐ ⏐ F1 !

detE,X

U1 × X −−−−→ ⏐ ⏐ F1X !

Pic ⏐ ⏐ π!

det

Pic ×X ⏐ ⏐ πX !

U × X −−−−→ M(1) × X −−−−→ XGm

E → M(1) U −−−−

The induced morphism U1 × X −→ XGm is Φ det(E) . Hence, we have the following diagram on U1 × X: LU1 ×X/X ⏐ ⏐

←−−−−

det∗E,X LPic ×X/X ⏐ ⏐

∗ ∗ LU ×X/X ←−−−− det∗E,X πX LM(1)×X/X ←−−−− Φ(det(E))∗ LXGm /X F1X

Therefore, we obtain the following diagram on U1 : LU1 ⏐ ⏐

←−−−−

det∗E LPic ⏐ ⏐

(5.11)

F1∗ LU ←−−−− det∗E π ∗ LM(1) ←−−−− Obd (V• ) We put Obrel (V• , ρ) := Cone Obd (V• ) −→ det∗E LPic . Then, we obtain the following commutative diagram: Obrel (V• , ρ)[−1] −−−−→ Obd (V• ) ⏐ ⏐ ⏐ ⏐ obrel (V• ,ρ)! ! LU1 /U [−1]

−−−−→

F1∗ LU

5.2 Relative Obstruction Theory for Orientations

155

Recall the diagram (5.3). Then, we obtain the following commutative diagram: Obrel (V• , ρ)[−1] −−−−→ Ob(V• ) ⏐ ⏐ ⏐ ⏐ ! ! LU1 /U [−1]

(5.12)

−−−−→ F1∗ LU

The following lemma can be shown by an argument similar to Lemma 5.1.1. Lemma 5.2.1 The diagram (5.12) depends only on (E, ρ) in the derived category D(U1 ). The following lemma is easy to show by using the argument in Subsection 5.1.5. Lemma 5.2.2 Obrel (V• , ρ) is (non-canonically) isomorphic to H 0 (X, O)∨ ⊗ OU1 [0] ⊕ H 2 (X, O)∨ ⊗ OU1 [2] . The composite of the morphisms τ≤−2 Obrel (V• , ρ) [−1] −→ Ob(V• ) −→ LU1 is trivial.

5.2.2 Relative Obstruction Property For any U -scheme g : T −→ U , let F1 (T ) denote the set of orientations of g ∗ E. Then, we obtain the functor F1 of the category of U -schemes to the category of sets. The functor F1 is representable by Or(E)∗ . (See Subsection 3.1.1 for the orientation bundle Or(E).) Let π denote the projection Or(E)∗ −→ U . On Or(E)∗ × X, we have the universal orientation ρu of π ∗ E. From the resolution V• and the orientation ρu , we obtain a morphism: obrel (V• , ρu ) : Obrel (V• , ρu ) −→ LOr(E)∗ /U Lemma 5.2.3 The morphism obrel (V• , ρu ) gives a relative obstruction theory for Or(E)∗ over U . Proof We have only to show that H0 (obrel (V• , ρu )) is an isomorphism. From the diagram (5.11), we obtain the following morphisms: ϕ1 ϕ2 Obrel (V• , ρu ) −−−−→ det∗E Cone π ∗ LM(1) −→ LPic −−−−→ LOr(E)∗ /U Because Or(E)∗ U ×M(1) Pic, the morphism ϕ2 is an isomorphism. Note H1 (det∗E LPic ) = 0. We have the following commutative diagram:

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5 Obstruction Theories of Moduli Stacks and Master Spaces

H0 (Obd ) ⏐ ⏐ a1 !

−−−−→ det∗E ΩPic −−−−→ H0 Obrel (V• , ρu ) ⏐ ⏐ ⏐ ⏐ a2 ! !

H0 (det∗E π ∗ LM(1) ) −−−−→ det∗E ΩPic −−−−→ −−−−→

det∗E LPic /M(1)

H1 (Obd ) ⏐ ⏐ a3 !

−−−−→ 0 ⏐ ⏐ !

(5.13)

−−−−→ det∗E π ∗ H1 (LM(1) ) −−−−→ 0 The morphisms ai (i = 1, 3) are isomorphisms, by applying Corollary 5.1.4 to M(1). Thus, a2 is an isomorphism and the lemma is proved.

5.3 Relative Obstruction Theory for L-Sections 5.3.1 Construction of a Complex We use the notation in Subsection 5.1.1. Let L be a line bundle on X. Let ∂L P0 L P• := P−1 −→ be a locally free resolution,where P 0 stands in the degree 0. We have the natural right GL(W• )-action on N Pi , Wj given by (g−1 , g0 ) · f = gj−1 ◦ f . It induces GL(W• )-actions on the varieties N P−1 , W0 X , X, N (W−1 X , W0 X ) ×X N (P0 , W0 X ) ×X N (P−1 , W−1 X ). The quotient stacks are denoted by Y0 (W• , P• ), Y1 (W• , P• ), and Y2 (W• , P• ), respectively. We have the equivariant map h : N (W−1 X , W0 X )×X N (P−1 , W−1 X )×X N (P0 , W0 X ) −→ N (P−1 , W0 X ) given by h(f, a−1 , a0 ) = f ◦ a−1 − a0 ◦ ∂L . Since ∂L|x is injective for any point x ∈ X, the map h is smooth. It induces a smooth morphism of Y2 (W• , P• ) to Y0 (W• , P• ). We also have the morphism Y1 (W• , P• ) −→ Y0 (W• , P• ) induced by the 0-section X −→ N (P−1 , W0,X ). We denote the fiber product Y1 ×Y0 Y2 by Y (W• , P• ). Let F2 : U2 −→ U be an algebraic stack over U . Assume that we are given an L-section φ : p∗U2 (L) −→ F2∗X (E), and a morphism of complexes φ• = (φ−1 , φ0 ) : p∗U2 (P• ) −→ F2∗X (V• )

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157

which induces φ in the cohomology level. Such a φ• is called a lift of φ. We set ∨ ∗ grel (V• , φ• ) := Hom p∗U2 P• , F2,X V• We have the naturally induced morphisms γ(φ• ) : grel (V• , φ• )[−1] −→ g(V• ) and γ(φ• )≤1 : grel (V• , φ• )[−1]≤1 −→ g(V• )≤1 . We have the map Φ(V• , φ• ) : U2 × X −→ Y (W• , P• ) which induces the classifying maps Φi (V• , φ• ) : U2 × X −→ Yi (W• , P• ). Lemma 5.3.1 Φ(V• , φ• )∗ LY (W• ,P• )/X is represented by Cone γ(φ• )≤1 . Proof We have the induced morphisms κi : Φ0 (V• , φ• )∗ LY0 (W• ,P• )/X −→ Φi (V• , φ• )∗ LYi (W• ,P• )/X ,

(i = 1, 2).

Since Y2 (W• , P• ) −→ Y0 (W• , P• ) is smooth, Φ(V• , φ• )∗ LY (W• ,P• )/X is isomorphic to the cone of the induced morphism Φi (V• , φ• )∗ LYi (W• ,P• )/X . Φ0 (V• , φ• )∗ LY0 (W• ,P• )/X −→ i=1,2

Let us look at Φi (V• , φ• )∗ LYi (W• ,P• )/X . We use the argument explained in Subsection 2.3.2. We will omit to denote p∗U2 and F2∗X . In the case i = 2, it is represented by the following complex: Hom(V0 , V−1 )⊕ −→ Hom(V−1 , P−1 ) ⊕ Hom(V0 , P0 ) (b, c−1 , c0 )

Hom(V0 , V0 ) ⊕ Hom(V−1 , V−1 )

−→ f ◦ b + φ0 ◦ c0 , −b ◦ f + φ−1 ◦ c−1 (5.14)

Here, the first term stands in the degree 0. In the case i = 0, it is represented by the following complex: Hom(V0 , P−1 ) −→ Hom V0 , V0 ⊕ Hom V−1 , V−1 , c2 −→ 0, 0 (5.15) Again, the first term stands in the degree 0. In the case i = 1, it is represented by the following: (5.16) 0 −→ Hom(V0 , V0 ) ⊕ Hom(V−1 , V−1 ) Here the term 0 stands in the degree 0. For the description (5.14) and (5.15), the degree 0-part of κ2 is given as follows: Hom(V0 , P−1 ) −→ Hom(V0 , V−1 ) ⊕ Hom(V−1 , P−1 ) ⊕ Hom(V0 , P0 ), a −→ φ−1 ◦ a, a ◦ f, −∂L ◦ a .

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The degree 1-part of κ2 is given by the identity. On the other hand, the morphism κ1 is the obvious one for the descriptions (5.15) and (5.16). Then, the claim of the lemma can be checked directly. We put g(V• , φ• ) := Cone γ(φ• ) . Then, we obtain the following commutative diagram: ∗ ∗ LU ×X/X g(V• ) −−−−→ Φ F2∗X V• LY (W• )/X −−−−→ F2,X ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! ! g(V• , φ• ) −−−−→ Φ(V• , φ• )∗ LY (W• ,P• )/X −−−−→

LU2 ×X/X

We obtain the following morphism of the distinguished triangles on U2 × X: −→ g(V• , φ• ) −→ grel (V• , φ• ) −→ g(V• )[1] g(V• ) ↓ ↓ ↓ ↓ F2∗X LU ×X/X −→ LU2 ×X/X −→ LU2 ×X/U ×X −→ F2∗X LU ×X/X [1]. We put Obrel V• , φ• := RpX∗ grel (V• , φ• ) ⊗ ωX . Then, we obtain the following diagram on U2 : Obrel V• , φ• [−1] −−−−→ Ob(V• ) ⏐ ⏐ ⏐ ⏐ (5.17) ! ! LU2 /U [−1]

−−−−→ F2∗X LU/k

Lemma 5.3.2 The diagram (5.17) depends only on (E, φ) in the derived category D(U2 ) . (1) (1) (1) (2) Proof Let V• , P• , φ• be another choice. We take the resolution V• of E as (2) (1) (2) (2) in the proof of Lemma 5.1.1. We put P0 = P0 ⊕P0 and P−1 = ker(P0 −→ L). (2) (2) (2) (1) ∗ The lift φ• : P• −→ F2,X V• is naturally obtained from the lifts φ• and φ• . (2)

(2)

We have the compatible inclusions P• −→ P• and V• −→ V• . Then, we can show the claim of the lemma by using filtered objects as in Lemma 5.1.1.

5.3.2 Relative Obstruction Property ∗ For any U -scheme g : T −→ U , let F(T ) denote the set of L-sections of gX E. Thus, we obtain the functor of the category of U -schemes to the category of sets. The functor F is representable by a U -scheme M (L). Let π : M (L) −→ U denote ∗ E. Assume the projection. On M (L) × X, we have the universal L-section φu of πX that we are given a locally free resolution P• of L for which we have a lift of φu : ∗ V• φ• : p∗M (L) P• −→ πX

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159

We obtain a morphism: obrel (V• , φ• ) : Obrel (V• , φ• ) −→ LM (L)/U Lemma 5.3.3 obrel V• , φ• gives a relative obstruction theory for M (L) over U . Proof We have only to show the claim on any sufficiently small open subsets U of ) is independent of the choice of P• and φ• . ThereM (L). Recall that obrel (V • , φ•−1 i fore, we may assume H X, P0 ⊗ Vl | {u}×X = 0 for i = 0, 1, l = 0, −1 and any u ∈ U. For simplicity of description, we set C0 := grel (V• , φ• ),

C1 := grel (V• , φ• )≤0 .

We have the obvious map C0 −→ C1 . By the argument in Subsection 2.3.2, it can be ∗ LY (W ,P )/Y (W ) . Thus, we have shown that the complex C1 represents Φ(V• , φ) • • • the natural morphism C1 −→ LU ×X/U ×X . We put Ob1 := RpX ∗ C1 ⊗ωX . Then, −→ Ob1 and ob1 : Ob1 −→ LU/U . we have the induced morphisms Obrel (V• , φ) We use the It is easy to see that the composite of them is the same as obrel (V• , φ). following lemma. −→ H0 Ob1 is an isomorphism, Lemma 5.3.4 The morphism H0 Obrel (V• , φ) −→ H1 Ob1 is surjective. and H1 Obrel (V• , φ) Proof We have the exact sequence of the complexes 0 −→ Hom V−1 , P0 [−1] −→ C0 −→ C1 −→ 0. By our choice of P• , we have H i X, P0−1 ⊗ V−1 | {u}×X = 0 for any u ∈ U. Then, the claim can be easily shown. Therefore, we have only to show that ob1 gives an obstruction theory for U over U . For that purpose, we have only to check the conditions (A1) and (A2) in Proposition 2.4.2. We use the argument in the proof of Lemma 2.4.11. Let T be a scheme with a morphism h : T −→ U. Then, we have the following commutative diagram, for any coherent sheaf J on T : φ −−−−→ Exti h∗ Ob1 , J Exti h∗ LU /U , J ⏐ ⏐ ⏐ ⏐ ! Exti h∗X LU ×X/U ×X , JX −−−−→ Exti h∗X C1 , JX

(5.18)

Let T be a scheme such that T is a closed subscheme of T whose corresponding ideal sheaf J is square-zero. Let h be a morphism T −→ U such that the restriction h |T is π ◦ h, where π denotes the projection U −→ U . We have the obstruction class o(h, h ) ∈ Ext1 h∗ LU /U , J .

160

We set

5 Obstruction Theories of Moduli Stacks and Master Spaces

hX := Φ(V• , φ• ) ◦ hX : T × X −→ Y (W• , P• )

h X := Φ(V• ) ◦ h X : T × X −→ Y (W• )

Since the complex C1 represents Φ(V• , φ• )∗ LY (W• ,P• )/Y (W• ) , we have the obstruction class h X ) ∈ Ext1 h∗X C1 , JX . o(

hX ,

By the functoriality, o(h, h ) is mapped to o(

hX ,

h X ) in the diagram (5.18). Hence, φ o(h, h ) = 0 implies that

hX is extendable to a morphism

h1,X : T × X −→ Y (W• , P• ), which is a lift of

h X . Hence, we obtain an L-section of h X∗ E. Then, we obtain an extension of h to a morphism h1 : T −→ U which is a lift of h , by the universal property of M (L). Thus, the condition (A1) is checked. The condition (A2) can also be checked easily, and the proof of Lemma 5.3.3 is finished.

5.3.3 Preparation for Obstruction Theory of Master Space We will use the notation in Subsections 5.1.3 and 5.3.1. We have the naturally de fined right GL(W0 )-action on N OX , W0,X . The quotient stack is denoted by A(W• , P• ). Assume that we are given a non-trivial morphism ι : OX −→ P0 . We have the induced morphisms Y1 (W• , P• ) −→ A(W• , P• ) and ΓL : Y (W• , P• ) −→ ∗ A(W• , P• ). From (F2X V• , φ• ) on U2 × X and ι, we obtain a morphism φ : ∨ ∗ ∗ ∗ V• . We have pU2 OX −→ F2X V0 . We put hrel (V• , φ• ) := Hom p∗U2 OX , F2X the induced map γ(φ) : hrel (V• , φ• )[−1] −→ h(V• ). We put Ψ (V• , φ• ) := ΓL ◦ Φ(V• , φ• ) that is the classifying map of F2∗X V0 , φ , and then Ψ (V• , φ• )∗ LA(W• ,P• )/X is represented by Cone γ(φ)≤1 . The induced morphism Ψ (V• , φ• )∗ LA(W• ,P• )/X −→ Φ(V• , φ• )∗ LY (W• ,P• )/X is represented by the naturally given morphism Cone γ(φ)≤1 −→ Cone γ(φ• )≤1 . We put h(V• , φ• ) := Cone γ(φ) . We obtain the following commutative diagram: h(V• , φ• ) −−−−→ g(V• , φ• ) −−−−→ LU2 ×X/X ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ −−−−→ g(V• ) −−−−→ F2∗X LU ×X/X We put ObG rel (V• , φ• ) := RpX ∗ hrel (V• , φ• ) ⊗ ωX . We have the following commutative diagram on U2 : h(V• )

5.3 Relative Obstruction Theory for L-Sections

161

LU2 /U [−1] ←−−−− Obrel (V• , φ• )[−1] ←−−−− ObG rel (V• , φ• )[−1] ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! ! F2∗ LU/k

←−−−−

Ob(V• )

←−−−−

(5.19)

ObG (V• )

Now, we assume the condition (C) in Subsection 5.1.3. We have the natural right GL(W0 )-action on N (k, W0 ) = W0 . The quotient stack is denoted by B(W• , P• ). We have the natural isomorphism JL : B(W• , P• ) × X −→ A(W• , P• ). From φ : p∗U2 OX −→ V0 , we obtain the classifying map Ξ(V• , φ) : U2 −→ B(W• , P• ). Note that the composite of Ξ(V• , φ)X and JL equals Ψ (V• , φ• ). Therefore, we have the following commutative diagram: U2 × X −−−−→ B(W• , P• ) × X −−−−→ A(W• , P• ) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! ! U × X −−−−→

B(W• ) × X

−−−−→

A(W• )

Thus, we obtain the following commutative diagram: F2∗ LU/k ⏐ ⏐

←−−−−

Φ(V• )∗ LB/k ⏐ ⏐

τ

1 ←−− −−

ObG (V• ) ⏐ ⏐

τ2 LU2 /U [−1] ←−−−− Ξ(V• , φ)∗ LB(P• )/B [−1] ←−− −− ObG rel (V• , φ• )[−1]

Here, W• is omitted to denote, and see Subsection 5.1.3 for τ1 . Lemma 5.3.5 The morphism τ2 is an isomorphism. Proof The complex hrel (V• , φ• ) is quasi-isomorphic to Hom(V0 , O) → Hom(V−1 , O) , where the first term stands at the degree 0. And, the degree 0-part of the morphism hrel (V• , φ• ) −→ Ξ(V, φ)∗X LB(W• ,P• )/B(W• ) Hom(V0 , O) is given by the iden tity of Hom(V0 , O). Hence, the claim can be shown as in Lemma 5.1.2.

5.3.4 Preparation for Proposition 6.2.1 We have the morphism of the complexes f : grel (V• , φ• ) −→ OU2 ×X given as follows: ∗ ∗ V0 , p∗U2 P0 ) ⊕ Hom(F2X V−1 , p∗U2 P−1 ) −→ OU2 ×X , Hom(F2X

(a0 , a−1 ) −→ tr φ0 ◦ a0 + tr φ−1 ◦ a−1

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5 Obstruction Theories of Moduli Stacks and Master Spaces

It is easy to check that the following diagram is commutative: grel (V• , φ• )[−1] −−−−→ ⏐ ⏐ f! OU2 ×X [−1]

g(V• ) ⏐ ⏐ tr!

id

−−−−→ OU2 ×X [−1]

It induces the following commutative diagram: Obrel (V• , φ• )[−1] −−−−→ Ob(V• ) ⏐ ⏐ ⏐ ⏐ ! ! OU2 [−1]

(5.20)

−−−−→ OU2 [−1]

5.3.5 Another Construction ∗ Let φ : p∗U2 L −→ F2X E be an L-morphism. We will explain a construction of a complex Obrel (V• , φ) with a morphism obrel (V• , φ) : Obrel (V• , φ) −→ LU2 /U . ∗ V• of φ, then obrel (V• , φ) If we have a resolution P• and a lift φ• : p∗U2 P• −→ F2X is the same as obrel (V• , φ• ) in the derived category.

Remark 5.3.6 Since we do not have to take a resolution P• , this construction may give a simplification. Let Z(V• , L) be the quotient stack of N (p∗U L, V0 ) by the naturally induced action of N (p∗U L, V−1 ). From φ, we obtain the following commutative diagram: Φ(φ)

U2 × X −−−−→ Z(V• , L) ⏐ ⏐ ⏐ ⏐ ! ! =

U × X −−−−→ U × X By an argument in Subsection 2.3.2, we can show that Φ(φ)∗ LZ(V• ,L)/U ×X is rep∨ resented by grel (V• , φ) := Hom L, V• . Hence, we obtain a morphism r1 : grel (V• , φ) −→ LU2 ×X/U ×X We set Obrel (V• , φ) := RpX∗ grel (V• , φ) ⊗ ωX . Then, we obtain a morphism obrel (V• , φ) : Obrel (V• , φ) −→ LU2 /U . ∗ V• , which Now, assume that we are given a morphism φ• : p∗U2 P• −→ F2X ∗ induces φ : L −→ F2X E. By construction, we have the natural quasi-isomorphism

5.3 Relative Obstruction Theory for L-Sections

163

s : grel (V• , φ• ) = Hom(P• , V• )∨ −−−−→ grel (V• , φ) = Hom(L, V• )∨ Hence, we have a quasi-isomorphism λ : Obrel (V• , φ• ) −→ Obrel (V• , φ). Lemma 5.3.7 The composite obrel (V• , φ) ◦ λ is the same as obrel (V• , φ• ). Proof Let ∂V denote the morphism V−1 −→ V0 . We have the morphism h : Z2 (V• , P• ) := N (p∗U P−1 , V−1 )×U ×X N (p∗U P0 , V0 ) −→ Z0 := N (p∗U P−1 , V0 ) given by h(a−1 , a0 ) := ∂V ◦a−1 −a0 ◦∂L . Set Z1 (V• , P• ) := U ×X. The 0-section gives Z1 (V• , P• ) −→ Z0 (V• , P• ). We put Z(V• , P• ) := Z1 ×Z0 Z2 . From φ• , we obtain the following commutative diagram: • ) Φ (V• ,φ

U2 × X −−−−−−→ Zi (V• , P• ) −−−−→ Yi (W• , P• ) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! ! U ×X

−−−−→

U ×X

−−−−→

Yi (W• )

Hence, we have the following:

r2 LU2 ×X/U ×X Φ(V• , φ• )∗ LY (W• ,P• )/Y (W• ) −→ Φ (V• , φ• )∗ LZ(W• ,P• )/U ×X −→

Note that Φ (V• , φ• )∗ LZ(W• ,P• )/U ×X is represented by grel (V• , φ• )≤0 . Hence, we obtain the induced morphism r2 : grel (V• , φ• ) −→ LU2 ×X/U ×X . We have only to show that r1 ◦ s = r2 . We have the naturally defined action of N (P0 , V−1 ) on Z(V• , P• ) The quotient stack is denoted by Z(V• , P• ). We have the following morphisms: U2 × X ⏐ ⏐ !

−−−−→ Z(V• , L) ⏐ ⏐ !

Z(V• , P• ) −−−−→ Z(V• , P• ) It induces the following morphisms: LU2 ×X/U ×X ⏐ e1 ⏐

r

2 ←−− −− grel (V• , φ) ⏐ ⏐

2 ∗ LZ(V ,P )/U ×X ←−e− Φ (V• , φ) −− grel (V• , φ• ) • •

By construction, we have e2 ◦ e1 = r1 . Thus, we are done.

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5 Obstruction Theories of Moduli Stacks and Master Spaces

5.4 Relative Obstruction Theory for Reduced L-Sections 5.4.1 Construction of a Complex We use the notation in Subsection 5.3.1. The weight (−1)-action of a torus Gm on Pi induces the Gm -action on Yi (W• , P• ) (i = 0, 1, 2) and Y (W• , P• ). The quotient stacks are denoted by Yi (W• , [P• ]) (i = 0, 1, 2) and Y (W• , [P• ]). We have the morphism π3 : Y (W• , [P• ]) −→ XGm induced by Y (W• , P• ) −→ X. Let F3 : U3 −→ U be a U -stack with a reduced L-section [φ] : p∗U3 (L) ⊗ p∗X (M ) −→ F3∗X (E), where M denotes a line bundle on U3 . Let ΦM : U3 −→ Spec(k)Gm denote the classifying map for M . Notation 5.4.1 A morphism g : U3 −→ T and ΦM induce the morphism U3 −→ T × Spec(k)Gm , which is denoted by gM in the following argument. Assume that we are given a morphism of complexes [φ• ] = [φ−1 ], [φ0 ] : p∗U3 (P• ) ⊗ p∗X M −→ F3∗X (V• ) which gives [φ] in the cohomology level. Such [φ• ] is called a lift of [φ]. We set = Hom p∗ P• ⊗ p∗ M, F ∗ V• ∨ g rel (V• , [φ]) U3 X 3,X We have the naturally induced morphisms γ[φ• ] : g rel (V• , [φ• ])[−1] −→ g(V• ) and γ[φ• ]≤1 : g rel (V• , [φ• ])[−1]≤1 −→ g(V• )≤1 . We have the classifying map Φ V• , [φ• ] : U3 × X −→ Y (W• , [P• ]). We consider the trivial Gm -actions on U × X, Y (W• ) and X. The quotient stacks are denoted by (U × X)Gm , Y (W• )Gm and XGm respectively. We have the following commutative diagram: U3 × X ⏐ ⏐ F3XM ! (U × X)Gm

• ]) Φ(V• ,[φ

−−−−−−→ Y (W• , [P• ]) −−−−→ XGm ⏐ ⏐ ⏐ ⏐ π3 ! ! −−−−→

Y (W• )Gm −−−−→ XGm

∗ The composite π3 ◦ Φ(V• , [φ• ]) is the same as Φ(F3,X V• )M . The following lemma can be shown by an argument in the proof of Lemma 5.3.1. Lemma 5.4.2 Φ(V• , [φ• ])∗ LY (W• ,[P• ])/XGm is represented by Cone γ[φ• ]≤1 . We put g (V• , [φ• ]) := Cone γ[φ• ] . We obtain the following commutative diagram:

5.4 Relative Obstruction Theory for Reduced L-Sections

165

∗ ∗ g(V• ) −→ Φ F3∗X V• M LY (W• )Gm /XGm −→ F3,X,M L(U ×X)Gm /XGm ↓ ↓ ↓ g (V• , [φ• ]) −→ Φ(V• , φ• )∗ LY (W• ,[P• ])/XGm −→ LU3 ×X/XGm ∗ ∗ We remark F3,X,M L(U ×X)Gm/XGm and Φ F3∗X V• M LY (W• )Gm/XGm are naturally ∗ isomorphic to F3∗X,M LU ×X/X and Φ F3∗X V• LY (W• )/X , respectively. Then, we obtain the following morphism of distinguished triangles on U3 × X: g(V• ) ⏐ ⏐ !

−−−−→ g (V• , [φ• ]) −−−−→ ⏐ ⏐ !

g rel (V• , [φ• ]) ⏐ ⏐ !

∗ F3,X,M L(U ×X)Gm /XGm −−−−→ LU3 ×X/XGm −−−−→ LU3 ×X/(U ×X)Gm

−−−−→

g(V• )[1] ⏐ ⏐ !

(5.21)

∗ L(U ×X)Gm /XGm [1]. −−−−→ F3,X,M

We put Ob rel V• , [φ• ] := RpX ∗ g rel (V• , [φ• ])⊗ωX . We obtain the following diagram on U3 : Ob rel V• , [φ• ] [−1] −−−−→ LU3 /UGm [−1] ⏐ ⏐ ⏐ ⏐ (5.22) ! ! Ob(V• )

∗ ∗ −−−−→ F3,M LUGm /k −−−−→ F3,M LUGm /U

∗ LUGm /U as the composite We obtain the morphism Ob rel V• , [φ• ] [−1] −→ F3,M of the morphisms in the above diagram. We put ∗ LUGm /U [1] [−1]. Obrel (V• , [φ• ]) := Cone Ob rel (V• , [φ• ]) −→ F3,M We obtain the following morphism of distinguished triangles: ∗ LUGm /U −−−−→ Obrel (V• , [φ• ]) −−−−→ Ob rel (V• , [φ• ]) F3,M ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! ! ∗ LUGm /U −−−−→ F3,M

LU3 /U

−−−−→

LU3 /UGm

∗ −−−−→ F3,M LUGm /U [1] ⏐ ⏐ ! (5.23) ∗ −−−−→ F3,M LUGm /U [1]

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5 Obstruction Theories of Moduli Stacks and Master Spaces

Therefore, we obtain the following commutative diagram on U3 : Obrel V• , [φ• ] [−1] −−−−→ Ob(V• ) ⏐ ⏐ ⏐ ⏐ • ])! obrel (V• ,[φ !

(5.24)

−−−−→ F3∗ LU .

LU3 /U [−1]

Lemma 5.4.3 The diagram (5.24) depends only on (E, [φ]). Proof By an argument in the proof of Lemma 5.3.2, we can show that the diagram (5.22) is independent of the choice of (V• , [P• ], [φ• ]). Then, the diagram (5.24) is also independent.

5.4.2 Relative Obstruction Property For any U -scheme g : T −→ U , let F(T ) denote the set of reduced L-sections ∗ E, which are non-trivial everywhere. Thus, we obtain the functor F of the of gX category of U -schemes to the category of sets. It is representable by a U -scheme M [L]. Let π : M [L] −→ U denote the projection. We have the line bundle I on ∗ E over M [L], and the universal reduced L-section [φu ] : p∗X I ⊗ p∗M [L] L −→ πX M [L] × X. Assume that we have a locally free resolution P• of L for which we ∗ have a lift [φ• ] : p∗X I ⊗ p∗M [L] P• −→ πX V• of [φu ]. Then, we obtain a morphism obrel V• , [φ• ] : Obrel V• , [φ• ] −→ LM [L]/U . We also have a complex Ob rel V• , [φ• ] with a morphism Ob rel V• , [φ• ] −→ LM [L]/UGm . Let M (L) be as in Subsection 5.3.2. It is easy to observe that M (L) is isomorphic to I ∗ . We have the smooth projection π2 : M (L) −→ M [L]. The pull back of [φ• ] via π2 X is denoted by φ• . Lemma 5.4.4 We have the following commutative diagram: π2∗ Ob rel (V, [φ• ]) −−−−→ Obrel (V, φ• ) ⏐ ⏐ ⏐ ⏐ ! !

π2∗ LM [L]/UGm

−−−−→

LM (L)/U

Proof We have the following commutative diagram:

(5.25)

5.4 Relative Obstruction Theory for Reduced L-Sections • ) Φ(V• ,φ

M (L) × X −−−−−−→ ⏐ ⏐ !

ψ

Y (W• , P• ) −−−−→ ⏐ ⏐ !

• ]) Φ(V• ,[φ

167

Y (W• ) ⏐ ⏐ !

−−−−→

X ⏐ ⏐ !

ψ

M [L] × X −−−−−−→ Y (W• , [P• ]) −−−−→ Y (W• )Gm −−−−→ XGm Therefore, we obtain the following: ←− Φ(φ)∗ LY (W• ,P• )/Y (W• ) ←− grel (V• , φ• ) LM (L)×X/U ×X ↑ ↑ ↑ ∗ ∗ LM [L]×X/U ×XGm ←− Φ([φ])∗ LY (W• ,[P• ])/Y (W• )Gm ←− π2,X g rel (V• , [φ• ]) π2,X Then the claim is clear.

We have the morphisms Obrel (V• , φ• ) −→ LM (L)/U −→ LM (L)/M [L] , where the latter is the projection. Lemma 5.4.5 We have the following distinguished triangle: π2∗ Obrel V• , [φ• ] −→ Obrel (V• , φ• ) −→ LM (L)/M [L] −→ π2∗ Obrel (V• , [φ• ])[1] (5.26) In particular, π2∗ Obrel (V• , [φ• ]) Cone Obrel (V• , φ• ) −→ LM (L)/M [L] [1]. Moreover, we have the following morphism of distinguished triangles: π2∗ Obrel V• , [φ• ] −→ Obrel (V• , φ• ) −→ LM (L)/M [L] −→ π2∗ Obrel (V• , [φ• ])[1] ↓ ↓ ↓ ↓ −→ LM (L)/U −→ LM (L)/M [L] −→ π2∗ LM [L]/U [1]. π2∗ LM [L]/U (5.27) Proof Let F3,I denote the morphism M [L] −→ UGm , induced by I. Then, we have the following isomorphism of the distinguished triangles on M (L): ∗ LUGm /U [1] −→ π2∗ LM [L]/U [1] π2∗ LM [L]/U −→ π2∗ LM [L]/UGm −→ π2∗ F3,I =↓ ↓ ↓ =↓ LM (L)/M [L] −→ π2∗ LM [L]/U [1] π2∗ LM [L]/U −→ LM (L)/U −→ (5.28)

We obtain the distinguished triangle (5.26) from (5.25) and (5.28). We also obtain (5.27). Lemma 5.4.6 obrel V• , [φ• ] gives a relative obstruction theory for M [L] over U . Proof The complex LM (L)/M [L] is quasi-isomorphic to its 0-th cohomology sheaf, and we know that obrel (V• , φ• ) gives an obstruction theory of M (L) over U , according to Lemma 5.3.3. Then, we obtain the claim of the lemma from the diagram (5.27).

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5 Obstruction Theories of Moduli Stacks and Master Spaces

5.4.3 Preparation for Obstruction Theory of Master Space We will use the notation in Subsections 5.1.3, 5.3.3 and 5.4.1. We have the weight (−1)-action of a torus Gm on OX . It induces a Gm -action on A(W• , P• ). The quotient stack is denoted by A(W• , [P• ]). We have the induced morphism Γ[L] : Y (W• , [P• ]) −→ A(W• , [P• ]). ∗ V• , [φ• ]) on U3 × X and [ ι], we obtain the morphism From (F3,X ∗ V0 . [φ] : p∗U3 OX ⊗ p∗X M −→ F3,X

∨ ∗ We put h rel (V• , [φ• ]) := Hom p∗U3 OX ⊗ p∗X M, F3X V• . Then, we have the naturally induced maps γ([φ]) : h rel (V• , [φ• ])[−1] −→ h(V• ) and γ([φ])≤1 : h rel (V• , [φ• ])[−1]≤1 −→ h(V• )≤1 . We put Ψ (V• , [φ• ]) := Γ[L] ◦ Φ(V• , [φ• ]), which give the classifying map of ∗ V , [φ]). Then, Ψ (V• , [φ• ])∗ LA(W• ,[P• ])/XGm is represented by the the tuple (F3,X 0 complex Cone γ([φ])≤1 . The naturally defined morphism Cone γ([φ])≤1 −→ Cone γ([φ• ])≤1 represents Ψ (V• , [φ• ])∗ LA(W• ,[P• ])/XGm −→ Φ(V• , [φ• ])∗ LY (W• ,[P• ])/XGm . We set h (V• , [φ• ]) := Cone γ([φ]) . Then, we obtain the following commutative diagram: h (V• , [φ• ]) −−−−→ g (V• , [φ• ]) −−−−→ LU3 ×X/XGm ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ h(V• )

−−−−→

g(V• )

−−−−→ F3∗X LU ×X/X

G We put Ob rel (V• , [φ• ]) := RpX ∗ h rel (V• , [φ• ]) ⊗ ωX . Then, we have the following diagram on U3 : G (V• , [φ• ])[−1] LU3 /UGm [−1] ←−−−− Ob rel (V• , [φ• ])[−1] ←−−−− Ob rel ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ (5.29) ! ! !

F3∗ LU/k

←−−−−

Ob(V• )

←−−−−

ObG (V• )

We obtain the morphisms G ∗ Ob rel (V• , [φ• ])[−1] −→ LU3 /UGm [−1] −→ F3,M LUGm /U .

5.4 Relative Obstruction Theory for Reduced L-Sections

169

Let ObG rel (V• , [φ• ]) denote the cone of the composite. Then, we obtain the following diagram: LU3 /U [−1] ←−−−− Obrel (V• , [φ• ])[−1] ←−−−− ObG rel (V• , [φ• ])[−1] ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! ! F3∗ LU/k

←−−−−

Ob(V• )

←−−−−

(5.30)

ObG (V• )

Now, we assume the condition (C) in Subsection 5.1.3. We have the weight (−1)-action of Gm on the one dimensional vector space k. It induces the Gm action on B(W• , P• ). The quotient stack is denoted by B(W• , [P• ]). We have the natural isomorphism J[L] : B(W• , [P• ]) × X −→ A(W• , [P• ]). From [φ] : p∗X M ⊗p∗U3 OX −→ V0 , we obtain the morphism Ξ(V• , [φ]) : U3 −→ B(W• , [P• ]). Note that the composite of Ξ(V• , [φ])X and J[L] is the same as Ψ (V• , [φ]). Therefore, we have the following commutative diagram: U3 × X ⏐ ⏐ ! (U × X)Gm

−−−−→ B(W• , [P• ]) × X −−−−→ A(W• , [P• ]) ⏐ ⏐ ⏐ ⏐ ! ! −−−−→ B(W• ) × X G −−−−→ A(W• )Gm m

We obtain the following commutative diagram: ∗ F3,M LUGm /kGm ←− ↑

Φ(V• )∗M LBGm /kGm ↑

←−

ObG (V• ) ↑

τ3 G Ob rel (V• , [φ• ])[−1] LU3 /UGm [−1] ←− Ξ(V• , [φ])∗ LB([P• ])/BGm [−1] ←−

Here, W• is omitted to denote. The following lemma is similar to Lemma 5.3.5. Lemma 5.4.7 The morphism τ3 is an isomorphism.

By the standard modification, we obtain the following commutative diagram: F3∗ LU/k ←− ↓

Φ(V• )∗ LB/k ↓

←−

ObG (V• ) ↓ τ3 G ∗ LU3 /U [−1] ←− Ξ(V• , [φ]) LB([P• ])/B [−1] ←− Obrel (V• , [φ• ])[−1] The following lemma immediately follows from the previous lemma. Lemma 5.4.8 The morphism τ3 is an isomorphism.

(5.31)

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5 Obstruction Theories of Moduli Stacks and Master Spaces

5.4.4 Preparation for Proposition 6.2.1 We use the notation in Subsection 5.4.1. As a preparation for the proof of Proposition 6.2.1, let us look at the morphism ϕ1 : Ob rel (V• , [φ• ]) −→ F3∗ LUGm /U [1] on U3 , more closely. We recall Φ(V, [φ• ])∗ LY (W• ,[P• ])/Y (W• )Gm g rel (V• , [φ• ])≤0 . Lemma 5.4.9 We have the following commutative diagram: ∗ V• )∗M LY (W• )Gm /Y (W• ) [1] Φ(V, [φ• ])∗ LY (W• ,[P• ])/Y (W• )Gm −−−−→ Φ(F3,X ⏐ ⏐ ⏐ ⏐

g rel (V• , [φ• ])

f

−−−−→

O

Here, the morphism f is given as follows: ∗ ∗ V0 , p∗U P0 ⊗ p∗X M ⊕ Hom F3,X V−1 , p∗U P−1 ⊗ p∗X M −→ O, Hom F3,X (a0 , a−1 ) −→ tr [φ0 ] ◦ a0 + tr [φ−1 ] ◦ a−1 Proof It follows from Corollary 2.3.4.

The morphism f induces the following morphism: Ob rel (V• , [φ• ]) = RpX ∗ grel (V• , [φ• ]) ⊗ ωX −→ RpX ∗ p∗U3 ωX −→ OU3 It is easy to check that the composite is equal to ϕ1 .

5.5 Relative Obstruction Theory for Parabolic Structures 5.5.1 Construction of a Complex We use the notation in Subsection 5.1.1. Let D be a smooth hypersurface of X. Let F4 : U4 −→ U be a U -scheme, and let F∗ be a quasi-parabolic structure of (h) ∗ F4∗X E at D. We denote the kernel of F4,X V0 | D −→ Cokh−1 (E) by VD , which (1)

(2)

(l+1)

are locally free. The filtered vector bundle VD ⊃ VD ⊃ · · · ⊃ VD is denoted by VD∗ . We put W (1) := W0 and W (l+1) := W−1 . Let W (h) (h = 2, . . . , l) be vector (h) spaces over k such that dim W (h) = rank VD . We denote W (h) ⊗ OD and Wi ⊗ (h) OD by WD and Wi D , respectively. We have the natural right GL(W• )-action on N (W−1,D , W0,D ) given by (g0 , g−1 ) · f = g0−1 ◦ f ◦ g−1 .

5.5 Relative Obstruction Theory for Parabolic Structures

171

The quotient stack is denoted by YD (W• ). Similarly, we have the natural right (h+1) (h) , WD ) given by GL(W (h) ) × GL(W (h+1) )-action on N (WD (g (h) , g (h+1) ) · f = (g (h) )−1 ◦ f ◦ g (h+1) . (h+1) .l .l+1 (h) , WD , where the We obtain a right h=1 GL(W (h) )-action on h=1 N WD latter fiber product is taken over D. The quotient stack is denoted by YD (W• , W ∗ ). Composition of the morphisms induces the map l

(h+1) (h) N WD , WD −→ N (W−1 D , W0 D ).

h=1

It induces the morphism YD (W• , W ∗ ) −→ YD (W• ). We have the classifying map ΦD (V• , F∗ ) : U4 × D −→ YD (W• , W ∗ ) over D associated to the tuple VD∗ . We also have the classifying map Φ(V•|D ) : U × D −→ YD (W• ) associated to V•|D . Thus, we obtain the following diagram on U4 × D: LU4 ×D/D ⏐ ⏐ ∗ F4,D LU ×D/D

←−−−− ΦD (V• , F∗ )∗ LYD (W• ,W ∗ )/D ⏐ ⏐ ∗ ←−−−− Φ F4∗D (V•|D ) LYD (W• )/D .

We use the notation in Subsection 2.1.6. We put gD (V• , F∗ ) := C1 (VD∗ , VD∗ )∨ [−1],

grel (V• , F∗ ) := C2 (VD∗ , VD∗ )∨ [−1].

We also put g(V•|D ) := Hom(V•|D , V•|D )∨ [−1]. We have the morphism γD : g(V•|D ) −→ gD (V• , F∗ ) induced by ϕ given in (2.3). It is easy to see that grel (V• , F∗ ) is quasi-isomorphic to Cone(γD ). By using an argument explained in Subsection 2.3.2, we can show that ΦD (V• , F∗ )∗ LYD (W• ,W ∗ )/D ,

and Φ(V•|D )∗ LYD (W• )/D

are represented by gD (V• , F∗ )≤1 and g(V•|D )≤1 . Under the identification, the natural morphism ΦD (V• , F∗ )∗ LYD (W• ,W ∗ )/D −→ Φ(V• | D )∗ LYD (W• )/D is given by the restriction γD≤1 . We obtain the following commutative diagram: g(V•|D ) ⏐ ⏐ γD !

−−−−→

g(V•|D )≤1 ⏐ γD≤1 ⏐ !

−−−−→ F4∗ LU/k ⏐ ⏐ !

gD (V• , F∗ ) −−−−→ gD (V• , F∗ )≤1 −−−−→ LU4 /k

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5 Obstruction Theories of Moduli Stacks and Master Spaces

We obtain the following morphism of distinguished triangles on U4 × D: g(V•|D ) −→ gD (V• , F∗ ) −→ grel (V•|D , F∗ ) −→ g(V•|D )[1] ↓ ↓ ↓ ↓ F4∗X LU ×D/D −→ LU4 ×D/D −→ LU4 ×D/U ×D −→ F4∗X LU ×D/D Let ωD denote the dualizing complex of D. We put Obrel (V• , F∗ ) := RpD ∗ grel V• , F∗ ⊗ ωD , Ob(V• | D ) := RpD ∗ g(V• | D ) ⊗ ωD . We obtain the following commutative diagram: obrel (V• ,F∗ ) Obrel V• , F∗ [−1] −−−−−−−−→ LU4 /U [−1] ⏐ ⏐ ⏐ ⏐ ! ! Ob(V•|D )

ob(V•|D )

−−−−−−→

(5.32)

F4∗ LU/k .

We have the following exact sequence of complexes on U4 × X: 0 −→ g(V• ) ⊗ ωX −→ g(V• ) ⊗ ωX (D) −→ g(V•|D ) ⊗ ωD [1] −→ 0. Thus, we obtain a morphism RpD ∗ g(V•|D ) ⊗ ωD −→ RpX ∗ g(V• ) ⊗ ωX , that is, η : Ob(V•|D ) −→ Ob(V• ). Recall we have the morphism ob(V• ) : Ob(V• ) −→ F4∗ LU . Hence, we obtain the composite ob(V• ) ◦ η : Ob(V•|D ) −→ F4∗ LU . On the other hand, we have the morphism ob(V•|D ) : Ob(V•|D ) −→ F4∗ LU in the diagram (5.32). Lemma 5.5.1 Two morphisms ob(V• )◦η and ob(V• | D ) are the same in the derived category. ∗ LU ×D/D is the restriction of Proof It is easy to observe that g(V•|D ) −→ F4,D ∗ g(V• ) −→ F4,X LU ×X/X to U4 × D. Then, the coincidence ob(V• ) ◦ η = ob(V•|D ) follows from the compatibility of of the trace maps for the dualizing complexes ωX and ωD . (See [1], for example).

Thus, we obtain the following commutative diagram: Obrel V• , F∗ [−1] −−−−→ Ob(V•|D ) −−−−→ Ob(V• ) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ obrel (V• ,F∗ )! ! ! LU4 /U [−1]

−−−−→ F4∗ LU/k −−−−→ F4∗ LU/k .

(5.33)

5.5 Relative Obstruction Theory for Parabolic Structures

173

The following lemma can be shown by using filtered objects as in the proof of Lemma 5.1.1. Lemma 5.5.2 The diagram (5.33) depends only on E, F∗ (E) .

5.5.2 Relative Obstruction Property For any U -scheme g : T −→ U , let F(T ) denote the set of the parabolic structure ∗ E of a fixed type. Thus, we obtain the functor F of the category of U -schemes of gX to the category of sets. We can show that F is representable by a U -scheme M by using quot schemes. Let π : M −→ U denote the projection. On M × X, we have the universal ∗ E. From the resolution V• and F∗u , we obtain a comparabolic structure F∗u of πX u plex Obrel (V• , F∗ ) with a morphism obrel (V• , F∗u ) : Obrel (V• , F∗u ) −→ LM/U . Lemma 5.5.3 obrel (V• , F∗u ) gives an obstruction theory of M over U . Proof It follows from Lemma 2.4.17. Note that grel (V• , F∗u ) is naturally isomorphic to the complex considered there.

5.5.3 Decomposition into the Trace-Free Part and the Diagonal Part We use the setting in Subsection 5.5.1. Lemma 5.5.4 The morphism Obrel (V, F∗ )[−1] −→ Ob(V• ) factors through the trace free part Ob◦ (V• ). Proof We have the trace map tr : gD (V• , F∗ ) −→ OD [−1] given as follows: l+1

(i)

(i)

Hom(VD , VD ) −→ O,

(fi ) −→

tr(fi )

i=1

We also have the map i : OD [−1] −→ gD (V, F∗ ) given as follows: O −→

l+1

(i)

(i)

Hom(VD , VD ),

t −→ (t · idV (1) , 0, . . . , 0, −t · idV (l+1) ).

i=1

We put g◦D (V• , F∗ ) := Ker(tr) and gdD (V• , F∗ ) := Im(i). We also have the decomposition g(V•|D ) = g◦ (V•|D ) ⊕ gd (V•|D ) as in Subsection 5.1.2. We have the following commutative diagrams:

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5 Obstruction Theories of Moduli Stacks and Master Spaces

O[−1] −−−−→ ⏐ ⏐ i!

O[−1] ⏐ ⏐ i!

g(V•|D ) −−−−→ gD (V• , F∗ )

g(V•|D ) −−−−→ gD (V• , F∗ ) ⏐ ⏐ ⏐ ⏐ tr! tr! O[−1] −−−−→

O[−1]

Therefore, the decomposition is compatible with g(V•|D ) −→ gD (V• , F∗ ). It follows from grel (V• , F∗ )[−1] −→ g(V•|D ) factors through g◦ (V•|D ). Therefore, Obrel (V, F∗ )[−1] −→ Ob(V•|D ) factors through Ob◦ (V•|D ). It is easy to see that the morphism Ob(V•|D ) −→ Ob(V• ) is compatible with the decomposition into the trace-free part and the diagonal part. Thus we are done. As an immediate corollary, we obtain the decomposition of the cone: Cone Obrel (V• , F∗ ) −→ Ob(V• ) Cone Obrel (V• , F∗ ) −→ Ob◦ (V• ) ⊕ Obdrel (V• )

(5.34)

5.6 Obstruction Theory for Moduli Stacks of Stable Objects 5.6.1 Relative Complexes Let y be an element of H ∗ (X). Let y be an element of T ype whose H ∗ (X)component is y. Let M(m, y) be the open subset of M(y) determined by the condition Om . We have the natural morphism p1 : M(m, y) −→ M(m, y). On M(m, y) × X, we have the universal sheaf p∗1 X E u over M(m, y) × X with the parabolic structure F∗u at D. Applying the construction in Subsection 5.5.1 to the resolution V• of E u (m) and the parabolic structure F∗u , we obtain a complex Obrel (V• , F∗u ) with a morphism: obrel (m, y) : Obrel (m, y) −→ LM(m,y)/M(m,y) Let M(m, y, L) denote the open subset of M(y, L) determined by the condition Om . The natural morphism M(m, y, L) −→ M(m, y) is denoted by p2 . We have the universal L-section φu of p∗2 X E u . It induces the L(m)-section of p∗2 X E u (m), which is also denoted by φu . We fix an inclusion ι : O(−m) −→ L. If m is sufficiently large, we may assume that L(m) has a locally free resolution P• = (P−1 → P0 ) such that P0 is a direct sum of some OX . Since we have the isomorphism pX ∗ E u (m) pX ∗ (V0 ), the L(m)-section φu is canonically lifted to the morphism φu• : p∗M(m,y,L) P• −→ p∗2 X V• . Then, we obtain a complex Obrel (m, y, L) with a morphism by applying the construction in Subsection 5.3.1: obrel (m, y, L) : Obrel (m, y, L) −→ LM(m,y,L)/M(m,y)

5.6 Obstruction Theory for Moduli Stacks of Stable Objects

175

Let M(m, y, [L]) denote the open subset of M(y, [L]) determined by the condition Om . The natural morphism M(m, y, [L]) −→ M(m, y) is denoted by p3 . Then, we have the universal reduced L-section [φu ] of p∗3 X E u over M(m, y, [L]) × X. As before, we obtain a complex Obrel m, y, [L] with a morphism, by using the construction in Subsection 5.4.1: obrel m, y, [L] : Obrel m, y, [L] −→ LM(m,y,[L])/M(m,y) . Let M(m, y ) denote the open subset of M(

y ) determined by the condition Om . The natural morphism M(m, y ) −→ M(m, y) is denoted by p4 . Then, we have the universal orientation ρu of p∗4 X E u over M(m, y ) × X. From the resolution V• and the orientation ρu , we obtain a complex Obrel (m, y ) with a morphism by using the construction in Subsection 5.2.1: obrel (m, y ) : Obrel (m, y ) −→ LM(m, y)/M(m,y) .

5.6.2 Construction of Complexes and Morphisms

, [L]) to M(m, y), M(m, y), We have the naturally defined morphisms of M(m, y M(m, y, [L]) and M(m, y ). The pull back of the complexes Ob(m, y), Obrel m, y , Obrel m, y, [L] , Obrel m, y

, [L]) denote the cone of the followare denoted by the same symbols. Let Ob(m, y

, [L]): ing morphism on M(m, y Obrel m, y [−1] ⊕ Obrel m, y, [L] [−1] ⊕ Obrel m, y [−1] −→ Ob m, y Proposition 5.6.1 We have a naturally defined morphism

, [L]) : Ob(m, y

, [L]) −→ LM(m, y,[L]) . ob(m, y Proof We put C := Obrel m, y ⊕ Obrel m, y, [L] ⊕ Obrel m, y . According to the diagrams (5.12), (5.24) and (5.33), we have the following commutative diagram: C[−1] −−−−→ Ob(V• ) ⏐ ⏐ ⏐ ⏐ ! ! LM(m, y,[L])/M(m,y) [−1] −−−−→ LM(m,y)/k It induces the desired morphism.

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5 Obstruction Theories of Moduli Stacks and Master Spaces

Similarly, on the moduli stack M(m, y, L), we put Ob m, y, L :=

Cone Obrel (m, y, L)[−1] ⊕ Obrel m, y [−1] −→ Ob m, y .

Then, we obtain a morphism ob m, y, L : Ob m, y, L −→ LM(m,y,L)/k .

), we put On M(m, y

:= Cone Obrel (m, y )[−1] ⊕ Obrel (m, y)[−1] −→ Ob(m, y) . Ob m, y

: Ob m, y

−→ LM(m, y)/k . Then, we obtain a morphism ob m, y

, [L] denote the Let L = (L1 , L2 ) be a tuple of line bundles on X. Let Ob m, y

, [L]): cone of the following morphism on M(m, y Obrel (m, y, [Li ])[−1] −→ Ob(m, y) Obrel (m, y )[−1]⊕Obrel (m, y)[−1]⊕ i=1,2

, [L] : Ob m, y

, [L] −→ LM(m, y,[L])/k . Then, we obtain a morphism ob m, y

5.6.3 Obstruction Theories of Quot Schemes and Moduli Stacks

, [L]) be as in Subsection 3.6.7. Let Let Q◦ (m, y

, [L]) −→ M(m, y

, [L]) π : Q◦ (m, y

, [L]) denote the cone of the following mordenote the projection. Let ObQ (m, y

, [L]): phism on Q◦ (m, y π ∗ Obrel (m, y ) ⊕ π ∗ Obrel (m, y, [L]) ⊕ π ∗ Obrel (m, y) −→ Ob(V−1 , f ) Then, we obtain a morphism

, [L]) : ObQ (m, y

, [L]) −→ LQ◦ (m, y,[L])/k obQ (m, y by an argument in Subsection 5.6.2.

, [L]) gives an obstruction theory of Q◦ (m, y

, [L]). Proposition 5.6.2 obQ (m, y Proof It follows from Proposition 2.4.14, Remark 2.4.16, Lemma 5.5.3, Lemma 5.4.6, and Lemma 5.2.3.

, [L]) gives an obstruction theory of M(m, y

, [L]) Proposition 5.6.3 ob(m, y over k.

, [L]) Hi ObQ (m, y

, [L]) for i < 0. Note Proof We have π ∗ Hi Ob(m, y H1 ObQ (m, y, [L]) = H1 LQ◦ (m,y,[L])/k = 0 and H−1 (LQ◦ (m)/M(m) ) = 0.

5.6 Obstruction Theory for Moduli Stacks of Stable Objects

177

We have the following morphism of exact sequences from Lemma 5.1.3 and the construction of the complexes:

, [L]) −−−−→ H0 ObQ (m, y

, [L]) −−−−→ 0 −−−−→ π ∗ H0 Ob(m, y ⏐ ⏐ ⏐ ⏐ ! ! 0 −−−−→ π ∗ H0 (LM(m, y,[L])/k ) −−−−→ H0 (LQ◦ (m, y,[L])/k ) −−−−→

, [L]) −−−−→ 0 Hom(V , V ) −−−−→ π ∗ H1 Ob(m, y ⏐ ⏐ ⏐ ⏐ ! ! Hom(V , V ) −−−−→ π ∗ H1 (LM(m, y,[L])/k ) −−−−→ 0 Therefore, the claim follows from Proposition 5.6.2.

By a similar argument, we obtain the following:

) and ob(m, y

, [L]) give obstruction theoProposition 5.6.4 ob(m, y, L), ob(m, y

) and M(m, y

, [L]), respectively. ries of M(m, y, L), M(m, y

5.6.4 Obstruction Theories of Moduli Stacks of Stable Objects Let α∗ be a system of weight, and let δ ∈ P br . Take a sufficiently large integer m.

, [L]). y , [L], α∗ , δ) is an open substack of M(m, y Then, Ms (

, [L]) gives an obstruction theory of Proposition 5.6.5 The restriction of ob(m, y y , [L], δ, α∗ ). It is independent of a choice of m up to isomorphisms in the Ms (

derived category. Proof The first claim follows from Proposition 5.6.3. The second claim follows from Lemma 5.2.1, Lemma 5.4.3 and Lemma 5.5.2. By the same argument, we can show the following proposition: Proposition 5.6.6 Let y ∈ T ype. Let α∗ be a system of weights. If m is sufficiently large, the following holds:

) : Ob(m, y

) −→ LMs ( y,α∗ )/k gives an obstruction • The morphism ob(m, y theory. • Let δ ∈ P br . The morphism ob m, y, L : Ob(m, y, L) −→ LMs (y,L,α∗ ,δ) gives an obstruction theory.

, [L] : Ob(m, y

, [L]) −→ • Let δ = (δ1 , δ2 ) ∈ P br . The morphism ob m, y LMs ( y,[L],α∗ ,δ)/k gives an obstruction theory. The obstruction theories are independent of the choice of m.

178

5 Obstruction Theories of Moduli Stacks and Master Spaces

5.7 Obstruction Theory for Enhanced Master Spaces and Related Stacks 5.7.1 Enhanced Master Space Let us continue to use the setting in Section 5.6. Take a sufficiently large number m. vector space Vm . We put Pm := We put N := Hy (m). We take an N -dimensional P(Vm∨ ). We put Z1 := P OPm (0) ⊕ OPm (1) over Pm . We have the natural right GL(Vm )-action on Z2 := Z1 × Flag(Vm , N ), where Flag(Vm , N ) denotes the full We put W0 := flag variety of Vm as in (4.15). The quotient stack is denoted by Q. ∗ Vm . We set B (W• , [P• ]) := (Pm )GL(Vm ) . It is an open substack of B(W• , [P• ]) in Subsection 5.4.3, Let us fix an inclusion ι : O(−m) −→ L. Since reduced L-sections induce reduced O(−m)-sections, we obtain the morphism

, [L]) −→ B ∗ (W• , [P• ]). Ξ(V• , [φ]) : M(m, y For simplicity of description, we use the symbol Ψ1 instead of Ξ(V• , [φ]). We use the following lemma in our construction of a deformation theory of the enhanced master space. Lemma 5.7.1 The naturally defined morphism Ψ1∗ LB ∗ (W• ,[P• ])/k → LM(m, y,[L])/k is factorized as follows: ob(m, y,[L]) ϕ

, [L] −−−−−−−→ LM(m, y,[L])/k Ψ1∗ LB ∗ (W• ,[P• ])/k −−−−→ Ob m, y Proof Recall the complexes ObG (V• ) and ObG rel (V• , [φ• ]) constructed in Subsection 5.1.3 and Subsection 5.4.3 respectively. We set G ObG (V• , [φ• ]) := Cone ObG rel (V• , [φ• ])[−1] −→ Ob (V• ) . Then, we obtain the following commutative diagram from (5.30) and (5.31):

, [L] ObG (V• , [φ• ]) −−−−→ Ob m, y ⏐ ⏐ ⏐ ⏐ ϕ1 ! ! Ψ1∗ LB ∗ (W• ,[P• ])/k −−−−→ LM(m, y,[L])/k . It is easy to show that ϕ1 is an isomorphism by using Lemma 5.1.2 and Lemma 5.4.8. is denoted by N . By construc , [L]) ×B ∗ (W• ,[P• ]) Q The fiber product M(m, y

tion, the enhanced master space M is an open subset of N . The induced morphism

to is denoted by Ψ2 . Let p denote the naturally defined morphism of M N −→ Q

:

, [L]). We obtain the following morphism of distinguished triangles on M M(m, y

5.7 Obstruction Theory for Enhanced Master Spaces and Related Stacks

179

Ψ2∗ LQ/B −−−→ p∗ Ψ1∗ LB ∗ (W• ,[P• ])/k −−−−→ Ψ2∗ LQ/k ∗ (W ,[P ]) [−1] − ⏐• • ⏐ ⏐ ⏐ ⏐ ⏐ ! ! ! LM [−1]

/M(m,

y ,[L])

−−−−→

p∗ LM(m, y,[L])/k

−−−−→

LM

/k

−−−−→ Ψ2∗ LQ/B ∗ (W ,[P ]) ⏐ • • ⏐ ! −−−−→

(5.35)

LM

/M(m,

y ,[L])

From Lemma 5.7.1 and the diagram (5.35), we obtain the morphism ∗ Ψ2∗ LQ/B ∗ (W ,[P ]) [−1] −→ p Ob(m, y, [L]). • •

). We have the induced morphism The cone is denoted by Ob(M

) : Ob(M

) −→ L . ob(M M /k

) gives an obstruction theory of the master space M

. Proposition 5.7.2 ob(M

, [L]). By construction, we have the following morProof We put M := M(m, y phism of distinguished triangles: ∗

) −−−−→ L

, [L]) −−−−→ Ob(M LM

/M [−1] −−−−→ p Ob(m, y M /M ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! (5.36) ! !

LM

/M [−1] −−−−→

p∗ LM/k

−−−−→ LM

/k −−−−→ LM

/M

Then, the claim follows from Proposition 5.6.3.

∗ 5.7.2 Substack M We have the natural GL(Vm )-action on Vm∗ × Flag(Vm , N ). The quotient stack ∗ . We put B ∗ (W• , P• ) := (Vm∗ )GL(V ) , which is an open subis denoted by Q m ∗ −→ stack of B(W• , P• ) in Subsection 5.3.3. We have the natural morphism Q ∗ ∗ B (W• , P• ) which is a full flag bundle. Since Vm naturally gives the open subset ∗ naturally gives the open subset Z1 − P OP (0) ∪ P OP (1) of Z1 , the stack Q of Q.

∗ is an open subset of

∗ := M

− M

1 ∪ M

2 (Section 4.3). The stack M Recall M ∗ by construction. We have the commutative diagrams:

, L) ×B ∗ (W• ,P• ) Q M(m, y

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5 Obstruction Theories of Moduli Stacks and Master Spaces

∗ M ⏐ ⏐ p2 !

∗ Q ⏐ ⏐ !

Ψ

−−−4−→

Ψ

, L) −−−3−→ B ∗ (W• , P• ) M(m, y Ψ4∗ LQ ∗ /B ∗ (W• ,P• ) [−1] −−−−→ p∗2 Ψ3∗ LB ∗ (W• ,P• ) ⏐ ⏐ ⏐ ⏐ ϕ! ! LM [−1]

∗ /M(m,

y ,L)

p∗2 LM(m, y,L)

−−−−→

, L). Lemma 5.7.3 ϕ factors through p∗2 Ob(m, y Proof Recall the complexes ObG (V• ) and ObG rel (V• , φ• ) constructed in Subsection 5.1.3 and Subsection 5.3.3, respectively. We put G ObG (V• , φ• ) := Cone ObG rel (V• , φ• )[−1] −→ Ob (V• ) . We obtain the following commutative diagram on M(m, y, L): ObG (V• , φ• ) ⏐ ⏐ ϕ2 !

−−−−→ Ob(m, y, L) ⏐ ⏐ !

Ψ3∗ LB ∗ (W• ,P• )/k −−−−→ LM(m,y,L) It is easy to show that ϕ2 is an isomorphism by using Lemma 5.1.2 and Lemma 5.3.5.

, L), we obtain a morphism Since ϕ factors through Ob(m, y

, L). Ψ4∗ LQ/B ∗ (W ,P ) [−1] −→ Ob(m, y • •

∗ ). Then, we obtain a morphism The cone is denoted by Ob(M

∗ ) : Ob(M

∗ ) −→ L ∗ . ob(M M

∗ )

) ∗ −→ Ob(M Lemma 5.7.4 There exists a quasi-isomorphism u : Ob(M |M

∗ ) ◦ u = ob(M

). such that ob(M

, L) −→ M(m, y

, [L]). By Proof Let π1 denote the natural morphism M(m, y construction of ObG (V• , [φ• ]) and ObG (V• , φ• ), we have the following commuta , L): tive diagram on M(m, y Φ∗2 LB ∗ (W,P )/k ⏐ ψ1 ⏐

←−−−−

ObG (V, φ• ) ⏐ ψ2 ⏐

−−−−→

, L) Ob(m, y ⏐ ψ3 ⏐

, [L]) π1∗ Φ∗1 LB ∗ (W,[P ])/k ←−−−− π1∗ ObG (V, [φ• ]) −−−−→ π1∗ Ob(m, y

(5.37)

5.7 Obstruction Theory for Enhanced Master Spaces and Related Stacks

181

Moreover, the induced morphisms Cone(ψ2 ) −→ Cone(ψi ) (i = 1, 3) are isomor ∗ : phisms. Hence, we obtain the following commutative diagram on M

, L) Ψ4∗ LQ ∗ /B ∗ (W,P ) [−1] −−−−→ p∗2 Ψ3∗ LB ∗ (W,P ) −−−−→ p∗2 Ob(m, y ⏐ ⏐ ⏐ μ1 ⏐ μ2 ⏐ μ3 ⏐

, [L]) Ψ4∗ LQ ∗ /B ∗ (W,[P ]) [−1] −−−−→ p∗ Ψ1∗ LB ∗ (W,[P ]) −−−−→ p∗ Ob(m, y −−−−→ p∗2 LM(m, y,L) ⏐ ⏐ −−−−→ p∗ LM(m, y,[L]) Moreover, the induced morphisms Cone(μ1 ) −→ Cone(μ2 ) −→ Cone(μ3 ) are quasi-isomorphisms Then, the claim of the lemma is clear.

∗ ). Because Vm = W0 , we have B(W• ) = We have another description of Ob(M Spec(k)GL(Vm ) . We put F := Flag(Vm , N )GL(Vm ) . We have the following commutative diagram:

∗ M ⏐ ⏐ p2 !

Ψ

−−−4−→

∗ Q ⏐ ⏐ !

Γ

−−−1−→

Ψ

F ⏐ ⏐ !

(5.38)

Γ

, L) −−−2−→ B ∗ (W• , P• ) −−−2−→ B(W• ) M(m, y Then, we have the isomorphism Ψ4∗ LQ ∗ /B ∗ (W• ,P• ) Ψ4∗ Γ1∗ LF /B(W• ) . We also obtain the following morphisms:

, L) Ψ2∗ Γ2∗ LB(W• ) −−−−→ Ψ2∗ LB ∗ (W• ,P• ) −−−−→ Ob(m, y

, L). Therefore, we obtain a morphism α : Ψ4∗ Γ1∗ LF /B(W• ) [−1] −→ p∗2 Ob(m, y We naturally obtain the following quasi-isomorphism:

∗ ) Cone(α) Ob(M

(5.39)

, [L]) 5.7.3 Moduli Stack M(m, y

, [L]) be a moduli stack of tuples (E∗ , ρ, [φ], F) as follows: Let M(m, y • (E∗ , ρ, [φ]) is an oriented reduced L-Bradlow pair of type y, satisfying the condition Om . • F is a full flag of H 0 (X, E(m)).

182

5 Obstruction Theories of Moduli Stacks and Master Spaces

By an argument in Subsection 5.7.1, we can obtain an obstruction theory of and M to denote the stacks M(m,

, [L]). We use the symbols M

, [L]) M(m, y y

, [L]), respectively. We have B(W• ) = Spec(k)GL(Vm ) , and we put and M(m, y F := Flag(Vm )GL(Vm ) as in Subsection 5.7.2. The following diagram is Cartesian: Ψ

, [L]) −−−11 −→ M(m, y ⏐ ⏐ p1 !

F ⏐ ⏐ !

Ψ

, [L]) −−−12 −→ B(W• ) M(m, y

, [L]): We obtain the following morphism of distinguished triangles on M(m, y ∗ ∗ ∗ ∗ Ψ11 LF /B(W• ) [−1] −→ p∗1 Ψ12 LB(W• )/k −→ Ψ11 LF /k −→ Ψ11 LF /B(W• ) ↓ ϕ↓ ↓ ↓ [−1] −→ p∗1 LM/k −→ LM/k −→ LM/M LM/M

, [L]) Lemma 5.7.5 ϕ factors through p∗1 Ob(m, y Proof It can be shown by using the complex ObG (V• ) and the argument in the proof of Lemma 5.7.1. ∗

, [L]). The cone Thus, we obtain a morphism Ψ11 LF /B(W• ) [−1] −→ p∗1 Ob(m, y

, [L]). We have the following naturally defined morphism: is denoted by Ob(m, y

m, y

, [L] : Ob(m,

, [L]) −→ LM(m,

ob y y ,[L])/k By an argument in the proof of Proposition 5.7.2, we can show the following. m, y m, y, [L] .

, [L] gives an obstruction theory of M Proposition 5.7.6 ob We have the equivalent obstruction theory. Let Q1 denote the quotient stack of Pm × Flag(V, N ) via the natural GL(Vm )-action. Then, the following diagram is Cartesian: Ψ

, [L]) −−−13 −→ Q1 M(m, y ⏐ ⏐ ⏐ ⏐ p1 ! ! Ψ

, [L]) −−−14 −→ B ∗ (W, [P ]) M(m, y

, [L]): We have the following morphism of distinguished triangles on M(m, y ∗ ∗ ∗ ∗ LQ1 /B(W,[P ]) [−1] −→ p∗1 Ψ14 LB(W,[P ]) −→ Ψ13 LQ1 /k −→ Ψ13 LQ1 /B(W,[P ]) Ψ13 ↓ ϕ1 ↓ ↓ ↓ ∗ [−1] −→ p L −→ L −→ L LM/M 1 M/k M/k M/M

By the argument used in the proof of Lemma 5.7.1, we can show that ϕ1 factors 2 (m, y

, [L]). Let Ob

, [L]) denote the cone of the morphism through p∗1 Ob(m, y

5.7 Obstruction Theory for Enhanced Master Spaces and Related Stacks

183

∗

, [L]), Ψ13 LQ1 /B ∗ (W,[P ]) [−1] −→ p∗1 Ob(m, y

and then we have the following naturally defined morphism: 2 (m, y 2 (m, y

, [L]) : Ob

, [L]) −→ LM(m,

ob y ,[L]) Lemma 5.7.7 We have a natural quasi-isomorphism 2 (m, y

, [L])

, [L]) −→ Ob ψ : Ob(m, y 2 (m, y

, [L]) ◦ ψ is equal to ob(m,

, [L]). such that the composite ob y

, [L]). We obtain Proof Let V• be the canonical resolution of E u (m) over M(m, y • ]) by the constructions in Subsection the complexes ObG (V• ) and ObG (V , [ φ • rel 5.1.3 and Subsection 5.4.3. Let ObG (V• , [φ• ]) denote the cone of the morphism G ObG rel (V• , [φ• ])[−1] −→ Ob (V• ). We have the following commutative diagram

, [L]): on M(m, y

∗ LB(W ) Ψ12 ⏐ ⏐ !

←−−−−

ObG (V• ) ⏐ ⏐ !

) Ob(m, y ⏐ ⏐ !

−−−−→

∗

, [L]) LB ∗ (W,[P ]) ←−−−− ObG (V• , [φ• ]) −−−−→ Ob(m, y Ψ14

We have the commutative diagram:

, [L]) −−−−→ M(m, y ⏐ ⏐ !

Q1 ⏐ ⏐ !

−−−−→

F ⏐ ⏐ !

, [L]) −−−−→ B ∗ (W, [P ]) −−−−→ B(W ) M(m, y Hence, the following diagram is commutative:

∗ ∗ LF /B(W ) [3pt] −→ Ψ13 LQ1 /B(W,[P ]) [3pt] −→ LM(m,

Ψ11 y ,[L])/M(m,

y ,[L])

↓

, [L]) Ob(m, y

=

−→

Then, the claim is clear.

↓

↓

, [L]) Ob(m, y

p∗1 LM(m, y,[L])

−→

s y

, [L], α∗ , (δ, ) of (δ, )-stable objects is the Recall that the moduli stack M

, [L]). (See Subsection 3.3.3.) By restricting ob(m,

,[L]), open substack of M(m, y y s y

, [L], α∗ , (δ, ) . we obtain an obstruction theory of M

184

5 Obstruction Theories of Moduli Stacks and Master Spaces

5.7.4 Moduli Stack M(m, y, L) Let M(m, y, L) be a moduli stack of tuples (E∗ , φ, F) as follows: • (E∗ , φ) is an L-Bradlow pair of type y, satisfying the condition Om . • F is a full flag of H 0 (X, E(m)). We use the notation in Subsection 5.7.3. We have the following Cartesian diagram: Ψ M(m, y, L) −−−11 −→ F ⏐ ⏐ ⏐ ⏐ p1 ! ! Ψ

−→ B(W• ) M(m, y, L) −−−12 By using the construction in Subsection 5.7.3, we obtain an obstruction theory: ob(m, y, L) : Ob(m, y, L) −→ LM(m,y,L) ss y, L, α∗ , (δ, ) (Subsection 3.3.3) is the open substack The moduli stack M of M(m, y, L). By restricting ob(m, y, L), we obtain an obstruction theory of ss M y, L, α∗ , (δ, ) .

s (

5.7.5 Moduli Stack M y , α∗ , +)

) denote a moduli stack of objects (E∗ , F) as follows: Let M(m, y • E∗ is a parabolic torsion-free sheaf satisfying the condition Om . • F is a full flag of H 0 (X, E(m)). We use the same notation in Subsection 5.7.3. In this case, we have the following Cartesian diagram: Ψ

) −−−11 M(m, y −→ F ⏐ ⏐ ⏐ ⏐ ! ! Ψ

) −−−12 −→ B(W• ). M(m, y By using the construction in Subsection 5.7.3, we obtain an obstruction theory:

) : Ob(m,

) −→ LM(m,

ob(m, y y y) s (

Recall M y , α∗ , +) denotes a moduli stack of the objects (E∗ , F) as follows (Subsection 4.6.1):

5.7 Obstruction Theory for Enhanced Master Spaces and Related Stacks

185

• E∗ is a parabolic torsion-free sheaf of type y with weight α∗ . • F is a full flag of H 0 (X, E(m)). • (E∗ , Fmin ) is -semistable reduced O(−m)-Bradlow pair, where denotes any sufficiently small positive number. s (

), we obtain an obstruction y , α∗ , +) is the open substack of M(m, y Since M s

). theory of M (

y , α∗ , +) by restricting ob(m, y

5.7.6 Case in Which a 2-Stability Condition is Satisfied Let us construct an obstruction theory of the master space in the case that the 2-stability condition is satisfied for (y, L, α∗ , δ). Since it can be done by the way given in Subsection 5.7.1, we give only an indication. We use the notation in Subsections 4.7.1 and5.7.1. We have the natural GL(Vm ) action on the P1 -bundle P OPm (0) ⊕ OPm (1) over Pm . The quotient stack is de is an

, [L]) −→ B ∗ (W• , [P• ]), and M noted by Q. We have the map Ψ2 : M(m, y ∗

, [L]) ×B (W• ,[P• ]) Q: open substack of M(m, y

M ⏐ ⏐ p!

Ψ

−−−1−→

Q ⏐ ⏐ !

Ψ

, [L]) −−−2−→ B ∗ (W• , [P• ]) M(m, y

: We obtain the following morphism of distinguished triangles on M Ψ1∗ LQ/B ∗ (W,[P ]) [−1] −→ p∗ Ψ2∗ LB ∗ (W,[P ]) −→ Ψ1∗ LQ −→ Ψ1∗ LQ/B ∗ (W,[P ]) ↓

ϕ↓

↓

↓

∗

[−1] −→ p LM(m, y,[L]) −→ LM LM

/M(m,

−→ LM

/M(m,

y ,[L]) y ,[L])

, [L]) (Lemma 5.7.1), we obtain a morphism Since ϕ factors through p∗ Ob(m, y

). We have

, [L]). The cone is denoted by Ob(M Ψ1∗ LQ/B ∗ (W,[P ]) [−1] −→ Ob(m, y

the naturally defined morphism ob(M ) : Ob(M ) −→ LM

. By an argument used in

the proof of Proposition 5.7.2, we can show that ob(M ) gives an obstruction theory

. of M

−M

1 ∪ M

2 . It is an open substack of M(m, y

∗ := M

, L). We put Recall M ∗

) := ob(m, y

, L), and then we have an obstruction theory Ob(M

∗ ) : Ob(M

∗ ) −→ L ∗ . ob(M M

186

5 Obstruction Theories of Moduli Stacks and Master Spaces

) ∗ −→ Ob(M

∗ ) Lemma 5.7.8 There exists a quasi-isomorphism u : Ob(M |M

∗ ) ◦ u = ob(M

). such that ob(M Proof We have the commutative diagram:

∗ M ⏐ ⏐ π!

Ψ

−−−1−→ B ∗ (W• , P• ) ⏐ ⏐ ! Ψ

, [L]) −−−2−→ B ∗ (W• , [P• ]) M(m, y From the diagram (5.37), we obtain the following diagram: Ψ1∗ LB ∗ (W,P ) ⏐ μ1 ⏐

−−−−→

, L) Ob(m, y ⏐ μ2 ⏐

−−−−→

LM

∗ ⏐ μ3 ⏐

, [L]) −−−−→ π ∗ LM(m, y,[L]) π ∗ Ψ2∗ LB ∗ (W,[P ]) −−−−→ π ∗ Ob(m, y Moreover, the induced morphisms Cone(μ1 ) −→ Cone(μ2 ) −→ Cone(μ3 ) are quasi-isomorphisms. Then, the claim of the lemma is clear.

5.7.7 Oriented Reduced L-Bradlow Pairs Let L = (L1 , L2 ) be a pair of line bundles over X. Let us construct an obstruction theory of master spaces for moduli stacks of the oriented reduced L-Bradlow pairs, under the setting in Subsection 4.7.2. We give only an indication. We use the notation in Subsection 5.7.6.

in Subsection 4.7.2. We fix an inclusion We have constructed the master space M ι1 : O(−m) −→ L1 . The universal reduced L1 -section [φu1 ] induces a reduced OX , [L]) −→ B ∗ (W, [P ]). section [φ1 ]. Hence, we obtain the morphism Ψ2 : M(m, y

is an open subset of M(m, y

, [L]) ×B ∗ (W,[P ]) Q: By construction, M

M ⏐ ⏐ p!

Ψ

−−−1−→

Q ⏐ ⏐ !

Ψ

, [L]) −−−2−→ B ∗ (W, [P ]) M(m, y

: We obtain the following morphism of distinguished triangles on M Ψ1∗ LQ/B ∗ (W,[P ]) [−1] −→ p∗ Ψ2∗ LB ∗ (W,[P ]) −→ Ψ1∗ LQ −→ Ψ1∗ LQ/B ∗ (W,[P ]) ↓ ϕ↓ ↓ ↓ ∗ [−1] −→ p L −→ L −→ L LM

/M(m,

M(m,

y ,[L])

/M(m,

y ,[L]) M M y ,[L])

5.8 Moduli Theoretic Obstruction Theory of Fixed Point Set

187

By an argument similar to the proof of Lemma 5.7.1, it can be shown that ϕ factors

, [L]). Hence, we obtain a morphism through p∗ Ob(m, y

, [L]). Ψ1∗ LQ/B ∗ (W,[P ]) [−1] −→ Ob(m, y

). We have the naturally defined morphism The cone is denoted by Ob(M

) : Ob(M

) −→ L . ob(M M

) gives an By an argument in the proof of Proposition 5.7.2, we can show that ob(M

obstruction theory of M .

∗ := M

−M

1 ∪ M

2 . It is an open substack of the moduli stack Recall M

, L1 , [L2 ]), we have the fol , L1 , [L2 ]). (See Subsection 4.7.2.) On M(m, y M(m, y lowing morphism:

Obrel (m, y) ⊕ Obrel (m, y, L1 ) ⊕ Obrel (m, y, [L2 ]) ⊕ Obrel (m, y ) [−1] −→ Ob(m, y)

, L1 , [L2 ]). As in Subsection 5.6.2, we can naturally The cone is denoted by Ob(m, y construct a morphism

, L1 , [L2 ]) −→ LM(m, y,L1 ,[L2 ]) .

, L1 , [L2 ]) : Ob(m, y ob(m, y

, L1 , [L2 ]) gives an obstruction theory by using an arguWe can show that ob(m, y

∗ ) := Ob(m, y

, L1 , [L2 ]) and ment in the proof of Proposition 5.6.5. We set Ob(M ∗

, L1 , [L2 ]). ob(M ) := ob(m, y By using the argument in the proof of Lemma 5.7.8, we can show that there exists

∗ ) satisfying

) ∗ −→ Ob(M a quasi-isomorphism u : Ob(M |M

∗ ) ◦ u = ob(M

). ob(M

5.8 Moduli Theoretic Obstruction Theory of Fixed Point Set 5.8.1 Statement Let I = (y 1 , y 2 , I1 , I2 ) be a decomposition type as in Definition 4.4.2. We use the notation in Subsection 4.6.1. We put ss y 1 , L, α∗ , (δ, k0 ) × M ss (

split := M y 2 , α∗ , +) M

188

5 Obstruction Theories of Moduli Stacks and Master Spaces

ss y 1 , L, α∗ , (δ, k0 ) We constructed the obstruction theory Ob(m, y 1 , L) of M ss (

y 2 , α∗ , +) (Subsection 5.7.4) and the obstruction theory Ob(m, y 2 ) of M (Subsection 5.7.5). The direct sum Ob(Msplit ) gives an obstruction theory of split . The affine line Spec k[t] is denoted by A1 . M Proposition 5.8.1 We have obstruction theories

Gm (I)) : Ob(M

Gm (I)) −→ L G ob(M M m (I) M M

Gm (I)) −→ L G

Gm (I)) : Ob( ob( M m (I)×A1 /A1 with the following property: • We have the following commutative diagram:

) ϕ∗ ob(M

) ϕ∗I Ob(M ⏐ ⏐ !

−−I−−−−→

M ob(

Gm

ϕ∗I LM

⏐ ⏐ !

(5.40)

Gm (I)) −−−−−−−−→ L G Ob(M M m (I) (I))

a (M a (M

Gm (I)) : Ob

Gm (I)) −→ L G • Let ob denote the specialization of M m (I) 1 (M M

Gm (I)) = ob(M

Gm (I)). At

Gm (I)) at t = a. At t = 1, we have ob ob( t = 0, we have the following commutative diagram in the diagram (4.35): 0 (M

Gm (I)) −−−−→ F ∗ L G F ∗ Ob M m (I) ⏐ ⏐ ⏐ ⏐ ! ! split ) G ∗ Ob(M

(5.41)

−−−−→ G ∗ LM split

On each a, we have the following distinguished triangle: a (M

Gm (I)) −→ G ∗ Ob(m,

2 ) y 1 , L) −→ F ∗ Ob y G ∗ Ob(m, −→ G ∗ Ob(m, y 1 , L)[1]

(5.42)

We will prove Proposition 5.8.1 in Subsection 5.8.6, after some preparation.

5.8.2 Moduli Stack of Split Objects with Orientations For simplicity of description, we set M1 := M(m, y 1 , L) and M2 := M(m, y 2 ).

3 of tuples We put M3 := M1 × M2 . Let us consider a moduli stack M (E1 , F1 ∗ , φ, E2 , F2 ∗ , ρ) as follows:

5.8 Moduli Theoretic Obstruction Theory of Fixed Point Set

189

• (E1 , F1 ∗ , φ) ∈ M1 and (E2 , F2 ∗ ) ∈ M2 . • ρ is an orientation of E1 ⊕ E2 . We have the obstruction theory Ob(M3 ) := Ob(m, y 1 , L) ⊕ Ob(m, y 2 ) of

3 over M3 is constructed in the standard M3 . The relative obstruction theory of M

3 −→ M3 denote the promanner, which we explain in the following. Let π : M u u u u ) jection. We have the universal objects (E1 , F1∗ , φ ) over M1 × X and (E2u , F2∗ (i) over M2 × X. We also have the canonical resolutions V• of Eiu (m). We denote

3 × X by the same symbols. We have the orientation of the induced objects over M u u

3 × X. We obtain the following diagram: E1 ⊕ E2 over M (1)

(2)

gd (V• ⊕ V• ) ⏐ ⏐ !

(1)

(2)

−−−−→ g(V• ) ⊕ g(V• ) ⏐ ⏐ !

det∗E1u ⊕E2u ,X LPic ×X/X −−−−→

LM

3 ×X/X

We obtain the following: (1)

(2)

Obd (V• ⊕ V• ) −−−−→ Ob(m, y 1 , L) ⊕ Ob(m, y 2 ) ⏐ ⏐ ⏐ ⏐ ! ! det∗E1u ⊕E2u LPic

−−−−→

LM

3

3 /M3 ) := Cone Obd (V•(1) ⊕ V•(2) ) −→ det∗E u ⊕E u LPic . It is We put Obrel (M 1 2 equipped with a natural morphism

3 /M3 ) : Obrel (M

3 /M3 ) −→ L . obrel (M M3 /M3 We have the following natural morphism:

3 /M3 )[−1] −→ Ob(m, y 1 , L) ⊕ Ob(m, y 2 ) = π ∗ Ob(M3 ) γ : Obrel (M

3 ). Then, we have the natural morphism The cone of γ is denoted by Ob(M

3 ) : Ob(M

3 ) −→ L . ob(M M3 We have the following commutative diagram as in Subsection 5.2.1: LM

3 ⏐ ⏐

←−−−−

det∗E1u ⊕E2u LPic ⏐ ⏐

∗ (1) (2) π ∗ LM3 ←−−−− Φ det(E1u ⊕ E2u ) π1∗ LM(1) ←−−−− Obd (V• ⊕ V• )

190

5 Obstruction Theories of Moduli Stacks and Master Spaces

Here, π1 denotes the projection Pic −→ M(1). Therefore, we have the following commutative diagram: γ

3 /M3 )[−1] −−− −→ π ∗ Ob(M3 ) Obrel (M ⏐ ⏐ ⏐ ⏐

3 /M3 )! obrel (M !

−−−−→

LM

3 /M3 [−1]

π ∗ LM3

3 /M3 ) By the argument in the proof of Lemma 5.2.3, we can show that obrel (M

gives a relative obstruction theory of M3 over M3 . Therefore, Ob(M3 ) gives an

3 . obstruction theory of M

5.8.3 Embedding into Moduli Stack of Non-Split Objects

0 := M(m, y

, L). For simplicity of description, we put M0 := M(m, y, L) and M

Let π0 denote the projection M0 −→ M0 . Let V• denote the canonical resolution

(i) of pX ∗ E u (m) on M0 × X. We put V0 := pX ∗ V0 = pX ∗ E u (m) and V0 := (i) pX ∗ V0 . We have the naturally defined morphism f : M3 −→ M0 . We have the (1) (2)

(1)

(2) decomposition f∗X E u = E1u ⊕ E2u , f∗X V• = V• ⊕ V• and f∗ V0 = V0 ⊕ V0 . We have the naturally defined projections: f∗X g(V• ) −→ g(V• ) ⊕ g(V• ), (1)

(2)

(1) f∗X grel (V• , φ• ) −→ grel (V• , φ• ),

f∗X g(V•|D ) −→ g(V•|D ) ⊕ g(V•|D ), (1)

f∗X grel (V• , F∗u ) −→ grel (V• , F∗ (1)

u (1)

(2)

(2)

u (2)

) ⊕ grel (V• , F∗

).

They induce the following morphisms: f∗ Ob(V• ) −→ Ob(V• ) ⊕ Ob(V• ), (1)

(2)

f∗ Obrel (V• , φ• ) −→ Obrel (V (1) , φ• ) f∗ Ob(V•|D ) −→ Ob(V•|D ) ⊕ Ob(V•|D ), (1)

f∗ Ob(V• , F∗u ) −→ Obrel (V• , F∗ (1)

u (1)

(2)

(2)

u (2)

) ⊕ Obrel (V• , F∗

Therefore, we obtain the induced morphism: μ1 : f∗ Ob(m, y, L) −→ Ob(m, y 1 , L) ⊕ Ob(m, y 2 )

)

5.8 Moduli Theoretic Obstruction Theory of Fixed Point Set

191

Lemma 5.8.2 The following diagram is commutative: f∗ Ob(m, y, L) ⏐ ⏐ !

−−−−→ f∗ LM0 ⏐ ⏐ !

(5.43)

Ob(m, y 1 , L) ⊕ Ob(m, y 2 ) −−−−→ LM3 Proof Although this is almost obvious, we give a rather detailed argument. We take an Hy (m)-dimensional vector space W0 = Vm with a decomposition W0 = (1) (2) (i) W0 ⊕ W0 , where dim W0 = Hyi (m). We also take a Hy (m) − rank(y) (1) (2) dimensional vector space W−1 with a decomposition W−1 = W−1 ⊕ W−1 , where (i) dim W−1 = Hyi (m) − rank(y i ). We put (1)

(2)

(1)

(2)

Y (W• , W• ) := Y (W• ) × Y (W• ). (1)

(2)

We have the naturally defined morphism Y (W• , W• ) −→ Y (W• ). By consider(1) (2) ing the classifying map of V• and V• ⊕ V• , we obtain the following commutative diagram: Φ(V• ) M0 × X −−−−→ Y (W• ) ⏐ ⏐ fX ⏐ ⏐ (1)

(2)

Φ(V• ,V• )

(1)

(2)

M3 × X −−−−−−−−→ Y (W• , W• ) By the argument in Subsection 2.3.2, we can show that f∗X g(V• )≤1 −→ g(V• )≤1 ⊕ g(V• )≤1 (1)

(2)

represents the morphism f∗X Φ(V• )∗ LY (W• )/X → Φ(V• , V• )∗ LY (W (1) ,W (2) )/X . • • Therefore, we obtain the following commutative diagram: (1)

f∗X LM0 ×X/X ←−−−− ⏐ ⏐ ! LM3 ×X/X

f∗X Φ(V• )∗ LY (W• )/X ⏐ ⏐ !

(2)

←−−−−

g(V• ) ⏐ ⏐ !

←−−−− Φ(V• , V• )∗ LY (W (1) ,W (2) )/X ←−−−− g(V• )⊕g(V• ) (1)

(2)

(1)

•

(2)

•

Therefore, we obtain the following commutative diagram: f∗ LM0 ←−−−− ⏐ ⏐ !

f∗ Ob(m, y) ⏐ ⏐ !

(5.44)

LM3 ←−−−− Ob(m, y1 ) ⊕ Ob(m, y2 ). Let P• be a locally free resolution of L(m), as in Subsection 5.6.1. We put (1) (2) (1) (2) Y (W• , W• , P• ) := Y (W• , P• ) × Y (W• ). We have the naturally defined

192

5 Obstruction Theories of Moduli Stacks and Master Spaces (1)

(2)

morphism Y (W• , W• , P• ) −→ Y (W• , P• ), and the following commutative diagram: • ) Φ(V• ,φ −−−−−−→ Y (W• , P• ) M0 × X ⏐ ⏐ fX ⏐ ⏐ • ) Φ(V• ,V• ,φ (1)

(2)

(1)

(2)

M3 × X −−−−−−−−−−→ Y (W• , W• , P• ) Then, we obtain the following commutative diagram: f∗X LM0 ×X/X ←− f∗X Φ(V• , φ• )∗ LY (W• ,P• )/X ←− f∗X g(V• , φ• ) ↓ ↓ ↓ (1) (2) (1) (2) LM3 ×X/X ←− Φ(V• , V• , φ• )∗ LY (W (1) ,W (2) ,P• ) ←− g(V• , φ• ) ⊕ g(V• ) •

•

It is easy to see that RpX ∗ g(V• , φ• )⊗ωX is naturally isomorphic to Ob(m, y, L). Then, we obtain the following diagram: f∗ LM0 ←−−−− ⏐ ⏐ !

f∗ Ob(m, y, L) ⏐ ⏐ !

(5.45)

LM3 ←−−−− Ob(m, y1 , L) ⊕ Ob(m, y2 ) We have the natural morphisms Ob(m, y) −→ Ob(m, y, L), Ob(m, y1 ) ⊕ Ob(m, y2 ) −→ Ob(m, y1 , L) ⊕ Ob(m, y2 ). The diagrams (5.44) and (5.45) are compatible for the natural morphisms in the sense of Subsection 2.1.8. (i) (i)(j) (i) := Ker V0|D −→ Cokj−1 which are locally free For i = 1, 2, we put VD (i)∗

(i)

sheaves on M3 × D. Let VD denote the vector bundles VD with the filtrations (i)(1) (i)(2) (i)(l+1) ∗ VD ⊃ VD · · · ⊃ VD . Similarly, we have the filtered vector bundle VD on M0 × D. (j) (j) We put W (1)(j) := W0 and W (l+1)(j) = W−1 for j = 1, 2. We take vector spaces W (i) (i = 2, . . . , l) with decompositions W (i) = W (i)(1) ⊕ W (i)(2) such (i) (i)(j) that rank W (i) = rank VD and rank W (i)(j) = rank VD . We use the nota(1) tion in Subsection 5.5.1. We put YD (W• , W (1) ∗ , W (2) ∗ ) := YD (W• , W (1)∗ ) × (2) (1) (2) (1) (2) YD (W• , W (2)∗ ) and YD (W• , W• ) := YD (W• ) × YD (W• ). We have the naturally defined commutative diagram: YD (W• , W ∗ ) ⏐ ⏐

−−−−→

YD (W• ) ⏐ ⏐ (1)

(2)

YD (W• , W (1)∗ , W (2)∗ ) −−−−→ YD (W• , W• )

5.8 Moduli Theoretic Obstruction Theory of Fixed Point Set

193

∗ By considering the classifying maps for VD and (VD , VD ), we obtain the following commutative diagram: (1)∗

ΦD (V ∗ ,F u )

∗ → M0 × D −−−−−−− ⏐ ⏐

M3 × D

Φ

1 −−−− →

YD (W• , W ∗ ) ⏐ ⏐

(2)∗

−−−−→

YD (W• ) ⏐ ⏐ (1)

(2)

YD (W• , W (1)∗ , W (2)∗ ) −−−−→ YD (W• , W• )

We obtain the following: f∗D LM0 ×D/D ←−−−− f∗D ΦD (V• , F∗u )∗ LYD (W• ,W ∗ )/D ⏐ ⏐ ⏐ ⏐ ! ! LM3 ×D/D

←−−−−

Φ∗1 LYD (W• ,W (1)∗ ,W (2)∗ )/D ←−−−−

f∗D Φ(V•|D )∗ LYD (W• )/D ⏐ ⏐ !

←−−−− Φ(V•|D ⊕ V•|D )∗ LYD (W (1) ,W (2) )/D (1)

(2)

•

•

Then, we obtain the following diagram on M3 × D: f∗D gD (V• , F∗u ) ←− f∗D g(V•|D ) f∗D LM0 ×D/D ←− ↓ ↓ ↓ (1) u(1) (2) u(2) (1) (2) LM3 ×D/D ←− gD (V• , F∗ )⊕gD (V• , F∗ ) ←− g(V•|D )⊕g(V•|D ) We put

ObD (V, F∗u ) := RpD ∗ gD (V• , F∗u ) ⊗ ωD . u(i)

Similarly, we have ObD (V (i) , F∗ diagram:

). Then, we obtain the following commutative

f∗ LM0 ←− f∗ ObD (V• , F∗u ) ←− f∗ Ob(V•|D ) ↓ ↓ ↓ (1) u(1) (2) u(2) (1) (2) LM3 ←− ObD (V• , F∗ ) ⊕ ObD (V• , F∗ ) ←− Ob(V•|D ) ⊕ Ob(V•|D ) We remark that the cone of Ob(V•|D ) −→ Ob(V• ) ⊕ ObD (V, F∗ ) is naturally isomorphic to Ob(m, y). Thus, we obtain the following commutative diagram, which is compatible with (5.44): f∗ LM0 ←−−−− ⏐ ⏐ !

f∗ Ob(m, y) ⏐ ⏐ !

(5.46)

LM3 ←−−−− Ob(m, y 1 ) ⊕ Ob(m, y 2 ) From (5.44), (5.45) and (5.46), we obtain the desired diagram (5.43). Thus, the proof of Lemma 5.8.2 is finished.

194

5 Obstruction Theories of Moduli Stacks and Master Spaces

3 −→ M

0 . By construction of We have the naturally defined morphism f : M the obstruction theories, we have the following commutative diagram:

f∗ π ∗ Ob(m, y, L) 0 ⏐ ⏐ μ1 !

f∗ Obrel (m, y )[−1] −−−−→ ⏐ ⏐ !

3 /M3 )[−1] −−−−→ π ∗ Ob(m, y 1 , L) ⊕ Ob(m, y 2 ) Obrel (M Therefore, we obtain the following commutative diagram: μ2

f∗ Ob(m, y

3 )

, L) −−−−→ Ob(M ⏐ ⏐ ⏐ ⏐ ! !

f∗ L M0

−−−−→

LM

3

5.8.4 Some Compatibility (1)

(2)

We put B(W• , W• ) := Spec(k)GL(W (1) )×GL(W (2) ) . We have the classifying 0 0 map

(1)

(2) (1) (2) Φ(V0 , V0 ) : M3 −→ B(W• , W• ).

(1)

(2)

It induces a morphism ϕ : Φ(V0 , V0 )∗ LB(W (1) ,W (2) ) −→ LM3 on M3 . •

•

Lemma 5.8.3 The morphism ϕ factors through Ob(m, y1 , L) ⊕ Ob(m, y2 ). In particular, it factors through Ob(m, y 1 , L) ⊕ Ob(m, y 2 ). Proof We have the following commutative diagram: (1)

(2)

(1)

g(V• ) ⊕ g(V• ) ←−−−− ⏐ ⏐ ! LM3 ×X

(2)

h(V• ) ⊕ h(V• ) ⏐ ⏐ !

(1)

(2)

←−−−− Φ(V0 , V0 )∗ LB(W (1) ,W (2) )×X/X •

•

It induces the following diagram: Ob(m, y1 ) ⊕ Ob(m, y2 ) ←−−−− ⏐ ⏐ ! LM3

(1)

(2)

ObG (V• ) ⊕ ObG (V• ) ⏐ ⏐ τ2 !

(1)

←−−−− Φ(V0

(2) ∗

, V0

) LB(W (1) ,W (2) ) •

•

We can check that τ2 is an isomorphism by using Lemma 5.1.2. Then, the claim of the lemma is clear. We set B(W• ) := Spec(k)GL(W0 ) as in Subsection 5.1.3. By considering the

(i) classifying maps for the vector bundles V0 and V0 , we obtain the following commutative diagram:

5.8 Moduli Theoretic Obstruction Theory of Fixed Point Set π

3 −−− M −→ ⏐ ⏐

f!

Φ(V

(1)

,V

(2)

195

)

(1)

(2)

0 0 M3 −−−− −−−− −→ B(W• , W• ) ⏐ ⏐ ⏐ ⏐ f! !

Φ(V )

π2

0 −−− −→ M0 M

0 −−−− →

(5.47)

B(W• )

Lemma 5.8.4 We have the following commutative diagram:

3 ) Ob(M ⏐ μ2 ⏐

LM ←−−−−

-3 ⏐ ⏐

(1)

(2)

←−−−− π ∗ Φ(V0 , V0 )∗ LB(W (1) ,W (2) ) • • ⏐ ⏐

f∗ L ←−−−− f∗ Ob(m, y

, L) ←−−−− M0

f∗ π ∗ Φ(V )∗ LB(W ) 2 0 • (5.48)

Here, the composite of the horizontal arrows are the naturally defined morphisms from the diagram (5.47). Proof We have the following commutative diagram: f∗X h(V• ) ⏐ ⏐ ! (1)

f∗X g(V• ) ⏐ ⏐ !

−−−−→

(2)

(1)

(2)

h(V• ) ⊕ h(V• ) −−−−→ g(V• ) ⊕ g(V• ) Therefore, we obtain the following: f∗ ObG (V• ) ⏐ ⏐ !

f∗ Ob(m, y) ⏐ ⏐ !

−−−−→

(1)

(2)

ObG (V• ) ⊕ ObG (V• ) −−−−→ Ob(m, y1 ) ⊕ Ob(m, y2 ) On the other hand, we have the following commutative diagram: f∗X h(V• ) ⏐ ⏐ !

f∗X Φ(V0 )∗X LB(W• )×X/X ⏐ ⏐ !

−−−−→

(1)

(2)

h(V• ) ⊕ h(V• ) −−−−→ Φ(V0 , V0 )∗X LB(W (1) ,W (2) )×X/X (1)

(2)

•

•

Therefore, we obtain the following: τ

f∗ ObG (V• ) ⏐ ⏐ ! (1)

f∗ Φ(V0 )∗ LB(W• ) ⏐ ⏐ !

−−−1−→

(2)

τ

(1)

ObG (V• ) ⊕ ObG (V• ) −−−2−→ Φ(V0

(2) ∗

, V0

) LB(W (1) ,W (2) ) •

•

The morphisms τi are isomorphisms. Thus, we obtain the claim of the lemma.

196

5 Obstruction Theories of Moduli Stacks and Master Spaces

5.8.5 Deformation As explained in Subsection 5.5.3, we have the decomposition Ob(m, y 2 ) = Ob◦ (m, y 2 ) ⊕ Obd (m, y 2 ). We put Ob(m, y 2 ) := Ob◦ (m, y 2 ) ⊕ τ≤−1 Obd (m, y 2 ). Recall Ob(M3 ) = Ob(m, y 1 , L) ⊕ Ob(m, y 2 ). We have the following commutative diagram:

3 /M3 )[−1] −−−−→ Ob(m, y 1 , L) ⊕ Ob(m, y 2 ) Obrel (M ⏐ ⏐ ⏐ ⏐ (1)

(2)

λ

τ≤−1 Obd (V• ⊕ V• ) −−−1−→ Ob(m, y 1 , L) ⊕ Ob(m, y 2 )

3 ) := Cone λ1 . Then, we obtain the morphisms We put Ob1 (M

3 ) −→ Ob(M

3 ) −→ L . Ob1 (M M3

3 ) gives an Since the first morphism is a quasi-isomorphism, the composite ob1 (M

obstruction theory of M3 . Remark that we have the following commutative diagram (1) (2) (We put C1 := τ≤−1 Obd (V• ⊕ V• ) and C2 := Ob(m, y 1 , L) ⊕ Ob(m, y 2 ) in the diagram, to save the space.): H−1 C1 −−−−→ ⏐ ⏐ ! 0

3 ) −−−−→ −−−−→ H−1 Ob1 (M ⏐ ⏐ ! −−−−→ H−1 LM −−−−→ H−1 (π ∗ LM3 ) −−−−→

3

H0 (C2 ) ⏐ ⏐ !

H−1 (C2 ) ⏐ ⏐ !

3 ) −−−−→ −−−−→ H0 Ob1 (M ⏐ ⏐ !

H0 (π ∗ LM3 ) −−−−→ −−−−→

H0 (LM

3 ) H1 (C2 ) ⏐ ⏐ !

0 ⏐ ⏐ ! (5.49) 0

0 ⏐ ⏐ !

−−−−→ H0 (LM

3 /M3 )

3 )) −−−−→ −−−−→ H1 (Ob1 (M ⏐ ⏐ !

−−−−→ H1 (π ∗ LM3 ) −−−−→

H1 (LM

3 )

0 ⏐ ⏐ !

(5.50)

−−−−→ 0

3 ). Let i1 , i2 and η denote the following natuWe would like to deform ob1 (M rally defined morphisms:

5.8 Moduli Theoretic Obstruction Theory of Fixed Point Set

i1 : τ≤−1 Obd (V1 ) −→ Ob(m, y 1 , L),

197

i2 : τ≤−1 Obd (V2 ) −→ Ob(m, y 2 ),

η : τ≤−1 Obd (V1 ⊕ V2 ) −→ τ≤−1 Obd (V1 ) ⊕ τ≤−1 Obd (V2 ) The following is a special case of Lemma 5.2.2. Lemma 5.8.5 The composite i1 ◦ ob(m, y 1 , L) and i2 ◦ ob(m, y 2 ) are trivial. For any a ∈ k, let ϕa : τ≤−1 Ob be the morphism given by

d

(1) (2) (V• ⊕ V• )

−→ Ob(m, y 1 , L) ⊕ Ob(m, y 2 )

ϕa := a·i1 , i2 ◦ η. Then, the following diagram is commutative for any a, according to Lemma 5.8.5: (1)

(2)

ϕa

τ≤−1 Obd (V• ⊕ V• ) −−−−→ Ob(m, y 1 , L) ⊕ Ob(m, y 2 ) ⏐ ⏐ ⏐ ⏐ ! ! LM

3 /M3 [−1]

(5.51)

π ∗ LM3

−−−−→

a (M

3 ) := Cone(ϕa ). From the commutativity of the diagram (5.51), We put Ob a (M a (M

3 ) : Ob

3 ) −→ L for any a. The choice we obtain morphisms ob M3 of a does not have any effect on the diagram (5.50). Hence, it is easy to observe M

3 ) −→ that oba (M3 ) a ∈ k gives an obstruction theory ob(M3 ) : Ob( 1 1

L 1 1 of M3 × A over A . At a = 1, Ob1 (M3 ) is the same as Ob(M3 ) in M3 ×A /A

the derived category.

2 := M(m, y

2 ). There exists an algebraic stack S with Lemma 5.8.6 We put M the following diagram:

3 M ⏐ ⏐

G1 F

2 ←−−1−− S −−−− → M1 × M ⏐ ⏐ ⏐ ⏐

G

Gm (I) ←−F−−− S −−− M −→

split M

Here the bottom horizontal arrows are given in (4.35). The morphisms F1 and G 1 are etale proper of degree (r1 · r2 )−1 . Proof Similar to Proposition 4.6.1 and Corollary 4.6.2.

Lemma 5.8.7 For each a, we have the following distinguished triangle: a (M

3 ) −→ G ∗ Ob(m, y

2 ) G 1∗ Ob(m, y 1 , L) −→ F1∗ Ob 1 −→ G 1∗ Ob(m, y 1 , L)[1] 0 (M

3 ) = G ∗ Ob(m, y 1 , L) ⊕ G ∗ Ob(m, y

2 ). We also have F1∗ Ob 1 1

(5.52)

198

5 Obstruction Theories of Moduli Stacks and Master Spaces

3 : Proof We have the following naturally defined distinguished triangle on M a (M

3 ) −→ Cone(i2 ) −→ Ob(m, y 1 , L)[1] Ob(m, y 1 , L) −→ Ob

(5.53)

In the case a = 0, it splits. It is easy to see F1∗ Ob(m, y 1 , L) G 1∗ Ob(m, y 1 , L)

2 ). Hence, we obtain (5.52) from (5.53). and F1∗ Cone(i2 ) G ∗ 1 Ob(m, y Lemma 5.8.8 The composite of the following morphisms is independent of the choice of a, and it is the same as the naturally defined one: (1) (2) π ∗ Φ(V0 ⊕ V0 )∗ LB(W (1) ,W (2) ) −→ π ∗ Ob(m, y 1 , L) ⊕ Ob(m, y 2 ) •

•

ϕa

3 ) −→ LM −→ Oba (M

3

Proof It is clear from the construction.

(5.54)

5.8.6 Proof of Proposition 5.8.1

Gm (I). We take Hy (m)-dimensional Let us construct an obstruction theory of M i (i) (i) (1) vector spaces Vm . Let Fi denote the full flag variety of Vm . We set Vm := Vm ⊕ (2) (i) (i) Vm . We use identifications Vm = W0 and Vm = W0 (i = 1, 2). We put F i :=

Gm (I) is an open subset of (F 1 × F 2 ) ×

(Fi )GL(V (i) ) . Then, M (1) (2) M3 . B(W• ,W• ) m Hence, we have the following commutative diagram:

Gm (I) −−−g−→ M ⏐ ⏐ π1 !

3 M

F1 × F2 ⏐ ⏐ q! (1)

(5.55)

(2)

−−−−→ B(W• , W• )

We have the isomorphism g ∗ LF 1 ×F 2 /B(W (1) ,W (2) ) LM

3 . According to

Gm (I)/M • • Lemma 5.8.3, we have the following morphisms:

3 ) g ∗ LF 1 ×F 2 /B(W (1) ,W (2) ) [−1] −−−1−→ g ∗ q ∗ LB(W (1) ,W (2) ) −−−2−→ π1∗ Ob(M c

c

•

•

•

•

G

m (I) := Cone(c2 ◦ c1 ). Then, we obtain the morphism We put Ob M Gm Gm

(I) −→ L G

(I) : Ob M . ob M M m (I) G

m (I) gives By the argument used in Subsection 5.7.1, it can be shown that ob M

Gm (I). an obstruction theory of M (i) (i) (i) Let Flag(Vm , Ii ) denote the moduli of filtrations F∗ of Vm indexed by N as follows:

5.8 Moduli Theoretic Obstruction Theory of Fixed Point Set

199

(i)

(i)

(i)

F1 ⊂ F2 ⊂ · · · ⊂ FN = Vm(i) ,

(i)

(i)

dim Fj /Fj−1 =

1 (j ∈ Ii ) 0 (j ∈

Ii )

Recall that Flag(Vm , N ) denotes the full flag variety of Vm . We have the natu(1) (2) rally defined inclusion Flag(Vm , I1 ) × Flag(Vm , I2 ) −→ Flag(Vm , N ). Clearly, (i) Flag(Vm , Ii ) is naturally isomorphic Fi . We use the notation in Subsection 5.7.2. Then, the diagram (5.55) is compatible with the following diagram, which is given by (5.38):

∗ −−−−→ F M ⏐ ⏐ ⏐ ⏐ ! !

0 −−−−→ B(W• ) M

∗ ), we also obtain (5.40) from (5.48). By using the description (5.39) of Ob(M G M

m (I) . Corresponding to the equivalence We would like to construct Ob 1 (M

3 ) Ob(M

3 ), we have the equivalent obstruction theory Ob Gm Gm 1 M

(I) Ob M

(I) Ob By Lemma 5.8.8, we obtain the deformation Gm Gm M M

(I) −→ L G

(I) : Ob ob M m (I)×A1 /A1 M M

3 ) : Ob(

3 ) −→ L from ob( . We also obtain the distinguished triM3 ×A1 /A1 angle (5.42) and the splitting (5.41) at t = 0 from Lemma 5.8.7. Thus, the proof of Proposition 5.8.1 is finished.

5.8.7 Case in Which a 2-Stability Condition is Satisfied Let us describe the obstruction theory of the fixed point set of the master space, in the case that a 2-stability condition is satisfied. We use the notation in Subsection 4.7.1. We give only the statement. We put y 2 , α∗ ). Msplit := Ms (y 1 , L, α∗ , δ) × Ms (

Proposition 5.8.9 We have the obstruction theories

Gm (I)) −→ L G

Gm (I)) : Ob(M ob(M M m (I) M M

Gm (I)) −→ L G

Gm (I)) : Ob( ob( M m (I)×A1 /A1

200

5 Obstruction Theories of Moduli Stacks and Master Spaces

with the following property: • We have the following commutative diagram:

) ϕ∗I Ob(M ⏐ ⏐ !

−−−−→

ϕ∗I LM

⏐ ⏐ !

(5.56)

Gm (I)) −−−−→ L G Ob(M M m (I) a (M a (M

Gm (I)) : Ob

Gm (I)) −→ L G denote the specialization of • Let ob M m (I) Gm Gm

Gm (I)). At

ob(M (I)) at t = a. At t = 1, we have ob1 (M (I)) = ob(M t = 0, we have the following commutative diagram in (4.38): 0 (M

Gm (I)) −−−−→ F ∗ L G F ∗ Ob M m (I) ⏐ ⏐ ⏐ ⏐ ! ! G ∗ Ob(Msplit )

(5.57)

−−−−→ G ∗ LMsplit

On each a, we have the following distinguished triangle: a (M

Gm (I)) −→ G ∗ Ob(m, y

2 ) G ∗ Ob(m, y 1 , L) −→ F ∗ Ob −→ G ∗ Ob(m, y 1 , L)[1] Proof It can be shown by an argument used in the proof of Proposition 5.8.1. In

3 . The obstruction theories ob(M

3 ) :

Gm (I) is an open substack of M this case, M Gm

Ob(M3 ) −→ M (I) and ob(M3 ) : Ob(M3 ) −→ LM

Gm (I)×A1 /A1 are the desired objects.

5.8.8 Oriented Reduced L-Bradlow Pairs Let L = (L1 , L2 ) be a pair of line bundles over X. Let us describe the obstruction theory of the fixed point set of the master space for moduli stacks of oriented L-Bradlow pairs, under the setting in Subsection 4.7.2. We give only the indication. We put y 2 , [L2 ], α∗ , δ2 ). Msplit := Ms (y 1 , L1 , α∗ , δ1 ) × Ms (

Proposition 5.8.10 We have the obstruction theories

Gm (I)) : Ob(M

Gm (I)) −→ L G ob(M M m (I) M M

Gm (I)) −→ L G

Gm (I)) : Ob( ob( M m (I)×A1 /A1

5.9 Equivariant Obstruction Theory of Master Space

201

with the following property: • We have the following commutative diagram:

) ϕ∗I Ob(M ⏐ ⏐ !

−−−−→

ϕ∗I LM

⏐ ⏐ !

(5.58)

Gm (I)) −−−−→ L G Ob(M M m (I) a (M a (M

Gm (I)) : Ob

Gm (I)) −→ L G denote the specialization of • Let ob M m (I) G G 1 (M M

m (I)) = ob(M

Gm (I)). At

m (I)) at t = a. At t = 1, we have ob ob( t = 0, we have the following commutative diagram in (4.38): 0 (M

Gm (I)) −−−−→ F ∗ L G F ∗ Ob M m (I) ⏐ ⏐ ⏐ ⏐ ! ! G ∗ Ob(Msplit )

(5.59)

−−−−→ G ∗ LMsplit

On each a, we have the following distinguished triangle: a (M

Gm (I)) −→ G ∗ Ob(m, y

2 , [L2 ]) G ∗ Ob(m, y 1 , L1 ) −→ F ∗ Ob

∗ −→ G Ob(m, y 1 , L1 )[1] (5.60)

2 , [L2 ]) and M3 := M1 × Proof We put M1 := M(m, y 1 , L1 ), M2 := M(m, y

3 of objects (E1 , F1 ∗ , φ1 , E2 , F2 ∗ , [φ2 ], ρ) as M2 . We consider a moduli stack M follows: • (E1 , F1 ∗ , φ1 ) ∈ M1 and (E2 , F2 ∗ , [φ2 ]) ∈ M2 . • ρ denotes an orientation of E1 ⊕ E2 .

Gm (I) is an open substack of M

3 . Then, M

3 ) : Ob(M

3 ) −→ L by using We can construct the obstruction theory ob(M M3 M

3 ) : the argument in Subsection 5.8.2. We can also construct a deformation ob(

Ob(M3 ) −→ L G 1 1 by using the argument in Subsection 5.8.5. We M

m (I)×A

/A

can check that they are the desired objects by using an argument in the proof of Proposition 5.8.1.

5.9 Equivariant Obstruction Theory of Master Space 5.9.1 Statements Let Gm be a one-dimensional torus. Recall that we have the natural Gm -action

. on M

202

5 Obstruction Theories of Moduli Stacks and Master Spaces

). Proposition 5.9.1 We have a Gm -equivariant lift of the obstruction theory ob(M

∗ ) The construction is explained in Subsections 5.9.2–5.9.3. We explain that Ob(M can also be lifted equivariantly, in Subsection 5.9.4. Later, we will apply the localization formula of Graber and Pandharipande [55]. For that purpose, we have to study the induced obstruction theory and the virtual nor i −→ M

denote the inclusion (i = 0, 1). mal bundle at the fixed point set. Let ιi : M ∗

)inv and We have the decomposition of ιi Ob(M ) into the invariant part ι∗i Ob(M ∨ ∗ ∗ mov mov

)

i ) := ι Ob(M

) the moving part ιi Ob(M . We put N(M [1], which is i

i ) : called the virtual normal bundle. We have the induced obstruction theory ob1 (M ∗ inv

) −→ L [55]. We will prove the following proposition in Subsection ιi Ob(M Mi 5.9.5

i ) is equivalent to the Proposition 5.9.2 The induced obstruction theory ob1 (M

moduli theoretic obstruction theory ob(Mi ) : Ob(Mi ) −→ LM

i given in Sub

section 5.7.3. The virtual normal bundles N(Mi ) are isomorphic to Orel (−1)i−1 , and the weight of the induced Gm -action is (−1)i .

Gm (I) −→ M

Let I = (y 1 , y 2 , I1 , I2 ) be a decomposition type. Let ϕI : M ∗

) into the denote the inclusion. Similarly, we have the decomposition of ϕI Ob(M ∗ inv ∗ mov

. We obtain the invariant part ϕI Ob(M ) and the moving part ϕI Ob(M )

)mov ∨ [1], and the obstruction

Gm (I)) := ϕ∗ Ob(M virtual normal bundle N(M I

Gm (I)) : ϕ∗ Ob(M

)inv −→ L G . theory ob1 (M I M m (I) Gm

To describe N(M (I)), we prepare some notation. We have the following vir Gm (I). (See Subsection 2.1.7 for virtual vector bundles): tual vector bundles on M

N(EiM , EjM ) := −

(−1)l Rl pX∗ RHom EiM , EjM

(5.61)

l=0,1,2

N L, E2M := (−1)l Rl pX ∗ Hom L, E2M

(5.62)

l=0,1,2

M

M ND EiM∗ , EjM∗ := − (−1)l Rl pD ∗ RHom 2 Ei|D ∗ , Ej|D ∗

(5.63)

l=0,1

M M Here, Ei|D

Gm (I)×D with the induced filtrations. ∗ denote the restriction Ei ⊗ OM

3 be as in Subsections 5.8.2– (See Subsection 2.1.6 for RHom2 .) Let M0 and M G m

(I) −→ M

× M

3 is a regular embedding. 5.8.3. It is easy to observe that M M0 The normal bundle is denoted by N0 . It is naturally equipped with the Gm -action. We will prove the following proposition in Subsection 5.9.6.

Proposition 5.9.3

∗ )inv is isomorphic to Ob(M

Gm (I)), and the induced obstruction • ϕ∗I Ob(M theory

5.9 Equivariant Obstruction Theory of Master Space

203

∗ )inv −→ L G ϕ∗I Ob(M M m (I)

Gm (I)) : Ob(M

Gm (I)) −→ L G is equivalent to ob(M given in Section M m (I) 5.8.

Gm (I)) are equal to • We put A := 1 + r1 /r2 . The virtual normal bundles N(M the following virtual Gm -equivariant vector bundles:

N E1M , E2M ⊗ IA + N E2M , E1M ⊗ I−A + N L, E2M ⊗ IA M M

M M ⊗ IA + ND E2∗ ⊗ I−A + N0 + ND E1∗ , E2∗ , E1∗

(5.64)

Gm (I) with the Gm -action of Here, In denote the trivial line bundles on M weight n. Before going into a proof, we give a remark. Recall the diagram (4.33). We put

Gm (I) or S, A := 1 + r1 /r2 as above. Let O1,rel (1) denote the line bundle on M

1 , [L], α∗ , (δ, k) . By Proposition induced by the tautological line bundle on M y 4.6.1, we have the following relations on S: u u

1 , E

2 ⊗ O1,rel (−A) F ∗ N E1M , E2M = G∗ N E

(5.65)

u u

,E

⊗ O1,rel (A) F ∗ N E2M , E1M = G∗ N E 2 1

(5.66)

2u ⊗ O1,rel (−r1 /r2 ) F ∗ N L, E2M = G∗ N L, E

(5.67)

M M

u , E

u ) ⊗ O1,rel (−A) , F ∗ ND (E1∗ , E2∗ ) = G∗ ND (E 1∗ 2∗

(5.68)

u u M M

2∗ F ∗ ND (E2∗ , E1∗ ) = G∗ ND (E , E1∗ ) ⊗ O1,rel (A) .

(5.69)

u u

u ), N(L, E

u ) and ND (E

a∗

au , E , Eb∗ ) are the virtual vector bundles on Here, N(E 2 b

1 , [L], α∗ , (δ, ) × M(

y 2 , α∗ ) given as follows: M y

ju ) := −

iu , E N(E

iu , E

ju ) (−1)l Rl pX∗ RHom(E

(5.70)

u ) (−1)l Rl pX ∗ Hom(L, E 2

(5.71)

u

u (−1)l Rl pD ∗ RHom 2 E i|D ∗ , Ej|D ∗

(5.72)

l=0,1,2

u ) := N(L, E 2

l=0,1,2

u , E

u ) := − ND (E i∗ j∗

l=0,1

u

u Here E

Gm (I)×D with the induced filtrations. i|D ∗ denote the restriction Ei ⊗ OM

204

5 Obstruction Theories of Moduli Stacks and Master Spaces

) and ob(M

) 5.9.2 Gm -Equivariant Lift of Ob(M

). We use the notaWe would like to obtain the Gm -equivariant structure of Ob(M tion in Subsection 5.7.1. We have the following commutative diagram:

M ⏐ ⏐ p!

Ψ

−−−2−→

Q ⏐ ⏐ p !

Ψ

, [L]) −−−1−→ B(W• , [P• ]) M(m, y We have the Gm -action on Z1 given by t · [u0 : u1 ] = [t · u0 : u1 ]. It induces We remark that Ψ2 is Gm -equivariant. We have the natural the Gm -action on Q.

, [L]) and Ψ2∗ LQ/B(W . Gm -equivariant structure of p∗ Ob(m, y • ,[P• ])

Let Ci (i = 1, 2) be bounded Gm -complexes on M . We have the induced Gm action on Ext0 (C1 , C2 ), where Ext0 (C1 , C2 ) denotes the vector space of the homo ). In the following, we say that morphisms of C1 to C2 in the derived category D(M a morphism ϕ : C1 −→ C2 is contained in the Gm -invariant part of the Ext0 -group, if ϕ is contained in the Gm -invariant part of Ext0 (C1 , C2 ).

, [L]) is conLemma 5.9.4 The morphism Ψ2∗ LQ/B(W [−1] −→ p∗ Ob(m, y • ,[P• ]) 0 tained in the Gm -invariant part of the Ext -group. Proof By using Remark 2.3.6, we obtain the natural Gm -equivariant representa C (B(W• , [P• ])) and C3 of tives C (Q), Ψ2∗ LQ ,

Ψ2∗ p ∗ LB(W• ,[P• ]) ,

Ψ2∗ LQ/B(W . • ,[P• ])

We regard them as Gm -equivariant complexes. We have the natural Gm -equivariant We put morphism α : C(B(W• , [P• ])) −→ C(Q). C Q/B(W • , [P• ]) := Cone(α). We obtain the Gm -equivariant quasi-isomorphism C3 C (Q/B(W • , [P• ])). ∗ ∗ [−1] −→ Ψ p L can be repreThen, the morphism Ψ2∗ LQ/B(W B(W• ,[P• ]) 2 • ,[P• ]) sented by a Gm -equivariant morphism

C (Q/B(W • , [P• ]))[−1] −→ C (B(W• , [P• ])).

, [L]) On the other hand, the morphism p∗ u : p∗ Ψ2∗ LB(W• ,[P• ]) −→ p∗ Ob(m, y 0 is contained in the Gm -invariant part of the Ext -group. Then, the claim of the lemma is clear. Applying the general non-sense in Subsection 2.5.2, we obtain a Gm -equivariant

). representative of Ob(M

5.9 Equivariant Obstruction Theory of Master Space

205

Recall that we need only the (−1)-truncated cotangent complexes for the conis isomorphic to its struction of virtual fundamental classes. Since LM

/M(m,

y ,[L]) 0-th cohomology sheaf, we have the distinguished triangle: τ≥ −1 p∗ LM(m, y,[L]) −→ τ≥ −1 LM

−→ LM

/M(m,

y ,[L]) −→ τ≥ −1 p∗ LM(m, y,[L]) [1] (5.73)

, [L]) −→ τ≥−1 p∗ LM(m, y,[L]) is contained in Since the morphism p∗ Ob(m, y the Gm -invariant part of the Ext0 -group, the morphism LM [−1] −→

/M(m,

y ,[L])

τ≥−1 p∗ LM(m, y,[L]) is also contained in the Gm -invariant part of the Ext0 -group, according to Lemma 5.9.4. Thus, we obtain a Gm -equivariant structure of τ≥ −1 LM

.

By construction, ob(M ) : Ob(M ) −→ τ≥ −1 LM

is contained in the Gm invariant part of the Ext0 -group. Therefore, we have obtained Gm -equivariant rep ).

), τ≥ −1 L and ob(M resentatives of Ob(M M

5.9.3 Comparison of Gm -Equivariant Structures of τ≥−1 LM

We also have a Gm -equivariant structure of τ≥ −1 LM

induced by a Gm -equivariant embedding into a smooth Deligne-Mumford stack. We use the notation in Subsections 4.2.1, 4.3.1 and 4.5.1.

m . We have the

m be the vector bundle as in (3.12). We put B := B ×Zm Z Let Z ss

−→ B natural morphism TH which is GL(Vm ) × Gm -equivariant. We obtain a

is Deligne-Mumford, Gm -equivariant immersion ι : M −→ B/ GL(Vm ). Since M

we can take a smooth Deligne-Mumford open substack P of B/ GL(Vm ), which

. contains M

. We put Let I denote the ideal sheaf of P corresponding to M

) := Cone I/I 2 −→ ι∗ C(P ) C(M

, where C(P ) := ΩP/k . It is naturally a Gm -complex, and it is a representaon M tive of the (−1)-truncated cotangent complex τ≥ −1 LM

/k . Recall that the complex

) is used in the localization theory due to Graber and Pandharipande [55]. C(M Lemma 5.9.5 The above two Gm -equivariant structures of τ≥ −1 LM

are equivalent.

:= Cone(Ω −→ Ω Proof We have the complex C (B) B

B/P

)[−1] on B. It is

naturally GL(Vm ) × Gm -equivariant, and thus it induces Gm -equivariant complex C (P ) on P . We have the natural Gm -equivariant quasi-isomorphism C(P ) −→

) := Cone I/I 2 −→ ι∗ C (P ) on M

, then we have the C (P ) on P . We put C (M

) −→ C (M

). natural Gm -equivariant quasi-isomorphism C(M

206

5 Obstruction Theories of Moduli Stacks and Master Spaces

The quotient

m . We have the natural GL(Vm )-action on A. We put A := A×Zm Z stack is denoted by Q. By Proposition 4.1.2, we have a GL(Vm )-equivariant map

and hence M(m, y

, [L]) −→ A,

, [L]) −→ Q. Q◦ (m, y

:= Cone p∗ Ω −→ Let p3 : B −→ A denote the projection. We put C (A) 3 A [−1]. The complex is provided with the natural GL(Vm )-action. Hence, p∗3 ΩA/Q

it induces the complex C (Q) on P . We have the natural morphism C (Q) −→ C (P ). It is clear that the morphism I/I 2 −→ ι∗ C (P ) factors through ι∗ C (Q). We

2 put C (M) := Cone I/I −→ ι∗ C (Q) . We have the exact sequences of the Gm -equivariant complexes: 0 −→ C (Q) −→ C (P ) −→ ΩP/Q −→ 0

on P

) −→ Ω

−→ 0 on M 0 −→ C (M) −→ C (M M /M We put C (P/Q) := Cone C (Q) −→ C (P ) . We have the Gm -equivariant morphism C (P/Q)[−1] −→ C (Q) on P . We have the natural Gm -equivariant morphism ι∗ C (P/Q)[−1] −→ ι∗ C (Q) −→ C (M) of the Gm -complexes on

. We put C1 (M

) := Cone ι∗ C(P/Q)[−1] −→ C (M) . M

) is Gm -equivariantly quasi-isomorphic to C (M

). We Let us show that C1 (M 2 2 put C0 := Cone(I/I −→ I/I ). We have the composite of the morphisms

) −→ I/I 2 [1] −→ C0 [1]. We have the naturally defined Gm -equivariant C (M quasi-isomorphism

) → C0 [1] [−1] −→ C (M

). Cone C (M The morphism I/I 2 −→ ι∗ C (Q) induces the Gm -equivariant quasi-isomorphism

) → C0 [1] [−1] −→ C1 (M

). Cone C (M We have the following commutative diagram:

−−−−→ M ⏐ ⏐ !

P −−−−→ ⏐ ⏐ !

Q ⏐ ⏐ !

M −−−−→ Q −−−−→ B(W• , [P• ]) It induces the following commutative diagram: C (Q) ⏐ ⏐

−−−−→ C (P ) ⏐ ⏐

C (B(W• , [P• ])) −−−−→ C (Q)

5.9 Equivariant Obstruction Theory of Master Space

207

And, we obtain the isomorphism C (Q/B(W • , [P• ])) C (P/Q). We also have

∗ C (M) p τ≤−1 LM . Hence, two Gm -equivariant structures are equivalent.

∗) 5.9.4 Gm -Equivariant structure of Ob(M

) ∗ Ob(M

∗ ), we have already Since we have the quasi-isomorphism Ob(M |M

∗ ). We give another description of obtained the Gm -equivariant structure of Ob(M the Gm -equivariant structure. We use the notation in Subsection 5.7.2.

M

× X. We have the natural Gm -equivariant structure on the sheaves E over M ∗

, L). On the other hand, the It induces the Gm -equivariant structure on p2 Ob(m, y morphisms Ψ3 and p2 are Gm -equivariant. Therefore, we have the Gm -equivariant representative of p∗2 Ψ3∗ LB(W• ,P• ) .

, L) is Gm -equivariant. Lemma 5.9.6 p∗2 Ψ3∗ LB(W• ,P• ) −→ p∗2 Ob(m, y

M (m). By Remark 2.3.6, Proof Let V• denote the canonical resolution of E ∗ h(V• , φ• )≤1 gives a Gm -equivariant representative of Ψ3X LB(W• ,P• )×X/X , where the Gm -equivariant structure of h(V• , φ• ) is induced by the Gm -equivariant struc

M . Therefore, a Gm -equivariant representative of Ψ3∗ LB(W• ,P• ) is given ture of E

, L) by ObG (V• , φ• ). On the other hand, the morphism ObG (V• , φ• ) −→ Ob(m, y is Gm -equivariant, because their Gm -equivariant structures are induced by that of

M E . Thus we are done.

, L) is contained in Then, the morphism Ψ4∗ LQ ∗ /B(W• ,P• ) [−1] −→ p∗2 Ob(m, y 0 the Gm -invariant part of the Ext -group. Therefore, we can take the Gm -equivariant

∗ ). representative of Ob(M

∗ ) are equivalent. Lemma 5.9.7 Two Gm -equivariant structures of Ob(M Proof We have the following diagram:

, L) Ψ4∗ LQ ∗ /Y (W• ,P• ) [−1] −−−−→ Ob(m, y ⏐ ⏐ a⏐ b⏐

, [L]) Ψ4∗ LQ ∗ /Y (W• ,[P• ]) [−1] −−−−→ Ob(m, y The morphisms are contained in the Gm -invariant part of the Ext0 -groups. Therefore, the induced Gm -equivariant structures on Cone(a) and Cone(b) are equivalent.

208

5 Obstruction Theories of Moduli Stacks and Master Spaces

5.9.5 Proof of Proposition 5.9.2 We use the notation in Section 4.5 and Section 5.7. Put F := Flag(Vm , N )GL(Vm ) = Z 1 ×B(W ) F , we have L and Z 1 := (Z1 )GL(Vm ) . Because Q • Q/B(W,[P ]) = LZ 1 /B(W,[P ]) ⊕ LF /B(W ) .

) is given by Cone Ψ ∗ L

, [L]) . The −→ p∗ Ob(m, y Recall Ob(M 2

Q/B(W,[P ])

)mov is given by the following: moving part ι∗i Ob(M

i ι∗i Ψ2∗ LZ 1 /B(W,[P ]) [1] ι∗i LM

Orel (−1) [1]

/M

i [1] LM

i /M

Then, it is easy see that the virtual normal bundle NM

(Mi ) is given by the line toi−1 , and that the weight of the induced Gm -action is (−1)i . By bundle Orel (−1) construction, we have the following:

)inv := Cone Ψ ∗ L

i )

, [L]) Ob(M ι∗i Ob(M 13 F /B(W ) [−1] −→ Ob(m, y It is easy to observe that we have the following commutative diagram: ϕ1

) −−−

i ) −→ Ob(M ι∗i Ob(M ⏐ ⏐ ⏐ ⏐

i )! ι∗ ob(M i ob(M )!

ι∗i LM

ϕ2

−−−−→

LM

i

Here, ϕ2 is the naturally defined one, and ϕ1 is the projection onto the invariant

i ). part. Thus, the induced obstruction theory is given by ob(M

5.9.6 Proof of Proposition 5.9.3 We use the notation in Subsections 5.7.2 and 5.8.4. Recall the expression of

∗ ) as in (5.39). Let us describe the decomposition of Ψ ∗ Γ ∗ L Ob(M 4 F /B(W0 ) . We (1)

(2)

take a decomposition W0 = W0 ⊕ W0 as in Subsection 5.8.4. We put F i :=

(i) Flag(W0 , Ii )GL(W (i) ) . We also put F := Flag(W0 , N )GL(W (1) )×GL(W (2) ) . Then, 0

0

0

we have the regular immersion F 1 × F 2 −→ F . We have the following Cartesian diagrams:

Gm (I) −−Ψ−21 −→ F1 × F2 −−−−→ F M ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! !

3 M

(1)

(2)

−−−−→ B(W• , W• ) −−−−→ B(W• )

Therefore, we have the isomorphism:

5.9 Equivariant Obstruction Theory of Master Space

209

∗ Ψ4∗ Γ ∗ LF /B(W• ) Ψ21 LF /B(W (1) ,W (2) ) 0

0

The invariant part of Ψ4∗ Γ ∗ LF /B(W• ) is isomorphic to the pull back of the relative (1)

(2)

cotangent bundle of F 1 × F 2 over B(W0 , W0 ). The moving part is the same as

the pull back of the conormal bundle of F 1 ×F 2 in F , which is naturally isomorphic ∨ to N0 [1]. Let us look at the decomposition of Ob(m, y, L). Corresponding to the decom

M = E1M ⊕ E2M , we obtain the decomposition of the resolution position ϕ∗I E V• = V1 • ⊕ V2 • . It induces the following decompositions: (1)

(2)

g(V• )inv = g(V• ) ⊕ g(V• ), (1) (2) ∨ (2) (1) ∨ [−1] ⊕ Hom V• , V• [−1] g(V• )mov = Hom V• , V• (1)

grel (V• , φ)inv = grel (V• , φ), (2) ∨ grel (V• , φ)mov = Hom P• , V• (1)

(2)

gD (V, F∗ )inv = gD (V (1) , F∗ ) ⊕ gD (V (2) , F∗ ), (1)∗ (2)∗ ∨ (2)∗ (1)∗ ∨ [−1] ⊕ C1 VD , VD [−1] gD (V, F∗ )mov = C1 VD , VD (See Subsection 2.1.6 for C1 .) (1)

(2)

g(V• | D )inv = g(V•|D ) ⊕ g(V•|D ), g(V•|D )mov = HomOD (V•|D , V•|D )∨ [−1] ⊕ HomOD (V•|D , V•|D )∨ [−1] (1)

(2)

(2)

(1)

We also have gd (V• )inv = gd (V• ). The contribution to the virtual normal bundle can be calculated formally. We can also easily observe that Ob(m, y, L)inv is naturally

3 ) given in Subsection 5.8.2. Then, ι∗ Ob(M

∗ )inv is obtained isomorphic to Ob(M as the cone of the composite of the following morphisms:

3 ) LF 1 ×F 2 /B(W (1) ,W (2) ) [−1] −−−−→ LB(W (1) ,W (2) ) −−−−→ Ob(M

Gm (I)). (See Subsection 5.8.6.) We also have the Namely, it is the same as Ob(M diagram (5.40). Then, it is easy to observe that the induced obstruction theory is

Gm (I)). given by ob(M

5.9.7 Case in Which a 2-Stability Condition is Satisfied We give only the statement about the Gm -equivariant obstruction theory of the master space in the case that a 2-stability condition is satisfied. The proof is similar to those of Proposition 5.9.1, Proposition 5.9.2 and Proposition 5.9.3.

210

5 Obstruction Theories of Moduli Stacks and Master Spaces

Proposition 5.9.8 Under the setting in Subsection 4.7.1, the following claims hold:

. • We have a Gm -equivariant obstruction theory of the master space M Gm

• The induced obstruction theory of the fixed point sets Mi and M (I) are equivalent to the moduli theoretic obstruction theory.

i in M

are given by Orel (−1)i−1 with

i ) of M • The virtual normal bundles N(M the Gm -action of the weight (−1)i .

Gm (I)) of M

Gm (I) in • We put A = 1 + r1 /r2 The virtual normal bundles N(M

M are equal to the following virtual Gm -equivariant vector bundles

N(E1M , E2M ) ⊗ IA + N(E2M , E1M ) ⊗ I−A + N(L, E2M ) ⊗ IA

+ ND (E1M∗ , E2M∗ ) ⊗ IA + ND (E2M∗ , E1M∗ ) ⊗ I−A

(5.74)

Here, Iw denotes the trivial line bundle with the Gm -action of weight w, and the terms are as in (5.61), (5.62) and (5.63). (See also (5.65)–(5.69).)

5.9.8 Oriented Reduced L-Bradlow Pairs We give only the statement on the Gm -equivariant obstruction theory of master spaces in the case of oriented reduced L-Bradlow pairs, under the setting of Subsection 4.7.2. We prepare some notation.

For any decomposition type I, we have the virtual vector bundles N(EiM , EjM ),

ND (EiM∗ , EjM∗ ) and N(L1 , E2M ) given as in (5.61), (5.62) and (5.63). (See also

−→ M(m, y

, [L]) denote the naturally defined mor(5.65)–(5.69).) Let ϕ : M (2) (2) := ϕ∗ Orel (−1), which is naturally provided with the Gm phism. We put I

Gm (I): action. We have the following virtual vector bundle on M

N(L2 ⊗ I (2) , E1M ) :=

(−1)i Ri pX ∗ Hom L2 ⊗ I (2) , E1M

i=0,1,2

Lemma 5.9.9 Under the setting in Subsection 4.7.1, the following holds:

. • We have a Gm -equivariant obstruction theory of the master space M G

m (I) are equiv i and M • The induced obstruction theory of the fixed point sets M alent to the moduli theoretic obstruction theory.

i ) of M

i in M

are given by O(1) (−1)i−1 with • The virtual normal bundles N(M rel the Gm -action of the weight (−1)i . G

m (I) of M

Gm (I) in • We set A := 1 + r1 /r2 . The virtual normal bundles N M

are given by the following: M

5.9 Equivariant Obstruction Theory of Master Space

211

N(E1M , E2M ) ⊗ IA + N(E2M , E1M ) ⊗ I−A

+ N(L1 , E2M ) ⊗ IA + N(L2 ⊗ I (2) , E1M ) ⊗ I−A

+ ND (E1M∗ , E2M∗ ) ⊗ IA + ND (E2M∗ , E1M∗ ) ⊗ I−A • We have the following equality on S:

F ∗ N(L2 ⊗ I (2) , E1M ) = G∗ N(L2 , E1u ) ⊗ O1,rel (A) ⊗ O2,rel (1) Here, we put N(L2 , E1u ) := RpX ∗ Hom(L2 , E1u ).

(5.75)

Chapter 6

Virtual Fundamental Classes

We have constructed obstruction theories of some stacks in Chapter 5. Under some assumption, they are perfect and give virtual fundamental classes. We will study their property in this chapter. In Section 6.1, we obtain virtual fundamental classes for some stacks, by showing the perfectness of the obstruction theories. We compare the virtual fundamental classes of moduli stacks of δ-stable oriented reduced L-Bradlow pairs and δ-stable L-Bradlow pairs in Section 6.2. Although the moduli stacks are isomorphic up to e´ tale proper morphisms, the obstruction theories are not the same in general, and we obtain the vanishing of the virtual fundamental class of the moduli of δ-stable L-Bradlow pairs in the case pg = dim H 2 (X, OX ) > 0. In Section 6.3, we study the virtual fundamental classes of moduli stacks of objects with rank one. In Subsection 6.3.1, we look at a moduli of L-abelian pairs. In particular, we give a detailed description of the virtual fundamental class when H 2 (X, O) = 0 and H 1 (X, O) = 0 are satisfied. In Subsection 6.3.3, we study the obstruction theory of parabolic Hilbert schemes. In the rest of this section, we show the splitting stated in Proposition 6.3.8. In Sections 6.4–6.6, we give some relations of the virtual fundamental classes of some moduli stacks.

6.1 Perfectness of Obstruction Theories 6.1.1 Moduli Stacks of Semistable Objects Let y be an element of T ype, and let α∗ be a system of weights. Let L be a line bundle on X. We use the notation in Section 5.6. We will prove the following propositions in Subsection 6.1.4 after some preparation (Subsection 6.1.3). The expected dimensions can be calculated formally, which will be given in Subsection 6.1.5.

T. Mochizuki, Donaldson Type Invariants for Algebraic Surfaces: Transition 213 of Moduli Stacks, Lecture Notes in Mathematics 1972, DOI: 10.1007/978-3-540-93913-9 6, c Springer-Verlag Berlin Heidelberg 2009

214

6 Virtual Fundamental Classes

Proposition 6.1.1 Let m be a sufficiently large integer.

) of Mss (

• The obstruction theory Ob(m, y y , α∗ ) is perfect in the sense of Definition 2.4.3. • Let δ ∈ P br . The obstruction theory Ob(m, y, L) of Mss (y, L, α∗ , δ) is perfect. Proposition 6.1.2 Assume rank(y) > 1. Let δ ∈ P br . Let m be a sufficiently large integer.

, L) of Mss (

y , L, α∗ , δ) is perfect. • The obstruction theory Ob(m, y

, [L]) of Mss (

y , [L], α∗ , δ) is perfect. • The obstruction theory Ob(m, y Proposition 6.1.3 Assume rank(y) = 1. Let m be a sufficiently large integer.

) unless i = 0. In particular, the moduli • We have the vanishing Hi Ob(m, y M(

y ) is smooth. • Assume that the 2-vanishing condition is satisfied for (y, L). Then, the ob , L) of M(

struction theory Ob(m, y y , L) is perfect, and the obstruction theory

, [L]) of M(

Ob(m, y y , [L]) is perfect. We remark Ob(m, y, L) is always perfect (Proposition 6.1.1). Proposition 6.1.4 Let L = (L1 , L2 ) be a pair of line bundles on X. Let δ = (δ1 , δ2 ) be a pair of sufficiently small parameters as in Lemma 3.5.16. Let α∗ be a system of weights. Assume that the 2-vanishing condition holds for (y, L2 , α∗ ).

, [L]) is perfect on Mss (

y , [L], α∗ , δ). Then, the obstruction theory Ob(m, y Notation 6.1.5 By Propositions 6.1.1–6.1.4, we have the perfect obstruction theories of the moduli stacks M of corresponding stable objects, which induces virtual fundamental classes according to Proposition 2.4.4. They are denoted by [M]. We use the symbol M Φ to denote the evaluation of a cohomology class Φ via the virtual fundamental class [M]. (See Section 7.1.) Remark 6.1.6 It may be more standard to use the symbols [M]vir and [M]vir Φ. Since we almost always use virtual fundamental classes for evaluation, we prefer to omitting “vir”.

6.1.2 Master Spaces and Some Related Stacks

denote the master spaces explained in Subsections 4.5.1, 4.7.1 and 4.7.2. ReLet M

) for Ob(M

) (Subsections call we have constructed the obstruction theories Ob(M 5.7.1, 5.7.6 and 5.7.7).

) are perfect. Proposition 6.1.7 Ob(M

) explained in Subsection 5.7.1. The other cases Proof We consider only Ob(M can be argued in similar ways. We have the naturally defined smooth morphism

−→ M(m, y

, [L]). We remark that the image of p is contained in the open p:M

6.1 Perfectness of Obstruction Theories

215

substack M := Mss (

y , [L], α∗ , δ). Then, the claim immediately follows from the diagram (5.36) and Proposition 6.1.2. The following claims can be shown by a similar argument. Proposition 6.1.8 Let m be a sufficiently large integer. s (

) of M y , α∗ , +) is perfect. (See Subsection • The obstruction theory Ob(m, y 5.7.5.) m, y, L of M s y

, L, α∗ , (δ, ) is perfect. (See • The obstruction theory Ob Subsections 3.3.3 and 5.7.4.) • Assume one of the following: – rank(y) > 1. – rank(y) = 1, and the 2-vanishing condition holds for (y, L, α∗ , δ). s y

, [L] of M

, [L], α∗ , (δ, ) is perfect. Then, the obstruction theory Ob m, y (See Subsections 3.3.3 and 5.7.3.) M

Gm (I) × A1 over

Gm (I)) of M Recall that we obtained obstruction theory Ob( a (M

Gm (I)). A in Proposition 5.8.1. The specialization at t = a is denoted by Ob 1

a (M

Gm (I)) are perfect for any Proposition 6.1.9 The obstruction theories Ob a ∈ k. Proof By the distinguished triangle (5.42), we have only to show that Ob(m, y 1 , L)

2 ) are perfect, which is given in Proposition 6.1.8. and Ob(m, y

Gm (I) with respect Notation 6.1.10 We obtain the virtual fundamental classes of M a (M

Gm (I)) for each a ∈ k. It is independent of the to the obstruction theories Ob choice of a, according to Proposition 7.2 of [6] and Proposition 6.1.9. Therefore,

Gm (I)]. we denote it by [M Proposition 6.1.11 We use the notation in Subsection 4.6.1. In the diagram (4.35), we have the relation " Gm # " ss #

(I)] = G ∗ M ss y 1 , L, α∗ , (δ, k0 ) × M (

y 2 , α∗ , +) . F ∗ [M Proof It follows from the diagram (5.41).

We can show the following propositions by the same argument. Proposition 6.1.12 Under the situation of Subsection 5.8.7, the following claims hold. a (M

Gm (I)) in Proposition 5.8.9 are perfect for any • The obstruction theories Ob a ∈ k. a (M

Gm (I)).

Gm (I)] be the virtual fundamental class associated to Ob • Let [M Then, we have the relation

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6 Virtual Fundamental Classes

Gm

(I)] = G ∗ [Ms (y 1 , L, α∗ , δ) × Ms (

F ∗ [M y 2 , α∗ )] in the diagram (4.38).

Proposition 6.1.13 Under the situation of Subsection 5.8.8, the following claims hold. a (M

Gm (I)) in Proposition 5.8.10 are perfect for • The obstruction theories Ob any a ∈ k. a (M

Gm (I)).

Gm (I)] be the virtual fundamental class associated to Ob • Let [M Then, we have the relation Gm

(I)] = G ∗ [Ms (y 1 , L, α∗ , δ) × Ms (

y 2 , α∗ )] F ∗ [M in the diagram (4.7.9).

6.1.3 Vanishing of Some Cohomology Groups We use the notation in Chapter 5. Let E be a torsion-free sheaf on X. Let V• = (V−1 → V0 ) be a locally free resolution of E. Then, we put C(E) := g∨ (V• ), where g∨ (V• ) denotes the dual of g(V• ) as OX -complexes. We have the decomposition C(E) = C ◦ (E) ⊕ C d (E) corresponding to g(V• ) = g◦ (V• ) ⊕ gd (V• ). Let F∗ be a quasi-parabolic structure of E at D. We have the complexes g(V•|D ) and gD (V• , F∗ ) with the natural morphism gD (V• , F∗ ) −→ g(V•|D ) on D. (See Subsection 5.5.1.) Let g∨ (V•|D ) and g∨ D (V• , F∗ ) denote the dual complexes as OD complexes. We have the natural morphism C(E) −→ g∨ (V• )|D = g∨ (V•|D ). ∨ Thus, we obtain the morphism α : C(E) ⊕ g∨ D (V• , F∗ ) −→ g (V•|D ), and we set C(E, F∗ ) := Cone(α)[−1]. As explained in Subsection 5.5.3, we have the decompositions g(V•|D ) = g◦ (V•|D ) ⊕ gd (V•|D ) and gD (V• , F∗ ) = g◦D (V• , F∗ ) ⊕ gdD (V• , F∗ ). We have the corresponding decomposition C(E, F∗ ) = C ◦ (E, F∗ ) ⊕ C d (E, F∗ ). It is easy to see C d (E, F∗ ) = C d (E). Lemma 6.1.14 The hyper-cohomology group Hi X, C ◦ (E, F∗ ) vanishes unless i = −1, 0, 1. If (E, F∗ ) is stable with respect to some system of weights α∗ , we also have H−1 (X, C ◦ (E, F∗ )) = 0. Proof The i-th cohomology sheaf Hi C ◦ (V• , F∗ ) vanishes unless −2 ≤ i ≤ 1 ◦ −1 by construction. It is easy to see that the morphisms C ◦ (E)−2 −→ ◦ C (E) and ◦∨ −2 ◦∨ −1 −2 gD (V• , F∗ ) −→ gD (V• , F∗ ) are injective, and hence H C (E, F∗) = 0. Since C ◦ (E)1 −→ g◦∨ (V•|D )1 is surjective, we have H1 C ◦ (E, F∗ ) = 0. LetU ⊂ X − D be a Zariski open subset on which E is locally free. Then, H0 C ◦ (E, F∗ ) |U = 0. Therefore, the support of the sheaf H0 C ◦ (E, F∗ ) is at most 1-dimensional.

6.1 Perfectness of Obstruction Theories

217

Then, we obtain the vanishing Hi X, C ◦ (E, F∗) unless i = −1, 0, 1 by using the spectral sequence. It is easy to show that H−1 X, C ◦ (E, F∗ ) is the set of ento be stable with domorphisms of (E, F∗ ) whose trace is 0.If (E, F∗ ) is assumed respect to some weight α∗ , we obtain H−1 X, C ◦ (E, F∗ ) = 0. Let us consider the rank one case. Lemma 6.1.15 Let (E, F∗ ) be as above, with rank(E) = 1. Then, we have the vanishings Hi X, C ◦ (E, F∗ ) = 0 unless i = 0. Proof Let x be a point of X satisfying one of the following conditions: • x is contained in X − D, and E is locally free around x. • x is contained in D, E is locally free around x as OX -module, and Coki (E) are locally free around x as OD -modules. Around such a point, we can compute the cohomology sheaves of C ◦ (E, F∗ ) in the case V0 = E and V−1 = 0. Hence, it is easy to check C ◦ (E, F∗ ) 0 around such a point. We know Hi C ◦ (E, F∗ ) = 0 unless i = −1, 0, as explained in the proof of Lemma 6.1.14. By the above consideration, we know that the supports of Hi C ◦ (E, F∗ ) are at most 0-dimensional. Hence, we obtain Hi X, C ◦ (E, F∗ ) = 0 unless (E, F∗ ) is always stable in the case rank(E) = 1, we have i = −1, 0. Since H−1 X, C ◦ (E, F∗ ) = 0. Thus we are done. Let L be a line bundle on X. Let φ be an L-section of E. We take a locally free resolution P• = (P−1 −→ P0 ) of the line bundle L so that we have a lift φ• : P• −→ V• of φ. Let g∨ rel (V• , φ• ) denote the dual of grel (V• , φ• ) = Hom(P• , V• )∨ . Let C(E) −→ g∨ rel (V• , φ• )[1] be the morphism induced by the natural one grel (V• , φ• )[−1] −→ g(V ). It induces the morphisms: αL : C(E, F∗ ) −→ g∨ rel (V• , φ• )[1],

◦ αL : C ◦ (E, F∗ ) −→ g∨ rel (V• , φ• )[1]

◦ )[−1]. It We put C(E, F∗ , φ) := Cone(αL )[−1] and C ◦ (E, F∗ , φ) := Cone(αL ◦ is easy to see that C(E, F∗ , φ) and C (E, F∗ , φ) are well defined in the derived category D(X).

Assume φ = 0. Then, the hyperLemma 6.1.16 Let (E, F∗ , φ) be as above. cohomology groups Hi X, C(E, F∗ , φ) vanish unless i = −1, 0, 1. If moreover (E, F∗ , φ) is (α∗ , δ)-stable for some δ ∈ P br and some system of weights α∗ , then we have H−1 X, C(E, F∗ , φ) = 0. Proof We have Hi C(E, F∗ , φ) = 0 unless −2 ≤ i ≤ 1 by construction. The vanishing H−2 C(E, F∗ , φ) = 0 can be shown by an argument employed in the proof of Lemma 6.1.14. Since the morphism Hom(P0 , V0 ) −→ Hom(P−1 , V0 ) 1 C(E, F∗ , φ) = 0. It is easy to show the is surjective, weobtain the vanishing H vanishing of H0 C(E, F∗ , φ) around any point x ∈ X−D with φ(x) = 0 such that E is locally free around x. Therefore, the support of H0 C(E, F∗ , φ) is at most 1

218

6 Virtual Fundamental Classes

dimensional. Then, we obtain Hi X, C(E, F∗ , φ) = 0 unless i = −1, 0, 1 by using the spectral sequence. Since H−1 X, C(E, F∗ , φ) is the set of endomorphisms of (E, F∗ , φ), we obtain H−1 X, C(E, F∗ , φ) = 0 from the stability assumption of (E, F∗ , φ). Lemma 6.1.17 Let (E, F∗, φ) be as above. We assume φ = 0 and rank(E) > 1. Then, Hi X, C ◦ (E, F∗ , φ) vanish unless i = −1, 0, 1. Proof The i-th cohomology sheaf Hi C ◦ (E, F∗ , φ) vanish unless −2 ≤ i ≤ 1 by construction. The vanishings Hj C ◦ (E, F∗ , φ) for j = −2, 1 can be shown by the argument employed in the proof of Lemma 6.1.16. Let x be a point of X − D with φ(x) = 0 such that E is locally free around x. Under our assumption rank(E) > 1, we can easily check the vanishing H0 C ◦ (E, F∗ , φ) = 0 around such a point x. Thus, the claim can be shown by using the spectral sequence. Let us consider the rank one case. Lemma 6.1.18 Let (E, F∗, φ) be as above. = 0 and rank(E) We assume φ = 1. Moreover, we assume H 2 X, L−1 ⊗ E = 0. Then, Hi X, C ◦ (E, F∗ , φ) vanish unless i = 0, 1. Proof We have the following distinguished triangle: ◦ ◦ ∨ g∨ rel (V, φ• ) −→ C (E, F∗ , φ) −→ C (E, F∗ ) −→ grel (V, φ• )[1]

Hence, we obtain the following long exact sequence: · · · −→ H i X, L−1 ⊗ E −→ Hi X, C ◦ (E, F∗ , φ) −→ Hi X, C(E, F∗ ) −→ H i+1 X, L−1 ⊗ E −→ · · · Then, the claim follows from the assumption and Lemma 6.1.15.

Let L = (L1 , L2 ) be a pair of line bundles on X. Let φj be Lj -sections of E. (j) (j) By taking appropriate resolutions P• of Lj and lifts φj• : P• −→ V• of φj , we ◦ obtain the complexes C(E, F∗ , φ1 , φ2 ) and C (E, F∗ , φ1 , φ2 ) as above. They are well defined in the derived category D(X). ⊗ E = 0 and Lemma 6.1.19 Assume H 2 X, L−1 2 φj = 0 for j = 1, 2. We also assume rank(E) > 1. Then, we have Hi X, C ◦ (E, F∗ , φ1 , φ2 ) = 0 unless i = −1, 0, 1. If (E, F∗ , φ1 , φ2 ) is (α∗ , δ1 , δ2 )-stable, then we also have H−1 X, C ◦ (E, F∗ , φ1 , φ2 ) = 0. Proof We have the exact sequence: 0 −→ Hom P (2) , V• −→ C(E, F∗ , φ1 , φ2 ) −→ C(E, F∗ , φ1 ) −→ 0 Then, the claims can be reduced to Lemma 6.1.16 and Lemma 6.1.17.

6.1 Perfectness of Obstruction Theories

219

6.1.4 Proof of the Propositions in Subsection 6.1.1 Proposition 6.1.1 immediately follows from the following lemma.

) of M(m, y

) is perfect in the Lemma 6.1.20 The obstruction theory Ob(m, y sense of Definition 2.4.3. The obstruction theory Ob(m, y, L) of M(m, y, L) is perfect.

). We would like to show that Ob(m, y

) is quasiProof Let us consider Ob(m, y

). isomorphic to a complex of locally free sheaves E −1 → E 0 → E 1 on M(m, y

) is obtained from the push-forward of perfect complexes on Note that Ob(m, y

) × X and M(m, y

) × D. Hence, it is easy to show that Ob(m, y

) is quasiM(m, y isomorphic to a bounded complex of locally free sheaves by using the projectivity

) = 0 unless i = of X and D. Therefore, we have only to check H i i∗z Ob(m, y

). Let (E, F∗ , ρ) be the parabolic −1, 0, 1 for any closed point iz : z ∈ M(m, y

) oriented torsion-free sheaf corresponding to z. Then, the dual of Hi i∗z Ob(m, y −i ◦ is isomorphic to H X, C (E, F∗ ) in the case i = 0, or H0 X, C ◦ (E, F∗ ) ⊕

) follows H 1 (X, O)[0] in the case i = 0. Therefore, the perfectness of Ob(m, y from Lemma 6.1.14. The perfectness of Ob(m, y, L) can be shown by using a similar argument with Lemma 6.1.16.

, L) of Lemma 6.1.21 Assume rank(y) > 1. The obstruction theory Ob(m, y

, L) is perfect. The obstruction theory Ob(m, y

, [L]) of M(m, y

, [L]) is M(m, y perfect. Proof We have the following commutative diagram: Obrel (m, y )[−1] −−−−→ ⏐ ⏐ ! Obd (m, y) We set

−−−−→

Obrel (m, y, L)⊕ −−−−→ Obrel (m, y, L)[−1] Obrel (m, y )[−1] ⏐ ⏐ (6.1) ⏐ ⏐ ! ! Ob(m, y)

−−−−→

Ob◦ (m, y)

C1 := Cone Obrel (m, y )[−1] −→ Obd (m, y) , C2 := Cone Obrel (m, y, L)[−1] −→ Ob◦ (m, y) .

Thanks to (6.1), we have only to show that C1 and C2 are perfect of amplitude in [−1, 1]. It is easy to see that C1 is isomorphic to H 1 (X, O)∨ ⊗ O[0]. To check the claim for C2 , wehave only to consider i∗z C2 as in the proof of Lemma 6.1.20. The dual of Hi i∗z C2 are isomorphic to H−i X, C ◦ (E, F∗ , φ) . Thus, the claim for C2 follows from Lemma 6.1.17. Therefore, we obtain the first claim of the lemma.

, [L]) = 0 for To show the second claim, we have only to show Hi Ob(m, y i < −1. Let π : Ms (

y , L) −→ Ms (

y , [L]) be the natural morphism. Since π is

220

6 Virtual Fundamental Classes

, [L]) = 0 for i < −1. Because smooth, we have only to show Hi π ∗ Ob(m, y

, L) for i < −1, the claim follows from

, [L]) Hi Ob(m, y Hi π ∗ Ob(m, y Lemma 6.1.20. Let us show Proposition 6.1.3. The first claim can be shown by using Lemma 6.1.15 and the argument in the proof of Lemma 6.1.20. The second and third claims can be shown by using Lemma 6.1.18 and the argument in the proof of Lemma 6.1.21. Proposition 6.1.4 can be shown by using Lemma 6.1.19 and the argument in the proof of Lemma 6.1.21.

6.1.5 Expected Dimension We put pg := dim H 2 (X, O), and let χ(OX ) denote the Euler number of OX . It is easy to obtain the formulas of the expected dimension in the non-parabolic case. Let y be an element of H ev (X). The H 2i (X)-component of y is denoted by yi . We put 2 y1 − 2y0 · y2 ) − y02 · χ(OX ). d(y) := X

For a line bundle L, we set drel (y, L) :=

ch(L−1 ) · y · Td(X)

X

Proposition 6.1.22 For moduli of non-parabolic objects, the expected dimensions are as follows: • • • •

dimf M(

y ) = d(y) + 1 + pg . dimf M(y, L, δ) = d(y) + drel (y, L). y , L, δ) = d(y) + 1 + pg + drel (y, L). dimf M(

y , [L], δ) = d(y) + pg + drel (y, L). dimf M(

Proof Let us consider the first case. For y ), ∈ M(

the virtual tanany (E, ρ) gent space is K-theoretically given by i (−1)i Hi X, C ◦ (E) + H 1 (X, OX ). The Euler number can be easily calculated, and it is given by d(y) + χ(OX ) + dim H 1 (X, OX ) = d(y) + 1 + pg . The second and third cases can be argued similarly. Let p denote the natural morphism M(

y , L, δ) −→ M(

y , [L], δ). We have the distinguished triangle LM(m, y,L)/M(m, y,[L]) [−1] −→ p∗ Ob(m, y , [L]) −→ Ob(m, y , L) −→ LM(m, y,L)/M(m, y,[L]) on M(m, y, L). Then, the fourth claim follows from the third claim. The contribution of the parabolic structure can also be calculated formally. Let y = (y, y1 , y2 , . . . , yl ) ∈T ype. In this case, yi can be regarded as elements of H ev (D). We decompose j≤i yj = si + wi , where si ∈ H 0 (D) and wi ∈ H 2 (D).

6.2 Comparison of Oriented Reduced Case and Unoriented Unreduced Case

221

Let g denote the genus of D. Let drel (y, y) denote the Euler number of the complex grel (V• , F∗ ) for (E, F∗ ) of type y. We have the following equality: drel (y, y) = (1 − g)

l−1

l−1 si+1 · wi − si · wi+1 si+1 (si+1 − si ) +

i=1

i=1

D

Proposition 6.1.23 y , α∗ ) = d(y) + 1 + pg + drel (y, y) • dimf M(

y , [L], α∗ , δ) = d(y) + pg + drel (y, y) + drel (y, L). • dimf M(

6.2 Comparison of Oriented Reduced Case and Unoriented Unreduced Case 6.2.1 Statements

, [L]) −→ M(m, y, L) which is e´ tale We have the natural morphism κ : M(m, y proper of degree rank(y)−1 . But the obstruction theories are not the same in the case pg := dim H 2 (X, O) > 0. We would like to compare them. Proposition 6.2.1 We have the following commutative diagram in the derived cat , [L])): egory D(M(m, y

, [L]) κ∗ Ob(m, y, L) −−−−→ Ob(m, y ⏐ ⏐ ⏐ ⏐ ! !

κ∗ LM(m,y,L) −−−−→ LM(m, y,[L]) We also have the following distinguished triangle:

, [L]) κ∗ Ob(m, y, L) −→ Ob(m, y −→ H 2 (X, O)∨ ⊗ OM(m, y,[L]) [1] −→ κ∗ Ob(m, y, L)[1] A proof will be given in the next subsection. Before going into a proof, we give some consequences. y , [L], α∗ , δ) −→ Ms (y, L, α∗ , δ) which is We have the natural morphism Ms (

−1 e´ tale proper of degree rank(y) , which is also denoted by κ. Proposition 6.2.2 Assume rank(y) > 1.

" # • In the case pg > 0, we have the vanishing Ms (y, L, α∗ , δ) = 0. • In the case pg = 0, we have the following relation: " # " # y , [L], α∗ , δ) κ∗ Ms (y, L, α∗ , δ) = Ms (

222

6 Virtual Fundamental Classes

, [L]) of Proof Under the assumption rank(y) > 1, the obstruction theory ob(m, y y , [L], α∗ , δ) is perfect. From Proposition 6.2.1, we have the relation: Ms (

" " # # y , [L], α∗ , δ) = κ∗ Ms (y, L, α∗ , δ) Eu H 2 (X, O) ⊗ O ∩ Ms (

Then the claim is clear.

If rank(y) = 1, the obstruction theory of M(

y , [L]) is not necessarily perfect. Hence, a similar vanishing result is not always correct. However, we obtain the following proposition by the same argument. Since the stability condition is trivial, we omit to distinguish “s”, α∗ and δ. Proposition 6.2.3 Let y be an element of T ype such that rank(y) = 1. " # • In the case pg > 0, we have M(y, L) = 0, if H 2 X, L−1 ⊗ E = 0 for any (E∗ , φ) ∈ M(y, L).

, [L] ]. • In the case pg = 0, we have the relation κ∗ [M(y, L)] = [M y We remark that the assumption in the first claim is always satisfied in the case pg = 0. s (y, L, α∗ , δ, ) denote the natus (

y , [L], α∗ , δ, ) −→ M Similarly, let κ : M rally induced morphism which is e´ tale proper of degree rank(y)−1 . By the same argument, we obtain the following propositions. Proposition 6.2.4 Assume rank(y) > 1. s (y, L, α∗ , δ, )] = 0. • In the case pg > 0, [M " s # " s # (y, L, α∗ , δ, ) = M (

• In the case pg = 0, κ∗ M y , [L], α∗ , δ, ) .

Proposition 6.2.5 Assume rank(y) = 1.

s (y, L, α∗ , δ, )] = 0 if H 2 X, L−1 ⊗ E = 0 • In the case pg > 0, we have [M for any (E∗ , φ) ∈ M(y, L). " s # " s # (y, L, α∗ , δ, ) = M (

• In the case pg = 0, κ∗ M y , [L], α∗ , δ, ) . As a corollary, we obtain the following vanishing result in the case pg > 0 for the virtual fundamental classes of the fixed point sets of master spaces. Proposition 6.2.6 Let I = (y 1 , y 2 , I1 , I2 ) be a decomposition type (Definition 4.4.2). Assume pg > 0. Then, we have [M Gm (I)] = 0 if one of the following conditions is satisfied. • rank(y 1 ) > 1. • rank(y 1 ) = 1 and H 2 X, L−1 ⊗ E = 0 for any torsion-free sheaf E of type y. Proof It follows from Propositions 6.1.11, 6.2.4 and 6.2.5.

6.2 Comparison of Oriented Reduced Case and Unoriented Unreduced Case

223

6.2.2 Proof of Proposition 6.2.1

Let V[L], V(L) and V[L] denote the canonical resolutions of the universal sheaves

, [L]) × X, respectively. on M(m, y, [L]) × X, M(m, y, L) × X and M(m, y Let us look at Ob(m, y, L). From (5.20), we obtain the following commutative diagram on M(m, y, L): Obrel (V(L), φ• )[−1] −−−−→ Ob(m, y) ⏐ ⏐ ⏐ ⏐ f1 ! ! O[−1] We set

−−−−→

(6.2)

O[−1]

Obrel (V(L), φ• ) := Cone Obrel (V(L), φ• ) −→ O [−1], Ob(m, y) := Cone Ob(m, y) −→ O[−1] [−1].

We obtain the following commutative diagram on M(m, y, L): Obrel (V(L), φ• )[−1] −→ Obrel (V(L), φ• )[−1] −→ LM(m,y,L)/M(m,y) [−1] ↓ ↓ ↓ Ob(m, y) −→ Ob(m, y) −→ LM(m,y) We put Ob(m, y, L) := Cone Obrel (V(L), φ• )[−1] −→ Ob(m, y) . Then, we obtain the morphisms on M(m, y, L): Ob(m, y, L) −→ Ob(m, y, L) −→ LM(m,y,L) Since the first morphism is a quasi-isomorphism, the composite gives an equivalent obstruction theory. Let π2 : M(m, y, L) −→ M(m, y, [L]) denote the natural projection. By Lemma 5.4.9, we have the following commutative diagram on M(m, y, L) from the diagram (6.2): π2∗ Ob rel (V[L], [φ• ])[−1] −−−−→ Obrel (V(L), φ• )[−1] −−−−→ Ob(m, y) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! !

LM(m,y)Gm /M(m,y)

−−−−→

O[−1]

−−−−→

O[−1]

In particular, we have the following isomorphism: π2∗ Obrel (V[L], [φ• ]) Obrel (V(L), φ• )

(6.3)

224

6 Virtual Fundamental Classes

, [L]), we have Obrel (V[L], On M(m, y ρu ) := Cone Obd (V[L]) −→ LPic , and the trace map induces the following commutative diagram:

ρu )[−1] −−−−→ Ob(m, y) Obrel (V[L], ⏐ ⏐ ⏐ ⏐ ! ! O[−1] −−−−→ O[−1]

ρu ) := Cone Obrel (V[L], ρu ) −→ O [−1]. Then, we have We put Obrel (V[L], the following commutative diagram:

ρu ) −−−−→ Ob(m, y) κ∗ π2∗ Obrel (V[L], [φ• ]) ⊕ Obrel (V[L], ⏐ ⏐ ⏐ ⏐ ! !

Obrel (V[L], [φ• ]) ⊕ Obrel (V[L], ρu )

−−−−→ Ob(m, y)

We put

, [L]) := Cone κ∗ π2∗ Obrel (V[L], [φ• ])⊕Obrel (V[L], Ob(m, y ρu ) −→ Ob(m, y) .

, [L]): Then, we have the following morphisms on M(m, y

, [L]) −→ Ob(m, y

, [L]) −→ LM(m, y,[L]) Ob(m, y Since the first morphism is a quasi-isomorphism, the composite gives the equivalent obstruction theory. By the construction, we have the following isomorphism:

Obrel (V[L], ρu ) H 2 (X, O)∨ ⊗ O[2] Then, the claim of the proposition immediately follows.

6.3 Rank One Case 6.3.1 Moduli of L-Abelian Pairs Let L be a line bundle on X. An L-Bradlow pair (E, φ) is called an L-abelian pair, if E is a line bundle. Similarly, a reduced L-Bradlow pair (E, [φ]) is called a reduced L-abelian pair, if E is a line bundle. Let c ∈ H 2 (X). We denote by M (c, L) c, [L]) a moduli of L-abelian pairs (E, φ) such that c1 (E) = c. We denote by M (

a moduli of oriented reduced L-abelian pairs (E, [φ], ρ) such that c1 (E) = c. Both of them are schemes, and they are naturally isomorphic. The natural projection M (c, L) −→ Pic(c) is a projectivization of a cone over Pic(c).

6.3 Rank One Case

225

The universal object on M (

c, [L]) × X is denoted by (L u , [φu ]). The line bundle L is the pull back of the Poincar´e bundle on Pic(c) × X. The universal object on M (c, L) × X is denoted by (Lu , φu ). We have the relation Lu = L u ⊗ O rel (1). The obstruction theories of M (c, L) and M (

c , [L]) are denoted by Ob M (c, L) and Ob M (

c, [L]) respectively. They are not equivalent in general. By Proposition 6.1.1, Ob M (c, L) is always perfect. On the other hand, Ob M (

c, [L]) is not perfect in the case pg > 0, if H 2 X, L−1 ⊗ L = 0 for some (L, φ) ∈ M (c, L) with φ = 0. In the case H 1 (X, OX ) = 0, we have a simple description of the moduli of abelian pairs. For simplicity, we work on the complex number field. Let c ∈ H 2 (X, Z) be of (1, 1)-type, and let L be a line bundle on X such that c1 (L) = c. By the assumption H 1 (X, OX ) = 0, any line bundle L with c1 (L ) is isomorphic to L. Hence, the moduli M (c, L) is isomorphic to the projective space P H 0 (X, L−1 ⊗ L)∨ . Since the moduli is smooth, the obstruction theory of M (c, L) is given by the obstruction bundle O(c, L), and the virtual fundamental class is the Euler class of O(c, L). The vector bundle O(c, L) is obtained as follows: We have the universal sheaf Lu = L ⊗ Orel (1) and the universal L-section φ : p∗M (c,L) L −→ Lu over M (c, L) × X. We have the cokernel sheaf C := Cok(φ) ⊗ L−1 . Then, the sheaves pX∗ (C) and R1 pX∗ (C) are locally free OM (c,L) -modules. The vector bundle pX ∗ (C) gives the tangent bundle of the moduli, and R1 pX ∗ (C) gives the obstruction bundle O(c, L). The virtual fundamental class is given by the Euler class Eu(O(c, L)) ∩ [M (c, L)]0 , where [M (c, L)]0 is the ordinary fundamental class of the smooth variety M (c, L). Let d(c, L) denote the dimension of the smooth variety M (c, L), which is equal to dim H 0 (X, L−1 ⊗ L) − 1. Let χ(L−1 ⊗ L) denote the Euler number of L−1 ⊗ L.

u

Proposition 6.3.1 Assume H 1 (X, OX ) = 0 and pg = dim H 2 (X, OX ) > 0. Moreover, we assume that the virtual fundamental class of M (c, L) is not 0. Let L be a line bundle such that c1 (L) = c. • The expected dimension of M (c, L) is 0. • χ(L−1 ⊗ L) = 0. Equivalently, c − c1 (L) · c − c1 (L) − c1 (KX ) = 0. Here KX is the canonical line bundle of X. • H i (X, L−1 ⊗ L) = 0 for i = 0, 2, and H 1 (X, L−1 ⊗ L) = 0. d(c,L)−pg . • The total Chern class of O(c, L) is 1 + c1 (Orel (1)) • The virtual fundamental class is SW(c, L)·[p], where [p] denotes the cohomology class of a point M (c, L), and SW(c, L) denotes the following number: .d(c,L) SW(c, L) :=

(i − pg ) = (−1)d(c,L) d(c, L)!

i=1

'

pg − 1 d(c, L)

( (6.4)

226

6 Virtual Fundamental Classes

Proof Since we have assumed H 1 (X, OX ) = 0, the obstruction bundle O(c, L) sits in the following exact sequence on M (c, L): 0 −→ H 1 (X, L−1 ⊗ L) ⊗ Orel (1) −→ O(c, L) −→ H 2 (X, O) ⊗ OM (c,L) −→ H 2 (X, L−1 ⊗ L) ⊗ Orel (1) −→ 0 Thus, the total Chern class of the vector bundle O(c, L) is dim H 1 (X,L−1 ⊗L)−dim H 2 (X,L−1 ⊗L) . 1 + c1 (Orel (1)) Note that the rank of O(c, L) is dim H 1 (X, L−1 ⊗ L) − dim H 2 (X, L−1 ⊗ L) + pg . Thus, if dim H 1 (X, L−1 ⊗ L) − dim H 2 (X, L−1 ⊗ L) ≥ 0, the Euler class of O(c, L) is 0 by our assumption pg > 0. This contradicts with our assumption that the virtual fundamental class is not 0. Therefore, −e := dim H 1 (X, L−1 ⊗ L) − dim H 2 (X, L−1 ⊗ L) < 0 In particular, H 2 (X, L−1 ⊗ L) = 0. −e d(c,L) in the polynomial 1 + c1 (Orel (1)) is The coefficient of c1 (Orel(1)) not 0. Hence, the rank of O c must be d(c, L) = dim H 0 (X, L−1 ⊗ L) − 1. Therefore, we obtain χ(OX ) = χ(L−1 ⊗ L), and the expected dimension of the moduli is 0. By a formal calculation, we obtain (6.4). Since M (c, L) = ∅, we have H 0 (X,L−1 ⊗ L) = 0. We have already seen 2 H (X, L−1 ⊗ L) = 0, which implies H 0 X, L ⊗ L−1 ⊗ KX = 0 by Serre duality. Here, KX denote the canonical bundle of X. Note that the following naturally induced rational morphism is injective: ∨ ∨ −→ P H 0 X, KX P H 0 (X, L−1 ⊗ L)∨ × P H 0 X, L ⊗ L−1 ⊗ KX It implies dim H 0 (X, L−1 ⊗ L) + dim H 2 (X, L−1 ⊗ L) ≤ 1 + pg From (6.5) and χ(OX ) = χ(L ⊗ L−1 ), we obtain H 1 X, L−1 ⊗ L = 0.

(6.5)

Remark 6.3.2 The computation of SW(c, L) was essentially done by Friedman and Morgan in [38]. We continue to assume H 1 (X, OX ) = 0. Let L be a line bundle such that c, [L]), assumc1 (L) = c. Let us consider the virtual fundamental class of M (

L) = 0. We have the natural isomorphisms ing the vanishing H 2 (X, L−1 ⊗ M (

c, [L]) M (c, L) P H 0 (X, L−1 ⊗ L)∨ for such a line bundle L. The obstruction theory of M (

c, [L]) is perfect (Proposition 6.1.3). Since the moduli is

L), and the virtual fundamental class smooth, we have the obstruction bundle O(c,

6.3 Rank One Case

227

L). The following proposition can be checked directly is the Euler class of O(c, from the construction of the obstruction theory. 1 Proposition Let L 6.3.3 be a line bundle such that c1 (L) = c. Assume H (X, O) = 2 −1 0 and H X, L ⊗ L = 0.

• The expected dimension of M (

c, [L]) is χ(L−1 ⊗ L) − 1.

L) is H 1 (X, L−1 ⊗ L) ⊗ Orel (1). The total Chern • The obstruction bundle O(c, 1

L) is 1 + c1 (Orel (1)) dim H (X,L) . class of O(c, • In particular, the virtual fundamental class is 1

∩ [M (

c, [L])]0 . " # Here M (

c, [L]) 0 denotes the ordinary fundamental class. c1 (Orel (1))dim H

(X,L)

6.3.2 Involutivity and Relation with Seiberg-Witten Invariants Let us give some more complement on the numbers SW(c, L), assuming that X is a smooth simply connected complex projective surface. Since we consider only the case L = OX , we omit to distinguish L, i.e., SW(c, L) is denoted by SW(c). Proposition 6.3.4 We have the involutive relation: SW(c) = (−1)pg −1 SW −c + c1 (KX ) Proof Let L denote a holomorphic line bundle such that c1 (L) = c. By using H 1 (X, L) = 0, χ L = χ(OX ) and Serre duality, we obtain the following equality: dim H 0 (X, L) + dim H 0 X, L−1 ⊗ KX = pg + 1 Then, we obtain the desired equality by a direct computation.

We recall the relation between the numbers SW(c) and the Seiberg-Witten invariants of X. Let ω denote the first Chern class of a polarization OX (1) of X. We will not distinguish ω and a Kahler form representing ω, and we regard (X, ω) as a Kahler manifold. Take a ∈ H 2 (X, Z), and let Q be a C ∞ -line bundle on X such that c1 (Q) = a. Assume that Q is the determinant bundle of some Spinc -structure ξ. Let SW(a) denote the Seiberg-Witten invariant associated to ξ. (See [130]. See also [21].) If a is not of (1, 1)-type, then the associated Seiberg-Witten equation has no solution, and hence SW(a) = 0. Assume a is of (1, 1)-type, and we take an essentially unique holomorphic structure of Q. Let J be the intersection product ω · a. For simplicity, we assume J = 0. It is satisfied if ω is general. Let L be a holomorphic line bundle such that L2 Q ⊗ KX . Let c := c1 (L) = c1 (Q) + c1 (KX ) /2. Proposition 6.3.5 (Friedman-Morgan) We have the relation SW(a) = SW(c).

228

6 Virtual Fundamental Classes

Proof Let M SW (a) denote the moduli space of solutions of the associated SeibergWitten equation associated to ξ. According to Witten [130], it is known that M SW (Q) is naturally isomorphic to M (c, O) in the case J < 0. Both of the moduli spaces are equipped with the obstruction theories, and they are equivalent according to the result of Friedman and Morgan [38]. Hence, we obtain SW(a) = SW(c) in the case J < 0. Let us consider the case J > 0. Recall the relation SW(−a) = (−1)1+pg SW(a). −1 (See Corollary 6.8.4 of [95], for example.) Because of (L ⊗ KX )2 Q−1 ⊗ KX and the result for J < 0, we obtain SW(−a) = SW −c + c1 (KX ) . Then, SW(a) = SW(c) follows from Proposition 6.3.4.

6.3.3 Parabolic Hilbert Schemes Let y be an element of T ype such that rank(y) = 1. We assume that the first Chern class of y is trivial. Then, let X [y] denote a moduli of oriented parabolic sheaves E∗ of rank one such that det(E) = OX . In other words, X [y] denotes a moduli space of ideal sheaves of 0-dimensional schemes with parabolic structure of an appropriate type. We call X [y] a parabolic Hilbert scheme of type y. The universal sheaf I u over X [y] × X can be regarded as the ideal sheaf of the relatively 0-dimensional scheme Z(y). The relative length is given by −y2 , where y2 denotes the H 4 (X)-component of y. Proposition 6.3.6 When D is smooth, the parabolic Hilbert scheme is smooth. The expected dimension is equal to the ordinary dimension. y ) −→ Pic. Hence, the claim Proof X [y] is the fiber of the smooth morphism M(

follows from the first claim of Proposition 6.1.3. Thanks to Proposition 6.3.6, we have the natural obstruction theory of X [y] . But, we give another expression of the obstruction theory of X [y] for later use. We use the notation in Section 5.5. Let V• = (V−1 → V0 ) be a locally free resolution of the universal sheaf E over X [y] × X. We take vector spaces Wi such that rank Wi = rank Vi . We set SGL(W• ) := (g−1 , g0 ) ∈ GL(W• ), det(g−1 ) det(g0 ) = 1 . We put Y (W• ) := N (W−1 X , W0 X )SGL(W• ) . Then, we have the classifying map: Φ(E) : X [y] × X −→ Y (W• ) It induces the morphism Φ(E)∗ LY (W• )/X −→ LX [y] ×X/X on X [y] × X. We can show that Φ(E)∗ LY (W• )/X is represented by g◦ (V• )≤1 . Therefore, we obtain a morphism g◦ (V• ) −→ LX [y] ×X/X , which induces a morphism on X [y]

6.3 Rank One Case

229

Ob◦ (V• ) −→ LX [y] . Since it is uniquely determined in the derived category D(X [y] ), we denote it by ob◦ (y) : Ob◦ (y) −→ LX [y] . We put Y D (W• ) := N (W−1 D , W0 D )SGL(W• ) . We have the naturally defined .l .l right action of i=2 GL(W (i) ) × SGL(W• ) on i=1 N W (i+1) , W (i) . The quotient stack is denoted by Y D (W• , W ∗ ). We have the naturally defined morphism π : Y D (W• , W ∗ ) −→ Y D (W• ). We have the classifying morphisms: Φ(V•|D ) : X [y] × D −→ Y D (W• ) ΦD (V•|D , VD∗ ) : X [y] × D −→ Y D (W• , W ∗ ) Because π ◦ ΦD (V•|D , V ∗ ) = Φ(V•|D ), we obtain the morphisms on X [y] × D: Φ(V•|D )∗ LY D (W• )/D −→ ΦD (V•|D , VD∗ )∗ LY D (W• ,W ∗ )/D −→ LX [y] ×D/D It is easy to observe that ΦD (V•|D , VD∗ )∗ LY D (W• ,W ∗ )/D and Φ(V•|D )∗ LY D (W• )/D are represented by g◦D (V• , F∗ )≤1 and g◦ (V•|D )≤1 . Thus, we obtain the morphisms on X [y] × D: g◦ (V•|D ) −→ g◦D (V• , F∗ ) −→ LX [y] ×D/D On X [y] , we set Ob◦D (y) := RpD ∗ g◦D (V• , F∗ ) ⊗ ωD ,

Ob◦D (y) := RpD ∗ g◦ (V•|D ) ⊗ ωD .

They are independent of the choice of V• in the derived category D(X). Then, we obtain the following commutative diagram on X [y] : Ob◦D (y) −−−1−→ Ob◦ (y) ⏐ ⏐ ⏐ ⏐ i2 ! ! i

Ob◦D (y) −−−−→ LX [y] The cone of (i1 , −i2 ) : Ob◦D (y) −→ Ob◦ (y) ⊕ Ob◦D (y) is denoted by Ob◦ (y). We obtain the morphism ob◦ (y) : Ob◦ (y) −→ LX [y] . Lemma 6.3.7 The morphism ob◦ (y) is a quasi-isomorphism. Proof We have the inclusion X [y] −→ M(

y ). We take a locally free resolution of the universal sheaf on M(

y ) × X. The restriction to X [y] × X gives a locally free resolution of the universal sheaf on X [y] × X. We obtain the following naturally defined commutative diagram:

230

6 Virtual Fundamental Classes j1

j2

X [y] × X −−−−→ M(

y ) × X −−−−→ Pic ×X ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! ! Y (W• )

−−−−→

Y (W• )

−−−−→

XGm

We obtain the following commutative diagram on X [y] : LX [y] ←−−−− j1∗ LM( y) ←−−−− j1∗ j2∗ LPic ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Ob◦ (y) ←−−−−

Ob(y)

←−−−− Obd (y)

By the construction of Ob(

y ), we can obtain the following: LX [y] ←−−−− j1∗ LM( y) ←−−−− j1∗ j2∗ LPic ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Ob◦ (y) ←−−−−

Ob(

y)

←−−−− j1∗ j2∗ LPic

It is easy to observe that both horizontal rows are distinguished. We also have the following diagram: X [y] × X −−−−→ Y D (W• , W ∗ ) −−−−→ Y D (W• ) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! ! M(

y ) × X −−−−→ YD (W• , W ∗ ) −−−−→ YD (W• ) We obtain the following commutative diagram by construction of the complexes: LX [y] ⏐ ⏐

←−−−− Ob◦D (y) ←−−−− Ob◦D (y) ⏐ ⏐ ⏐ ⏐

j1∗ LM( y) ←−−−− ObD (y) ←−−−− ObD (y) Here, ObD (y) := RpD ∗ gD (V, F∗ )⊗ω D and ObD (y) := RpD ∗ g(V•|D )⊗ωD y ) . Then, we on M(

y ). We remark Ob(

y ) Cone ObD (y) −→ ObD (y) ⊕ Ob(

obtain the following morphism of the distinguished triangles: LX [y] ⏐ ⏐

←−−−− j1∗ LM( y) ←−−−− j1∗ j2∗ LPic ⏐ ⏐ a1 ⏐ a2 ⏐

Ob◦ (y) ←−−−− Ob(

y ) ←−−−− j1∗ j2∗ LPic The morphisms ai (i = 1, 2) are isomorphisms. Therefore, the claim of the lemma follows.

6.3 Rank One Case

231

6.3.4 Splitting into Moduli of Abelian Pairs and Parabolic Hilbert Schemes Let y be an element of T ype such that rank(y) = 1. We have the following description of M y, L . Let c denote the H 2 (X)-component of y, and y(−c) := y ·exp(−c). We have the universal line bundle Lc on M (c, L)×X. We also have the structure sheaf OZ(y(−c)) of the universal subscheme Z(y(−c)) ⊂ X [y(−c)] × X. We use the same symbols to denote the pull back of them via the projection M (c, L) × X [y(−c)] × X onto M (c, L) × X and X [y(−c)] × X. We put K := Luc ⊗ L−1 ⊗ OZ(y(−c)) . It is easy to see that V := pX ∗ K is a locally free sheaf on M (c, L) × X [y(−c)] . We have the naturally induced morphism pX ∗ (Luc ⊗ L−1 ) −→ V. We also have the natural section of pX ∗ (L−1 ⊗ Luc ) induced by the universal section φu for M (c, L). Therefore, we obtain the section ψ : M (c, L) × X [y(−c)] −→ V. It is easy to observe that ψ −1 (0) is isomorphic to the moduli M(y, L). We have the following Cartesian diagram: M(y, L) ⏐ ⏐ j!

F

−−−−→ M (c, L) × X [y(−c)] ⏐ ⏐ψ ! i

M (c, L) × X [y(−c)] −−−−→

(6.6)

V

Here i denotes the 0-section. Since i is a regular embedding, we can define the Gysin map i! . Proposition 6.3.8 • We have the following relations among the virtual fundamental classes: i! [M (c, L)] × [X [y(−c)] ] = [M(y, L)] i∗ [M(y, L)] = Eu(V) ∩ [M (c, L)] × [X [y(−c)] ] Here, Eu(V) denotes the Euler class of the vector bundle V. • Assume H 2 X, L = 0 for any L ∈ Pic(c) . Note the natural identifications M (c, L) M (

c, [L]) and M(y, L) M(

y , [L]). Then, " # c, L)] × [X [y(−c)] ] = M(

y , [L]) i! [M (

y , [L])] = Eu(V) ∩ [M (

c, [L])] × [X [y(−c)] ] i∗ [M(

A proof will be given in Subsections 6.3.5–6.3.8. Before going#into a proof, we remark that the study of the virtual fundamental " class M(y, L) is reduced to the study of the intersection theory on X [y(−c)] by

232

6 Virtual Fundamental Classes

Proposition 6.3.1 and Proposition 6.3.8, if H 1 (X, O) = 0 and pg > 0 are satisfied. We are mainly interested "in the cap# product of some cohomology class Φ and the virtual fundamental class M(y, L) . In the interesting cases, the cohomology class [y(−c)] , and thus we have only to consider the product Φ is defined"on M (c, L) # ×X of Φ and i∗ M(y, L) . Let L be any line bundle on X such that c1 (L) = c. We := OZ( y(−c)) ⊗ L ⊗ L−1 on X [y(−c)] × X. We obtain the vector bundle put K " # := pX ∗ K on X [y(−c)] . We have the following description of i∗ M(y, L) . V Proposition 6.3.9 Assume H 1 (X, O) = 0 and pg > 0. We have the following formula in the cohomology ring H ∗ M (c, L) × X [y(−c)] H ∗ M (c, L) ⊗ H ∗ (X [y(−c)] ): " # ∩ [X [y(−c)] ] (6.7) i∗ M(y, L) = SW(c, L) · [p] × Eu(V) Here, [p] denotes the cohomology class of a point of M (c, L). Proof If [M(y, L)] = 0, the expected dimension of M (c, L) is 0, and [M (c, L)] is equal to SW(c, L) · [p] according to Proposition 6.3.1. Therefore, we have only to consider Eu(V) ∩ [p] × [X [y(−c)] ] . Moreover, we can replace V with |{p}×X [y(−c)] . Then, we obtain the formula (6.7). V|{p}×X [y(−c)] = V

6.3.5 Morphism to Moduli of Abelian Pairs We use the notation in Subsection 5.3.1. We would like to obtain a diagram similar to (5.3) from an L-Bradlow pair (E, φ) in the case rank(E) = 1. Since E is contained For simplicity, in det(E), the morphism φ : L −→ det(E) is naturally induced. assume that U is connected. Then, c := c1 det(E)|{u}×X is independent of the choice of u ∈ U . We obtain the induced morphism det E,φ : U2 −→ M (c, L). Proposition 6.3.10 We have the following commutative diagram on U2 : LU2 ⏐ ⏐ det∗E,φ LM (c,L)

Ob(V• , φ• ) ⏐ ⏐ ∗ ←−−−− detE,φ Ob M (c, L) ←−−−−

(6.8)

Here, we put Ob(V• , φ• ) := RpX ∗ g(V• , φ• ) ⊗ ωX , and Ob M (c, L) denotes the obstruction theory of M (c, L). The diagram (6.8) is compatible with (5.3). Proof Let us begin with a general non-sense. Let V0 and V−1 be locally free sheaves on a stack Z with a morphism f : V−1 −→ V0 . Assume rank V0 − rank V−1 = 1. Let v1 , . . . , vl be a local frame of V−1 , and put ω := v1 ∧ · · · ∧ vl which is a local section of det(V−1 ). For a local section u of V0 , we put Λf (u) := u ∧ f(ω) ⊗ ω −1

6.3 Rank One Case

233

which is a local section of det(V• ) = det(V0 ) ⊗ det(V−1 )−1 . It is easy to see Λf (u) is independent of the choice of a frame v1 , . . . , vl . Therefore, we obtain a morphism Λf : V0 −→ det(V• ). It is easy to see Λf ◦ f = 0. Hence, we have the morphism of the complexes Cone(f) −→ det(V• ). f

Applying the above construction to V−1 −→ V0 , we obtain a morphism Λf : V• −→ det(V• ) det(E). We also obtain the morphism Λf ◦ φ0 : P0 −→ det(E). Because Λf ◦ φ0 (P−1 ) = 0, it naturally induces a morphism L −→ det(E), which is the same as φ . We put ∨ grel := Hom P• , det(V• ) . A morphism grel [−1] −→ O[−1] is naturally induced by Λf ◦ φ0 . The cone is denoted by g det(V• ), φ . It is convenient to make a minor change in the construction of Ob(V• , φ• ). We have the natural right GL(W• )-action on N (W−1 X , W0 X ) × N (P−1 , W0 X ). The quotient stack is denoted by Y0 (W• , P• ). And we set Y1 (W• , P• ) := Y (W• ),

Y2 (W• , P• ) := Y2 (W• , P• ),

Y (W• , P• ) := Y1 (W• , P• ) ×Y0 (W• ,P• ) Y2 (W• , P• ). We have the natural morphisms Yi (W• , P• ) −→ Yi (W• , P• ), which induce the isomorphism Y (W• , P• ) Y (W• , P• ). : U2 × X −→ Y (W• , P• ) and From (V• , φ• ), we have the classifying map Φ i : U2 × X −→ Yi (W• , P• ). In the construction of Subsection the induced maps Φ ∗ L 5.3.1, we can replace Φ(V• , φ• )∗ LY (W• ,P• )/X with Φ Y (W• ,P• )/X . We have the action of G on det(W ) by multiplication, which induces Gm m • actions on N Pi , det(W• X ) (i = 0, 1). We set Z0 (W• , P• ) := N P−1 , det(W• X ) G , Z1 (W• , P• ) := XGm , m

Z2 (W• , P• ) := N P0 , det(W• X ) G . m

Let Z(W• , P• ) denote the fiber product Z1 (W• , P• )×Z0 (W• ,P• ) Z2 (W• , P• ), which is isomorphic to N (L, det(W• X ))Gm . From the section Λf ◦ φ0 : P0 −→ det(V• ), we obtain the classifying map Ψ : U2 × X −→ Z(W• , P• ) and the induced maps Ψi : U2 × X −→ Zi (W• , P• ) (i = 0, 1, 2). We have the morphism Γ2 : N (W−1X , W0X )×N (P−1 , W−1X )×N (P0 , W0X ) −→ N P0 , det(W•X ) given by Γ2 (e, a−1 , a0 ) := Λe ◦ a0 . We have the morphism GL(W• ) −→ Gm given by (g−1 , g0 ) −→ det(g−1 )−1 · det(g0 ). Then, Γ2 is equivariant with respect to the actions of GL(W• ) and Gm . Thus, we obtain a morphism Γ2 : Y2 (W• , P• ) −→ Z2 (W• , P• ).

234

6 Virtual Fundamental Classes

Similarly, we have the equivariant morphism Γ0 : N (W−1X , W0X ) × N (P−1 , W0 ) −→ N P−1 , det(W• X ) given by Γ0 (e, a) = Λe ◦ a, which induces Γ0 : Y0 (W• , P• ) −→ Z0 (W• , P• ). We also have the obvious map Γ1 : Y1 (W• , P• ) −→ Z1 (W• , P• ). By construction, i = Ψi for i = 0, 1, 2. Thus, we obtain Γ : Y (W• , P• ) −→ Z(W• , P• ) Γi ◦ Φ = Ψ. satisfying Γ ◦ Φ We have the universal L-abelian pair Lu , φu over M (c, L)×X, which induces the classifying map Φi (Lu , φu ) : M (c, L) × X −→ Zi (W• , P• ). We obtain the following commutative diagram: Φ

U2 × X ⏐ ⏐ detE,φ,X !

i −−−− →

Yi (W• , P• ) ⏐ ⏐ Γi !

(6.9)

Φi (Lu ,φu )

M (c, L) × X −−−−−−−→ Zi (W• , P• ) Thus, we obtain the following morphism on U2 × X: ϕ : Cone Ψ0∗ LZ0 (W• ,P• )/X → Ψi∗ LZi (W• ,P• )/X −→ i=1,2

∗0 L ∗i L Cone Φ → Φ Y0 (W• ,P• )/X Yi (W• ,P• )/X i=1,2

By using the argument in Subsection 2.3.2, we can showthat ϕ is represented by the morphism of the complexes g det(V• ), φ −→ Cone γ(φ• )≤1 . Lemma 6.3.11 The morphism ϕ is naturally factorized into g det(V• ), φ −→ g V• , φ −→ Cone γ(φ• )≤1 . Proof We give only an indication. The following diagram is commutative: O −−−−→ Hom(V0 , V0 ) ⊕ Hom(V−1 , V−1 ) ⏐ ⏐ ⏐ ⏐ ! ! 0 −−−−→

Hom(V−1 , V0 )

In fact, it is a part of the morphism of the complexes O −→ Hom(V• , V• )∨ . And, the following diagram is commutative: Hom det(V• ), P0 −−−−→ Hom(V−1 , P−1 ) ⊕ Hom(V0 , P0 ) ⏐ ⏐ ⏐ ⏐ ! ! 0

−−−−→

Hom(V−1 , P0 )

6.3 Rank One Case

235

∨ In fact, it is a part of the morphism of the complexes Hom P• , det(V• ) −→ ∨ Hom P• , V• . Then, the claim of the lemma can be checked easily. Let us finish the proof of Proposition 6.3.10. We obtain the following commutative diagram on U2 × X from (6.9) and Lemma 6.3.11: g det(V• ), φ

g(V• , φ• ) ←−−−− ⏐ ⏐ ⏐ ⏐ ! ! LU2 ×X/X ←−−−− det∗E,φ,X LM (c,L)×X/X It is easy to observe Ob M (c, L) = RpX ∗ g det(V• ), φ ⊗ωX . Thus, we obtain the desired commutative diagram (6.8). The compatibility of (6.8) and (5.3) follows from the compatibility of (5.2) and (6.9). By a similar argument, we obtain a similar commutative diagram in the reduced case. We use the notation in Subsection 5.4.1. Assume rank(E) = 1. The reduced L-Bradlow pair det(E), [φ ] is induced, and we obtain a morphism detE,[φ] : U3 −→ M (c, [L]). Lemma 6.3.12 We have the following commutative diagram: Ob(V• , [φ• ]) ⏐ ⏐ ←−−−− Ob M (c, [L]) ←−−−−

L U3 ⏐ ⏐ det∗E,[φ] LM (c,[L])

(6.10)

Here, Ob M (c, [L]) denotes the obstruction theory of M (c, [L]), and we set Ob(V• , [φ• ]) := Cone Obrel (V• , [φ• ])[−1] −→ Ob(V• ) . Proof We indicate only an outline. From the weight (−1)-actions of Gm on P• and OX , we obtain the Gm -actions on Yi (W• , P• ) and Zi (W• , P• ). The quotient stacks are denoted by Yi (W• , [P• ]) and Zi (W• , [P• ]). Natural morphisms Γi : Yi (W• , [P• ]) −→ Zi (W• , [P• ]) are induced. : U3 × X −→ Yi (W• , [P• ]). From (V• , [φ• ]), we obtain the classifying maps Φ i

We set Ψi := Γi ◦ Φi . We also have the classifying maps Φi (Lu , [φu ]) : M (c, [L]) × X −→ Zi (W• , [P• ]) for the universal objects on M (c, [L]) × X. The following diagram is commutative: U3 × X ⏐ ⏐ detE,[φ],X !

Φ

i −−−− →

Φi (Lu ,[φu ])

Yi (W• , [P• ]) −−−−→ XGm ⏐ ⏐ ⏐ ⏐ =! Γi !

M (c, [L]) × X −−−−−−−→ Zi (W• , [P• ]) −−−−→ XGm

236

6 Virtual Fundamental Classes

We have the following induced morphism on U3 × X: ϕ : Cone Ψ0 ∗ LZ0 (W• ,[P• ])/XGm → Ψi ∗ LZi (W• ,[P• ])/XGm −→ i=1,2

0∗ L Cone Φ Y0 (W• ,[P• ])/XG

m

→

i ∗ L Φ Yi (W• ,[P• ])/XG

m

i=1,2

We can show that ϕ is represented by the morphism of the complexes g (det(V• ), [φ ]) −→ Cone(γ[φ• ]≤0 [−1]). It factors through g (V• , [φ• ]). Therefore, we have the following diagram: g (V• , [φ• ]) ←−−−− ⏐ ⏐ !

g det(V• ), [φ ] ⏐ ⏐ !

LU3 ×X/XGm ←−−−− det∗E,[φ],X LM (c,[L])×X/XGm We obtain the following diagram: ϕ1 ϕ2 −→ LU3 /kGm −→ LkGm /k [1] RpX ∗ g (V• , [φ• ]) ⊗ ωX ↑ ↑ ↑ ψ1 ψ2 ∗

RpX ∗ g det(V• ), [φ ] ⊗ ωX −→ detE,[φ] LM (c,[L])/kGm −→ LkGm /k [1] It is easy to observe Ob(V• , [φ• ]) Cone ϕ2 ◦ ϕ1 and det∗E,[φ] Ob M (c, [L]) Cone ψ2 ◦ ψ1 . Thus, we obtain the desired diagram (6.10). When E has an orientation ρ, we have morphisms Obrel (V• , ρ)[−1] −→ Ob(V• ) −→ Ob(V• , [φ• ]). Let Ob(V• , ρ, [φ• ]) denote the cone of the composite of the morphisms. Proposition 6.3.13 Let detE,[φ],ρ : U3 −→ M (

c, [L]) denote the naturally induced morphism. We have the following commutative diagram: L U3 ⏐ ⏐ det∗E,[φ],ρ LM ( c,[L])

Ob(V• , ρ, [φ• ]) ⏐ ⏐ ←−−−− det∗E,[φ],ρ Ob M (

c, [L]) ←−−−−

Here, Ob M (

c, [L]) denotes the obstruction theory of M

c, [L] .

6.3 Rank One Case

237

Proof The orientation induces the morphisms U3 −→ M (

c, [L]) −→ Pic(c), and we obtain the following commutative diagram: Obd (V• ) ⏐ ⏐ !

−−−−→

Obd (V• ) ⏐ ⏐ !

−−−−→ Ob(V• ) ⏐ ⏐ !

det∗E,ρ LPic −−−−→ det∗E,[φ],ρ LM ( c,[L]) −−−−→

LU3

Then, the claim follows from the construction of the relative obstruction complexes for orientations.

6.3.6 Morphism to Parabolic Hilbert Scheme Let U be a k-scheme. Let (E, F∗ ) be a quasi-parabolic torsion free sheaf of type y. Assume rank(y) = 1. Let c be the H 2 (X)-component of y. Then, we put y(−c) := −1 y · exp(−c). We put I(E) := det(E) ⊗ E, which has the induced quasi-parabolic structure F∗ . The types of I(E), F∗ are y(−c). Thus, we obtain the morphism Ξ(E) : U −→ X [y(−c)] . Lemma 6.3.14 We have the following commutative diagram: ←−−−−

LU ⏐ ⏐ Ξ(E)∗ LX [y(−c)]

Ob(V• , F∗ ) ⏐ ϕ⏐ ←−−−− Ξ(E)∗ Ob X [y(−c)]

(6.11)

Here, Ob(V• , F∗ ) := Cone Obrel (V• , F∗ )[−1] → Ob(V• ) , and Ob X [y(−c)] denotes the obstruction theory of X [y(−c)] . Moreover, ϕ factors through the tracefree part Ob◦ (V• , F∗ ). Proof Let I u denote the universal sheaf on X [y(−c)] × X. We take a locally free resolution V• of I u . It is easy to observe that V• := Ξ(E)∗X V• ⊗ det(E) is a locally free resolution of E. We have g(V• ) = Ξ(E)∗X g(V• ), g(V•|D ) = Ξ(E)∗D g(V•|D ) and gD (V• , F∗ ) = Ξ(E)∗D g(V• , F∗ ). We take vector spaces Wi (i = −1, 0) such that rank Wi = rank Vi . We have rank W0 − rank W−1 = 1. In that case, we have the homomorphism GL(W• ) −→ SGL(W• ) given as follows: g−1 , g0 −→ det(g• )−1 ·g−1 , det(g• )−1 ·g0 Here, det(g• ) denotes det(g0 ) · det(g−1 )−1 . Therefore, we have the isomorphism Y (W• ) Y (W• ) ×X XGm . In particular, we have the morphism: w1 : Y (W• ) −→ Y (W• )

238

6 Virtual Fundamental Classes

Then, we obtain the following commutative diagram: U ×X ⏐ ⏐ !

−−−−→ Y (W• ) ⏐ ⏐ !

X [y(−c)] × X −−−−→ Y (W• ) We obtain the following commutative diagram on U × X: ←−−−−

LU ×X/X ⏐ ⏐

Φ(V• )∗ LY (W• ) ⏐ ⏐

←−−−−

g(V• ) ⏐ ⏐

Ξ(E)∗X LX [y(−c)] ×X/X ←−−−− Ξ(E)∗X Φ(V• )∗ LY (W• ) ←−−−− Ξ(E)∗ g◦ (V• ) Hence, we obtain the following commutative diagram: LU ⏐ ⏐

←−−−−

Ob(V• ) ⏐ ⏐

Ξ(E)∗ LX [y(−c)] ←−−−− Ξ(E)∗ Ob◦ (V• ) A similar diagram is induced by V•|D and V•|D . Moreover, we obtain the following diagram by the argument of Lemma 5.5.1: ←−−−−

LU ⏐ ⏐

Ob(V• ) ⏐ ⏐

←−−−−

Ob(V•|D ) ⏐ ⏐

Ξ(E)∗ LX [y(−c)] ←−−−− Ξ(E)∗ Ob◦ (V• ) ←−−−− Ξ(E)∗ Ob◦ (V•|D ) We also have the following commutative diagram: LU ↑ ∗

Ξ(E) LX [y(−c)]

←− Ob(V•|D ) RpD ∗ gD (V• , F∗ ) ⊗ ωD ↑ ↑ ◦ ∗ ∗ ←− Ξ(E) RpD ∗ gD (V• , F∗ ) ⊗ ωD ←− Ξ(E) Ob◦ (V•|D ) ←−

We remark the following quasi-isomorphisms: Ob(V, F∗ ) Cone Ob(V•|D ) −→ Ob(V• ) ⊕ RpD ∗ gD (V• , F∗ ) ⊗ ωD Ob(X [y(−c)] ) Cone Ob◦ (V•|D ) −→ Ob◦ (V• ) ⊕ RpD ∗ g◦D (V• , F∗ ) ⊗ ωD Hence, we obtain the desired commutative diagram (6.11).

6.3 Rank One Case

239

6.3.7 Mixed Case Let (E, F∗ , φ) be a quasi-parabolic L-Bradlow pair of type y over U × X. Assume rank(y) = 1. Let c denote the H 2 (X)-component of y, and we put y(−c) := y · exp(−c). Then, we have the morphisms detE,φ : U −→ M (c, L) and Ξ(E) : U −→ X [y(−c)] . Assume that we have a locally free resolution V• of E, a locally free resolution P• of L, and a lift φ• : P• −→ V• of φ. We have the natural morphisms i1 : Ob(V• ) −→ Ob(V• , φ• ) and i2 : Ob(V• ) −→ Ob(V• , F∗ ). By using them, we set (i1 ,−i2 ) Ob(V• , F∗ , φ• ) := Cone Ob(V• ) −→ Ob(V• , F∗ ) ⊕ Ob(V• , φ• ) . We have the induced map Ob(V• , F∗ , φ• ) −→ LU . Lemma 6.3.15 We have the following commutative diagram: det∗E,φ Ob M (c, L) ⊕ Ξ(E)∗ Ob(X [y(−c)] ) −−−−→ Ob(V• , F∗ , φ• ) ⏐ ⏐ ⏐ ⏐ ! ! det∗E,φ LM (c,L) ⊕ Ξ(E)∗ LX [y(−c)]

−−−−→

Proof It follows from Proposition 6.3.10 and Lemma 6.3.14.

LU

Let (E, F∗ , ρ, [φ]) be an oriented quasi-parabolic reduced L-Bradlow pair of type c, [L]) and Ξ(E) : y over U × X. We have the morphism detE,ρ,[φ] : U −→ M (

[y(−c)] U −→ X . Assume that we have a locally free resolution V• of E, a locally free resolution P• of L, and a lift [φ• ] of [φ]. We have the natural morphisms i1 : Ob(V• ) −→ Ob(V• , ρ, [φ• ]) and i2 : Ob(V• ) −→ Ob(V• , F∗ ). Then, we set (i1 ,−i2 ) Ob(V• , F∗ , [φ• ], ρ) := Cone Ob(V• ) −→ Ob(V• , [φ• ], ρ) ⊕ Ob(V• , F∗ ) . Lemma 6.3.16 We have the following commutative diagram: c, [L]) ⊕ Ξ(E)∗ Ob(X [y(−c)] ) −−−−→ Ob(V• , F∗ , [φ• ], ρ) det∗E,ρ,[φ] Ob M (

⏐ ⏐ ⏐ ⏐ ! ! det∗E,ρ,[φ] LM ( c,[L]) ⊕ Ξ(E)∗ LX [y(−c)]

−−−−→

Proof It follows from Proposition 6.3.13 and Lemma 6.3.14.

LU

240

6 Virtual Fundamental Classes

6.3.8 Proof of Proposition 6.3.8 Applying the construction in Subsection 6.3.7 to the universal object (E u , φ) on M(y, L), we obtain a morphism F : M(y, L) −→ M (c, L) × X [y(−c)] . It is the same as the inclusion given in Subsection 6.3.4. Let Ob(y, L) denote the obstruction theory of M(y, L). We have the following commutative diagram, due to Lemma 6.3.15: F ∗ Ob M (c, L) ⊕ Ob(X [y(−c)] ) −−−−→ Ob y, L ⏐ ⏐ ⏐ ⏐ (6.12) ! ! −−−−→ LM(y,L) F ∗ LM (c,L)×X [y(−c)] Let Luc denote the universal line bundle on M (c, L) × X. We have the natural inclusion E u ⊗L−1 −→ F ∗ Luc ⊗L−1 , and the quotient is isomorphic to K|M(y,L)×X . Thus, we have the following: " # Cone F ∗ Ob X [y(−c)] ⊕ Ob M (c, L) −→ Ob(y, L) " # RpX ∗ Cone RHom F ∗ Luc , L → RHom E u , L ⊗ ωX ∨ RpX ∗ K |M(y,L) [1] V∨ [1]|M(y,L) j ∗ LM (c,L)×X [y(−c)] /V

(6.13)

From the diagram (6.12), we obtain a morphism ν : j ∗ LM (c,L)×X [y(−c)] /V −→ LM(y,L)/M (c,L) . Lemma 6.3.17 ν is the same as the morphism obtained from the diagram (6.6). In particular, the obstruction theories of M (c, L) × X [y(−c)] and M(y, L) are compatible over the morphism i. ∗ u ∨ Proof We set M0 := M (c, L) × X [y(−c)] . We set grel (c, L) := Hom(L, F Lc ) on M0 × X, and Obrel (c, L) := RpX∗ grel (c, L) ⊗ ωX on M0 . Let E0u denote the universal sheaf on M(y)×X. The pull back to M0 ×X is also denoted by the same notation. The restriction to M(y, L) × X is the same as E u . We take a locally free resolution V• of L−1 ⊗ E0u on M0 × X. We set grel (y, L) := i∗ Hom(OX , V• )∨ on M(y, L) × X, and Obrel (y, L) := RpX∗ grel (y, L) ⊗ ωX on M(y, L). Recall the construction of relative obstruction theory for L-sections in Subsection 5.3.5. By the compatibility of the diagrams (6.8) and (5.3), we have the following commutative diagram:

i∗ Obrel (c, L) −−−−→ Obrel (y, L) ⏐ ⏐ ⏐ ⏐ obrel(c,L) ! obrel (y,L)! i∗ LM0 /M(y) −−−−→ LM(y,L)/M(y)

(6.14)

6.3 Rank One Case

241

We set E1◦ := Luc , E0◦ := E0u and ◦ := Luc , V1,0

◦ V1,−1 := 0,

◦ V0,0 := V0 ,

◦ V0,−1 := V−1 .

We have the following commutative diagram on M0 × X: ◦ ◦ −−−−→ V1,−1 V0,−1 ⏐ ⏐ ⏐ ⏐ ! ! ◦ ◦ V0,0 −−−−→ V1,0 ⏐ ⏐ ⏐ ⏐ ! !

E0◦

−−−−→

E1◦

−−−−→ K

The universal L-section of Luc induces sections φ◦ and ϕ of E1◦ and K on M0 × X. On M(y, L) × X, we have the induced section φ◦0 of E0◦ . Let k(K, ϕ) denote the cone of the natural morphism i∗ grel (c, L) −→ grel (y, L) on M(y, L) × X. We obtain Ob(K, ϕ) := RpX∗ k(K, ϕ) ⊗ ωX . From the commutativity of (6.14), we obtain the following morphism of distinguished triangles: i∗ Obrel (c, L) −−−−→ Obrel (y, L) ⏐ ⏐ ⏐ ⏐ obrel (y,L)! i∗ obrel (c,L)!

−−−−→

Ob(K, ϕ) ⏐ ⏐ ob(K,ϕ)!

i∗ LM0 /M(y) −−−−→ LM(y,L)/M(y) −−−−→ LM(y,L)/M0 By construction ob(K, ϕ) is the same as ν. According to Proposition 2.3.13, ob(K, ϕ) is the same as the morphism j ∗ LM0 /V −→ LM(y,L)/M0 . Hence, we obtain Lemma 6.3.17. Recall that X [y] is smooth. Let m be a sufficiently large integer such that H X, L(m) =0 (i = 1, 2) for anyline bundleL such that c1 (L) = c. Then, the moduli stacks M y, O(−m) and M c, O(−m) are smooth. Letι : O(−m) −→ L be an inclusion. We have the induced inclusions M(y, L) −→ M y, O(−m) and M (c, L) −→ M c, O(−m) . On M c, O(−m) × X [y(−c)] × X, we put i

K := Luc ⊗ L−1 ⊗ OZ(y(−c)) ,

V := pX ∗ K .

Then, V gives the vector bundle over M c, O(−m) × X [y(−c)] such that V |M (c,L)×X [y(−c)] = V. Therefore, we obtain the following Cartesian diagrams: M(y,L) ⏐ ⏐ !

−−−−→ M (c,L)×X [y(−c)] −−−−→ M c,O(−m) ×X [y(−c)] ⏐ ⏐ ⏐ ⏐ i1 ! i!

M (c,L)×X [y(−c)] −−−−→

V

−−−−→

V

242

6 Virtual Fundamental Classes

Here, 0-sections. We have i! = i!1 . Therefore, we obtain the relation i and i1 are the ! [y] i [M (c, L)] × [X ] = [M(y, L)] from (6.13), according to Proposition 2.4.8. c, [L])]×[X [y] ] = [M(

y , [L])] can be shown by using a similar The relation i! [M (

argument. Thus, we finish the proof of Proposition 6.3.8.

6.4 Bradlow Perturbation 6.4.1 Statements Let L be a line bundle on X. If δ ∈ P br is sufficiently small, we have the projective morphism: y , [L], α∗ , δ) −→ Mss (

y , α∗ ) (6.15) F : Ms (

To compare the virtual fundamental classes through F, let us consider the following condition for (y, L, α∗ ) and i = 1, 2: (i-vanishing condition): We have H j X, E ⊗ L−1 = 0 for any j ≥ i and for any E∗ ∈ Mss (y, α∗ ). Obviously, a 1-vanishing condition implies a 2-vanishing condition. Proposition 6.4.1 Assume that the 1-vanishing condition holds for (y, L, α∗ ). • The morphism F : Ms (

y , [L], α∗ , δ) −→ Mss (

y , α∗ ) is smooth. • Assume moreover that the 1-stability condition holds for (y, α∗ ). Then, we have the following relation: " # y , α∗ )] = Ms (

y , [L], α∗ , δ) F∗ [Ms (

Proof We give only an outline. The smoothness of F is clear. We put M1 := y , α∗ ) and M2 := Ms (

y , [L], α∗ , δ). It is easy to obtain the following morMss (

phism of distinguished triangles: −−−−→ LM2 /k −−−−→ LM2 /M1 F∗ LM1 /k ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ F∗ Ob(m, y) −−−−→ Ob(m, y, [L]) −−−−→ Obrel (m, y, [L]) By Proposition 2.4.8, we obtain the equality F∗ [M1 ] = [M2 ].

Let L = (L1 , L2 ) be a pair of line bundles on X. We take δi ∈ P br (i = 1, 2) such that both δi are sufficiently small. If δ2 is sufficiently smaller than δ1 , we have the projective morphism: y , [L], α∗ , δ) −→ Ms (

y , [L1 ], α∗ , δ1 ) F1 : Ms (

6.4 Bradlow Perturbation

243

Proposition 6.4.2 Assume that the 1-vanishing condition holds for (y, α∗ , L2 ). The morphism F1 is smooth, and we have the following relation: " # " # y , [L1 ], α∗ , δ1 ) = Ms (

y , [L], α∗ , δ) F∗1 Ms (

Proof It can be shown by using an argument similar to that employed in the proof of Proposition 6.4.1. If a 1-vanishing condition does not hold, F1 is not necessarily smooth. The following proposition will be proved in Subsections 6.4.2–6.4.4. Proposition 6.4.3 Assume that the 2-vanishing condition holds for (y, α∗ , L2 ). y , [L1 ], α∗ , δ1 ) with a vecThen, there exists a Deligne-Mumford stack B over Ms (

tor bundle V and a section ψ such that the following holds: y , [L1 ], α∗ , δ1 ) is smooth. • The morphism G : B −→ Ms (

y , [L], α∗ , δ) is ψ −1 (0). Namely, the following diagram is Cartesian: • Ms (

Ms (

y , [L1 ], α∗ , δ1 ) −−−−→ ⏐ ⏐ !

B ⏐ ⏐ψ !

(6.16)

i

−−−−→ V

B

Here, i denotes the 0-section. • We have the following relation: y , [L1 ], α∗ , δ1 )] = [Ms (

y , [L], α∗ , δ)] i! G∗ [Ms (

Here, i! denotes the Gysin map induced by the diagram (6.16). Before going into a proof of Proposition 6.4.3, we give a corollary. Let P be any y , [L1 ], α∗ , δ1 ). The fiber G−1 (P ) is smooth, and F−1 point of Ms (

1 (P ) is the 0-set of the section ψ|G−1 (P ) of V|G−1 (P ) . Therefore, we obtain the following. y , [L1 ], α∗ , δ), the fiber F−1 Corollary 6.4.4 For any k-valued point P ∈ Ms (

1 (P ) is provided with the perfect obstruction theory. We also have the following formula: Φ·Ψ = Ψ· Φ Ms (

y ,[L],α∗ ,δ)

F−1 1 (P )

Ms (

y ,[L1 ],α∗ ,δ1 )

Here, Φ and Ψ are cohomology classes on Ms (

y ,[L1 ],α∗ ,δ1 ) and Ms (

y ,[L],α∗ ,δ), f y , [L1 ], α∗ , δ1 ), (ii) Ψ can be exrespectively, such that (i) deg(Φ) = 2 dim Ms (

tended on B. (See Section 7.1 for cohomology classes and evaluations considered in this monograph.) Remark 6.4.5 Similar claims also hold for the morphism F in (6.15), if the 2 vanishing condition holds for (y, L, α∗ ). A proof is similar.

244

6 Virtual Fundamental Classes

6.4.2 Construction of B Let m be a sufficiently large number such that the 1-vanishing condition holds for (y, O(−m), α∗ ). We put L 2 := O(−m) and L 2 := (L1 , L 2 ). We define y , [L ], α∗ , δ). B := Ms (

y , [L1 ], α∗ , δ1 ). Then, we have the natural smooth morphism G : B −→ Ms (

We take an inclusion ι : L 2 −→ L2 such that the cokernel L2 /L 2 is a line bundle on some smooth divisor of X. It naturally induces the morphism −→ L 2−1 whose cokernel is denoted by Cok. We also obtain the inclusion L−1 2 s y , [L], α∗ , δ) −→ B. M (

y , [L], α∗ , δ) −→ M(

y , [Li ]) We have the naturally defined morphisms Ms (

(i = 1, 2). The pull back of the relative tautological line bundles are denoted by (i) y , [L1 ]) and M(

y , [L 2 ]). Orel (1). Similarly, we have the morphisms of B to M(

(1) The pull back of the relative tautological bundles are also denoted by Orel (1) and (2) Orel (1), respectively.

u denote the universal sheaf over B × X. We have the universal reduced Let E (2)

u . We put E u := E

u ⊗ O(2) (1). We [L2 ]-section [φ2 ] : Orel (−1) ⊗ L 2 −→ E rel set u V := pX ∗ E ⊗ Cok . It is easy to show Ri pX ∗ E u ⊗ Cok = 0 for i = 1, 2 by using the 2-vanishing condition for (y, α∗ , L2 ). Hence, V is a locally free sheaf on B. The universal reduced L 2 -section [φu2 ] induces the section ψ of V over B. It is easy to observe y , [L], α∗ , δ). that ψ −1 (0) is isomorphic to Ms (

Thus, we obtain the following Cartesian diagram: i

y , [L], α∗ , δ) −−−1−→ Ms (

⏐ ⏐ j! B

B ⏐ ⏐ψ !

(6.17)

i

−−−−→ V

Here, i denotes the 0-section. For the proof of Proposition 6.4.3, we have only to show i! [B] = [Ms (

y , [L], α∗ , δ)].

6.4.3 Compatibility of the Obstruction Theories We will show the following lemma in this subsection. Lemma 6.4.6 The obstruction theories of Ms (

y , [L], α∗ , δ) and B are compatible over i in the diagram (6.17).

6.4 Bradlow Perturbation

245

Proof We consider the issue in some more general situation. Let L be a line bundle on X. We take a sufficiently large integer m. We take an inclusion ι : O(−m) −→ L. It naturally induces the morphism L−1 −→ O(m). We assume that the cokernel Cok is a line bundle on some smooth divisor of X.

, [L]) and M(m, y

, [O(−m)]), respectively. Let M1 and M2 denote M(m, y

u denote the universal sheaf The inclusion i1 : M1 −→ M2 is induced by ι. Let E 2

u ⊗Orel (1) and V := pX ∗ E u ⊗Cok . The universal over M2 ×X. We set E2u := E 2 2

u induces the section ψ of V. We have M1 ψ −1 (0) reduced O(−m)-section of E 2 and the following commutative diagram: i

M1 −−−1−→ ⏐ ⏐ j!

M2 ⏐ ⏐ ψ!

(6.18)

i

M2 −−−−→ V To prove the claim of the lemma, we have only to show that the obstruction theories of Mi are compatible over i in the diagram (6.18).

u denote the universal sheaf on M1 × X, and let [φu ] denote the universal Let E 1 1

u (m), and let V1,−1 denote the kernel reduced L-section. We put V1,0 := p∗X pX ∗ E 1

u (m). We take a locally free resolution P• of the natural surjection V1,0 −→ E 1 of L(m) such that P0 is a direct sum of OX . We have the canonical lift [φ1• ] : p∗M1 P• ⊗ p∗X Orel (−1) −→ V1,• . Recall we set ∨ g rel (V1,• , [φ1• ]) = Hom p∗M1 P• ⊗ p∗X Orel (−1), V1,• , ∨ g(V1,• ) = Hom V1,• , V1,• [−1].

2 (m) on Similarly, we take a locally free resolution V2 • of the universal sheaf E

2 . We have the M2 × X. Let [φ2 ] denote the universal reduced O(−m)-section of E ∗ ∗ canonical lift [φ2 ] : pM2 OX ⊗ pX Orel (−1) −→ V2,0 of [φ2 ]. We regard it as the morphism of complexes, and denote it by [φ2• ]. In this case, we have ∨ g rel (V2• , [φ2• ]) = Hom p∗M2 O ⊗ p∗X Orel (1), V2,• , ∨ g(V2 • ) = Hom V2 • , V2 • [−1]. The inclusion ι : O −→ L(m) has the canonical lift OX −→ P• . Therefore, we have the following commutative diagram on M1 : i∗1 g rel V2 • , [φ2• ] [−1] −−−−→ g rel V1 • , [φ1• ] [−1] ⏐ ⏐ ⏐ ⏐ (6.19) ! ! i∗1 g(V2 • ) We set M0 := M(m, y).

−−−−→

g(V1 • )

246

6 Virtual Fundamental Classes

Lemma 6.4.7 The diagram (6.19) is compatible with the morphisms to the cotangent complexes: i∗1 LM2 ×X/(M0 ×X)Gm [−1] −−−−→ LM1 ×X/(M0 ×X)Gm [−1] ⏐ ⏐ ⏐ ⏐ ! ! i∗1 π2∗ L(M0 ×X)Gm /XGm

−−−−→

π1∗ L(M0 ×X)Gm /XGm

Here, πi denote the natural morphisms of Mi to M0 . (See Subsection 2.1.8 for compatibility of diagrams.) Proof We give only an indication. We use the notation in Subsection 5.4.1. We construct the stack Y (W• , [OX ]) by replacing P• with the complex (0 → OX ). We have the following commutative diagram: M1 × X ⏐ ⏐ !

−−−−→ Y (W• , [P• ]) −−−−→ XGm ⏐ ⏐ ⏐ ⏐ ! !

M2 × X ⏐ ⏐ !

−−−−→ Y (W• , [OX ]) −−−−→ XGm ⏐ ⏐ ⏐ ⏐ ! !

(M0 × X)Gm −−−−→

Y (W• )Gm

(6.20)

−−−−→ XGm

The desired compatibility follows from the construction of the complexes. Thus, we obtain Lemma 6.4.7. Lemma 6.4.8 We have the following commutative diagram: i∗1 Ob m, y, [O(−m)] −−−−→ Ob(m, y, [L]) ⏐ ⏐ ⏐ ⏐ ! ! i∗1 LM2

−−−−→

(6.21)

LM1

Proof We obtain the following commutative diagram from (6.19): s i∗1 Ob rel V2 • , [φ2• ] [−1] −−−3−→ Ob rel V1• , [φ1• ] [−1] ⏐ ⏐ ⏐ ⏐ s1 ! s2 ! −−−−→ Ob V1 • i∗1 Ob V2,•

(6.22)

It is compatible with the morphisms to the cotangent complexes according to Lemma 6.4.7: i∗1 LM2 /M0,Gm [−1] −−−−→ LM1 /M0,Gm [−1] ⏐ ⏐ ⏐ ⏐ ! ! i∗1 π2∗ LM0,Gm

−−−−→

π1∗ LM0,Gm

6.4 Bradlow Perturbation

247

By a modification as in Subsection 5.4.1, we obtain the commutative diagram from (6.22): i∗1 Obrel V2 • , [φ2• ] [−1] −−−−→ Obrel V1• , [φ1• ] [−1] ⏐ ⏐ ⏐ ⏐ ! ! −−−−→ Ob V1 • i∗1 Ob V2,• It is compatible with the following commutative diagram: i∗1 LM2 /M0 [−1] −−−−→ LM1 /M0 [−1] ⏐ ⏐ ⏐ ⏐ ! ! i∗1 π2∗ LM0

−−−−→

π1∗ LM0

Then, we obtain the desired commutative diagram (6.21) by construction. Thus, Lemma 6.4.8 is proved. By Lemma 6.4.8, we obtain an induced morphism: Obrel := Cone i∗1 Ob(m, y, [O(−m)]) −→ Ob(m, y, [L]) −→ LM1 /M2 (6.23) It is easy to observe that we have a natural isomorphism Obrel V∨ [1] j ∗ LM2 /V . (See also the argument below.) Let us check that the induced morphism V∨ [1] −→ LM1 /M2

(6.24)

is the same as the one obtained from the diagram (6.18). It is convenient to use the construction of relative obstruction theory for Lsections in Subsection 5.3.5. We put E1◦ := E2u ⊗ O(m), E0◦ := E2u ⊗ L−1 and ◦ := V2,i ⊗ Orel (1), V1,i

◦ ◦ V0,i := V2,i ⊗ OX (−m) ⊗ L−1 .

Then, we obtain the following diagram: ◦ ◦ −−−−→ V1,−1 V0,−1 ⏐ ⏐ ⏐ ⏐ ! ! ◦ ◦ −−−−→ V1,0 V0,0 ⏐ ⏐ ⏐ ⏐ ! !

E0◦

−−−−→

E1◦

−−−−→ E2u ⊗ Cok

248

6 Virtual Fundamental Classes

u induces sections ϕ◦ and φ◦ of E u ⊗ Cok and A reduced OX (−m)-section of E 2 2 ◦ E1 on M2 × X, respectively. On M1 × X, we have the induced section φ◦0 of E0◦ ,

u . which corresponds to the universal reduced L-section of E 1 ∨ ◦ on Mi × X for i = 1, 2. Let We set g rel (Vi• , [φui ]) := Hom OX , Vi−1,• ◦ , φ◦ ) denote the cone of the naturally induced morphism k(E•◦ , V•,• i∗1 g rel (V2,• , [φu2 ]) −→ g rel (V1,• , [φu1 ]) Note that it is the same as the complex considered in Subsection 2.3.3. As argued in Subsections 2.3.3 and Subsection 5.3.5, we have the following: i∗1 g rel (V2• , [φu2 ]) ⏐ ⏐ !

−−−−→

g rel (V1• , [φu1 ]) ⏐ ⏐ !

◦ −−−−→ k(E•◦ , V•,• , φ◦ ) ⏐ ⏐ ◦ r(E•◦ ,V•,• ,φ◦ )!

i∗1 LM2 ×X/M0,Gm ×X −−−−→ LM1 ×X/M0,Gm ×X −−−−→ LM1 ×X/M2 ×X (6.25) The diagram (6.25) induces the following: t i∗1 Ob rel (m, [O(−m)]) −−−3−→ Ob rel (m, [L]) −−−−→ Ob E2 ⊗ Cok, ϕ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ t1 ! t2 ! ob(E2 ⊗Cok,ϕ)! i∗1 LM2 /M0,Gm

−−−−→ LM1 /M0,Gm −−−−→

LM1 /M2

By Lemma 5.3.7, ti (i = 1, 2) are the same as si in the diagram (6.22). It is easy to see t3 = s3 by construction. Hence, the morphism ob(E2 ⊗ Cok, ϕ) is the same as (6.23). Then, (6.24) is the same as the morphism obtained from the diagram (6.18) thanks to Proposition 2.3.13. Thus, the proof of Lemma 6.4.6 is finished.

6.4.4 Ambient Smooth Stack Since B is not necessarily smooth, we shall construct an ambient smooth stack to use Proposition 2.4.8. Let C ⊂ X denote the support of Cok. Recall that C is smooth and that Cok is isomorphic to a line bundle LC on C, by our choice of ι. For a k-scheme T , let F (T ) denote the set of quotients q : p∗T Vm,C −→ E over T × C, satisfying the following conditions: • E isflat over T , and the type of E is y|C · ch(LC ). • H 1 C, E|{u}×C = 0 for any point u ∈ T . We obtain a functor F of the category of k-schemes to the category of sets. The functor F is representable by a scheme Q◦1 . Lemma 6.4.9 The scheme Q◦1 is smooth.

6.4 Bradlow Perturbation

249

Proof We have the perfect obstruction theory Ob Q◦1 of Q◦1 (Proposition 2.4.14). ◦ Let z = (q, E) be a point of Q◦1 , and let iz denote the inclusion {z} −→◦ Q1 . Let K −1 ∗ i Ob(Q1 ) = 0. denote the kernel of Vm,C −→ E. We have only to show H z ∨ The dual Hi i∗z Ob(Q◦1 ) is isomorphic to Ext−i K, E . We have the exact sequence: Ext1 Vm,C , E −→ Ext1 K, E −→ Ext2 E, E We have the vanishing of the first term by definition of Q◦1 . Since C is a smooth curve, we also have the vanishing of the third term. Therefore, we obtain the desired vanishing, and hence Q◦1 is smooth. We have the universal quotient sheaf p∗Q◦ Vm,C −→ C on Q◦1 × C. The push1 forward pC ∗ C gives the vector bundle on Q◦1 , which is denoted by V1 . We use the notation in Subsection 4.1.1. We set

) := Q◦ (m, y) ×Q(m,y) Q(m, y ). Q◦ (m, y Let Z m be as in (3.12). We have the GL(Vm )-closed immersion:

) −→ Z m × Q◦ (m, y Gm,i i

(See Subsection 3.6.3 for Gm,i .) We also have the naturally defined GL(Vm )equivariant morphism

) −→ Q◦1 Q◦ (m, y by the correspondence (q, E, F∗ , ρ) −→ (q , E(−m) ⊗ Cok), where q denotes the naturally induced map Vm,C −→ E(−m) ⊗ Cok. (2)

m × . Gm,i × Q◦ × P(1) We have the natural right GL(Vm )-action on Z m × Pm . 1 i . The bundle V1 induces the GL(Vm )-vector The quotient stack is denoted by B . (1) (2) ◦

m × bundle on Z i Gm,i × Pm × Pm × Q1 , and hence the vector bundle V on B . We have the GL(Vm )-equivariant immersion (2)

, [L1 ]) × P(2) Q◦ (m, y Gm,i × Q◦1 × P(1) m −→ Zm × m × Pm , i

. Since B is Deligne-Mumford, we can which induces the immersion B −→ B take an open neighbourhood B of B in B , which is Deligne-Mumford and smooth. to B is denoted by the same symbol. By the construction, the The restriction of V y , [L], α∗ , δ). Then, we obtain the restriction of V to B is V. We set M := Mss (

following commutative diagram: M −−−−→ ⏐ ⏐ i1 !

B −−−−→ B ⏐ ⏐ ⏐ ⏐ i2 ! i!

ψ B −−−−→ V −−−−→ V

250

6 Virtual Fundamental Classes

Here i and i2 denote the 0-section. We have i! = i!2 . The obstruction theories of M and B are compatible (Lemma 6.4.6). Then, we obtain i! [M2 ] = [M1 ] thanks to Proposition 2.4.8. Thus, the proof of Proposition 6.4.1 is finished.

6.5 Comparison with Full Flag Bundles Let y ∈ T ype, and let α∗ be a system of weights. Let L be a line bundle on X. We s (

y , [L], α∗ , δ) denote the full flag bundle take a sufficiently large integer m. Let M s

u (m). We have the natural over M (

y , [L], α∗ , δ) associated to the bundle pX ∗ E smooth morphism: s (

y , [L], α∗ , δ) −→ Ms (

y , [L], α∗ , δ) F1 : M Lemma 6.5.1 We have the following relation: s (

y , [L], α∗ , δ)] = [M y , [L], α∗ , δ)] F∗ [Ms (

1

, [L]) −→ M(m, y

, [L]). By the conProof Let π denote the projection M(m, y struction in Subsection 5.7.3, we have the following morphism of distinguished tri

, [L]): angles on M(m, y

, [L]) −−−−→ Ob(m,

, [L]) −−−−→ LM(m,

y π ∗ Ob(m, y y ,[L])/M(m,

y ,[L]) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! ! π ∗ LM(m, y,[L])

−−−−→ LM(m,

−−−−→ LM(m,

y ,[L]) y ,[L])/M(m,

y ,[L])

, [L]) and Hence, we obtain the compatibility of the obstruction theories Ob(m, y s (

, [L]) of Ms (

Ob(m, y y , [L], α∗ , δ) and M y , [L], α∗ , δ). Then, the claim follows from Proposition 2.4.8. s (

y , α∗ , +) Let y and α∗ be as above. Take a sufficiently large integer m. Let M be as in Subsection 5.7.5. Let be a sufficiently small positive number. Then, we have the naturally defined morphism: s (

y , α∗ , +) −→ Ms (

y , [O(−m)], α∗ , ) F2 : M Lemma 6.5.2 We have the following relation: s (

y , [O(−m)], α∗ , )] = [M y , α∗ , +)] F∗2 [Ms (

(6.26)

Proof In this case, we use the other way of construction for the obstruction the , [O(−m)]) on M(m, y

, [O(−m)]) explained in Subsection 5.7.1. ory Ob(m, y Let Vm be an Hy (m)-dimensional vector space. We put B(W• ) := kGL(Vm ) and

6.5 Comparison with Full Flag Bundles

251

u denote B ∗ (W• , [P• ]) := P(Vm∨ )GL(Vm ) as in Subsections 5.1.3 and 5.7.1. Let E

) × X. Then, M(m, y

, [O(−m)]) is the projecthe universal sheaf over M(m, y

u (m)). Hence, we have the following naturally defined Cartetivization of pX ∗ (E sian diagram: Ψ

, [O(−m)]) −−−−→ B ∗ (W• , [P• ]) M(m, y ⏐ ⏐ ⏐ ⏐ π! !

) M(m, y

ϕ

−−−−→

B(W• )

∗

), we obtain the following Since ϕ LB(W• ) −→ LM(m, y) factors through Ob(m, y morphisms:

) Ψ ∗ LB ∗ (W• ,[P• ])/B(W• ) [−1] −−−−→ π ∗ ϕ∗ LB(W• ) −−−−→ π ∗ Ob(m, y

, [O(−m)]). Then, we obtain the The cone of the composite is denoted by Ob2 (m, y morphism:

, [O(−m)]) : Ob2 (m, y

, [O(−m)]) −→ LM(m, y,[O(−m)]) ob2 (m, y We use the following lemma.

, [O(−m)]) = ob(m, y

, [O(−m)]) in the derived category Lemma 6.5.3 ob2 (m, y

, [O(−m)])). D(M(m, y Proof We use the result in Subsection 5.4.3. We locally free take uthe canonical

(m) , and let V−1 denote

u (m). Namely, we put V0 := p∗ pX ∗ E resolution of E X

u (m). The reduced OX -section the kernel of the canonical morphism V0 −→ E u

[φ] of E (m) is canonically lifted to the reduced OX -section [φ• ] of V0 . We set

) and M2 := M(m, y

, [L]). In this case, the diagram (5.30) can M1 := M(m, y be rewritten as follows: LM2 /M1 [−1] ←−−−− Obrel (V• , [φ• ])[−1] ←−−−− ObG rel (V• , [φ• ])[−1] a1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! ! π ∗ LM1

←−−−−

Ob(V• )

←−−−−

ObG (V• )

The diagram (5.31) can be rewritten as follows: π ∗ LM1 ⏐ ⏐

←−−−−

Φ(V• )∗ LB(W )/k ⏐ ⏐

τ

1 ←−− −−

ObG (V• ) ⏐ ⏐

3 LM2 /M1 [−1] ←−−−− Ψ ∗ LB ∗ (W,[P ])/B(W ) [−1] ←−− −− ObG rel (V• , [φ• ])[−1]

τ

According to Lemma 5.1.2 and Lemma 5.4.8, τ1 and τ3 are isomorphisms. In this case, a1 is also an isomorphism. Then, the claim of Lemma 6.5.3 immediately follows.

252

6 Virtual Fundamental Classes

) and It is easy to see the compatibility of the obstruction theories ob(m, y s s 2 (m, y (

, [O(−m)]) on M ob y , α∗ , +) and M (

y , [O(−m)], α∗ , ). Hence, we obtain (6.26) thanks to Proposition 2.4.8, and thus the proof of Lemma 6.5.2 is finished.

6.6 Parabolic Perturbation Since we will not use the result in this section later, the reader can skip here.

6.6.1 Statement Let y be an element of H ∗ (X). Let y be an element of T ype whose H ∗ (X)component is y, and let α∗ denote a system of weights. Let us compare the virtual y ) and Ms (

y , α∗ ). fundamental classes of the moduli stacks Ms (

We assume (i) any semistable torsion-free sheaf of type y is also μ-stable, (ii) αi are sufficiently close to 1. Then, we have the naturally defined morphism: y , α∗ ) −→ Ms (

y ). F : Ms (

y) Proposition 6.6.1 There exists a Deligne-Mumford stack B(

y , α∗ ) over Ms (

with a vector bundle V and a section ψ, such that the following holds: y ) is smooth. • The morphism G : B(

y , α∗ ) −→ Ms (

s y , α∗ ) is isomorphic to ψ −1 (0). Namely, the following diagram is Cartesian: • M (

Ms (

y , α∗ ) −−−−→ B(

y , α∗ ) ⏐ ⏐ ⏐ ⏐ ψ! ! i

B(

y , α∗ ) −−−−→

(6.27)

V

Here, i denotes the 0-section. • We have the following relation: " # y )] = Ms (

y , α∗ ) i! G∗ [Ms (

(6.28)

Here, i! denotes the Gysin map induced by the diagram (6.27). A proof will be given in the next subsections. Before going into a proof, we give some remarks. Corollary 6.6.2 Let P be any point of Ms (

y ). The fiber G−1 (P ) is smooth, and −1 F (P ) is the 0-set of the section ψ|G−1 (P ) of V|G−1 (P ) in the situation of Propo sition 6.6.1. Therefore, we obtain a perfect obstruction theory of F−1 (P ).

6.6 Parabolic Perturbation

253

We are mainly interested in the cap product of some cohomology classes and the virtual fundamental classes. Corollary 6.6.3 Let Φ be a cohomology class on Ms (

y ), and let Ψ be a cohoy , α∗ ). Assume (i) deg(Φ) = 2 dimf Ms (

y ), (ii) Ψ can be mology class on Ms (

extended on B(

y , α∗ ). Proposition 6.6.1 implies the following relation for any y ): k-valued point P of Ms (

Φ·Ψ = Ψ× Φ (6.29) Ms (

y ,α∗ )

F−1 (P )

Ms (

y)

(See Section 7.1 for cohomology classes and evaluations considered in this monograph.) Let L be a line bundle on X, and let δ be an element of P br . We can consider y , [L], δ) and Ms (

y , [L], α∗ , δ). We assume that any δa similar relation for Ms (

semistable L-Bradlow pair is μ-δ-stable, and that αi are sufficiently close to 1. Then, y , [L], α∗ , δ) −→ Ms (

y , [L], δ). we have the morphism FL : Ms (

Proposition 6.6.4 There exists a Ms (

y , [L], δ)-scheme B(

y , [L], α∗ , δ) with a vector bundle V and its section ψ, such that the following holds: y , [L], α∗ , δ) −→ Ms (

y , [L], δ) is smooth. • The morphism GL : B(

s y , [L], α∗ , δ) is isomorphic to ψ −1 (0). Namely, the following diagram is • M (

Cartesian: y , [L], α∗ , δ) −−−−→ B(

y , [L], α∗ , δ) Ms (

⏐ ⏐ ⏐ ⏐ ψ! (6.30) ! i

B(

y , [L], α∗ , δ) −−−−→

V

Here, i denotes the 0-section. • We have the following relation: " # " # y , [L], δ) = Ms (

y , [L], α∗ , δ) i! G∗L Ms (

Here i! denotes the Gysin map induced by the diagram (6.30). y , [L], δ): As a result, the following formula holds for any k-valued point P of Ms (

Φ·Ψ = Ψ· Φ Ms (

y ,[L],α∗ ,δ)

G−1 L (P )

Ms (

y ,[L],δ)

Here Φ and Ψ denote cohomology classes on Ms (

y , L, δ) and Ms (

y , L, α∗ , δ) f y , [L], δ), (ii) Ψ can be exrespectively, and we assume (i) deg(Φ) = 2 dim Ms (

tended on B(

y , [L], α∗ , δ). We will give the proof of Proposition 6.6.1 in the next subsections. A proof of Proposition 6.6.4 is similar, and hence we omit it.

254

6 Virtual Fundamental Classes

6.6.2 Construction of a Stack B with an Obstruction Theory Let m be a sufficiently large integer. Let E u denote the universal sheaf on M(m, y ). We put V0 := p∗X pX ∗ E u (m), and the kernel of the natural morphism V0 −→ E u (m) is denoted by V−1 . We obtain the vector bundle V0 | D on M(m, y ) × D. Let g : T −→ M(m, y ) be a morphism. Let F (T ) denote the set of sequences ∗ V0 | D = Cl+1 → Cl → Cl−1 → · · · → C2 → C1 satisfying the of the quotients gD following conditions: • Ci are flat over T . • The induced morphisms H 0 D, Cl+1 | {u}×D −→ H 0 D, Ci | {u}×D are surjective for any i = 1, . . . , l and any point u ∈ T . • H 1 D, Ci | {u}×D = 0 for any u ∈ T and for any i = 1, . . . , l. • The type of Ci is j≤i yj (m). Then, F gives the functor of the category of M(m, y )-schemes to the category of sets. The functor is representable by an M(m, y )-scheme, which is denoted by B. Let π : B −→ M(m, y ) denote the natural projection. (1) ∗ V0|D on B × D. Let us consider the obstruction theory of B. Set VD := πD (1) (i) u We have the universal quotients VD −→ Ci (i = 1, . . . , l). We put VD := (1) u for i = 2, . . . , l + 1. We also have the locally free sheaf Ker VD −→ Ci−1 V−1 |D . By changing slightly the construction in Subsection 2.1.6, we consider the ∗ ) given as follows: complex C(V•|D , VD d−1

Hom(V0|D , V−1|D ) −→

l+1

(i)

(i)

Hom(VD , VD ) ⊕ Hom(V−1|D , V−1|D )

i=1 d0

−→

l

(i+1)

Hom(VD

(i)

, VD ) ⊕ Hom(V−1|D , V0|D )

i=1

The first term stands in the degree −1. The morphism d−1 is the composite of the following morphisms: a

1 Hom(V0|D , V0|D ) ⊕ Hom(V−1|D , V−1|D ) Hom(V0|D , V−1|D ) −→

a

2 −→

l+1

(i)

(i)

Hom(VD , VD ) ⊕ Hom(V−1|D , V−1|D )

i=1

Here a1 is the differential of the complex Hom(V•|D , V•|D ), and a2 is the inclusion (1) via V0|D = VD . The morphism d0 is made of the following maps bi (i = 1, 2): b1 :

l+1 i=1

(i)

(i)

Hom(VD , VD ) −→

l i=1

(i+1)

Hom(VD

(i)

, VD )

6.6 Parabolic Perturbation (1)

255 (1)

b2 : Hom(VD , VD ) ⊕ Hom(V−1|D , V−1|D ) −→ Hom(V−1|D , V0|D ) Here b1 is given in (2.2), and b2 is the differential of the complex Hom(V•|D , V•|D ). ∗ ∗ ∨ ) := C(V•|D , VD ) [−1]. We have the naturally defined morWe put gD (V• , VD ∗ phism C(V•|D , VD ) −→ Hom(V•|D , V•|D ), which induces the morphism: ∗ ) g(V•|D ) −→ gD (V• , VD (i)

We take vector spaces W (i) over k such that rank W (i) = rank VD . We also take a vector space W−1 over k such that rank W−1 = rank V−1 , and we put (i) W0 := W (1) . We put WD := W (i) ⊗ OD and Wi D := Wi ⊗ OD . We have (i+1) .l+1 .l (i) the naturally defined right i=1 GL(W (i) )-action on i=1 N WD , WD . We also have the natural right GL(W (1) ) × GL(W−1 )-action on N (W−1 D , W0 D ) by the identification W0 = W (1) . Therefore, we have the naturally defined right action (i+1) .l+1 .l (i) of i=1 GL(W (i) ) × GL(W−1 ) on i=1 N WD , WD × N W−1 D , W0 D , where the latter fiber products are taken over D. The quotient stack is denoted by Y D (W• , W ∗ ). (We remark that we used the symbol Y D (W• , W ∗ ) in a different meaning in Section 6.3.) Moreover, we use the stack YD (W• ) introduced in Subsection 5.5.1. The morphism Y D (W• , W ∗ ) −→ YD (W• ) is induced by the natu.l+1 .l (i+1) (i) ral projections i=1 N (WD , WD ) × N (W−1 D , W0 D ) and i=1 GL(W (i) ) × GL(W−1 ) onto N (W−1 D , W0 D ) and GL(W0 ) × GL(W−1 ), respectively. (i+1) ∗ and V−1|D , we have the classifying map Φ(V• , VD ) : B × D −→ From VD ∗ Y D (W• , W ). We also have the classifying map Φ(V•|D ) : M(m, y ) × D −→ YD (W• ). They give the following commutative diagram: B×D ⏐ ⏐ !

−−−−→ Y D (W• , W ∗ ) ⏐ ⏐ !

M(m, y ) × D −−−−→

(6.31)

YD (W• )

∗ ∗ ∗ ) LY D (W• ,W ∗ ) is represented by gD (V• , VD )≤1 . It can be shown that Φ(V• , VD Moreover, the diagram (6.31) induces the following commutative diagram:

LB×D/D ⏐ ⏐

∗ ∗ ∗ ←−−−− Φ(V• , VD ) LY D (W• ,W ∗ ) ←−−−− gD (V, VD ) ⏐ ⏐ ⏐ ⏐

∗ πD LM(m, y)×D/D ←−−−−

Φ(V•|D )∗ LYD (W• )

←−−−−

g(V•|D )

∗ ∗ ) := Cone g(V•|D ) −→ gD (V• , VD ) . We set We put grel (V• , VD ∗ Ob(V•|D ) := RpD ∗ g(V•|D ) ⊗ ωD , Obrel (B) := RpD ∗ grel (V• , VD ) ⊗ ωD .

256

6 Virtual Fundamental Classes

We obtain the following commutative diagram: π ∗ LM(m, y) ⏐ ⏐

←−−−−

Ob(V•|D ) ⏐ ⏐

LB/M(m, y) [−1] ←−−−− Obrel (B)[−1] Therefore, we obtain the following commutative diagram: Obrel (B)[−1] −−−−→ π ∗ Ob(m, y ) ⏐ ⏐ ⏐ ⏐ ! ! LB/M(m, y) [−1] −−−−→ π ∗ LM(m, y) We put Ob(B) := Cone Obrel (B) −→ π ∗ Ob(m, y ) . We obtain a morphism Ob(B) −→ LB . Proposition 6.6.5 The morphism Obrel (B) −→ LB/M(m, y) gives a relative obstruction theory of B over M(m, y ). The complex Obrel (B) is quasi-isomorphic to its 0-th cohomology sheaf. Proof The first claim follows from Lemma 2.4.17. Let us show the second claim. By an argument in the proof of Proposition 6.1.1, we have only to check H i i∗z Obrel (B) = 0 ∗ for i = 0 and any point iz : {z} −→ B. Let (E, VD ) denote the tuple correi ∗ sponding to z. Then, H iz Obrel (B) is the dual of the hyper-cohomology group H−i (D, Q), where Q is the following complex: l i=1

(i+1)

Hom(VD

(i+1)

, VD

) −→

l

(i+1)

Hom(VD

(i)

, VD )

i=1

Here, the first term stands in the degree −1. We use the following lemma. (i+1) (i) (i+1) = 0. , VD /VD Lemma 6.6.6 H 1 D, Hom VD (1) (i) Proof By definition of B, we have H 1 D, VD /VD = 0 for any i = 1, . . . , l+1. We have the following exact sequence: (1) (i+1) ν1 (1) (i) (i) (i+1) H 0 D, VD /VD −→ H 0 D, VD /VD −→ H 1 D, VD /VD (1) (i+1) =0 −→ H 1 D, VD /VD

6.6 Parabolic Perturbation

257

By definition of B, we have the surjectivity of ν1 . Hence, we obtain (i) (i+1) H 1 D, VD /VD = 0. (i+1)

From the exact sequence, VD ing exact sequence:

(1)

(1)

(i+1)

−→ VD −→ VD /VD

, we have the follow-

(1) (i) (i+1) (i+1) (i) (i+1) −→ Ext1 VD Ext1 VD , VD /VD , VD /VD (1) (i+1) (i) (i+1) −→ Ext2 VD /VD , VD /VD

(6.32)

(1)

Recall that VD is a direct sum of OD , and hence we have the vanishing of the first term in (6.32). Since the divisor D is smooth, we have the vanishing of the third term in (6.32). Therefore, we obtain the desired vanishing. From Lemma 6.6.6, we can easily obtain the vanishing of Hi D, Q unless i = 0. Thus, the proof of Proposition 6.6.5 is finished. As a result, Ob(B) −→ LB is an obstruction theory of B, and the morphism B −→ M(m, y ) is smooth.

6.6.3 Compatibility of the Obstruction Theories On B × D, we have the filtration V (l+1) ⊂ V (l) ⊂ · · · ⊂ V (1) . We put ∗ u E (m)|D V := pD ∗ Hom V (l+1) , πD We have the canonical section ψ, which is given by the composite V (l+1) ⊂ ∗ u E (m)|D . V (1) −→ πD Lemma 6.6.7 V is a locally free sheaf on B. Proof Let z = (E, VD∗ ) be any point of B. We have only to check (l+1) Ext1 VD , E(m)|D = 0. We have the following exact sequence: (1) (l+1) Ext1 VD , E(m)|D −→ Ext1 VD , E(m)|D

(1) (l+1) −→ Ext2 VD /VD , E(m)|D

(1)

Since VD is a direct sum of OD , the first term vanishes. Since D is smooth, the last term vanishes. Therefore, we obtain the desired vanishing.

258

6 Virtual Fundamental Classes

). We have the following Cartesian It is easy to observe ψ −1 (0) = M(m, y diagram: i

) −−−1−→ B M(m, y ⏐ ⏐ ⏐ ⏐ (6.33) j! ψ! B

i

−−−−→ V

Here i is the 0-section.

). We take an isoLet us compare the obstruction theories of B and M(m, y morphism I : W (l+1) W−1 . It induces the morphism YD (W• , W ∗ ) −→ Y D (W• , W ∗ ). Then, we obtain the following commutative diagram:

) M(m, y ⏐ ⏐ !

−−−−→

B ⏐ ⏐ !

−−−−→ M(m, y ) ⏐ ⏐ !

YD (W• , W ∗ ) −−−−→ Y D (W• , W ∗ ) −−−−→ YD (W• )

): It induces the following commutative diagram on M(m, y LM(m, y) ⏐ ⏐

←−−−−

LB ⏐ ⏐

←−−−− LM(m, y) ⏐ ⏐

LYD (W• ,W ∗ ) ←−−−− LY D (W• ,W ∗ ) ←−−−− LYD (W• ) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ∗ ∗ g(V•|D , VD ) ←−−−− g(V•|D , VD ) ←−−−− g(V•|D )

): Therefore, we obtain the following commutative diagram on M(m, y β

) i∗1 Ob(B) −−−−→ Ob(m, y ⏐ ⏐ ⏐ ⏐ ! ! i∗1 LB

(6.34)

−−−−→ LM(m, y)

Lemma 6.6.8 The cone of β is isomorphic to j ∗ LB/V . The morphism Cone(β) −→ LM(m, y)/B obtained from (6.34) is the same as the morphism obtained from the

) and B are comdiagram (6.33). In particular, the obstruction theories of M(m, y patible over i. ∨ Proof We put grel := Hom V−1 | D [1], V•|D . Then, it is easy to see the following: Cone g(V•|D , V ∗ ) −→ g(V•|D , V ∗ ) LYD (W• ,W ∗ )/Y D (W• ,W ∗ ) grel We put Obrel := RpD ∗ grel ⊗ ωD . We have the induced morphism: a : Obrel −→ LM(m, y)/B

6.6 Parabolic Perturbation

259

) , and hence we obtain a morWe have Obrel Cone i∗1 Ob(B) −→ Ob(m, y phism b : Obrel −→ LM(m, y)/B from the diagram (6.34). It is easy to observe a = b. We have the natural GL(W−1 ) × GL(W (1) ) × GL(W (l+1) )-action on (l+1)

N (WD

(1)

(1)

, WD ) ×D N (W−1D , WD ).

The quotient stack is denoted by Y1 . The isomorphisms W−1 W (l+1) and W0 W (1) induce YD (W• ) −→ Y1 . Then, we obtain the following diagram: γ1

M(m, y) × D −−−−→ YD (W• , W ∗ ) −−−−→ YD (W• ) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ! ! ! −−−−→ Y D (W• , W ∗ ) −−−−→

B×D

Y1

It induces the isomorphism: ∗ ∗ ∗ ∗ ∗ ) γ1 LYD (W• )/Y1 −→ Φ(V•|D , VD ) LYD (W• ,W ∗ )/Y D (W• ,W ∗ ) Φ(V•|D , VD

We have the natural GL(W−1 ) × GL(W0 ) × GL(W (l+1) )-action on (l+1)

N (W−1 D , W0 D ) ×D N (WD

, W−1 D ).

The quotient stack is denoted by Y0 . The isomorphism I : W (l+1) W−1 induces the following map: N (W−1 , W0 ) −→ N (W−1 , W0 ) × N (W (l+1) , W−1 ),

f −→ (f, I)

(6.35)

We also have the homomorphism GL(W−1 ) −→ GL(W0 ) × GL(W (l+1) ) induced by I. The morphism (6.35) is equivariant with respect to the actions. Therefore, we obtain a morphism YD (W• ) −→ Y0 . It is easy to observe that the morphism is an open immersion. Hence, we have the following diagram: open

M(m, y) × D −−−−→ YD (W• ) −−−−→ ⏐ ⏐ ⏐ ⏐ ! ! B×D

−−−−→

Y1

Y0 ⏐ ⏐ !

=

−−−−→ Y1

Put Wi := N (W (l+1) , Wi ) (i = −1, 0). We have the natural right GL(W−1 ) × GL(W0 )-action on N (W−1 , W0 ) × N (k, W−1 ) and N (W−1 , W0 ) × N (k, W0 ). The quotient stacks are denoted by Z 0 and Z 1 , respectively. We have the naturally induced map Yi −→ Z i , and we obtain the following commutative diagram: a

M(m, y) × D −−−1−→ ⏐ ⏐ ! B

a

Y0 −−−2−→ ⏐ ⏐ !

Z0 ⏐ ⏐ !

−−−−→ Y1 −−−−→ Z 1

(6.36)

260

6 Virtual Fundamental Classes

Thus, we obtain the following:

a∗1 a∗2 LZ 0 /Z 1 −→ a∗1 LY0 /Y1 −→ LYD (W• ,W ∗ )/Y D (W• ,W ∗ ) −→ LM(m,y)/B We would like to use the result in the latter 2.3.3. We half part of Subsection have the resolution of Hom V (l+1) , E u (m) given by Hom V (l+1) , V−1 −→ Hom V (l+1) , V0 on B × D. We set V0 := Hom V (l+1) , V−1 ,

V1 := Hom V (l+1) , V0 .

We have the naturally defined map φ : O −→ V1 on B × D, and the lift φ : O −→ (l+1) u , E (m) is naturally V0 on M(m, y) × D. The section ϕ of G := Hom V induced by φ. We put Zi = N (O, Vi ) (i = 0, 1). Then, we have the following commutative diagram: M(m, y) × D −−−−→ ⏐ ⏐ ! B×D

Z0 −−−−→ ⏐ ⏐ !

Z0 ⏐ ⏐ !

(6.37)

−−−−→ Z1 −−−−→ Z 1

We have the coincidence of the composite of the horizontal arrows in the diagrams (6.36) and (6.37). We have Obrel = Ob(G, ϕ), and the induced morphism Obrel −→ LM(m,y)/B is the same as that induced by the diagram (6.33) by Proposition 2.3.14. Thus, the proof of Lemma 6.6.8 is finished.

6.6.4 Smooth Ambient Stack Let Vm be an H(m)-dimensional vector space, where H denotes the Hilbert polynomial associated to y. For any k-scheme T , let F1 (T ) denote the set of sequences of quotients p∗T Vm,D = Cl+1 → Cl → Cl−1 → · · · → C2 → C1 satisfying the following conditions: • Ci and F are flat over T . • The induced morphisms H 0 D, Cl+1 | {u}×D −→ H 0 D, Ci | {u}×D are surjective u ∈ T . The induced morphism for any i = 1, . . .0,l and any point −→ H D, F is also surjective. H 0 D, C1 | {u}×D |{u}×D • We have H 1 D, F1 | {u}×D = 0 and H 1 D, Ci | {u}×D = 0 for any i = 1, . . . , l and any point u ∈ T . • The types of Ci are j≤i yj (m). The type of F is j yj (m). Let F2 (T ) denote the set of quotients p∗T Vm,D −→ F satisfying the following conditions: • F is flat over T .

6.6 Parabolic Perturbation

261

• For any point u ∈ T , the induced morphism Vm −→ H 0 (D, F|{u}×D ) is surjective. • H 1 D, F|{u}×D = 0 for any u ∈ T . • The type of F is j yj (m). Then, we obtain functors Fi (i = 1, 2) of the category of k-schemes to the category of sets. The functors Fi are representable by k-schemes, which we denote by Bi . We have the natural right GL(Vm )-action on Q◦ (m, y ) × B1 . The quotient stack is isomorphic to B. By restricting to D, we obtain a natural morphism Q◦ (m, y ) −→ B2 , which is GL(Vm )-equivariant.

m be as in (3.12). Then, we have the natural right GL(Vm )-action on Z

m × Let Z B1 × B2 . The quotient stack is denoted by B. We have the GL(Vm )-equivariant

m × B1 × B2 , which is immersion. Therefore, we morphism Q◦ (m, y ) × B1 −→ Z obtain an immersion B −→ B.

m × B1 × B2 × D: We have the universal filtration on Z (l+1)

VD

(l)

(2)

(1)

⊂ VD ⊂ · · · ⊂ VD ⊂ VD = Vm ⊗ OZ m ×B1 ×B2 ×D

m × B1 × B2 × D: We also have the universal subsheaf on Z V−1 D ⊂ V0 D = Vm ⊗ OZ×B

1 ×B2 ×D

m × B1 × B2 is naturally lifted to the actions on them. The GL(Vm )-action on Z (i) The descents are denoted by VD and V−1 D . We set := pD ∗ Hom V (l+1) , V (1) /V−1 D . V D D is locally By using an argument in the proof of Lemma 6.6.7, it can be shown that V free. We also have V|B V. Therefore, we obtain the following diagram:

) −−−−→ M(m, y ⏐ ⏐ i1 ! B

B −−−−→ B ⏐ ⏐ ⏐ ⏐ i2 ! i!

ψ −−−−→ V −−−−→ V

Here i and i2 denote the 0-sections.

6.6.5 Proof of Proposition 6.6.1 Let us finish the proof of Proposition 6.6.1. We take a sufficiently large integer m y , α∗ ). Then, we have such that the condition Om holds for any (E, F∗ , ρ) ∈ Ms (

) and Ms (

y , α∗ ) −→ M(m, y y ) −→ M(m, y). We the open immersions Ms (

262

6 Virtual Fundamental Classes

take the stack B as in Subsection 6.6.2. We put B(

y , α∗ ) := B ×M(m, y) Ms (

y ). s y ). The restrictions of V and ψ to By Proposition 6.6.5, it is smooth over M (

y , α∗ ). B(

y , α∗ ) are denoted by the same symbols. It is clear ψ −1 (0) = Ms (

Since B(

We have the immersion B(

y , α∗ ) −→ B. y , α∗ ) is Deligne-Mumford, y , α∗ ) of B(

which is Delignethere exists an open neighbourhood B(

y , α∗ ) in B, Mumford and smooth. The restriction of V to B(

y , α∗ ) are denoted by the same symbols. Then, we obtain the following diagram: y , α∗ ) y , α∗ ) −−−−→ B(

y , α∗ ) −−−−→ B(

Ms (

⏐ ⏐ ⏐ ⏐ ⏐ ⏐i ! !i !2 B(

y , α∗ ) −−−−→

V

−−−−→

V

Thanks to Lemma 6.6.8, the obstruction theories of Ms (

y , α∗ ) and y , α∗#) are " B(

y , α∗ )] = Ms (

y , α∗ ) accompatible. Therefore, we obtain the relation i! [B(

y )] = [B(

y , α∗ )]. cording to Proposition 2.4.8. We also have the relation G! [Ms (

Thus, we obtain the relation (6.28).

Chapter 7

Invariants

We construct our invariants, and study their transitions. For simplicity, the ground field is assumed to be the complex number field C in this chapter. Let H ∗ (A) and H∗ (A) denote the singular cohomology and homology groups of a topological space A with Q-coefficient. They are naturally Z/2Z-graded. Let X be a smooth connected complex projective surface with a base point, and let D be a smooth hypersurface of X. We denote the Picard variety of X by Pic. We often use the natural identification H 4 (X) = Q. The intersection product of a, b ∈ H 2 (X) is denoted by a · b. In Section 7.1, we explain a way of evaluation for our argument. In Section 7.2, we show transition formulas in simpler cases. By using them, we construct our invariants in Section 7.3. They are also sufficiently useful for the transition problems in the rank 2 case, which we study in Section 7.4. In Section 7.5, we give transition formulas for the case pg = dim H 2 (X, OX )! > 0. They are formally the same as those in the simpler case. In Section 7.6, we study transition formulas for the case pg = 0. By using it, we obtain a weak wall crossing formula in Section 7.7. We write down the weak wall crossing formula and a weak intersection rounding formula for the rank 3 case in Subsection 7.7.3–7.7.4. We also give a transition formula for variation of parabolic weights in Subsection 7.7.5. In Section 7.8, we derive weak intersection rounding formulas from weak wall crossing formulas.

7.1 Preliminary Naively, we would like to define and investigate Φ

(7.1)

Mss (

y ,α∗ )

T. Mochizuki, Donaldson Type Invariants for Algebraic Surfaces: Transition 263 of Moduli Stacks, Lecture Notes in Mathematics 1972, DOI: 10.1007/978-3-540-93913-9 7, c Springer-Verlag Berlin Heidelberg 2009

264

7 Invariants

for cohomology classes Φ on Mss (

y , α∗ ). If the 1-stability condition holds for (y, α∗ ), it should be the evaluation of Φ over the virtual fundamental classes y , α∗ )], obtained in Section 6.1. But, we would like to consider (7.1), even [Ms (

if the 1-stability condition does not hold for (y, α∗ ). It will be argued in Subsection 7.3.1, but we consider only cohomology classes which are obtained as polynomials in the slant products of cohomology classes of X and the Chern character of the universal sheaves. We introduce some universal ring in which a formal calculation of such cohomology classes can be done, and we will explain how to take evaluations.

7.1.1 Ring Rl Let Mapf Z2≥ 0 , H ∗ (X) denote the set of maps ϕ : Z2≥ 0 −→ H ∗ (X) such that (n1 , n2 ) ϕ(n1 , n2 ) = 0 is finite. We use the symbol Mapf Z3≥ 0 , H ∗ (D) in a similar meaning. The sets Mapf Z2≥ 0 , H ∗ (X) and Mapf Z3≥ 0 , H ∗ (D) are naturally vector spaces over Q. They are also naturally Z/2Z-graded. In general, when we are given a Z/2Z-graded vector space V , let Sym(V ) denote the super symmetric product of V , which is a super commutative algebra. For any l ≥ 1, we define ∗ ∗ ⊗ Sym Mapf Z3l R l := Sym Mapf Z2l ≥ 0 , H (X) ≥ 0 , H (D) Rl := H ∗ (Pic) ⊗ R l In the case l = 1, we prefer to using the symbol R instead of R1 . An element of Rl is described as the sum of the elements of the following form: P =c·

m1

(ai , V i ) ·

i=1

m2

(bj , U j )

j=1

• c ∈ H ∗ (Pic). (h) (h) • ai ∈ H ∗ (X) and V i = vi (1), vi (2) h = 1, . . . , l ∈ Z2l ≥ 0 . We identify 2l ∗ (ai , V i ) with the map ϕi : Z≥ 0 −→ H (X): ϕi (V ) =

ai (V = V i ) 0 (V =

V i)

(h) (h) (h) • bj ∈ H ∗ (D) and U j = uj (1), uj (2), uj (3) ∈ Z3l ≥ 0 . We identify ∗ (bj , U j ) with the map ψj : Z3l −→ H (D): ≥0 ψj (U ) =

bj (U = U j ) 0 (U =

Uj)

7.1 Preliminary

265

For a given P as above, we set d1 (P ) :=

m1 h

(h)

(h)

vi (1) · vi (2) +

m2

i=1

(h)

(h)

uj (1) · uj (2) − 2m1 − m2 .

h j=1

If ai , bj and c are homogeneous, we put d2 (P ) := deg(ai )/2 + deg(bj )/2 + deg(c)/2 i

j

and d(P ) := d1 (P ) + d2 (P ). Let (a, V ) ∈ H ∗ (X) × Z2l ≥ 0 be as above. We regard V as a tuple: (v (1) , . . . , v (l) ) ∈ (Z2≥ 0 )l Let ΔlX denote the diagonal map X −→ X l , and ΔlX ∗ denotes the Gysin map H ∗ (X) −→ H ∗ (X l ) = H ∗ (X)⊗ l . We have the following expression: ΔlX ∗ (a) =

l

αi,h

h i=1

Then, we set α1,h , v (1) ⊗ · · · ⊗ αl,h , v (l) q l (a, V ) = h

Let (b, U ) ∈ H ∗ (D) × Z3l ≥ 0 be as above. We regard U as a tuple: (u(1) , . . . , u(l) ) ∈ (Z3≥ 0 )l Let ΔlD denote the diagonal map D −→ Dl . We have the following expression: ΔlD ∗ (b) =

l

βi,h

h i=1

Then, we set β1,h , u(1) ⊗ · · · ⊗ βl,h , u(l) q l (b, U ) = h

They induce the algebra homomorphism q l : R l −→ R 1⊗ l . We also have the morphism ql : H ∗ (Pic) −→ H ∗ (Pic)⊗ l induced by the group structure of Pic. Then, we obtain the algebra homomorphism: ql : Rl −→ R⊗ l

(7.2)

266

7 Invariants

We have the naturally defined algebra homomorphism R ⊗ l −→ R . Hence, ql induces the algebra homomorphism rl : Rl −→ R.

7.1.2 Homomorphisms Let Y be an algebraic stack over Pic. When we are given a tuple of parabolic sheaves E ∗ = (E1 ∗ , . . . , El ∗ ) over Y × (X, D), we put R(E ∗ ) := Rl . In that case, (a, V ) and (b, U ) are symbolically denoted as follows: l

B a,

v (h) (2)

l

chv(h) (1) (Eh )

h=1

B u(h) (2) chu(h) (1) Gru(h) (3) (Eh ) b

(7.3)

h=1

In particular, we use the symbol R(E∗ ) in the case l = 1. We will often omit to denote the parabolic structure if there are no risk of confusion, i.e., R(E) and R(E) are used instead of R(E ∗ ) and R(E∗ ). When we are given a direct sum E∗ = E1 ∗ ⊕ E2 ∗ , we have the algebra homomorphism 1∗ ,E2∗ : R(E∗ ) −→ R(E1 ∗ , E2 ∗ ) ϕE E∗ induced by the following correspondence: j 1∗ ,E2∗ chi (E)/a ϕE E∗

j ' ( B j h chi (E1 ) · chj−h = (E ) a 2 i h h=0

1∗ ,E2∗ ϕE E∗

chji Grk (E) b

=

j ' ( j

h

h=0

B Grk (E2 ) chhi Grk (E1 ) chj−h b i

' ( j 1∗ ,E2∗ and q2 , denote the binomial coefficients. As the composite of ϕE E∗ h we obtain the algebra morphism Here,

R(E∗ ) −→ R(E1∗ ) ⊗ R(E2 ∗ ). For an element P (E) ∈ R(E∗ ), the image is denoted by P (E1 ⊕ E2 ) ∈ R(E1 ∗ ) ⊗ R(E2 ∗ ). The algebra homomorphism rl gives l

) *+ , rl : R(E∗ , E∗ , . . . , E∗ ) −→ R(E∗ ). Let t be a formal variable, and let w ∈ Q. The element chji (E ⊗ ewt )/a ∈ R(E)[t] is given by

7.1 Preliminary

267

chji (E ⊗ ewt )/a := i k=0

.i

jk =j

j!

k=0 jk !

ri+1

i

B

chjkk (E)

a

k=0

(wt)ji− jk ·k · .i jk k=0 (i − k)!

(7.4)

Similarly, we have the element chji (Grh (E) ⊗ ewt )/b ∈ R(E)[t] given as follows: chji Grh (E) ⊗ ewt b := i B (wt)ji− jk ·k j! jk Gr ch (E) b · r .i jk i+1 h .i k i k=0 jk ! k=0 k=0 (i − k)! j =j k=0

(7.5)

k

By the correspondences chji (E)/a −→ chji (E · ewt )/a and chji (Grh (E))/b −→ chji (Grh (E) · ewt )/b, we obtain the algebra isomorphism R(E) −→ R(E)[t]. The image of P (E) ∈ R(E) is denoted by P (E · ewt ). Remark 7.1.1 The formula (7.4) is formally obtained from a development of

i

h=0

(wt)i−h chh (E) · (i − h)!

The formula (7.5) is obtained similarly.

j .

7.1.3 Evaluation Let Y be a proper Deligne-Mumford stack over Pic. Assume that we are given a parabolic sheaf E∗ over Y × (X, D) with a parabolic structure. Let A∗ (Y) denote the rational Chow group of Y. Let P ∈ R(E∗ ) be of the following form: P =c·

m2 u (2) v (2) chvii (1) (E)/ai · chujj (1) Gruj (3) (E) bj

m1 i=1

j=1

For simplicity, we assume that c ∈ H ∗ (Pic), ai ∈ H ∗ (X) and bj ∈ H ∗ (D) are homogeneous. We would like to construct a linear morphism of A∗ (Y) to Q. Let πX,i denote the projection of Y × X m1 × Dm2 onto the product of Y and the i-th X. Let πD,j denote the projection of Y × X m1 × Dm2 onto the product of Y and the j-th D. Let p denote the natural morphism Y × X m1 × Dm2 to Pic ×X m1 × Dm2 . Let Z be a d-dimensional algebraic cycle on Y. We have the induced cycle m1 ,m2 := Z × X m1 × Dm2 . We obtain the following element of the Chow group ZX,D A∗ Pic ×X m1 × Dm2 of Pic ×X m1 × Dm2 with rational coefficient:

268

7 Invariants

ΛP (E∗ , Z) := ⎞ ⎛ m2 m1 ∗ ∗ vi (2) uj (2) " m1 ,m2 # chuj (1) πD,j Gruj (3) (E) ∩ ZX,D ⎠ p∗ ⎝ chvi (1) πX,i E i=1

j=1

Thus, we obtain a linear map ΛP (E∗ , ·) : Ad (Y) −→ Ad−d1 (P ) Pic ×X m1 × Dm2 . The cycle ΛP (E∗ , Z) determines a homology class cycl ΛP (E∗ , Z) ∈ H2(d−d1 (P )) Pic ×X m1 × Dm2 Let π∗ denote the push-forward for Pic ×X m1 ×Dm2 to a point pt. Then, we obtain the following: m1 m2 deg P (E∗ ) ∩ [Z] := π∗ c · ai · bj ∩ cycl ΛP (E∗ , Z) ∈ H2(d−d(P )) (pt) i=1

j=1

It is trivial unless d = d(P ). We identify H0 (pt) Q. Thus, we obtain a linear map deg P (E∗ ) ∩ • : A∗ (Y) −→ Q. Let A∗ (Y) denote the bivariant theory A∗ (Y → Y). (See [41] and [129].) Let Z be an algebraic cycle of Y, and let F ∈ A∗ (Y). Then, we obtain the following number deg P (E∗ ) ∩ F [Z] . ∗ Notation 7.1.2 Let Y and E∗ be as above. ∗ ,Y) := R(E∗ ) ⊗ A (Y). We put R(E ∗ We can naturally regard R(E∗ , Y) as an R(E∗ ), A (Y) -bimodule.

We obtain a linear morphism R(E∗ , Y)⊗A∗ (Y)−→Q by the above construction. Remark 7.1.3 Formally, deg P (E∗ ) ∩ [Z] is the following number: Z

c·

m1 i=1

m2 vi (2) uj (2) chvi (1) E chuj (1) Gruj (3) (E) bj /ai ·

(7.6)

j=1

The author does now know an appropriate reference for cohomology and homology theories of Deligne-Mumford stacks with good cycle maps from the Chow groups, Gm -localization theory and any other expected properties. That is the reason to avoid (7.6) as the definition. We are particularly interested in the following examples. Example 7.1. Let y ∈ T ype, and let α∗ be a system of weights. We have a universal

u over Mss (

y , α∗ ) × X with the parabolic structure at Mss (

y , α∗ ) × D. sheaf E

∗u ) = R, it is denoted by P (E

u ) ∈ R(E

∗u ). Let P ∈ R. With the identification R(E

7.1 Preliminary

269

Assume that the 1-stability condition holds for (y, α∗ ). By using the virtual funday , α∗ )] in Subsection 6.1.1, we define mental class [Mss (

u ) := deg P (E

u ) ∩ [Mss (

P (E y , α∗ )] . Mss (

y ,α∗ )

In other words, we obtain a linear map Mss ( y,α∗ ) : R −→ Q, under the assumption that the 1-stability condition holds for (y, α∗ ). We will later discuss how to obtain such a morphism in the general case. Example 7.2. Let y and α∗ be as above. Let L be a line bundle on X, and let δ ∈ condition holds for (y, L, α∗ , δ). We denote by ω ∈ P br such that the 1-stability y , [L], α∗ , δ) the first Chern class of the tautological line bundle Orel (1) A∗ Ms (

y , [L], α∗ , δ). For any P ∈ R, we have the element on Ms (

u

u ) · ωk ∈ R E

, Ms (

P (E y , [L], α∗ , δ) . " # By using the virtual fundamental class Ms (

y , [L], α∗ , δ) in Subsection 6.1.1, we obtain the following number: " #

u ) · ω k := deg P (E

u ) · ω k ∩ Ms (

P (E y , [L], α∗ , δ) Ms (

y ,[L],α∗ ,δ)

If the 1-vanishing condition holds for (y, L, α∗ , δ), moreover, then we have the y , [L], α∗ , δ) −→ M(

y , [L]). relative tangent bundle Θrel of the smooth map Ms (

Let Eu(Θrel ) denote the Euler class of Θrel . For any P ∈ R, we obtain the integral

u ) · Eu(Θrel ) ∈ Q. P (E Ms (

y ,[L],α∗ ,δ) The above evaluation procedure is naturally extended to the case that we are given a tuple of parabolic sheaves E ∗ = (E1 ∗ , . . . , El ∗ ) on Y × (X, D). Let P ∈ Rl (E ∗ ) be of the following form: l m l m1 v(h) (2) 2 u(h) (2) j i Gru(h) (3) (Eh ) bj P =c· ch (h) (Eh )/ai · ch (h) i=1

h=1

vi

(1)

j=1

h=1

(1)

uj

j

We assume that c ∈ H ∗ (Pic), ai ∈ H ∗ (X) and bj ∈ H ∗ (D) are homogeneous for simplicity. Let Z be a d-dimensional algebraic cycle on Y. Then, we obtain the following element of the Chow group A∗ Pic ×X m1 × Dm2 : ΛP (E ∗ , Z) := p∗

m1 l h=1 i=1 m2 j=1

∗ v(h) (2) chv(h) (1) πX,i Eh i · i

u(h) j (2)

∗ chu(h) (1) πD,j Gru(h) (3) (Eh ) j

j

⎞ " m1 ,m2 # ∩ ZX,D ⎠

270

7 Invariants

Thus, we obtain a linear map ΛP (E∗ , ·) : Ad (Y) −→ Ad−d1 (P ) Pic ×X m1 × Dm2 . The cycle ΛP (E∗ , Z) determines the homology class cycl(ΛP (E∗ , Z)) ∈ H2(d−d1 (P )) (Pic ×X m1 × Dm2 ). We obtain the following: m2 m1 ai bj ∩ cycl ΛP (E∗ , Z) deg P (E ∗ ) ∩ [Z] := π∗ c i=1

j=1

∈ H2(d−d(P )) (pt). By identifying H0 (pt) = Q, we obtain a linear map deg P (E ∗ ) ∩ · : A∗ (Y) −→ Q. Notation 7.1.4 We put R(E ∗ , Y) := R(E ∗ ) ⊗ A∗ (Y). We obtain a linear map R(E ∗ , Y) ⊗ A∗ (Y) −→ Q by the above construction. We have the following commutative diagram, which we will use implicitly. l

) *+ , R(E∗ , . . . , E∗ , Y) −−−−→ Hom(A∗ (Y), Q) ⏐ ⏐ ⏐r ⏐= !l ! R(E∗ , Y)

−−−−→ Hom(A∗ (Y), Q)

.l Assume that we have a decomposition Y = i=1 Yi such that Ei ∗ are the pull back of parabolic sheaves over Yi × (X, D) via the natural.projections, where Yi are stacks over Pic and the map Y −→ Pic is induced by Yi −→ Picl and the group structure of Pic. We have the naturally defined morphism: l l R(Ei ∗ , Yi ) −→ HomQ A∗ (Yi ), Q Γ1 : i=1

i=1

We also have the following morphism: Γ2 : R(E ∗ ) ⊗

l

∗

A (Yi ) −→ HomQ

i=1

The algebra homomorphism ql in (7.2) induces

l i=1

A∗ (Yi ), Q

7.1 Preliminary

271

R(E ∗ ) ⊗

l

A∗ (Yi ) −→

i=1

l

R(Ei∗ , Yi ),

i=1

which is also denoted by ql . We will use the following lemma implicitly, which can be checked by a formal calculation. Lemma 7.1.5 We have Γ1 ◦ ql = Γ2 .

7.1.4 Equivariant Case We continue to use the setting in Subsection 7.1.3. We recall equivariant Chow groups with respect to a torus action. See [23] and [55] for more details and precise. l Let T denote an l-dimensional torus m(Gl m ) . The T -equivariant bivariant group of ∗ a point R(T ) is the limit of A (P ) with respect to the morphisms induced by inclusions. Let ti denote the first Chern class of the tautological line bundle of the i-th Pm , equipped with degree 2. Then, we have the isomorphism R(T ) Q[t1 , . . . , tl ] of graded algebras. It is convenient to use the symbol ew·ti to denote a trivial line bundle with a T -action induced by the action of the i-th Gm with weight w ∈ Q. Note an action with rational weight makes sense, by taking a covering Gm −→ Gm . Assume that Y is provided with a T -action. Let Ap+1 denote the (p + 1)dimensional linear space. We have the Gm -action on Ap+1 ∗ := Ap+1 −{(0, . . . , 0)} l given by component-wise multiplication. It induces the T -action on Ap+1 ∗ × Y. The quotient stack is denoted by Y (p) . The T -equivariant Chow group of ATd (Y) is defined to be Ad+lp (Y (p) ) for a sufficiently large p in this case. We take a linear inclusion ι : Ap+1 −→ Ap+2 , which induces the regular embedding Y (p) −→ Y (p+1) . Thus, we obtain the morphism Ad+l(p+1) (Y (p+1) ) −→ Ad+lp (Y (p) ), which is independent of the choice of ι. It is an isomorphism if p is sufficiently large for d. Hence, ATd (Y) is well defined. We have the naturally defined morphism Y (p) −→ (Pp )l . Let O(i) (1) denote the tautological of the i-th Pp . We have the action of the first Chern class (i) line bundle(p) c1 O (1) : Ad+lp (Y ) −→ Ad−1+lp (Y (p) ), which induces the action ti : ATd (Y) −→ ATd−1 (Y). Thus, we can naturally regard AT∗ (Y) as an R(T )-module. If Y is a variety over C, we also have the T -equivariant homology group HdT (Y) given by Hd+lp (Y (p) ), where p is sufficiently large for d. We can naturally regard T (Y) is an H∗T (Y) as an R(T )-module, and the cycle map cycl : ATd (Y) −→ H2d R(T )-homomorphism. We have the equivariant version of the evaluation procedure explained in Subsection 7.1.3. Assume that E∗ is provided with a T -action. Let P ∈ R(E∗ ). We (p) naturally obtain the parabolic sheaf E∗ on Y (p) . Let Z ∈ ATd (Y), and let Z (p)

272

7 Invariants

be the corresponding Ad+lp (Y(p) ). We obtain the following element of p l element of m1 Ad−d1 (P )+lp (P ) × Pic ×X × Dm2 : (p)

ΛP (E∗ , Z (p) ) := m2 m1 ∗ vi (2) ∗ uj (2) chvi (1) πX,i E (p) chuj (1) πD,j Gruj (3) (E (p) ) p∗ i=1

j=1

# " ∩ Z (p) × X m1 × Dm2

(p) It is easy to observe that the family ΛP (E∗ , Z (p) ) determines an element ΛTP (E∗ , Z) of ATd−d1 (P ) (Pic ×X m1 × Dm2 ). Thus, we obtain a linear map ΛTP (E∗ , ·) : ATd (Y) −→ ATd−d1 (P ) (Pic ×X m1 × Dm2 ). We obtain the equivariant homology class T Pic ×X m1 × Dm2 cycl ΛTP (E∗ , ·) ∈ H2(d−d 1 (P )) Since the T -action on Pic ×X m1 × Dm2 are trivial, c, ai and bj naturally give the equivariant cohomology classes. Let π∗T denote the equivariant Gysin map for Pic ×X m1 × Dm2 −→ pt. We obtain the following: m1 m2 degT P (E∗ ) ∩ [Z] := π∗T c · ai · bj ∩ cycl ΛTP (E, Z) i=1

j=1 T ∈ H2(d−d(P )) (pt)

Since H∗T (pt) is a free R(T )-module of rank one with the special base 1 ∈ H0T (pt), we identify it with R(T ). The above procedure is compatible with the operations of ti . Thus, we obtain an R(T )-homomorphism degT P (E∗ ) ∩ • : AT∗ (Y) −→ R(T ).

Notation 7.1.6 Let A∗T (Y) denote the T -equivariant bivariant theory of Y. We put RT (E∗ , Y) := R(E∗ ) ⊗Q A∗T (Y). By the above construction, we obtain an R(T )-homomorphism degT : RT (E∗ , Y) −→ HomR(T ) AT∗ (Y), R(T ) We are particularly interested in the following example:

be a master space as in Subsections 4.5.1, 4.7.1 or 4.7.2. We Example 7.3. Let M

M

× X with the parabolic structure at M

× D. We have the

have the sheaf E on M

7.1 Preliminary

273

, which is naturally lifted to that on E

M Gm -action ρ on M . As explained in Section

is lifted to the Gm -equivariant obstruction 5.9, the perfect obstruction theory of M

] ∈ AGm (M

). For any Φ(E

M

theory. Thus, we have [M ∗ ) ∈ RT (E, M ), we obtain ∗

Gm

M

M

Φ(E Φ(E ∗ ) := deg ∗ ) ∩ [M ] ∈ R(Gm ).

M

According to [23], we have the isomorphism: −1 m ] AG ∗ (M ) ⊗Q[t] Q[t, t

Gm (I) ⊗Q Q[t, t−1 ]

i ) ⊗Q Q[t, t−1 ] ⊕ A∗ (M A∗ M i=1,2

I

−1 m ] due to GraberWe have the following decomposition in AG ∗ (M ) ⊗Q[t] Q[t, t Pandharipande [55]:

i ) −1 ∩ [M

i ] +

Gm (I)) −1 ∩ [M

Gm (I)]

] = Eu N(M Eu N(M [M i=1,2

I

Gm (I)] are the virtual fundamental classes with respect to the

i ] and [M Here [M

i ) and N(M

Gm (I)) are the virtual normal induced obstruction theories, and N(M bundles. Then, we obtain the following equality in A∗ (Pic ×X m1 × Dm2 )[t, t−1 ]: m ΛG P

∗M

] Φ(E ), [M

=

m ΛG P

i=1,2

+

Φ

m ΛG P

i ] [M

i ) Eu(N M

∗M E

i , |M

Φ

M E

Gm (I) , ∗ |M

I

Gm (I)] [M

Gm (I)) Eu N(M

We obtain the following equality in Q[t, t−1 ]:

M

M Φ(E ∗ )

=

i=1,2

i M

M

) Φ (E ∗ |Mi

i )) Eu(N(M

+

I

Gm (I) M

M

) G Φ (E ∗ |M m (I)

Gm (I))) Eu(N(M

In particular, we obtain the equality for the coefficients of t0 .

7.1.5 Ring RCH By using the bivariant theories of X, D and Pic, we set RCH := Sym Mapf Z2≥ 0 , A∗ (X) ⊗ Sym Mapf Z3≥ 0 , A∗ (D) ⊗ A∗ (Pic)

274

7 Invariants

An element of RCH is described as in the case of R. When a parabolic sheaf E∗ is given, we put RCH (E∗ ) := RCH . Similarly, Rl,CH are defined for each l ≥ 1. When a tuple of parabolic sheaves E ∗ = (E1 ∗ , . . . , El, ∗ ) is given, we set RCH (E ∗ ) := Rl,CH . We use a convention (7.3) to denote elements of RCH (E ∗ ). The maps A∗ (X) −→ H 2 ∗ (X) and A∗ (D) −→ H 2 ∗ (D) induce the algebra homomorphisms T : RCH (E ∗ ) −→ R(E ∗ ). .l v (h) (2) Let Z ∈ A∗ (Y). For h=1 chv(h) (1) (Eh ) a, we set l v (h) (2) Q chv(h) (1) (Eh ) a ∩ [Z] := h=1

pX ∗

l

chv(h) (1) (Eh )v

(h)

(2) ∗ pY (a)

∩ [Z × X] ∈ A∗ (Y).

h=1

For

.l h=1

u(h) (2) chu(h) (1) Gru(h) (3) (Eh ) b, we put

l u(h) (2) Q chu(h) (1) Gru(h) (3) (Eh ) b ∩ [Z] := h=1

pD ∗

l

u(h) (2) ∗ chu(h) (1) Gru(h) (3) (Eh ) pY (b) ∩ [Z × D] ∈ A∗ (Y)

h=1

They induce an algebra homomorphism Q : RCH (E ∗ ) −→ A∗ (Y). The following lemma can be checked by a formal calculation. We will use it implicitly. Lemma 7.1.7 Let Q ∈ RCH (E ∗ ),P ∈ R(E ∗ , Y) and Z ∈ A∗ (Y). Then, we have the equality deg (T(Q) · P ) ∩ [Z] = deg P ∩ (Q(Q) · Z) .

7.1.6 Equivariant Euler Class −1 For an algebra R, let R[[t series aj · tj such that , t] denote the algebra of power the sets j > 0|aj = 0 are finite. We put R(t) := Q[[t−1 , t]. Inductively, we set R(t1 , t2 , . . . , tk ) := R(t2 , . . . , tk )[[t−1 1 , t1 ] Let Ei ∗ (i = 1, 2) be parabolic sheaves over Y × (X, D). Assume that we are given the Gm -actions on Ei ∗ of weights wi . Then, we obtain the following virtual vector bundles on Y equipped with the Gm -action of weight w := w2 − w1 : F1 := RpX ∗ RHom E1 , E2 , F2 := RpD ∗ RHom 2 E1|D ∗ , E2|D ∗

7.1 Preliminary

275

(See Subsection 2.1.6 for the symbol RHom 2 .) We have the Gm -equivariant Euler class Eu(Fa ) ∈ A∗Gm (Y) ⊗Q[t] Q[t, t−1 ] of Fa (a = 1, 2). They are formally d(a)−i ∈ A∗ (Y) ⊗ R(t), where d(a) denote the expected ranks i≥0 ci (Fa ) · w · t of Fa . Thanks to the Grothendieck-Riemann-Rochtheorem,we have the elements Eu (Fa ) ∈ RCH (E1 ∗ , E2 ∗ ) ⊗ R(t) such that Q Eu (Fa ) = Eu(Fa ). For abbreviation, we use the symbols Eu(Fa ) to denote Eu (Fa ) and their image via the composite of the following morphisms: q2

T

RCH (E1 ∗ , E2 ∗ ) ⊗ R(t) −→ R(E1 ∗ , E2 ∗ ) ⊗ R(t) −→ R(E1 ∗ ) ⊗ R(E2 ∗ ) ⊗ R(t)

(7.7)

More generally, let T denote an l-dimensional torus. If Ei ∗ (i = 1, 2) are provided with T -actions, then we have the T -action on the virtual vector bundles Fa (a = 1, 2) above. Therefore, we obtain the T -equivariant Euler classes Eu(Fa ).

Thanks to the Grothendieck-Riemann-Roch theorem, we have the element Eu (Fa )

of RCH (E1 ∗ , E2 ∗ ) ⊗ R(t1 , . . . , tl ) such that Q Eu(Fa ) = Eu(Fa ). For abbreviation, we use the symbols Eu(Fa ) to denote Eu(Fa ) and their image via the composite of the morphisms in (7.7) replaced R(t) with R(t1 , . . . , tl ).

7.1.7 Twist by Line Bundle Let L be a line bundle on Y. We put ω := c1 (L). Formally, we often use the symbol eω to denote L, if there are no risk of confusion. Let E∗ be a parabolic sheaf over Y × (X, D). Then, we have the natural isomorphism R(E∗ , Y) R E∗ ⊗ eω , Y given by the following correspondence: chji (E ⊗ eω )/a := i k=0

jk =j

.i

j!

k=0 jk !

ri+1

i k=0

B

chjkk (E)

a

ω ji− jk ·k jk k=0 (i − k)!

· .i

chji Grh (E) ⊗ eω b := i B ω ji− jk ·k j! jk chk Grh (E) b · .i ri+1 .i jk i j ! k k=0 k=0 k=0 (i − k)! j =j k=0

(7.8)

(7.9)

k

Thus, we can naturally regard P (E⊗eω ) ∈ R(E⊗eω , Y) as an element of R(E, Y). Remark 7.1.8 We will often use “·” instead of “⊗” to save spaces.

276

7 Invariants

Let us consider the case Y = Y1 × Y2 . Assume that L comes from a line bundle on Y1 , and that E comes from a parabolic sheaf on Y2 × (X, D). The formulas (7.8) and (7.9) determine an element P (E · eω ) ∈ A∗ (Y1 ) ⊗ R(E).

7.2 Transition Formulas in Simpler Cases 7.2.1 Basic Case Let y ∈ T ype, and let α∗ be a system of weights. Let L be a line bundle on X. Let δ ∈ P br . Let P be an element of R, and let k be a non-negative integer. Let ω denote y , [L], α∗ , δ). We the first Chern class of the tautological line bundle Orel (1) on Ms (

obtain the element u

, Ms (

u ) · ωk ∈ R E y , [L], α∗ , δ) . P (E If δ is not critical, we obtain the following number by the procedure explained in Subsection 7.1.3:

u ) · ωk ∈ Q P (E Φ(

y , [L], α∗ , δ) := Ms (

y ,[L],α∗ ,δ)

Let δ be critical. We take parameters δ− < δ < δ+ such that δκ (κ = ±) are sufy , [L], α∗ , δ− ) ficiently close to δ. We would like to describe Φ(

y , [L], α∗ , δ+ ) − Φ(

as the sum of integrals over the products of moduli stacks of objects with lower ranks. Such a description is called a transition formula. For that purpose, we prepare some notation. Let S(y, α∗ , δ) denote the set of the decomposition types: 2 y 1 + y 2 = y, S(y, α∗ , δ) := I = (y 1 , y 2 ) ∈ T ype α∗ ,δ Py1 = Pyα2∗ = Pyα∗ ,δ For a given (y 1 , y 2 ) ∈ S(y, α∗ , δ), we put ri := rank y i . We also set

2 , L, α∗ , δ) := Mss (y 1 , L, α∗ , δ) × Mss (

y 2 , α∗ ). M(y 1 , y Note that the 2-stability condition for (y, L, α∗ , δ) implies the 1-stability conditions

2 , L, α∗ , δ) × X for (y 1 , L, α∗ , δ) and (y 2 , α∗ ). Let E1u be the sheaf on M(y 1 , y which is the pull back of the universal sheaf over Ms (y 1 , L, α∗ , δ) × X via the

u in a similar meaning. We set natural morphism. We use the symbol E 2 ω1 := c1 (Or(E1u ))/ rank r1 . We use the symbol ew·ω1 to denote Or(E1u )w/r1 .

7.2 Transition Formulas in Simpler Cases

277

Let Gm be a one dimensional torus. Let R(Gm ) be the Gm -equivariant bivariant theory of a point. It is isomorphic to the limit of A∗ (PN ), and we use the identification R(Gm ) = Q[t], where t corresponds to the first Chern class of the tautological line bundles. It is convenient to use the symbol ew·t to denote the trivial line bundle with a Gm -action of weight w ∈ Q. Note that “an action of rational weight” makes sense by considering a covering of Gm . We have the following virtual Gm 2 , L, α∗ , δ): equivariant vector bundle on M(y 1 , y

2u · er1 (t−ω1 )/r2 N0 (y 1 , y 2 ) := −RpX ∗ RHom E1u · e−t , E u r1 (t−ω1 )/r2

2 · e −RpX ∗ RHom E , E1u · e−t u −t u r1 (t−ω1 )/r2 −RpD ∗ RHom 2 E1|D · e , E · e ∗ 2|D ∗ u

r1 (t−ω1 )/r2 u −RpD ∗ RHom2 E2|D ∗ · e , E1|D ∗ · e−t

2u · er1 (t−ω1 )/r2 +RpX ∗ Hom L · e−t , E (See Subsection 2.1.6 for the symbol RHom 2 .) We have the equivariant Euler class

2 )[[t−1 , t]. (See Subsection Eu N0 (y 1 , y 2 ) ∈ R E1u , Ms (y 1 , L, α∗ , δ) ⊗ R(E 7.1.6 for our convention on equivariant Euler classes.) By the homomorphisms in Subsection 7.1.2 and the twist in Subsection 7.1.7, we have the element:

u · er1 (t−ω1 )/r2 ∈ R E u , Ms (y 1 , L, α∗ , δ) ⊗ R(E

u )[t] P E1u · e−t ⊕ E 2 1 2

u )[[t−1 , t]: We obtain the following element of R E1u , Ms (y 1 , L, α∗ , δ) ⊗ R(E 2

u · er1 (t−ω1 )/r2 · tk P E1u · e−t ⊕ E 2 Eu N0 (y 1 , y 2 )

u ), by Then, we obtain the following element of R E1u , Ms (y 1 , L, α∗ , δ) ⊗ R(E 2 −1 taking the residue with respect to t, i.e., the coefficient of t :

u · er1 (t−ω1 )/r2 · tk P E1u · e−t ⊕ E 2 (7.10) Ψ (y 1 , y 2 ) = Res t=0 Eu N0 (y 1 , y 2 ) Theorem 7.2.1 Assume that the 2-stability condition holds for (y, L, α∗ , δ). Then, we have the following equality: y , [L], α∗ , δ− ) = Φ(

y , [L], α∗ , δ+ ) − Φ(

(y 1 ,y 2 )∈S(y,α∗ ,δ)

M(y 1 ,

y 2 ,L,α∗ ,δ)

Ψ (y 1 , y 2 )

(7.11)

278

7 Invariants

Moreover, the contribution of (y 1 , y 2 ) ∈ S(y, α∗ , δ) vanishes if pg > 0 and rank(y 1 ) = 1,

denote the master space Proof We use the notation in Subsection 4.7.1. Let M connecting the moduli stacks y , [L], α∗ , δ+ ), Ms (

Ms (

y , [L], α∗ , δ− )

−→ M(m, y

, [L]) be the naturally defined as in Subsection 4.7.1. Let ϕ : M

, [L]) with the Gm morphism. Let T (1) denote the trivial line bundle on M(m, y

M

action of weight 1. We have the natural Gm -actions on E and ϕ∗ Orel (1). We M

, M

: consider the following elements of RGm E k

M ) · c1 ϕ∗ Orel (1) , Φt := P (E

t := Φt · c1 ϕ∗ T (1) . Φ

Recall Proposition 5.9.8. Then, we obtain the polynomial M

Φt in the variable t, as explained in Example 7.3. When we forget the Gm -action, we have c1 ϕ∗ T (1) = 0. Hence, we have M

Φt|t=0 = 0. On the other hand, we have the following equality −1 in Q[t , t], thanks to the localization of the virtual fundamental classes [55]:

t = Φ

M

i=1,2

i M

t Φ +

i ) Eu N(M

I∈S(y,α∗ ,δ)

Gm (I) M

t Φ

Gm (I)) Eu N(M

Gm (I)) denote the virtual normal bundles with the Gm i ) and N(M Here, N(M action given in Proposition 5.9.8. We have c1 ϕ∗ T (1) |M

i = t,

c1 ϕ∗ T (1) |M

Gm (I) = t.

Therefore, we obtain the following equality in Q[t, t−1 ]: i=1,2

Res

i t=0 M

Φt

i ) Eu N(M

+

I

Res

Gm (I) t=0 M

Φt

Gm (I)) Eu N(M

= 0. (7.12)

u

M

i . We have ι∗ E =E Let us look at the contributions from the components M i ∗ ∗ and ιi ϕ Orel (1) = Orel (1) with the trivial Gm -action. By Proposition 5.9.8, we have the following equality: ∞ j ω

i ) −1 = (−1)i · (t − ω)−1 = (−1)i · 1 · Eu N(M t j=0 t

7.2 Transition Formulas in Simpler Cases

279

Therefore, we obtain the following: i=1,2

Res

Φt

i )) Eu(N(M

i t=0 M

=

(−1)i ·

i=1,2

i M

Φ

y , [L], α∗ , δ− ) = −Φ(

y , [L], α∗ , δ+ ) + Φ(

(7.13)

Gm (I). We remark Let us calculate the contributions from the components M ∗ ∗ that ϕ Orel (1)|M

∗ and ϕ T (1)|M

∗ are naturally isomorphic as Gm -equivariant line

u in Proposition 4.7.4. Then, we bundles. We use the relation among EiM , E1u and E ∗ u ∗ u 2

, S [t]: obtain the following equality in R G E , G E 1

2

2u · er1 (t−ω1 )/r2 · tk F ∗ Φt = G ∗ P E1u · e−t ⊕ E

2 [[t−1 , t]: We also have the following equality in R G ∗ E1u ⊗ R G ∗ E

Gm (I)) = G ∗ Eu N0 (y 1 , y 2 ) F ∗ Eu N(M We have the equality of the virtual fundamental classes in Proposition 6.1.12. Thus, we obtain the following equality:

Φt Res =

Gm (I))

Gm (I) t=0 Eu N(M M

u · er1 (t−ω1 )/r2 · tk P E1u · e−t ⊕ E 2 (7.14) Res Eu N0 (y 1 , y 2 ) M(y 1 ,

y 2 ,L,α∗ ,δ) t=0

The desired equality (7.11) follows from (7.12), (7.13) and (7.14). The second claim of the theorem immediately follows from Proposition 6.2.2 and the first claim. Corollary 7.2.2 Assume pg > 0. Assume that the 2-stability condition and the 2vanishing condition hold for (y, L, α∗ , δ). Then, we have y , [L], α∗ , δ− ). Φ(

y , [L], α∗ , δ+ ) = Φ(

Proof It immediately follows from Theorem 7.2.1 and Proposition 6.1.3.

7.2.2 Twist with the Euler Class of the Relative Tangent Bundle Assume that the 1-vanishing condition holds for (y, L, α∗ , δ). Let Θrel denote the y , [L], α∗ , δ) −→ M(

y ). relative tangent bundle of the smooth morphism Ms (

We put

280

7 Invariants

Td(X) · y · ch(L−1 ).

NL (y) :=

(7.15)

X

For a non-critical δ, we have the following integral: Φ(

y , [L], α∗ , δ) :=

M(

y ,[L],α∗ ,δ)

Eu(Θrel )

u ) · P (E NL (y)

For a critical δ, we take parameters δ− < δ < δ+ such that δκ (κ = ±) are sufficiently close to δ. The transition formula for Φ(

y , [L], α∗ , δ) is rather simple, if the 1-vanishing condition and the 2-stability condition hold for (

y , L, α∗ , δ). Although we will prove a more general formula later, we give it as an explanation of the argument. It is sufficiently useful in the rank 2 case. If pg > 0, the problem is much easier. Proposition 7.2.3 Assume pg > 0. Assume that the 1-vanishing condition and the 2-stability condition hold for (y, L, α∗ , δ). Then, we have the equality: Φ(

y , [L], α∗ , δ− ) = Φ(

y , [L], α∗ , δ+ ) Proof By using the argument in the proof of Theorem 7.2.1, we can express

Gm (I). Under y , [L], α∗ , δ− ) as the sum of integrals over M Φ(

y , [L], α∗ , δ+ ) − Φ(

Gm (I)] = 0 By Proposition 6.1.12, the assumption of the proposition, we have [M Proposition 6.2.2 and Proposition 6.2.3. Thus we are done. Let us consider the case pg = 0. For any (y 1 , y 2 ) ∈ S(y, α∗ , δ), we define

2 , [L], α∗ , δ) := Ms (

M(

y1 , y y 1 , [L], α∗ , δ) × Ms (

y 2 , α∗ ).

2 , [L], α∗ , δ), for which it is conWe introduce some cohomology classes M(

y1 , y venient to use the terminology of Gm -equivariant cohomology. We use the notation in Subsection 7.2.1. We introduce the variable s := r1 · t. (It corresponds to taking the covering Gm −→ Gm of degree r1 .) We have the following virtual Gm 2 , [L], α∗ , δ) (see Subsection 5.9.1 for the equivariant vector bundle on M(

y1 , y notation):

u ) · es/r1 +s/r2 + N(E

u , E

u ) · e−s/r1 −s/r2

1u , E N(E 2 2 1

1u∗ , E

2u∗ ) · es/r1 +s/r2 + ND (E

2u∗ , E

1u∗ ) · e−s/r1 −s/r2 + ND (E

(7.16)

2 · es/r2 ) ∈ R(E

1 ) ⊗ R(E

2 )[[s−1 , s] denote the equivariant

1 · e−s/r1 , E Let Q(E Euler class of (7.16). Theorem 7.2.4 Assume that the 2-stability condition and the 1-vanishing condition hold for (y, L, α∗ , δ). In the case pg = 0, we have the following equality:

7.2 Transition Formulas in Simpler Cases

281

Φ(

y , [L], α∗ , δ+ ) − Φ(

y , [L], α∗ , δ− ) = (y 1 ,y 2 )∈S(y,α∗ ,δ)

NL (y1 ) NL (y)

M(

y 1 ,

y 2 ,[L],α∗ ,δ)

Ψ (y 1 , y 2 )

(7.17)

u u

, Ms (

are given as y 1 , [L], α∗ , δ) ⊗ R E The elements Ψ (y 1 , y 2 ) ∈ R E 1 2 follows:

u · e−s/r1 ⊕ E

u · es/r2 P E Eu(Θ1,rel ) 1 2 (7.18) Ψ (y 1 , y 2 ) = Res · u u −s/r s/r

s=0 1 2 NL (y1 ) Q E ·e , E ·e 1

2

2 , [L], α∗ , δ) induced by the relHere, Θ1,rel denotes the vector bundle on M(

y1 , y y 1 , [L], α∗ , δ) −→ M(

y 1 ). ative tangent bundle of the smooth morphism Ms (

Proof We use the notation in Subsection 4.7.1. By using the argument in the proof of Theorem 7.2.1, we can reduce the problem to the calculation of the contribu Gm (I) for I = (y 1 , y 2 ) ∈ S(y, α∗ , δ). Let ϕI denote the inclusion tions from M Gm

. Recall the relation of the virtual fundamental classes in the case M (I) −→ M pg = 0 in Proposition 6.2.2. Thus, we set Gm " #

(I)] = G∗ M(

2 , [L], α∗ , δ) . y1 , y [S] := F ∗ [M We have the following decomposition of the equivariant vector bundles on S:

2u ⊗O1,rel (−r1 /r2 ) ⊗ e(1+r1 /r2 )t F ∗ ϕ∗I Θrel = G∗ Θ1 rel ⊕ G∗ pX ∗ Hom L, E Therefore, we have the following equality in A∗Gm (S): F ∗ ϕ∗I Eu Θrel =

2u · e(1+r1 /r2 )·t−r1 ω1 /r2 G∗ Eu(Θ1 rel ) · G∗ Eu pX ∗ Hom L, E The second term in the right hand side also appears in Eu(N0 (y 1 , y 2 )), and hence they are cancelled out in the evaluation of [S]. Then, we obtain the following equality in Q[t−1 , t]:

F S

∗

M

· Eu(Θrel ) ϕ∗I P E =

Gm (I)) Eu N(M

u · e−t+ω1 ⊕ E

u · er1 (t−ω1 )/r2 · Eu(Θ1 rel ) P E 1 2 G (7.19) u

· e−t+ω1 , E

u · er1 (t−ω1 )/r2 Q E S 1 2

∗

u −t+ω 1

·e

u · er1 (t−ω1 )/r2 is the Euler class of the following virtual Here, Q E ,E 1 2

2 , [L], α∗ , δ) for A = 1 + r1 /r2 : Gm -equivariant vector bundle on M(

y1, y

282

7 Invariants

1u , E

2u ) · eA(t−ω1 ) + N(E

2u , E

1u ) · e−A(t−ω1 ) N(E

1u∗ , E

2u∗ ) · eA(t−ω1 ) + ND (E

2u∗ , E

1u∗ ) · e−A(t−ω1 ) + ND (E We remark that the integrand of the right hand side of (7.19) is of the form Aj · (t − ω1 )j , j

u u

, M(

. By a direct calculation, we can where Aj ∈ R E y 1 , [L], α∗ , δ) ⊗ R E 1 2 check the following: 1 (j = −1) Res (t − ω1 )j = 0 (j = −1) t=0 Hence, we have Rest=0 (t − ω1 )j = Rest=0 tj for any j. In particular, we have the following equality: Aj · (t − ω1 )j = Res Aj · tj Res t=0

j

t=0

j

Thus, we obtain the following equality:

u · e−t+ω1 ⊕ E

u · er1 (t−ω1 )/r2 · Eu(Θ1 rel ) P E 1 2 Res G u

· e−t+ω1 , E

u · er1 (t−ω1 )/r2 Q E S t=0 1 2

u · e−t ⊕ E

u · er1 t/r2 · Eu(Θ1 rel ) P E 1 2 ∗ = Res G u

· e−t , E

u · er1 t/r2 Q E S t=0

∗

1

2

Therefore, we obtain the following:

M ) · Eu(Θrel ) P (E Res =

Gm (I)

Gm (I) t=0 Eu(N M M

u · e−t ⊕ E

u · er1 t/r2 · Eu(Θ1,rel ) P E 1 2 Res r1 · u

· e−t , E

u · er1 t/r2 Q E M(

y 1 ,

y 2 ,[L],α∗ ,δ) t=0 1 2

Because r1 · t = s, we obtain the desired formula (7.18).

7.2.3 Oriented Reduced L-Bradlow Pairs Let L = (L1 , L2 ) be a pair of line bundles on X. Assume (i) the 1-vanishing condition holds for (y, L1 , α∗ ), (ii) the 2-vanishing condition holds for (y, L2 , α∗ ), in the sense of Subsection 6.4.1. We study a transition formula under the situation of Subsection 4.7.2.

7.2 Transition Formulas in Simpler Cases

283

Let δ = (δ1 , δ2 ) ∈ (P br )2 . Assume that both δi are sufficiently small as in Subsection 4.7.2. Recall that the 1-stability condition does not hold for (y, L, α∗ , δ), if and only if the following conditions are satisfied: • δ1 /r1 = δ2 /r2 holds for a pair of positive integers (r1 , r2 ) such that r1 + r2 = r. • There exists a (y 1 , y 2 ) ∈ T ype2 such that (i) y 1 + y 2 = y, (ii) rank y i = ri , (iii) Pyα1∗ = Pyα2∗ . Assume that the 1-stability condition does not hold for (y, L, α∗ , δ). Take δκ ∈ P br (κ = ±) such that (i) δ− < δ1 < δ+ , (ii) |δκ − δ1 | (κ = ±) are (1) sufficiently small. We set δ κ := (δκ , δ2 ) for κ = ±. Let Θrel denote the relative s y , [L], α∗ , δ κ ) −→ M(

y , [L2 ]). Let tangent bundle of the smooth morphism M (

(i) y , [Li ]) via the Orel (1) denote the pull back of the tautological line bundle on M(

(i) morphism Mss (

y , [L], α∗ , δ) −→ M(

y , [Li ]). We put ω (i) := c1 Orel (1) . Let NL1 (y) be given as in (7.15). For κ = ±, we define Φ(

y , [L], α∗ , δ κ ) :=

Ms (

y ,[L],α∗ ,δ κ )

Eu(Θrel )

u ) · ω (2) k · P (E NL1 (y) (1)

Let us argue a transition formula for the difference between Φ(

y , [L], α∗ , δ ± ). Proposition 7.2.5 In the case pg > 0, we have the equality: y , [L], α∗ , δ − ) Φ(

y , [L], α∗ , δ + ) = Φ(

−→ M(m, y

, [L]) denote the naturally defined morphism. Proof Let ϕ : M

, [L]) with the Gm -action of Let T (1) denote the trivial line bundle on M(m, y (i) weight 1. We put Ii := ϕ∗ Orel (−1). Let us consider the Gm -equivariant cohomology class: (1) Eu ϕ∗ Θrel u

· c1 (I2−1 )k , Φt := P (ϕ E ) · NL1 (y) ∗

t := Φt · c1 ϕ∗ T (1) Φ

By applying the argument in the proof of Theorem 7.2.1 to M

Φt , we can obtain a y , [L], α∗ , δ − ) as the sum of integrals description to express Φ(

y , [L], α∗ , δ + ) − Φ(

Gm (I). Under the assumption of the proposition, we have over the fixed point sets M Gm

[M (I)] = 0, By Proposition 6.1.13, Proposition 6.2.2 and Proposition 6.2.3. Thus we are done. To study the case pg = 0, we prepare some notation. We put S(y, α∗ , δ) := (y 1 , y 2 ) ∈ T ype2 Pyα1∗ = Pyα2∗ , δ1 /r1 = δ2 /r2 . For any (y 1 , y 2 ) ∈ S(y, α∗ , δ), we set

2 , [L], α∗ , δ) := Ms (

y 1 , [L1 ], α∗ , δ1 ) × Ms (

y 2 , [L2 ], α∗ , δ2 ). M(

y1 , y

284

7 Invariants

2 , [L], α∗ , δ), we use Gm To introduce the cohomology classes on M(

y1, y equivariant cohomology classes. Let t and s be the variables as in Subsections 7.2.1 and 7.2.2. We have the following virtual Gm -equivariant vector bundle on

2 , [L], α∗ , δ): M(

y1 , y

1u , E

2u ) · es/r1 +s/r2 + N(E

2u , E

1u ) · e−s/r1 −s/r2 N(E

u , E

u ) · es/r1 +s/r2 + ND (E

u , E

u ) · e−s/r1 −s/r2 + ND (E 1∗ 2∗ 2∗ 1∗

u · es/r1 , E

u · es/r2 ) ∈ R(E

1 ) ⊗ R(E

2 )[[s−1 , s] denote its equivariant Let Q(E 1 2 Euler class. Let O2,rel (1) denote the tautological line bundle on M(

y 2 , [L2 ]). The pull back is also denoted by the same symbol. We put ω2 := c1 O2,rel (1) , and we use the symbol ew·ω2 to denote O2,rel (w). We alsohave the following virtual

2 , [L], α∗ , δ) : y1, y Gm -equivariant vector bundle on M(

RpX ∗ Hom(L2 , E1u ) · e−s/r1 −s/r2 +ω2 Its equivariant Euler class is denoted by

u · e−s/r1 ) ∈ R(E

1 ) ⊗ A∗ Ms (

y 2 , [L2 ], α∗ , δ) [[s−1 , s]. R(L2 · e−ω2 +s/r2 , E 1 Proposition 7.2.6 In the case pg = 0, the following equality holds: y , [L], α∗ , δ − ) = Φ(

y , [L], α∗ , δ + ) − Φ(

NL1 (y1 ) Ψ (y 1 , y 2 ) NL1 (y) M( y1 , y2 ,[L],α∗ ,δ) (y 1 ,y 2 )∈S(y,α∗ ,δ)

The elements

1 , Ms (

2 , Ms (

y 1 , [L1 ], α∗ , δ1 ) ⊗ R E y 2 , [L2 ], α∗ , δ2 ) Ψ (y 1 , y 2 ) ∈ R E are given as follows: Eu(Θ1,rel ) × NL1 (y1 ) u −s/r 1

·e

u ·es/r2 · (ω2 − s/r2 )k P E ⊕E 1 2 Res (7.20) u

· e−s/r1 , E

u · es/r2 · R L2 ·e−ω2 +s/r2 , E

u ·e−s/r1 s=0 Q E

Ψ (y 1 , y 2 ) :=

1

2

1

2 , [L], α∗ , δ) induced by the relative y1 , y Here, Θ1,rel denotes the bundle on M(

1 ). y 1 , [L1 ], α∗ , δ) −→ M(m, y tangent bundle of the smooth morphism Ms (

Proof The proof is essentially the same as that of Theorem 7.2.4. Applying the argument in the proof of Theorem 7.2.1 to M

Φt in the proof of Proposition 7.2.5, y , [L], α∗ , δ − ): we obtain the following expression of Φ(

y , [L], α∗ , δ + ) − Φ(

7.2 Transition Formulas in Simpler Cases

285

Φ(

y , [L], α∗ , δ + ) − Φ(

y , [L], α∗ , δ − ) = I∈S(y,α∗ ,δ)

Res

Gm (I) t=0 M

Φt

Gm (I)) Eu N(M

We have the following Gm -equivariant isomorphism: F ∗ ϕ∗I I2−1 G∗ O2,rel (1) ⊗ O1,rel (r1 /r2 ) ⊗ e−r1 ·t/r2 . Therefore, we have the equality F ∗ c1 I2−1 = G∗ ω2 + r1 · (ω1 − t)/r2 . Hence, ∗

1 , G∗ E

2 , S [t]: we have the following equality in R G E 1 NL1 (y) u ω1 −t ∗

2u · e−r1 (ω1 −t)/r2 · G∗ ω2 + r1 (ω1 − t)/r2 k × G P E1 · e ⊕E

u · e−r1 (ω1 −t)/r2 × G∗ Eu(Θ1,rel ) · G∗ R L1 · e−t , E 2

F ∗ ϕ∗I (Φt ) =

u · e−r1 (ω1 −t)/r2 denotes the equivariant Euler class of the Here, R L1 · e−t , E 2 following Gm -vector bundle:

2u ) · e−r1 ω1 /r2 +(1+r2 /r1 )t pX∗ Hom(L1 , E

u , G∗ E

u , S)[[t−1 , t]: We also have the following equality in R(G∗ E 1 2 u ω1 −t u −r1 (ω1 −t)/r2

Gm (I)) = G∗ Q E

1 · e

2 · e ,E F ∗ Eu N(M

u · e−r1 (ω1 −t)/r2 × G∗ R L1 · e−t , E 2

1u · e−r1 (ω1 −t)/r2 × G∗ R L2 · e−ω2 −r1 (ω1 −t)/r2 , E

(7.21)

We put t − ω1 =: t. Then, we obtain the following: r1 Φt Res Ψ (y , y ) =

Gm (I)) NL1 (y) M( y1 , y2 ,[L],α∗ ,δ) 1 2

Gm (I) t=0 Eu N(M M (7.22) u −t

·e ⊕E

u · er1 t/r2 · ω2 − r1 t/r2 k · Eu(Θ(1) ) P E 1 2 1,rel Ψ (y 1 , y 2 ) := Res

u · e−t , E

u · er1 t/r2 · R L2 · e−ω2 +r1 t/r2 , E

u · e−t t=0 Q E

1

2

1

By an argument in the proof of Theorem 7.2.4, we can replace t with t in (7.22). We put s = r1 · t, and then we can show that the right hand side of (7.22) is the integral

2 , [L], α∗ , δ). y1 , y of Ψ (y 1 , y 2 ) over M(

Let δ be critical, and δ κ (κ = ±) be as above. We consider the following integrals, assuming that the 1-vanishing condition holds for (y, L2 , α∗ ):

286

7 Invariants

Φ1 (

y , [L], α∗ , δ ± ) :=

(1)

Ms (

y ,[L],α∗ ,δ ± )

(2)

Eu(Θrel ) Eu(Θrel )

u ) · · P (E NL1 (y) NL2 (y)

(7.23)

(2)

Here, NL2 (y) be given as in (7.15), and Θrel denotes the relative tangent bundle of y , [L], α∗ , δ κ ) −→ M(

y , [L1 ]). the smooth morphism Ms (

Lemma 7.2.7 If the 1-vanishing condition holds for (y, L2 , α∗ ), the following holds: y , [L], α∗ , δ + ) − Φ1 (

y , [L], α∗ , δ − ) = Φ1 (

NL (y1 ) · NL (y2 ) 1

(y 1 ,y 2 )∈S(y,α∗ ,δ)

2

NL (y1 ) · NL (y2 )

M(

y 1 ,

y 2 ,[L],α∗ ,δ)

Ψ1 (y 1 , y 2 )

The elements u u

, Ms (

, Ms (

y 1 , [L1 ], α∗ , δ1 ) ⊗ R E y 2 , [L2 ], α∗ , δ2 ) Ψ1 (y 1 , y 2 ) ∈ R E 1 2 are given as follows:

u · e−s/r1 ⊕ E

u · es/r2 P E Eu(Θ1,rel ) Eu(Θ2,rel ) 1 2 · · Res u u −s/r s/r

s=0 1 2 NL1 (y1 ) NL2 (y2 ) Q E1 · e , E2 · e

2 , [L], α∗ , δ) obtained from the y1 , y Here, Θ2,rel denotes the vector bundle on M(

y 2 , [L2 ], α∗ , δ2 ) −→ M(

y 2 ). relative tangent bundle of the smooth morphism Ms (

Proof We put t := t − ω1 . Then, we formally have the following decomposition: (2)

u · e−t F ∗ ϕ∗I Eu(Θrel ) = G∗ Eu(Θ2,rel ) · G∗ R L2 · e−ω2 +r1 ·t/r2 , E 1

u ·e−t also appears in the decomposition (7.21) The term G∗ R L2 ·e−ω2 +r1 ·t/r2 , E 1

Gm (I)) , which are cancelled out in the evaluation of S. Then, the of F ∗ Eu N(M claim can be easily deduced.

7.3 Invariants 7.3.1 Construction Let y ∈ T ype, and let α∗ be a system of weights. Let P ∈ R. If uthe 1-stability

). We would condition holds for (y, α∗ ), we obtain the number Ms ( y,α∗ ) P (E like to obtain such a number even in the case that a 1-stability condition does not hold. We take a line bundle L on X such that the 1-vanishing condition holds for (y, L, α∗ ) in the sense of Subsection 6.4.1. Let δ ∈ P br be non-critical and

7.3 Invariants

287

sufficiently small such that there are no critical value smaller than δ. Let Θrel denote y , [L], α∗ , δ) −→ M(

y ). the relative tangent bundle of the smooth morphism Ms (

Then, we obtain the following number:

u ) · Eu(Θrel ) P (E Φ(L) := NL (y) s M (

y ,[L],α∗ ,δ) Proposition 7.3.1 • In the case pg > 0, the number Φ(L) is independent of the choice of L. • In general, let L and L be line bundles on X. We assume that L−1 is ample. Then, there exists the limit limm→∞ Φ(L ⊗ Lm ), and it is independent of the choice of L and L. • Assume that the equality PE1 = PE2 holds for any E1 ∗ ⊕ E2 ∗ ∈ Mss (y, α∗ ). Then, Φ O(−m) is independent of the choice of any sufficiently large m. Proof Let L 1 , L1 and L2 be line bundles on X. Assume (i) L−1 1 is ample, (ii) the (m)

1-vanishing condition holds for (y, L2 , α∗ ). We put L1 := L1 ⊗ Lm 1 . If we take a (m) sufficiently large integer m, the 1-vanishing condition holds for (y, L1 , α∗ ). We (m) put L(m) := (L1 , L2 ). We consider the following number: (m)

g(L1 , L2 , δ1 , δ2 ) :=

Ms (

y ,[L(m) ],α∗ ,δ)

u ) · P (E

(1)

(2)

Eu(Θrel ) Eu(Θrel ) NL(m) (y) NL2 (y) 1

(1)

Here, Θrel denotes the relative tangent bundle of the naturally defined morphism (2) Ms (

y , [L(m) ], α∗ , δ) −→ M(

y , [L2 ]). We use the symbol Θrel in a similar meaning. We assume that both δi are sufficiently small. When δ1 is sufficiently smaller than δ2 , we have the following equality according to Proposition 6.4.2: (m)

g(L1 , L2 , δ1 , δ2 ) = Φ(L2 )

(7.24)

Similarly, when δ2 is sufficiently smaller than δ1 , we have the following equality: (m)

(m)

g(L1 , L2 , δ1 , δ2 ) = Φ(L1 )

(7.25)

(m)

We fix δ2 and move δ1 . Transition of the values g(L1 , L2 , δ1 , δ2 ) occurs if the following holds: • δ1 = δ2 · r1 /r2 holds for some pair of positive integers (r1 , r2 ). • There exists a (y 1 , y 2 ) ∈ T ype2 such that (i) y 1 + y 2 = y, (ii) rank y i = ri , (iii) Pyα1∗ = Pyα2∗ . We set

S(y, α∗ ) := (y 1 , y 2 ) Pyα1∗ = Pyα2∗ , y 1 + y 2 = y (m)

Therefore, we have the following expression of Φ(L1 ) − Φ(L2 ) from (7.24) and (7.25):

288

7 Invariants

NL(m) (y1 ) · NL2 (y2 )

(y 1 ,y 2 )∈S(y,α∗ )

NL(m) (y) · NL2 (y)

(m)

Φ(L1 ) − Φ(L2 ) =

1

· G(m) (y 1 , y 2 ) (7.26)

1

By Proposition 7.2.5, the contributions G(y 1 , y 2 ) are trivial in the case pg > 0. Hence, we obtain the first claim. Let us show the second claim. We use an induction on rank(y). By Proposition 7.2.6 and Lemma 7.2.7, G(m) (y 1 , y 2 ) is expressed as follows:

G(m) (y 1 , y 2 ) =

M(

y 1 ,

y 2 ,[L(m) ],α∗ ,δ )

Res s=0

u es/r2 )

1u e−s/r1 ⊕ E P (E 2

u e−s/r1 , E

u es/r2 ) Q(E 1

2

Eu(Θ1,rel ) Eu(Θ2,rel ) NL(m) (y1 ) NL2 (y2 ) 1

Here δ = (δ1 , δ2 ) is any element of (P br )2 such that δi are sufficiently small. We have rank(y 1 ) < rank(y). Recall (m)

2 , [L(m) ], α∗ , δ) = Ms (

y 1 , [L1 ], α∗ , δ1 ) × Ms (

y 2 , [L2 ], α∗ , δ2 ). M(

y1 , y By the hypothesis of the induction, we have the limit limm→∞ G(m) (y 1 , y 2 ). Moreover, it is independent of the choices of L1 and L 1 . We obviously have the limit: lim

NL(m) (y1 ) · NL2 (y2 ) 1

m→∞

NL(m) (y) · NL2 (y) 1

=

r1 · NL2 (y2 ) r · NL2 (y)

(m) Therefore, there exists the limit of the sequence Φ(L1 ) , and it is independent of the choice of L1 . Hence, the second claim is shown. Let us show the third claim. We use an induction on rank y. We put L 1 = OX and L1 = OX (−1). Then, we have NL(m) (y) = Hy (m) and NL(m) (y1 ) = 1

1

Hy1 (m). By the hypothesis of the induction, G(m) (y 1 , y 2 ) are independent of the choice of m. By the assumption Py1 = Py , we also have the following equality: r1 · NL2 (y2 ) Hy1 (m) · NL2 (y2 ) = Hy (m) · NL2 (y) r · NL2 (y) Therefore, we obtain the desired independence.

Definition 7.3.2 Let P be an element of R. We take a line bundle L such that L−1 is ample, and we take a sufficiently small δ ∈ P br . Then, we put

u ) := lim

u ) · Eu(Θrel ) (7.27) P (E P (E m→∞ Ms (

NLm (y) Mss (

y ,α∗ ) y ,[Lm ],α∗ ,δ) It is well defined by Proposition 7.3.1. Thus, we obtain a linear map Mss ( y,α∗ ) : R −→ Q.

7.3 Invariants

289

Remark 7.3.3 Although we have chosen a base point x0 of X to take a Poincar´e bundle, the linear map Mss ( y,α∗ ) is independent of the choice of x0 . To distinguish the dependence on x0 , we use the symbol Mss (

y , [L], α∗ , δ, x0 ). It is easy to construct the family π : M −→ X with the following property: • π −1 (x) Mss (

y , [L], α∗ , δ, x). • It is provided with a relative perfect obstruction theory Ob(M) −→ LM/X such that the restriction to π −1 (x) is the same as the obstruction theory of y , [L], α∗ , δ, x). Mss (

Then, the claim follows from Proposition 2.4.6. See also Subsection 7.3.5.

7.3.2 Easy Property We do not have to take a limit in some cases, contrast to Definition 7.3.2. Lemma 7.3.4 Assume pg > 0. We take a line bundle L such that the 1-vanishing condition holds for (y, L, α∗ ). Then, the following equality holds:

u ) =

u ) · Eu(Θrel ) P (E P (E NL (y) ss s M (

y ,α∗ ) M (

y ,[L],α∗ ,δ) Proof It follows from Proposition 7.3.1.

The following lemma is clear from the construction and Proposition 6.4.1. Lemma 7.3.5 If the 1-stability condition holds for (y, α∗ ), Definition 7.3.2 is compatible with the ordinary definition. Proposition 7.3.6 Assume that the equality PE1 = PE2 holds for any E1 ∗ ⊕E2 ∗ ∈ Mss (y, α∗ ). Then, we have the following equality for any sufficiently large m: u

u ) · Eu(Θrel ) P (E ) = P (E Hy (m) ss s M (

y ,α∗ ) M (

y ,[O(−m)],α∗ ,δ) Proof It immediately follows from the third claim of Proposition 7.3.1.

We say that a system of weights α∗ is not critical, if Mss (y, α∗ ) = Mss (y, α∗ ) for any α∗ such that |αi − αi | are sufficiently small. Corollary 7.3.7 Assume one of the following: • α∗ is not critical. • The parabolic part of y is trivial. Then, we have the following equality for any sufficiently large m:

u ) =

u ) · Eu(Θrel ) P (E P (E Hy (m) Mss (

y ,α∗ ) Ms (

y ,[O(−m)],α∗ ,δ)

290

7 Invariants

Proof We have only to check the condition in Proposition 7.3.6. If the parabolic part of y is trivial, the condition is trivial. Let us show the second claim. Assume that α∗ is not critical, and take E1 ∗ ⊕ E2 ∗ ∈ Mss (y, α∗ ). Let y i be the types of α α Ei ∗ . Then, we have Py1∗ = Py2∗ for any α∗ which are sufficiently close to α∗ . It implies Py1 = Py2 , i.e., PE1 = PE2 . Thus we are done.

ss (

7.3.3 Integrals over M y , α∗ , +) ss (

Let M y , α∗ , +) be as in Subsection 4.6.1. (Note that it depends on the choice rel denote the relative tangent bundle of a sufficiently large integer m.) Let Θ ss y , α∗ , +) −→ M(

y ). Recall the description of of the smooth morphism M (

ss (

M y , α∗ , +) as a flag bundle over Ms (

y , [O(−m)], α∗ , ) for any sufficiently small positive number . We also remark the equality of the virtual fundamental classes in Lemma 6.5.2. Then, we can easily obtain the following equality: rel ) Eu(Θ u

u ) · Eu(Θrel ) (7.28) = P (E ) · P (E H (m)! Hy (m) ss (

y y ,α∗ ,+) Ms (

y ,[O(−m)],α∗ ,) M

Here, Θrel denotes the relative tangent bundle of Ms (

y , [O(−m)], α∗ , ) −→

). M(m, y Lemma 7.3.8 Assume one of the following: • pg > 0 • The condition in Proposition 7.3.6 is satisfied. Then, we have the following equality: ss (

y ,α∗ ,+) M

u ) · P (E

rel ) Eu(Θ = Hy (m)!

u ) P (E

(7.29)

Mss (

y ,α∗ )

In particular, if one of the conditions in Corollary 7.3.7 is satisfied, the equality (7.29) holds.

7.3.4 Another Expression In the case pg > 0, we have another way to express the integral (7.27). Lemma 7.3.9 Assume thatthe 2-vanishing condition holds for (y, L, α∗ ). We also assume pg > 0 and d := χ y · ch(L)−1 − 1 ≥ 0. Let P ∈ R. Then, the following equality holds for any sufficiently small δ ∈ P br :

7.3 Invariants

291

Ms (

y ,[L],α∗ ,δ)

u ) · ωd = P (E

u ) P (E

(7.30)

Mss (

y ,α∗ )

Proof We use the argument in the proof of Proposition 7.3.1. We also use the notation in Subsection 7.2.3. We take a line bundle L1 on X such that the 1-vanishing condition holds for (y, L1 , α∗ ). The pair (L1 , L) is denoted by L. We set g(L1 , L, δ1 , δ) :=

(1)

u ) · Eu(Θrel ) · ω (2) d . P (E N (y) s L1 M (

y ,[L],α∗ ,δ)

When δ1 is sufficiently smaller than δ, we have the following: g(L1 , L, δ1 , δ) =

Ms (

y ,[L],α∗ ,δ)

u ) · ω (2) d P (E

When δ is sufficiently smaller than δ1 , we have the following equality, by Corollary 6.4.4:

u ) g(L1 , L, δ1 , δ) = P (E Mss (

y ,α∗ )

We move δ1 , and then the transitions are trivial according to Proposition 7.2.5. Thus, we obtain the desired equality (7.30). Recall that a 2-vanishing condition can be controlled numerically for later use. Lemma 7.3.10 Let y be an element of T ype◦ . Assume Py (t) > PK (t) for any sufficiently large t, where PK denotes the reduced Hilbert polynomial of the canonical line bundle of X. Then, the 2-vanishing condition holds for (y, O). Proof Take E ∈ Mss (y). If H 2 (X, E) = 0, we have a non-trivial morphism E −→ KX . It implies Py (t) ≤ PK (t), which contradicts the assumption.

7.3.5 Deformation Invariance Let S be a variety over C. Let X be a smooth projective surface over S provided with a relatively ample line bundle O(1). For simplicity, we assume that (i) S is smooth and affine over C, (ii) we are given a section S −→ X . (The assumptions are not essential for our deformation invariance.) Let D be a divisor of X smooth over S. For each closed point s ∈ S, let (Xs , Ds ) denote the fiber of (X , D) over s. For any closed points si (i = 1, 2), we have the natural identifications H i (Xs1 ) H i (Xs2 ) (i = 0, 4) and H i (Ds1 ) H i (Ds2 ) (i = 0, 2). Let s0 be a closed point of S. Let y be an element of T ype for (Xs0 , Ds0 ). For simplicity, we assume that the H 2 (Xs0 )-component c1 (y) of y is monodromy invariant. For example, it is satisfied if there exists a line bundle M on X such

292

7 Invariants

that c1 (M|Xs ) = c1 (y). Let α∗ be a system of weights. We use the symbol y , α∗ , s) to denote a moduli stack of parabolic sheaves on (Xs , Ds ) of type y Mss (

u (s) to denote the with weight α∗ . Note that Mss (

y , α∗ , s) may be empty. Let E ss y , α∗ , s) × (Xs , Ds ). From (Xs , Ds ) and universal quasi-parabolic sheaf on M (

u

u (s), we obtain the ring R E

(s) . E Let s be any closed point of S. We fix a path γ connecting s0 and s in the classical topology. Then, we obtain the identifications H ∗ (Xs ) H ∗ (Xs0 ) and H ∗ (Ds ) u u

(s) R E

(s0 ) . For P (E

u (s0 )) ∈ H ∗ (Ds0 ). It induces the identification R E u u u

(s0 ) , let P (E

(s)) ∈ R E

(s) denote the corresponding element. R E Proposition 7.3.11 Under the above setting, we have the following equality: u u

(s0 ) =

(s) P E P E (7.31) Mss (

y ,α∗ ,s0 )

Mss (

y ,α∗ ,s)

Proof We give only an outline. Let m be any large integer. Let Os (−m) := O(−m)|Xs . For δ ∈ P br and s ∈ S, let Ms denote moduli stacks of δ-stable parabolic reduced Os (−m)-Bradlow pairs with weight α∗ of type y on (Xs , Ds ).

u denote the universal sheaf on Ms × Xs . Note that Ms may be empty. Let E s Let us consider the family version. By the section S −→ X , we have the Picard variety PicX /S and the Poincar´e bundle on PicX /S ×S X . Hence, we have the naturally defined notion of orientation for a U -coherent sheaf on U ×S X . We also have the naturally defined notion of reduced O(−m)-sections of a U -coherent sheaf on U ×S X . Let M be the moduli stack of oriented parabolic reduced O(−m)-Bradlow

∗u denote the universal family on M×S X . Let π : M −→ S pairs with type y. Let E be the natural projection. Then, π −1 (s) is isomorphic to Ms for each s ∈ S, and

u is E

u . the pull back of E ∗ s∗ If δ is sufficiently small, the 1-stability condition holds for (y, α∗ , Os (−m), δ) and for any s ∈ S. Then, M is proper over S. Let ιs denote the inclusion {s} −→ S. By making the construction of Chapter 5 in family, we obtain a perfect obstruction theory Ob(M/S) −→ LM/S such that the induced morphism ι∗s Ob(M/S) −→ LMs is equivalent to the obstruction theory of Ms given in Chapter 5. By Proposition 2.4.4, we obtain a virtual fundamental class [M] of M. Thanks to Proposition 2.4.6, we obtain ι∗s [M] = [Ms ]. Let M denote a moduli stack of oriented parabolic sheaves on (X , D) of type y with weight α∗ . Let Θrel denote the relative tangent bundle of M −→ M. The restrictions of Θrel to Ms are denoted by the same symbol. By making the procedure of Subsection 7.1.3 in family, we obtain u

∗ , Eu(Θrel ) · [M] ∈ H∗ PicX /S ×S X m1 ×S Dm2 Hy (m)−1 ΛP E u

(s), Eu(Θrel )·[Ms ] . The specialization to PicXs ×Xsm1 ×Dsm2 is Hy (m)−1 ΛP E ∗ By making the evaluation in family along the path γ, we obtain (7.31).

7.4 Rank Two Case

293

7.4 Rank Two Case 7.4.1 Reduction to the Sum of Integrals over the Products of Hilbert Schemes For simplicity, we assume H 1 (X, O) = 0 in this subsection. Let y be an element of T ype◦ with rank(y) = 2. In the following, the H 2 (X)-component of y is denoted by a, and the H 4 (X)-component of y is denoted by b. The second Chern class corresponding to y is denoted by n. We have the relation b = a2 /2 − n. We assume for y and the canonical line bundle (i) Py > PK for reduced Hilbert polynomials K, (ii) χ(y) − 1 ≥ 0, where χ(y) := X ch(y) · Td(X). For any P ∈ R, we would

u ) as the sum of integrals over the products of Hilbert like to express Mss ( y) P (E schemes. Let N S(X) denote the subgroup of H 2 (X, Z) generated by the 1-cycles on X. For any element a1 ∈ N S(X), we put a2 := a−a1 . Let eai denote the holomorphic line bundle whose first Chern class is ai . Since we have assumed H 1 (X, O) = 0, it is uniquely determined up to isomorphisms. For any non-negative integer l, let X [l] denote a Hilbert scheme of 0-schemes of X with length l. Let Iiu denote the universal ideal sheaves on X [ni ] × X. Let Zi denote the universal 0-scheme of X [ni ] × X. Let Ξi denote pX ∗ OZi ⊗ eai . We use the same symbol to denote the pull back of them via appropriate morphisms. Let Gm denote a one dimensional torus. We introduce some cohomology classes on X [n1 ] × X [n2 ] , for which we use Gm -equivariant cohomology classes. The Gm equivariant bivariant theory of a point is the limit lim A∗ (PN ) with respect to in←− clusions PN ⊂ PN +1 . Let s correspond to the first Chern class of the tautological line bundle. It is convenient to use the symbol ew·s to denote the trivial line bundle with a Gm -action of weight w. Then, we have the following virtual Gm -equivariant vector bundle on X [n1 ] × X [n2 ] : −RpX ∗ RHom I1u ea1 −s , I2u ea2 +s − RpX ∗ RHom I2u ea2 +s , I1u ea1 −s Its equivariant Euler class is denoted by Q I1u · ea1 −s , I2u · ea2 +s ∈ R(I1u · ea1 ) ⊗ R(I2u · ea2 )[[s−1 , s]. We have P I1u · ea1 −s , I2u · ea2 +s ∈ R(I1u · ea1 ) ⊗ R(I2u · ea2 )[s], which is induced as explained in Subsection 7.1.2. We also have the equivariant Euler class Eu(Ξ2 ·e2s ) and the ordinary Euler class Eu(Ξ1 ). They can be regarded as elements of R(I2u ·ea2 )[[s−1 , s] and R(I1u ·ea1 ) by the Grothendieck-Riemann-Roch theorem. Thus, we obtain the following element of R(I1u · ea1 ) ⊗ R(I2u · ea2 ): P I1u · ea1 −s ⊕ I2u · ea2 +s Eu(Ξ1 ) · Eu(Ξ2 · e2s ) Ψ (a1 , n1 , a2 , n2 ) := Res s=0 (2s)n1 +n2 −pg Q I1u · ea1 −s , I2u · ea2 +s

294

7 Invariants

In the case c1 (O(1)) · a1 < c1 (O(1)) · a2 , we set A(a1 , y) := Ψ (a1 , n1 , a2 , n2 ) n1 +n2 =n−a1 ·a2

X [n1 ] ×X [n2 ]

In the case c1 (O(1)) · a1 = c1 (O(1)) · a2 , we set A(a1 , y) := n1 +n2 =n−a1 ·a2 n1 >n2

X [n1 ] ×X [n2 ]

Ψ (a1 , n1 , a2 , n2 )

We put as follows: SW(X, y) :=

" # M(ea1 , O) = 0, a1 ∈ N S(X) 2a1 · c1 (OX (1)) ≤ a · c1 (OX (1))

Recall that the expected dimension of M(ea1 , O) is 0 if [M(ea1 , O)] = 0 (Proposition 6.3.1). Therefore, we can regard [M(ea1 , O)] as the number, which is denoted by SW(a1 ). Theorem 7.4.1 Assume pg > 0 and H 1 (X, O) = 0. Assume Py > PK and χ(y) − 1 ≥ 0. We have the following equality:

u ) + P (E SW(a1 ) · 21−χ(y) · A(a1 , y) = 0. Mss (

y)

a1 ∈SW(X,y)

Proof Let δ ∈ P br be non-critical. Let ω denote the first Chern class of the relative y , [O], δ). We set d := χ(y) − 1. Then, we put tautological line bundle on Ms (

u ) · ωd Φ(δ) := P (E Ms (

y ,[O],δ)

For a critical δ ∈ P br , we put S(y, δ) := (y1 , y2 ) ∈ T ype2 y1 + y2 = y, Hy1 + δ = Hy2 We take parameters δ− < δ < δ+ which are sufficiently close to δ. Let E1u de u denote a universal sheaf over note a universal sheaf over M(y1 , O) × X, and E 1 y1 , [O]), M(

y1 , [O]) × X. Recall we have the isomorphism M(y1 , O) M(

although it does not preserve their virtual fundamental classes. We identify the

u · eω1 , moduli spaces via the isomorphism. Then, we have the relation E1u = E 1 where ω1 denotes the first Chern class of the relative tautological line bundle of y1 , [O]). We set M(y1 , O) M(

u ω −s

·e 1 ⊕E

u · es−ω1 · sd P E 1 2 Ψ (y1 , y2 ) := Res u

· eω1 −s , E

u · es−ω1 · Eu RpX ∗ E

u · e2s−ω1 s=0 Q E 1

2

2

7.4 Rank Two Case

295

According to Theorem 7.2.1, we have the following equality: Ψ (y1 , y2 ) Φ(δ+ ) − Φ(δ− ) = (y1 ,y2 )∈S(y,δ)

M(y1 ,O)×M(

y2 )

We set S(y) := (y1 , y2 ) ∈ T ype2 y1 + y2 = y, Py1 < Py2 . Recall Mss (

y , [O], δ) = ∅ for any sufficiently large δ (Proposition 3.4.5). Because Φ(δ) =

u ) for any sufficiently small δ (Lemma 7.3.9), we obtain the following P (E Mss (

y) equality: u

P (E ) + Ψ (y1 , y2 ) (7.32) 0= Mss (

y)

(y1 ,y2 )∈S(y)

M(y1 ,O)×M(

y2 )

If [M(y1 , O)] = 0, the expected dimension of M(ea1 , O) is 0 (Proposition 6.3.1). Hence, we can omit ω1 when we consider the evaluation of Ψ (y1 , y2 ) on y2 ). By using Proposition 6.3.8 and the virtual fundamental class M(y1 , O) × M(

u = I u · eai , we obtain the following: E i i M(y1 ,O)×M(

y2 )

Ψ (y1 , y2 ) = SW(a1 )×

Res

X [n1 ] ×X [n2 ] s=0

P I1u · ea1 −s ⊕ I2u · ea2 +s · sd · Eu(Ξ1 ) Q I1u · ea1 −s , I2 · ea2 +s · Eu RpX ∗ (I2u · ea2 +2s )

For simplicity of description, we set χ(ai ) := χ(eai ). By a formal calculation, we obtain the following: Eu(Ξ2 · e2s ) 1 = u (2s)χ(a2 ) Eu RpX ∗ (I2 · ea2 +2s ) Because [M(ea1 , O)] = 0, we have χ(a1 ) = 1 + pg (Proposition 6.3.1). Hence, we have the following: χ(a2 ) = χ(y2 ) + n2 = χ(y) − χ(y1 ) + n2 = χ(y) − 1 − pg + n1 + n2 . Then, the desired equality can be obtained by a direct calculation.

Remark 7.4.2 Theorem 7.4.1 is motivated by the Witten conjecture. (See [130] and [21]. It is also mentioned in the preface of this monograph.) The author learned the idea in [21] and [42]. Let L be an sufficiently ample line bundle on X. We have the equality u

u ) P (E ) = P (E Mss (

y)

Mss (y·ch(L))

We also have SW X, y · ch(L) = a1 ∈ N S(X) [M(ea1 , O)] = 0 . See Subsection 6.3.2 for the relation between the numbers SW(a1 ) and the Seiberg-Witten

296

7 Invariants

invariants. Hence, Theorem 7.4.1 gives some relations between Donaldson invariants and Seiberg-Witten invariants. However, the author does not know whether we can deduce the formula in the Witten conjecture from Theorem 7.4.1.

7.4.2 Dependence on Polarizations Let y ∈ T ype◦ with rank(y) = 2. We use a similar convention as in Subsection 7.4.1. To distinguish the dependence on a polarization H, we use the symbol y ) to denote a moduli stack of torsion-free sheaves of type y which are MH (

semistable with respect to H. Let C denote the ample cone in N S(X) ⊗ R, which is the cone generated by the first Chern class of ample line bundles. For a given line bundle L, the first Chern class c1 (L) ∈ N S(X) is also denoted by L for the simplicity of description. Let ξ be an element of N S(X) such that ξ +a is divisible by 2 in N S(X). We c · ξ = 0 , which is called also assume a2 − 4n ≤ ξ 2 < 0. We put W ξ := c ∈ C the wall determined by ξ. Connected components of C − W ξ are called chambers. y ) depends only on the chambers to which For general H, it is well known that MH (

H belongs. Let P ∈ R. We set

u ) y ) := P (E (7.33) ΦH (

MH (

y)

(In general, (7.33) should be understood in the sense of Definition 7.3.2. But note Lemma 7.3.5, Proposition 7.3.6 and Corollary 7.3.7.) We would like to study how y ) varies when H crosses a wall W ξ . We put as follows: ΦH (

y + y1 = y, S(y, ξ) = (y0 , y1 ) ∈ T ype2 0 a0 − a1 = m · ξ (m > 0) in N S(X)

i denote y0 , y 1 ) := M(

y0 ) × M(

y1 ). Let E For each (y0 , y1 ) ∈ S(y, ξ), we put M(

the sheaf over M(

y0 , y 1 ) × X which is the pull back of the universal sheaves over M(

yi ) × X via the appropriate projections. We have the following virtual Gm equivariant vector bundle on M(

y0 , y 1 ):

0 · e−s , E

1 · es − RpX ∗ RHom E

1 · es , E

0 · e−s −RpX ∗ RHom E The equivariant Euler class is denoted by

0 ) ⊗ R(E

1 · es ∈ R(E

1 )[[s−1 , s].

0 · e−s , E Q E

0 ) ⊗ R(E

1 · es ∈ R(E

1 )[s] by the homomorphisms

0 · e−s ⊕ E We also have P E in Subsection 7.1.2.

7.4 Rank Two Case

297

Theorem 7.4.3 Let C+ and C− be chambers which are divided by the wall W ξ . Let H+ and H− be ample line bundles contained in C+ and C− , respectively. We assume H− · ξ < 0 < H+ · ξ. y ) = ΦH− (

y ). Namely, the invariants do not • In the case pg > 0, we have ΦH+ (

depend on the choices of generic polarizations. • In the case pg = 0, we have the following equality:

u · e−s ⊕ E

u · es P E 0 1 y ) − ΦH− (

y) = Res ΦH+ (

0 · e−s , E

1 · es Q E M(

y0 ,

y1 ) s=0 (y0 ,y1 )∈S(y,ξ)

(7.34) We give two arguments to prove Theorem 7.4.3. Both of them are based on the following observation. Lemma 7.4.4 ([26], [88]) Let H be an ample line bundle contained in W ξ . We can take Hκ ∈ Cκ (κ = ±) satisfying the following: • There exists a very ample curve C such that H+ = H ⊗ O(C) and H− = H ⊗ O(−C). • The following holds for a torsion-free sheaf E of type y: – E is H+ -semistable if and only if E(C) is H-semistable. – E is H− -semistable if and only if E(−C) is H-semistable.

Ellingsrud and G¨ottsche proved the formula (7.34) for Donaldson invariants under the assumption that the wall W ξ is good. (See [26]. See also the work of Friedman and Qin [36].) They used the parabolic structure E(−C) ⊂ E(C) y , α) denote a moduli stack of with weight α for torsion-free sheaves E. Let Mss (

torsion-free sheaves with the parabolic structure as above, which are semistable with y , 1) = MH+ (

y) respect to the polarization H. By Lemma 7.4.4, we have Mss (

y , 0) = MH− (

y ). We say that α is critical, if Mss (

y , α) = Mss (

y , α ) and Mss (

for any sufficiently close α = α. We can easily show that there are only finitely many critical values, by using some boundedness result. By investigating transition of invariants at critical parabolic weights, they obtained the formula (7.34). Using their framework and our transition formula (7.17), we will show the formula (7.34) without assuming that the wall is good, in Subsection 7.4.3. We will give another argument for the proof of Theorem 7.4.3 in Subsection 7.4.4. It seems a little more suitable when we discuss a similar problem in the higher rank case. We give some preliminary for the argument. In the rest of this section, we use the polarization H contained in the wall W ξ , but not contained in any other walls. Semistability conditions and μ-semistability conditions are considered with respect to H. Let S denote the family of μ-semistable torsion-free sheaves of type y. Let S denote the family of torsion-free sheaves E of rank one with the following property: • μ(E ) = μ(y). (See Subsection 3.2.4 for μ(y).) • There is a member E of S, such that E is a saturated subsheaf of E.

298

7 Invariants

The families S and S are bounded. We take a sufficiently large integer m such that the family S satisfies the condition Om . In the rest of this section, the degree of δ ∈ P br is 0.

7.4.3 Proof of Theorem 7.4.3 (I) For a torsion-free sheaf E, we denote by F (α) the parabolic structure E(−C) ⊂ E(C), α . The following lemma is clear. Lemma 7.4.5 If (E, F (α) , φ) is a δ-semistable parabolic L-Bradlow pair, E is μsemistable. (Recall δ is assumed to be a polynomial of degree 0.) ss Let M y , [L], α, δ denote a moduli stack of the oriented parabolic reduced L-Bradlow pairs of type y with weight α, where the parabolic structure is given as above. Let α be any real number, and let E be any member of S. We take E ∈ S such that E is a saturated subsheaf of E. We put E

:= E/E . Then, we put fα (E ) := PEα − PEα . The number is determined by ch(E ), and hence it is independent of the choice of E. When we fix α, the function fα : S −→ R has finitely many values, by the boundedness of S. It is clear that α is critical if and only if there exists a member E ∈ S such that fα (E ) = 0. Let α0 be critical, and let be any sufficiently small positive number. We can take a small positive number η > 0 such that the following holds for any α with |α − α0 | < η: • fα0 (E ) > 0 ⇐⇒ fα (E ) > • fα0 (E ) < 0 ⇐⇒ fα (E ) < − • fα0 (E ) = 0 ⇐⇒ |fα (E)| < Lemma 7.4.6 We have Mss y , [L], α0 , Mss y , [L], α , for any α such y , [L], α0 , ) = Ms (

y , [L], α0 , ). that |α0 − α | < η. Moreover, we have Mss (

Proof Let (E, ρ, F (α ) , [φ]) be an oriented parabolic reduced L-Bradlow pair, such that E is μ-semistable. Let E ⊂ E be a member of S. We put E

:= E/E . We have PEα0 + < PEα0 if and only if PEα + < PEα . We have PEα0 < PEα0 + if and only if PEα < PEα + . Then, the first claim of the lemma is clear. The second claim is also easy to show. y ) and ΦH− (

y), we would like to consider the invariants obTo compare ΦH+ (

tained from the moduli stacks Ms y , [O(−m)], α, δ . We remark that the divisor of the parabolic structure is not reduced in this case, contrast to that we assumed the smoothness of the divisor in Chapter 5. But, we do not have to think the contribution of the parabolic structure to the obstruction theory, because the filtration is

7.4 Rank Two Case

299

canonically determined by the sheaf. In fact, Ms (

y , [L], α, δ) is an open substack of M(m, y , [L]) for a sufficiently large m, in this case. Thus, we obtain the perfect obstruction theory Ob(m, y , [L]) and the virtual fundamental class. Therefore, we can freely apply our previous results. Let Θrel denote the relative tangent bundle of the naturally defined morphism y , [O(−m)], α, δ) −→ M(m, y ). For any δ ∈ P br which is not critical with Mss (

respect to (y, O(−m), α), we obtain the following number:

u ) · Eu(Θrel ) . P (E Φ(α, δ) := Hy (m) Ms (

y ,[O(−m)],α,δ) The following lemma is the special case of Lemma 7.3.5. δ) is independent of the Lemma 7.4.7 Assume that α is not critical. Then, Φ(α, choice of m, if δ is sufficiently small. The number is denoted by Φ(α). We have Φ(1) = ΦH+ (y) and Φ(0) = ΦH− (y). Therefore, we have only to see the transition of the invariants at critical weights. For a critical α, we put as follows: S(y, ξ, α) := (y0 , y1 ) ∈ S(y, ξ) Pyα0 = Pyα1 Proposition 7.4.8 We take real numbers α+ > α > α− such that |ακ − α| < η. • In the case pg > 0, we have Φ(α+ ) − Φ(α− ) = 0. • In the case pg = 0, we have the following formula: Φ(α+ ) − Φ(α− ) =

(y0 ,y1 )∈S(y,ξ,α)

Res

u · e−s ⊕ E

u · es P E 0

1

u · e−s , E

u · es ) Q(E 0 1 (7.35)

M(

y0 ,

y1 ) s=0

Proof We have Ms (

y , [O(−m)], α+ , ) Ms (

y , [O(−m)], α− , ) by Lemma 7.4.6. Therefore, we obtain the following equality: (7.36) Φ(α+ ) − Φ(α− ) = Φ(α− , ) − Φ(α− ) − Φ(α+ , ) − Φ(α+ ) To investigate the terms appearing in the right hand side of (7.36), let us look at transitions of Φ(ακ , δ) for κ = ±, when δ is varied from 0 to . Note that a transition of moduli stacks occurs at δ, if Pyα0κ +δ = Pyα1κ holds for some (y0 , y1 ) ∈ S(y, ξ, α). Let us consider the case pg > 0. Since the transitions of the invariants are trivial according to Theorem 7.2.4, we have Φ(ακ , ) − Φ(ακ ) = 0 for κ = ±. Hence, we obtain Φ(α+ ) − Φ(α− ) = 0. Let us consider the case pg = 0. For any (y0, y1 ) ∈ S(y, ξ, α), we put M(

y0 , y 1 , [O(−m)]) := M y 0 , [O(−m)] × M y 1 . We set

u · e−s ⊕ E

u · es P E 0 1 . Ψ0 (y0 , y1 ) := Res

u · e−s , E

u · es ) s=0 Q(E 0

1

300

7 Invariants

Then, we obtain the following equality: Φ(α− , ) − Φ(α− ) = Eu(Θ0,rel ) Hy0 (m) · Ψ0 (y0 , y1 ) · Hy (m) Hy0 (m) M(

y0 ,

y1 ,[O(−m)]) (y0 ,y1 )∈S(y,ξ,α) Hy0 (m) · Ψ0 (y0 , y1 ) (7.37) = Hy (m) M(

y0 ,

y1 ) (y0 ,y1 )∈S(y,ξ,α)

Similarly, we have the following equality in the case pg = 0: Φ(α+ , ) − Φ(α+ ) =

(y0 ,y1 )∈S(y,ξ,α)

u · e−s ⊕ E

u · es P E 1 0 Res

u · e−s , E

u · es ) Q(E M(

y1 ,

y0 ) s=0 1 0 Hy (m) · − 1 Ψ (y0 , y1 ) (7.38) Hy (m) M(

y0 ,

y1 )

Hy1 (m) · Hy (m) =

(y0 ,y1 )∈S(y,ξ,α)

Therefore, we obtain the following: Φ(α+ ) − Φ(α− ) = (y0 ,y1 )∈S(y,ξ,α)

'

Hy0 (m) Hy1 (m) + Hy (m) Hy (m)

( ·

M(

y0 ,

y1 )

=

(y0 ,y1 )∈S(y,ξ,α)

Thus, the proof of Proposition 7.4.8 is finished.

Ψ (y0 , y1 )

M(

y0 ,

y1 )

Ψ (y0 , y1 )

The first claim of Theorem 7.4.3 obviously follows from the first claim of Proposition 7.4.8. Since the intersection pairing C · ξ is sufficiently large, we have / S(y, ξ) = S(y, ξ, α) 0<α<1

Then, the formula (7.34) immediately follows from (7.35). Thus, the first proof of Theorem 7.4.3 is finished.

7.4.4 Proof of Theorem 7.4.3 (II) We put y(C) := y · ch(O(C)). We use the symbol y(−C) in a similar meaning. We regard them as elements of T ype◦ . We put O(−m, C) := O(−m) ⊗ O(C) and O(−m, −C) := O(−m) ⊗ O(−C). From a reduced O(−m)-Bradlow pair

7.4 Rank Two Case

301

(E, [φ]) such that φ = 0, we naturally obtain the reduced O(−m, C)-Bradlow pair (E(−C), [φ pair (E(C), [φC ]), and the reduced O(−m, −C)-Bradlow −C ]). s y

(C), [O(−m, C)], δ Let Θrel denote the relative tangent bundle of M −→ Ms (m, y (C)). We have a similar bundle on Ms y (−C), [O(−m, −C)], δ . For any non-critical δ, we set Eu(Θrel ) C (δ) := (7.39) Φ· Φ Hy (m) Ms (

y (C),[O(−m,C)],δ) −C (δ) := Φ

Ms (

y (−C),[O(−m,−C)],δ)

Φ·

Eu(Θrel ) Hy (m)

(7.40)

C (δ) = ΦH (y) and Φ (δ) = ΦH (y). When δ is sufficiently small, we have Φ + −

C

∗ Let T ype(S) denote the set ch(E ) ∈ H (X) E ∈ S . For any y0 ∈ S, we put y1 := y − y0 , and then y1 is also an element of T ype(S). We put yi (C) := yi · ch O(C) . We use the symbol yi (−C) in a similar meaning. We remark that Py0 (C) − Py1 (C) and Py0 (−C) − Py1 (−C) are polynomials of degree 0. Let ai , bi and ni denote the first Chern class, the second Chern character and the second Chern class corresponding to yi . We have a0 · H = a1 · H. Since H is a generic element of W ξ , we have a0 − a1 = A · ξ for some A ∈ Q in H 2 (X). Since C · ξ is assumed to be sufficiently large, we have Py0 (C) − Py1 (C) = 0 unless y0 = y1 . We also have Py0 (−C) − Py1 (−C) = 0 unless y0 = y1 . We set S(y, C) := (y0 , y1 ) y0 + y1 = y, yi ∈ T ype(S), Py0 (C) < Py1 (C) , S(y, −C) := (y0 , y1 ) y0 + y1 = y, yi ∈ T ype(S), Py0 (−C) < Py1 (−C) . We take a positive constant δ0 satisfying the following inequalities: δ0 > max |Py0 (C) − Py1 (C) | (y0 , y1 ) ∈ S(y, C) δ0 > max |Py (−C) − Py (−C) | (y0 , y1 ) ∈ S(y, −C) 0

1

Lemma 7.4.9 (E(C), φC ) is δ0 -semistable, if and only if the following conditions hold: • E is μ-semistable. • For any subobject (E , φ ) ⊂ (E, φ) such that φ = 0, we have μ(E ) < μ(E). Moreover, (E(C), φC ) is δ0 -stable, if it is δ0 -semistable. Proof Assume that (E(C), φC ) is δ0 -semistable. Since δ0 is a polynomial of degree 0, it is easy to see that E(C) is μ-semistable. Hence, the first condition holds. Let (E , φ ) ⊂ (E, φ) be a subobject such that φ = 0. We put E

= E/E . Assume μ(E ) = μ(E). Then, E is a member of S, and hence we have |PE (C) (t) − PE (C) (t)| < δ0 , by our choice of δ0 . Therefore, we ob tain PEδ0 (C) (t) > PE (C) (t), which contradicts the δ0 -semistability of E(C), φC . Hence, the second condition holds.

302

7 Invariants

Let us assume that the two conditions are satisfied. Let (E , φ ) ⊂ (E, φ) be a δ0 subobject such that φ = 0. Because μ(E ) < μ(E), the inequality P(E (C),φ ) (t) < C

δ0 P(E(C),φ holds for any sufficiently large t. Take a subobject E ⊂ (E, φ). We have C) the inequality μ(E ) ≤ μ(E). When the strict inequality holds, we obviously have δ0 for any sufficiently large t. Assume μ(E ) = μ(E). Then, PE (C) (t) < P(E(C),φ C)

E is a member of S. We put E

= E/E . Then, we have PE (C) (t) < PEδ0 (C) by our choice of δ0 . Thus, we obtain the δ0 -semistability of (E(C), φC ). δ0 δ0 From the above argument, we also obtain that P(E (C),φ ) = P(E(C),φ ) cannot C C hold. Therefore, we obtain the second claim.

Lemma 7.4.10 (E(C), φC ) is δ0 -semistable, if and only if (E(−C), φ−C ) is δ0 semistable. Moreover, the 1-stability condition holds for (y(C), O(−m, C), δ0 ) and (y(−C), O(−m, −C), δ0 ). Proof By the argument used in the proof of Lemma 7.4.9, we can show that (E(−C), φ−C ) is δ0 -semistable if and only if the two conditions in Lemma 7.4.9 hold. Then, the first claim immediately follows. The second claim can be shown similarly. −C (δ0 ). Therefore, we C (δ0 ) = Φ By Lemma 7.4.10, we obtain the equality Φ obtain the following equality: −C (δ0 ) − ΦH (

C (δ0 ) − ΦH (

y ) − ΦH− (

y) = Φ y) − Φ y) (7.41) ΦH+ (

− + To investigate the first term in the right hand side of (7.41), let us look at transi−C (δ) when δ is varied from 0 to δ0 . Note that a transition occurs when tions of Φ Py0 (−C) +δ = Py1 (−C) holds for some (y0 , y1 ) ∈ S(y, −C). In the case pg > 0, the transitions of the invariants are trivial according to Theorem 7.2.4. Hence, we obtain −C (δ0 ) − ΦH (

y ) = 0. In the case pg = 0, we obtain the following equality, as in Φ − the equality (7.37): −C (δ0 ) − ΦH (

y) = Φ − (y0 ,y1 )∈S(y,−C)

Hy0 (m) · Hy (m)

Res

M(

y0 ,

y1 ) s=0

u · es )

u · e−s ⊕ E P (E 0 1 u u −s

Q(E · e , E · es ) 0

1

C (δ0 ) − ΦH (

Similarly, we obtain Φ y ) = 0 in the case pg > 0, and we have the + following equality in the case pg = 0: C (δ0 ) − ΦH (

Φ y) = + (y0 ,y1 )∈S(y,C)

Hy0 (m) · Hy (m)

Res

M(

y0 ,

y1 ) s=0

u · es )

0u · e−s ⊕ E P (E 1

u · e−s , E

u · es ) Q(E 0

1

Now, we have already obtained ΦH+ (

y ) − ΦH− (

y ) = 0 in the case pg > 0. Namely the first claim of Theorem 7.4.3 is proved. To show the claim in the case pg = 0, we compare the sets S(y, C) and S(y, −C). We set

7.5 Higher Rank Case (pg > 0)

303

y ∈ T ype(S), y0 + y1 = y, . S1 := (y0 , y1 ) i r0 = r1 = 1, a0 = a1 , b0 < b1 We also put S (y, ξ) := (y0 , y1 ) (y1 , y0 ) ∈ S(y, ξ) . Then, it is easy to observe S(y, −C) = S(y, ξ) S1 and S(y, C) = S (y, ξ) S1 . We remark the equality (7.38). Therefore, we obtain the following equalities: y ) − ΦH− (

y) ΦH+ (

=

(y0 ,y1 )∈S(y,ξ)

+

(y0 ,y1 )∈S1

Hy0 (m) · Hy (m)

+

(y0 ,y1 )∈S(y,ξ)

−

(y0 ,y1 )∈S1

Res

M(

y0 ,

y1 ) s=0

Res

M(

y0 ,

y1 ) s=0

=

Res

Res

1

u · es )

u · e−s ⊕ E P (E 0 1

u · e−s , E

u · es ) Q(E 0

M(

y0 ,

y1 ) s=0

(y0 ,y1 )∈S(y,ξ)

Thus we are done.

0

M(

y0 ,

y1 ) s=0

0

Res

M(

y0 ,

y1 ) s=0

1

u · es )

u · e−s ⊕ E P (E 0 1

u · e−s , E

u · es ) Q(E

1

u · es )

0u · e−s ⊕ E P (E 1

u · e−s , E

u · es ) Q(E

u · es )

u · e−s ⊕ E P (E 0 1 u u −s

Q(E · e , E · es ) 0

Hy1 (m) · Hy (m)

Hy0 (m) · Hy (m)

Hy0 (m) · Hy (m)

1

u · es )

0u · e−s ⊕ E P (E 1

u · e−s , E

u · es ) Q(E 0

1

7.5 Higher Rank Case (pg > 0) 7.5.1 Transition Formula in the Case pg > 0 Let y ∈ T ype, and let α∗ be a system of weights. Let L be a line bundle on X. Let δ ∈ P br be non-critical with respect to (y, L, α∗ ). We denote by ω the first Chern y , [L], α∗ , δ). Let P ∈ R. When the 1-stability condition class of Orel (1) on Ms (

holds for (y, L, α∗ , δ), we put

u ) · ωk Φ(

y , [L], α∗ , δ) := P (E Ms (

y ,[L],α∗ ,δ)

We would like to obtain a transition formula, when the 2-stability condition does not necessarily hold for (y, L, α∗ , δ). In the case pg > 0, the problem is simple. Actually, we will obtain the same formula as that in Theorem 7.2.1.

304

7 Invariants

We use the notation in Subsection 7.2.1. We put S1 (y, α∗ , δ) := (y 1 , y 2 ) ∈ S(y, α∗ , δ) rank y 1 = 1 For (y 1 , y 2 ) ∈ S1 (y, α∗ , δ), we have M(y 1 , L) = Ms (y 1 , L, α∗ , δ), because stability conditions are trivial in the rank one case. Theorem 7.5.1 Assume pg = dim H 2 (X, OX ) > 0. The following equality holds: y , [L], α∗ , δ− ) = Φ(

y , [L], α∗ , δ+ ) − Φ(

(y 1 ,y 2 )∈S1 (y,α∗ ,δ)

M(y 1 ,

y 2 ,L,α∗ ,δ)

Ψ (y 1 , y 2 )

(7.42)

u ) are given as in (7.10): The elements Ψ (y 1 , y 2 ) ∈ R E1u , M(y 1 , L) ⊗ R(E 2

u · e(t−ω1 )/(r−1) · tk P E1u · e−t ⊕ E 2 Ψ (y 1 , y 2 ) = Res t=0 Eu N0 (y 1 , y 2 ) Here, we put ω1 := c1 Or(E1u ) . κ ) denote M ss Proof Take a sufficiently large m, and let M(δ y , [L], α∗ , δκ ). (See m (

rel denote the relative tangent bundle of the smooth morSubsection 3.3.1.) Let Θ κ) ⊂

, [L]) to M(m, y

, [L]). We have the open embedding M(δ phism of M(m, y

, [L]). The restriction of Θrel is denoted by the same symbol. M(m, y

be the master space connecting M(δ + ) and M(δ − ) constructed in SubLet M

, [L]) denote the naturally defined morphism. section 4.5.1. Let ϕ : M −→ M(m, y

, [L]) with the Gm -action of Let T (1) denote the trivial line bundle on M(m, y M

, M

: weight 1. We consider the following elements of RGm E t := P ϕ∗ E

u · c1 ϕ∗ Orel (1) k · Eu(Θrel ) , Φ Hy (m)!

t · c1 ϕ∗ T (1) . Φt := Φ

By using Proposition 5.9.1, we obtain the polynomial M

Φt ∈ Q[t]. When we ∗ forget the Gm -action, we have c1 ϕ T (1) = 0. Hence, we have M

Φt|t=0 = 0. On the other hand, we have the following equality in Q[t−1 , t], by the localization of the virtual fundamental classes [55]: Φt Φt Φt = +

Gm (I) Eu N(M Gm (I)) M i=1,2 Mi Eu N(Mi ) I∈Dec(m,y,α∗ ,δ) M Here, Dec(m, y, α∗ , δ) denotes the set of the decomposition types (Definition

i ) and N(M

Gm (I)) denote the virtual normal bundles with the 4.4.2), and N(M Gm -action given in Proposition 5.9.2 and Proposition 5.9.3. We remark

7.5 Higher Rank Case (pg > 0)

305

c1 ϕ∗ T (1) |M

= t, i

c1 ϕ∗ T (1) |M

Gm (I) = t

Therefore, we obtain the following equality: t t Φ Φ Res Res + = 0.

i )

Gm (I))

i t=0

Gm (I) t=0 Eu N(M Eu N(M M M i=1,2

I

(7.43) As in the proof of Theorem 7.2.1, the first term of the left hand side of (7.43) can be rewritten as follows: + =− + Φ Φ Φ Φ (7.44) −

1 M

2 M

+) M(δ

−) M(δ

κ ) is the full flag bundle over Mss y

, [L], α∗ , δκ , and we have the Recall M(δ equality of the virtual fundamental classes as in Lemma 6.5.1. Hence, we obtain the following equality: t Φ Res y , [L], α∗ , δ− ) y , [L], α∗ , δ+ ) + Φ(

= −Φ(

i )

t=0 Eu N(M i=1,2 Mi

Gm (I) can be calculated by the arguments in the The contributions from M proof of Theorem 7.2.1 and Theorem 7.2.4. For any decomposition type I = (y 1 , y 2 , I1 , I2 ) ∈ Dec(m, y, α∗ , δ), we set ss y 1 , L, α∗ , δ, k(I) × M ss (

M(I) := M y 2 , α∗ , +) 1,rel denote the vector bundle over M(I) induced by the relative tangent bunLet Θ ss 2,rel de dle of the smooth map M (y 1 , L, α∗ , δ, k(I)) −→ M(m, y 1 , L). Let Θ note the vector bundle over M(I) induced by the relative tangent bundle of the ss

2 ). Let N0 be as in Subsection 5.9.1. smooth map M (

y 2 , α∗ , +) −→ M(m, y Then, we obtain the following decomposition of the vector bundles: 1,rel ⊕ Θ 2,rel ⊕ N0 . rel = Θ ϕ∗I Θ ∗ We remark that ϕ∗ Orel (1)|M

∗ and ϕ T (1)|M

∗ are naturally isomorphic as Gm

u in Corollary 4.6.2. equivariant line bundles. We use the relation of EiM , E1u and E 2

∗ u

∗ u Then, we obtain the following equality in R(G E , G E , S)[t]: 1

t = F ∗Φ

2

Hy1 (m)!Hy2 (m)!

2u · er1 (t−ω1 )/r2 · tk · G ∗ P E1u · e−t ⊕ E Hy (m)! 1,rel ) Eu(Θ 2,rel ) Eu(Θ · Eu(N0 ) × Hy1 (m)! Hy2 (m)!

306

7 Invariants

We also have the following equality of the equivariant Euler classes in A∗ (S)[[t−1 , t]:

Gm (I)) = G ∗ Eu N0 (y 1 , y 2 ) · Eu(N0 ) F ∗ Eu N(M Recall that we have the equality of the virtual fundamental classes in Proposition

Gm (I) is as follows: 6.1.11. Therefore, the contribution from M Hy1 (m)! · Hy2 (m)! × Hy (m)!

u · er1 (t−ω1 )/r2 · tk Eu(Θ 2,rel ) 1,rel ) Eu(Θ P E1u · e−t ⊕ E 2 Res t=0 Hy1 (m)! Hy2 (m)! Eu N0 (y 1 , y 2 ) M(I) (7.45) We remark that the virtual fundamental class of M(I) is 0, and hence (7.45) vanishes, if the conditions rank(y 1 ) > 1 and pg > 0 are satisfied (Proposition 6.2.2). In the case rank(y 1 ) = 1, the (δ, )-semistability condition is trivial. We also remark ss (y 1 , L, α∗ , δ, k(I)) ⊗ that the integrand of (7.45) is the element of R E1u , M u ss (

2 , M y 2 , α∗ , +) , and hence we have only to consider the component-wise R E integration. By using Lemma 7.3.8, we can rewrite (7.45) as follows: Hy1 (m)! · Hy2 (m)! × Hy (m)!

u · e(t−ω1 )/(r−1) · tk P E1u · e−t ⊕ E 2 Res Eu N0 (y 1 , y 2 ) M(y 1 ,

y 2 ,L,α∗ ,δ) t=0

The number of the decompositions (I1 , I2 ) of {1, . . . , Hy (m)} satisfying |Ii | = Hyi (m) is Hy (m)! · Hy1 (m)!−1 · Hy2 (m)!−1 . Therefore, the second term in the left hand side of (7.43) is equal to the left hand side of (7.42). Thus, we obtain the desired formula.

7.5.2 Reduction to the Sum of Integrals over the Products of Hilbert Schemes We assume pg > 0 and dim H 1 (X, O) = 0. Let y be an element of T ype◦r . Assume Py (t) > PK (t) for any sufficiently large t, where PK denotes the reduced Hilbert polynomial of the canonical line bundle of X. We also assume χ(y) − 1 ≥ (r − 2) · (1 + pg ), where χ(y) := X T d(X) · y. We give a straightforward generalization of Theorem 7.4.1. For a given element yi ∈ T ype, we use the symbols ri , ai , bi , and ni to denote the rank, the first Chern class, the second Chern character and the second Chern class of yi , in the following argument. For the simplicity of description, we set χ(ai ) := χ(eai ).

7.5 Higher Rank Case (pg > 0)

307

Let S(y) denote the set of tuples (y1 , y2 , . . . , yr ) ∈ T ype◦1 r satisfying the following conditions: yi = y, Hyi < (r − i)−1 Hyj , χ(ai ) = 1 + pg (i < r) j>i

.r Let (y1 , . . . , yr ) ∈ S(y). We put X [n] := i=1 X [ni ] . The universal ideal sheaves over X [ni ] ×X are denoted by Iiu . Let Zi denote the universal scheme of X [ni ] ×X with length ni . We also use the same symbol to denote the pull back of Iiu and Zi via the projection X [n] ×X −→ X [ni ] ×X. Let eai denote the holomorphic line bundle corresponding to ai . Since we have assumed H 1 (X, OX ) = 0 in this subsection, such a holomorphic line bundle is uniquely determined up to isomorphisms. Let G denote an (r − 1)-dimensional torus (Gm )r−1 . We introduce some cohomology classes on X [n] , for which we use G-equivariant cohomology classes. The G-equivariant bivariant theory of a point is identified with Q[t1 , . . . , tr−1 ] as in Subsection 7.1.4, and it is included in R(t1 , . . . , tr−1 ). (See Subsection 7.1.6 for the ring R(t1 , . . . , tr−1 ).) We set Ti :=

tj − ti , (i = 1, . . . , r − 1), r−j j
Tr :=

tj r−j j
We use the symbol ew·ti to denote a trivial line bundle with a G-action induced by the action of the i-th Gm with weight w ∈ Q. We obtain the G-equivariant sheaf Iiu · eai +Ti . We use the symbols Q(Iiu · eai +Ti , Iju · eaj +Tj ) to denote the G-equivariant Euler classes of the following virtual G-equivariant vector bundles on X [n] : − RpX ∗ RHom Iiu · eai +Ti , Iju · eaj +Tj − RpX ∗ RHom Iju · eaj +Tj , Iiu · eai +Ti We regard them as elements of

1r i=1

R Iiu · eai ⊗Q R(t1 , . . . , tr−1 ). We set

Q I1u · ea1 +T1 , I2u · ea2 +T2 , . . . Iru · ear +Tr := Q Iiu · eai +Ti , Iju · eaj +Tj i<j

We also have the induced element P

r i=1

r Iiu · eai +Ti ∈ R(Iiu · eai )[t1 , . . . , tr−1 ] i=1

by the in Subsection 7.1.2. Let Ξi denote bundle homomorphisms 1r the vector pX ∗ OZi ⊗ eai . Then, we obtain the following element of i=1 R Iiu · eai :

308

7 Invariants

Ψ (y1 , . . . , yr ) =

r−1

Eu(Ξi )×

i=1

⎛ ⎞ .r−1 j≥i χ(yj )−1 r u ai +Ti Tj −Ti P I e t Eu(Ξj e )⎠ i=1 i i=1 i Res · · · Res ⎝ χ(aj ) tr−1 t1 (T − T ) Q I1u ea1 +T1 , . . . , Iru ear +Tr j i i<j (7.46) Here, Restj denote the operations to take the coefficient of t−1 k . Theorem 7.5.2 Assume pg > 0 and H 1 (X, O) = 0. We also assume χ(y) − 1 ≥ (r − 2) · (1 + pg ) and Py > PK . We have the following formula:

u ) = (−1)r−1 P (E

M(

y)

r−1

SW(ai ) ·

Ψ (y1 , . . . , yr ) X [n]

(y1 ,...,yr )∈S(y) i=1

Proof We set S1 (y) := (y1 , y2 ) ∈ T ype1 × T yper−1 y1 + y2 = y, Py1 < Py2 By using the transition formula (7.42) and the argument in the proof of Theorem 7.4.1, we obtain the following equality:

u ) = − P (E SW(a1 ) Ψ (y1 , y2 ) (7.47) M(

y)

X [n1 ] ×M(

y2 )

(y1 ,y2 )∈S1 (y)

Ψ (y1 , y2 ) := Res t=0

u et/(r−1) · tχ(y)−1 P I1u ea1 −t ⊕ E 2

u et/(r−1) ) · R OX e−t , E

u et/(r−1) Q(I u e−t , E 1

2

2

u ·et/(r−1) denotes the equivariant Euler class of the following Here, R OX ·e−t , E 2 virtual equivariant vector bundle:

2u · et/(r−1) ) . pX ∗ Hom(OX · e−t , E We remark rank(y2 ) = r − 1, χ(y2 ) − 1 ≥ 0, Py2 > Py > PK and χ(y2 ) − 1 ≥ (r − 3)(1 + pg ). We also remark that Ψ (y1 , y2 ) of (7.47) is the element of

2 , and hence we have only to consider the component-wise R I1 · ea1 ⊗ R E integration. Thus, we may use (7.47) inductively. We put T i := j
y)

u ) + (−1)r P (E

r−1

(y1 ,...,yr )∈S(y) i=1

The elements Ψ1 (y1 , . . . , yr ) ∈

1r i=1

SW(ai ) ·

Ψ1 (y1 , . . . , yr ) = 0 X [n]

R(Iiu · eai ) are given as follows:

7.5 Higher Rank Case (pg > 0)

309

Ψ1 (y1 , . . . , yr ) = ⎛ ⎞ u a +T .r−1 j≥i χ(yj )−1 .r−1 P Eu(Ξ ) Ii e i i · i=1 ti i i=1 Res · · · Res ⎝ . ⎠ u ai −ti , I u eaj +Tj −T i · R O e−ti , I u eaj +Tj −T i tr−1 t1 Q I e X i j j i<j We have Q Iiu · eai −ti , Iju · eaj +Tj −T i = Q Iiu · eai +Ti , Iju · eaj +Tj . We can also show the following equality by a formal calculation: (Tj − Ti )χ(aj ) R OX e−ti , Iju eTj −T i = Eu Ξj eTj −Ti Hence, we obtain Ψ1 (y1 , . . . , yr ) = Ψ (y1 , . . . , yr ). Thus we are done.

7.5.3 Independence from Polarizations in the Case pg > 0 Let y ∈ T ype◦r . We use the symbol MH (

y ) to denote a moduli stack of torsion-free sheaves of type y, which are semistable with respect to a polarization H. For any P ∈ R, we define

u ). y ) := P (E ΦH (

MH (

y)

Theorem 7.5.3 Assume pg > 0. Then, the invariant ΦH (

y ) is independent of the choice of a generic polarization H. Proof Let C denote the ample cone in N S(X) ⊗ R. For a given line bundle L, the first Chern class c1 (L) is denoted by L. For any ξ ∈ N S(X), let W ξ denote {c ∈ C | c · ξ = 0}, which is called the wall corresponding to ξ. According to [88], thereis the locally finite subset U of {ξ ∈ N S(X) | ξ 2 < 0} such that if y ) depends only on the chamber to which H H ∈ C − ξ∈U W ξ , the moduli MH (

belongs. Here, chamber means a connected component of C − ξ∈U W ξ . Let C+ and C− be chambers which are divided by a wall W ξ . Let H+ and H− be ample line bundles contained in C+ and C− respectively. We assume ξ · H+ > 0 > y ) = ΦH− (

y ). We take an ample line bundle H ξ · H− . We have only to show ΦH+ (

which is generic in W ξ . As in Lemma 7.4.4, we may assume the following: • H+ = H ⊗O(C) and H− = H ⊗O(−C) for some sufficiently ample divisor C. • A torsion free sheaf E of type y is H+ -semistable, if and only if E(C) is Hsemistable. • E is H− -semistable, if and only if E(−C) is H-semistable. We use the notation and the argument in Subsection 7.4.4. Let S be the family of torsion-free sheaves of type y, which are μ-semistable. Let S be the family of torsion-free sheaves E with the following properties:

310

7 Invariants

• μ(E ) = μ(y) and rank(E ) < r. • There exists a member E of S such that E is a saturated subsheaf of E. We take a large integer m such that the condition Om holds for the family S. As in Subsection 7.4.4, we consider the integrals over M y

(C), [O(−m, C)], δ and M y (−C), [O(−m, −C)], δ given by (7.39) and (7.40). When δ is sufficiently −C (δ) = ΦH (

C (δ) = ΦH (

y ) and Φ y ). small, we have Φ + − Let T ype(S) denote the set of the types of members of S. For each y0 ∈ T ype(S), we have y1 := y − y0 ∈ T ype(S). We remark that Py0 (C) − Py1 (C) and Py0 (−C) − Py1 (−C) are polynomials of degree 0. Let ri , ai , bi and ni denote the rank, the first Chern class, the second Chern character and the second Chern class corresponding to yi . We have μ(y0 ) = μ(y1 ). Since H is assumed to be generic in W ξ , we have a0 /r0 − a1 /r1 = A · ξ for some A ∈ Q in H 2 (X). Since the intersection number C · ξ is assumed to be sufficiently large, we have Py0 (C) − Py1 (C) = 0 unless y0 /r0 = y1 /r1 . We also have Py0 (−C) − Py1 (−C) = 0 unless y0 /r0 = y1 /r1 . We set S(y, C) := (y0 , y1 ) y0 + y1 = y, yi ∈ T ype(S), Py0 (C) < Py1 (C) S(y, −C) := (y0 , y1 ) y0 + y1 = y, yi ∈ T ype(S), Py0 (−C) < Py1 (−C) . We take a positive constant δ0 satisfying the following inequalities: δ0 > max |Py0 (C) − Py1 (C) | (y0 , y1 ) ∈ S(y, C) δ0 > max |Py0 (−C) − Py1 (−C) | (y0 , y1 ) ∈ S(y, −C) . The following lemma can be shown by using the argument in the proof of Lemma 7.4.9 and Lemma 7.4.10. Lemma 7.5.4 (E(C), φC ) is δ0 -semistable, if and only if (E(−C), φ−C ) is δ0 semistable. Moreover, the δ0 -semistability implies the δ0 -stability in both cases. By using the lemma, we obtain the following: −C (δ0 ) − ΦH (

C (δ0 ) − ΦH (

y ) − ΦH− (

y) = Φ y) − Φ y) ΦH+ (

− + C (δ) when the parameter −C (δ) and Φ Let us look at transitions of the invariants Φ δ is varied from 0 to δ0 . We use Theorem 7.5.1. Since the condition Om holds for S, it is easy to see that the contributions from the decomposition types (y0 , y1 ) are trivial even in the case rank(y0 ) = 1 (Proposition 6.2.3). Hence, we obtain −C (δ0 ) − ΦH (

C (δ0 ) − ΦH (

Φ y ) = 0 and Φ y ) = 0. Thus, the proof of Theorem − + 7.5.3 is finished.

7.6 Transition Formula (pg = 0)

311

7.6 Transition Formula (pg = 0) 7.6.1 Statement Let us argue a transition formula in the case pg = 0, when a 2-stability condition does not necessarily hold. In this section, we consider only the case that the 1vanishing condition holds for (y, L, α∗ , δ). Let P ∈ R. For a non critical parameter δ, we set

u ) · Eu(Θrel ) P (E Φ(δ) := NL (y) Ms (

y ,[L],α∗ ,δ) Let δ be critical. We take δ− < δ < δ+ such that δκ (κ = ±) are sufficiently close to δ. We would like to express Φ(δ+ ) − Φ(δ− ) as the sum of integrals over the products of moduli stacks of objects with lower ranks. We impose the following condition to (y, L, α∗ , δ): Condition 7.6.1 For any (E1∗ , φ) ⊕ E2 ∗ ⊕ E3 ∗ ∈ Mss (y, L, α∗ , δ), the equality PE2 = PE3 holds for the reduced Hilbert polynomials of E2 and E3 . For each positive integer k, we set Sk (y, δ) := Y = (y 1 , . . . , y k ) ∈ T ypek Pyαi∗ = Pyα∗ ,δ (i = 1, . . . , k) . For each element Y = (y 1 , . . . , y k ) ∈ Sk (y, δ), we put |Y | := define k rank(y i ) W (Y ) := 1≤j≤i rank(y j ) i=1

k i=1

y i . We also

We set S k (y, δ) := (y 0 , Y ) ∈ T ype × Sk (y, δ) y 0 + |Y | = y , and S(y, δ) :=

S k (y, δ).

k

For any (y 0 , Y ) ∈ S k (y, δ), we put k

0 , [L], α∗ , δ− × M(

y 0 , Y , [L]) := Ms y Mss (

y i , α∗ ). i=1

u denote the sheaf over M(

y 0 , Y , [L]) × X which is obtained as the pull back Let E 0 y 0 , [L], α∗ , δ− ) × X via the natural projection. We of the universal sheaf over Ms (

u in similar meanings. use the symbols E i When (y 0 , Y ) ∈ S k (y, δ) is given, we define some cohomology classes on M(

y , Y , [L]). For that purpose, it is convenient to use Gkm -equivariant cohomology classes, where Gkm is a k-dimensional torus. As explained in Subsection 7.1.4, the

312

7 Invariants

Gkm -equivariant bivariant theory of a point is identified with Q[t1 , . . . , tk ], which is included in R(tk , . . . , t1 ). We set

T0 := −

1≤j≤k

Ti := −

tj 0≤h<j rank(y h )

ti tj + . rank(y i ) 0≤h<j rank(y h )

1≤j≤i−1

We use the symbol ew·ti to denote a trivial line bundle with a Gkm -action induced by the action of the i-th Gm with weight w ∈ Q. We have the following Gkm -equivariant vector bundles on M(

y 0 , Y , [L]): u Ti u Tj

e ,E

u eTj − RpX ∗ RHom E

e ,E

u eTi − RpX ∗ RHom E i j j i u u Ti u Tj Tj u Ti

−RpD ∗ RHom 2 E −RpD ∗ RHom 2 E i|D ∗ e , Ej|D ∗ e j|D ∗ e , Ei|D ∗ e (See Subsection 2.1.6 for the symbol RHom 2 .) Their equivariant Euler classes are denoted by k u Ti u Tj

·e

u ) ⊗ R(tk , tk−1 , . . . , t2 , t1 ).

i · e , E ∈ R(E Q E j i i=0

Then, we set u T0 u T1

· e ,E

· e ,...,E

u · eTi :=

u · eTi , E

u · eTj Q E Q E 0 1 k i j

(7.48)

i<j

Let Θ0 rel denote the vector bundle over M(

y 0 , Y , [L]) obtained from the relative

, [L]). Then, y 0 , [L], α∗ , δ− ) −→ M(m, y tangent bundle of the smooth map Ms (

1k0 s

i :

y 0 , [L], α∗ , δ− ) ⊗ i=1 R E we have the following element of R E0 , M (

Ψ (y 0 , Y ) := Res · · · Res t1 =0

tk =0

k

u Ti P Eu(Θ0,rel ) i=0 Ei · e · u u T T

· e 0, . . . , E

·e k NL (y0 ) Q E 0 k

Theorem 7.6.2 Assume that the 1-vanishing condition and Condition 7.6.1 hold for (y, L, α∗ , δ). Then, we have the following transition formula: NL (y0 ) · W (Y ) · Φ(δ+ ) − Φ(δ− ) = Ψ (y 0 , Y ) (7.49) NL (y)

,[L]) M(

y 0 ,Y (y 0 ,Y )∈S(y,δ)

A proof will be given in the next subsections 7.6.2–7.6.4.

7.6 Transition Formula (pg = 0)

313

7.6.2 Step 1 ) := M ss Take a sufficiently large m. We put M(δ, m y, [L], α∗ , (δ, ) for a positive integer . We also set 0) := M ss M(δ, m (y, [L], α∗ , δ+ ),

− ) := M ss M(δ m (y, [L], α∗ , δ− )

rel denote the relative tangent bundle of the smooth morphism of M(m,

, [L]) Let Θ y ) and M(δ − ) are substacks of M(m,

, [L]). The stacks M(δ,

, [L]). to M(m, y y rel to M(δ, ) and M(δ − ). We use the same symbol to denote the restriction of Θ We set := P (E

u ) · Eu(Θrel ) · Eu(Θrel ) (7.50) Φ NL (y) Hy (m)! (κ = ±) Φ(δ, ) := Φ Φ(δκ ) := Φ M(δ, )

κ) M(δ

− ) = Φ(δ− ). 0) = Φ(δ+ ) and Φ(δ Then, we have Φ(δ, Recall that Dec(m, y, α∗ , δ) denotes the set of decomposition types (Definition 4.4.2). We set S( ) := I = (y 1 , y 2 , I1 , I2 ) ∈ Dec(m, y, α∗ , δ) ⊂ I1 . For any decomposition type I = (y 1 , y 2 , I1 , I2 ) ∈ Dec(m, y, α∗ , δ), we set

ss y

1 , [L], α∗ , δ, k(I) × Mss (

M( I) := M y 2 , α∗ )

1,rel denote the vector bundle over M( I) induced by the relative tangent bunLet Θ ss

1 , [L]). Let Θ1,rel dle of the smooth map M y 1 , [L], α∗ , δ, k(I) −→ M(m, y

denote the vector bundle over M( I) obtained from the relative tangent bundle of

1 ).

1 , [L]) −→ M(m, y the smooth morphism M(m, y Proposition 7.6.3 We have the following equality: NL (y1 ) Hy (m)! · Hy (m)! 1 2 Ψ (I). NL (y) Hy (m)!

I) M(

) − Φ(δ− ) = Φ(δ,

(7.51)

I∈S( )

ss (

2 ) are given as fol 1 , M y 1 , [L], α∗ , δ, k(I)) ⊗ R(E The elements Ψ (I) ∈ R E lows:

u · e−s/r1 ⊕ E

u · es/r2 1,rel ) P E Eu(Θ1,rel ) Eu(Θ 1 2 · (7.52) Ψ (I) = Res

u · e−s/r1 , E

u · es/r2 ) s=0 NL (y1 ) Hy1 (m)! Q(E 1

2

Proof Since the argument is essentially the same as those employed in the proof

denote the of Theorem 7.5.1 and Theorem 7.2.4, we give only an indication. Let M

314

7 Invariants

) and M(δ − ) constructed in Subsection 4.5.1. By master space connecting M(δ,

and the argument in the proof of Theorem 7.5.1, we obtain the following using M expression:

) − Φ(δ− ) = Φ(δ,

I∈S( )

Gm (I) M

t Φ

Gm (I)) Eu N(M

Gm (I) can be calculated by the arguments in the proof The contributions from M of Theorem 7.5.1 and Theorem 7.2.4. We remark that we can use Lemma 7.3.8, thanks to Condition 7.6.1. Then, we arrive at the formulas (7.51) and (7.52) For our later argument, we reword Proposition 7.6.3. Let I be a finite subset of Z>0 such that |I| = Hy (m). We naturally regard I as the totally ordered set. Let ss y

, [L], α∗ , (δ, i0 ), I be the moduli stack of the i0 be any element of I. Let M objects (E∗ , [φ], ρ, F) as follows: • (E∗ , [φ], ρ) is a δ-semistable oriented reduced L-Bradlow pairs. • F is a full flag of H 0 (X, E(m)) indexed by I. • The tuple (E∗ , [φ], ρ, F) is (δ, i0 )-semistable in the following sense. We take a partial Jordan-H¨older filtration of (E∗ , [φ]): (1)

E∗

(2)

E∗

(j−1)

· · · E∗

(j)

(E∗ , [φ])

(j+1)

(E∗

(k−1)

, [φ]) · · · (E∗

(k)

, [φ]) (E∗ , [φ])

Then, Fi0 ∩ H 0 X, E (j−1) (m) = {0} and Fi0 ⊂ H 0 X, E (k−1) (m) . We have the bijection ϕ : I −→ {1, . . . , Hy (m)} preserving the order. Then, we ss y ss y

, [L], α∗ , (δ, i0 ), I M

, [L], α∗ , (δ, ϕ(i0 )) , have the isomorphism M ss y

, [L], α∗ , (δ, i0 ), I is naturally identified with an open substack and hence M be rel is denoted by the same symbol. Let Φ

, [L]). The restriction of Θ of M(m, y as in (7.50). We set i0 , I) := Φ. Φ(δ, ss (

y ,[L],α∗ ,(δ,i0 ),I) M

Let Dec(m, y, α∗ , δ, I) denote the set of tuples I = (y 1 , y 2 , I1 , I2 ) satisfying the following conditions: y 1 + y 2 = y,

Pyα∗ ,δ = Pyα1∗ ,δ = Pyα2∗ ,

I1 I2 = I,

For any I = (y 1 , y 2 , I1 , I2 ) ∈ Dec(m, y, α∗ , δ, I), we set k(I) := max i ∈ I1 i < min(I2 ) .

|Ii | = Hyi (m)

7.6 Transition Formula (pg = 0)

315

We also put ss y

1 , [L], α∗ , (δ, k(I)), I1 × Mss (

y 2 , α∗ ). M(I) := M We put S(i0 , I) := I ∈ Dec(m, y, α∗ , δ, I) k(I) ≥ i0 . Then, Proposition 7.6.3 can be reworded as follows. Proposition 7.6.4 We have the following equality: i0 , I) − Φ(δ− ) = Φ(δ,

I∈S(i0 ,I)

Here, Ψ (I) are given as in (7.52).

NL (y1 ) Hy1 (m)! · Hy2 (m)! NL (y) Hy (m)!

I) M(

Ψ (I).

7.6.3 Step 2 We define the set Dec(j) (m, y, α∗ , δ) inductively on j. Put Dec(1) (m, y, α∗ , δ) := Dec(m, y, α∗ , δ). Assume that Dec(j−1) (m, y, α∗ , δ) is already given. Let Dec(j) (m, y, α∗ , δ) be the (j) (j) (j) (j) set of tuples I(j) = (y 1 , Y 2 , I1 , I 2 ) as follows: (j)

(j)

(j−1)

(1)

, . . . , y 2 ) of Sj (y, δ). • Y 2 denotes an element (y 2 , y 2 (j) (j) (j−1) (1) , . . . , I2 ) of subsets of {1, . . . , Hy (m)}. As• I 2 denotes a tuple (I2 , I2 (i) (i−1) sume min(I2 ) > min(I2 ) for i = 2, . . . , j Aj (j) (i) • We assume {1, . . . , Hy (m)} = I1 i=1 I2 . (j−1) (j) (j) (j−1) (j) (j) (j) (j) (j) (j) • Put y 1 := y 1 + y 2 and I1 := I1 I2 . Then, (y 1 , y 2 , I1 , I2 ) (j−1) (j−1) is an element of Dec(m, y 1 , α∗ , I1 ) in the sense of Subsection 7.6.2. (j−1) (j−1) (1) (j−1) (j−1) (1) • We put Y 2 := (y 2 , . . . , y 2 ) and I 2 = (I2 , . . . , I2 ). Then, (j−1) (j−1) (j−1) (j−1) (y 1 ,Y 2 , I1 , I2 ) is an element of Dec(j−1) (m, y, α∗ , δ). (j)

(j)

(j)

(j)

Let I(j) = (y 1 , Y 2 , I1 , I 2 ) be an element of Dec(j) (m, y, α∗ , δ). We put (j) (j) k(I(j) ) := max i ∈ I1 i < min(I2 ) We also put j (j) (j) (i) ss y (j) ) := M

1 , [L], α∗ , (δ, k(I(j) )), I1 × Mss (

y 2 , α∗ ) M(I i=1 j (j) (i)

1 , [L], α∗ , δ− × M− (I(j) ) := Mss y Mss (

y 2 , α∗ ) i=1

316

7 Invariants

(j) (j)

(j) be the universal sheaves on M ss y

1 , [L], α∗ , (δ, k(I(j) )), I1 × X Let E 1 (j)

(i) (i = 1, . . . , j) denote the universal

1 , [L], α∗ , δ− × X. Let E and Mss y 2 (i)

2 , α∗ × X. The appropriate pull backs are denoted by the same sheaves on Mss y symbols. (j) (j) (j)

1 , [L] → M m, y

1 . Let Θ1,rel be the relative tangent bundle of M m, y (j) (j) (j) be the relative tangent bundle of M(m,

1 , [L]) → M(m, y

1 , [L]). Let Θ y 1,rel The appropriate pull backs are denoted by the same symbols. (j) ) and M− (I(j) ). For that We introduce some cohomology classes on M(I j purpose, it is convenient to use Gm -equivariant cohomology classes, where Gjm denotes a j-dimensional torus. As explained in Subsection 7.1.4, the Gjm -equivariant bivariant theory of a point is identified with Q[s(1) , . . . , s(j) ]. We set (j)

T1

:= −

s(h) 1≤h≤j

(i)

(i)

T2 := −

, (h)

r1

(i)

(h)

Here, r2 := rank y 2 and r1

s(h) (h)

1≤h
r1

(h)

:= rank y 1 +

+

s(i) (i)

(i = 1, . . . , j).

r2

(p)

h
r2 . We use the symbol

(h)

ew·s to denote a trivial line bundle with the Gjm -action induced by the action of the h-th torus with weight w. For i, l ∈ {1, 2} and a, b ∈ {1, . . . , j}, we have the y 0 , Y , [L]): following virtual Gjm -equivariant vector bundles on M(

(a) T (a) (b)

e i ,E

(b) eTl − RpX ∗ RHom E i l (b) T (b) (a)

e l ,E

(a) eTi − RpX ∗ RHom E i l (a) T (a) (b)

(b) eTl i − RpD ∗ RHom E ,E i|D ∗ e l|D ∗ (b) T (b) (a)

(a) eTi . l ,E − RpD ∗ RHom E l|D ∗ e i|D ∗ (a) T (a) (b) T (b)

e i ,E

e l , which are Their equivariant Euler classes are denoted by Q E i l regarded as elements of

(j) ) ⊗ R(E 1

j

(h) ) ⊗ R s(1) , s(2) , . . . , s(j−1) , s(j) . R(E 2

h=1

We put (j) T (j) (j) T (j) (j−1) T (j−1) (1)

e 2 ,E

(1) eT2 :=

e 1 ,E e 2 ,...,E Q E 1 2 2 2 (h) T (h) (i) T (i) (j) T (j) (h) T (h)

e 2 ,E

e 2 ×

e 1 ,E

e 2 . Q E Q E 2 2 1 2 h
h≤j

7.6 Transition Formula (pg = 0)

317

(j)

, M− (I(j) ) ⊗ 1j R(E

(i) ): Then, we obtain the following element of R E 1 2 i=1 (j)

(j)

Ψ (j) (y 1 , Y 2 ) := Res · · · Res s(j)

s(1)

(j) T (j) k (j)

·e 1 ⊕

(i) T2(i) Eu(Θ1,rel ) P E 1 i=0 E2 · e · (j) T (j) (j) T (j) (j)

· e 1 ,E

· e 2 ,...,E

(1) · eT2(1) NL (y1 ) Q E 1 2 2

(j) (j) ) ⊗ 1j R(E

(j) ):

, M(I We also obtain the following element of R E 1 2 i=1 (j) (j) (j) (j) Ψ(j) (y 1 , Y 2 ) := Ψ (j) (y 1 , Y 2 ) ·

Eu(Θ 1,rel ) (j)

Hy(j) (m)! 1

Lemma 7.6.5 For each j, we have the following formula: Φ(δ+ ) − Φ(δ− ) = .i NL (y (i) ) Hy(i) (m)! h=1 Hy(h) (m)! (i) (i) 1 1 2 Ψ (i) (y 1 , Y 2 ) N (y) H (m)! (i) ) L y M (I − i<j I(i) ∈Dec(i) (m,y,α∗ ,δ)

+

.j (j) NL (y1 ) Hy1(j) (m)! h=1 Hy2(h) (m)! (j) (j) Ψ(j) (y 1 , Y 2 ) NL (y) Hy (m)! (j) M(I )

I(j) ∈Dec(j) (m,y,α∗ ,δ)

(7.53) Proof We use an induction on j. In the case j = 1, the claim is proven in Proposition 7.6.3 (the case = 0). Assume that the formula (7.53) holds for j, and we will prove it for j + 1. By definition, we have the naturally defined morphism πj : Dec(j) −→ Dec(j−1) . By Proposition 7.6.4, we obtain the following equality: (j) ) M(I

(j) (j) Ψ(j) (y 1 , Y 2 ) −

(j)

M−

I(j+1) ∈Dec(j+1) (m,y,α∗ ,δ) πj (I(j+1) )=I(j)

(I(j) )

(j)

Ψ (j) (y 1 , Y 2 ) =

) Hy1(j+1) (m)Hy2(j+1) (m) × (j) Hy(j) (m)! NL (y ) (j+1)

NL (y1

1

1

(j+1) ) M(I

(j+1) (j+1) Ψ (j+1) (y 1 ,Y 2 )

318

7 Invariants

(j+1) (j+1) Here, the elements Ψ (j+1) (y 1 ,Y 2 ) are given as follows:

(j+1) (j+1) ,Y 2 )= Ψ (j+1) (y 1

Eu(Θ 1,rel ) Eu(Θ1,rel ) (j+1)

(j+1)

Hy(j+1) (m)! NL (y (j+1) ) 1

×

1

1 (j+1) (j+1) (j+1) (j+1) −s(j+1) /r (j+1) /r1 s(j+1) s(j) s(1) 2 Q E1 e−s , E2 e (j+1) T (j+1) j (i) (i) e 1 ⊕ i=1 E2 · eT2 P E1 (j+1) T (j+1) (h) T (h) . (h) T (h) (i) T (i) . 2 e 1 , E2 e 2 , E2 e 2 h<j+1 Q E1 h
(j+1) (j+1) (j+1) (j+1) ,Y 2 ) = Ψ(j+1) (y 1 ,Y 2 ). Thus, the It is easy to observe Ψ (j+1) (y 1 formula (7.53) holds for j + 1.

7.6.4 Step 3 When j is sufficiently large, we have Dec(j) (m, y, α∗ , δ) = ∅. Therefore, we obtain the following: Φ(δ+ ) − Φ(δ− ) =

.k (k) NL (y1 ) Hy1(k) (m)! h=1 Hy2(h) (m)! (k) (k) Ψ (k) (y 1 , Y 2 ) NL (y) Hy (m)! M− (I(k) )

k I(k) ∈Dec(k) (m,y,α∗ ,δ)

(7.54) We have the map ρk : Dec(k) (m, y, α∗ , δ) −→ S k (y, δ) given as follows: (k) (k) (k−1) (1) , . . . , y 2 =: (y 0 , y 1 , y 2 , . . . , y k ) ρk (I(k) ) = y 1 , y 2 , y 2 Then, (7.54) can be rewritten as follows: Φ(δ+ ) − Φ(δ− ) =

NL (y0 ) NL (y)

.k

Hyi (m)! −1 #ρk (y 0 , Y ) Hy (m)!

i=0

k (y 0 ,Y )∈S k (y,δ)

Ψ (y 0 , Y ) (7.55)

,[L]) M(

y 0 ,Y

Lemma 7.6.6 Under Condition 7.6.1, we have the following equality: .k

Hyi (m)! · #ρ−1 k (y 0 , Y ) = W (Y ) Hy (m)!

i=0

7.7 Weak Wall Crossing Formula

319

Proof It is easy to observe that ρ−1 k (y 0 , Y ) is bijective to the following set: Ak I = {1, . . . , Hy (m)}, |Ii | = Hyi (m) , (I0 , I1 , . . . , Ik ) i=0 i min(Ik ) < min(Ik−1 ) < · · · < min(I1 ) (k)

(k)

(1)

The correspondence is given by (I1 , I2 , . . . , I2 ) = (I0 , I1 , . . . , Ik ). We put N := Hy (m) and Ni := Hyi (m). We have the following equality: N! (N − N0 − 1)! × N0 !(N − N0 )! (Nk − 1)!(N − N0 − Nk )! (N − N0 − Nk − 1)! × ··· (Nk−1 − 1)!(N − N0 − Nk − Nk−1 )! k (N − N0 − j>i Nj − 1)! N! · = N0 !(N − N0 )! i=1 (Ni − 1)!(N − N0 − j≥i Nj )! .k N! i=1 Ni = .k · .k i=0 Ni ! i=1 1≤j≤i Nj

#ρ−1 k (y 0 , Y ) =

Under Condition 7.6.1, we have Ni /ri = Nj /rj for 1 ≤ i, j ≤ k. Therefore, we obtain the following: #ρ−1 k (y 0 , Y ) ·

.k .k Ni ! i=1 Ni i=1 ri = .k = .k N! N i=1 i=1 1≤j≤i j 1≤j≤i rj

.k

i=0

Thus, the proof of Lemma 7.6.6 is finished.

We immediately obtain (7.49) from (7.55) and Lemma 7.6.6.

7.7 Weak Wall Crossing Formula 7.7.1 Statement Let y ∈ T ype◦r . We use the symbol MH (

y ) to denote the moduli stack of torsionfree sheaves of type y, which are semistable with respect to a polarization H. For P ∈ R, we set

u ). y ) := P (E (7.56) ΦH (

MH (

y)

Let C be the ample cone of X. For ξ ∈ N S(X), let W ξ := c ∈ C c · ξ = 0 , which is called the wall corresponding to ξ. According to [88], thereis the locally finite subset U of {ξ ∈ N S(X) | ξ 2 < 0} such that if H ∈ C − ξ∈U W ξ , the

320

7 Invariants

moduli MH (

y ) depends only on the chamber to which H belongs. Here, chamber means a connected component of C − ξ∈U W ξ . Let C+ and C− be chambers which are divided by a wall W ξ . Let H+ and H− be ample line bundles contained in H+ and H− respectively. Their first Chern classes are also denoted by the same symbols. We assume ξ · H+ > ξ · H− . We would like to express ΦH+ − ΦH− as the sum of integrals over the products of moduli stacks of objects with lower ranks. We have to prepare some notation. In the following, ri , ai and bi denote the rank, the first Chern class and the second Chern character of a given yi ∈ T ype◦ , respectively. We take a generic H in the intersection of the closures of C+ and C− . Let S+,k denote the set of y = (y0 , y1 , . . . , yk ) ∈ (T ype◦ )k+1 satisfying the following conditions: • There exist Ai ≥ 0 (i = 1, . . . , k) such that ai /ri − ai−1 /ri−1 = Ai · ξ in H 2 (X). • In the case A1 = 0, the inequality b0 /r0 < b1 /r1 holds. • In the case Ai = 0 for some i ≥ 2, we have bi /ri ≤ bi+1 /ri+1 . Let S−,k denote the set of y = (y0 , y1 , . . . , yk ) ∈ (T ype◦ )k+1 satisfying the following conditions: • There exist Ai ≤ 0 (i = 1, . . . , k) such that ai /ri − ai−1 /ri−1 = Ai · ξ. • In the case A1 = 0, the inequality b0 /r0 < b1 /r1 holds. • In the case Ai = 0 for some i ≥ 2, we have bi /ri ≤ bi+1 /ri+1 . Then, we set S+ :=

/

S+,k ,

S− :=

k

/

S−,k .

k

Let y = (y0 , . . . , yk ) be an element of S+,k or S−,k . Then, the integers 1 = i(1) < i(2) < · · · < i(s) ≤ k are determined by the following condition: Pyi(j)−1 = Pyi(j) = · · · = Pyi(j+1)−1 = Pyi(j+1) We formally put i(s + 1) := k + 1. We set B(y) :=

r0 B(y), r

.k

B(y) := .s j=1

For κ = ±, we put y) = MHκ (

k

i=1 ri .i(j+1)−1 h=i(j)

i(j)≤l≤h rl

MHκ (

yi ).

i=0

u denote the pull back of the universal sheaf over M(

Let E yi ) × X. i

.

(7.57)

7.7 Weak Wall Crossing Formula

321

When y is given, we introduce some cohomology classes on MHκ (

y). For that purpose, it is convenient to use Gkm -equivariant cohomology classes, where Gkm is a k-dimensional torus. As explained in Subsection 7.1.4, the Gkm -equivariant bivariant theory of a point is identified with Q[t1 , . . . , tk ]. We set T0 := −

j>0

Ti := −

j>i

tj 0≤h<j rank(y h )

ti tj + . rank(y i ) 0≤h<j rank(y h )

w·ti

We use the symbol e to denote a trivial line bundle with the Gkm -action induced by the action of the i-th torus with weight w. We have the following virtual Gkm equivariant vector bundles M(

y) as in Subsection 7.6.1:

i · eTi , E

j · eTj , E

j · eTj − RpX ∗ RHom E

i · eTi . −RpX ∗ RHom E

i ·eTi , E

j ·eTj . We regarded them Their equivariant Euler class are denoted by Q E 1k

i ) ⊗ R(tk , . . . , t1 ). We put as elements of i=0 R(E

1 · eT1 , . . . , E

k · eTk :=

i · eTi , E

j · eTj .

0 · eT0 , E Q E Q E i<j

Then, we define Ψ (y) := Res · · · Res t1

tk

k

u Ti P i=0 Ei · e u

· eT0 , . . . , E

u · eTk Q E 0

(7.58)

k

Theorem 7.7.1 We have the following formula: y ) − ΦH− (

y) = ΦH+ (

− B(y) · y∈S+

MH+ (

y)

Ψ (y) +

y∈S−

B(y) ·

Ψ (y)

(7.59)

MH− (

y)

We will write down the formula for the rank 3 case in Subsection 7.7.3.

7.7.2 Proof of Theorem 7.7.1 We use the argument and the notation in the proof of Theorem 7.5.3. Lemma 7.7.2 Let y = (y0 , . . . , yk ) be an element of (T ype◦ )k+1 such that yi = y and μ(yi ) = μ(y). We also assume that there are μ-semistable torsionfree sheaves of type yi (i = 0, . . . , k). Then, it is contained in S+,k if and only if the

322

7 Invariants

following inequality holds for any sufficiently large t: Py0 (C) (t) < Py1 (C) (t) ≤ Py2 (C) (t) ≤ · · · ≤ Pyk (C) (t)

(7.60)

Similarly, y is contained in S−,k if and only if the following inequality holds for any sufficiently large t: Py0 (−C) (t) < Py1 (−C) (t) ≤ Py2 (−C) (t) ≤ · · · ≤ Pyk (−C) (t) Proof We remark the finiteness of y satisfying the assumption of the lemma, by the boundedness of S. The condition (7.60) is equivalent to the following: b1 + a1 · C b2 + a2 · C bk + ak · C b0 + a0 · C < ≤ ≤ ··· ≤ r0 r1 r2 rk

(7.61)

We have (ai+1 · H)/ri+1 = (ai · H)/ri , and H is generic in W ξ . Hence, we have ai+1 /ri+1 − ai /ri = Ai · ξ for some rational numbers Ai , and we have (ai+1 ·C)/ri+1 −(ai ·C)/ri = Ai ·(ξ·C). Since ξ · C is sufficiently large, the condition (7.61) is equivalent to y ∈ S+,k . Thus, the first claim is proved. The second claim can be shown similarly. For any y = (y0 , . . . , yk ) ∈ S+ , we put k M

y(C), [O(−m, C)] := Ms y 0 (C), [O(−m, C)], × Mss y i (C) . i=1

Here, denotes any sufficiently small positive number. Similarly, we put as follows for any y ∈ S− : M

y(−C), [O(−m, −C)] := k Mss y i (−C) Ms y 0 (−C), [O(−m, −C)], × i=1

y(C),[O(−m, C)] induced by the relative tanLet Θrel denote the bundle over M

gent bundle of the smooth morphism Ms y (C), [O(−m, C)], −→ M(m, y (C)).

Let Θrel also denote the bundle over M y (−C), [O(−m, −C)] induced by the relative tangent bundle of Ms y (−C), [O(−m, −C)], −→ M(m, y (−C)). C (δ0 ) − Φ(H+ ) and Let δ0 be as in the proof of Theorem 7.5.3. To investigate Φ Φ−C (δ0 ) − Φ(H− ), we use the transition formula (Theorem 7.6.2) inductively. By Lemma 7.7.2, we obtain the following equality: Hy (m) Eu(Θrel ) 0 B(y) y) + Ψ (y) ΦC (δ0 ) = ΦH+ (

Hy (m) Hy0 (m) M(

y(C),[O(−m,C)]) y∈S+

7.7 Weak Wall Crossing Formula

323

We also obtain the following equality: −C (δ0 ) = ΦH (

y) + Φ −

Hy (m) Eu(Θrel ) 0 B(y) Ψ (y) Hy (m) Hy0 (m) M(

y(−C),[O(−m,−C)])

y∈S−

−C (δ0 ) = Φ C (δ0 ) (Lemma 7.5.4), we obtain the following From the equality Φ equality: Hy (m) Eu(Θrel ) 0 · B(y) · Ψ (y) · Hy (m) Hy0 (m) M(

y(C),[O(−m,C)]) y∈S+ Hy (m) Eu(Θrel ) 0 · B(y) · y) + Ψ (y) · = ΦH− (

Hy (m) Hy0 (m) M(

y(−C),[O(−m,−C)])

y) + ΦH+ (

y∈S−

Taking the limit for m → ∞, we obtain the desired equality (7.59), according to Proposition 7.3.1.

7.7.3 Weak Wall Crossing Formula in the Rank 3 Case As an example, we write down the formula (7.59) in the rank 3 case. In the following, ai and bi denote the first Chern class and the second Chern character of a given yi . We set U1 := (y1 , y2 ) ∈ T ype1 × T ype2 y1 + y2 = y U2 := (y1 , y2 ) ∈ T ype2 × T ype1 y1 + y2 = y U3 := (y1 , y2 , y3 ) ∈ T ype31 y1 + y2 + y3 = y Then, we consider the following subsets: (1) S1,+ := (y1 , y2 ) ∈ U1 a2 − 2a1 = A · ξ (A > 0) (2) S1,+ := (y1 , y2 ) ∈ U1 2a1 = a2 , 2b1 < b2 (1) S2,+ := (y1 , y2 ) ∈ U2 2a2 − a1 = A · ξ (A > 0) (2) S2,+ := (y1 , y2 ) ∈ U2 a1 = 2a2 , b1 < 2b2 (1) S3,+ := (y1 , y2 , y3 ) ∈ U3 a2 − a1 = A · ξ, a3 − a2 = B · ξ (A, B > 0) (2) S3,+ := (y1 , y2 , y3 ) ∈ U3 a1 + A · ξ = a2 = a3 (A > 0), b2 < b3 (3) S3,+ := (y1 , y2 , y3 ) ∈ U3 a1 = a2 = a3 − A · ξ (A > 0), b1 < b2 (4) S3+ := (y1 , y2 , y3 ) ∈ U3 a1 = a2 = a3 , b1 < b2 < b3 (1) S4,+ := (y1 , y2 , y3 ) ∈ U3 a1 + A · ξ = a2 = a3 (A > 0), b2 = b3 (2) S := (y1 , y2 , y3 ) ∈ U3 a1 = a2 = a3 , b1 < b2 = b3 4,+

324

7 Invariants

We also consider the following subsets: (1) S1,− := (y1 , y2 ) ∈ U1 a2 − 2a1 = A · ξ (A < 0) (2)

= (y1 , y2 ) ∈ U1 2a1 = a2 , 2b1 < b2 S1,− := S1,+ (1) S2,− := (y1 , y2 ) ∈ U2 2a2 − a1 = A · ξ (A < 0) (2) (2) S2,− := S2,+ = (y1 , y2 ) ∈ U2 a1 = 2a2 , b1 < 2b2 (1) S3,− := (y1 , y2 , y3 ) ∈ U3 a2 − a1 = A · ξ, a3 − a2 = B · ξ (A, B < 0) (2) S3,− := (y1 , y2 , y3 ) ∈ U3 a1 + A · ξ = a2 = a3 (A < 0), b2 < b3 (3) S3,− := (y1 , y2 , y3 ) ∈ U3 a1 = a2 = a3 − A · ξ (A < 0), b1 < b2 (4) (4) S3,− := S3,+ = (y1 , y2 , y3 ) ∈ U3 a1 = a2 = a3 , b1 < b2 < b3 (1) S4,− := (y1 , y2 , y3 ) ∈ U3 a1 + A · ξ = a2 = a3 (A < 0), b2 = b3 (2) (2) S4,− := S4,+ = (y1 , y2 , y3 ) ∈ U3 a1 = a2 = a3 , b1 < b2 = b3 We use the symbol M(

y , Hκ ) (κ = ±) to denote the moduli stack of oriented torsion-free sheaves of type y which are semistable with respect to Hκ . In the case rank(y) = 1, we omit to denote Hκ . To introduce some cohomology classes, it is convenient to use equivariant cohomology classes. Let Gm denote a one dimensional torus. The Gm -equivariant bivariant theory of a point is identified with Q[t] as explained in Subsection 7.1.4. Similarly, the (Gm )2 -equivariant bivariant theory of a point is identified with Q[t1 , t2 ]. We use the symbol ew·ti to denote a trivial line bundle with (Gm )2 action induced by the action of the i-th Gm with weight w. We use ew·t in a similar meaning. Let (y0 , y1 ) be an element of (T ype◦ )2 . From Gm -equivariant coherent sheaves y1 , Hκ ) × M(

y2 , Hκ ) × X, which is assumed to be flat over Ei (i = 1, 2) on M(

y2 , Hκ ), we obtain the following virtual Gm -equivariant vector M(

y1 , Hκ ) × M(

y2 , Hκ ): bundle on M(

y1 , Hκ ) × M(

−RpX ∗ RHom(E1 , E2 ) − RpX ∗ RHom(E2 , E1 ) The Gm -equivariant Euler class is denoted by Q(E1 , E2 ). Let P ∈ R. We set P(E1 , E2 ) := P(E1 , E2 ) :=

P (E1 ⊕ E2 ) ∈ R(E1 ) ⊗ R(E2 ) ⊗ R(t) Q(E1 , E2 )

(See Subsection 7.1.6 for R(t).) Let (y1 , y2 , y3 ) be an element of (T ype◦ )3 . Similarly, we obtain the follow2 ing . sheaves Ei (i = 1, 2, 3) on . element from given (Gm ) -equivariant coherent yi ): M(

yi ) × X, which are assumed to be flat over M(

P (E1 ⊕ E2 ⊕ E3 ) ∈ R(E1 ) ⊗ R(E2 ) ⊗ R(E3 ) ⊗ R(t2 , t1 ). P(E1 , E2 , E3 ) := . i<j Q(Ei , Ej )

7.7 Weak Wall Crossing Formula

325

For κ = ±, let Λ(Hκ ) denote the following:

y) + ΦHκ (

1 3

i=1,2 (i) (y1 ,y2 )∈S1,κ

+

2 3

M(

y1 )×M(

y2 ,Hκ )

M(

y1 ,Hκ )×M(

y2 )

i=1,2 (i) (y1 ,y2 )∈S2,κ

+

1 3

i=1,2,3,4 (i) (y1 ,y2 ,y3 )∈S3,κ

+

1 6

i=1,2 (i) (y1 ,y2 ,y3 )∈S4,κ

. i

u ·e−t , E

u ·et/2 Res P E 1 2 t

u ·e−t/2 , E

u ·et Res P E 1 2 t

u e−t1 −t2 /2 , E

u et1 −t2 /2 , E

u et2 Res Res P E 1 2 3

M(

yi ) t1

t2

1u e−t1 −t2 /2 , E

2u et1 −t2 /2 , E

3u et2 Res Res P E

. i

M(

yi ) t1

t2

(1)

Then, the formula (7.59) says Λ(H+ ) = Λ(H− ). The contributions from S3,κ (2)

and S4,κ are cancelled out.

7.7.4 Weak Intersection Rounding Formula in the Rank 3 Case The weak wall crossing formula (7.59) is not so easy to treat, even in the rank 3 case. We would like to obtain a more accessible quantity from our invariants. For that purpose, we consider “intersection rounding phenomena”, or “wall-crossing of wall-crossing”. Later, we will study it in the general case. (Section 7.8). However, the general statement is a little complicated. Hence, we give the formula for the rank 3 case in this subsection. W ξ2 H−+ q H−−

q

qH++ q

W ξ1

H+−

We take an element ξ = (ξ1 , ξ2 ) ∈ N S(X)2 such that ξ1 and ξ2 are linearly independent, and we put W ξ := W ξ1 ∩ W ξ2 . A connected component T of W ξ \ W =W ξi W is called a tile. For each tile T , there exists four chambers whose closures contain T . Let C be such a chamber, and let HC be an ample line bundle contained in C. Then, the map ϕC : {1, 2} −→ {±} is determined by ϕC (i) = sign(HC · ξi ). We have the bijective correspondence between such

326

7 Invariants

chambers and the maps, and we denote by C(ϕ) the chamber corresponding to a map ϕ : {1, 2} −→ {±}. We put k(ϕ) := #{i | ϕ(i) = −}. We take any ample line bundle Hϕ ∈ C(ϕ). Then, we put as follows: y ) := (−1)k(ϕ) ΦHϕ (

y) DξT Φ(

ϕ

We would like to show that DξT Φ(

y ) is independent of a choice of T , and we would T like to express Dξ Φ(

y ) as the sum of integrals over the products of Hilbert schemes. Let S(2, 1) be the set of a = (a0 , a1 , a2 ) ∈ N S(X)3 with the following property: • (a0 + a1 )/2 − a2 = A1 · ξ1 and a0 − a1 = A2 · ξ2 for some Ai > 0. Let S(1, 2) be the set of a = (a0 , a1 , a2 ) ∈ N S(X)3 with the following property: • a0 − (a1 + a2 )/2 = A1 · ξ1 and a1 − a2 = A2 · ξ2 for some Ai > 0. For each a, we set N (y, a) = n + X(y, a) :=

a20 + a21 + a22 − a2 2 2

(X [ni ] × Pic(ai ))

n0 +n1 +n2 =N (y,a) i=0

u Here, a and n denote . the first Chern class and the second Chern class of y. Let E i [ni ] denote the sheaf over (X ×Pic(ai ))×X which is the pull back of the universal ideal sheaf over X [ni ] × Pic(ai ) × X. Let G denote a two dimensional torus (Gm )2 . The G-equivariant bivariant theory of a point is identified with Q[t1 , t2 ], as explained in Subsection 7.1.4. It is included in R(t1 , t2 )..(See Subsection 7.1.6.) Let Ej (j = 1, 2, 3) be G-equivariant coherent sheaves on (X [ni ] × Pic(ai )) × X. Then, Q(Ej , Ek ) denotes the G-equivariant Euler class of the following: −RpX ∗ RHom(Ej , Ek ) − RpX ∗ RHom(Ek , Ej ) We use the following symbol: P E1 ⊕ E2 ⊕ E3 ∈ R(E1 ) ⊗ R(E2 ) ⊗ R(E3 ) ⊗ R(t1 , t2 ) P(E1 , E2 , E3 ) := . i<j Q(Ei , Ej ) Proposition 7.7.3 y ) is independent of the choice of T . Therefore, we can omit to denote T . • DξT Φ(

• The following equality holds:

7.7 Weak Wall Crossing Formula

Dξ Φ(

y) =

a∈S(2,1)

0u e−t1 , E

1 et1 /2−t2 , E

2u et1 /2+t2 ) Res Res P(E

X(y,a) t2

a∈S(1,2)

+

327

t1

0u e−t1 /2−t2 , E

1u e−t1 /2+t2 , E

2u et1 Res Res P E

X(y,a) t2

t1

(7.62) Proof We use the notation in Subsection 7.7.3. We identify ϕ and (ϕ(1), ϕ(2)). We have the equality: Λ(H+,+ ) − Λ(H+,− ) = Λ(H−,+ ) − Λ(H−,− ) = 0 We put as follows: (1) S1,+ := (y1 , y2 ) ∈ U1 a2 − 2a1 = A · ξ1 (A > 0) (2) S1,+ := (y1 , y2 ) ∈ U1 2a1 = a2 , 2b1 < b2 (1) S2,+ := (y1 , y2 ) ∈ U2 2a2 − a1 = A · ξ1 (A > 0) (2) := (y1 , y2 ) ∈ U2 a1 = 2a2 , b1 < 2b2 S 2,+

We also put as follows: (1) S1,− := (y1 , y2 ) ∈ U1 a2 − 2a1 = A · ξ1 (A < 0) , (1) S2,− := (y1 , y2 ) ∈ U2 2a2 − a1 = A · ξ1 (A < 0) ,

S1,− := S1,+ (2) (2)

(2)

S2,− := S2,+

Then, we have the following equality for κ = ±: Λ(Hκ,+ ) − Λ(Hκ,− ) = Φ(Hκ,+ ) − Φ(Hκ,− ) 1

u · e−t , E

u · et/2 + Res P E 1 2 3 M( y1 )×M( y2 ,Hκ,+ ) t (i) i=1,2 (y1 ,y2 )∈S1,κ

−

i=1,2 (y ,y )∈S (i) 1 2 1,κ

+

i=1,2 (y ,y )∈S (i) 1 2 2,κ

−

1 3 2 3

i=1,2 (y ,y )∈S (i) 1 2 2,κ

M(

y1 )×M(

y2 ,Hκ,− )

M(

y1 ,Hκ,+ )×M(

y2 )

2 3

1u · e−t , E

2u · et/2 Res P E t

1u · e−t/2 , E

2u · et Res P E t

M(

y1 ,Hκ,− )×M(

y2 )

1u · e−t/2 , E

2u · et Res P E t

328

7 Invariants

By a formal calculations using the weak wall crossing formula in the rank 2 (Theorem 7.4.3), we can show the following equalities:

M(

y1 )×M(

y2 ,H−+ )

(1)

(y1 ,y2 )∈S1,−

u −t u t/2

1 · e , E

2 · e Res P E t

− =

M(

y1 )×M(

y2 ,H−− )

t2

t1

M(

y1 ,H−+ )×M(

y2 )

(1)

(y1 ,y2 )∈S2,−

t

1 · e−t/2 , E

2 · et Res P E t

− =

t

0 · e−t1 /2−t2 , E

1 · e−t1 /2+t2 , E

2 · et1 Res Res P E

X(y,a) t2

a∈S(2,1)

1 · e−t/2 , E

2 · et Res P E

M(

y1 ,H−− )×M(

y2 )

(7.63)

X(y,a)

u −t1 u t1 /2−t2 u t1 /2+t2

0 · e , E

1 · e

2 · e P E ,E

Res Res

a∈S(1,2)

u −t u t/2

1 · e , E

2 · e Res P E

t1

(7.64)

We also have the following:

M(

y1 )×M(

y2 ,H++ )

(1)

(y1 ,y2 )∈S1,+

u −t u t/2

1 · e , E

2 e Res P E t

−

M(

y1 )×M(

y2 ,H+− )

M(

y2 ,H++ )×M(

y1 )

(1)

(y1 ,y2 )∈S1,+

− =−

a∈S(2,1)

M(

y2 ,H+− )×M(

y1 )

t

=−

u −t u t/2

1 · e , E

2 e Res P E

u −t/2 u t

e

·e Res P E ,E 2 1 t

u t/2 u −t

e ,E

·e Res P E 2 1

t

0 · e−t1 /2−t2 , E

1 · e−t1 /2+t2 , E

2 · et1 Res Res P E

X(y,a) t2

t1

(7.65)

7.7 Weak Wall Crossing Formula

329

M(

y1 ,H++ )×M(

y2 )

(1)

(y1 ,y2 )∈S2,+

1 · e−t/2 , E

u · et Res P E 2 t

−

M(

y1 ,H+− )×M(

y2 )

=−

(1)

(y1 ,y2 )∈S2,+

1 · e−t/2 , E

2u · et Res P E t

M(

y2 )×M(

y1 ,H++ )

− =−

M(

y2 )×M(

y1 ,H+− )

a∈S(1,2) (2)

t1

Res P t

u −t

2 · e , E

1 · et/2 Res P E

u E 2

t

1 · e · e ,E t

−t/2

u −t1 u t1 /2−t2 u t1 /2+t2

· e ,E

·e

·e P E ,E 0 1 2

Res Res t2

(7.66)

X(y,a)

(2)

Recall S1,+ = S1,− . Then, it is also easy to observe the following equalities by using the weak wall crossing formula in the rank 2 case:

(2)

(y1 ,y2 )∈S1,+

−

M(

y1 )×M(

y2 ,H+,+ )

M(

y1 )×M(

y2 ,H+,− )

(2)

M(

y1 )×M(

y2 ,H−,+ )

(2)

(y1 ,y2 )∈S1,−

−

(2)

(y1 ,y2 )∈S1,−

(2)

(y1 ,y2 )∈S2,+

−

M(

y1 )×M(

y2 ,H−,− )

M(

y1 ,H+,+ )×M(

y2 )

M(

y1 ,H−,+ )×M(

y2 )

(2)

(y1 ,y2 )∈S2,−

−

(2) (y1 ,y2 )∈S2,−

t

1u · e−t , E

2u · et/2 Res P E t

1u · e−t , E

2u · et/2 Res P E t

(7.67)

t

M(

y1 ,H+,− )×M(

y2 )

(2)

1u · e−t , E

2u · et/2 Res P E

1u · e−t/2 , E

2u · et Res P E

(y1 ,y2 )∈S2,+

=

t

(y1 ,y2 )∈S1,+

=

1u · e−t , E

2u · et/2 Res P E

M(

y1 ,H−,− )×M(

y2 )

1u · e−t/2 , E

2u · et Res P E t

u · e−t/2 , E

u · et Res P E 1 2 t

u · e−t/2 , E

u · et Res P E 1 2 t

Then, we obtain (7.62) by a formal calculation.

(7.68)

330

7 Invariants

7.7.5 Transition for a Critical Parabolic Weight We give a generalization of Proposition 7.4.8 in the higher rank case. The argument is the essentially same as that employed in the proof of Theorem 7.7.1. Hence, we state the result without a proof. Let α be a critical parabolic weight. We take α− < α < α+ sufficiently closely. For P ∈ R, we put

u ) P (E Φ(

y , ακ ) := M(

y ,ακ )

Let S+,k (y, ξ, α) be the set of y = (y0 , . . . , yk ) ∈ N S(X)k+1 with the following properties: • yi = y and Pyαi = Pyα . • We have a0 /r0 − a1 /r1 = m0 · ξ for some m0 < 0. • We have ai /ri − ai+1 /ri+1 = mi · ξ for some mi ≤ 0. In the case ai /ri = ai+1 /ri+1 , we have yi /ri = yi+1 /ri+1 . Here ri and ai denote the rank and the first Chern class of yi , as usual. Let S−,k (y, ξ, α) denote the set of y = (y0 , . . . , yk ) ∈ N S(X)k+1 with the following properties: • yi = y and Pyαi = Pyα . • We have a0 /r0 − a1 /r1 = m0 · ξ for some m0 > 0. • We have ai /ri − ai+1 /ri+1 = mi · ξ for some mi ≥ 0. In the case ai /ri = ai+1 /ri+1 , we have yi /ri = yi+1 /ri+1 . For κ = ±, we put as follows: /

Sκ (y, ξ, α) :=

Sκ,k (y, ξ, α)

k

For each y = (y0 , . . . , yk ) ∈ Sκ,k (y, ξ, α), we put as follows: k

M(

y, ακ ) :=

M(

yi , ακ )

i=0

Proposition 7.7.4 The following equality holds: y , α− ) = Φ(

y , α+ ) − Φ(

− B(y) · y∈S+ (y,ξ,α)

Ψ (y) +

M(

y,α+ )

Here, Ψ (y) denote the elements of the numbers given by (7.57).

y∈S− (y,ξ,α)

1k i=0

B(y) ·

Ψ (y)

(7.69)

M(

y,α− )

u ) given by (7.58), and B(y) denote R(E i

As an example, we write down the formula (7.69) in the rank 3 case. We use the notation in Subsection 7.7.3. We set

7.8 Weak Intersection Rounding Formula

331

Ui (α) := (y1 , y2 ) ∈ Ui Pyαi = Pyα (i = 1, 2) U3 (α) := (y1 , y2 , y3 ) ∈ U3 Pyαi = Pyα S1+ := (y1 , y2 ) ∈ U1 (α) 2a1 − a2 = m · ξ (m < 0) S2,+ := (y1 , y2 ) ∈ U2 (α) a1 − 2a2 = m · ξ (m < 0) S3+ := (y1 , y2 , y3 ) ∈ U3 (α) a1 − a2 = m1 · ξ, a2 − a3 = m2 · ξ(mi < 0) S4+ := (y1 , y2 , y3 ) ∈ U3 (α) a1 − a2 = m · ξ (m < 0), y2 = y3 Here, ai denotes the first Chern class of yi . We also put S1− := (y1 , y2 ) ∈ U1 (α) 2a1 − a2 = m · ξ (m > 0) S2,− := (y1 , y2 ) ∈ U2 (α) a1 − 2a2 = m · ξ (m > 0) S3− := (y1 , y2 , y3 ) ∈ U3 (α) a1 − a2 = m1 · ξ, a2 − a3 = m2 · ξ(mi > 0) S4− := (y1 , y2 , y3 ) ∈ U3 (α) a1 − a2 = m · ξ (m > 0), y2 = y3 For κ = ±, we set y , ακ ) Λ(ακ ) := Φ(

+ (y1 ,y2 )∈S1,κ

+

(y1 ,y2 )∈S2,κ

1 3 2 3

M(

y1 )×M(

y2 ,ακ )

M(

y1 ,ακ )×M(

y2 )

u e−t , E

u et/2 Res P E 1 2 t

u e−t/2 , E

u et Res P E 1 2 t

1

u e−t1 −t2 /2 , E

u et1 −t2 /2 , E

u et2 P E Res Res 1 2 3 3 . i M( yi ) t1 t2 (y1 ,y2 ,y3 )∈S3,κ 1

u e−t1 −t2 /2 , E

u et1 −t2 /2 , E

u et2 + P E Res Res 1 2 3 6 . i M( yi ) t1 t2 +

(y1 ,y2 ,y3 )∈S4,κ

Then, Proposition 7.7.4 says Λ(α+ ) = Λ(α− ).

7.8 Weak Intersection Rounding Formula 7.8.1 Preliminary We prepare some terminology to state the theorem. An oriented tree is a tree provided with an orientation for each edge. For an oriented tree R, let V (R) denote the set of the vertices of R. We have a natural order ≤R on V (R). Let Vmax (R) denote the set of the maximal vertices of R, and we put V ◦ (R) := V (R) − Vmax (R). An oriented tree R is called an indexed rooted oriented binary plane tree of rank r, if the following conditions are satisfied (see the pictures below):

332

7 Invariants

• R is embedded in R × R≥ 0 , and it intersects with R × {0} transversally. The intersection R ∩ (R × {0}) consists of the maximal points. • There exists the unique minimal vertex o with respect to the order induced by the orientation. The vertex is called the root. • The maximal verticesare {(0, 0), (1, 0), . . . , (r − 1, 0)}. • For any v ∈ V (R) − {o} ∪ Vmax (R) , there are three edges which contain v. • A bijective map ϕ : V ◦ (R) −→ {1, . . . , r − 1} is provided, which preserves the orders on the sets V ◦ (R) and {1, . . . , r − 1}. r=2

r (0, 0)

rϕ = 1 @ @ @ Rr @ (1, 0)

r=3 rϕ = 1 @ r ϕ = 2@ @ @ r Rr @ Rr @ (0, 0) (1, 0) (2, 0)

r=4

r (0, 0)

rϕ = 1 @ Rrϕ = 2 @ @ Rrϕ = 3 @ @ r r Rr @ (1, 0) (2, 0) (3, 0)

rϕ = 2 @ r Rr @ (0, 0) (1, 0)

rϕ = 1 @ @ @

Rrϕ = 3 @ @ r Rr @ (2, 0) (3, 0)

r (0, 0)

r (0, 0)

rϕ = 1 @ Rrϕ = 2 @ @ r Rr @ (1, 0) (2, 0)

rϕ = 1 @ Rrϕ = 2 @ @ r ϕ = 3@ @ @ r Rr @ Rr @ (1, 0) (2, 0) (3, 0)

rϕ = 1 @ @ @ r ϕ = 3 Rrϕ = 2 @ @ @ r Rr @ r Rr @ (0, 0) (1, 0) (2, 0) (3, 0)

7.8 Weak Intersection Rounding Formula

333

rϕ = 1 @ r ϕ = 2@ @ @ r ϕ = 3@ @ @ @ @ r Rr @ Rr @ Rr @ (0, 0) (1, 0) (2, 0) (3, 0)

r (0, 0)

rϕ = 1 @ r ϕ = 2@ @ @ Rr ϕ = 3@ @ @ @ r Rr @ Rr @ (1, 0) (2, 0) (3, 0)

We have a natural notion of isotopy. Let T(r) denote the set of the isotropy classes of an indexed rooted oriented binary plane trees of rank r. For example, #T(1) = 1, #T(2) = 1, #T(3) = 2 and #T(4) = 6. Remark 7.8.1 T(r) parameterizes the set of composition rules. It is easy to show #T(n) = (n − 1)!, which was pointed out by the referee. Let R be an indexed rooted oriented binary plane tree of rank r. For any v ∈ V (R), we set r(v) := # u ∈ Vmax (R) u ≥R v Take v ∈ V ◦ (R). The vertices v b , v l and v r are given as in the following picture: r vb ? rv @

vr @ Rr @

vrl

(If v is the root, v b is not given.) We set 1 (v = (v b )r ) sign(v) := −1 (v = (v b )l ) Let t1 , . . . , tr−1 be formal variables, and we put as follows for i = 0, . . . , r − 1: TRi :=

sign(v)

v≤R (i,0) v=o

tϕ(vb ) r(v)

(7.70)

We identify (i, 0) ∈ Vmax (R) and i. For any a = (a0 , . . . , ar−1 ) ∈ N S(X) and each v ∈ V (R), we put as follows: au av := u∈Vmax (R) u≥R v

334

7 Invariants

Let a be an element of N S(X). Let ξ = (ξ1 , . . . , ξr−1 ) be an element of N S(X)r−1 such that ξi are linearly independent. We denote by S(r, a, ξ, R) the set of a = (a0 , . . . , ar−1 ) ∈ N S(X)r satisfying ai = a and the following condition: • There exists a positive rational number A such that the following equality holds in N S(X) ⊗ Q for any v ∈ V ◦ (R): avr avl − = Av · ξϕ(v) r(v l ) r(v r ) Let a be any element of S(r, a, ξ, R). Let y be an element of T ype◦ such that rank(y) = r and det(y) = a. Let n denote the second Chern class of y. We set a2 a2i + 2 2 i=1 r−1

N (a, y) := n −

M(a, y) :=

r−1

r−1 i=0

X [ni ] × Pic(ai )

ni =N (a,y) i=0

u which is the pull back of the universal On M(a, y) × X, we have the sheaf E i [ni ] × Pic(ai ) × X via the naturally defined projection. sheaf on X Let (Gm )r−1 denote an (r − 1)-dimensional torus. We use (Gm )r−1 -equivariant cohomology classes to introduce some cohomology classes on M(a, y). As explained in Subsection 7.1.4, the (Gm )r−1 -equivariant bivariant theory of a point is identified with Q[t1 , . . . , tr−1 ]. For an indexed rooted oriented binary plane tree R of rank r, let TRi (i = 0, . . . , r − 1) be given as in (7.70). We have the following virtual (Gm )r−1 -equivariant vector bundle on M(a, y): u Ti u Tj

· e R, E

u · eTRj − RpX ∗ RHom E

· e R, E

u · eTRi −RpX ∗ RHom E i j j i u Ti u Tj

· e R, E

· e R . We regard Their equivariant Euler classes are denoted by Q E i j 1r−1 u

them as the elements of i=0 R(Ei ) ⊗ R(t1 , . . . , tr−1 ). We set u T0 u T1

· e R, E

· e R, . . . , E

u · eTRr−1 :=

u · eTRi , E

j · eTRj Q E Q E 0 1 r−1 i i<j

r−1 u T i 1r−1

u

R ∈ Let P ∈ R. We obtain P i=0 Ei · e i=0 R(Ei )[t1 , . . . , tr−1 ] by the homomorphisms in Subsection 7.1.2. Then, we obtain the following element of 1r−1

u ): R( E i i=0 r−1 u T i

R P i=0 Ei e Ψ (a, R) := Res Res · · · Res (7.71) u T0 u T1 tr−1 tr−2 t1

e R, E

e R, . . . , E

u eTRr−1 Q E 0

1

r−1

7.8 Weak Intersection Rounding Formula

335

7.8.2 Statement We take an element ξ = (ξ1 , ξ2 , . . . , ξl ) ∈ N S(X)l such that ξ1 , . . . , ξl are Cl ξi linearly independent. We put W ξ := i=1 W . A connected component of ξ W \ W =W ξi W is called a tile. Let ϕ : {1, . . . , l} −→ {±} be a map. We have the chamber C(ϕ) satisfying the following conditions: • The closure of C(ϕ) contains T . • Let Hϕ be any ample line bundle contained in C(ϕ). Then, sign ξi · H = ϕ(i). We take line bundles Hϕ ∈ C(ϕ). We put k(ϕ) := #{i | ϕ(i) = −}. For Φ =

u ) ∈ R(E

u ), we define P (E DξT Φ(

y ) := (−1)k(ϕ) · ΦHϕ (

y) ϕ

y ) is given as in (7.56). Here, ΦHϕ (

Theorem 7.8.2 Assume pg = 0. Let y ∈ T ype◦ , and let ξ = (ξ1 , . . . , ξl ) and T be as above. y ) = 0. • In the case l ≥ rank(y), we have DξT Φ(

• In the case l = rank(y) − 1, the number DξT Φ(

y ) is independent of the choice of a tile T , and the following equality holds: y) = Ψ (a, R) (7.72) Dξ Φ(

R∈T(r) a∈S(r,a,ξ,R)

M(a,y)

Here, Ψ (a, R) are given in (7.71). The theorem will be proved in Subsection 7.8.4 after the combinatorial preparation. We have already written down the formula (7.72) for the rank 3 case in Subsection 7.7.4.

7.8.3 Preparation from Combinatorics Let S be a finite set. Let Map(S, {±}) denote the set of maps of S to {±}. For any ϕ ∈ Map(S, {±}), let k(ϕ) denote the number of i ϕ(i) = − . Let I be a + subset of S. For any ϕI ∈ Map(I, {±}), let Ind− S (ϕI ) and IndS ϕI be elements of Map(S, {±}) determined as follows: ϕI (i) (i ∈ I) ϕI (i) (i ∈ I) − + IndS ϕI (i) := IndS ϕI (i) := − (i ∈ I) + (i ∈ I)

336

7 Invariants

Lemma 7.8.3 For any functions Fi : Map(S, {±}) −→ C (i = 1, 2), we have the following equality: (−1)k(ϕ) F1 (ϕ) · F2 (ϕ) = B(I, J) (7.73) IJ=S

ϕ∈Map(S,{±})

Here, B(I, J) are given as follows: ⎛

⎞ ⎠ (−1)k(ϕI ) F1 Ind− S (ϕI )

B(I, J) := ⎝

ϕI ∈Map(I,{±})

⎛ ×⎝

⎞

⎠ (−1)k(ϕJ ) F2 Ind+ S (ϕJ )

ϕJ ∈Map(J,{±})

Proof We use an induction on |S|. In the case |S| = 1, we have the natural identification Map(S, {±}) = {±}, and the claim is the following obvious equality: F1 (+) · F2 (+) − F1 (−) · F2 (−) = F1 (+) − F1 (−) · F2 (+) + F1 (−) · F2 (+) − F2 (−) Assume that the claim holds in the case |S| = k − 1, and we will show the claim in the case |S| = k. We take any element s ∈ S, and we put S := S − {s}. The left hand side of (7.73) can be rewritten as follows:

(−1)k(ϕ) · F1 (ϕ) · F2 (ϕ) +

ϕ∈Map(S,{±}) ϕ(s)=+

=

(−1)k(ϕ) · F1 (ϕ) · F2 (ϕ)

ϕ∈Map(S,{±}) ϕ(s)=−

+

(−1)k(ϕ ) F1 Ind+ S (ϕ ) · F2 IndS (ϕ )

ϕ ∈Map(S ,{±})

−

−

(−1)k(ϕ ) F1 Ind− S (ϕ ) · F2 IndS (ϕ )

(7.74)

ϕ ∈Map(S ,{±})

By using the hypothesis of the induction, the right hand side of (7.74) can be rewritten as follows:

+ − + (−1)k(ϕI )+k(ϕJ ) F1 Ind+ S (IndS (ϕI )) F2 IndS IndS (ϕJ )

IJ=S ϕI ∈Map(I,{±}) ϕJ ∈Map(J,{±})

−

− − + (−1)k(ϕI )+k(ϕJ ) F1 Ind− S (IndS (ϕI )) · F2 IndS (IndS (ϕJ ))

IJ=S ϕI ∈Map(I,{±}) ϕJ ∈Map(J,{±})

(7.75)

7.8 Weak Intersection Rounding Formula

337

It is equal to the following: (−1)k(ϕI )+k(ϕJ ) F1 Ind+ (Ind− (ϕI )) −F1 Ind− (Ind− (ϕI )) S S S S IJ=S + ×F2 Ind+ ϕI ∈Map(I,{±}) S IndS (ϕJ ) ϕJ ∈Map(J,{±})

+

(−1)k(ϕI )+k(ϕJ )

IJ=S ϕI ∈Map(I,{±}) ϕJ ∈Map(J,{±})

− + + Ind F2 Ind+ (Ind (ϕ )) −F (Ind (ϕ )) J 2 J S S S S − (Ind (ϕ ×F1 Ind− I )) S S

It can be rewritten as follows:

+ (−1)k(ϕI )+k(ϕJ ) F1 Ind− S (ϕI ) · F2 IndS (ϕJ )

IJ=S s∈I

+

+ (−1)k(ϕI )+k(ϕJ ) F1 Ind− S (ϕI ) · F2 IndS (ϕJ )

IJ=S s∈J

=

+ (−1)k(ϕI )+k(ϕJ ) F1 Ind− S (ϕI ) · F2 IndS (ϕJ )

IJ=S

Thus we are done.

7.8.4 Proof of Theorem 7.8.2 We consider the following statements: P (r): Assume rank(y) ≤ r. For any ξ = (ξ1 , . . . , ξl ) such that l ≥ rank(y), we y ) = 0. have DξT Φ(

Q(r): Assume rank(y) ≤ r. For any ξ = (ξ1 , . . . , ξl ) such that l = rank(y) − 1, y ) is independent of the choice of the tile T . Moreover, the the number DξT Φ(

equality (7.72) holds. It is easy to see Q(r) implies P (r). We have already known that Q(1) and Q(2) hold. We will prove Q(r), and hence P (r), by an induction on r.

i ). We Let yi (i = 1, . . . , s) be elements of T ype, and let Φi be elements of R(E set s s T Φi (

yi ) := (−1)k(ϕ) Φi,Hϕ (

yi ). Dξ i=1

ϕ

i=1

For a given ξ = (ξ1 , . . . , ξl ) and a subset I ⊂ {1, . . . , l}, we put ξ I := ξi | i ∈ I . For a tile T in W ξ , let T−I be the tile of W ξI such that (i) H · ξj < 0 for any j ∈ {1, . . . , l} \ I and any element H ∈ T−I , (ii) T is contained in the closure of T−I . Similarly, let T+I be the tile of W ξI such that (i) H · ξj > 0 for any j ∈ {1, , . . . , l} \ I and any element H ∈ T+I , (ii) T is contained in the closure of T+I .

338

7 Invariants

We set S := {1, . . . , l}. Then, we obtain the following equality from Lemma 7.8.3: s s I J T T Φi (

yi ) = DξI− Φ1 (

y1 ) · DξJ+ Φi (

yi ) (7.76) DξT i=1

i=2

IJ=S

Lemma 7.8.4 Assume that P (r − 1) holds. Let y1 , . . . , ys (s ≥ 1) be elements of T ype◦ such that rank yi ≤ .r − 1. For ξ = (ξ1 , . . . , ξl ) such that l > s s T yi ) = 0. i=1 rank(yi ) − s, we have Dξ i=1 Φi (

Proof We use an induction on s. In the case s = 1, the claim follows from the assumption P (r − 1). Assume that the claim holds in the case s − 1, and we will show the claim in the case s. If the inequality |I| ≥ rank(y1 ) holds, the contribution from (I, J) is 0 in (7.76), s by the assumption P (r − 1). If |I| ≤ rank(y1 ) − 1 holds, the inequality |J| > i=2 rank yi −s+1 holds. Therefore, we obtain the vanishing of the contribution by the hypothesis of the induction. Thus we are done. Lemma 7.8.5 Assume that Q(r − 1) holds. Let yi (i = 1, 2) be elements of T ype◦ such that rank y1 + rank y2 = r. For ξ = (ξ1 , . . . , ξr−2 ), the number DξT (Φ1 (

y1 ) · y2 )) is independent of the choice of a tile T , and the following equality holds: Φ2 (

Dξ Φ1 (

y1 ) · Φ2 (

y2 ) = DξI1 Φ1 (

y1 ) · DξI2 Φ2 (

y2 ) I1 I2 ={1,...,r−2} |Ii |=rank yi −1

Proof It can be shown by an argument similar to the proof of Lemma 7.8.4.

Let us show the claim Q(r) assuming Q(r − 1). We put S := {2, . . . , l}. For a map λ : S −→ {±}, let T (λ) denote the tile of W ξ1 determined by the following conditions: • The closure of T (λ) contains T . • Let H be any ample line bundle contained in T (λ). Then, we have sign(H ·ξi ) = λ(i). y ), we obtain the following equality: From the definition of DξT Φ(

T (λ) y) = Φ(

y) DξT Φ(

(−1)k(λ) · Dξ1

(7.77)

λ

Let Hκ,λ denote an ample line bundle contained in the chamber corresponding to IndκS (λ). We can rewrite (7.77) as follows by using Theorem 7.7.1: y) DξT Φ(

+

λ

k(λ)

(−1)

y∈S+

B(y) ·

Ψ (y) MH+λ (

y)

k(λ) (−1) B(y) · − λ

y∈S−

Ψ (y) = 0 (7.78) MH−λ (

y)

7.8 Weak Intersection Rounding Formula

339

We put ξ := (ξ2 , . . . , ξl ). Let Tκξ (κ = ±) be the tiles of W ξ determined by the following conditions:

• The closure of Tκξ contains T . • Let Hκ be any element of Tκξ . Then, we have H− · ξ1 < 0 < H+ · ξ1 . For y = (y0 , . . . , yk ), we define Tξ Tξ y) := Dξκ Ψ (y)(

y0 , . . . , y k ). Dξκ Ψ (

Then, we can rewrite (7.78) as follows: Tξ Tξ y) + y) − y) = 0 B(y) · Dξ+ Ψ (

B(y) · Dξ− Ψ (

DξT Φ(

y∈S+

(7.79)

y∈S−

By Lemma 7.8.4, the contribution from y = (y0 , . . . , yk ) is 0, if k ≥ 2 holds. In the case k = 1, the contributions do not depend on Tκξ . So, we can omit to denote them in the following argument. We set S1 := (y0 , y1 ) ∈ T ype2 y0 + y1 = y, a0 /r0 − a1 /r1 = A · ξ1 (A < 0) , S2 := (y0 , y1 ) ∈ T ype2 y0 + y1 = y, a0 /r0 − a1 /r1 = A · ξ1 (A > 0) S3 := (y0 , y1 ) ∈ T ype2 y0 + y1 = y, a0 /r0 = a1 /r1 , b0 < b1 Then, we obtain the following equality from (7.79) and Lemma 7.8.5: y) + 0 = DξT Φ(

(y0 ,y1 )∈S1

−

(y0 ,y1 )∈S2

r0 Dξ Ψ (

y0 , y 1 ) + r r0 Dξ Ψ (

y0 , y 1 ) − r

(y0 ,y1 )∈S3

(y0 ,y1 )∈S3

r0 Dξ Ψ (

y0 , y 1 ) r r0 Dξ Ψ (

y0 , y 1 ) r

The contributions from S3 are cancelled out. We have the bijection S1 −→ S2 given by (y0 , y1 ) −→ (y1 , y0 ). We also have Dξ Ψ (

y0 , y 1 ) = −Dξ Ψ (

y1 , y 0 ) for (y0 , y1 ) ∈ S1 and (y1 , y0 ) ∈ S2 . Therefore, we obtain the following equality: DξT Φ(

y) = Dξ Ψ (

y0 , y 1 ) (7.80) (y0 ,y1 )∈S2

By using (7.80) and Lemma 7.8.5 inductively, we can obtain the formula (7.72). Thus, the claim Q(r) is proved, and the proof of Theorem 7.8.2 is finished.

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Glossary

This is not a complete list of the symbols. In principle, we try to extract some of them which are used in different sections.

Subsection 2.1.1 gY : the morphism T ×S Y −→ U ×S Y induced by g : T −→ U pX : the projection X ×S U −→ U , forgetting the X-component

Subsection 2.1.2 Hom(V1 , V2 ), N (V1 , V2 ): For given vector bundles Vi (i = 1, 2), Hom(V1 , V2 ) denotes the sheaf of homomorphisms from V1 to V2 , and N (V1 , V2 ) denotes the corresponding vector bundle F ∗ , F ∨ , P(F ∨ ), PF : Let F be a vector bundle on Y . The complement of the image of the 0-section in F is denoted by F ∗ , and the dual bundle of F is denoted by F ∨ . The projectivization is denoted by P(F ∨ ) or PF .

Subsection 2.1.3 U -coherent sheaf: a coherent sheaf over U × X, which is flat over U . We will often omit to denote “U -”, if there are no risk of confusion. U -torsion-free sheaf: a U -coherent sheaf E such that E|{u}×X is torsion-free for each u ∈ U . We will often omit to denote “U -”, if there are no risk of confusion. E(m): E ⊗ p∗U OX (m) on U × X for a given polarization OX (1)

347

348

Glossary

Subsection 2.1.4 ZG , Z/G: quotient stack of Z with an action of G

Subsection 2.1.5 Hom(C • , D • ), C • ∨ , Hom(C • , D • )∨ : complexes induced by C • and D•

Subsection 2.1.6 C1 (V1∗ , V2∗ ), C2 (V1∗ , V2∗ ): complexes induced by filtered vector bundles Vi∗ (i = 1, 2). RHom1 (E1∗ , E2∗ ), RHom2 (E1∗ , E2∗ ): complexes (objects in the derived category) induced by filtered sheaves Ei∗ (i = 1, 2)

Subsection 2.2.1 Y s (L), Y ss (L): set of (semi)stable points with respect to a G-polarization L Y ss //G: categorical quotient

Subsection 2.2.2 μλ (P, L): weight of the action of λ on the fiber of L over limt→0 λ(t) · P Gl (V ): Grassmann variety of l-dimensional subspaces of a vector space V Gl (V ): Grassmann variety of l-dimensional quotients of a vector space V Subsection 2.3.1 LX /Y , Lf : cotangent complex of f : X −→ Y o(h): obstruction class ΩX /Y , Ωf : sheaf of Kahler differentials of f : X −→ Y Subsection 2.3.2 ΘY /S : the relative tangent bundle of a smooth morphism Y −→ S Y (W• ): quotient stack N (W−1 , W0 )GL(W−1 )×GL(W0 )

Glossary

349

Subsection 2.3.3 ∨ k(E• , V•,• , φ): cone of Hom(OU1 ×X , V1• )∨ −→ Hom OU1 ×X , V0• r(E• , V•,• , φ): induced morphism k(E• , V•,• , φ) −→ LU0 ×X/U1 ×X Ob(E• , V•,• , φ): pX∗ k(E• , V•,• , φ) ⊗ ωX ob(E• , V•,• , φ): induced morphism Ob(E• , V•,• , φ) −→ LU0 /U1 k(V, ϕ), r(V, ϕ): We set k(V, ϕ) := k(E• , V•,• , φ) and r(V, ϕ) := r(E• , V•,• , φ) in the derived category. Ob(F , ϕ), ob(F , ϕ): We set Ob(F, ϕ) := Ob(E• , V•,• , φ) and ob(F, ϕ) := ob(E• , V•,• , φ) in the derived category.

Second half of Subsection 2.3.3 k(E• , ϕ), r(E• , ϕ): We set k(E• , ϕ) := Hom(OU0 ×D , V•|U0 ×D )∨ , and let r(E• , ϕ) : k(E• , ϕ) −→ LU0 ×D/U1 ×D be the induced morphism. Ob(F , ϕ), ob(F , ϕ): We set RpD ∗ k(F, ϕ) ⊗ ωD , and let ob(F, ϕ) be the induced morphism Ob(F, ϕ) −→ LU0 /U1 .

Subsection 2.4.1 A∗ (X ): Chow group of an algebraic stack X with rational coefficient [X , φ], [X ]: virtual fundamental class associated to an obstruction theory φ. If there are no risk of confusion, we use [X ] instead of [X , φ].

Subsection 2.4.2 g(f ), Ob(f ), ob(f ): Let f : p∗U F −→ p∗U V on U × X. Then, g(f ) := p∗U Hom(V, F ) and Ob(f ) := RpX∗ g(f ) ⊗ ωX . And, ob(f ) denotes the induced morphism Ob(f ) −→ LU .

Subsection 2.4.3 ∗ g(F, f ), Ob(F, f ), ob(F, f ): For a given f : F −→ pUV , we set g(F,f ) := Hom(p∗U V, F ) −→ Hom(F, F ) and Ob(F, f ) := RpX∗ g(F, f ) ⊗ ωX . And ob(F, f ) denotes the induced morphism Ob(F, f ) −→ LU .

350

Glossary

Subsection 2.4.4 g(V ∗ ), Ob(V ∗ ), ob(V ∗ ): Let V, F be vector bundles on D/S. Let g : D ×S T −→ D be a morphism induced by g : T −→ S. For a filtration V = V (1) ⊃ V (2) ⊃ · · · ⊃ V (l) ⊃ V (l+1) ⊃ g∗ F, we set g(V ∗ ) := C2 (V ∗ , V ∗ )∨ [−1] and Ob(V ∗ ) := RpX∗ g(V ∗ ) ⊗ ωX . And ob(V ∗ ) denotes the induced morphism Ob(V ∗ ) −→ LT /S .

Subsections 2.6.1, 2.6.2 fj , y(j), x(i, j): vectors in U =

N

Q · ei N (1)

i=1

y (2) (j), x1 (j), x2 (j): vectors in U =

i=1

(1)

Q · ei

⊕

N (2) i=1

(2)

Q · ei

Subsection 3.1.1 PicX : Picard variety PoinX : Poincare bundle detE : morphism U −→ PoinX induced by the determinant bundle of a U coherent sheaf E on U × X Or(E): orientation bundle

Subsection 3.1.2 F∗ (E): quasi-parabolic structure Gri (E): Fi (E)/Fi+1 (E) Coki (E): E/Fi+1 (E)

Subsection 3.1.3 [φ]: reduced L-section L = (L1 , L2 ): pair of line bundles [φ] = ([φ1 ], [φ2 ]): pair of reduced L-sections

Glossary

351

Subsection 3.1.4 H ev (X): the even part of H ∗ (X) T ype: set of types T ype◦ : set of types whose parabolic part is trivial T yper : set of types whose ranks are r T ype◦r : T ype◦ ∩ T yper type(E, F∗ ): type of (E, F∗ ) depth(y): depth of y, max{i | yi = 0} rank(y): the H 0 (X)-component of y χ(y): X Td(X) · y M(y): moduli stack of quasi-parabolic sheaves of type y M(

y ): moduli stack of oriented quasi-parabolic sheaves of type y M y, L : moduli stack of quasi-parabolic L-Bradlow pairs of type y whose Lsections are non-trivial everywhere

, L : moduli stack of oriented quasi-parabolic L-Bradlow pairs of type y M y whose L-sections are non-trivial everywhere M y, [L] : moduli stack of quasi-parabolic reduced L-Bradlow pairs of type y

, [L] : moduli stack of oriented quasi-parabolic reduced L-Bradlow pairs of M y type y

, [L] : moduli stack of oriented quasi-parabolic reduced L-Bradlow pairs of M y type y M(m, y): substack of M(y) determined by the condition Om

): substack of M(

M(m, y y ) determined by the condition Om

): substack of M(

M(m, y y ) determined by the condition Om M(m, y, L): substack of M(y, L) determined by the condition Om

, L) substack of M(

M(m, y y , L) determined by the condition Om M(m, y, [L]): substack of M(y, [L]) determined by the condition Om

, [L]): substack of M(y, [L]) determined by the condition Om M(m, y

, [L]): substack of M(

M(m, y y , [L]) determined by the condition Om

u : universal sheaf in the oriented case E E u : universal sheaf in the unoriented case

352

Glossary

Subsection 3.1.5 Orel (1): relative tautological line bundle on M(

y , [L]). Pull backs are also denoted by the same notation.

Subsection 3.2.1 HE : Hilbert polynomial of E PE : reduced Hilbert polynomial h0 (E) dimension of H 0 (X, E)

Subsection 3.2.2 i : We set i := αi+1 − αi for a given system of weights α∗ . HE∗ : Hilbert polynomial of a parabolic sheaf E∗ PE∗ : reduced Hilbert polynomial of E∗ par-deg(E∗ ): degree of E∗ μ(E∗ ): slope of E∗

h0 (E∗ ): We set h0 (E∗ ) := α1 h0 E(−D) + i h0 Fi+1 (E) .

Subsection 3.2.3 P br : set of polynomials δ with R-coefficients such that (i) deg(δ) ≤ dim X − 1, (ii) δ(t) > 0 for any sufficiently large t. δtop : For any δ ∈ P br , the coefficient of td−1 in δ is denoted by δtop , which may be 0. H δ (E∗ , φ): δ-Hilbert polynomial of (E∗ , φ) δ : reduced δ-Hilbert polynomial of (E∗ , φ) P(E ∗ ,φ)

μδ (E∗ , φ): slope of (E∗ , φ) with δ δ H(E : δ-Hilbert polynomial of (E∗ , φ) ∗ ,φ) δ P(E : reduced δ-Hilbert polynomial of (E∗ , φ) ∗ ,φ)

Glossary

353

Subsection 3.2.4 Hy (t), Hy,i (t), Hy (t): polynomials associated to y. Hyα∗ : Hilbert polynomial associated to (y, α∗ ) Pyα∗ : reduced Hilbert polynomial associated to (y, α∗ ) deg(y, α∗ ): degree associated to (y, α∗ ) μ(y, α∗ ): slope associated to (y, α∗ ) deg(y): degree associated to y μ(y): slope associated to y Hyα∗ ,δ : δ-Hilbert polynomial associated to (y, α∗ ) Pyα∗ ,δ : reduced δ-Hilbert polynomial associated to (y, α∗ ) Subsection 3.3.1 Ms (y, α∗ ): moduli stack of stable parabolic sheaves of type y with weight α∗ Ms (

y , α∗ ): moduli stack of stable oriented parabolic sheaves of type y with weight α∗ Ms (y, L, α∗ , δ): moduli stack of δ-stable L-Bradlow pairs of type y with weight α∗ , whose L-sections are non-trivial everywhere Ms (

y , L, α∗ , δ): moduli stack of δ-stable oriented L-Bradlow pairs of type y with weight α∗ , whose L-sections are non-trivial everywhere Ms (y, [L], α∗ , δ): moduli stack of δ-stable reduced L-Bradlow pairs of type y with weight α∗ Ms (

y , [L], α∗ , δ): moduli stack of δ-stable oriented reduced L-Bradlow pairs of type y with weight α∗ Ms (

y , [L], α∗ , δ): moduli stack of δ-stable oriented reduced L-Bradlow pairs of type y with weight α∗ We replace the superscript “s” with “ss” to denote moduli stacks of semistable objects. s y , α∗ ): full flag bundle over Ms (

s (

y , α∗ ) associated to the M m y , α∗ ), M (

u

vector bundle pX∗ E (m) s y , [L], α∗ , δ): full flag bundle on Ms (

s (

y , [L], α∗ , δ) M m y , [L], α∗, δ), M (

u

(m) associated to pX∗ E

354

Glossary

s y , [L], α∗ , δ): full flag bundle on Ms (

s (

y , [L], α∗ , δ) M m y , [L], α∗, δ), M (

u

(m) associated to pX∗ E

Subsection 3.3.3. ss y, [L], α∗ , (δ, ) : moduli stack of (δ, )ss y, [L], α∗ , (δ, ) , M M m semistable (E, [φ], F) ss y ss y

, [L], α∗ , (δ, ) : moduli stack of (δ, )M m , [L], α∗ , (δ, ) , M semistable (E, ρ, [φ], F)

Subsection 3.4.1 μmax (E), μmin (E): We denote the slope of the first (resp. last) term of the Harder-Narasimhan filtration by μmax (E) (resp. μmin (E)).

Subsection 3.4.2 SS(y, L, α∗ , δ (0) ): a family of parabolic L-Bradlow pairs with type y, weight α∗ , and non-vanishing L-section which are δ-μ-semistable for some δ ≤ δ (0) in P br SS y, L, α∗ : δ∈P br SS y, L, α∗ , δ SS(y, L, α∗ , δ (0) ): a family of parabolic L-Bradlow pairs with a similar property

Subsection 3.4.3 YOK(m, K, y, L, δ): See Definition 3.4.10 YOK(N, K, y, L, δ): m≥N YOK(m, K, y, L, δ) YOK(N, K, y, L, δ): Yokogawa family

Subsections 3.5.2–3.5.3 Cr(y, α∗ , L): set of critical values for (y, α∗ , L) br 2 Cr(y, α∗ , L): set of δ ∈ (P ) such that the 1-stability conditions do not hold for y, α∗ , L, δ) br Cr(y, of δ2 ∈ P such that the 1-stability conditions do not hold α∗ , L, δ1 ): set for y, α∗ , L, (δ1 , δ2 )

Glossary

355

Subsection 3.6.1 y(m): y(m) := y · ch O(m) det(y): the H 2 (X)-component of y Subsection 3.6.2 Vm,X : vector bundle Vm ⊗ OX Q(m, y): moduli scheme of quotient sheaves of Vm,X with type y E u : universal quotient sheaf Q◦ (m, y): moduli of quotient sheaves of Vm,X with type y satisfying (TFV)condition

moduli of quotient oriented sheaves of Vm,X with type y Q(m, y): Q◦ (m, y ): moduli of quotient oriented sheaves of Vm,X with type y satisfying (TFV)-condition Subsection 3.6.3 Q◦ (m, y): moduli of quotient parabolic sheaves of Vm,X with type y satisfying (TFV)-condition

m , Zm : See (3.12). Z Zm : Gieseker space Gm,i : Grassmann variety of Hy,i (m)-dimensional quotients of Vm Subsections 3.6.4, 3.6.5, 3.6.6 Q◦ (m, y, L): moduli of quotient quasi-parabolic L-Bradlow pairs of Vm,X with type y satisfying (TFV)-condition Q◦ (m, y, [L]): moduli of quotient quasi-parabolic reduced L-Bradlow pairs of Vm,X with type y satisfying (TFV)-condition Q◦ (m, y, [L]): moduli of quotient quasi-parabolic reduced L-Bradlow pairs

Subsection 3.6.7

, [L]): moduli of quotient quasi-parabolic oriented reduced L-Bradlow Q◦ (m, y pairs of Vm,X with type y satisfying (TFV)-condition.

), Q◦ (m, y

, L), Q◦ (m, y

, [L]), etc., in similar meanings. We use the symbols Q◦ (m, y

356

Glossary

Subsection 4.1.1 Am (y): Am (y) := Zm ×

. i

Gm,i

Am (y, [L]): Am (y, [L]) := Am (y) × Pm (1)

(2)

Am (y, [L]): Am (y, [L]) := Am (y) × Pm × Pm

Ly (A, B∗ ), Ly,L (A, B∗ , C), Ly,L (A, B∗ , C1 , C2 ): We set Ly (A, B∗ ) := OZm (A) ⊗

l

OGm,i (Bi ),

on Am (y)

i=1

Ly,L (A, B∗ , C) := Ly (A, B∗ ) ⊗ OPm (C)

on Am (y, L),

Ly,L (A, B∗ , C1 , C2 ) := Ly (A, B∗ ) ⊗ OP(1) (C1 ) ⊗ OP(2) (C2 ) m

m

on Am (y, L).

ss ss Ass m (y, A, B∗ ), Am (y, L, A, B∗ , C), Am (y, L, A, B∗ , C1 , C2 ): ss Let Am (y, A, B∗ ) denote the set of semistable points of Am (y) with respect to Ly (A, B∗ ). ss We use the symbols Ass m (y, L, A, B∗ , C) and Am (y, L, A, B∗ , C1 , C2 ) in similar ways.

Qss (m, y, α∗ ), Qs (m, y, α∗ ): maximal open subset of Q◦ (m, y) which consists of the points (q, E, F∗ ) such that the parabolic sheaf (E(−m), F∗ , α∗ ) is (semi)stable We use the following symbols in similar ways: Qss (m, y, [L], α∗ , δ), Qs (m, y, [L], α∗ , δ), Qss (m, y, [L], α∗ , δ),

Qs (m, y, [L], α∗ , δ).

Subsection 4.1.2 V /W∗ : Vm /Wi | i = 1, . . . , l : See (4.1). Ψ q, E∗ , φ, W∗ , [φ]

Subsection 4.2.1 V , Q, L, A: We set V := Vm , Q := Qss (m, y, [L], α∗ , δ) L := Ly,L Pyδ,α∗ (m), ∗ , δ(m) , A := Am (y, [L])

Glossary

357

Lγ : Take a sufficiently large k such that L⊗ k is an actual line bundle, and we set Lγ := L⊗ k ⊗ OPm (γ). (Note that k is not the ground field.) Flag(V, N ): full flag variety of V A: We set Q := Q × Flag(V, N ) and A := A × Flag(V, N ). Q, m : induced morphism Q −→ A Ψ 1N L(γ, n∗ ): Lγ ⊗ i=1 ρ∗i OGi (V ) (ni ). s (γ, n∗ ): set of (semi)stable points with respect to L(γ, ss (γ, n∗ ), A n∗ ) A

Subsections 4.2.2, 4.2.3 ss m, y, [L], α∗ , δ± Qss ±: Q

ss (δ, ): maximal subset of Q, which consists of the points q, E∗ , [φ], F such Q that (E∗ , [φ], F) is (δ, )-semistable

Subsection 4.3.1 2 : Let k be a number such that k · (γ1 − γ2 ) = 1, and we set 1 , L L 1 , n∗ )⊗ k , L1 := L(γ

2 , n∗ )⊗ k . L2 := L(γ

B1 , B2 : Let π1 : A −→ Pm denote the projection, and we set B, B := P π1∗ OPm (0) ⊕ π1∗ OPm (1) . We set B1 := P π ∗ OP (0) and B2 := P π ∗ OP (1) . That is a P1 -bundle over A. 1 1 m m OB (1), Bss , Bs : We set OB (1) := OP (1) ⊗ L1 . Let Bss (resp. Bs ) denote the set of the semistable (resp. stable) points with respect to OB (1). THi := Bi × Q and TH, TH1 , TH2 , TH∗ , THss : We put TH := B ×A Q, A ∗ ss ss TH := TH − TH1 ∪ TH2 . We also put TH := B ×A Q. Subsection 4.4.1 ρ, ρ: naturally defined torus actions. (Please do not confuse with orientations.) I = (y 1 , y 2 , I1 , I2 ) decomposition data. See Definition 4.4.2. Dec(m, y, α∗ , δ): set of decomposition data for (m, y, α∗ , δ)

358

Glossary

Subsection 4.4.2 Q(1) , Q(2) : Q(1) := Qss m, y 1 , [L], α∗ , δ and Q(2) := Qss (m, y 2 , α∗ ) Flag(V (i) , Ii ), Flag(i) : For i = 1, 2, we set @ ? filtration indexed by N , (i) (i) . Flag(i) = Flag(V (i) , Ii ) := F∗ dim GrF = 1 (j ∈ Ii ), or = 0 (j ∈ Ii ) j split (I): We set Q (i) , Q (i) := Q(1) × Flag(i) and Q split (I) := Q (1) × Q (2) . Q THsplit (I), TH∗split (I), THi,split (I): We set split (I), THsplit (I) := TH ×Q Q

split (I) TH∗split (I) := TH∗ ×Q Q

split (I). THi,split (I) := THi ×Q Q Subsection 4.5.1

∗ ss

Q, Q, TH, TH , TH : We set

:= Qss (

y , [L], α∗ , δ), Q

:= TH × Q, TH Q

∗

× Flag(V, N ), Q := Q

:= TH∗ ×TH TH, TH

TH

ss

= THss ×TH TH.

: We set M

:= THss / GL(V ), that is the enhanced master space in the oriented M case.

M , F M , [φM ], ρM ): universal object induced on M

×X (E ∗

M

associated to pX∗ E (m) F M : full flag bundles on M

ρ: torus action on M

Subsection 4.5.2 ss

ss ss ss

i : We set TH ,M := THi ×TH TH

i := TH / GL(V ). and M TH i i i

Subsection 4.5.3

∗ : M

− M

1 ∪ M

2 M (1) (δ, k): maximal open subset of Q (1) determined by the (δ, k)-semistability Q

Glossary

359

(2) : maximal open subset of Q (2) determined by the condition that the underlyQ + ing reduced O(−m)-Bradlow pair is -stable for any sufficiently small > 0.

(1) (δ, k) × Q (2) and Q over Q Q split (I): fiber product of Q + ss ∗

TH split (I): TH × Qsplit (I) Q

ss

Gm (I): quotient stack of TH split (I) by a natural GL(V (1) )×GL(V (2) )-action M

Gm (I) −→ M

ϕI : naturally defined morphism M Subsection 4.5.5

ss B : Z m ×Zm Bss

ss B : quotient stack of B by a natural GL(Vm )-action

in B

B: Gm -equivariant open neighbourhood of M Subsection 4.5.6

ss

ss

ss

ss Bi , Bi : We set Bi := Bi ×B B . The quotient stack Bi / GL(V ) is denoted by Bi . Q: decomposition type for A

m A: A ×Zm Z

C1 (Q): locally closed regular subvariety of A associated to Q C2 (Q), C3 (Q): We set C2 (Q) := B∗ ×A C1 (Q),

C3 (Q) := C2 (Q)/ GL(V ) ∩ B.

Subsection 4.5.7 m BG 0 :

A I∈S(m,y)

C3 Q(I)

ψ1 : naturally defined morphism

A I∈S(m,y)

×B BGm

Gm (I) −→ M M 0

Subsection 4.6.1 (1) ss y

1 , [L], α∗ , (δ, k) : moduli of (δ, k)-semistable (E∗ , [φ(1) ], ρ(1) , F (1) ) M with type y 1

360

Glossary

u , φu , ρu , F u : universal object E 1∗ 1 1 1 ss y , L, α∗ , (δ, k) : moduli of (δ, k)-stable (E∗(1) , φ(1) , F (1) ) with type y M 1 1 u (E1u∗ , φu 1 , F1 ): universal object (2) ss y

2 , α∗ , + : moduli of (E∗ , ρ(2) , F (2) ) such that the associated reduced M (2) (2) OX (−m)-Bradlow pair E∗ , Fmin is -stable for any sufficiently small > 0

u , ρu , F u ): universal object (E 2 2 2 algebraic stack S and line bundle Orel (1/r2 ): See Proposition 4.6.1.

Subsection 4.7.1

Li : Take k such that k · (γ2 − γ1 ) = 1, and we put Li := Lkγi . 1 B, Bi : We set B := P OPm (0)⊕ OPm (1) , which is a P -bundle over A. We put B1 = P OPm (0) and B2 = P OPm (1) , which are naturally regarded as closed subschemes of B. OB (1), Bss : We put OB (1) := OP (1) ⊗ L1 , where OP (1) is the tautological line bundle. Let B ss denote the set of the semistable points of B with respect to OB (1). ss

ss

, TH : We set THss := Q ×A B ss and TH THss , TH i ss ss also put THi := Bi ×A TH . ss

ss

We := THss ×Q Q.

ss

∗ : We put M

, M

i , M

:= TH / GL(V ), M

i := TH i / GL(V ) and M

∗ := M

− M

1 ∪ M

2 . M I: decomposition type S(y, α∗ , δ): set of decomposition types for (y, α∗ , δ)

Gm (I): moduli of tuples (E∗(1) , φ), E∗(2) , ρ , where (i) (E∗(1) , φ) is δ-stable M (2) L-Bradlow pair of type y 1 , (ii) E∗ is stable of type y 2 , (iii) ρ is an orientation of (1) (2) E ⊕E

M

×X (E , [φM ], ρM ): universal object on M

S, Orel (1/r2 ): See Proposition 4.7.4 for the case in which a 2-stability condition is satisfied.

Subsection 4.7.2 A, L, Lγ : We set A := Am (y, [L]), L := Ly,L Pyα∗ ,δ (m), ∗ , δ(m) and Lγ := L ⊗ OP(1) (1) (γ) m

Glossary

361

Ass (Lγ ): open subset of semistable points with respect to Lγ ss ss ss Q, Qss ± : We set Q := Q (m, y, [L], α∗ , δ) and Q± := Q (m, y, [L], α∗ , δ ± ).

Q(

Q: y , [L], α∗ , δ) Li : line bundle L⊗k γi

B, Bi : We put B := P OP(1) (0) ⊕ OP(1) (1) which is a P1 -bundle over A. We m m put B1 = P OP(1) (0) and B2 = P OP(1) (1) , which are naturally regarded as the m m closed subschemes of B. OB (1), Bss : We put OB (1) := OP (1) ⊗ L1 , where OP (1) is the tautological line bundle. Let B ss denote the set of the semistable points of B with respect to OB (1). ss

ss

, TH : We put THss := Q ×A B ss , TH THss , TH i ss ss := Bi ×A TH . TH i

ss

and := THss ×Q Q,

ss ss

∗ : We define M

, M

i , M

:= TH / GL(V ), M

i := TH / GL(V ) and M

∗ := M i

M − M1 ∪ M2 .

I: decomposition type S(y, α∗ , δ): set of decomposition types

Gm (I): moduli stack of tuples E∗(1) , φ1 , E∗(2) , [φ2 ], ρ such that (i) (E∗(1) , φ1 ) M (2) is a δ1 -semistable L1 -Bradlow pair of type y 1 , (ii) (E∗ , [φ2 ]) is a δ2 -semistable reduced L2 -Bradlow pair of type y 2 , (iii) ρ is an orientation of E (1) ⊕ E (2)

M M

M

(E ∗ , [φ1 ], [φ2 ], ρ) universal object on M × X (i)

, [L]) which are the pull back of the tautoOrel (1): the line bundles on M(m, y

, [L]) −→

, [Li ]) via the natural morphism M(m, y logical line bundles on M(m, y

, [Li ]) M(m, y S, Oi,rel (1), Oi,rel (1/r2 ): See Proposition 4.7.9 for the case of oriented reduced L-Bradlow pairs.

Subsection 5.1.1 ∨ g(V• ), Ob(V• ), ob(V • ): For V• on U × X, we set g(V• ) := Hom(V• , V• ) [−1] and Ob(V• ) := RpX∗ g(V• ) ⊗ ωX . Let ob(V• ) denote the naturally defined morphism Ob(V• ) −→ LU .

W• , GL(W• ), Wi X : Let Wi (i = −1, 0) be vector spaces with dim Wi = rank Vi . We set GL(W• ) := GL(W−1 ) × GL(W0 ) and Wi X := Wi ⊗ OX .

362

Glossary

Y (W• ), Φ(V• ): Let Y (W• ) denote the quotient stack of N (W−1 X , W0 X ) by a natural action of GL(W• ). The classifying map U × X −→ Y (W• ) is denoted by Φ(V• ). Φ(E), Ob(E), ob(E): If E is a vector bundle of rank R, the classifying map U × X −→ XGL(R) is denoted by Φ(E). We set Ob(E) := RpX∗ Hom(E, E) ⊗ ωX . The naturally induced morphism Ob(E) −→ LU is denoted by ob(E).

Subsection 5.1.2 w: naturally induced morphism Y (W• ) −→ XGm Φ(det(E)): classifying map U × X −→ XGm of det(E) i : O[−1] −→ g(V• ), tr : g(V• ) −→ O[−1]: naturally defined maps g◦ (V• ): trace-free part Ker(tr) gd (V• ): diagonal part Im(i)

◦ d ◦ d Ob◦ (V •d), Ob (V• ): We set Ob (V• ) := RpX∗ g (V• ) ⊗ ωX and Ob (V• ) := RpX∗ g (V• ) ⊗ ωX .

Subsection 5.1.3 A(W• ), B(W• ): We set A(W• ) := XGL(W0 ) and B(W• ) := Spec(k)GL(W0 ) . Γ , Ψ (V• ): The natural morphism Y (W• ) −→ A(W• ) is denoted by Γ . We set Ψ (V• ) := Γ ◦ Φ(V• ). h(V• ): Hom(V0 , V• )∨ [−1] ObG (V• ): RpX∗ h(V• ) ⊗ ωX

Subsection 5.1.4 H: Hilbert polynomial Hy associated to a type y E u : universal sheaf on M(m, y) × X V• : canonical resolution of E u (m) V : pX∗ V0 Ob(m, y), ob(m, y): Ob(m, y) := Ob(V• ) and ob(m, y) := ob(V• )

Glossary

363

Subsection 5.1.5 g(Poin): Φ(Poin)∗ LXGm /X Ob(Poin): RpX∗ g(Poin) ⊗ ωX

Subsection 5.2.1 Obrel (V• , ρ): cone of the natural morphism Obd (V• ) −→ det∗E LPic obrel (V• , ρ): naturally defined morphism Obrel (V• , ρ) −→ LU1 /U

Subsection 5.2.2 Obrel (V• , ρu ), obrel (V• , ρu ): Let π denote the projection Or(E)∗ −→ U . A complex Obrel (V• , ρu ) is induced by the universal orientation ρu of π ∗ E with a morphism obrel (V• , ρu ) : Obrel (V• , ρu ) −→ LOr(E)∗ /U . Subsection 5.3.1 P• L: a resolution by locally free sheaves Y0 (W• , P• ): quotient stack of N (P−1 , W0 X ) by a natural action of GL(W• ) Y1 (W• , P• ): quotient stack of X by a natural action of GL(W• ) Y2 (W• , P• ): quotient stack of N (W−1 X , W0 X ) ×X N (P0 , W0 X ) ×X N (P−1 , W−1 X ) by a natural action of GL(W• ) • ): Hom p∗ P• , F ∗ V• ∨ grel (V• , φ U2 2,X • ): naturally defined morphism grel (V• , φ)[−1] γ(φ −→ g(V• ) • ): cone of γ(φ• ) g(V• , φ • ): induced morphism U2 × X −→ Y (W• , P• ) Φ(V• , φ • ): induced morphism U2 × X −→ Yi (W• , P• ) Φi (V• , φ • : RpX∗ grel (V• , φ• ) ⊗ ωX Obrel V• , φ

364

Glossary

Subsection 5.3.3 A(W• , P• ): quotient stack of N (OX , W0 X ) by a natural GL(W0 )-action B(W• , P• ): quotient stack of W0 by a natural GL(W0 )-action. We have a natural isomorphism JL : B(W• , P• ) A(W• , P• ). ΓL : natural morphism Y (W• , P• ) −→ A(W• , P• ) • ): composite ΓL ◦ Φ(V• , φ• ) : U2 × X −→ A(W• , P• ) Ψ (V• , φ Ξ(V• , φ): induced map U2 −→ B(W• , P• ) • ): Hom p∗ OX , F ∗ V• ∨ hrel (V• , φ U2 2X γ(φ): naturally induced morphism hrel (V• , φ• )[−1] −→ h(V• ) • ): cone of γ(φ) h(V• , φ ObG rel (V• , φ• ): RpX∗ hrel (V• , φ• ) ⊗ ωX Subsection 5.3.5 Z(V• , L): quotient stack of N (p∗U L, V0 ) by a natural action of N (p∗U L, V−1 ) grel (V• , φ): Hom(L, V• )∨

Obrel (V• , φ), obrel (V• , φ): We set Obrel (V• , φ) := RpX∗ grel (V• , φ) ⊗ ωX . A naturally induced morphism Obrel (V• , φ) −→ LU2 /U is denoted by obrel (V• , φ).

Subsection 5.4.1 gM : Let M be a line bundle on U3 . Let ΦM : U3 −→ Spec(k)Gm denote the classifying map for M . A morphism g : U3 −→ T and ΦM induce the morphism U3 −→ T × Spec(k)Gm , which is denoted by gM . Y (W• , [P• ]): quotient stack Y (W• , P• ) by Gm Yi (W• , [P• ]) (i = 0, 1, 2): quotient stack Yi (W• , P• ) by Gm • ]): Hom p∗ P• ⊗ p∗ M, F ∗ V• ∨ grel (V• , [φ U3 X 3X • ] : induced morphism U3 × X −→ Y (W• , [P• ]) Φ V• , [φ (U × X)Gm , Y (W• )Gm , XGm : quotient stacks by the trivial Gm -actions • ]): naturally induced morphism g (V• , [φ• ])[−1] −→ g(V• ) γ([φ rel • ]): cone of γ([φ• ]) g (V• , [φ • ] : RpX∗ g (V• , [φ• ]) ⊗ ωX Obrel V• , [φ rel

Glossary

365

• ]), obrel (V• , [φ • ]): We set Obrel (V• , [φ ∗ LUGm /U [1] [−1]. Obrel (V• , [φ• ]) := Cone Ob rel (V• , [φ• ]) −→ F3,M The induced morphism Obrel (V• , [φ• ]) −→ LU3 /U is denoted by obrel (V• , [φ• ]).

Subsection 5.4.3 A(W• , [P• ]): quotient stack of A(W• , P• ) by a natural Gm -action B(W• , [P• ]): quotient stack of B(W• , P• ) by a natural Gm -action. We have a natural isomorphism J[L] : B(W• , [P• ]) × X A(W• , [P• ]). • ]): Hom p∗ OX ⊗ p∗ M, F ∗ V• ∨ hrel (V• , [φ U3 X 3X G Obrel (V• , [φ]): RpX∗ h rel (V• , [φ• ]) ⊗ ωX ObG rel (V• , [φ]): cone of the induced morphism G ∗ (V• , [φ• ]) −→ F3,M LUGm /U [1] Ob rel

Γ[L] : natural morphism Y (W• , [P• ]) −→ A(W• , [P• ]) • ]) composite Γ[L] ◦ Φ(V• , [φ• ]) : U3 × X −→ A(W• , [P• ]). Ψ (V• , [φ Ξ(V• , [φ]): induced morphism U3 −→ B(W• , [P• ])

Subsection 5.5.1 VD∗ : filtered vector bundle VD ⊃ VD ⊃ · · · ⊃ VD ∗ E parabolic structure of F4X (1)

(2)

(l+1)

on U4 × D, induced by a

vector spaces W (h) : We set W (1) := W0 and W (l+1) := W−1 . We take vector (h) spaces W (h) (h = 2, . . . , l) such that dim W (h) = rank VD . (h)

(h)

Wi D , WD : We set Wi D := Wi ⊗ OD and WD := W (h) ⊗ OD . YD (W• ): quotient stack of N (W−1 D , W0,D ) by a natural action of GL(W• ) .l (h+1) (h) , WD ) by a natural action of YD (W• , W ∗ ): quotient stack of h=1 N (WD .l+1 (h) ) h=1 GL(W ΦD (V• , F∗ ): induced morphism U4 × D −→ YD (W• , W ∗ ) Φ(V•|D ): induced morphism U4 × D −→ YD (W• ) gD (V• , F∗ ): C1 (VD∗ , VD∗ )∨ [−1]

366

Glossary

grel (V• , F∗ ): C2 (VD∗ , VD∗ )∨ [−1] g(V•|D ): Hom(V•|D , V•|D )∨ [−1]

Obrel (V• , F∗ ), obrel (V• , F∗ ): We set Obrel (V• , F∗ ) := RpD∗ grel (V• , F∗ ) ⊗ ωD . An induced morphism Obrel (V• , F∗ ) −→ LU4 /U is denoted by obrel (V• , F∗ ). Ob(V• | D ), ob(V•|D ): We set Ob(V• | D ) := RpD∗ g(V•|D ) ⊗ ωD . The induced morphism Ob(V• | D ) −→ F4∗ LU is denoted by ob(V• | D ). Subsection 5.6.1 V• : canonical resolution of E u (m) on M(m, y) × X P• : canonical resolution of L(m) for a sufficiently large m Obrel (m, y), obrel (m, y): From V• , we obtain obrel (m, y) : Obrel (m, y) −→ LM(m,y)/M(m,y) . Obrel (m, y, L), obrel (m, y, L): From V• and P• , we obtain obrel (m, y, L) : Obrel (m, y, L) −→ LM(m,y,L)/M(m,y) . Obrel (m, y, [L]), obrel (m, y, [L]): From V• and P• , we obtain obrel (m, y, [L]) : Obrel (m, y, [L]) −→ LM(m,y,[L])/M(m,y) .

From V• , we obtain Obrel (m, y ), obrel (m, y): obrel (m, y ) : Obrel (m, y ) −→ LM(m, y)/M(m,y) . Subsection 5.6.2

, [L]), ob(m, y

, [L]): We obtain a morphism Ob(m, y

, [L]) : Ob(m, y

, [L]) −→ LM(m, y,[L]) ob(m, y from ob(m, y), obrel (m, y), obrel (m, [L]) and obrel (m, y ). Similarly, we obtain ob(m, y, L) : Ob(m, y, L) −→ LM(m,y,L)/k

) : Ob(m, y

) −→ LM(m, y)/k ob(m, y

, [L]) : Ob(m, y

, [L]) −→ LM(m, y,[L])/k ob(m, y

Glossary

367

Subsection 5.7.1 Pm : P(Vm∨ ) Z1 : P OPm (0) ⊕ OPm (1) that is a P1 -bundle over Pm Z2 : Z1 × Flag(Vm , N ) quotient stack of Z2 by a natural action of GL(Vm ) Q: W0 : We often discuss under the setting W0 = Vm . B ∗ (W• , [P• ]): (Pm )GL(Vm ) Ψ1 : Ξ(V• , [φ]) • ]): cone of ObG (V• , [φ• ])[−1] −→ ObG (V• ) ObG (V• , [φ rel

, [L]) ×B ∗ (W• ,[P• ]) Q N : M(m, y Ψ2 : induced morphism M −→ Q

−→ M(m, y

, [L]) p: projection M ∗

): cone of Ψ ∗ L ∗ Ob(M 2 Q/B (W• ,[P• ]) [−1] −→ p Ob(m, y, [L])

): induced morphism Ob(M

) −→ L ob(M M Subsection 5.7.2 ∗ : quotient stack of V ∗ × Flag(Vm , N ) by a natural GL(Vm )-action Q m B ∗ (W• , P• ): quotient stack of Vm∗ by a natural GL(Vm )-action • ): cone of ObG (V• , φ• )[−1] −→ ObG (V• ) ObG (V• , φ rel

∗ −→ Q ∗ Ψ4 : naturally induced morphism M ∗ −→ F . F : Flag(Vm , N )GL(Vm ) . We have a naturally defined morphism Γ1 : Q

∗ ): cone of Ψ ∗ L ∗

, L). It is isomorphic to Ob(M 4 Q/B (W• ,P• ) [−1] −→ Ob(m, y ∗ ∗ ∗

, L) the cone of Ψ4 Γ1 LF /B(W• ) [−1] −→ p2 Ob(m, y

∗ ): naturally induced morphism Ob(M

∗ ) −→ L ∗ ob(M M Subsection 5.7.3

u (m) over M(m, y

, [L]). We

, [L]): full flag bundle associated to pX∗ E M(m, y have a naturally defined morphisms

, [L]) −→ F , y Ψ11 : M(m,

, [L]) −→ M(m, y

, [L]) p1 : M(m, y

368

Glossary

∗

, [L])

, [L]): cone of Ψ11 LF /B(W• ) [−1] −→ p∗1 Ob(m, y Ob(m, y

, [L]): induced morphism Ob(m,

, [L]) −→ LM(m,

ob(m, y y y ,[L]) Q1 : quotient stack of Pm × Flag(V, N ) by a natural GL(Vm )-action. We have a naturally defined morphism

, [L]) −→ Q1 . y Ψ13 : M(m, ∗ 2 (m, y

, [L]): cone of Ψ13

, [L]) Ob LQ1 /B ∗ (W,[P ]) [−1] −→ p∗1 Ob(m, y

2 (m, y 2 (m, y

, [L]): naturally induced morphism Ob

, [L]) −→ LM(m,

ob y ,[L]) Subsection 5.7.4 M(m, y, L): full flag bundle associated to pX∗ E u (m) over M(m, y, L). We have naturally defined morphisms y, L) −→ F , Ψ11 : M(m,

p1 : M(m, y, L) −→ M(m, y, L)

∗ Ob(m, y, L): cone of Ψ11 LF /B(W• ) [−1] −→ p∗1 Ob(m, y, L)

ob(m, y, L): induced morphism Ob(m, y, L) −→ LM(m,y,L) Subsection 5.7.5

u (m) over M(m, y

). We have nat ): full flag bundle associated to pX∗ E M(m, y urally defined morphisms

) −→ M(m, y

), y p1 : M(m,

) −→ F Ψ11 : M(m, y

∗

): cone of Ψ11

) Ob(m, y LF /B(W• ) [−1] −→ p∗1 Ob(m, y

): naturally induced morphism Ob(m,

) −→ LM(m,

ob(m, y y y) Subsection 5.7.6 Q: quotient stack of P OPm (0) ⊕ OPm (1) by a natural GL(Vm )-action. We have

−→ Q. a natural morphism Ψ1 : M

): cone of Ψ ∗ LQ/B ∗ (W,[P ]) [−1] −→ p∗ Ob(m, y

, [L]) Ob(M 1

): naturally induced morphism Ob(M

) −→ L ob(M M

Glossary

369

∗ ), ob(M

∗ ): We set Ob(M

∗ ) := ob(m, y

, L). The naturally induced Ob(M ∗

) −→ L ∗ is denoted by ob(M

∗ ). morphism Ob(M M Subsection 5.7.7

), ob(M

): Let Ob(M

) be the cone of Ob(M

, [L]). Ψ1∗ LQ/B ∗ (W,[P ]) [−1] −→ Ob(m, y

) −→ L is denoted by ob(M

). The naturally induced morphism Ob(M M

∗ ), ob(M

∗ ): We set Ob(M

∗ ) := Ob(m, y

, L1 , [L2 ]). The naturally inOb(M ∗

) −→ L ∗ is denoted by ob(M

∗ ). duced morphism Ob(M M Subsection 5.8.1 split : M ss (y 1 , L, α∗ , (δ, k0 )) × M ss (

y 2 , α∗ , +) M

Gm (I)), ob(M

Gm (I)): obstruction Ob(M Proposition 5.8.1.

theory

of

Gm (I). M

See

M M

Gm (I)): obstruction theory of M

Gm (I)), ob(

Gm (I) × A1 over A1 . Ob( See Proposition 5.8.1. a (M a (M

Gm (I)), ob

Gm (I)): obstruction theory of M

Gm (I). See Ob Proposition 5.8.1.

Subsection 5.8.2 Mi (i = 1, 2, 3) We set M1 := M(m, y 1 , L), M2 := M(m, y 2 ) and M3 := M1 × M2 . They are equipped with the obstruction theories Ob(Mi ).

3 : moduli of (E1 , F1 ∗ , φ, E2 , F2 ∗ , ρ) such that (i) (E1 , F1 ∗ , φ) ∈ M1 , (ii) M (E2 , F2 ∗ ) ∈ M2 , (iii) ρ is an orientation of E1 ⊕ E2 . (i)

Eiu , V• : Let Eiu be universal sheaves on Mi × X (i = 1, 2). Let V (i) be the canonical resolution of Eiu .

3 /M3 ), Ob(M

3 ), ob(M

3 ): complex and morphisms induced by Obrel (M the universal orientation.

370

Glossary

Subsection 5.8.3

0 : We set M0 := M(m, y, L) and M

0 := M(m, y

, L). We have a M0 , M naturally defined morphism F : M3 −→ M0 . (i) (i)(j) (i) VD : Ker V|D −→ Cokj−1 . (a)

(1)

(2)

(1)

(2)

Wb : We take decompositions W0 = W0 ⊕ W0 and W−1 = W−1 ⊕ W−1 , (i) (i) where dim W0 = Hyi (m) and dim W−1 = Hyi (m) − rank(y i ). (1)

(2)

(1)

(2)

(1)

(2)

Y (W• , W• ): Y (W• ) × Y (W• ). (1)

(2)

Y (W• , W• , P• ): Y (W• , P• ) × Y (W• ). (i)(j)

W (i)(j) : vector spaces such that dim W (i)(j) = rank VD

YD (W• , W (1) ∗ , W (2) ): YD (W• , W (1)∗ ) × YD (W• , W (2)∗ ) (1)

(1)

(2)

(2)

(1)

(2)

YD (W• , W• ): YD (W• ) × YD (W• ) ObD (V, F∗u ): RpD ∗ gD (V• , F∗u ) ⊗ ωD (i) u(i) ObD (V (i) , F∗u(i) ): RpD ∗ gD (V• , F∗ ) ⊗ ωD Subsection 5.8.4 (1)

(2)

B(W• , W• ): Spec(k)GL(W0 )×GL(W−1 ) (1)

Φ(V0

(2)

, V0

(1)

(2)

): induced morphism M3 −→ B(W• , W• )

Subsection 5.8.5 Ob(m, y 2 ): Ob◦ (m, y 2 ) ⊕ τ≤−1 Obd (m, y 2 )

3 ): cone of the morphism of τ≤−1 Obd V•(1) ⊕ V•(2) to Ob(m, y 1 , L) ⊕ Ob1 (M Ob(m, y 2 ) M a (M a (M

3 ), Ob(

3 ): Let Ob

3 ) be the cone of ϕa . The family version is Ob

Ob(M3 ).

2 , S , F1 , G : See Lemma 5.8.6. M 1 Subsection 5.8.6 (i)

Fi , F i : Fi denotes the full flag of Vm , and F i denotes the quotient of Fi by a (i) natural GL(Vm )-action.

Glossary

371

Gm (I)): We have naturally defined morphisms: Ob(M

Gm (I) −→ F 1 × F 2 , g:M

Gm (I) −→ M

3 π1 : M

Gm (I)) denotes the cone of the morphism Then, Ob(M

3 ). g ∗ LF 1 ×F 2 /B(W (1) ,W (2) ) [−1] −→ π1∗ Ob(M •

•

(i) Flag(Vm , Ii ): full flag of V (i) indexed by Ii .

Subsections 5.8.7–5.8.8 M M

Gm ), Ob(M

Gm ), ob(

Gm ), Ob(

Gm ): Msplit , ob(M See Subsection 5.8.7 for the case in which a 2-stability condition is satisfied, and Subsection 5.8.8 for the case of oriented reduced L-Bradlow pairs.

Subsection 5.9.1

)inv , ob1 (M

i ): Gm -invariant of ι∗ Ob(M

), where ιi : M

i −→ M

. ι∗i Ob(M i ∗ inv

i ) : ι Ob(M

) −→ L . We have the induced morphism ob1 (M i Mi

)mov : moving part of ι∗ Ob(M

) ι∗i Ob(M i

)inv , ob1 (M

Gm (I)): Gm -invariant part of ϕ∗ Ob(M

), where ϕI : ϕ∗I Ob(M I G ∗ inv

. The induced morphism ϕ Ob(M

) −→ L G

m (I) −→ M is denoted M I M m (I) G

m (I)). by ob1 (M

)mov : moving part of ϕ∗ Ob(M

) ϕ∗I Ob(M I

i ): normal bundle of M

i in M

, isomorphic to Orel (−1)i−1 . The weight of N(M the induced Gm -action is (−1)i .

Gm (I)): virtual normal bundle of M

Gm (I) in M

N(M

N(EiM , EjM ): − l=0,1,2 (−1)l Rl pX∗ RHom EiM , EjM

l l M N(L, E2M ): l=0,1,2 (−1) R pX ∗ Hom L, E2 M

M ND (EiM∗ , EjM∗ ): − l=0,1 (−1)l Rl pD ∗ RHom 2 Ei|D ∗ , Ej|D ∗

Gm (I) ⊂ M

× M

3 N0 : normal bundle of M M0 l l

u, E

u ): −

u u N(E l=0,1,2 (−1) R pX∗ RHom(Ei , Ej ) i j

372

Glossary

l l

u ):

u N(L, E l=0,1,2 (−1) R pX ∗ Hom(L, E2 ) 2

u l l

u , E

u ): −

u ND (E l=0,1 (−1) R pD ∗ RHom2 Ei|D ∗ , Ej|D ∗ i∗ j∗

Subsections 6.1.1, 6.1.2 [M], [M]vir : virtual fundamental class of the moduli stacks M Φ: evaluation of a cohomology class Φ via the virtual fundamental class [M] M

Gm (I)]: virtual fundamental class of M

Gm (I) [M Subsection 6.3.1 M (c, L): moduli of L-abelian pairs (E, φ) such that c1 (E) = c (Lu , φu ): universal object on M (c, L) × X M (

c, [L]): moduli space of oriented reduced L-abelian pairs (E, [φ], ρ) such that c1 (E) = c

u , [φu ], ρu ): universal object on M (

c, [L]) × X (L L: line bundle such that c1 (L) = c ∈ H 2 (X, Z). If H 1 (X, OX ) = 0, it is determined uniquely up to isomorphisms. d(c, L): dim H 0 (X, L−1 ⊗ L) − 1 = dim M (c, L) l l −1 χ(L−1 ⊗ L): ⊗ L) l=0,1,2 (−1) dim H (X, L O(c, L), [M (c, L)]0 : If H 1 (X, OX ) = 0 and pg > 0, M (c, L) is smooth. The actual dimension is denoted by O(c, L). The naive fundamental class is denoted by [M (c, L)]0 . We have [M (c, L)] = Eu O(c, L) ∩ [M (c, L)]0 . pg , KX : We set pg := dim H 2 (X, OX ). Let KX denote the canonical line bundle of X. SW(c, L), SW(c): Assume H 1 (X, OX ) = 0 and pg > 0. If [M (c, L)] = 0, the expected dimension of M (c, L) is 0. Hence, we can regard [M (c, L)] as a number, which is denoted by SW(c, L). In the case L = OX , it is also denoted by SW(c). Ob(

c), [M (

c, [L])]0 : Assume H 1 (X, OX ) = 0 and H 2 (X, L−1 ⊗ L) = 0. Then, M (

c, [L]) is smooth and equipped with a perfect obstruction theory. The obstruction bundle is denoted by Ob(

c). The naive fundamental class is denoted by [M (

c, [L])]0 . Subsection 6.3.2 SW(a): Seiberg-Witten invariant associated to a Spinc -structure ξ with det (ξ) = a

Glossary

373

Subsection 6.3.3 X [y] : parabolic Hilbert scheme of ideals with type y SGL(W• ): (g−1 , g0 ) ∈ GL(W• ), det(g−1 ) det(g0 ) = 1 Y (W• ): quotient stack N (W−1 X , W0 X )SGL(W• ) Y D (W• ): quotient stack N (W−1 D , W0 D )SGL(W• ) .l Y D (W• , W ∗ ): quotient stack of i=1 N W (i+1) , W (i) by a natural action of .l (i) ) × SGL(W• ) i=2 GL(W ◦ ObD (y): RpD ∗ g◦D (V• , F∗ ) ⊗ ωD Ob◦D (y): RpD ∗ g◦ (V•|D ) ⊗ ωD Ob(y): cone of Ob◦D (y) −→ Ob◦ (y) ⊕ Ob◦D (y) ob(y): naturally induced morphism Ob(y) −→ LX [y] Subsection 6.3.4 y(−c): For y ∈ T ype, we set y(−c) := y · exp(−c), where c is the H 2 (X)component of y. Z(y(−c)): universal 0-scheme of X [y(−c)] × X K: Luc ⊗ L−1 ⊗ OZ(y(−c)) on M (c, L) × X y(−c) × X V: pX∗ K Assume H 1 (X, OX ) = 0. Take L with c1 (L) = c and we set K V: := K, −1 [y(−c)] [y(−c)] OZ(y(−c)) ⊗ L ⊗ L on X × X, and V := pX∗ K on X . Subsection 6.3.5 φ : L-section of det(E) induced by φ of E detE,φ : morphism U2 −→ M (c, L) induced by (det(E), φ ) • ): RpX∗ g(V• , φ• ) ⊗ ωX Ob(V• , φ Ob(M (c, L)): obstruction theory of M (c, L) ∨ grel , g(det(V• ), φ ): We set grel := Hom P• , det(V• ) . Let g(det(V• , φ )) denote the cone of the morphism grel [−1] −→ O[−1]. Yi (W• , P• ): Let Y0 (W• , P• ) be a quotient stack of N (W−1 X , W0 X ) × N (P−1 , W0 X )

374

Glossary

by a natural GL(W• )-action. We set Y1 (W• , P• ) = Y (W• ) and Y2 (W• , P• ) := Y2 (W• , P• ). Y (W• ): Fiber product of Yi (W• , P• ) (i = 1, 2) over Y0 (W• , P• ) is denoted by Y (W• , P• ). Zi (W• , P• ): We set Z0 (W• , P• ) := N P−1 , det(W•,X ) G , Z1 (W• , P• ) := m XGm and Z2 (W• , P• ) := N P0 , det(W•, X ) Gm . Z(W• , P• ): Fiber product of Z1 (W• , P• ) (i = 1, 2) over Z0 (W• , P• ) is denoted by Z(W• , P• ). Ob(M (c, [L])) obstruction theory of M (c, [L]) • ]): cone of Obrel (V• , [φ• ])[−1] −→ Ob(V• ) Ob(V• , [φ • ]): cone of Obrel (V• , ρ)[−1] −→ Ob(V• , [φ• ]) Ob(V• , ρ, [φ Ob(M (

c, [L])): obstruction theory of M (

c, [L])

Subsection 6.3.6 I(E): det(E)−1 ⊗ E Ξ(E): morphism U −→ X [y(−c)] induced by I(E) with the induced parabolic structure. Ob(V• , F∗ ): cone of Obrel (V• , F∗ )[−1] −→ Ob(V• ). Ob(X [y(−c)] ): obstruction theory of X [y(−c)] , that is the same as LX [y(−c)] = ΩX [y(−c)] Ob◦ (V• , F∗ ): trace-free part of Ob(V• , F∗ )

Subsection 6.3.7 • ): cone of Ob(V• ) −→ Ob(V• , F∗ ) ⊕ Ob(V• , φ• ) Ob(V• , F∗ , φ • ], ρ): cone of Ob(V• ) −→ Ob(V• , [φ• ], ρ) ⊕ Ob(V• , F∗ ) Ob(V• , F∗ , [φ

Subsection 7.1.1 Mapf Z2≥ 0 , H ∗ (X) , Mapf Z3≥ 0 , H ∗ (D) : Let Mapf Z2≥ 0 , H ∗ (X) de note the set of maps ϕ : Z2≥ 0 −→ H ∗ (X) such that (n1 , n2 ) ϕ(n1 , n2 ) = 0 is 2 finite. We use Mapf Z≥ 0 , H ∗ (X) in a similar meaning. ∗ ∗ ⊗ Sym Mapf Z3l Rl : Sym Mapf Z2l ≥ 0 , H (X) ≥ 0 , H (D)

Glossary

375

Rl , R: We set Rl := H ∗ (Pic) ⊗ R l and R := R1 . ql : homomorphism of algebras Rl −→ R⊗ l rl : homomorphism of algebras Rl −→ R

Subsection 7.1.2 R(E ∗ ), R(E): When we are given E ∗ = (E1∗ , . . . , El∗ ), we set R(E ∗ ) := Rl . It is also denoted by R(E). R(E∗ ), R(E): When we are given E∗ , we set R(E∗ ) := R. It is also denoted by R(E). P (E · et ): image of P ∈ R(E) via the homomorphism R(E) −→ R(E)[t] given in Subsection 7.1.2.

Subsection 7.1.3 A∗ (Y): bivariant theory A∗ (Y → Y) R(E ∗ , Y): R(E ∗ ) ⊗ A∗ (Y) deg P (E ∗ ) ∩ F ([Z]) : evaluation of P (E ∗ ) · F ∈ R(E ∗ , Y) over [Z]

u ): If the 1-stability condition holds for (y, α∗ ), we define P (E Mss (

y ,α∗ ) Mss (

y ,α∗ )

Ms (

y ,[L],α∗ ,δ)

u ) := deg P (E

u ) ∩ [Mss (

P (E y , α∗ )] .

u ) · ω k : If the 1-stability condition holds for (y, L, α∗ , δ), P (E

we set

" #

u ) · ω k := deg P (E

u ) · ω k ∩ Ms (

P (E y , [L], α∗ , δ)

Ms (

y ,[L],α∗ ,δ)

Subsection 7.1.4 T : l-dimensional torus (Gm )l R(T ): T -equivariant bivariant theory of a point ew·ti : trivial line bundle with the T -action induced by the action of i-th Gm with weight w AT ∗ (Y): T -equivariant Chow group of Y H∗T (Y): T -equivariant homology group of Y

376

Glossary

degT P (E∗ ) ∩ [Z] ∈ R(T ): evaluation of P (E∗ ) over [Z] ∈ AT∗ (Y) RT (E∗ , Y): R(E∗ ) ⊗ A∗T (Y)

Gm

M

M

Φ(E

Φ(E∗ ): deg ∗ ) ∩ [M ] M Subsection 7.1.5 RCH : Sym Mapf Z2≥ 0 , A∗ (X) ⊗ Sym Mapf Z3≥ 0 , A∗ (D) ⊗ A∗ (Pic) RCH (E∗ ): When we are given E∗ , we set RCH (E∗ ) := RCH . Rl,CH , RCH (E ∗ ): Similar Q: natural homomorphism RCH (E ∗ ) −→ A∗ (Y) Subsection 7.1.6 R[[t−1 , t]: algebra of power series

aj · tj such that j > 0|aj = 0 are finite.

R(t): Q[[t−1 , t] R(t1 , . . . , tk ): R(t1 , . . . , tk ) := R(t2 , . . . , tk )[[t−1 1 , t1 ] Eu(Fa ): equivariant Euler class of Fa Subsection 7.1.7 eω : a line bundle L such that c1 (L) = ω. P (E ⊗ eω ): image of P (E) ∈ R(E) via the naturally defined morphism R(E) −→ R(E ⊗ eω , Y). Subsection 7.2.1 S(y, α∗ , δ): set of (y 1 , y 2 ) ∈ (T ype)2 such that (i) y 1 + y 2 = y, (ii) Pyα1∗ ,δ = Pyα2∗ = Pyα∗ ,δ

2 , L, α∗ , δ): Mss (y 1 , L, α∗ , δ) × Mss (

M(y 1 , y y 2 , α∗ )

2 , L, α∗ , δ) × X obtained as the pull back of the universal E1u : sheaf on M(y 1 , y sheaf Mss (y 1 , L, α∗ , δ) × X

u : sheaf on M(y , y E 1 2 , L, α∗ , δ) × X obtained as the pull back of the universal 2 y 2 , α∗ ) × X sheaf Mss (

ω1 : c1 Or(E1u ) / rank y 1 ew·t : trivial line bundle with the Gm -action of weight w

Glossary

377

Subsection 7.2.2 Θrel : relative tangent bundle of Ms (

y , [L], α∗ , δ) −→ M(

y ), if the 1-vanishing condition is satisfied for (

y , [L], α∗ , δ) NL (y): X Td(X) · y · ch(L−1 )

2 · es/r2 ): equivariant Euler class of the virtual vector bundle

1 · e−s/r1 , E Q(E (7.16)

Subsection 7.2.3 (i)

(1)

Θrel : Let Θrel be the relative tangent bundle of the smooth morphism y , [L], α∗ , δ ± ) −→ M(

y , [L2 ]). Ms (

(2)

We use the symbol Θrel in a similar meaning. (i)

y , [Li ]) via the morphism Orel (1): pull back of the tautological line bundle on M(

y , [L], α∗ , δ) −→ M(

y , [Li ]). Mss (

S(y, α∗ , δ): set of (y 1 , y 2 ) ∈ (T ype)2 such that Pyα1∗ = Pyα2∗ ,

δ1 / rank y 1 = δ2 / rank y 2

2 , [L], α∗ , δ): Ms (

M(

y1 , y y 1 , [L1 ], α∗ , δ1 ) × Ms (

y 2 , [L2 ], α∗ , δ2 ) y 2 , [L2 ]). The Oi,rel (1): Let O2,rel (1) denote the tautological line bundle on M(

pull back is also denoted by the same symbol. We use the symbol O1,rel (1) in a similar meaning. ωi , ew·ωi : We set ωi := c1 (Oi,rel (1)) and ew·ωi := Oi,rel (w).

Subsection 7.3.1 Mss (

y ,α∗ )

: a linear map R −→ Q. See Definition 7.3.2.

Subsection 7.4.1 y, a, b, n: For y ∈ T ype, we have the decomposition y = rank(y) + a + b, where a ∈ H 2 (X) and b ∈ H 4 (X). The number n = a2 /2 − n corresponds to the second Chern class. N S(X): subgroup of H 2 (X, Z) generated by algebraic 1-cycles on X X [l] : Hilbert scheme of l-points

378

Glossary

Subsection 7.4.2 MH (

y ): moduli stack of torsion-free sheaves of type y which are semistable with respect to H C: ample cone in N S(X) ⊗ R ξ: element of N S(X) W ξ : wall determined by ξ C± , H± : For a given ξ, let C± be chambers which are divided by the wall W ξ . Let H± ∈ C± . We assume H− · ξ < 0 < H+ · ξ. M(

y0 , y 1 ): M(

y0 ) × M(

y1 )

i : pull back of the universal sheaf on M(

yi ) × X E y , α): moduli stack of torsion-free sheaves with trivial quasi-parabolic Mss (

structure and a weight α S(y, ξ): set of (y0 , y1 ) ∈ (T ype)2 such that (i) y0 + y1 = y, (ii) a0 − a1 = mξ for some m > 0 S: family of μ-semistable torsion-free sheaves of type y S: family of torsion-free sheaves E of rank one such that (i) μ(E ) = μ(y), (ii) there is a member E of S, such that E is a saturated subsheaf of E.

Index

1-stability condition, 84, 85, 88 1-vanishing condition, 242 2-stability condition, 84, 85, 88 2-vanishing condition, 242 H ev (X)-component, 68 L-Bradlow pair, 66 L-abelian pair, 224 L-section, 66 U -coherent sheaf, coherent sheaf, 26 U -torsion-free sheaf, torsion-free sheaf, 26 μ-semistable, δ-μ-semistable, 74 μ-stable, δ-μ-stable, 74 L-Bradlow pair, 68 (TFV)-condition, 89, 90 ample cone, 296 bivariant theory A∗ (Y), 268 category Dqcoh (X ), 33 Chow group A∗ (X ), 45 Chow group A∗ (Y), 267 classifying map Φ(V•|D ), 171 • ] , 164 classifying map Φ V• , [φ classifying map Φ(E), 148 classifying map Φ(V• ), 146 classifying map ΦD (V• , F∗ ), 171 compatibility of obstruction theory, 45 compatible diagrams, 29 • ]), 164 complex g (V• , [φ complex g(V•|D ), 171 complex gD (V• , F∗ ), 171 • ), 157 complex grel (V• , φ complex grel (V• , F∗ ), 171 complex Ob(E• , V•,• , φ), 41 • ]), 178 complex ObG (V• , [φ

complex ObG rel (V• , [φ• ]), 169 • ), 160 (V , φ complex ObG • rel /M ), 189 complex Obrel (M 3 3 • , 158 complex Obrel V• , φ ∗ )inv , 202 complex ϕI Ob(M ∗ )mov , 202 complex ϕI Ob(M • ]), 164 complex grel (V• , [φ complex g(Poin), 153 complex g(f ), 46 complex g(F, f ), 48 complex g(V ∗ ), 52 complex g(V• ), 146 complex gd (V• ), 149 complex g◦ (V• ), 149 complex gd D (V• , F∗ ), 173 complex g◦ D (V• , F∗ ), 173 • ]), 168 complex hrel (V• , [φ complex h(V• ), 150 • ), 160 complex h(V• , φ • ), 160 complex hrel (V• , φ complex k(F , ϕ), 43 complex k(V, ϕ), 41 complex k(E• , ϕ), 43 complex k(E• , V•,• , φ), 40 complex k(V•u ), 152 complex V• , 151 complex Hom(C • , D• ), 26 complex Hom(C • , D• )∨ , 27 complex RHom1 (E1∗ , E2∗ ), 28 complex RHom2 (E1∗ , E2∗ ), 28 ), 179, 185, 187 complex Ob(M ∗ ), 185, 187 complex Ob(M Gm (I)), 188, 199, 200 complex Ob(M complex Ob(F , ϕ), 41, 43 complex Ob(M3 ), 189

379

380 3 ), 189 complex Ob(M complex Ob(Poin), 153 complex Ob(E), 148 complex Ob(f ), 47 complex Ob(F, f ), 48 , [L]), 175 complex Ob(m, y complex Ob(m, y), 153 complex Ob(V ∗ ), 52 complex Ob(V•|D ), 172 complex Ob(V • ), 146 , [L] , 176 complex Ob m, y , 176 complex Ob m, y complex Obd (V• ), 149 • ), 180 complex ObG (V• , φ complex ObG (V• ), 150 complex Ob◦ (V• ), 149 G • ]), 168 complex Obrel (V• , [φ G • ]), 169 complex Obrel (V• , [φ • ] , 165 complex Obrel V• , [φ complex Obrel (m, y), 174 complex Obrel (m, y), 175 complex Obrel (m, y, [L]), 175 complex Obrel (m, y, L), 174 • ]), 165 complex Obrel (V• , [φ complex Obrel (V• , ρ), 154 complex Obrel (V• , ρu ), 155 complex Obrel (V• , F∗ ), 172 M Gm (I)), 188, 200, 201 complex Ob( y, L), 184 complex Ob(m, ), 184 y complex Ob(m, , [L]), 182 y complex Ob(m, 2 (m, y , [L]), 182 complex Ob a (M Gm (I)), 188, 200, 201 complex Ob complex C • ∨ , 27 complex C1 (V1∗ , V2∗ ), 28 complex C2 (V1∗ V2∗ ), 28 condition Om , 66 Condition (C), 150 cotangent complex, LX /Y , Lf , 32 critical, 85, 88, 289, 297

decomposition type, 121 degree deg(y, α∗ ), 73 degree deg(y), 73 degree par-deg(E∗ ), 72 depth of y, depth(y), 68 depth of parabolic structure, 65 enhanced master space, 129 equivariant Chow group AT ∗ (Y), 271 equivariant Euler class, 274 equivariant Euler class Eu(Fa ), 275

Index equivariant homology group H∗T (Y), 271 equivariant line bundle ew·ti , 271 equivariant line bundle ew·t , 277 equivariant representative, 56 equivariant virtual bundle N0 (y 1 , y 2 ), 277 Euler class Eu(Θrel ), 269 expected dimension, 44 family SS(y, L, α∗ , δ (0) ), 79 family YOK(N, K, y, L, δ), 81 family YOK(N, K, y, L, δ), 81 family YOK(m, K, y, L, δ), 81 ∗ , 170 filtered bundle VD full flag variety Flag(V, N ), 107 general linear group, GL(R), 35 Gieseker space Zm , 91 group GL(W• ), 146 group N S(X), 293 Harder-Narasimhan filtration, 76 Hilbert polynomial, 71

integral [M]vir , 214 integral M , 214 inv , 202 invariant part ι∗ i Ob(M ) Jordan-H¨older filtration, 76 lift, 157, 164 line bundle I (2) , 210 (i) line bundle Orel (1), 283 line bundle O2,rel (1), 284 line bundle eω , 275 line bundle K, 293 , 269, 288 linear map Mss (y,α ∗) map ql , 265 map rl , 266 , 129, 139, 141 master space M moduli stack M(y, L), 69 moduli stack M(m, y), 69 moduli stack Ms (y, [L], α∗ , δ), 75 moduli stack Ms (y, L, α∗ δ), 75 , [L], α∗ , δ), 75 moduli stack Ms (y , L, α∗ , δ), 75 moduli stack Ms (y moduli stack Mss (y, α∗ ), 75 moduli stack Mss (y, L, α∗ , δ), 75 moduli stack Ms (y), 75 s (y , [L], α∗ , δ), 75 moduli stack M s (y moduli stack M m , [L], α∗ , δ), 75 ss (y, [L], α∗ , (δ, )), 77 moduli stack M

Index ss (y , [L], α∗ , (δ, )), 77 moduli stack M s (y , α∗ ), 75 moduli stack M s (y moduli stack M m , α∗ ), 75 ss (y, [L], α∗ , (δ, )), 77 moduli stack M m moduli stack M(y), 69 ), 69 moduli stack M(y , [L]), 69 moduli stack M(y , [L]), 69 moduli stack M(y , L), 69 moduli stack M(y moduli stack M(y), 69 moduli stack M(m, y, [L]), 69 moduli stack M(m, y, L), 69 ), 69 moduli stack M(m, y , [L]), 69 moduli stack M(m, y , [L]), 69 moduli stack M(m, y , L), 69 moduli stack M(m, y moduli stack M(m, y), 69 moduli stack M(m, y), 69 moduli stack M(y), 69 moduli stack Ms (y, α∗ ), 75 , α∗ ), 75 moduli stack Ms (y , [L], α∗ , δ), 75 moduli stack Mss (y , α∗ ), 75 moduli stack Mss (y moduli stack Ms (y, [L], δ), 75 moduli stack Ms (y, L, δ), 75 moduli stack MH (y), 296, 309 moduli stack M(m, y, L), 184 ), 184 moduli stack M(m, y , [L]), 181 moduli stack M(m, y ss y, L, α∗ , (δ, ) , 77 moduli stack M s (y , [L], α∗ , δ), 75 moduli stack M s (y moduli stack M m , [L], α∗ , δ), 75 ss y, L, α∗ , (δ, ) , 77 moduli stack M m ss (y moduli stacks M m , [L], α∗ , (δ, )), 77 morphism degT P (E∗ ∩ •) , 272 morphism r(E• , ϕ), 43 morphism r(E• , V•,• , φ), 40 morphism ob(E• , V•,• , φ), 41 morphism degT , 272 morphism detE , det, 64 E , 149 morphism det morphism Γ , 150 morphism γ([φ]), 168 morphism γ(φ), 160 morphism γD , 171 morphism ΓL , 160 morphism Γ[L] , 168 morphism r(F , ϕ), 43 morphism r(V, ϕ), 41 morphism w, 149 ), 179, 185, 187 morphism ob(M ∗ ), 185, 187 morphism ob(M

381 Gm (I)), 188, 199, 200 morphism ob(M morphism ob(F , ϕ), 43 3 ), 189 morphism ob(M morphism ob(V, ϕ), 41 morphism ob(E), 148 morphism ob(f ), 47 , [L]), 175 morphism ob(m, y morphism ob(m, y), 153 morphism ob(V ∗ ), 52 morphism ob(V•|D ), 172 morphism ob(V • ), 146 , [L] , 176 morphism ob m, y , 176 morphism ob m, y Gm (I)), 202 morphism ob1 (M i ), 202 morphism ob1 (M morphism obrel (m, y), 174 morphism obrel (m, y), 175 morphism obrel (m, y, [L]), 175 morphism obrel (m, y, L), 174 morphism obrel (V• , ρ), 154 morphism obrel (V• , ρu ), 155 morphism obrel (V• , F∗ ), 172 M Gm (I)), 188, 200, 201 morphism ob( morphism ob(m, y, L), 184 ), 184 y morphism ob(m, , [L]), 182 y morphism ob(m, 2 (m, y , [L]), 183 morphism ob a (M Gm (I)), 188, 200, 201 morphism ob morphism Φ(det(E)), 149 morphism Ψ (V• ), 150 • ]), 168 morphism Ψ (V• , [φ • ), 160 morphism Ψ (V• , φ morphism Ψ1 , 178 morphism Ψ2 , 178 morphism tr, 149 morphism Ξ(V• , [φ]), 169 morphism gY , 25 mov , 202 moving part ι∗ i Ob(M ) Mumford-Hilbert criterion, 30

non-trivial everywhere, 66, 67 normal bundleN0 , 202 number degT P (E∗ ) ∩ [Z] , 272 number χ(y), 69, 293 number χ(ai ), 295 number deg P (E∗ ∩ •) , 268 , [L], α∗ , δ), 280 number Φ(y number SW(c), 227 number SW(c, L), 225 number h0 (E), 71 number h0 (E∗ ), 72 number NL (y), 279

382 obstruction class, 33 obstruction theory, 44 orientation, 64 orientation bundle Or(E), 64 oriented L-Bradlow pair, 66 oriented U -coherent sheaf, 64 oriented coherent sheaf, 64 oriented parabolic reduced L-Bradlow pair, 68 oriented parabolic reduced L-Bradlow pair, 67 oriented reduced L-Bradlow pair, 68 oriented reduced L-Bradlow pair, 67 parabolic L-Bradlow pair, 68 parabolic part, 68 parabolic reduced L-Bradlow pair, 67 parabolic sheaf, 65 parabolic structure, 65 partial Jordan-H¨older filtration, 76 perfect obstruction theory, 44 Picard variety, PicX , 63 Poincar´e bundle, PoinX , 63 δ , 72 polynomial H(E ∗ ,φ) δ polynomial P(E∗ ,φ) , 72 δ polynomial P(E , 72 ∗ ,φ) δ polynomial H(E , 72 ∗ ,φ) polynomial HE , 71 polynomial HE∗ , 71 polynomial PE , 71 polynomial PK , 291 polynomial PE∗ , 71 polynomials Hyα∗ ,δ , Pyα∗ ,δ , 73 polynomials Hyα∗ , Pyα∗ , 73 polynomials Hy (t), Hy,i (t), Hy (t), 73 projection pX , 26 quasi-parabolic structure, 65 quotient object of parabolic sheaves, 65 quotient stack B(W• , [P• ]), 169 quotient stack B ∗ (W• , [P• ]), 178 quotient stack B ∗ (W• , P• ), 179 quotient stack (U × X)Gm , 164 quotient stack F , 181 quotient stack Q, 185 178 quotient stack Q, ∗ , 179 quotient stack Q quotient stack Yquo (W• ), 48 quotient stack A(W• ), 150 quotient stack A(W• , [P• ]), 168 quotient stack A(W• , P• ), 160 quotient stack B(W• ), 150 quotient stack B(W• , P• ), 161 quotient stack kGL(R) , 35 quotient stack XGm , 164

Index quotient stack Y (W• ), 38, 146 quotient stack Y (W• )Gm , 164 quotient stack YD (W• ), 171 quotient stack YD (W• , W ∗ ), 171 quotient stack ZG , Z/G, 26 quotient stacks Yi (W• , P• ), 156 quotient stacks Yi (W• , P• ) (i = 0, 1, 2), 164 rank of y, rank(y), rank(y), 69 reduced L-abelian pair, 224 reduced L-Bradlow pair, 67 reduced L-section, 67 relative tangent bundle ΘY /S , 37 relative tautological line bundle Orel (1), 70 residue Res, 277 (i) resolution V• , 189 resolution P• , 156 ring R, 264 ring R(E ∗ ), R(E), 266 ring R(E ∗ , Y), 270 ring R(E∗ ), R(E), 266 ring R(E∗ , Y), 268 ring Rl , 264 ring RT (E∗ , Y), 272 ring RCH , 273 ring RCH (E ∗ ), 274 ring RCH (E∗ ), 274 ring Rl,CH , 274 ring R(T ), 271 ring R[[t−1 , t], 274 rings R(t), R(t1 , . . . , tk ), 274 scheme Qss (m, y, [L], α∗ , δ), 99 scheme Qss (m, y, [L], α∗ , δ), 99 scheme Qs (m, y, [L], α∗ , δ), 99 scheme Qs (m, y, [L], α∗ , δ), 99 scheme Q(m, y), 90 scheme Q(m, y), 89 scheme Qs (m, y, α∗ ), 99 scheme Q◦ (m, y), 90 scheme Q◦ (m, y, [L]), 94 scheme Q◦ (m, y, [L]), 93 scheme Q◦ (m, y, L), 92 , [L]), 94 scheme Q◦ (m, y scheme Q◦ (m, y), 90 scheme Q◦ (m, y), 89 scheme Qss (m, y, α∗ ), 99 scheme X [l] , 293 Seiberg-Witten invariant SW(a), 227 semistability, (δ, )-semistability, 77 semistable point, 29 semistable, δ-semistable, 73

Index

383

set Mapf Z3≥ 0 , H ∗ (D) , 264 set Mapf Z2≥ 0 , H ∗ (X) , 264 set Cr(y, α∗ , L), 85 set Dec(m, y, α∗ , δ), 121 set P br , 72 set S(i0 ), 58 set T ype, 68 set T ype, 68 set S(y, α∗ , δ), 276 set S(y, α∗ , δ), 283 sets T yper , T ype◦ , T ype◦ r , 69 sets Y s (L), Y ss (L), 29 sheaf Coki (E), 65 u , 276 sheaf E 2 sheaf E u , 151 sheaf Hom(V1 , V2 ), 26 sheaf E1u , 276 sheaf Gri (E), 65 sheaf of Kahler differentials, ΩX /Y , 34 slope μ(y, α∗ ), 73 slope μ(y, α∗ , δ), 73 slope μ(E∗ ), 72 slope μ(y), 73 slope μδ (E∗ , φ) of (E∗ , φ) with δ, 72 slopes μmax (E), μmin (E), 78 stable point, 29 stable, δ-stable, 73 1, y 2 , [L], α∗ , δ), 283 stack M(y ∗ , 130, 139, 141 stack M Gm (I), 130 stack M 2 , L, α∗ , δ), 276 stack M(y 1 , y 1, y 2 , [L], α∗ , δ), 280 stack M(y split , 187, 199, 200 stack M stack N , 178 stack Y (W• , [P• ]), 164 stack Y (W• , P• ), 156 1 , M 2 , 129, 139, 141 stacks M subobject of L-Bradlow pair, 66 subobject of parabolic sheaves, 65 system of weights, 65 tautological line bundle, Orel (1), 70 transition formula, 276

twist E∗ (m) for E∗ , 66 type, 69 uniform categorical quotient, Y ss //G, 29 up to e´ tale proper morphisms, 134 vector fj , 58 vector x(i1 , i2 ), 58 vector x1 (j), 60 vector x2 (j), 60 vector y(j), 58 vector y (2) (j), 60 vector bundle Θ1,rel , 284 (1) vector bundle Θrel , 283 vector bundle N (V1 , V2 ), 26 vector bundle WX , 48 (h) vector bundles WD and Wi D , 170 vector bundles W−1 X , W0 X , 146 vector spaces W (h) , 170 vector spaces W−1 , W0 , 146 virtual G-equivariant vector bundle, 28 virtual fundamental class, 45 Gm (I)], 215 virtual fundamental class [M virtual fundamental class [M], 214 virtual fundamental class [M]vir , 214 Gm (I)), 202 virtual normal bundle N(M i ), 202 virtual normal bundle N(M virtual vector bundle, 28 u, E u ), 203 virtual vector bundle N(E i j

virtual vector bundle N(EiM , EjM ), 202 u ), 203 virtual vector bundle N(L, E 2 virtual vector bundle N(L, E2M ), 202 virtual vector bundle N(L2 , E1u ), 211 virtual vector bundle N(L2 ⊗ I (2) , E1M ), 210 u , E u ), 203 virtual vector bundle ND (E i∗ j∗

virtual vector bundle ND (EiM∗ , EjM∗ ), 202 weight of parabolic structure, 65 Yokogawa data, 80 Yokogawa family, 81

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