Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens,
Groningen
B. Teissier, Paris
1638
Springer Berlin
Heidelberg New York Barcelona
Hong Kong London Milan Paris
Singapore Tokyo
Pol Vanhaecke
Integrable Systems in the realm
of Algebraic Second Edition
Y,Vkl
Springer "841
Geometry
Author Pol Vanhaecke
D6parternent de Math6matiques UFR Sciences SP2MI
Universit6 de Poitiers
T616port
2
Boulevard Marie et Pierre Curie BP 30179
86962
Futuroscope
E-mail:
Chasseneuil Cedex, France
[email protected]
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CIP-Einheitsaufnahme
Vanhaecke, Pol: Integrable systems
in the realm of algebraic geometry / Pol Vanhaecke. 2. Berlin ; Heidelberg New York ; Barcelona ; Hong Kong ; London Milan ; Paris ; Singapore Tokyo : Springer, 2001
ed..
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(Lecture notes in mathematics ; ISBN 3-540-42337-0 Mathematics
1638)
Subject Classification (2000): 14K20, 14H70, 17B63,
37J35
ISSN 0075- 8434 ISBN 3-540-42337-0 ISBN 3-540-61886-4
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(Ist edition) Springer-Verlag Berlin Heidelberg
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Preface to the second edition
book, five years after the first edition, has been spiced with naturally in the point of view that had been adapted in the original text and with some new examples and constructions that will help the reader to appreciate better our approach to integrable systems. The present edition of this
several recent results which fit
On this occasion I wish to thank my collaborators from the last five years, to wit Christina
Birkenhake, Peter Bueken, Rui Fernandes, Masoto Kimura, Vadim Kuznetsov, Marco Pedroni, Michael Penkava, Luis Piovan and Claude Roger for a fruitful interaction and for their warm friendship. Most of the results that have been added axe taken from, or are inspired by, joint work with some of them; I acknowledge their permission to add these, sometimes unpublished, results.
colleagues at my newest working environment, the University of Poitiers (aance), me a pleasant and stimulating working enviromnent. I wish to acknowledge the support of all of them. Special thanks go to Marc van Leeuwen, Claude Quitt6 and Patrice Tauvel for sharing their insights with me, which usually led to a real improvement of parts The
created for
of the text.
least, Yvette Kosmann-Schwambach, who was not acknowledged in the most probably because my gratitude to her was too big and too is thanked here in all possible superlatives, for her constant support and for her obvious! sincere friendship. Merci Yvette! Last but not
first version of this book -
-
Acknowledgments
indispensable for establishing and presenting the results Not enough credit can be given to those who created at home, at the Max-Planck-Institut in Bonn, at the University of Lille and finally at the University of California at Davis a pleasant and stimulating atmosphere. Even some people I don't know by name should be thanked here. The
which
help
are
of many
people
was
contained in this work.
Special thanks
are
due to Mark Adler and Pierre
van
Moerbeke, whose fundamental work
a.c.i. systems was the starting point for the research contained in this book. Stimulating discussions with them have led to an improvement of many of the results and to a better on
understanding of the subject. Also Michble Audin deserves a special plarce here for sharing insights with me through long discussions and letters. Extremely helpful for a thorough understanding were several algebraic-geometric explanations by Laurent Gruson. her
I wish to thank my collaborators Jos6 Bertin and Marco Pedroni for
a
fruitful interac-
tion. I have also benefited from discussions with my colleagues at Lille, in particular Jean d'Almeida, Robert Gergondey, Johannes Huebschmann, Rapha6l Freitas, Armando Treibich,
Gijs Taymnan and Alberto Verjowski and at UC Davis, in particular Josef Mattes, Mulase, Michael Penkava, Albert Schwarz and Craig Tracy.
Motohico,
I also acknowledge my other friends scattered around the globe, to wit, Christina Birkenhake, Robert Brouzet, Peter Bueken, Jan Denef, Paul Dhooghe, Jean Fastr6, Ljubomir Gavrilov, Luc Haine, Horst Knbrrer, Franco Magri, Askold Perelomov, Luis Piovan, Elisa Prato and Taka Shiota for their interest in my work and helpful related discussions. For useful comments
on
the manuscript I
referee and several students in my Last but not
this adventure.
graduate
least, special thanks
am
indebted to Mich6le
course
in UC Davis
Audin,
an
anonymous
(Spring 1996).
to my wife Lieve for her constant assistance
through
Table of Contents
1. Introduction IT.
.
.
.
.
.
.
.
Hamiltonian systems
Integrable
1. Introduction
.
.
.
.
.
.
.
.
2.1. Affine Poisson varieties 2.2.
.
.
.
2. Affine Poisson varieties and their
.
.
.
.
.
.
.
.
morphisms
.
.
.
.
.
3.
Morphisms of affine Poisson varieties
Decompositions
Integrable
Hamiltonian systems and their Hamiltonian systems
.
Integrable
.
.
on
.
.
systems
2.2. The
.
.
.
.
and their .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
case
.
.
3.1. The real and
.
complex level
3.2. The structure of the
.
.
.
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.
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.
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.
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.
.
Compactification
significance
.
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.
sets
1.
17. 17.
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19.
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19.
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26.
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28. 37.
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47. 47.
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54.
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57.
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62. 65.
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65.
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69.
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.
curves
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71.
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71.
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73.
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73.
-jwd
73. 78.
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.
.
complex level manifolds
of the
.
.
.
3.3. The structure of the real level manifolds 3.4.
.
I., -1d'
3. The geometry of the level manifolds
3.5. The
.
.
Poisson spaces
on
.
.
morphisms
structures
in involution for
hyperelliptic
.
.
integrability
compatible Poisson
PolynomiaJs
2.4. The
.
.
Hamiltonian systems and symmetric products of
2. 1. Notation
2.3.
.
.
.
other spaces
Integrable Hamiltonian systems
1. Introduction 2. The
.
.
.
Compatible and multi-Hamiltonian integrable systems
.
.
.
affine Poisson varieties
on
.
.
integrable Hamiltonian systems
Hamiltonian systems
.
.
Morphisms of integrable Hamiltonian systems
Integrable
.
.
.
Integrable
4.2.
.
.
3.2.
4.1. Poisson spaces
111.
.
3.1.
3.4.
.
and invariants of affine Poisson varieties
3.3. Constructions of
4.
.
.
2.3. Constructions of affine Poisson varieties
2.4.
.
affine Poisson varieties
on
.
.
.
.
complex level manifolds
of the Poisson structures viii
-j'Pd
.
.
83.
.
.
.
85.
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.
85.
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87.
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89.
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93.
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95.
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.
IV. Interludium: the
1. Introduction
.
.
geometry of Abelian varieties
.
.
.
.
2. Divisors and line bundles
2.1. Divisors
.
.
.
2.2. Line bundles
.
.
.
.
.
.
.
Hyperelliptic
3. Abelian varieties 3.1.
Complex
.
.
.
on
4. Jacobi varieties
.
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.
97.
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99.
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99.
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100.
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101.
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103.
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.
105.
in
projective
.
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106.
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108.
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108.
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109.
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111.
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114.
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.
algebraic
4.2. The
analytic/transcendental
.
.
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.
114.
Jacobian
.
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.
114.
4.3. Abel's Theorem and Jacobi inversion
4.4. Jacobi and Kummer surfaces
V.
generic
5.2. The
non-generic
(1,4)
.
.
case
.
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119.
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121.
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123.
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123.
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.
124.
Algebraic completely integrable
1. Introduction 2. A.c.i.
.
systems
3. Painlev6
Hamiltonian systems
.
.
.
.
127.
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127.
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129.
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135.
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138.
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.
140.
.
VI. The Mumford
.
.
analysis for a.c.i. systems
equations
.
.
.
4. The linearization of two-dimensional a.c.i.
5. Lax
space
.
Jacobian
case
97.
.
.
4.1. The
5.1. The
.
.
.
.
5. Abelian surfaces of type
.
.
Abelian varieties
3.3. Abelian surfaces
.
.
tori and Abelian varieties
3.2. Line bundles
.
.
embeddings
curves
.
.
2.4. The Riemann-Roch Theorem
2.6.
.
.
.
2.3. Sections of line bundles
2.5. Line bundles and
.
.
.
.
.
systems
.
.
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.
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.
.
systems .
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143.
.
1. Introduction
.
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143.
2. Genesis
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145.
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145.
2.1. The
.
.
algebra
of
pseudo-differential operators
.
.
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.
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.
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.
146.
2.3. The inverse construction
.
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.
150.
2.4. The KP vector fields
.
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.
152.
2.2. The matrix associated to two
.
commuting operators
ix
3. Multi-Hamiltonian structure and
3.1. The 3.2.
loop algebra
4. The odd and the
4.2. The
4.3.
(odd) even
.
.
.
.
general
Mumford system
case
.
.
.
.
VII. Two-dimensional a.c.i. 1. Introduction
.
.
Mumford systems
even
Mumford system
.
.
.
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.
.
2.2. The genus two
even
Application: generalized
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155.
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157.
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161.
potential
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161.
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163.
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164.
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.
168.
and Laurent solutions .
.
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.
.
Linearizing variables
5.3. The map M -+ M .
6. The H6non-Heiles
.
.
hierarchy
.
.
6.1. The cubic H6non-Heiles 6.2. The
.
.
.
.
7. The Toda lattice
.
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175.
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177.
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177.
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179.
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181.
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185.
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185.
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186. 190.
.
of order three
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196.
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196.
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202.
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206.
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211.
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216. 220.
.
.
(1,4)
to the genus 2
even
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220.
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222.
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226.
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230.
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230.
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232.
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233.
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235.
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.
235.
.
.
.
.
.
.
.
.
.
.
.
.
237.
.
.
.
.
.
.
.
.
.
.
.
.
240.
Mumford system
7.3. Toda and Abelian surfaces of type
References
.
explicit
.
7.1. Different forms of the Toda lattice
morphism,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
II
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
175.
.
potential
hierarchy
.
.
quartic H6non-Heiles potential
6.3. The H6non-Heiles
.
.
.
.
.
.
on SO(4) SO(4) for metric
.
.
.
potentials
on
.
.
integrable geodesic flow geodesic flow
.
.
4.4. The relation with the canonical Jacobian made
5.1. The
.
.
potential and its integrability
4.5. The central Garnier
.
.
system
.
.
.
automorphism
.
.
.
Kummer surfaces
.
.
.
Mumford system
an
.
.
4.3. The precise relation with the canonical Jacobian
.
.
.
.
.
4.2. Some moduli spaces of Abelian surfaces of type
Index
.
.
.
.
4.1. The Garnier
7.2. A
.
.
configuration on the Jacobian of r projective embedding of the generalised Kummer surface
4. The Gaxnier
5.2.
.
.
.
.
3.2. The 94
5. An
.
.
systems
with
curves
.
.
2.3. The Bechlivanidis-van Moerbeke system
3.1. Genus two
155.
.
.
.
.
2.1. The genus two odd Mumford,
3.3. A
.
.
systems and applications
.
2. The genus two Mumford.
3.
.
.
.
.
Algebraic complete integrability
5. The
.
.
.
the R-brackets and the vector field V
Pteducing
4.1. The
91,
symmetries
(1,3) .
.
x
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
243.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
253.
Chapter
II
Integrable Hamiltonian systems affine Poisson varieties
on
1. Introduction
In this
chapter
give the basic definitions and properties of integrable Hamiltonian morphisms. In Section 2 we define the notion of a Poisson bracket (or Poisson structure) on an affine algebraic variety. The Poisson bracket is precisely what is needed to define Hamiltonian mechanics on a space, as is well-known from the theory of symplectic and Poisson manifolds. We shortly describe the simplest Poisson structures (i.e., constant, linear, affine and quadratic Poisson structures; also general Poisson structures on C2 and C') and describe two natural decompositions of affine Poisson varieties, one is given by the algebra of Casimirs, the other comes from the notion of rank of a Poisson systems
on
we
affine Poisson varieties and their
structure
(at
from old
ones.
a
point).
We also describe several ways to build
new
affine Poisson varieties
Morphisms of affine Poisson variety are regular maps which preserve the Poisson bracket. Isomorphisms preserve the rank at each point, leading to a polynomial invariant for affine Poisson varieties. This invariant permits us on the one hand to distinguish many different affine Poisson varieties, on the other hand it allows us to display in a structured way the basic characteristics of the Poisson structure. It will be computed for many different examples and a
refinement of this invariant is also discussed. In Section 3
we turn to integrable Hamiltonian systems. We motivate our definition by propositions and (counter-) examples. The notions of super-integrability, compatibility and integrable multi-Hamiltonian systems fit very well into the picture and most of our propositions are easily adapted to the case that the integrable Hamiltonian systems under
several
discussion have
one
of these extra structures.
decomposition of the variety,
as
the
one
The notion of momentum map leads to it is much finer).
given by the Casimirs (however
17
P. Vanhaecke: LNM 1638, pp. 17 - 70, 1996, 2001 © Springer-Verlag Berlin Heidelberg 1996, 2001
a
Chapter We also define
11.
Integrable Hamiltonian systems
morphisms of integrable Hamiltonian systems; they
are
Poisson
mor-
algebra of functions in involution. It allows one to state precisely the relation between different integrable Hamiltonian systems, for example between new systems and the old ones from which they were constructed. Our discussion is parallel to the one of affine Poisson varieties (up to some modifications). Some really interesting examples of integrable Hamiltonian systems will be given in later chapters. phisms which
preserve the
The final section
(Section 4)
is devoted to
a
generalization
of
our
definitions to the
of other spaces. We draw special attention to the case of real Poisson manifolds. The main difference is that on the one hand the algebras we work with in the case of an affine case
variety are in general not finitely generated so that many constructions do not apply polynomial invariant), on the other hand many local constructions (e.g., Darboux coordinates, action-angle variables) which cannot be performed for affine Poisson varieties, play a dominant role in the study of some other Poisson spaces, including Poisson manifolds. Poisson
(e.g.,
the
Apart from Section
4
we
will in this
chapter always work
numbers.
18
over
the field of
complex
2. Affine Poisson varieties and their
2. Affine Poisson varieties and their
morphisms
morphisms
2.1. Affine Poisson varieties Phase space will closed subset of C'
always
(closed
be
an
affine
vaxiety
for the Zariski
in the
topology).
sense
Such
a
of
[Har], i.e.,
variety
an
irreducible
M C CI is the
zero
prime ideal Im of C[xi.... Xn], and its ring (or C-algebra) of regular functions denoted, resp. defined by
locus of
a
1
is
C[Xi'...' Xn]
O(M)
=
IM
integral domain (it has no zero divisors) and it is finitely generated; M can be reconstructed, up to isomorphism, from O(M) as SpecmO(M), the set of closed points in
O(M)
is
an
SpecO(M). The extra structure which Poisson bracket
on
a
Lie
algebra
algebra
Let M be
Definition 2.1 is
its
structure
we use
of
an
I-, j
to describe Hamiltonian
systems
on
M is
given by
a
fanctions.
regular
affine variety. A Poisson bracket or Poisson structure on M O(M), which is a bi-derivation, i.e., for any f G O(M) the
on
C-linear map
Xf:O(M)-+O(M) -+Ig,fl
g
is
a
derivation
(satisfies
the Leibniz
rule),
Xf (gh)
=
(2.1)
(Xf g) h + gXf h
for all g, h E O(M). The derivation Xf is called the Hamiltonian derivation associated to the Hamiltonian f and we write Ham (M, f -, -1) for the (vector) space
Ixf
=
I., f I I f
of Hamiltonian derivations. A function
Xf
=
0,
is called
a
Casimir
function
Cas
f
or a
(M, 1., -1)
E
E
O(M)
OMI whose Hamiltonian vector field is zero, we denote
Casimir and
=
If
O(M) I Xf
G
=
01
(vector) space of Casimirs; it is the center of the Lie bracket I-, j hence it is a Lie (O(M), I-, J). When no confusion can arise, either argument in Ham (M, I-, J) and Cas (M, f J) is omitted.
for the
ideal of
-
,
Remarks 2.2 1.
Xf being a
derivation may be refrased in
TM,
reason we
usually call
the elements
a
geometric
way
by saying
that it is
a
global
HO(M, Tm) (for the definition of the sheaf algebraic variety see [Hax] Section 11.8). For this Xf of Ham (M, 1., -1) Hamiltonian vector fields.
the tangent sheaf to M, i.e., of differentials and the tangent sheaf to an section of
Xf
E
Using the above mentionned correspondence between an affine variety and its algebra regular functions we have that affine Poisson varieties correspond to finitely generated Poisson algebras without zero divisors. 2.
of
19
Chapter 3.
Turning
upside down
the above definition
a
and its
subspace
one
gets
at the
following, equivalent definin
denote the vector space Hom(A 0 (M), 0 (M)) by C' (M) of skew-symmetric n-derivations by Der' (M). For every p, q > 0 a bilinear
Poisson bracket. Let
tion of
Hamiltonian systems
Integrable
Il.
us
map
F -1
:
,
is defined for P E
[Pj Q] (fl
CP(M), Q I... I
CP M
and for
Cq(M)
E
C, (M)
X
-+
CP+"- I (M)
fi,..., fp+,-i
E
O(M) by
fp+q-1)
o,ESq,p-i
1:
+
...
i
fa(p+q-1))
aESp,q_i where
o-(1)
Sp,
<
...
that if P E
(p,q) shuffles (permutations a of 11,...'p + qj such that < a(p + q); c(a) is the sign of a). It is easy to see a(p) and a(p + 1) < restricts to a DerP (M) and Q E Derq (M) then [P, Q] E Derp+q-1 (M). Thus denotes the set of
<
...
bracket
-Is
:
DerP (M)
called the Schouten bracket. For P
[P, P]s(f, g, h) so
2
Der
(M)
2(P(P(f, g), h)
we
-+
DerP+q- 1 (M),
have that
P(P(g, h), f)
+
+
P(P(h, f), g)),
skew-symmetric bi-derivations P such that following interesting interpretations. If P G defines a Poisson structure then the (graded) Jacobi identity for [-, -Is implies that becomes a complex when the coboundary operator
that Poisson structures
[P, PIS 2 Der (M) Der* (M)
=
E
Derq (M)
x
=
0.
This
can
also be defined
8p
:
as
to the
point of view leads
Derq (M)
-+
Derq+1 (M)
corresponding cohomology is called precisely the 0-cocycles and that the Hamiltonian vector fields are the 1-coboundaries. For X E Deri (M), 8p X -,CxP, where LX is the Lie derivative of P with respect to X, hence the 1-cocycles are the vector fields Q E Derq (M) by 8p (Q) cohomology. One observes that
[P, Q] s.
The
is defined for
=
Poisson
the Casimirs
are
=
that preserve the Poisson structure P (such vector fields are called Poisson vector fields). A similar interpretation of the 2-cocycles and the 2-coboundaxies will be given at the end of
this section. The
following properties follow
Proposition
(3.) (2) (3) (4) (5)
Let
2.3
at
(M, 1-, -1)
once
be
from Definition 2.1.
an
affine
Poisson
variety.
as Lie bracket); subalgebra of Der' (M) (with the commutator Ham(M) Ham(M) is however in general not an O(M)-module, as opposed to Der(M); Cas(M) is a subalgebra of O(M); The adjoint map ad : O(M) -+ Ham(M) which is defined by f i-+ -Xf is a Lie algebra homomorphism; For all f, g E O(M), ad(fg) f X, + gXf; f ad(g) + ad(f)g; equivalently, Xf, There is a short exact sequence of Lie algebras
is
a
Lie
=
=
0
Cas(M)
-,
20
O(M)
ad __
-o-
Ham(M)
0
2. Afflne Poisson varieties and their
By (2.1) the Hamiltonian any system of generators gi,
differential equations
where j
is
a
-
vector field -
Xf gi
=
convenient notation for
system of generators
completely determined by
.
.
.
(i
when
Xf gi
gj,
9
its action
on
system of first order polynomial
a
,
a
=
17
...
14
(2.2)
particular choice of f E O(M) has been rule, completely described
in view of the Leibniz 98
by the Poisson
matrix
Q9i19jD1
=
whose entries gij axe polynomials in gj, for any three functions in O(M) as soon of O(M), with i < j < k. .
Every vector field,
is
Igi, f
=
Similarly a Poisson structure I-, j is,
in terms of the
Xf
g8 of 0 (M). It leads to
,
O(M), namely
on
i
fixed.
-
morphisms
.
.
,
as
g,
-
Morevoer,
the Jacobi
it is satisfied for all
identity will be satisfied triples of generators gi, 9j, gk
in
particulax every Hamiltonian vector field, can be (non-uniquely) exthe ambient affine space of M. Namely, suppose that M C C8, let P1, g8 be the standard coordinates of C8 and denote the corresponding generators of O(M) by g.1, Fj (gi, 98. The vector field determined by j g,) then leads to a vector field on C8 determined by gi Fj (pl, of first orp,). Conversely, a system 9i Fj (gi, der polynomial differential equations on C8 defines a derivation on O(M) C[g.1, g.]Ilm precisely when j E Im for every f E Im C i.e., when the vector field is tangent to M as a subvariety of C". Similarly, a Poisson bracket on O(M) defines a skewsymmetric bi-derivation on C', but the latter is in general not a Poisson structure because the Jacobi identity need not hold. Explicitly, let #jj -gji denote any extensions of gij to C[gj .... g,] (I < ij:5 k) anddefine askew-symmetric bi-derivationP of C[gj,..-'9,] by tended to .
.
.
a
vector field
on
,
.
.
.
=
,
.
.
.
,
=
.
.
.
=
,
.
=
-
-
-,
=
a
P=I:gij -99i
a A
Then the Jar-obi
identity
(2.3)
api
i<j
for P is tantamount to the
following system
of
partial differentia.1
equations. a I
a
9jk
+
9ji
2 g=gki + 9ki 1-9ij 'g
E
(1
Em,
< i <
j
<
k <
(2.4)
I
showing that the in general.
Poisson matrix
We will often define
a
does not define
Poisson bracket
by specifying
a
Poisson bracket
its Poisson matrix
system of generators); necessary and sufficient conditions for structure are given by the following proposition.
21
on
a
C[gj,
(in
matrix to define
08]
terms of a
a
Poisson
Chapter 11. Integrable Hamiltonian systems
C[ 1,...' g,]/-E be an affine variety and let 91, Proposition 2.4 Let M g, denote the cormsponding system of generators of O(M). Let g be a skew-symmetric .5 x s matrix, with entries in O(M), and let p be any skew-symmetric matrix, whose entries are extensions Then g is the Poisson matrix (in terms of the corresponding elements in g to of gi, gj of a Poisson bracket on M if an only if =
-..,
.
.
.
,
(i) the (2) for
Jacobi
identity (2.4)
any G E
Im
is
satisfied for all triples (Pi, 9i 9k) (With i
C[p,.... g.]
C
I
"
one
i <
i
<
k);
has
OG
E_ gjj E IM,
U
=
(2.5)
1,-, 8).
Proof
(.t) is necessary; if G E Im then for any gj one has fgj, GI 0 implies (2.5) upon using (2.3). Let us show that (i) and (2) are also (in O(M)), sufficient. Define a skew-symmetric bi-derivation I I' on C [pi, 9, ] by (2.3). It is not necessarily a Poisson bracket since (i.) only guarantees that the Jacobi identity is satisfied in 0 (M), i.e., its right hand side belongs to Im. Condition (2) implies that for all F1, F2 E C[gj'...' #,] and for all G1, G2 Elm we have We have
seen
above that
=
which
-
-
-
,
IF, for
some
G3
E
Im hence
we
+
have
G1, F2 an
+
G2 11
=
IF,, F21'
induced bracket
I
-
-
,
I
+
on
-
-
,
G3 0 (M). It satisfies the Leibniz
original bracket does; because of condition (:L) it satisfies the Jacobi identity of generators hence for all regular functions. Hence it defines a Poisson bracket system
rule because the
for
a
on
O(M).
I
It is natural to ask if for every affine Poisson variety M one can find a system of generators gl,...,g, of O(M) (realizing M as a subvariety of C') and extensions of their Poisson brackets to C[gj,...' g,], such that the bi-derivation (2.3) is a Poisson bracket on C".
Remark 2.5
The
answer
to this
Definition 2.6
question is
Let
at
(M, 1-, -1)
present unknown. be
an
affine Poisson
rank of the Poisson structure at x, denoted one of the following:
(i) (ii)
The rank at
x
Rkxl., -1,
variety and let as
the
even
x
G M. We define the
number defined
representing 1., -1; linearly independent Hamiltonian derivations
by
either
of any Poisson matrix
The number of
The rank of the Poisson structure is defined
Rkj-, -1 and the co-rank dimM (or variety) is called regular if it
-
Rkj-, .1
is
at
x.
maxjRkxj-, -11 x E MI and is denoted by denoted by CoRkj-, -1. The Poisson structure as
has constant rank and trivial if its rank is
zero.
equivalence of the two definitions is easily established; since (ii) is intrinsic we see depend on the chosen system of generators of O(M). Notice that the rank is also defined at the singular points of M and that, in view of (i) it is maximal on a Zariski open subset of M (see also Paragraph 2.4 below). Moreover, since at the general point of M the number of independent Hamiltonian derivations is bounded by dimM we have that Rkj-, -1 < dimM. We give some first examples of affine Poisson varieties. The
that
(i)
does not
22
2. Affine Poisson varieties and their
morphisms
Example 2.7 Any constant skew-symmetric n x n matrix is the matrix of a Poisson structure on Cn, in terms of its standard coordinates, as follows from (2.4). We refer to such a Poisson structure as a constant Poisson structure. By the classification theorem for skewsymmetric bilinear forms the Poisson matrix takes the standard form 0
1,
-1,
0
(0 after alinear
change
of
such
a
a
in Im
degree
0
standard Poisson structure of rank 2r
or
(2.5)
constant bracket
bracket to define
inimal
0)
0
of coordinates. Here 2r is the rank of the Poisson matrix. Thisstructure
is often called the canonical case
0
a
on
Cn.
Since in the
degree deg G I we see that a necessary condition for bracket on an affine subvariety M of Cn is that all elements of is of
-
Casimirs.
are
Example 2.8 Let g be any finite-dimensional (complex) Lie algebra, with Lie bracket By Proposition 2.4, the Lie bracket extends to the symmetric algebra Symg of 0, making Sym. g into a Poisson algebra. Since Sym. g -5-- 0(g*) the dual space g* of g carries a natural Poisson structure. For f g E 0 (g*) it is given at E g* by ,
RR), Qb 1
If, g I W the hat
denoting
always
or
This Poisson structure is known as the -+ g. the Lie-Poisson structure of g*. The rank at the origin is
so it is never regular (unless g is commutative so that I -} is trivial). Notice resulting Poisson algebra is characterized by having independent generators with
zero,
that the
respect
pairing (,g*)*
the natural
canonical Poisson structure
(2.6)
-
,
to which the Poisson matrix is linear.
Choosing a non-degenerate bilinear defined by
form
on
g
we
have
an
isomorphism X
:
Sym g
O(q)
x
(" ej)
11(ei,
sym
It allows
f,g
E
us
0(g)
to transfer the Poisson structure to 9; it is easy to
at
x
E
g
V, 91W where the
see
that it is
given, for
by
gradient Vf (x)
of
f
E
E
0
(9(g)
=
at
(X, [Vf W, Vg(XT' x
is defined
by
d
Vy
Example
2.9
3-dimensional
For
Tf W, Y)
=
wt-lt=o
f (X
+
ty)-
quadratic Poisson structures (on C') a general theory is not known. A of quadratic structures on C' was given by Sklyanin (see [Skll]). Let linear coordinates on C4 and consider the following matrix:
family
XO; X1 I X2 and X3 be
0
(-b3XIX2 -b.IX2X3 -b2XIX3
bjX2X3
b2X1X3
b3XIX2
0
a3XOX3
9. a2 XOx-
-a3XOX3
0
aixox,
-a2XOX2
-aixox,
0
23
Chapter
II.
Integrable Hamiltonian systems
By Proposition 2.4 four checks of the Jacobi identity sufficeto show that this matrix and they are all satisfied if and only if alb,
a2b2 + a3b3
-
:::--
is
a
Poisson
0;
notice that this is also
equivalent to the vanishing of the determinant of the Poisson matrix This gives a 5-dimensional family of quadratic Poisson structures on C4. Except for the trivial structure they are all of rank two and two Casimirs are given 2 The Poisson structures given by Sklyanin and ajX20 b3X22 + b2X2. a2X 2 + a3X2 by ajX21 3 3 correspond to
(for
all values of the
xi).
-
-
al
=
-a2
=
bi + b2 + b3 We will to
see
in
a3,
0-
==
Example 2.52 that the 5-dimensional family given above than the family given by Sklyanin.
is
more
general (up
isomorphism)
Example 2.10 It is easy to describe afl Poisson structures on C2 since in this case the Ja, cobi identity is satisfied for any skew-symmetric bracket. Let x and y be coordinate functions then any polynomial V in two variables defines a Poisson bracket on C2 by jx, yj W(x, y) and conversely every Poisson bracket is of this form. It is regular if and only if W is constant, 0. otherwise the rank drops to zero along the algebraic curve W(x, y) =
=
Example
2.11
For C3 the Jacobi
identity
( will be
a
Poisson matrix if and
only
P
leading
matrix is
(F, G, H).
=
0
G
-F
a
-G) 0
if
For X and W
laxge number of given by
to
the matrix
F
-H
(V where
H
0
trivially satisfied. Indeed
is not
X
P) -0 -
==
(2.7)
0,
arbitrary polynomials j0 XVW is a solution to (2.7), on C3; explicitly for such .9 the Poisson =
Poisson structures
X
(
OW
0
OW Oy OW
-
OZ
OW 8Z
0
OX
E
-OW
Oy
OX
I
0
and W is seen to be a Casimir of this Poisson structure. The rank is two except at the zero locus of X and at the points where the gradient of o vanishes. Notice that not all solutions to
(2.7)
axe
of this
form, for example the Poisson
matrix
0
0
X
0
0
Y
(-X 0) -Y
is not of the form
XVW (with
X and W
regular)
-
24
2. Affine Poisson varieties and their
morphisms
In many cases, especially in the theory of integrable systems, the space comes naturally equipped with several Poisson structures, moreover they are often compatible in the following sense.
Let M be
Definition 2.12 on
M. Then
say that
we
vaxiety and let compatible if for any
affine
an
they
axe
1','Ia,b is
Poisson bracket
a
on
M.
Similaxly
any linear combination of them is
=
n
1-, -11
a
+ b
two Poisson brackets
I- J2 i
Poisson brackets
Poisson bracket
a
and
a, b E C the linear combination
on
on
M
are
said to be
compatible
if
M.
implies that 1-, -11 and {* '12 are compatible if and only if [aj-, -11 + bl. i'12]S 0, for all a, b E C, which is equivalent to [1-, -Ili J* *12]S 0This means that 1', *12 defines a 2-cocycle in the Poisson cohomology of 1-, .11. It is from this point of view a natural question to ask whether the cohomology class defined by 1., .12 is trivial, i.e., whether there exist a vector field X on M such that Lxf-, -11 f' *12- Such a vector field is said to have the deformation property with respect to 1-, -11. Notice that if P and LxP are both Poisson structures then they are automatically compatible Poisson Remark 2.2
bj- '12, al-, .11
+
=
=
7
=
1
structures since
[P, Lx P] S
=
Explicitly, the condition [1-, -11, 1- *121S 1
Iff, g1l, h12 jjfj 912, h1l for all
and h in
fig
_[p76PX] =
n
_62P X
0 takes the
f Ig, h1l, f12 + 11g, h12) f}1 +
O(M) (or, equivalently,
from the Schouten bracket that
=
brackets
+ +
=
0.
following form
ff h, fIll 912 j1h, fl21 911
+
(2.8) =
0,
for
are
It a system of generators). compatible if and only if they
is also cleax are
pairwise
compatible. Example
2.13
Constant Poisson structures
are
always compatible.
Examples 2.7 and 2.8 describe all possible constant and linear Poisson Cn, i.e., Poisson structures whose Poisson matrix (with respect to some, hence with respect to any system of linear coordinates for Cn) consists only of constant resp. linear elements. One may wonder about their combination, that is, Poisson brackets on Cn whose Poisson matrices (as above) have entries of degree at most one; let us call them affine Poisson brackets on Cn. Obviously both the constant and linear paxt of such Poisson structures are Poisson structures, hence a constant Poisson structure on Cn which is compatible with a Lie-Poisson structure on C' defines (by taking their sum) an affine Poisson structure on Cn
Example structures
2.14
on
and any affine Poisson structure on Cn is of this form. These affine structures If as modified canonical Poisson structures or modified Lie-Poisson structures. the Poisson bracket determined constant X1,
-
-
-,
part by
I-
-
,
jo
then
the linear part
by
we see
that condition
are
known
we
denote
by I-, -I, and the one determined by the (2.8) reduces in terms of linear coordinates
Xn for Cn to
JjXi, Xjjl XkJO
+
JjXj, Xkjl, XiJO 25
+
JjXki Xill, XjJ0
_-::
0-
(2.9)
Chapter
Using
the hat-notation from
11.
Integrable Hamiltonian systems
Example 2.8, by
we
look at Cn
as
the dual of
Lie
a
algebra 0
with
basis J i and Lie bracket determined
4--'-i I
I&i, &j Then the second bracket
(the
constant
C: 9 A 9
bracket) C
-+
:
determines
(.- j, &j)
F-+
a
linear map
Ixi, xj jo-
With this notation
jjXi7 Xjj1i XkJO hence
(2.9)
---
C(jXij XjJ17 -:4)
Q-'Ni &j], 41);
becomes
Q-'N,&j1i&'k)
C&ji&k1iM
+
+
Q&ki&d7: j):--:: 0-
This formula expresses that C is a 2-cocycle in the cohomology' of g associated with the trivial representation of 9 on C, giving another way to describe affine Poisson structures on vector space. As
an application of this point of view, recall that if g is semisimple the first cohomology groups are trivial; then C is a coboundary, C W', written out, C( tj,: j) C([., j,&j]) and we see that the affine Poisson bracket is nothing but the original Lie-Poisson bracket with xi replaced by xi + C'(&j), i.e., both brackets are the same up to an affine change of variables.
a
and second
=
=
2.2.
Morphisms
of affine Poisson varieties
We recall that a map 0 : M, -+ M2 of affine varieties is called a regular map or a morphism if O*O(M2) C O(MI), where 0*(f) f o 0 for functions f E O(M2); thus 0 induces and is uniquely defined by an algebra homomorphism 0* : O(M2) -+ O(MI). A regular map which has a regular inverse is called a biregular map or an isomorphism. =
Definition2.15 map
0: M,
-+
Let
(M1j*,'}1)
M2 is called
a
and
Poisson
(M2,1',*12)
morphism
or a
be two affine Poisson varieties, then a morphism of affine Poisson varieties if
(:L) 0 is a morphism, O*O(M2) c O(MI); (2) For all f,g E O(M2), O*Jf7912 JOV4*911=
Both conditions
are
conveniently summarized by the commutativity
O(M2) 0.
For
an
introduction to the
O(M2)
X
following diagram:
1',*12
O(M2)
X"*1
O(MI) 5
X
of the
1"* O(Mi)
cohomology
T-
of Lie
26
O(MI)
algebras
see
e.g.
Appendix
5 in
[LM3].
2. Affine Poisson varieties and their
morphisms
adjectives which axe used for morphisms of affine varietie's (e.g., injective, dominant, finite, -) may also be used for Poisson morphisms. A biregular map is a Poisson morphism if and only if its inverse is a Poisson morphism; we call such a map a (Poisson) isomorphism. When M, is an affine subvariety of M2 and the inclusion map (MI, 1-, .11) (W J* *12) is a Poisson morphism then M, is called an affine Poisson subvariety of M2. The standard
..
1
image of an affine variety by a morphism. needs not be an affine variety; consider example the image of the map (x, y) -+ (x, xy), defined on C2. If the image of a Poisson morphism is an affine subvaxiety (i.e., a (Zariski) closed subset) of the target spar-O then there is an induced Poisson structure on it, making it into an affine Poisson subvariety, as shown in the following proposition. The
for
Proposition2-16 If 0: M, -+ M2 is a morphism of affine Poisson varieties and the image O(MI) is an affine subvariety of M2 then O(MI) has a unique structure of an affine Poisson variety such that 0 can be factorized as the composition of a Su7jective and an injective Poisson morphism, as in the following diagram: M2
M,
O(MI)
Proof o , where M, -+ O(MI) is a surJective By assumption 0 can be factorized as 0 mOrphism (of affine varieties) and s : O(MI) -+ M2 is an inclusion map. Since 0 is surJective, O(MI) -+ M2 is an inclusion map, t* 0* : O(O(Mi)) -+ O(MI) is injective; since O(M2) -+ O(O(Mi)) is surJective. -
following definition of a Poisson bracket on O(MI). Let f g E.O(O(Ml)), z*f' and g =s*g. We by surJectivity of%* there exist f' and g' in O(M2) such that f
This leads to the then
,
=
define
If; 9} and
that it is
verify
suffices to show that
morphism
we
=:=
Z* lfi 91 12;
independent
*S* Ift 9'}2 depends only on f 5
::=
,
It follows from
(2.11)
(2.10)
can
that
f -, -1
also be written
that We
Poisson 6
is
a
now
and g; since 0* is injective it and g. Using the fact that 0 is a Poisson
Poisson
f *fi *911-
satisfies the Jacobi
(2.11)
identity and that
Z
are
a
Poisson
morphism.
*g} I,
morphism.
show how the rank of the Poisson structure at
morphism
is
as
g} so
f'
find
0*ZIf 9112 Now
(2.10)
of the choices made for
related;
Examples include closed,
it
implies equality for
proper and finite
27
an
a point and isomorphism.
morphisms,
see
[Har]
at its
Ch. II
4.
image by
a
11.
Chapter 2.17
Proposition M,
M2
-+
a
Let
Poisson
Integrable Hamiltonian systems
(Mi, I- ji) ,
(i 1, 2) Rk.,j-, -11 =
,
Then
morphism.
be two
:
affine Poisson varieties and let 0 Rk,5( &, *12 for any x E MI.
Proof Let gj,
.
.
be
completed
we
have
be a system of generators of 0 (M2). Then hi , g, into a system of generators hi, of 0 (MI). , h,+t
.
.
Rkx
I
==
Rk
> Rk =:
The
proposition implies that
in
.
.
O*gj, (i 1, s) can By definition of the rank
=
=
.
.
.
,
(I hi, hj (1hi, hj I (x)) I
Puko(m)l'i *12-
general
an
affine
subvariety
of
an
afline Poisson
variety
does not carry a Poisson structure which makes it into an affine Poisson subvariety. Necessary and sufficient conditions for this to happen will be given in Proposition 2.18.
2.3. Constructions of affine Poisson varieties are four (known) basic constructions of PoiSson brackets on finite dimensional (here taken to be affine varieties). The first one is that of the canonical Lie-Poisson structure (Example 2.8). In the most important examples, at least from the point of view of
There
spaces
integrable Hamiltonian systems, second
one
the relevant Lie bracket is
an
R-bracket,
consists of the canonical Poisson structure associated to
a
see
Section V.5. The
symplectic
structure
the prime example being here the one of cotangent bundles. Notice the Poisson structure is never regular while in the second case it is
(see Example 4.2 below), that in the first
always regular.
case
Both these
axe
very
classical,
as
opposed
to the other two which will not be
discussed in detail here. The first of these two deals with Poisson structures in
particular Lie-Poisson
in
[BCKII.
a
The last
one
on
Lie groups,
groups; an excellent account of this is given in Semenov's paper consists of the construction of higher order brackets, starting from
Lie bracket, also within the R-matrix approach (see [LP]). Apart from these four basic constructions there are also several constructions which allow
one
to build
new
Poisson brackets from
given
ones.
We will discuss these here in the context
of affine varieties, in fact we will show how the standard constructions by which new affine varieties are built from given ones, have their equivalents for afline Poisson varieties. We think here of the
(1) (2) (3)
an affine subvariety; product of two affine Poisson varieties; the quotient and fixed point set of an affine Poisson variety under the
the
(4) are
a
or
considered next in the above order:
Poisson structure to restrict to tion 2.38
action of
reductive group; removing a divisor. finite
They
following:
the restriction to
an
affine
we
start
subvariety (for
by giving a precise condition for a important example, see Proposi-
an
below).
Proposition 2.18 Let (M, I., -J) be an affine Poisson variety and affine subvariety of M. Then the following are equivalent.-
28
suppose that N is
an
2. Affine Poisson varieties and their
(i)
There exists
a
Poisson structure
on
N with
morphisms
respect to which N is
subvariety of M; (ii) The ideal of N is a Poisson ideal of O(M); (iii) The restriction of every Hamiltonian vector field Xf, f
E
an
O(M)
affine Poisson
to N is
tangent
to N.
Proof
equivalence of (ii) and (iii) is immediate from the definition of a Poisson ideal: an Poisson algebra is called a Pois8on ideal if it has the additional property of being a Lie ideal; thus, -EN, the ideal of regular functions vanishing on N, is a Poisson ideal if and only if I-EN, 0 (M) I C -TN which is in turn equivalent to The
ideal of
a
Xf g
=
1g, f I
vanishes at all points of
N,
for every Hamiltonian vector field If IN is
a
Poisson ideal
Xf, f E O(M) and every g E IN. of 0 (M) then for any f, g E 0 (M) and If
and
1',*IN
define
we can
at
n
+
IN, g
E N
+
-TN} (n)
=
n
c- N
I f gj (n),
(2.12)
,
by j%*fjZ*9jN(n)
=
If,gl(n),
where
z
:
N -+ M is
the inclusion map; clearly the latter becomes a Poisson morphism. This shows that (ii) implies (i). If z : N -+ M is a Poisson morphism then (2.12) holds, in particular JIN, gj
vanishes any g E
on
O(M). (i) implies (ii).
N for any g E
O(M),
so
Since IN is
a
prime ideal it follows that JIN, gj
C
IN for I
is an ideal of Example 2.19 Suppose that of Sym g generated by 0. For h E 0 and for gi,
a
Lie
algebra g
g,,, F-
g
a
and denote
direct
by (0) the ideal application of the Leibniz
rule shows that every term in
1h, g,
92'
gm
This shows on Sym g. isomorphism Symo 1--- 0(,g*) the ideal corresponds to the ideal of functions vanishing on . Therefore, Proposition 2.18 implies that the subspace of g* which consists of the elements of Z* that vanish on 0 is an affine Poisson subvariety of 9* with its Lie-Poisson structure.
belongs
that
to
is
(0),
a
way
or
denotes the canonical Poisson structure
Poisson ideal of
Remark 2.20 one
where
"
Symg.
Under the canonical
in There are, of course, other ways in which a subvariety may inherit a Poisson structure from its ambient affine Poisson variety. Think for
another
-
-
example of a proper symplectic subvariety of a symplectic, manifold, which is never a Poisson subva,riety, but still carries a natural Poisson structure "inherite&' by the symplectic 2-form. on its ambient manifold. This will be discussed later in this paragraph.
29
Chapter
Second,
Integrable Hamiltonian systems
products of affine Poisson
consider
we
11.
varieties.
Proposition 2.21 Let (Mi, 1., -11) and (M2, 1" '12) be two affine Poisson varieties, then the product M, x M2 has a natural structure of an affine Poisson variety such that the + projection maps iri : M, x M2 -+ Mi are Poisson morphisms. Moreover it has rank Rkf + Rk,,,,IRkf-,'121 the rank at (MI, M2) G M1 x M2 being given by Rk,,,, 7
Proof The
algebra
of
regular functions 0 (MI
on
M2)
X
the
=
product Mi
7r* 0 (MI)
0
x
M2 is given by
lr * 0 (M2)
(2.13)
1
hence the construction amounts to
making the tensor product of two Poisson algebras into algebra. Formula (2.13) implies that O(M., x M2) is generated by the functions ir,*L fl, fn is an arbitrary system of generators 7r2*g,,,, where fl, irl fn, 7r2*gl, of O(MI) and gl,...,g,,, for O(M2). In order for -7r, and ir2 to be Poisson morphisms it is Poisson
a
.
.
.
.
,
..
.
,
.
.
,
necessary and sufficient to define J1r1*fi,-7r1*fjJ = ?r1*Jfi,fjJ1 and J7r2*gi,7r;gjJ ir2J9i,9jJ2 2 for all i and j. A natural choice for the remaining brackets firl*fi, 7r2*gjl is to make them =
all
zero:
with this choice the Jacobi
identity
is
surely satisfied.
The Poisson matrix of
1., .1
with respect to the generators ?rif1,...'7r2*g,,, has a block form, hence Rk(,,,,,,,n,)I-, J Rk,n, I , I I + Rk,,,, I 1 '12 for any (tnl , M2) E M1 X M2 It is also easy to see that the algebra =
-
-
-
-
of Casimirs of
(Mi
x
M2,
given by ir,* Cas(MI)
is
The Poisson bracket on M, product bracket and is denoted by I-, -1m, xm,. Definition 2.22
Example
Suppose that (G, 1-, -1)
2.23
is
an
x
0
7r;2 Cas(M2).
M2 given by Proposition 2.21 is called the
affine Poisson variety and that G is
an
alge-
braic group with multiplication X: Gx G -* G. Then (G, 1-, .1) is called an affine Lie-Poisson group if X is a Poisson morphism, the Poisson bracket on G x G being the product bracket.
A related construction appears when having a family of affine Poisson depend on a (or several) parameter(s). More precisely we assume that there is given a dominant morphism 7r : P -+ N of affine varieties (N being the parameter space) and Define for f g E O(P) and p E P a Poisson bracket I* Jn on each non-empty fiber 7r-'(n)
Example
2.24
varieties which
.
i
ff'gl(p)
,
(p).
=
f,g c O(P) one has ff,gl E O(P) (roughly speaking, regularly with n Ei N) then 1-, .1 defines a Poisson bracket on P
If for any va,ry
f -In makes every irreducible component M x N, (P, I-, J). As a special case, let P
bracket of
take
-
i
of 7r-1 (n) into
=
ir as
projection
of the fibers of
.7r
on
the second factor.
which varies
regularly
coincides with the product bracket
Third,
we
on
Clearly x
E N.
The N is
n
consider the Poisson structure
on
an
an
this leads to
N, where
with
M
where M is
a
if the brackets
MIG
where G is
a
30
J.,-In
and for any n E N the affine Poisson subvariety
affine Poisson variety and Poisson structure on each
resulting Poisson structure on P given the trivial (zero) bracket.
the fixed point set and
the quotient equipped with
on
finite group or a reductive algebraic group which is Poisson structure (notice that it needs not be an affine Lie-Poisson group).
space a
(2.14)
2. Affine Poisson varieties and their
Before
doing thiswe recall
a
morphisms
few facts about group actions on affine varieties. All groups or reductive; moreover, when we want to consider Poisson
considered here will be either finite structures
on
reductive groups
we
will assume that
category of affine Poisson varieties. A (finite
they are affine varieties, algebraic) group G is said
or
M C C' if the action is the restriction to M of
so as
to
to act
stay in the
on an
affine
representation of G on Cn. When G is finite every representation on Cn is completely reducible, i.e., if the action of G leaves invariant some subspace of Cn then it leaves invariant a complementary subspace. For infinite groups the above property characterizes reductive groups (since we are working over C). We recall also that there is an induced action of G on O(M), given for g E G, f E O(M) and x E M by g*f (x) f (g-'x).
variety
a
=
If G is finite then
we
say that the action of G
on
M is
a
Poisson action if for every g E G
isomorphisin M -+ M defined by m i-+ gm is Poisson. If G is infinite, say G is an affine algebraic group, it may itself carry a Poisson structure and the proper generalization of the the
above notion of Poisson action is that the map G is given the product Poisson structure.
x
M -4 M is
a
Poisson map, where G
Proposition 2.25 Let (M, I-, Jm) and affine Poisson variety and let G be braic group acting on M.
(i) If there the
(2) If
is
a
Poisson structure
on
G
for which the
action is
a
an
Poisson
x
M
affine alge-
action, then
O(M)G of G-invariant functions is a Poisson subalgebra of O(M). moreover reductive or finite, then O(M)G is finitely generated, hence
algebra
G is
cor-
responds to an affine Poisson variety MIG, leading to the following commutative diagram of Poisson morphisms (7r2 is projection onto the second component). x
GxM
M
I
1r
1r2
(2.15)
MIG
M -,r
Proof
Clearly f
E
0 (M) is G-invariant if and
only
if the
following diagram
is commutative.
x
GxM
-
f
7r2
M
M
-
C
f
Thus,
if
f, g
E
O(M)G
and X is Poisson then
X*Jf19JM--`JX*f1X*gJ G xm --flr2Yi r2*91GXM:--:7r2*ffiglMl and
we see
subalgebra of
that the bracket of any two G-invariant functions is G-invariant. Therefore the of O(M) is aJso a Lie subaJgebra of O(M), i.e., it is a Poisson subalgebra
O(M)G
O(M). 31
Chapter Il. Integrable Hamiltonian systems Assume now that G is reductive or finite. Then O(M)G is finitely generated (see [Muml] or [Spr]) hence is the algebra of regulax functions on an affine variety, denoted MIG and called the (categorical) quotient; MIG is naturally identified with the orbit space of an open subset of M. The natural projection map M -+ MIG is regular and yields the commutative diagram (2.15). Granted (1) this proves (2). M e.g.
We next consider
generalization of the above proposition. We consider again a group G on M and leaving some subvariety N of M invariant. We will show that N may inherit a Poisson structure from M, even if N is not a Poisson subariety of M. Let us denote by O(M, N)G, the algebra of regular functions on M that restrict to G-invariant functions on N, and by p : O(M,N)G -+ O(N)G the natural map induced by the inclusion N M.
(assumed
finite
or
Definition 2.26 action and N
a
on
(M,
Let
-1)
subvaxiety of
Poisson-reducible if bracket
a
reductive) acting
O(N)G
be
an
affine Poisson
M which is G-stable.
O(M, N)G
is
Poisson
a
subalgebra of O(M)
(2.16) N
on
G
x
M
-+
M
triple (M, G, N)
and if there exists
Poisson
a
is called Poisson
a
one
PJFJ, F21
(2.16)
O(M, N)G.
holds for all F1, F2 E
Formula
:
X
such that
JP(FI)i P(F2)10(N)G
functions
vaxiety,
Then the
says that in order to compute the Poisson bracket of two G-invariant computes the Poisson bracket of any extensions to M and then restricts
(2.16) uniquely defines a bracket on O(N)G (if it exists) surJective. In the following proposition we give necessary and sufficient conditions for (M, G, N) to be Poisson-reducible (for a proof see [PV1]). the result to N. Notice also that
since p is
Proposition action and N
and
2.27 a
Let
(M, 1., -1)
subvariety of
affine
Poisson
variety, X: G x M -+ M a Poisson (M, G, N) is Poisson-reducible if
Then
a
f O(M, N)G7 I(N)}
in this condition that its
implicit In
of
an
only if p
it is
be
M which is G-stable.
slightly different
vein
a
left
=
0;
(2.17)
hand side makes
sense.
Poisson structure is also inherited
group action. This fact was first shown in is generahzed in Proposition 2.29 below. First a
[FV] we
in the
need
a
case
of
by a
the fixed point
variety
Poisson involution and
lemma about the ideal of
a
fixed
point variety. Let G be a finite or reductive group acting on an affine Poisson variety and fixed point variety. The ideal IN is generated by functions fj for which there
Lemma 2.28
let N be its
exist gj E G and
j
(E
C
11
such that
gj* fj
=
j fj
Proof
Since G is finite
or
reductive the representation space C' which contains M
decomposes
of spaces VO, V,, such that VO is the fixed point set of the action and , such that the action of G on the other Vi is irreducible. Then vo n m N and IN is as
a
direct
sum
.
.
.
=
generated by ?ri* Vi*)
where 7ri is the natural projection Cn
32
-+
Vi and i
=
1,
.
.
.
,
s.
Let i be
2. Affine Poisson varieties and their
morphisms
(between I and s) and take any non-zero element 0 E Vi*. Since the action of G on Vi is irreducible there exists g E G such that g*0 -A 0. Since G is reductive (or finite), G is generated by its semi-simple elements (see [Hum] p. 162). Therefore, let g be a semi-simple
fixed
element for which for
j
=
it follows that
Consider
<
u
the
now
Then
g*0 =54 0.
1,...,v with j v
0 for
=
and
we
we
j
=
have
linear basis
a
1,...,u and j :A
have at least
one
01,
0 for
function
V)
0,
...'
j
for
of
Vi*,
where
g*oj
=
joj
1,...,v. Since g*0 7 0 which g*'O O, with 7 1.
=
u
+
=
is the span of all functions fj E Vj* for which 1 such that gj*fj jfj- We have already established that Wj*
subspace Wj* of Vj* which
there exists gj E G and j : :4 contains a non-zero element. Therefore it suffice to =
Let
that
Wj*
that
Ejn-, cjg*fj
=
Vi*.
E
verify that Wj* is invariant to conclude f rjn-, cj fj (=- Wi*, with fj as above, and let g E G. We need to show Wi*. This follows at once from =
(g-,gjg)*g*fj since
=
g*gj*fj
=
6g*h,
1.
j :7
Proposition 2.29 Suppose that G is a finite or reductive group acting on an affine Poisson variety (M, 1-, -1). We assume that for every g E G the isomorphism -ID, : M -+ M which corresponds to the action of g is a Poisson map. Let N be the subvariety of M consisting of the fixed points of d) and denote the inclusion map N -+ M by i. Then N carries a (unique) Poisson Structure
j",jN
such that
Z*IFIIF2} for
all F1, F2 E 0 (M) that
=
jS*F1jZ*F2jN
G-invariant.
are
Proof
For
O(N)
we
reductive
we
fl, f2
G is finite
or
E
choose F1, F2 E O(M) such that f, %*P2. Since t*Fl and f2 define We G-invariant. and that are assume F, F2 may =
=
Iflif2IN and show that this definition is
=
(2.18)
Z*fFliF21
independent of the choice of F, and F2. To do this it is 0. We 0, then z* IF,, F21
sufficient to show that if F, and F2 are G-invariant, with z*Fl will actually show that if F2 is G-invariant and F1 E IN then
=
=
z*IFI, F21
=
0. Let
us
denote
system of generators of IN as given by the previous lemma. If F is G-invariant then the fact that (D, is a Poisson map implies
by fl,
.
.
.
,
ft
a
z* jfj,
showing
our
claim,
Fj
since
=
z*-I)* jfj, .9
j =A
Fj
=
z* 1-(D*fj, 9
- 9*F}
=
jz* jfj, FI,
1. Note also that the bracket of any two G-invariant functions (2.18) is independent of the choice of F, and F2
is G-invariant. In view of this and because we
have for any
fl, f2, f3
E
O(N)
that
jjf17f2jN)f3 IN= Z*jjF1iF2jiP3}i leading at once to the Jacobi identity symmetric biderivation follows.
Similarly the
for
33
fact that
j* JN I
is
an
anti-
Chapter
For
11.
Integrable Hamiltonian systems
We next give a few examples of Proposition 2.25 in the case examples which involve larger groups see Section VI.3.
Example
Consider the
2.30
following automorphism
?,1(P17P2)
of finite group actions
0,
Cn.
C2,
of
(-PliP2)i
corresponds to a diagonal action Of Z2 on C2 which has a line of fixed points. Let compute the algebras of invariants for the induced action and derive from it the Poisson structures on C2 which descend to the quotient. If we denote the standard coordinates on C2
which us
x, and X2 then the invariant functions are the
polynomials whose terms are even in x1, again C2 and the projection map is given by (PI P2) -+ (ql, q2) The map zi is a Poisson morphism, if and only if I-XI X21 (P2, Z*JX1i X21- If we denote I P2). 1 JX1, X21 F(X1 X2) then this condition means that -F(XI, X2) F(-XI, X2) i.e., F is odd in xjL and it follows that F can be factorized as x, times an invariant polynomial. Then the Poisson structure on the quotient is given by JY1 Y2 1 0 2y, G (yl, Y2) where G is defined by
by
hence the
C2/Z1
quotient
is
=
i
=
7
=
=
I
=
i
F(XI,X2)
xG(X2, I X2).
==
where the rank is
zero:
Notice that these Poisson structures
if non-trivial
they
on
Example 2.31 The only other possibility (up to isomorphism) on C2 corresponds to the following automorphism of C2: t2
(PI P2)
=
7
C2 and
C2/t,
have
a
line
regular.
are never
for Z2 to act
non-trivially
(-PI -P2) i
In the notations of the
previous example, a polynomial function is now invariant if and only only of terms which axe of even total degree in x, and X2. Therefore the algebra of invariants is generated by y, x 2, Y2 X2 2, with the single relation y22 XIX27 Y3 y1y3 1 and the quotient space is a quadratic cone. The projection map is then given by (plIP2) 1-+ (p2' 2) and 1.2 is a Poisson morphism. if and only if the polynomial F (ql, q2, 93) 1 PIP27 p2 which defines the bracket, JX1 X2} F(xi, yl) is even, F(X1, X2) F(-XI, -X2) In this case there exists a polynomial G(yl, Y21 Y3) for which G(X2, 2) F(X1, X2)- In terms 1 XIX2iX 2 of the generators Yli Y2 and y3 the Poisson structure on the quotient is then described by the following Poisson matrixif it consists
=
=
=
=
=
=
=
-
i
=
2G(y,
Even if the is
an
7
Y2 i
Y3)
0
YJ
2Y2
-Y1
0
Y3
-2Y2
-Y3
0
Poisson structure is regular (e.g., if F 1, in which case (C2 /221 1*;'10) subvariety of,01(2)*, with its standard Lie-Poisson structure) the quotient (if non-trivial) is never regular: it always has rank zero at the vertex of the
original
=
affine Poisson
Poisson structure cone.
Example
The two
2.32
fective actions of
a
cyclic
preceding examples are easily generalized, giving on C2 Namely let
group
Z3 (P1 i
Here p and q
corresponds
are
to
P2)
=
(TI 6P2) i
7
ep
=
6q
=
possible ef-
1.
6q 1. The map Z3 integers satisfying ep cyclic group of order 1 1.c.m.(p, q) and by the
assumed to be the smallest
an
all
.
effective action of the
=
=
34
=
2. Affine Poisson varieties and their
morphisms
remarks made above every such action is of this form. Suppose first that p and q are not 4 and Y3 Xq2 are invariant coprime, let d denote their g.c.d. and p p'd, q q'd. Then y, =
functions X2 are
and,
since
EP'
=
=
=
'
and dq
chosen such that Y2
are
primitive d-th
=4 1x" 2
roots of
unity,
is also invariant. Then the
we
may suppose that x, and
quotient is
C3 given
in
a cone
The bracket JX1, X21 yd. F(xl, X2) defines a Poisson bracket for which 23 is a y1y3 2 Poisson action if and only if flexi, 6X2) == e6flxi, X2)- If 6: :k 1 then F(xi, X2) is X1X2 times
by
=
=
Xp'Xq' an invariant polynomial and we may define G(yl, Y2i Y3) by XlX2G(xp, q) 2,X2 1, 1 It is easy to check that the Poisson structure on the quotient is then described
(
G(y,IY2iY3)
If
on
the other hand 6
may define G(yi, Y27 described by
=
1 then p
=
0
PY1Y2
pqy.lyf3
-PYIY2
0
qY2Y3
-pqyly3
-qY2Y3
0
q and
Y3) by G(4, XIX2i4)
(
pG(y17Y21Y3) Finally, algebra
a
=
X1X2-
Then
Y1
PYP2-1
0
Y3
-Y3
0
0 Y1 I
-PYP2
F(xl, X2)
is invariant and
The Poisson structure is in this
we
case
coprime then 4 and Xq2 generate the algebra of invariants, hence this polynomial algebra. As above F is divisible by XIX2 and we define G(u, v) by F(xi, X2) and find that the Poisson structure on the quotient is determined by
if p and q is
v
F(xi, X2).
=
=F(x,,X2)by
are
G(4, xq) 2 JYI Y210 =pqy1Y2G(y1,y2). =
Example
2.33
The
cyclic permutation
Z4(PliP27N)
=
(P27NiPl)
gives an action Of Z3 on C3 which will also appear later diagonal form; in order to diagonalize the induced action cubic root of unity and define
Then the action of I E Z3 2
6
U3.
The
algebra
on
=
X1 + X2 + X31
U2
=
X1 + 'EX2 +
U3
=
X1 +
6
2X2
6
(see
C [XI i X2 i
now
X31
VII.7).
It is not in
let
a
e
be
primitive
2X31
(2.19)
+ C:X3-
C[Ul, U2, U31
of invariants is
is given by T*,ul generated by ul, v
=
=
U1,
T*JU2
3
U21
W
=
=
15U2
3 U3 and t
and =
VJU3
U2U3 with
showing that the quotient C3 /Z4 is a cylinder over a cubic surface. projection map is given by (q q2, q3) (ql, q 3 q3, q2q3). in and 1 be and let and i to a U2 polynomials weight U3, all of whose ul, Assign X W ui terms have the same weight modulo 3. According to Example 2.11 these lead to a Poisson structure on C3; the above action Of Z3 will be a Poisson action if and only if all terms in the product XW have weight 0 modulo 3; equivalently X and W must both be weight homogeneous and the sum of their weights must be a multiple of 3. The resulting Poisson matrix for the quotient is easily written down. the
single relation t3
U1
Section
on
on
In terms of the
new
=
vw,
coordinates the
,
-
35
,
Chapter
11.
Integrable Hamiltonian systems
Example 2.34 As a final example let us consider the natural action of the symmetric Sd on Cd. It is well-known that the algebra of invariant functions for this action is freely generated by the elementary symmetric functions, in particular the invariant :ftmctions constitute a polynomial algebra and the quotient is just Cd, the projection map being
group
(pi By
I
...
1Pd)
transformation similar to
a
(PI +P2
-+
(2.19)
+
-
-
-
+Pd,
the action
brackets which descend to the quotient are the metric polynomials is a symmetric polynomial.
be
can
*
*Pd).
'
partly diagonalized. The
Poisson
for which the bracket of any two sym-
ones
The fourth and final construction is that of
variety. We show that the resulting
1P1P2
...
removing
a
divisor from
space still has the structure of
an
an
affine Poisson
affine Poisson
variety.
Proposition 2.35 Let (M, I-, -1m) be an affine Poisson variety and let f E O(M) be a regular function which is not constant. Then there exists an affine Poisson variety (N&,*JN) and a Poisson morphism N -+ M which is dominant, having the complement (in M) of the zero locus of f as image. Proof
Consider functions
a new
variable t and define
an
affine
variety
N
by
its
ring O(N) of regular
follows:
as
O(N)
=
O(M)[t] idl(f t 1) -
Let
denote the canonical
us
is the
complement of the
divisor of
N -+ M.
7r :
If
f.
7r
Clearly
is to be
a
ir
is dominant and its
Poisson
morphism, then
image we axe
Of O(M) (we made 19i , gj IN Jgi, gj I for a system of generators gi I I A notational identification between -7r*gi and gi). In view of Proposition 2.4 the only way to
forced to define a
projection by
zero
extend this to Thus
one
=
...
a
Poisson structure
on
N is upon
19i, tIN By the
using I f t
-
0
1, gi
(for
i
==
1,
.
.
.
,
k).
needs to add the brackets
same
erators gi,
.
.
.
_t21g,J f IN-
=
(2.20)
proposition it now suffices to check the Jacobi identity on the system of gengk, t; since we know it is valid for gi gk the following easily established ,
identity suffices:
119ii gjJ7 tIN
JJ9ii9j}itJN
(one
uses
This
gives the desired
=
-t2 ffgi
+
Jjgj t1i giIN
gjJ7 fIN7
Poisson bracket
on
+
JJt7 gili gjIN
which is
an
=
0
immediate consequence of f G Cas(M) then
(2.20)).
N. Note that if
Cas(N)
Cas(M)[t] idl(f t 1)
=
-
Remark 2.36 tions
be
on an
irreducible)
functions of
Another way to state the above result is that the algebra of rational funcvariety with poles only at some fixed divisor (which need not
affine Poisson
an
is also
a
(finitely generated)
affine Poisson
variety
can
in
Poisson a
algebra. Clearly, the
similar way be turned into
36
a
field of rational Poisson
algebra.
2. Affine Poisson varieties and their
morphisms
Decompositions and invariants of affine Poisson varieties
2.4.
There stitute of
are
three natural decompositions of an affine Poisson variety, two of which convarieties. The first one discussed here is a level decomposition by the
algebraic
the nonCasimirs, the second is a decomposition according to rank and the last one is the decomposition into symplectic leaves. Due to its non-algebraic nature, algebraic one the latter will only indirectly (via the other decompositions) be used in this book and is -
-
discussed in Section 4.
a
The Casimir
decomposition
decomposition of an affine Poisson variety which is naturally associated to its algebra of Casimirs applies (and will be applied) equally for other subalgebras of O(M), so let us introduce it for an arbitrary subaJgebra A (containing 1) of O(M). To each m E M we may associate an algebra homomorphism X,,, : A -+ C by f -+ X,,, (f f (m), allowing us to The
associate to
m
E M the ideal
If
-
x,,(f) I f
E
Al
A, which is a point in Spec A, the spectrum of A; we will denote this natural map M Spec A by irA. Another way to see how this map comes out is as follows. The inclusion map z : A c M allows one to associate to a prime ideal I of O(M) the prime ideal I n A of A, hence leading to a morphism of
?,*
:
Spec 0 (M)
-+
Spec A
of affine schemes. The space Spec O(M) contains M as the set of its closed points this seen as the maximal spectrum, the space of all maximal ideals, of O(M) -
set may also be
-
just the restriction of %* to M. We prefer to work with M rather than with Spec O(M) since that is the space we originally started from; however we like to keep Spec A, even when A is finitely generated so that its underlying variety is an affine variety, because we will also be interested in the fibers of z* over points which are not closed. Notice that the irreducible components of each fiber of 7rA are affine varieties and a complete set of equations (from which we may choose a finite generating set) for the fiber which contains m E M is given by Vf E A: f (x) xn(f). (2.21) and
our
map irA is
=
Often
our
statements will be about the
general fibers of
irA:
when
saying that
some
property holds for a general fiber we mean that it holds for the fiber over a general point, i.e., for all closed points which do not belong to a certain divisor. We denote the Krull dimension of A The
dim Spec A. by dim A; if A is finitely generated then it is a basic result that dim A following proposition relates the dimension of A to the dimension of the fibers of irA
(see [Sha]
=
Ch.
116).
Proposition2.37 All (non- empty) fibers of irA : M -+Spec A have co-dimension dim A and the general fiber has co-dimension precisely dim A.
at most
Let us apply this to the case where A is the algebra of Casimirs of an affine Poisson variety (M, 1-, -1), which we denoted by Cas(M). As an application of Proposition 2.18, the following proposition shows that the fibers of M -+ Spec Cas(M) inherit a Poisson structure, thereby giving each irreducible component the structure of an affine Poisson variety. 37
II.
Chapter
Integrable Hamiltonian systems
Proposition 2.38 Every (non-empty) fiber.F of -7rc,,(M) inherits a Poisson bracket from 1-, -1 and all Hamiltonian vector fields Xf, f E O(M) are tangent to these fibers. Moreover RkJ +F < RkJ I with equality for a general fiber .97. -
-
,
-
,
Proof It suffices to prove the property for a (non-empty) fiber any such fiber and m E F. In view of (2.21) the ideal of F is
If so
it is
Poisson ideal of
a
O(M)
to obtain the
same
equality for
a
x,,,(f) I f
E
we use
at every
item
(ii)
applies.
In order to determine the rank
of Definition 2.6. It shows that the rank
point. This leads
general fiber, just
closed point. Let F be
Cas(M)l
and Proposition 2.18
of the restricted Poisson bracket of both structures is the
-
over a
generated by
at
to the
once
inequality;
recall that the rank of
in order
is maximal
on a
Zariski open subset of M.
Referring
I
Example
to
from the Poisson structure
2.24
we see
that the Poisson structure
on
M
can
be reconstructed
the fibers of M -+
Spec Cas M, where Spec Cas M plays the role of the space of parameters. Therefore we will call Spec Cas M the parameter space and the map M -+ Spec Cas M the parameter map of the affine Poisson variety (M, 1-, .1). The fibers of the parameter map will also be called level sets of the Casimirs because picking a -fiber (resp. over a closed point) corresponds to fixing some (resp. all) Casimirs. The decomposition of M into the'fibers over closed points is called the Casimir decomposition. on
Remarks 2.39 1. A Poisson spaces
(for
2.
Poisson
a
morphism does not necessarily induce a map of the corresponding parameter counterexample, see Example 3.14 below).
Since
Cas(M) is a Poisson subalgebra, of O(M) (actually a Lie ideal), 17rCas(M) morphism, Spec Cas (M) having the trivial Poisson structure.
3. For special fibers F of the parameter strictly smaller, see Example 2.54 below. 2.40
Proposition
Let
(M, 1-, .1)
be any
is
a
map the rank of the Poisson structure may be
affine
dim Cas(M) :
Poisson
variety. Then
CoRkJ-, J.
Proof Let F be
a
general
structure. Then dimM
fiber of the parameter map and let I-, dim.F dimCas(M) and RkJ-, .1jr. =
-
=
RkJ-, jp) equals the number of independent of 0(,F) at a general point of 17 we find that dim Cas(M)
=
dimM
-
derivations
dim.F < dimM
38
-
-JY be the induced Poisson RkJ-, .1. Since dim,r (resp.
(resp.
RkJ-, .1
=
Hamiltonian
CoRkJ-, -1.
derivations)
2. Affine Poisson varieties and their
morphisms
studying integrable Hamiltonian systems we will exclusively be interested in affine inequality is an equality, because only in that case, the varieties or manifolds which are traced out by the flows of the integrable vector fields, can be affine (sub-) varieties (of the phase space). It motivates the following definition. When
Poisson varieties for which the above
Let
Definition 2.41
be
(M,
an
We say that its
affine Poisson variety.
algebra of
Casimirs is maximal if dim Cas(M)
i.e.,
if the
CoRkj-, J,
=
level set of the Casimirs has dimension
general
to the rank of the Poisson
equal
structure.
Following the above comment, we will want to know if maximality of the algebra of preserved by the different constructions we made (restriction, product, quotient and taking away a divisor). We show that maximality of the algebra of Casimirs is indeed preserved by restriction of the Poisson structure to a general level of (a subalgebra of) the algebra of Casimirs. Similar propositions for the other constructions which we have discussed are easier to obtain and axe not made explicit here. Casimirs is
Let (M, 1-, -1) be an affine Poisson variety whose algebra of Casimirs is for any subalgebra A of Cas(M) the induced Poisson structure on the general Spec A is also maximal. 2.42
Proposition
maximal. Then
fiber of M
-+
Proof
subalgebra of Cas(M) and
Let A be any M
-+
Spec A
so
that dim -T
=
dim M
-
let Y denote
dim A.
Obviously,
a
if
general fiber of the natural map f E Cas(M) then fly E Cas(.F)
and
dimCasY > Since
Cas(M)
dimCas(M)ly
is maximal and Jr is
dim.F
showing that for
-
a
dim Cas(.F)
general fiber
general
< dim M
F the
=
dimCas(M)
dimA.
-
it follows that
-
dim Cas(M)
algebra
=
Rkj-, j
of Casimirs of
I
-
-
,
IT
=
Rkj-, jy,
is maximal.
Example 2.43 The algebra of Casimirs needs not be maximal. The simplest counterexample is given by the following Poisson matrix, defining a Lie-Poisson structure on C3,
( coming from
a
solvable Lie
0
0
ax
0
0
-Y
-ax
y
0
algebra. F(x, y, z) 8F i9z
)1
will be
i9F
a
Casimir if and
only
if
9F
=ax5x -y-=0C9Y
easily solved giving F cxy" +d where c and d are integration constants. a regular function, if a 0 Q then the level sets of F are not even 0 algebraic varieties. It should however be remarked that for any a p1q E Q we can restrict C, the Poisson structure to a general level, which is an affine algebraic surface given by xqyP
These If
a
equations
are
=
N then F is not
=
=
where C E C.
39
Chapter
Example
Taking
up
Integrable Hamiltonian systems
Example
2.11
again
we see
that every
polynomial F(x,'Y, z)
C3. Namely, consider the Poisson C3 defined by the following Poisson matrix (which corresponds to X 1 and 0 as a
pears on
2.44
11.
Casimir for
some
Poisson structure
on
=
0
OF OZ
-OF
0
OZ OF
Then F is
=
F):
OF
OY OF Ox
OF
;9_Y
al>-
structure
0
_5X_
Casimir of this Poisson structure. All the fibers of the parameter map are twothey all have rank two, the singularities of these
a
dimensional and if F is non-constant then fibers
being precisely
the points where the rank
We also
Example
2.45
lying
closed points may have
over
give
an
drops
to
zero.
example to show that the fibers (of the parameter map) higher dimension. Consider on C4 the following Poisson
matrix:
0
0
0
X
0
0
0
-Y
0
0
0
-Z
-X
y
Z
0
It is easy to verify that the algebra of Casimirs is given by Cas(C4) C[xy,xz]. Thus the parameter space Spec CaS (C4) is isomorphic to C2 and the fibers of the parameter map b for a, b E C. For (a, b) 54 (0, 0) the fiber is two-dimensional, are given by xy a, xz =
=
=
however xy 0 has x 0 as a component, hence is three-dimensional. Notice that xz on this special fiber the algebra of Casimirs of the induced Poisson structure is not maximal =
=
=
anymore.
its
We prove one more proposition about the algebra of Casimirs which may be useful for computation; the type of axgument used in the proof will be used several times in the next
section.
Proposition 2.46 The algebra of Casimirs of an affine Poisson variety (M, grally closed in O(M).
is inte-
Proof
Suppose
F E
O(M) and Ein-0 ajF' Cas(M). Taking
need to show that F E
0
=
i.e.,
F E
O(M). Cas(M).
Since
0, with
Ig, ajF'j ( ' E i=O
for all g G
=
n was
ai E
Cas(M),
an
the bracket with any g G
=
n
E(i i=1
supposed minimal,
40
-
I)aiF'-')
it follows that
=
1 and
O(M)
we
n
minimal. We
find
Ig, F1, Ig, Fj
=
0 for all g EE
O(M),
2. Affine Poisson varieties and their
9
Another
The rank
decomposition to the rank of the affine Poisson
decomposition relates define for 0 < i < 1 2
(M,
morphisms
variety
at each
point. Given
Rkf -, -1 Mi
Then the components of each Mi
=
E M
lp
I Rkj-, -1
<
2ij.
affine varieties: let g be the Poisson matrix of I. , j can be described as a so-called
are
arbitrary system of generators, then Mi determinantal variety (see [ACGH] Ch. 2)
with respect to
an
Mi
lp
=
E M
I
all determinants of order 2i + I of g vanish at
pl,
7 description which also gives the equations defining these algebraic sets Obviously Mi C is regular if and Poisson structure and the M has for one also Mr r Rkj-, -1/2 Mi+,; 0; finally Mo M if and only if the bracket is trivial. We call the (singular) only if Mr-1 stratification by the Mi the mnk decomposition of M. a
.
=
=
=
=
The
algebraic
sion. Therefore
Mi may be reducible and their components may be of and j E N define for 0 < i < I 2 Rkj-, .1
sets
we
Mij Thus
Mij
E
p
-1)
(M,
(M, 1-, -1) be an affine Z[R, S] is defined by
Let E
p(M) The
Mi,
p E D and dim D
=
j
(finite) union of the j-dimensional irreducible components of Mi. This leads following polynomial invariant for an affine Poisson variety.
Definition 2.47
p(M)
comp. of
dimen-
is the
at once to the
=
Mi I 3D irred.
varying
polynomial
can
=
E pij RSj,
also be
pij
Poisson
variety.
Its invariant
irred. comp. of
polynomial
Mij.
represented by a matrix (of minimal size), called the invariant by taking as (i, j)-th entry the integer pij (labeling of matrix
matrix of the Poisson structure entries starts here from
Let
Proposition
2.48
there exists
biregular
a
zero). and
(MI, 1-, .11)
map
0: MI
-+
(M2, 1*) '12)
be two
affine
M2 which is Poisson, then
Poisson varieties.
p(MI)
=
If
p(M2).
Proof
biregular Poisson morphism is also Poisson. From points x, the rank at x equals the rank at O(x). of the subvarieties M, or Mr, is a biregular map onto the
We noticed that the inverse of
Proposition 2.17
a
it follows that for all
Thus the restriction of
0
to any
subvarieties N, and Nr,. It follows that
p(M)
=
p(N).
1
Strictly speaking the Mi are defined here as affine schemes, i.e., the ideal generated by general not reduced. The invariant, defined below, leads to another invariant when passing to the radical of this ideal, but all properties listed below also hold 7
these determinants is in for this
(coarser)
invariant.
41
Chapter
Proposition and let p
=
Let
2.49
p(R, S)
11.
(M, 1-, -1)
Integrable be
an
Hamiltonian systems
affine Poisson variety of rank
2r and dimension d
its invariant.
(i.) p(M) R'Sd(I + O(R-', S-')), (2) If M is non-singular the coefficients pij of p E pijRSj, satisfy joij 0 for 2i > j. (3) For any r < d/2, there exists an affine Poisson variety whose invariant is RrSd. =
=
=
Proof M we have Prd Since M is irreducible and Md 1, all other Mij have by definition being given as the intersection of hypersurfaces in M they also have lower =
=
lower rank and
dimension. This shows
(2)
As for
(iL).
rely on the symplectic foliation, described in Section 4 below; an algebraic proof which would allow to remove the assumption about M being non-singular is still missing (in view of Proposition 2.18 it would suffice to show that the irreducible components of the Mi are affine Poisson subvarieties of M). Through every point of M passes a leaf which inherits a symplectic structure from the Poisson structure, so on the one hand all Hamiltonian vector fields at this point (which span a subspace of dimension equal to the rank 2r of the Poisson structure at this point) are tangent to such a leaf, on the other hand such a leaf is entirely contained in the subset M2,; thus every irreducible component of M2, has dimension at least 2r showing (2). For
Before
(3)
we
we
need to
take the canonical Poisson structure of rank 2r
give
a
refinement of the
invariant, let
us
consider
on
C2d (Example 2.7).
some
first
0
examples.
Example 2.50 An affine Poisson variety is regular if and only if its invariant polynomial is a monomial, i.e., is of the form R'S', where 2r is the rank and s the dimension of the variety. In particular the invariant polynomial of the trivial structure on an afline Poisson variety of dimension s is S'.
Example 2.51 For the Poisson structures on C2, which axe defined by a single polynomial jx, yj, with W:A 0 we have p RS2 + kS, where k is the number of components of W(x, y) the plane curve defined by W(x, y) 0. Its invariant matrix is thus given by =
=
=
( It follows in
0
k
0
0
0). 1
particular that the polynomial invariant is not a complete invariant: all nonpolynomials W(x, y) lead to a Poisson structure on C2 with invariant
constant irreducible p
=
RS2
+ S.
Example
2.52
The
Sklyanin brackets and
their
generalizations (see Example 2.9) lead for
the various values of the parameters to a lot of different invariant polynomials, giving an easy proof that many of these Poisson structures are different. We give the different polynomials -
which
are
easily computed
-
in the
following
table
(the integers i, j, k
and range from 1 to 3; a dash means that the values of the parameters the relation alb, a2b2 + a3b3 0)-
:--
42
are
are
taken different
incompatible with
P
all b
=
0
2. Alfine Poisson varieties and their
morphisms
all
ak
a
=
0
s4
=
bj
=
0
RS4
+
2S3
bi
=
bk
=
0
RS4
+
2S3
bi
=
all b
0
RS4
2S3
+
RS4 + S3
RS4
+
S3 +,52
0
RS4
+
S3 + S2
=k
=
aj
0
=
S3 +
+
all
S2
RS4
a
+
: S'
0
+ S
3S2
S2
+
RS4 +,52 RS4 + S3 + S
RS4
RS4 +3S2
0
+
RS4
0
=
=
RS4
bi
bk
ai
RS4
+
+
RS4
2S2
s2 +2S
+
RS4
+ 2S
S2 + 2S + 4S
Table I
A
above there
precise description of
more
polynomial corresponds are
Spec Cas(M) by
E
affine Poisson
affine Poisson vaxieties. Then
components for each
c
an
variety
can
be
given by combining the
invariant with Proposition 2.38. We know from that proposition that to each point of the affine variety Spec Cas(M) a fiber whose irreducible
P,-(M)
=
P
we
may define
a
polynomial
invariant
p,(M)
(-7r-I (M) (C) Cas
assumption that the fiber over c is irreducible; if not then the right hand side in just replaced by the sum over all irreducible components. Thus we label each point of Spec Cas(M) by the invariant polynomial of the corresponding fiber over it and obtain in this way a more sensitive invariant for affine Poisson varieties. In the examples which follow we will only consider the fibers over closed points c.
under the
this definition is
Example
2.53
the dual of
a
of this space
The
simplest non-trivial example is given by the Lie-Poisson structure on semi-simple Lie algebra (see Example 2.8). A basis Ix, y, zJ be chosen such that the corresponding Poisson matrix takes the form
three-dimensional can
( The
algebra
zero,
we
-Z
Y
Z
0
X
-Y
-X
0
(2.22)
.
y2 Z2] hence Spec Cas(M) can be clearly given by C[X2 2 Z 2; we denote the corresponding by evaluation on the element X2 Y Since (2.22) has only rank zero at the origin, which lies in the fiber over
of Casimirs is
identified with C
coordinate
0
by
u.
conclude that P
=
,
RS3 + 1 and
Pc
RS2 RS2 +1
43
if if
U(c) :;:A 0, U(C) 0. =
Chapter It may also be
depictured
as
11.
Integrable Hamiltonian systems
follows.
0 X
RS2+1
U
RS2
2.54 For the Heisenberg algebra the Lie-Poisson structure can be written as x. As above one finds that the algebra of Casimirs is given 0, ly, zj jx, zj jxj yj by C[x], and again its spectrum can be identified with C (with coordinate u) by evaluation
Example =
on
=
the Casimir
=
The Poisson structure has
x.
entire level of the Casimirs
level
sets).
(showing
It follows that p
RS3
=
that
is
case
depictured
as
in
zero on the plane x Proposition 2.38 needs
=
0 which is
an
not hold for all
S2 and
+
f RS2
PC
This
rank
now
equaJity
(c) U(c)
if
S2
0,
U
if
0.
follows. 0 X
S2
Example
An
2.55
interesting example is found by taking the Lie-Poisson structure on following basis
Consider the
gf(2)*.
1
X
=
0
0
( 0)
for g and let x,
.
.
.
,
t be the
to
x
T=
0
generators of 0 (Z),
1
T. The
X,
(0 0), corresponding Poisson
0
Y
-Y
0
we
-
X
-Y
-
t
Y
0
-Z
Z
0
=
+ t and xt
the points
t
0
-Z X
Cas(q*) C[x+t, xt-yz]. It follows that Spec Cas(g*) is in this case isomorphic pick the isomorphism. such that the standard coordinates u and v on C2 correspond
have
we
C2 ;
0
0 0),
given by
0
to
Z=
0
Z
and
0
(0 1),
Y=
,
0
matrix is
U
RS2
on
-
yz
(in
the line y
that =
z
order). =
pe
0,
x
Since the rank of the Poisson structure is two except for RS4 + S and t, we find that in this case p
=
RS2 RS2 +I
=
if
U2(e)
if U2 (C)
44
4v(c), 4V (C).
2. Affine Poisson varieties and their
Example
2.56
(Section VII.7).
The
following example will
structure determined
by the
up later when
come
t6l
In terms of coordinates
morphisms
for C'
we
studying the
Toda lattice
consider the Lie-Poisson
Poisson matrix
0
-t2
tj
0
t3 -t3
-t1
t2
0
tT
0
(-T )
with T
0
(2.23)
C[t1t2t3j t4 + t5 + t6], so that (in Paragraph VII.7.1) that CaS(C6) C2, with coordinates u and v, corresponding to t1t2t3 and t4 + t5 + t6 (in that order). By computing a few determinants one sees that,the rank is zero 0 (1 < i < j ! 3) on the three-plane tj t2 0, two on the three four-planes ti t3 tj We will show later
Spec Cas (C6)
can
=
be identified with
=
=
=
=
and four elsewhere. From it
one
p=R2,56 PC
It is
==
=
easily obtains the following invariant polynomials:
f3R
+
3R84
+
S3,
R2S4 2S4 + 3RS3 + S2
if if
U(c) U(C)
0, 0.
represented by the following diagram.
;3+S2 u
Proposition 2.57 Let (M, I., .1m) and (N, I* JN) be two affine their product M x N be equipped with the product bracket. Then
p(M In
x
N)
Poisson varieties and let
p(M)p(N).
=
particular, if the invariant polynomial of an affine Poisson variety variety is not a product (with the product bracket).
is irreducible then this
Poisson
Proof We use as above Mi, Nj and (M x N)i as notation for the determinantal varieties associated to M, N and M x N respectively. The coefficients of the invariant polynomials and By Proposition 2.21, we have p(M), p(N) and p(M x N) are written as pi'.,
pi2j
(M
x
N)i
U k+l=i
45
pi'j.
Mk
x
N1.
Chapter
11.
Integrable
Hamiltonian systems
Using the fact that the irreducible components irreducible components,
pixj
we
#j-dim. irred.
E
#j-dim.
of
a
product
are
precisely
the
products of
find comp. of
(M
x
irred. comp. of
N)j Mk
x
N,
k+l=i
E 1:
(#m-dim.
irred. comp. of
Mk) (#n-dim.
irred. comp. of
NI)
k+l=i m+n=j
1: 1: PklrnPin
-
k+l=i m+n=j
This shows that
p(M
Remark 2.58
It would be interesting to determine the invariant(s) of the Lie-Poisson arbitrary semi-simple Lie algebra and to relate it to the theory of (co-)
structure of
an
x
N)
=
p(M)p(N).
adjoint orbits.
46
3.
3.
Integrable Hamiltonian systems and their morphisms
Integrable Hamiltonian systems
and their
morphisms
In the
study of semi-simple Lie algebras the notion of a Cartan subaJgebra plays a corresponding object for affine Poisson spaces is an integrable algebra: a maximal commutative (in this context called involutive) subaJgebra. An affine Poisson variety with a fixed choice of integrable algebra is what we call an integrable Hamiltonian system. The study of integrable Hamiltonian systems can be seen as a chapter in Poisson geometry; for example we will see that all propositions which we proved for affine Poisson varieties have their equivalents for integrable Hamiltonian systems. Our definition is an adaption of the classical definition of an integrable system on a symplectic manifold (see e.g., [AMI]) to the case of an affine Poisson variety. Notice that we do not ask that the rank of the Poisson variety be maximal (or constant). Another difference is that the classical definition demands for having the right number of independent functions in involution, while we ask for having a complete algebra (of the right dimension) of functions in involution, completeness meaning here that this algebra contains every function which is in involution with all the elements of this algebra. On the one hand this adaption is very natural, it is even inevitable if one wants to discuss morphisms and isomorphisms of integrable Hamiltonian systems. On the other hand it is not easy to verify completeness of an involutive algebra, e.g., the (polynomial) algebra generated by a maximal number of functions in involution needs not be complete. Accordingly we will also prove some propositions in this section which will be useful for describing and determining explicitly the integrable algebra in the case of concrete examples. dominant role. The
3.1.
Integrable Hamiltonian systems
Definition 3.1
one
has
f f, Al
(M, JA, Al
-1)
Let
called involutive if =
0 -#>
Hamiltonian system
f
0; c-
A.
be we
an
on
affine Poisson
say that it is
The
affine Poisson varieties
variety. A subalgebra A of O(M) is complete if moreover for any f E O(M)
triple (M,
A)
is called
a
(complete)
involutive
-
Lemma 3.2
Let (M, A) be an involutive Hamiltonian system. (i.) If A is complete then A is integrally closed in O(M); (2) The integral closure of A in O(M) is also involutive and is finitely generated
when
A is finitely generated. Proof The
proof of (i.)
goes in
exactly the same way as the proof of Proposition 2.46, replacing O(M) by g Ei A. It is well-known that if A is finitely generated then its integral closure in O(M) (defined as the set of all elements 0 of O(M) for which there exists a monic polynomial with coefficients in A, which has 0 as a root) is also a finitely generated algebra (see e.g., [AD] Ch. 5). To check that it is involutive, we first check that
Cas(M) by
A and g
E
every element of the integral closure of A is in involution with all elements of A. be an element of O(M) for which there exists a polynomial
p(X) for which
P(0)
=
Xn +
a1Xn-1
+
-
-
-
Thus,
let
0
+ an
0 and with all ai belonging to A; we For any f E A the equality f P(o), f J
that the polynomial is implies as in the proof of Proposition 2.46 that 10, f I 0, upon using the minimality of P. Using this, it can now be checked by a similar argument that any two functions in the integral closure are in involution.1 of minimal
=
degree.
=
47
assume =
0
Chapter II. Integrable Hamiltonian systems
Every involutive algebra is contained in an involutive algebra which is complete, but the general not unique. This is contained in the following lemma.
latter is in
integral
(3.) (2) (3)
(M, 1-, .1, A)
Let
Lemma 3.3
be
an
involutive Hamiltonian system and denote
of the field of fractions of A. The subalgebra An o(m) of O(M) is also involutive; A; If A is complete then A n O(M) A is contained in an involutive subalgebra B of O(M) which dim A. if dim B
by A the
closure
=
is
complete;
it is
unique
=
Proof
(e. g., from [AD] Ch. 5) that A n o (m) can be identified as the set of elements 0 of for which there exists a polynomial (which is not necessarily monic) with coefficients
Recall
O(M) in
A,
which has
0
root. if
as a
0
P(X)
E
A n O(M) and
=
aoXn
+
aXn-I
+
-
-
+ an
-
0, then polynomial of minimal degree (with coefficients ai in A) for which P(O) of in the P of the as proof 0 minimality (again using 0, upon implies 10, Al JP(O), Al Proposition 2.46). In turn this implies that if 0' is another element of An 0 (M) the equality JP(O), O'l 0 leads to 10, O'l 0. Thus A n O(M) is involutive, showing (i.); from it (2)
is
=
a
=
=
=
=
follows at
once.
A n O(M); if the latter is complete complete we pass to AO unique involutive subalgebra of O(M) which contains A and is complete. If not, we 0 and repeat the above construction to add ail element f E O(M) \ AO for which If, AO I dim AO + 1 we are done after a finite number of steps; because of obtain A,. Since dim A, the choice of f the algebra which is obtained is not unique in general (interesting examples 0 of this are given below). If A is involutive but not
=
it is the
=
==
only be interested in involutive algebras of the maximal possible proposition. We know from Lemma 3.3 that such an algebra A dimension, given by A I if A has a unique completion, which we will denote by Compl(A) (or by Complf fl, is generated by If,, A 1) In this text
will
we
the next
.
Proposition
3.4
.
.
-
,
Let
(M,
A)
be
an
involutive Hamiltonian system. Then 1
dim A ::' , dim M
-
2
(3.1)
Rkj-, .1.
Proof Consider map A C
a
general fiber.F of
the map M
-+
SpecA
which is induced
by
the inclusion
O(M). By Proposition 2.37, dim.F
=
dim M
-
dim A.
(3.2)
equals the number of independent derivations of O(Y) at a general point of F and involutivity of A implies that such derivations can be constructed using functions from A.
dim.F also
48
Integrable Hamiltonian systems and their morphisms
3.
To m
see
the
latter, recall that the ideal of F is generated by the functions f arbitrary but fixed and f ranges over A. For any g E A we have
E 97 is
Xg(f hence
X.
-
X-M)
is tangent to the locus defined
If, gj
=
=
by the ideal of F, i.e.,
to Y and
O(Y) using elements of A. Next we show that the dim Cas(M) independent derivations, giving a lower bound
-
nested sequence of
where
construct
we can
elements of A lead to for diM.F. Consider
a
subalgebras Cas
where dim Aj+j
X'-"(f)
0,
derivations of dim A
-
=
Ao
C
Ai
C
A2
C
c
...
A,
=
O(M),
dim A, + 1, in particular r Rkj 1. If ni denotes the number of independent vector fields on M coming from A, (i.e., having independent vectors at a general point) then obviously ni < ni+l :5 ni + 1, no 0 and n, r. It follows that ni i for all i. It gives the following lower bound =
=
-
-
,
=
dim.F > dim A
Combining (2.40), (3.2)
and
(3.3)
we
=
-
=
dim Cas (M).
(3.3)
find I
dimA
We
finally get
<
(dim M + dim Cas (M))
to the definition of an
<
dim M
-
2
Rkj-, .1.
integrable Hamiltonian system (on
(3.4)
an
affine Poisson
variety). Definition3.5 imal and A is
a
If
(Mj-,-j)
is
an
affine Poisson
complete involutive subalgebra dimA
=
dimM
of
-
variety whose algebra of Casimirs is O(M) then A is called integrable if
1Rkj-,
(3.5)
2
The
triple (M,
A)
is then called
an
max-
integrable Hamiltonian system and each
non-zero
vector field in
Ham(A) is called
=
JXf I f
E
Al
integrable vector field. The dimension of A is called the dimension or the degrees of freedom of the integrable Hamiltonian system. M is called its phase space and Spec A its an
base space. If A, and
A2
axe
two different
subalgebras
of 0 (M) which make 0 (M) into
Hamiltonian system then every non-zero vector field in the intersection is called a super-integrable vector field.
49
an
integrable
Ham(AiL) n Ham(,42)
Chapter
11.
Integrable Hamiltonian systems
Remarks 3.6 1. What
we
call
an
vector field is in the literature often called
integrable
an
integrable
system; the distinction we make is motivated by the fact that the datum of one integrable vector field Xf (or its corresponding Hamiltonian f) does not suffice in general to determine A (see Examples 3.10 and 3.11 below).
(3.5);
structure
it
-
in
the condition that the
added in the
was
hypotheses
algebra of Casimirs
to stress that it is
approach affine Poisson varieties integrable Hamiltonian systems.
our
maJ do not admit 3.
(3.4)
In view of
2.
from
whose
a
is maximal follows
condition
algebra
on
the Poisson
of Casimirs is not maxi-
Completeness of the integrable algebra A implies that Cas(M) c A and A can be intermediate involutive object between Cas(M) and O(M); for example, it follows
seen as an
from
(3.4)
and
(3-5)
that
I(dim M + dim Cas(M)),
dim A
2
which supports this assertion. The commutative
triangle of inclusions
OM I
\ A
induces,
as
Cas(M)
-
explained in Paragraph 2.4, the following commutative triangle of dominant (Pois-
son) morphisms. M "ro-(M 0",(M, -7rA
Spec A
-
-
7r
Spec Cas(M)
Thus the parameter map irc , m, which maps the phase space to the parameter space, can be factorized via the map 7rA : M -+ Spec A from the phase space to the base space; we call the latter map the momentum map. The irreducible components of the fibers of the momentum the level map axe affine varieties which will play a dominant role in this text. We call them sets
of
the
integrable Hamiltonian system
or
the level sets
of A
for short.
technicallity alluded to at the beginning of this section. We know from Lemma 3.3 abstractly how to complete an involutive algebra A (say of the maximal possible dimension), but it does not lead to an explicit description of the completion when studying concrete examples. The following proposition gives sufficient and checkable conditions for such an algebra A to be complete; it will be used several times when we get to the examples. We
now come
to the
50
Hamiltonian systems and their morphisms
Integrable
3.
Proposition 3.7 Let (M, I., J) subalgebra of O(M) of dimension
be
dimA
affine
an
=
dimM
Poisson
-
variety and let A be
involutive
an
IRkj-, J.
2
Then A is two
complete, hence integrable, if the fibers of 7rA properties the
(i.) (2)
general fiber is irreducible; fibers over all closed points have
the
the
same
M
:
Spec(A)
-+
have the
following
dimension.
Proof 0 for some f E O(M). We complete, i.e., f 0 A and If, Al algebra generated by f and the elements of A, which has by Proposition 3.4 the same dimension as A. By Lemma 3.3 f belongs to the integral closure of the quotient field of A. Thus f Ei O(M) is a root of a polynomial Q(t) E A[t]. Consider the following commutative diagram which is induced by the inclusion A C W. Let
denote
us
suppose that A is not
by A!
=
the
M IrAl
Spec A! If
Q(t)
has
degree
at least two then
z
is
a
Spec A
ramified
covering
map of
degree
at least
two, hence
the fiber of -7rA over a general point P has at least two components, which axe the fibers of -7rA, over the antecedents %--l (P). This is in conflict with assumption (L), hence Q(t) is of Since f E 0 (M) \ A neither p, nor P2 are constant. Therefore Spec A which corresponds to an algebra homomorphism onto C which sends both p, and P2 to 0. This closed point is the image under z of a point which is not closed, namely the corresponding algebra homomorphism. can take any value on f.
degree
one,
there is
a
Q (t)
closed
P1 t + P2
::::::
point
-
P in
Then the fibers of 7rA, over these points have dimension one less than the dimension of the dim A by assumption (2). Since A! has the same dimension which is dim M
7r. '(P)
fiber
-
all fibers of 7rA, have dimension at least dimM that A is complete. as
A,
We have
tangent
seen
in
Proposition
a
contradiction.
2.38 that all Hamiltonian vector fields
It follows I
Xf, f
E
O(M)
are
parameter map. Similarly we show now that all integrable vector twigent to all fibers of the momentum map; in addition they have the
Xf, f E A are special property to pairwise Proposition
3.8
nian vector fields in
they
commute.
Let (M, 1., -1, A) be an integrable Hamiltonian system. Then all HamiltoHam(A) are tangent to all fibers of the momentum map 7rA : M -+ Spec A
all commute; the irreducible components of these fibers are affine varieties and the of the general fiber is 12 RkI., -1, which coincides with the number of independent
dimension vector
dimA,
to all fibers of the
fields
and
-
fields
in
Ham(A).
51
II.
Chapter
Integrable Hamiltonian systems
A,) is another integrable Hamiltonian system, then super-integrable vector If (M, fields in Ham(A) n Ham(Al) are tangent to the (strictly smaller) intersection of the fibers of the corresponding maps irA and irA, Proof
0, hence X, if, gJ f E Ham(A) we have X,f Clearly these fibers are affine varieties and commutativity of the vector fields in Ham(A) follows from item (3) in Proposition 2.3. The dimension of a 1 in view of Proposition 2.37. Our claim about general fiber is dimM dimA 2 RkJ-, .1 0 super-integrable vector fields follows at once from the first paxt of the proposition. tangent
Then for any
Ham(A).
Let g E
is
=
-
We
=
=
to all fibers of 7rA.
get
now
to
examples of integrable Hamiltonian systems, super-integrable vector field.
first
some
examples of
will give two
a
in
particulax
we
Example3.9 If (M, I., -J) is anaffine Poisson variety of rank two whose algebra of Casimirs is maximal, then any function F which does not belong to Cas(M) leads to an integrable Hamiltonian system. Namely A ComplICas(M), F1 is obviously involutive and dim A dim M dim Cas (M) + I 1, hence A is integrable; clearly its level sets are just algebraic =
=
=
-
curves.
This well-known fact is often expressed by saying that in one degree of freedom all Hamiltonian systems are integrable (although the condition that the algebra of Casimirs should be maximal is never stated explicitly; when assuming implicitly that M has dimension two this condition is of
course
automatically satisfied).
Example 3.10 Another trivial class of integrable Hamiltonian systems is defined on Cn, with a regular Poisson bracket, by considering linear functions; the example shows that the integrable algebra is not always determined by just one of its (non-trivial) elements. For simplicity let us take the case n 4 with a constant Poisson structure of rank 4. As we know from Example 2.7 lineax coordinates q, p, q2 P2 on C4 may be picked such that Jqi, pj I 6ij 0- Take F and Jq1, q2} aql + bq2 + CP1 + dP2 with e.g. a =A 0 and look for a Jp1 P21 linear function G alql + b'q2 + 41 + dIP2 which is in involution with F. Replacing G by =
=
I
I
i
=
=
=
7
=
G
-
Fa'/a
if necessary
we
may
assume
G
=
b1q2
that a'
+
(db'
==
-
0 and
bd)pl
we
+
find
dP2
general solution (up to adding multiples of F). Here Y, d' EE C are arbitrary, so essentially a one-paxameter family of possibilities for G (paxametrized by d1b'), The Poisson bracket of two of these all leading to an integrable subalgebra A of O(C4) for is by G, given possibilities
as
the most
that
we
have
.
Jb'q2 + (dY which is tion 3.4.
Ham(A)
non-zero
The are
if
-
bd)pl
they
general
are
+
421 Vq2
different
their flow evolves
on a
(db"
-
bdll)Pl
+
dIP21
=
Yd'
-
db"
in agreement with Proposijust a plane and all integrable vector fields each plane. Clearly, all these vector fields axe super-
(i.e., non-proportional),
fiber of A is in this
constant when restricted to
integrable and
+
case
(straight)
52
line.
3.
Integrable Hamiltonian systems and their morphisms
The above
systems
examples are the most trivial classes of examples of integrable Hamiltonian apart from the really trivial class where affine Poisson spaces of rank zero are
-
considered. To increase rank and not constant may be considered and
complexity
may consider Poisson structures which
one
(in particular they are never regular),
also
are of higher polynomials of higher degree
ambient affine
variety of higher dimension. It turns out that in these integrable Hamiltonian systems. There are of course some trivial ways to obtain new systems from old ones, one may for example take the product of two integrable Hamiltonian systems or rewrite a simple system in a complicated way by changing variables (see Section 3.3), but these results are in reality often only interesting in the other sense, namely for reducing large or complicated integrable Hamiltonian systems to smaller or simpler ones. A general scheme for either constructing integrable Hamiltonian systems or for deciding whether a given Hamiltonian vector field is integrable is not known. We will come back to this in Chapters III and VI. it is
cases
a
Example
an
non-trivial matter to find
Let E be
3.11
a
compact oriented topological surface of genus g
>
1 with
fundamental group 7r, (E) and let G be a reductive algebraic group. Then Hom (7r, (E), G) is an affine variety on which G acts by conjugation, more precisely if p : 7r, (F,) --+ G and g E G then g-p is the homomorphism ir,(E) -+ G defined by g-P
for C E
?r,(E).
It turns out
(see [Gol]) M
(which
is
G
SL (n)
fc a
well-defined
O(M).
It
g(P(O)g
that the quotient
=
Hom(7r, (F,), G) IG
affine
explicitly
is
=
variety since G is reductive) has a natural Poisson structure which can be described for the classical groups. For simplicity let us consider the case in the standard representation. For a curve C G 7ri (E) the function
an
very =
M
was
:
M -+ G
:
p j-+
regular function on M and it by Goldman (see [Gol]) given by
shown
maximal rank is
I fc, fc, I
T ace(p(C))
can
that
(p; C, C)
be shown that these functions generate on such functions a Poisson bracket of
fc, C',
fc fc,
-
(3-6)
n
PEC#C1
The
sum runs over
intersect curves on
the intersection
transversally)
and
C and C' intersect at p, at p which is obtained
E, based A
points of C and C' (one may suppose that the curves is a sign which is determined by the way the (oriented) upon using the orientation of E. Finally, CpCp' is the curve
e(p; C, C')
large
involutive
by
first
following
C and then
for this bracket is obtained
following C'.
as follows. E can be decomposed (in algebra trinions; a trinion, also called a pair of pants, is just a three-holed sphere and such a decomposition will consist of 2g 2 trinions (in the case of genus two there exist precisely two such decompostions) Each trinion being bounded by three curves (which are identified two by two) one gets 3g 3 curves on E and what is important here is that they are non-intersecting. Calling these curves C, Gg-3 we find from Goldman's formula (3.6) that the functions fo in thus one obtains an involutive algebra are involution; ......
several
ways)
into so-called
-
-
I
...
53
I
Chapter 11. Integrable Hamiltonian systems
A
=
and its dimension is computed to be 3g maximal, A will be integrable if and only if
Compllfc...... fc,,, -j
the Poisson bracket is
-
3
=
dimM
3. Since the rank of
1
1
3g
-
Rkj-, -1
-
2
=
2
dimM,
6. Since iri(E) has a system of 2g generators, which are bound 6g i.e., for dim M 1) dim G, hence M has dimension by one relation, dim Hom(iriL (E), G) has dimension (2g (2g 2) dim G and A is integrable if and only if =
-
-
-
6g i.e., for dimG
=
Since
3.
we
-
6
(2g
=
-
2) dim G,
restricted ourselves to G
=
SL(n)
we
SL(2); it is clear from the above pictures that the integrable for G fields corresponding to all functions fc, are actually super-integrable. =
3.2.
find that A is
only
Hamiltonian vector
Morphisms of integrable Hamiltonian systems
In parallel with our discussion of morphisms morphisms of integrable Hamiltonian systems.
Definition3.12
we now
turn to
(M2&,'j2,A2) be two integrable Hamiltonian -+ (M2ij*i'j2iA2) is a morphism 0: M,
and
Let
of affine Poisson varieties
systems, then a morphism 0: (Mj,j-,-jj,Aj) M2 with the following properties
(j-) 0 is a Poisson morphism; (2) 0* CaS(M2) C Ca$(MI); (3) O*A2 CAI -
the map and
Schematically, regularity of
Cas(M2)
(2)
-
and
(3)
can
be
Cas(MI)
-
--------
Al
as
follows:
O(M2)
A2
(3.7)
0*
0*
0.
represented
-
O(Mi)
morphism 0: (M17j*7"j1iA1) -+ (W J'i *12, A2) which is biregular has an inverse which is automatically a morphism: we call such a map an isomorphism (it forces all inclusion maps in the diagram to be bijective).
A
Ikom the very definition it is clear that the composition of two morphisms is a morIt is also immediate that for any biregular map 0 : we have a category). Hamiltonian for and system (MI, 1- 7 -11, A,) there exists a unique -+ integrable M2 M, any Poisson bracket 1. -12 on M2 and a unique integrable algebra A2 C O(M2) such that
phism (hence
0: (Ml J* i'll IAI) 7
-+
(M2 I i
Ifi 912
*
i
j 2 A2) 1
is
an
isomorphism; explicitly A2
(0-1)* 10*f7 0*911 54
Vig
G
O(m2)-
A, and
3.
Conditions on
(i.)
Integrable Hamiltonian systems
and
(2)
axe
and their morphisms
conditions at the level of the Poisson structures, rather than
integrable algebras. Condition (2) resp. (3) implies that 0 induces a morcorresponding paxameter spaces resp. base spaces, as is shown in the following
the level of the
phism of the proposition. Proposition
3.13
Let
0: (MI, 1-, -11, A,) -+ (M2i I' *121 A2) 0 induces a morphism 1
be
a
morphism of integrable
Hamiltonian systems. Then
0: Spec Cas(MI) which makes the
-+
Spec Cas(M2)
following diagram commutative, M,
M2
7rc-(Ml)I
I7rC-(M2) Spec Cas(M2)
Spec Cas(MI)
as
well
as a
morphism :
which makes the
following diagram
Spec A,
Spec A2
commutative.
M,
M2
IrAjI
I-A2 Spec A2
Spec A,
If 0* Cas(M2)
=
Cas(MI) (resp. O*A2
=
Aj
then
(resp. )
is
injective.
Proof
The first assertions of
0* implies injectivity
are
diagram (3.7) by taking spectra; corresponding spectra.
immediate from
at the level of the
also
surjectivity 0
differently, condition (3) in Definition 3.12 implies that each level set of A, is a level set Of A2 and if O*A2 A, then different level sets of A, are mapped into different level sets of A2; condition (2) can be given a similax interpretation. We further illustrate the meaning and relations between the three conditions in Definition 3.12 in the following examples and propositions. Said
mapped
into
55
Chapter 11. Integrable Hamiltonian systems
Example 3.14 Let us show that in Definition 3.12 neither (2) nor (3) follow from (1). Consider C4 (with coordinates q1, q2 P1 P2) with the canonical Poisson structure Jqi, pj I I and q2 0, and C-3 (with coordinates q1, q21 PI) with Jq1, p, I fpi, pj I 8ij, Jqi, qj I as Casimir. We look at this C3 as the qlq2PI-plane in C4 and denote by 0 the projection map along P2. Then 0 is a Poisson morphism, however O*q2 is not a Casimir of C4 showing that (3.) does not imply (2). Notice that in this case 0 does not induce a map 0 as in Proposition 3.13. Taking two different functions on C2 (i.e., the algebras generated by them) shows that (i.) does not imply W7
=
=
i
=
=
,
of morphisms for which condition (2) in Definition 3.12 universally closed morphisms; these include the proper morphisms and, in particular, the finite morphisms (see [Har] pp. 95-105). We prove this in the following proposition, however we restrict ourselves to the case of finite morphisms, since we will only use the result in this case (the proof however generalizes verbatim to the case of universally closed morphisms). There is however
follows from
a
large class
(i.), namely
that of
Proposition 3.15 Let (MI, -11) and (M2, J*)'12) be two affine Poisson varieties and suppose that 0 : M, -4 M2 is a finite morphism (for example a (possibly ramified) covering map). If 0 is a Poisson morphism then 0* Cas(M2) C Cas(MI); if 0 is moreover dominant then Cas(Mi) is the integral closure of 0* Cas(M2) in O(Mi). Proof Let
show that if
us
elements of
morphisms
0
is finite then for any
f
E
Cas(M2), O*f
is in involution with all
The main property which is used about finite is that if 0: M, -+ M2 is such a morphism then O(MI) is
O(MI).
(or universally closed) integral over O*O(M2).
Thus any element g E O(MI) is a root of a monic polynomial P (of minimal coefficients in O*O(M2). As in the proof of Proposition 2.46 we find 0
10*f, PWI
=
=
desired. We have shown that
with
P,(g)lo*f gI ,
where P' denotes the derivative of the as
degree)
polynomial P. By minimality of P we find JO*f gJ 0* Cas(M2) C Cas(MI). ,
=
0
take an element g E Cas(MI) and call P its polynomial as above, with coefO*O(M2). We show that P has actually its coefficients in 0* Cas(M2), thereby proving that Cas(MI) is the integral closure of 0* Cas(M2). To do this, let O*f E O*O(M2) Next
we
ficients in be
arbitrary, then 0
=
10*f P(g)11
=
JO*f, g'
,
+
O*alg'-'
10*f, O*alllg'-'
O*Jf,a1J2gn-1
+... +
+
+
+'*'+
0 *a.11
10*f, O*a.11
O*Ifi anJ2-
polynomial has its coefficients in O*O(M2) and since P was supposed of minimal 0 for all i. Since 0 is dominant it follows that If, ai 0 degree, we find that 0* If, ai I for all f E O(M2), so that ai E Cas(M2) for i n. 0 1, Since this
=
=
-
56
-
.'
Integrable Hamiltonian systems and
3.
their
morphisms
It can be seen in a similar way that if 0 : (MI, {-, -11, A,) -+ (M21 A2) is a morphism of integrable Hamiltonian systems which is finite and dominant then A, is the integral closure Of O*A2 in O(Mi) (for a proof, use completeness of A,). It leads to the following corollary.
Corollary 3.16 Let (MI, I j 1, A,) -+ (M2 whose image is an affine subvariety of M2. Then 0 su7jective morphism. 1
21
A2)
is the
be a morphism which is finite and composition of an injective and a
Proof
Proposition 2.16 that, 0 s o . Define
We know from
(O(Mi), 1-, -1)
say
A For
f,g
A
E
involutive. If
*f
Then for an
E
as a
Poisson
morphism, 0
can
be
decomposed
via
=
have
we
If, A}
=
If
=
E
O(O(Mi)) I *f
E
Ai}
0; by injectivity of * we see that A is J *f, *g} *Jf,gj J *f, A,} 0 since A, is the integral closure of O*A in O(MI). =
=
0 then
=
A, by completeness of A, and A is also complete. Finally the dimension
O(MI) is the same as the one for M, since 0 is integrable Hamiltonian system. Clearly % and
finite. It follows that axe
is
integrable Hamiltonian
of
morphisms
count
(O(MI), 1-, -1, A)
3
systems.
If a Poisson morphism 0 : (MI, l'I'll) -+ (W 1'7 *12) is finite but not Cas(M.1) may be larger than the integral closure of 0* Cas(M2) in O(MI). Take for example for (M2, J* '}2) the Lie-Poisson structure for the Heisenberg algebra (Ex0 with the trivial Poisson structure and for 0 the inclusion ample 2.54), for M, the plane x C[xl hence 0* Cas(M2) C, while Cas(MI) O(MI). map. Then Cas(M2)
Example
3.17
dominant then
=
Even if
Example3.18
=
=
=
a
Poisson
morphism 0: (Mill* 7,11)
-+
(W J*,*12)
is finite and
Cas(MI) may be different from 0* Cas(M2). Take for example on C3 the Poisson structure from Example 3.14 and consider the finite covering map 0 : C3 -+ C' given (qj, pl, q22). Obviously this is a Poisson morphism; however the Casimir q2 by O(ql, pl, q2) dominant then
=
O*F for any function injective, being given by (q2)
is not of the form case
not
=
O(C3).
F E 2
q2
C
Notice that
A similar remark
.
applies
-+
C is in this
to condition
(3)
in
Definition 3.12.
3.3. Constructions of In Section 2.3 ones.
Using these
systems
on
we
integrable
Hamiltonian systems
gave several constructions to build new affine Poisson varieties from old give the corresponding constructions for integrable Hamiltonian
we now
them. We first show that
an
integrable
Hamiltonian system restricts to
a
general
fiber of the parameter map.
Proposition
3.19
Let
(M,
A)
is
an
integrable Hamiltonian system
AI.F)
and T
an
irre-
is an integrable ducible component of a general level of the Casimirs. Then (,F, f -, JI.F, Hamiltonian System and the inclusion map is a morphism. The property also holds for the
general levels of
any
subalgebra of
the Casimirs.
57
Chapter II. Integrable Hamiltonian systems Proof Let B be any
subalgebra of Cas(M) and let Y be an irreducible component of a general Spec B. We know already from Proposition 2.38 that Y has an induced Poisson structure and from Proposition 2.42 that the algebra of Casimirs of this structure is maximal. If we restrict A to Y then we get again an involutive algebra Ap which is complete since A is complete and Y is general. Thus it suffices to compute the dimension of A,77, fiber of M
-+
I
dimY
-
dim A
This shows that
dimM
Ay is
Definition3.20 sition 3.19 is called
dim B
-
-
(dim A
-
dim B)
=
2
RkJ-, .1
integrable algebra. Clearly the inclusion map
an
RkJ-, -I.F.
2
is
a
morphism.
Any integrable Hamiltonian system obtained from (M, a trivial subsystem.
One may think of
trivial
a
subsystem
being
as
obtained
2
A) by Propo-
by fixing the values
of
some
of
the Casimirs.
Example fiber.F
3.21
(i.e.,
In the
examples
in the choice of values
one
has however to be careful when
assigned
(some of)
to
the
picking
particular
a
Casimirs). Namely
one
has to
check that F is
general enough in the sense that both the dimension and rank of Y coincide with those of a general fiber. The dimension of a special fiber F may be higher and/or its rank may be lower; then dim.F
(F,
so
none
AI.F)
of the
integrable
is not
integrable
trivial, while
Proposition
3.22
>
dhnA
dimAly,
integrable Hamiltonian system.
an
Hamiltonian systems Hamiltonian system on the fiber
that fiber is
on x
Reconsider e.g.
Example
C' for this Poisson structure will lead =
0,
2.54: to
an
since the induced Poisson structure
on
Al, 54 0(.F) r
For i E
and let -7ri denote the natural
11, 21
let
projection
(MI is
Rkf-, .1y
-
X
(Mi, I., Ji, A,) map
M,
x
M2
be -+
M27 f'i -1m, xm2,-7r,*Al
an integrable Hamiltonian system Mi Then
0
*7r2*A2)
(3.8)
integrable Hamiltonian system and the projection maps 7ri are morphisms. Each level of the integrable Hamiltonian system is a product of a level set of (MI, f -, -11, A,) and a level Set Of (W J* i'}27 A2)an
set
Proof The
Poisson-part of this proposition
was
already given
in
Proposition
2.21.
involutivity,
firi Ai
(2)
7r2* A2 7r,*1 Al ,
(9
7r;2 A2 I mi
.
m,
58
-"::::
7ri* 1 J&
A111
+
1r*2JA2, A212 2
0-
As for
3.
Integrable Hamiltonian systems and their morphisms
We count dimensions: dim -7r,*Al 0
7r2*A2
=
dim A, + dim A2
=
dim Mi
1 2
dim(Mi
=
1
-
RkJ-, -11
M2)
X
+ dim M2
2
Rk 2
-
Itkf* 1'12
Q -, JM1 xM2)
Since ?r,*Al (8) ?r2*A2 is complete and involutive with respect to the product bracket, this computation shows that 7r,*Ai (8),7r2*A2 is integrable. Since for earch of the projection maps iri The fibers of the one has -7ri*Ai C 7r1*A1 0 7rM2, these projection maps are morphisms. momentum map are given by the fibers of M, x M2 -+ Spec(7r,*Al 0 lr2*A2), that is, of the product map M, x M2 -+ Spec A, x Spec A2 hence all fibers are products of level sets of A, and A2. I It is easy to show in addition that Ham(Ai) (or Ham(A2)) does.
Ham(-7r,* A, (9 7r2* A2)
contains
a
super-integrable vector
field if
We call
Definition3.23
(3-8)
the
product of (M1,J-,J1,A1) and (M2,J',*}2,A2)-
A construction which is related to
(but
which will be used several times in the next
different
chapters,
the product construction and dealing with integrable
from)
is obtained when
Hamiltonian systems which depend on parameters. By this we mean that we have an affine Poisson variety (M, I , J) and for all possible values c of a set of parameters we have an -
integrable algebra A, on it. This set of parameters is assumed here to be the points on an affine variety N and we assume that A, (i.e., its elements) depends regularly on c. Then we can build a big affine Poisson variety which contains all the integrable Hamiltonian systems (M, 1., .1, A,) as trivial subsystems. This is given by the following proposition.8 for each c r= N an integrable Hamiltois given on an affine Poisson variety (M, 1-, .1) then M x N has a structure of an affine Poisson variety (M x N, I-, J) and O(M x N) contains an integrable subalgebra A such that each (M, I-, Jm, A,) is isomorphic to a trivial subsystem of (M x N, 1-, -1, A) via the inclusion maps
Proposition nian
3.24
system (M,
If
N is
an
affine variety
and
I., Jm, A,.), depending regularly
0,:
M
-+
M
x
N:
on c
m i-+
(m,c).
Proof For N a
one
takes the trivial structure
Poisson manifold. The
is maximal and
so
algebra of Casimirs
that on
Cas(N)
this
=
O(N)
product
which makes M
is maximal since the
x
N into
one on
M
N) Cas(M) (9 O(N). The fact that A, depends regularly on c means that there exists a subalgebra A of O(M x N) which restricts to A, on the fiber over c of the projection p, : M x N -+ N. Clearly its dimension is given by dim A dim A, +dim N Cas(M
x
=
=
8
generalizes to the situation considered in Example 2.24, namely when morphism, for each n E N, I-, -In is a Poisson bracket on the fiber -7r(-) (n) and An is an involutive subaJgebra of 0 (-7r(- 1) (n)) which is integrable for general n; both I-, Jn and An axe supposed to depend regularly on n G N. Proposition 3.24 ir
:
The proposition
P -+ N is
corresponds
a
dominant
to the
special
case
P
=
M
x
N considered at the end of
59
Example
2.24.
Chapter so
that dim A
Since
O(N)
is
dim(M x N)
=
a
subalgebra
Integrable Hamiltonian systems
1
2 Rkf Cas(M
-
of
II.
since A is x
N)
the fiber
complete and involutive p is
over
the restriction of the Poisson structure which is
corresponds to the isomorphism when restricted to such a fiber.
an
a
it is
integrable.
level set of the Casimirs and
one on
M via the
morP hism.
The next construction we discuss is that of taking a quotient. This is of interest, because many of the classical integrable Hamiltoniau systems possess discrete or continuous symmetry groups. The algebraic setup which we use here has the virtue to allow to pass easily to the
quotient (one does so
not need to worry about the action
being free, picking regular
values and
on).
3.25 Let G be a finite or reductive group and consider a Poisson action M, where (M, 1-, -1) is an affine Poisson variety. If A is an involutive algebra such that for each g (=- G the biregular map X, : M -+ M defined by X(g, m) leaves X, (m) A invariant, i.e., X*A C A, then (MIG, j.'.10, AG) is an involutive Hamiltonian system 9 and the quotient map -7r is a morphism. Here 1., -10 is the quotient bracket on MIG given by Proposition 2.25. If G is finite then (MIG, f.,.}O, AG) is integrable.
Proposition X: G
x
M
-+
=
Proof
Involutivity of AG is immediate from Proposition 2.25. Suppose now that G is finite. completeness of A implies completeness of A n O(M)G. As for dimensions, since G is
Then a
finite group
we
have
dimAn
O(M)G
=
dim.A
=
dimM
1Rkf
-
2
-,
-1
I =
where one
A
n
we
dim M/G
-
2
Rkf -, -jo,
dim O(M) and A c O(M). Similarly equality that dim O(M)' algebra of Casimirs is maximal, being given by Cas(M) n O(M)G. Thus integrable; obviously -7r*(A n O(M)G) C A, hence the quotient map is a
used in the first
=
shows that the
O(M)G
is
morphism.
0
We will encounter
Example
A
3.26
O(M)G). Namely,
a
lot of examples later. Here
special
in this
case
case
occurs
the level sets of
(MIG, I
A similar result
applies for the level
Example 3.27
The
-
,
when A C
each level set of
j o A) 5
quotient
are
precisely
are some
O(M)G (which implies Cas(MIG)
(M, f -, J, A) the
construction leads to
a
(M
c
is stable for the action of G and
quotients of the level
sets of the Casimirs in
systems which look interesting. One may e.g. start with
(M, I-, -}, A) and consider its x M by interchanging the
first observations.
case
sets of
Cas(MIG)
C
(M, f
-
-
,
1, A).
O(M)G.
lot of an
new integrable Hamiltonian integrable Hamiltonian system
M, I-, -Imxm, A (9 A). The group Z2 acts on product. Obviously this is a Poisson action and the action leaves A (& A invariant, thereby leading to a quotient. The level sets which correspond to the diagonal are symmetric products of the original level sets.
M
square
x
factors in the
60
3.
Integrable Hamiltonian systems and their morphisms
Notice that the group G in
phism. group of M. For future quasi-automorphism.
Proposition 3.25 use we
be
can
seen as a
introduce also the
slightly
subgroup of the automormore general notion of a
(A I-, J,A) bean integrable Hamiltonian system. An automorphism -+ (M, I-, -}, A). More generally, if 1., -11 and J* *12 are two Poisson brackets on M then an isomorphism (M, -.11, A) -+ (A {-, '12, A) is called a quasi-automorphism. Definition3.28 is
Let
isomorphism (M, I-, -}, A)
an
The final construction is to
1
remove a
divisor from
phase
space.
Proposition 3.29 Let (M, 1-, -1, A) be an integrable Hamiltonian system and let f E O(M) be a function which is not constant. Then there exists an integrable Hamiltonian system (N, f"i'lN, AN) and a morphism (N, J* 7'IN7 AN) -+ (M, 1-, -1, A) which is dominant, having the complement (in M) of the zero locus of f as image. Proof
proof (the Poisson part) was given in Proposition 2.35 and we proposition. We start with the case f E A. If we define AN then AN is involutive since 7r is a Poisson morpbism and it has the right dimension to be integrable. We need to verify completeness. Let Ein-0 fiti EE O(N) then Most of the
notation of that
use
:--
the
7r*A[t]
in order
j-
n
fit', AN i=O
IN
n
0
:>
Effi, 7r*A[t]lNti
0
i=O n
Effii lr*AlNfn-i
0
i=O n
1:1& Alfn-i
0
i=O
E ffn-i, A
0
i=O n
E fjn-i E A i=O n
1: ffn-itn
G
AN
i=O n
1: fit'
CE
AN-
i=O
Since AN is involutive the last line also desired
implies the first line,
so we
have established the
equivalence.
an explicit description Of AN is still available if (M, I J,A) satisfies Spec 7r*A also satisfy the Proposition 3.7. In that case the fibers of N 7r*A. In general one has conditions of Proposition 3.7 hence -7r*A is complete and AN AN Compl(-7r*A) and a more explicit description is not available.
If
f
A then
-
,
the conditions of
=
61
Chapter
Compatible
3.4.
We
now
11.
Integrable Hamiltonian systems
and multi-Hamiltonian
introduce
a
integrable systems
few concepts which relate to
compatible integrable Hamiltonian
systems. Definit ion 3.30
brackets
Let
affine
i
variety M.
=-=
If
1,
n
be
(linearly independent) compatible
n
(M, I-, ji, A)
is
Poisson
integrable Hamiltonian system for each i n then these systems axe called compatible integrable Hamiltonian 1, systems. Any non-zero vector field Y on M which is integrable (in particular Hamiltonian) with respect to all Poisson structures i.e., for which there exist fl, f,, E A such that on an
an
=
.
.
.
,
.
Y
is called
a
multi-Hamiltonian
f., fill
=
= ...
(bi-Hamiltonian
many different ways; any of the an
=
.
.
,
1', Aln,
if n 2) vector field, since it is Hamiltonian in integrable Hamiltonian systems (M, I-, ji, A) is then called =
integrable multi-Hamiltonian system (bi-Hamiltonian when
Remark 3.31
We do not demand in the definition of
system that all the integrable satisfied in
an
=
2).
integrable multi-Hamiltonian
vector fields be multi-Hamiltonian.
3.33 and 3.34 it is far too restrictive in
Examples
n
Although
this condition is
general.
All propositions and basic constructions given above are easily adapted to the case of compatible or multi-Hamiltonian structures, but this will not be made explicit here. Just one example: an action of a reductive group which is a Poisson action with respect to both Poisson structures of two compatible integrable Hamiltonian systems yields on the quotient two compatible integrable Hamiltonian systems. Here are some properties which are specific to compatible integrable Hamiltonian systems.
Proposition 3.32 (1) Compatible integrable Hamiltonian systems have the same level sets; (2) The Poisson brackets of compatible integrable Hamiltonian systems have the same rank, which also equals the rank of a general linear combination of these Poisson structures
(3) If (M, I., -1j, A)
are
linear combination
integrable
compatible integrable Hamiltonian system then for
I-, +x of
the Poisson structures the system
(M,
general A) is an
a
Hamiltonian system.
Proof The
proof of (l.) is obvious since the level sets
determined
by A only. Since Rkf ji equal. To determine the rank of a linear combination of these structures one looks at the corresponding Poisson matrix (with respect to a system of generators of O(M)) which is given by the same linear combination of the Poisson matrices of the structures I-, ji. Now a general linear combination of invertible matrices is invertible, which applied to a non-singular minor of size Rkj-, ji leads to (2). 2 dimM-2 dimA
For
a
dimA
linear combination =
dimM
-
1L 2
are
find that the rank ofall structures
we
I., ji
is
0 and I-, .1,\ of (maximal) rank Rkj-, jj one has that JA, A},\ Rkj-, ji, hence (M, I-, +\, A) is an integrable Hamiltonian system, =
showing W-
62
Integrable Hamiltonian systems
3.
We will encounter in this text many
and their
morphisms
(non-trivial) examples
of
compatible integrable Here are two simple
Ha,miltonian systems and of integrable multi-Hamiltonian systems. examples of integrable bi-Hamiltonian systems.
Example
Consider the Poisson structures
3.33
qj, q2, p, and
P2)
defined
by the
1-, -11
and
1' J2 1
on
C4 (with coordinates
Poisson matrices
0
0
1
0
0
0
0
1
0
0
0
1
0
0
1
0
-1
0
0
0
0
-1
0
0
0
-1
0
0
-1
0
0
0
and
O(C4)
For A c
structures
are
take those functions which
compatible and
since their
I they
are
are independent of q, and q2. Then both integrable vector fields are of the form
a
f
Poisson
C9 + g
9ql
1 f,g
9q2
A
E
all bi-Hamiltonian.
Example
Recall from
3.34
Example 2.11 that the matrix OF Oz
-OF
0
OF Ox
OF
-OF
;9__V
TX_
0
0
U(
OF
5
OY
defines for any u and F in O(C') a Poisson structure on C3 F is assumed non-constant here in order to obtain a non-triviaJ Poisson structure. Let us denote this Poisson structure
by J* juF. 1
j','ju,F+G
If G is any other non-constant element of O(C3) then I-, Ju,F + l'i"ju,G 1' 1 *}u,F and J* , ju,G are compatible and, assuming that F and G are in__"
hence
dependent, A ComplIF, G} defines an integrable Hamiltonian system on (C3, J* ju,F)However, by interchanging the roles of F and G. we find that A also defines an integrable Hamiltonian system on (C3, J* ju,G) hence leading to a pair of compatible integrable Hamil=
,
I
,
tonian systems. Since structures
are
moreover
the Hamiltonian vector fields with respect to both Poisson
given by
fuoVF we
conclude that A defines
Closely
an
integrable
x
VG
10
c
Al
bi-Hamiltonian system
on
C3.
related to the concept of an integrable multi-Hamiltonian system is that of a hierarchy. Let us define this in the case of a bi-Hamiltonian hierarchy and
multi-Hamiltonian
explain
its
use.
Let
sequence of functions
1-, -11 and J",'}2 be jfj I i E ZI is called I-, fiJ2
The
following property
is
--::
two a
compatible
I' fi+111i
essentially due
Poisson brackets
bi-Hamiltonian
i
(i
to Lenaxd and
63
E
hierarchy
Z).
Magri.
if
on
M.
Then
a
Chapter
11.
Integrable Hamiltonian systems
All functions fi of a bi-Hamiltonian hierarchy jfj I i E Z} are in 3.35 involution with respect to both Poisson brackets (hence with respect to any linear combination). If one of these functions is a Casimir (for either of the structures) then all these fi are also
Proposition
in involution with the elements
of
any other bi-Hamiltonian
hierarchy.
Proof If
jfj I
i E
ZI
forms
a
hierarchy,
then for any i <
JA fj}l
j
E Z
Ifii fj-1}2 U41, fj-l}l
1h fib I
so
Ifi, fj}l
=
0
by skew-symmetry. They
ond bracket since bi-Hamiltonian
Jfi)fjj2
hierarchy
=
and
Jfjjj+jjj.
fk
is
a
are
Casimir,
Ifi; 9jj1
=
also in involution with respect to the secsame way, if jgj I j E Z} is another
In the
say of
1., .11
jfkj gi+j-k}l
=
then for any i, j E Z 0.
The above proposition leads to many interesting integrable Hamiltonian systems; said it can be used to give an elegant proof of the involutivity of many integrable
differently
Hamiltonian systems.
64
Integrable Hamiltonian systems
4.
Integrable
4.
Hamiltonian systems
on
on
other spaces
other spaces
In this section
we wish to consider briefly integrable Hamiltonian systems on spaces other algebraic vaxieties. One possible generalization is to consider spaces which are not necessarily algebraic, but have a differential structure (real or complex analytic), at least on a dense open subset. Examples include smooth manifolds, analytic varieties and orbifolds. Note however that extra generality comes also from the fact that one can often choose which algebra of functions on these spaces to consider, for example one may consider an affine vaxiety with its algebra of rational functions; however these algebras should be reasonably big in order to lead to integrable Hamiltonian system, as is cleax from the example of a projective algebraic vaxiety with its regular functions (which axe only the constant functions). Another possible generalization, closely related to the problem raised by the latter example is to consider (reasonable) ringed spaces or schemes. We will only consider the first generalization here.
than affine
4.1. Poisson spaces
At first and smooth
define
we
manifolds,
a
general class of spaces, which includes both affine algebraic vaxieties which it is possible to define the notion of an integrable Hamiltonian
on
system. Let M be
Definition 4.1
(or holomorphic)
a
smooth
is
big enough
holomorphic)
to
a
topological
space which has at least
Also let R be
structure.
distinguish (smooth) points
in
a
Poisson
The case
on a
dense open subset on M which
of functions
M, and whose elements
axe
-
algebra
J) (or (M, 1-, -1) CI(M) (resp. manifold (resp. analytic Poisson manifold). of Hamiltonian vector
of affine Poisson varieties.
fields
=
and the
algebra
for R
short) a Poisson Cw(M)) (M,
of Casimir8
The Hamiltonian vector fields
axe
a
space;
is
=
of
axe
defined
as
in
only (real or non-singular part
course
vector fields
on the non-singular paxt of the space. On this representing the Poisson bracket can be defined and also there is a of rank at a non-singular point. Notice that all this was in the case of affine Poisson even defined at the singular points.
holomorphic)
(resp.
smooth
dense open subset of M. A Poisson bracket on (M, R) is as in the case of Lie bracket 1-, -1 R x R -+ R : (f , g) i-+ If , gJ, which satisfies the Leibniz
rule in each of its arguments. We call (M, R, I-, in the special case that M is a manifold and R a
algebra
on a
affine varieties
called
an
Poisson tensor
notion spaces
example which originated the theory of Poisson brackets and Poissymplectic manifolds. A symplectic manifold (M,w) is a manifold equipped with a closed two-form w (a symplectic two-form) which is non-degenerate (as a bilinear form on each tangent space). A vector field XF is associated to any function f E C'(M) by df w(Xf,
Exarnple
4.2
The
son
manifolds is that of
and
a
skew-symmetric bracket
is defined
on
If, g1 definition of
new
of affine Poisson varieties.
Xf
by
W(Xf' Xg)-
is consistent with the definition
Notice that this gave in the
case
=
smooth functions
65
Xf
f I which
we
Chapter
11.
Integrable Hamiltonian systems
is a derivation in each of its arguments and the Jacobi identity for this Clearly equivalent to the fact that w, is closed. Thus a symplectic manifold is a Poisson manifold in a natural way. Such a Poisson manifold is regular and its dimension equals its rank (in particular it is even). Conversely every regular Poisson manifold of maximal rank is a symplectic manifold in a natural way. In turn, the main examples of symplectic manifolds are provided by the cotangent bundle to any manifold and by Khhler manifolds. The literature on symplectic manifolds is immense. See e.g. [AL], [AM1j and [LM3].
bracket is
A fundamental property of symplectic manifolds is that they admit locally so-called (the Daxboux Theorem). The following theorem provides the proper
canonical coordinates
generalization of this property a proof we refer to [CW].
to Poisson manifolds. This theorem is due to A.
Weinstein;
for
Theorem 4.3 a
coordinate
Let
(M, 1-, -1) V
neighborhood
be a Poisson manifold and letp E M be arbitrary. There exists of p with coordinates (qj, q,, pi.... Pr, Y1 y.) centered 7
....
at p, such that 8
I-, JV
A
aqi
+
-
2
api
A E Okl(Y)yYk
k'1=1
where the
functions Oki
The rank of
are
smooth
functions which vanish
yj
at p.
is 2r but is not necessarily constant on a neighborhood of p. When the neighborhood of p the neighborhood V can be chosen such that, on V, the functions Oki vanish, yielding the following canonical brackets for the above coordinates:
rank is constant
1., -1
on a
lqi)qjl
==
fPiiPjl
=
fqi7Ykl
=
fPi,Ykl
=
fYk7YI1
=
f%jpjj
0,
=
6ij7
(4.1)
where I < i, j < r and I < k, I < s. In this form Weinstein s Theorem is usually referred to as the Darboux Theorem and the above local coordinates are called Darboux coordinates or
canonical coordinates.
The Darboux Theorem may be refrased by saying that the rank a point where it is locally constant is the only local invariant of
of the Poisson manifold at a
Poisson manifold.
A stronger version of the Darboux. Theorem says that
a
collection of
independent functions (around the point) which satisfy canonical commutation relations can be extended to a complete set of canonical coordinates. In this stronger form the Darboux Theorem is false for affine Poisson variety, consider for example on C' the Poisson bracket Ix, yj x at a point not on the Y-axis and let the incomplete collection consist just of jyj. The only way to complete it with f such that If, yj 1, is to take f ln(x) which is not a regular function on any Zariski open subset of C'. Canonical coordinates (which are regular on a Zariski open subset) exist however for this bracket, for example one has 11, -yxj (clearly canonical coordinates which are regular on C' do not exist). It is unlikely that a set of independent regular (on an open subset) functions, satisfying commutation relations as in the Darboux Theorem, can be found for any affine Poisson variety, but a counterexample (if any) is missing. =
=
=
X
Although there in the
case
is
a
notion of rank at each
point of
a
Poisson
of affine Poisson varieties that the rank is constant
the Poisson manifold which may result in Consider the following example.
some
66
manifold,
on an
it is not true
as
open dense subset of
nasty behavior of the algebra of Casimirs.
4.
Example non-zero
W(x, y)
on
other spaces
a bump Poisson structure on the plane R1. Let W be a R2 whose support Supp(W) is compact and connected. Clearly Ix, yJ Poisson bracket on R2 and there is an open subset where the rank is two but
We first construct
4.4
function
defines
Integrable Hamiltonian systems
a
on
=
an open subset where the rank is zero. Moreover its algebra of Casimirs is non-trivial since it contains all functions whose support is disjoint from Supp(w). Thus Supp(w) is a
also
as every point in M \ Supp(W). The former level set is never a manifold (in might be a manifold with boundary, but it is in general singular as well). Of course all this is typical for the smooth case; when analytic brackets axe considered then the rank is constant on an open dense subset, the fibers of a (real or complex) analytic map will be analytic varieties and so on.
well
level set
as
the best
case
it
2.4 two decompositions of affine Poisson varieties, the decomposition. From what we said it is clear that the rank decomposition does not have its counterparts in a smooth setting. There is however in the case of Poisson manifolds another decomposition (singular foliation) the symplectic decomposition or symplectic foliation which is very useful. Its name stems from the fact that the Poisson structure restricts to a regular structure of maximal rank on each leaf, hence the Poisson structure permits to define a symplectic structure on each leaf. On an affine Poisson variety the leaves of the symplectic foliation need not be algebraic (as e.g. in the Example 2.43) and they (i.e., equations for them) are difficult to determine explicitly in general (for example it is a well-known result that in the Lie-Poisson case (see Example 2.8) the symplectic leaves coincide with the co-adjoint orbits, i.e., the orbits of the corresponding group G acting on 9* via the co-adjoint action; even in low dimensions these orbits may be We have discussed in
Paragraph
Casimir decomposition and the rank
very hard to
compute).
The easiest way to obtain the symplectic foliation is by Indeed, a subvariety of M around p is obtained by taking y,
using Weinstein's
=
= ...
y,
=
Theorem.
along this the only one
0 and
subvariety I -, J restricts to a symplectic structure and this (local) subvariety is containing p on which f-, -1 restricts to a Poisson bracket of maximal rank. Hence we may globalize this construction to find a unique symplectic leaf passing through each point. Notice that these leaves are immersed submanifolds and not closed submanifolds in general; each leaf may even be dense in M, as is shown in the following example (the example also shows that, even in the case of Poisson manifolds, the algebra of Casimirs needs not be maximal).
(1,a,,3) where 1,a and,6 Example4.5 Take on R3 an orthogonal basis el, e2, e3 with e3 are linearly independent over Q. The bivector el A e2 determines by parallel translation a ==
Poisson structure
All
symplectic
level set of the
on
R3 which descends
to
a
Poisson structure
1., -1
two-dimensional, but they are dense, hence Casimirs, such level sets being always closed.
leaves
are
the torus
on
none
of them
WIZ3.
can
be
a
As a final remark about the symplectic foliation, we wish to point out that Weinstein's proof is easily seen to be valid also in the holomorphic case, yielding a holomorphic symplectic foliation on any holomorphic Poisson manifold. For affine Poisson varieties this leads to a holomorphic symplectic foliation on its smooth part (which is a complex manifold). In the
following definition
we
generalize Definition
spaces.
67
2.15 to the
case
of
general
Poisson
11.
Chapter Definition 4.6 map
0: M,
-+
Let
(Ml, R1, J'7 *11)
M2 is called
'--
morphism which
In terms of
nian systems
integral
(as
has
curves
defined
and
(M2, R21 J* '12) be morphism if
Poisson
a a
(1) 0*7Z2 C R1, (2) 0*1figJ2 10*f,0*911i A Poisson
Integrable Hamiltonian systems
for all
an
f,g
two Poisson spaces, then
R2-
E
inverse is called
a
Poisson
isomorphism.
the relevance of Poisson is formulated
below)
a
by
the
morphisms for (integrable) following proposition.
Hamilto-
Proposition 4.7 Let (MI, 1-, -11) and (M2, J* *12) be two Poisson manifolds and suppose that 0: M, -+ M2 is a Poisson morphism. Then the integral curves of a Hamiltonian vector field XH, H E C'(M2) which intersect O(MI) are entirely contained in O(MI) and are the projections under 0 of the integral Curves Of XO-H. 1
Proof If -y is an integral curve of any local coordinates, then
(gi
o
O*H then 0 oy
0 oy)*
=
Igi
o
is
0, H o 01
an
oy
integral
=
curve
Igi, HI
0
o
of H.
Indeed, let
gi be
oy.
If P E
O(Ml) c M2, let Q E M, be lying over P, then the above computation shows that integral curve of H o 0 through Q projects (via 0) onto the (unique) integral curve of H in particular this integral curve cannot leave O(MI). 0 passing through P the
-
We wish to point out that a similar proposition, stating that all integral curves Of XH projections of integral curves Of XO*Hi is given in [Wei2] (Lemma 1.2 p. 528), but this cannot be true: it would imply surJectivity of the map 0 (at least onto the non-singular part). are
dealing with integrable Hamiltonian systems on symplectic manifolds one proposition consider Poisson morphisms rather than symplectic maps. It is seen from the following simple example that the two concepts do not agree in general and that the above proposition needs not hold for symplectic maps. Even when
should
by
the above
R4 (with coordinates X1 Y1 X2 Y2) and M, C R4 the plane Example 4.8 Take M2 0- On both M, and M2 we put the standard symplectic structure: given by X2 Y2 dxl A dy, + dX2 A dy2. Then there are obvious projection and dxl A dyl and W2 wl =
5
==-
i
i
==
=
=
inclusion maps 7r:
and it is easy to check that Poisson.
Example
4.9
Let
us
show
R4
now
the
symplectic
R
2
and
z:
R
2
_+
W,
-7r
is Poisson but not
by
a simple modification of the previous example that Proposymplectic maps. Instead of the obvious inclusion map we
sition 4.7 needs not be true for
consider
-+
symplectic and
z
is
symplectic
but not
map
0: R2 The function X2 on W has all included in the image of 0.
-+
W
integral
:
(xi, yi)
curves
-+
(xi, y,
parallel
68
,
x, ,
to the
0)
-
Y2 axis, hence
none
of them is
4.
Integrable Hamiltonian systems
on
other spaces
The
polynomial invariant which we associated to affine Poisson varieties does not genergeneral Poisson spaces since the rank decomposition may not lead to (a finite number of) reasonable spaces, so it may not be clear how to count "components". For analytic brackets our construction goes however over verbatim. A lot of attention has been given over the last few years to global invariants for symplectic manifolds, a good introduction and more references are given in [AL]. ahze to
4.2.
Integrable Hamiltonian systems As for
integrable Hamiltonian systems
Definition 3.5, but At
a
few modifications
Poisson spaces
on
on
general
Poisson spaces
we
would like to copy
needed.
axe
wish the rank of the Poisson space to be constant on some open dense subset, may run into complications such as in Example 4.4 in which at some open subset the level sets of the integrable Hamiltonian system are given by the levels of -the Casimirs
first,
otherwise and in such
we
we
some
other open subset
they
given by
axe
the level sets of the
integrable algebra.
In
the Poisson space can be split in two, so it is a mild assumption that the rank is constant on an open dense subset; this constant is then called the rank of the Poisson space. case
Second, the notions of spectrum and dimension for an algebra A C R need to be modified. algebras A have no spectrum nor a dimension; the dimension is naturally replaced by the number of independent functions (we say that a collection of functions is independent if their differentials axe independent at every point of some open dense subset). As for the spectrum, which we needed in order to define the momentum map, we could take Hom(A, R) (resp. Hom(A, C)) or the real spectrum (in the case of manifolds) but this may be a very complicated (and ugly) object; in particular we will not have a smooth or holomorphic projection map M -+ Hom(A, R); however for any system of generators fl,... fn as above, we will have a smooth (resp. holomorphic) map M -- Rn (resp. M _+ Cn). our
,
Third, it is not clear at all how to show for general Poisson spaces that some algebra is complete (in the sense of Definition 3.1). Recall that we insisted in having completeness in order not to call two systems non-isomorphic while their algebras have the same completion. A solution to this is not to insist on completeness in the definition of an integrable Hamiltonian system but to call two systems isomorphic when some involutive extension of their integrable algebras coincide. These remarks lead to the
Definition 4.10
following definition.
Let
be
open dense subset of M and whose
a
algebra
Poisson space which is of constant rank on an maximal, i.e., it contains dimM
of Casimirs is
-
CoRkJ-, -1 independent functions. An involutive subalgebra A of Z is called integrable if it 1 contains dimM independent functions. The quadruple (M, 7Z, I-, J, A) is then 2 RkJ-, -1 -
called
an
integrable Hamiltonian system and
Ham(A) is called
an
integrable
vector
=
each
non-zero
JXf I f
field.
69
E
A}
vector field in
Example standard
In its
4.11
symplectic
Chapter
11.
original
form the three
Integrable Hamiltonian systems
E dqi
structure
body Toda lattice is given on RI with the dpi by the algebra generated by the following two
A
smooth functions:
IE Pk+Ee 2
H=
2
k=1
k=1 3
PIP2P3
-
EPkeqk+l k=1
Since the translations
(qi, q2i q37P1 iP2i P3) define
a
-+
(ql
+ a, q2 + a, q3 + a,pi, P21 P3)
action, the quotient of RI by these translations, which is R5 inherits a on every hyperplane pi + P2 + P3 c (c E R any fixed constant)
Poisson
Poisson structure. It leads to
a
=
Since the group action leaves the functions H and I invariant descend to this quotient and since they are in involution they are also in involution on
symplectic
structure.
they the quotient. Clearly they integrable. In view of the on an
axe
also
hence the
independent,
exponentials this
is not what
affine Poisson space; it is however
we
called
algebra generated by
an
integrable
related to one,
closely
see
H and I is
Hamiltonian system
Section VII.7.
A second example is given by the elliptic Calogero-Moser system, studied 4.12 (especially from the point of view of algebraic geometry) by Treibich and Verdier (see (TV]). The setup is the same as for the Toda lattice above but the exponentials axe replaced by the Weierstrass p function. In the simplest case of three "particles" the involutive algebra is generated by the following two meromorphic functions (P is the Weierstrass function associated to a fixed elliptic curve)
Example in detail
1
H
=
3
3 2
2.EPk
-
1: p(qk+l
2
3
=
3- 1: Pk
-
E (Pk+l
2
there
case
Hamiltonian systems
Finally, here
on
the third
and
-
qk-1)
as
in that
case
they
are
all
Calogero system (rational, closely related to integrable
affine Poisson varieties.
is the definition of
spaces. Notice that in
Pk-I)P(qk+l
many different versions of the
axe
trigoniometric, relativistic, ...)
-
k=1
k=1
As in the Toda
qk-1),
3
3
K
-
k=1
k=1
property
(3)
a
morphism. of integrable Hamiltonian system on Poisson we do not ask that O*A2 C Al, in accordance with
below
remark, just before Definition 4.10.
Definition4.13
Let
nian systems, then
a
and
(M1,R1,J*,'J1,A1)
map
0 : M,
-+
M2 is
(1.) 0 is a Poisson morphism, (2) 0* CaS(M2) C CaS(MI); (3) O*A2 C A3, where A3 C Ri
is
an
a
(M2,7Z2&,*J2,A2)
morphism
involutive
70
if it has the
algebra which
be integrable Hamiltofollowing properties.
contains
A,.
III
Chapter
Integrable and
Hamiltonian systems
symmetric products
of
curves
1. Introduction
chapter is devoted to the construction and a geometric study of a big family of integrable Hamiltonian systems. The phase space is C2d equipped with an infinite dimenThis
,
sional vector space of Poisson structures: for each non-zero W E C[x,y) which makes (C2d, into Paragraph 2.2) a Poisson bracket I-, J1 d d
we an
construct
(in
affine poiSSon
vaxiety. Each of these brackets has maximal rank 2d (in paxticular the algebra of Casimirs is trivial) and they are all compatible. An explicit formula for all these brackets is given; they grow in complexity (i.e., degree) with W so that only the first members are (modified) Lie-Poisson structures.
What is
surprising is that
all these structures
(for fixed d)
have many
integrable algebras given by a very compact and simple formula. Namely there is one integrable algebra corresponding to each polynomial F(x, y) in two vaxiables (it is assumed here that the polynomial depends on y). The magical formula is given by in common;
system of generators of these algebras
moreover a
H(,X) in this formula
u(A)
is
a
monic
=
are
F(.\, v(A)) mod u(A); degree d and v(A) is a polynomial of degree polynomials are the coordinates on CU. The formula by taking
polynomial
of
less than d and the 2d coefficients of these two
integrable algebra is obtained from this
AF,d
=
C[HOi
...
1Hd-11i
where Hi is the coefficient of Xi in H(.X). It leads to many integrable Hamiltonian systems and for fixed F(x, y) they are all compatible; the integrable vector fields which correspond to them a,re
however different
so
that these do not
give integrable multi-Hamiltonia-n systems. Their 71
P. Vanhaecke: LNM 1638, pp. 71 - 96, 1996, 2001 © Springer-Verlag Berlin Heidelberg 1996, 2001
Chapter
integrability
Y2
_
f(X)
in
III.
Integrable Hamiltonian systems and symmetric products
of
curves
is shown in
Paragraph 2.3. We will look at the special case for which F(x, y) Paragraph 2.4; in this case we are able to write down Lax equations for the
vector fields.
A closer
study of the fibers
of the momentum map reveals the meaning of the polyWe describe the fiber FFd over (Ho,...' Hd-,) F(x, y). (0, ...' 0) and obtain a description of the other fibers by a slight change in F. If the algebraic curve IPF (in C2) nomial
=
defined
0 is non-singular then TFd is non-singular and we show that in this case by F(x, y) the fiber -FFd is isomorphic to an affine part of the d-fold symmetric product of the algebraic curve ]Pp (we also give an explicit description of the divisor which is missing). This shows =
that
basically all our systems (for different F) axe different and that the d-fold symmetric products of any curve (smoothly embedded in C2) appears as a level set of some integrable Hamiltonian system. We deduce from the description of the general fibers of the momentum 2 (when surfaces are obtained as level sets) the map a description of their real parts. For d description is easily visualized and shows at once that a large family of topological types is present. The level sets are described in Paragraph 3.2 and their real parts in Paragraph 3.3. =
The effect of changing the Poisson structure
(keeping F(x, y) and d fixed) manifests itself (the Poisson structure is not seen from the fibers of the momentwn map since these depend on F(x, y) and d only). These vector fields axe all tangent to the same fibers and span the tangent space at each (non-singular) point, only
at the level of the
integrable
vector fields
hence these vector fields must be
related; they are in the present example even related in a however these vector fields are different for all choices of V so that changing way, also different leads to (i.e., non-isomorphic) systems. The effect of varying the Poisson V
very
simple
structure is
given
in
Paragraph
Later in the text
we
3.5.
will refer
on
several occasions to the systems described in this
chapter. For a futher generalization of these systems, in which F(x, y) is replaced by a family of algebraic curves, we refer to [Van5]. For a more abstract, but less explicit, construction of these systems, where C2 is replaced by any Poisson surface, see [Bot].
72
systems and their integrability
2. The
2. The
systems and their integrability
In this section
of functions which is
C2d,
on
which is
polynomial F(x, y) an algebra compatible Poisson structures polynomials W(x, y) in two variables.
show how there is associated to every integrable with respect to a family of
we
the set of all
parametrized by
2.1. Notation
C2d is viewed throughout this chapter as the spare of pairs of polynomials (u(A), v(A)), with u(A) monic of degree d and v(,\) of degree less than d, via
so
U(A)
=
V(A)
=
the coefficients ui and vi
by denoting
Ud
=
Ad
+ Ud- 1Ad-1 +
+
UI'\
+ UO'
Vd-I Ad-I +... +
VIA
+ VO,
coordinates
serve as
...
on
(2.1)
C2d. Some formulas below
are
simplified
1-
For any rational function r(,\), we denote by [r(A)]+ its polynomial part and we let r(A) [r(,\)]+. If f (,\) is any polynomial and g(,\) is a monic polynomial, then than deg g (A), defined by mod f (,\) g (A) denotes the polynomial of degree less =
-
f (A) mod g (A)
=
g (A)
[ fg ((,\)1111
f (A) mod g(,\) + h(,\)g(,\) for a unique polynomial h(,\) and f (A) mod u(A) f (A) computed as the rest obtained by the Euclidean division algorithm.
so
=
2.2. The
,
compatible
is
easily
Poisson structures
W(x, y), hence also Any polynomial w(x, y) specifies a Poisson bracket on C2 by ly, xj C2 X X C2 (by taking the product bracket). Explicitly the cartesian product (C2)d =
on
=
fyi, Xj I
(xi, yi)
where on
are
=
...
the coordinates
the i-th
on
.
: --
I ((XI Y1)) (X2 Y2)
and consider the map S
((X1iYI)i (X21Y2)1
7
i
...
I
:
i
...
I
-+
lyi, Yj I
=
(2.2)
0,
factor, coming from the chosen coordinates
(Xdi Yd)) I
(C2)d \ A _4 C2d'
(Xd, Yd))
=
(C2)d defined by
C2 Let A denote the closed subset of A
1xi, Xj I
6ij W(Xj' yi),
xi
=
xj for
some
i
0 jJ,
given by
(U(A) V(,X)) i
73
(A_ Xi),
Yi
rl
A Xi
-
-
Xj
Xj
(2.3)
111.
Chapter
Integrable Hamiltonian systems
and
symmetric products of
This map can be interpreted as a morphism of affine Poisson varieties upon tion 11.2.35. This is done as follows. Define
I (XO (XI Y0
MI
I
I
I
...
(Xdi Yd))
I
X0
H(X, Xj)2 _
11
=
C
C
curves
using Proposi-
(C2)d,
X
i<j
and
M2 Then
we
have
a
=
J(t, u(A), v(A)) I
commutative
disc(u(A))
t
S
C2d
morphism between the affine varieties M, and M2.
Poisson bracket
j.'.1
using the relation
jx0' Xjj
(C2)d
on
x0
=
CU.
X
P2
(C2)d \A
upon
C
C
M2
P1
a
11
diagram M,
with S
=
leads to
11i<j (X,
_
Xj) 2
and
0,
a
Poisson bracket
=
1; namely
one
adds the brackets
I ]I(X,
2
1X0, A
By Proposition 11.2.35 the MI, also denoted by
on
-X0
_
Xj)2, A
i<j
the latter
being computed from
IX0, Yk} 11(xi
-
xj)
2
JI(X, Xj)2, y1c
+
_
i<j
i<j
The natural action of the permutation group since it is
a
Poisson action
on
(C2)
(unramified) covering morphism
Sd
on
(C2)
I
XO
=
0.
d
lifts to
a
free action of Sd
d
it is also
a
Poisson action
onto the affine
(MI, I-
on
M2 and
S
-
,
on
M, and
1) NowS is a d! .
:
I
is invariant for the action
variety MI, hence M2 may be identified with the quotient MIlSd. By Proposition 11.2.25 M2 has a unique Poisson structure such that S is a Poisson morphism. It will be denoted by I-, -J d' of
Sd
on
We would like to transport this Poisson structure on M2 to C2d by the morphism P2 CU. Of course this is in general impossible, however in the present case it turns
M2 _+
out that there does exist
that P2 is
a
Poisson
Proposition M2 _+ C2d is bracket is
2.1 a
a
(unique)
morphism.
It is
There exists
Poisson
a
morphism.
Poisson structure
given
in the
on
C2d,
also denoted
-1'*, d
by
such
following proposition.
(unique)
Poisson structure
I-, J'd
In terms
of the coordinates
U,' V,
on
C2d such that
P2
for O(CM)
the Poisson
0 <
1.
given by
IUN
I
IV(A),Vjld'
Uj Id
IU('\)'Vj}d'
=
1Uj1V(A)1d'
=
=
01
W(AIVN)
I U(11)
Aj+1
74
mod u (A),
j :5 d
-
(2.4)
Except for the compatible. As
a
zero
bracket
special and
most
then the Poisson structure Poisson bracket
(2.4)
2. The
systems and their integrability
1., -10, d
all Poisson brackets
important
I-, -I'd,
I-, J'd
the
of rank 2d and they
are
all
ify and x are canonical variables, i.e., V(x, y) 1, by 1"'Id; is regular; the nonzero part of the
case,
=
also denoted
reduces in this
to
case
[u`1)
1U(A)'Vj1d=1Uj1V(A)Id= and its matrix
are
(2.5)
),j+l
to the coordinate
of Poisson brackets with respect
functions
ui and vj, takes
form 0
1
I
Ud-1
1
U3
U2
I
Ud-1
U2
Ul
is
given by
(0
0
...
0
0
...
0
)
0
P=
(-U U)
U
where
0
In terms
of I') Jdy
the Poisson structure
1U(A)7 Adp MA) fl d
W(A7 VN) 1U(A)7 f1d mod u(A),
0
f
is any element
(2.6)
(P(Al VN) IVN f1d mod u(A), I
7
where
I-, -I'Pd
of 0 (CM).
Proof We compute explicitly on M2 the Poisson brackets of uo.... i Ud-li VOi Vd-1 (without that of with observe brackets and t; by Propothey are independent t) worrying about their ...
sition 11.2.4 this leads to
morp1hism (the unicity
a
f. .IVd on C2d which makes p2 into a Poisson immediate). Clearly Ju(A), u(p) J d' 0. If I < j :5 d,
Poisson bracket
of this bracket is
then
JUd-jMA)jd
(-I
ji1
A
Xil Xi2
yl
xij 1=1
-
11 X1
-
kol
A
1: i1
EjXi1Xi2-XiJ7y1 1W 11 X1 d
k961
1=1
Xk
Xk
Id
Xk Xk
i
A
i1
xij W(xit IYO
ist
Xil Xi2 t=1
kOit
A
(-W-1 d
k961 Xk
X1
Xk
I)j
-1
M=0
-
1=1 M=0
7
YO kol
75
-
Xk
X7nUd -j+m+l
j-1
E E Xm1Ud-j+M+1(P(X1
X1
j-1
A
Axt, YO k961
d
Xk
XilXi2
10fi1
(-W-1
-
X't
X1
-
Xk
Xk'
-
Xk
-
Xk
Chapter
M.
Substituting A=
Integrable Hamiltonian systems and symmetric products of
x, in the
right hand side one
mial in A of degree less than d which takes at A for I
1,...,d. As the
=
J-1
xj axe the zeros of
\M Ud-j+m+1W(A,
E M=O
v
(,\)) mod u(,\),
=
IUd-hV(,XWd'O
xj the value
u(A)
and
that
sees
is the
(unique) polyno=0 Xl"'Ud-j+m+l W (Xii V(X1))7 v(xl) the same is true for I
Ej
rn
and since yj
curves
=
find
we
j-1
jUd-j V(A) Id
1: 1\
=
,
M
Ud-j+m+lW(/\i V(,\)) mod u(A)
,n=O
which proves the second
equality
U(,\) I\d-j+l
W(A,V('M
=
(2.4).
in
mod u(,\),
For the first
I
(2.4),
notice that
W
d
IU001 V(A)ld
in
equality
HP\
X.)
-
YZ
Id
/Z
-
H X1
-
j961
Xj Xj
d
(A
=EW(X1,Y0jj 1=1
is
symmetric
in
and p, which leads at
Ivi, vj I'd
In order to show that
general
W the result then follows from
IV (/\), V (A) I d
E
=
(
A
i,k=l
where k
as
(i
well
++ as
k,
denotes
p
\ Xi
a
once
1UNIVild'
to us
xj Xj
-
)I
d
=
lUi,V( A
11 Xk
Yi,
X,
(i
-
-
by chosing
196k
++
term similar to the first one, obtained
Xi
-
Xi
Xj
corresponding
X1
-
k,
p ++
1; for
(2.7)
Xi
-
to
by exchanging i with k i axe given by =
X1,
which is
symmetric in X and p so that these cancel when substracting the symmetric (2.7). remaining terms, which correspond to i =A k can be rewritten as
term
The
in
Yk k96i
a
(p
X1
( -Xj) ( O-x,)r-
Yi
d*
construction
As for the first term, its terms
p with
xj)
the formulas
simplify
(2.6). By
-
-
jol
0, let
==
.) Ut
x
polynomial
terms in we
(2.7)
(
"
-
Xi
xi Xj
-
) (Ini
which evaluates to 0 for all are
of
96k,
(X, p)
degree less than d which
may conclude that
IV W
7
V
(11) Id
=
/,
X,
-
Xk -XI
=
)
(x,, xt)
agree
on
P
I
which is
an
easy consequence of
+
I -'J'd =I
(2.4). 76
W+1P
.
.
I
(Xi
Id
2
Xk
-
-
with
the d
0-
Compatibility pf the brackets derives from the formula
I- Td
P
Xk)21
=A t. It follows that both couples (X, p) (x,, xt) and s
=
2. The systems and their
Notice that it is also
O(CM) using
O(C2d):
__
monic
a
For W
j
=
0,
.
.
.
,
d
polynomial 1
=
seen
from formula
it suffices to
one
that if
use
(2.4)
integrability
-j'Pd really is
that
several
polynomial (in a polynomial (in
a
then the result is also
obtains
(2.5),
because the
map from
variables) variables).
by
is less than d for any
1, which also leads at once to the matrix representation of equals 1, it is regular of rank 2d. Note also that with the standard Poisson structure on CU. compatible
-
-
is not
X
all these
determinant of this matrix
1* "Id
O(CM)
is reduced
IU 1+
degree of
a
d
since the
if d > I then
fails to be maximal, we need to To see where the rank of the Poisson structure I-, J1 d Using elementary investigate the determinant of the matrix of Poisson brackets f ui, vj 1'. d finds that for values of determinants one Xdi properties any -ol, ...
det
(jui, v(xj+,)Idl) O
=
det
i
RUi1VjPd0) O
-
X0
(2.8)
k<1
Choosing
det
Xd to be
x,
the roots of
u(A),
we
get from (2.4)
)' +P1=0j+')
W(xj+,,v(xj+,)) RN V (Xj+l) I 0)0
Aj+1
O
d =
11
det
Ai+1
-\=xd +
+
W(X-' V(X.))
O
det
(N' V(Xj+l)}d) 0<,,j
d
M
( _1)[d/2]
JI(Xk
-
H
XI)
(i)
where in
we
used
(2.8)
for W
W(XM' V(XM))'
M=1
k<1
1. It follows that
(even
if
u(A)
has
multiple roots)
d
det
[d/2]
RUil VAd
11
W(XM' V(X.)),
M=1
on
all of
C2d,
which for
hence the Poisson structure is of lower rank
given
W and d is easy written
Finally, (2.6)
follows
immediately
as
on
the equation of
the lo cus
an
algebraic hypersurface
depends only on x and has degree at most d, then 1., -11d x1, 0 < n < (see Example 11.2.14). Explicitly, for W in taken order above the the to coordinates, respect Ud-1,
If W
=
...
by Pn
in
from the Leibniz property of Poisson brackets.
structure
with
d
rl.!_-, W(xj, v(xj))
0
0
Un
0
0
0
0
-Un'
Un
0
0
0
Un/
0
0
0
77
is
a
0, CU. 9
modified Lie-Poisson
d the Poisson matrix P,, iUO,Vd-1,
...
7
vo I is
given
Chapter 111. Integrable Hamiltonian systems and symmetric products of
curves
where
Un
In
0
0
0
=
d
2(d
1
I
Ud-1
if 0 <
n)
-
a
*
-
0
E
C)
Un+1
and
-
Ul
UO
Un-2
Un-3
-
UO
0
Ul
UO
UO
0
J
...
0
0
...
0
0/
a
for which
y)
is the
only
Ul
UO
Ud-3
UO
0
Ul
UO
...
0
0
UO
0
...
0
0/
Ud
Poisson matrix is
I
one
Ud-2
Ud-2 I
where
,
Ud-1
I
-
-
,
Id
which
is
a
gives
n < d it is easy to
E!
given by
%=
non-singular 0. By induction Un' uO =
a
Lie-Poisson structure.
compute the invariant polynomial
has co-rank at least k if and
In conclusion the rank is at most 2d
hyperplane
=
uO
0,
on
all of
C14,
it is of rank at most 2d
...'
-
zero
2n
the
on
(2d
=
since
if and
only if uO ul it is of rank at most 2d =
(defined
Namely, ...
=
only
Uk-1
-
2
=
on
n)-dimensional
-
in
Un
=
...
=
space
=
R dS2d +
=
Rd-nS2d-n (I
1dX'
Rd-ISM-l +---+Rd-nS2d-n + RS +
R2S2
+
-
-
-
+ Rn Sn)
Notice that the fact that pd,,, is reducible reflects the fact that the Poisson structure is product. The polynomial has the simple (d + 1) X (2d + 1) matrix representation
where I is the
polynomials
I F (x, y)
structures
on
identity
matrix of size
n
0
0
0
In+,
a
+ 1.
Polynomials in involution for I-, -j"d We will
that
0-
the
on
-
Pd,n
2.3.
if
the latter space it is regular. Thus we have established the un-i formula for the polynomial associated to the Poisson structure I-, (0 :5 n :5 d):
=
0 and
Notice that
OciPi.
modified Lie-Poisson structure and that
Poisson structure, which we will denote by Pd,n. it suffices to look at the matrix U' whose determinant is .
is
following
*
a
corresponding
0 <
=
Definition
UO =U1
Un-2
.
Xn, p(x, y) 11.2.47) of the
For
Un+2
Un+2
d then the bracket is
<
(0Ud' _Ud)
rXd' (c
Un+3
C2(d-n)
E!
=
Un-1
product bracket (of a regular Poisson structure non-regular Poisson structure of rank 2n on C2n) For Lie-Poisson bracket which is given by n
on
For V cixi the "=O these axe the only W (x,
,p(X, y)
)
Ud-1
0
Pd
=
1
and Un1
finds
one
0
...
:
-'--
paxticular,
of rank n
(0
now
show how
an
arbitrary polynomial F(x, y) leads
to
a
natural set of d
C2d which have the remarkable property to be in involution for all the Poisson 1. 1 d These polynomials generate a d-dimensional algebra (under the assuinption
on
-
-
,
is not
C2d for any
independent
structure
.1'. d
of
y),
hence
they define
Since all the brackets
78
an
are
integrable Hamiltonian system compatible this means that we
2. The
have for each
F(x, y) (which
systems and their integrability
is not
independent of y)
a
large
class of
Hamiltonian systems. They are however not multi-Hamiltonian below (see however also Paragraph VI.3 and [Van5]). Let
F(x, y)
C[x, y] \ C[x]
E
and let
us
view Cd
as
as we
the space of
compatible integrable will
see
in Section 3.5
polynomials (say
in
A)
of
d
degree less than d. Then there is a natural map flPd from (C2) \'6' to Cd, which assigns to d-tuple ((xl yl) (Xdi Yd)) the unique polynomial in C[A] of degree less than d, which takes for A xi the value F(xi,yi) (for i 1,...,d). We thereby arrive at the following commutative diagram a
,
I
...
I
=
=
M,
M2
PI
P2
(C2)d \A
S
(2.9)
C2d
\f1ii-111 fIll, d
cd in which the existence of the dotted
Lemma 2.2 in
(2.9)
There exists
is commutative.
a
Hpd
arrow
is
guaranteed by the following lemma.
(unique) morphism HFd is
C2d
_+
Cd such that the triangle
explicitly given by
HF,d(U(A), V(,\))
=
F(A, v(A)) mod u(A).
(2.10)
Proof
P*lfIF,d
:
Cd is a morphism which is invariant for the action of Sd hence it can be quotient MIISd which we identified with M2. This means that we have a M2 _4 Cd. It associates to (t, u(A), v(A)) E M2 the unique polynomial (in A)
MI
_+
factorized via the
morphism p3: of degree at most d
-
I whose value for A
compact formula for P3
can
be
P3 (t,
To check this at most d
formula,
=
xi, xi any root of
u(A),
is
given by F(xi, yi). A
given:
u(A), v (A))
note that the
right
=
F(A, v (A)) mod u(A).
hand side is
clearly
a
polynomial (in A)
I and for any xi which is aroot of u(A) it evaluates to F(xi, v(xi)) Since the map P3 does not depend on t it can be factorized in turn via P2) i.e., P3 and HFd is explicitly given by (2.10). -
=
of
degree
F(xi, yi). =
P*2HFd 0
The d components of the map HFd define d regular functions (polynomials) on C2d' which will be simply denoted by Hd-,,..., HO (omitting the dependence on F and d in the
notation), i.e., HFd(U(A), V(,X))
=
Hd-,Ad-1
The main result of this section is the
+
Hd-2,\d-2
following.
79
+
+
HO.
Chapter Ill. Integrable Hamiltonian systems and symmetric products of
curves
Proposition2.3 For any polynomial F(x,y) E C[X,y]\C[X], let AFd C[Ho, .,Hd-,.], where HO,...' Hd_j are the coefficients (in A) of H(A) F(A, v(A)) mod u(A). The triple AFd) defines for any non -zero W(x, y) E C[x, y] an integrable Hamiltonian sys(C2d, I., J, d tem and these systems are all compatible. =
-
=
Before
proving this proposition we
for the Hamiltonian vector fields
Letp(A), q(A)
Lemma2A
r(A)
(1)
XH ,
and
r(A)
[A-'q(A)] + mod q(A)
prove
=
a
key
be
polynomials,
r(A)
with
[A-'q(,\)] +
deg q
(2)
lemma and write down
explicit equations
I-, Hilld
=
-
degq(A) ! degr(A)
q(A)
and let i E N.
[A-'r(A)]
deg q
p(A) [A-lq(A)] +
p
(2.11)
A'-'p(p) [tt'q(p)] + mod q(M).
mod q(A)
Proof For the
proof of (i)
note that if
deg r(A) :! deg q(A)
then the
right hand
side of the
identity
r(A) is
[A-'q(A)] +
polynomial
a
may
assume
-
q(A)
[A-r(A)] +
=
-r(A)
[A-'q(A)]
+
q(A)
_
[A-'r(A)]
degree less than deg q(A), hence also the left hand side. To show (2) we deg p(A) < deg q(A) because the equality depends only on p(A) mod q(A).
of
that
Then deg q
deg q
Al-I
[M-lq(/.A)]+ mod q(p)
(p(p) [lj,-lq(p)]
-
+
q(p)
[IL-lp(p)] +)
deg q
(p(A) [A-1q(A)]
-
+
q(A)
[A-1p(A)] +
deg q
[A-lq(A)]+ mod q(A). applied part (i) of this lemma; the exchange property expanding the polynomials or by induction on deg q(X). In
(i)
we
in
(ii)
is proven at
once
M
The coefficients Hi ofF(A, v(A)) mod u(A) determined polynomial Proposition2.5 which are explicitly given by on C2d' fields XH 'i aF
XH U(A) XW Hiv(A)
=
=
W(AIV(X)) ,
W(A' VNI
ay
GX' v(A))
[F(A,v(A)) U(A)
80
U(A) I'Xi+1
by
vector
modu(A), (2.12)
1+
Ai+1
mod u(A).
2. The
systems and their integrability
Moreover, the following remarkable identities hold for all 0
jui, HjjId
=
JU31 H-l'd -
"
Ivi, Hjj'Pd
and
i, j :5 d
<
-
1:
jvj, HiJ'*d
=
(2.13)
Proof
Writing XH, we
obtain
as
as a
shorthand for
the coefficient of
/.jd-i
XHI,,
in
first compute
we
=
ju(A),Hjj4,
which
JU(,X)iH-Fd(U(P),V(/4))Id*
d-1
JU(A), HFd(U(/Z)) V(/'))Id
XH,u(A)
1: IUN
(9HF d
'
I
Vi Id
j=O
'
avi
(U(/')'V(/4))
d-I =
E
19HFd(U(/.,),V(,,))
[U(,\)] Aj+1
j=0 d-I =
avj
+
d-j-1
Ud-kA d-j-k-I
E E k=O
j=O
d-I d-I =
j=O
A,-,
OF
ay d
H(A) is
this leads to
found,
Ca, V(/Z))
exchange property (2.11)
(A, v(A))
IU(I11 ),I
mod u(p) +
modu(A) +
in the last step. Since
equation (2.12) for XHu(A) in
the computation of
IU(P)I A
aF
ay used the
(/,, v(p))pj mod u(p)
49F
d
we
i9y
Z E uj+,Al-l ay (/.z, v(IL))pj mod u(p) 1=1
where
aF
-OujLHFd(U(A), V(/I))
case
p(x, y)
is however
=
Hi is the coefficient of Ai in
1. In
more
a
similar way
involved: let 0
<
XHv(A) j
< d
then
a
a
(F (p, v (p)) mod u (p))
'9Uj
F(p, v(p)) U(m)
(u(t) (U(/Z) I
-U(/')
U(A)
/d
RM-) y
F(p, v(p)) U01)
I +)
[F(p,v(p))]+_ U(P) [F(g, v(p))
U(O
81
+]
modu(p). +
pi F(p,v(p))] U(P) U(A) I +)
-
1
Chapter In
(i)
we
III.
Integrable
used that if R R
Granted this,
we
R(p)
=
[Pj+
obtain
Hamiltonian systems and symmetric products of
as
-
and P
[RP]+
=
=
R
P(p)
[P]+
tit
=
d 1=0
obtained at
once
(2.12)
once
to the
for
upon
[R [P]+] +
-
d-1
the formulas
rational
functions,
with
[R]+
0, then
=
[R [P]+]
=
above
jV(A),HF,d(U(A)i which leads at
axe
curves
[U(,X) ] [F(A,U(X)v(A)) ] +
expression (2.12) for XHj v (X)
XHu(,\)
and
modu(X),
X1+1
XHv(X),
in
case
+
W(x, y)
the formulas for
=
1.
XH",u(,\)
Having obtained
and
XHI%v(X),
are
using (2.6).
Finally, the exchange property (2.11) implies that X and M are everywhere interchangeable in the above computations so we get JU(,X), HFd(U(A)i V(ltffld o JU(/1)7 HFd(U(X)i V(,\))Ilpd which is tantamount to the identity juj,Hjj' The second formula in (2.13) juj,Hjj'. d d =
I
=
follows in the
same
way.
Proof of Proposition 2.3
VNWd
We first prove that jHi, HPd(U(X) i 0 for 0 < i < d 1, which shows that is involutive. To make the proof more transparent, we use the following abbreviations:
aF
FV so
that
=
(2.12)
19Y
(,\,V(,\)),
is rewritten
F(U)
as
=
=
F(,\, v(,\)) U(A)
X"
Hj u(A)
=
-
and
Uj
u(A) [UiF,]
=
and
W(X, V (,X)) U(A)
XH v(A)
=
1U(11)I
I
Ai+1
u(A)
AFd
+
[Uj [F(u)] +]
Then
U(A) jHF,d(U(A)iv(A)),Hjj O=X.'Fj, d
I F(A,(A)v(A)) U
-
lp ) [F(u)]_+u(A) -=XHjUG
=
U(A)
[[UiF,]- [F(u)]
+ _
-) I XH' 0,F(A,v(A)) U(A) [Uj [F(u)] +1
FV
F(u)XH Ou(A)
I F(u) [UiFv]_] U(A)
-
-
-
_
W
=
=:
[- [UiF,] [F(u)] u(A) [[UiFv]+ [F(u)] +] u(A)
+ +
F, Uj
[F(u) ] + I
_
0.
0. polynomial, i.e., [F,,]_ of d coefficients the show that We now Hpd(U(1\), V(I\)) F(A, v(A)) mod u(A) are indedim C2d 1Rk I d the last d coefficients pendent, showing that dim AFd IClearly d in because f1d-1,...' f1i (it does f1d-1,...' AO of F(A, v(A)) are independent vi appears only appear since F(x,y) 0 C[x]). Reducing F(A,v(A)) modulo u(A) amounts to substracting from ki polynomials of lower degree in the variables vj, so it cannot make these functions dependent and the independence of JHO,...' Hd-11 follows. Finally A.Fd is also complete. Namely we will show (in Proposition 3.3) that the general fiber of M --+ Spec(AFd) is irreducible and (in Lemma 3.4) that all fibers have the same dimension (d). Thus AF,d satisfies all conditions of Proposition 11.3.7 which implies its
In
(i)
we
used the fact that
F,,
is
a
=
=
==
-
-
,
completeness.
82
2. The
systems and their integrability
Amplification 2.6 Suppose that F(x, V) and F(x, y) differ only by which is independent of y and is of degree less than d in x, say
a
polynomial c(x)
d-1
F(x, y)
=
F(x, y)
E cix',
+
i=O
then d-1
F(A, v(.X))
mod u(A)
=
F'(,X, v(A))
mod u(A) +
E ci) , i=O
hence the
=
by to
polynomials
find APd saying that
we
a
in involution which
they determine
are
up to constants the
same
and
API,d, that is both systems are isomorphic. We might reformulate our result for W(x, y) fixed we have associated an integrable Hamiltonian system -
-
family d-1
Fc (x, Y)
F (x,
=
y)
cix' I
+
ci E C
i=O
bigger family M is given, i.e., F,.(x, y) depends on one or several extra parameters which we suppose to parametrize an affine algebraic variety. One observes that the Hamiltonians Hl,...,Hd depend polynomially on the coefficients of F, hence also on the parameters c, so by Proposition 11.3.24 there is an integrable Hamiltonian system on M X C2d with O(M) as its algebra of Casimirs and with projection M x C2d -+ M Such that the fiber of this morphism over any closed point c E M is precisely our original system on C2d corresponding to the polynomial F,(x, y) where the parameters have been given the fixed value c. We will often prefer to work on these bigger systems, see Chapters VI and VII. For a further generalization, in which arbitrary families of algebraic curves are considered, Suppose
now
that
a
,,
see
[Van5].
2.4. The We
hyperelliptic
now
turn to
case
which will be important later: the case that F(x, y) y2 f(X) We call it the hyperelliptic case because F(x, y) 0 now defines a
a case
=
_
some polynomial f (x). hyperelliptic curve (see Paragraph IV.2.6). In the following proposition we give Lax equations for the hyperelliptic case (Lax equations will be explained in more detail in Section V.5; in this section a Lax equation is no more than a neat way to write down the differential equations describing an integrable vector field).
for
=
y2 f (X) for Some poly_ Proposition 2.7 If F(x, y) has the hyperelliptic form F(x, y) nomial f (x) then the differential equations describing the vector fields XHI, are written in the Lax form (with spectral parameter X) =
X '
Hi
A(,\)
=
[A(,\), [Bi (A)]+]
_
(2.14)
,
where
A(A)
=
V(A) (X)
w
U(A) V (X
Bi(A)
-
83
-
O(A'v(A)) U(,\)
[u(11) I Ai+'
A(A) +
III.
Chapter
Integrable
Hamiltonian systems and symmetric products of
curves
and
W(A)
rF.(/X, v F-R-A)l +'
--
-
0, which spectral curve det(A(A) M Id) is XHI',, given by t12 f (A) HPd(U(A), V(A)) The
preserved by the flow of the
is
=
-
vector
fields
=
_
-
Proof If
we
define the
XH ',u(A)
=
polynomial w(X)
as
2W(A,v(A))v(A)
2u(,\) W(A, v(A))
-
XH'jV(A)=-WGX1V(A))W(A)
above, then equations (2.12)
stated
[U1111
+
),i+l
1W
U(A)
(A, v W)
V(A) U(A)
axe
[U(A)I+l [U(),)]+]
rewritten
as
Ai+1
W(A) U(A)
Xi+1
+
using
upon
(9F
5yRom on
F
(2.15)
we can
(A, v (A))
2v (A).
=
P w(A); observe how in this calculation the explicit dependence compute X Hj
disappears completely!
9 XH
i
WOO
=
(A)) U(A)
F (A,
-X3
j
1
2
+
XV
-2
+
41
u
=
v
I v(A)) XH Ou(A ) V(A) v(A) 1 [F(A,U(A) UN I l (A) 1 1 w(A) U(A) [v(A) [O(A) O()k'v(A)) U(A) 'P
=
+
+
=
This leads at
2w (A)
once
I
-
Aj+j
I
W (A,
v
(A))
VN U(A)
to the above Lax
[ (I` 1
+
U
Ai+l
+
1
2v(.\) WO,, v(A))
WN U(A)
[U(),) 1 Ai+1
+
The associated
equations.
+
spectral
curve
is
+
1
-
+
computed
as
follows:
det(A(A)
-
p
Id)
2 =
/4 _
2
_V
For
example,
v(A)
=
if
we
I
U( A )
v2 (,\)
IA2
_
=
/42
_
U(,\) f (A)
restrict ourselves to d
=
-
1
_
f (A)
U(A)
+
I 2(A)U(A)f(A) _
HFd(U(A), V(A))(i.e.,
one
degree
of
freedom),
then
u(A)
vo and
HPl (UO'
VO)
=
=
=
and
is the standard bracket
one
of freedom
degree
+
V
=
_
A + uO,
(A)
on
(V2 (V20 V20
C',
corresponds exactly
f (A)) -
_
mod u (A)
f (A)) mod(A
+
uo)
f(_UO)'
so we
84
hyperelliptic case in polynomial potentials on the line.
find that for W
to the
case
of
=
1 the
3. The
3. The
geometry of the level manifolds
geometry of the level manifolds
In this section
we
determine the nature of the
(general)
level sets of
(C2d, J.'.JW AFd) ,
SpecAPd where, as dwe have Cd. is In to isomorphic seen, SpecAPd particulax they axe independent of the polynomial V(x, y) which dictates the Poisson structure (the impact of W (x, y) presents itself only at the level of the integrable vector fields and is discussed in Section 3.5). It was generally believed that the general level set of an integrable Hamiltonian system with polynomial invariants is an (affine part of) a complex torus (Abelian variety) or an extensions of a complex torus by C*n (see Chapters IV and V). It will turn out that the level sets encountered in these examples are of a different nature. We will also look at the real parts of the smooth fibers: whereas the real parts of Abelian varieties axe quite special (see [Sil]), it will turn out that we find here a very rich class of topological types which appear as real parts of the fibers of These level sets
the fibers of the momentum map
are
C2d
_+
the momentum map.
3.1. The real and
complex level
sets
Since Spec(APd) is isomorphic to Cd and since the functions Ho,..., Hd_1 are independent, the fibers over closed points are given by the level sets of HO, Hd- I or equivalently by the level sets of Hpd(U('\), V(,\))- Since HPd(U(,\), V(A)) is defined as F(A, v(A)) mod u(,\), the fiber over an arbitrary polynomial c(,\) of degree smaller than d is the same as the fiber .
over
0 for
HFI,di
lr(X7 Y)
Where
0 for all
lying TP,d is given by over
conjugation
map
axe
,
-
(U(A), V (.X))
defined
C2d
r :
=
_+
C2d
C
as
follows:
: z
-+ 2 as
TF d (TF,d, T)
.
F(x, y) c(x). Therefore it suffices to describe the fiber polynomials F(x, y). We denote this fiber by "T'Fd; thus, by definition,
TF,d
The real level sets
.
=
F(A, v (A)) u (,\)
C2d
I
0 -
I
(3.1)
-
denote the fixed point set of the Fix(,r) and we define we
Fix(7-)
n
complex
FPd-
(3.2)
real
algebraic variety (see [Sil]), whose real Part is TP d; in fact, if F(x, y) is a polynomial Y7d are nothing but the level sets of the corresponding real integrable Hamiltonian system ( btained by replacing in all definitions C by R, see Paragraph 11.4-2). We determine the non-singular real and complex fibers in the following is
a
real
two
then the level sets
propositions.
Proposition then the fiber
3.1
): d
If the algebraic curve rp C C2 defined by F(x, y) C2d is also non-singular.
=
0 is
non-singular,
C
Proof
Y Fd only
will be smooth if and
only
if
H.Fd
is submersive at each
point Of )7
dl
if Rk
(9Ud-1"'*' 8UO'avd-1'***' 5vO)O
d,
along
-FF d,
i.e.,
if and
Chapter
III.
Integrable Hamiltonian systems and symmetric products of
curves
the i-th and (d + i)-th columns proof of Proposition 2.3 and the definition of -FRd, F, this matrix are respectively given by
From the of
Ai
F(A, v (A)) modu(A) UN
Aj
and
OF
VY
(,\,v(,X))modu(,\).
It is therefore sufficient to show that if r, is smooth then the dimension of the linear space
R,(A)F(X'v('\)) U(A)
equals d. Let
OF
+R2(A)
Oy
(A, v
(M)
A, be the distinct
roots of
F(Ai, v(,\i)) U(,\i)
and
mod u (A),
deg R (,X)
<
(3.3)
d,
u(,\), Ai having multiplicity
si.
We claim
that
c9F =
0
19Y
(,\i, v
(3.4)
0
simultaneously if IPF is smooth. For otherwise (xi, yi) '9F 0, singular point of rp: if (3.4) holds then clearly 6-Y (xi, yi)
cannot hold a
'9-F
67X (xi,
yi)
=
(Ai, v(Ai)) would F(xi, yi)
=
=
0 because in this
case
F(x, yi)
has
a
double
zero
at
x
=
be
but also
xi.
investigated by using the fact that for any polynomial p(,\), 1 derivatives just p('Xi), and the values of the first si AX) of p(,\) mod u(A) at Xi are given by the values of the corresponding derivatives of p(,X) at \i (si is the multiplicity of Ai in p(A)). Let us suppose that the different roots of u(A) are ordered such that A, ......\t are also zeros of !M- (A, v, (A)), while At+,, A, are not. As a OY 1 derivatives vanish at Xi first restriction, let Rl(,X) (resp. R2(,\)) be such that its first si The dimension of
(3.3)
is
now
mod u(,\) at \i is
the value of
-
.
.
.
,
-
(resp. I < i < t). As a further restriction it is (by the first restriction and as (3.4) cannot happen) now easy to see that R, (,\) (resp. R2 (,X)) can be determined such I derivatives of (3.3) take any given that the polynomial given by (3.3) and the first si values at Xi for 1 < i < t (resp. t + I < i < r). These d conditions are independent, hence I the dimension of (3.3) equals d and )7& is smooth. for t + I < i <
r
-
The
Proposition 3.2 is non-singular.
curve
r,
C
C2 is non-singular if and only if the fiber TFd
C
C2d
Proof If rF has
a
singular point P,
=
(xi, VI),
choose for i
=
2,.
.
.'
d
-
1
an
extra
point
E .97,vd. All polynomials Pi (xi, yi) on IPF and define (u(A), v(A)) S((xi, yi), 7 (Xd , Yd)) xi, hence they span a linear space of dimension less than d. Thus given by (3.3) vanish for X HF,d is not submersive at (u(A), v(,\)) and AFd is singular at this point. This shows the if part =
=
...
=
of the
proposition; the only if part
It will be
seen
that
a
clear
(for IPF smooth), leads also
to
is proven verbatim
as
in the real
case
(Proposition 3.1). 0
understanding of the structure of the complex precise description of their real parts )7 d*
a
86
level sets 97Fd
3. The
3.2. The structure of the
17F
C
C2. Recall (e.g. from
d
product Sym rp
affine part of the d-fold symmetric
an
that
[Gun])
permutation group Sd
action of the
level manifolds
complex
-FFd is
We will show that
geometry of the level manifolds
SyMd rF
is defined
the cartesian
on
of
the orbit space of the natural
as
product rdF
rF
x
X
l7p (d factors),
i.e., S ym,dr.F
Symd rF
inherits its structure
rdF ISd
=
from the
algebraic variety
as an
structure of
algebraic
r,
d Moreover if IFF is smooth then the same holds true for Sym IPF: namely each point P I, (with all Pi different; mi is the multiplicity of Pi in P) has (PMJ,..., p.,n,) C I
Symd
r
a
neighborhood of ((Pm1),...' (Pm-)) in Sym7l IPF x neighborhood the diagonal of symm, rF admits local coordinates on and a point (Pim') x sym- rF, given by the mi elementary symmetric functions of the mi coordinate functions on 17'pl. which is
isomorphic
to
a
...
Proposition
0 is non-singular, If the algebraic curve rF in C', defined by F(x, y) d biholomorphic to the (Zariski) open subset of Sym ]pF, obtained by removing 3.3
=
then -FFd is from it the divisor
DF,d= In
f
(
(P]L,...'Pd)J3iJ:1
particular FPd
x(Pi) Pi
=
x(Pj)
=
is
Pj
a
=A Pj,
with Pi
or
ramification point of x
is irreducible.
Proof Construction
0
Given
a
of the
map
point (u(A),v(A))
every root
Ai of u(A)
one
OFd : -T-7Fd
E
has
TFd,
a
-+
SYMdrF \ DFd
point
F(Ai, v(,Xi))
=
SymdrF
in
0, because
follows: for
as
each root
I
is associated to it
F(A,v(X)) U(A)
I
=
0,
so
\i of
-
point (Ai, v(Ai)) on r, Thus there corresponds to (u(A), v(A)) E FFd an d unordered set of d points (P.1, Pd) E Sym r.F, where Pi is defined by (x (Pi), y (Pffl Pj; therefore, to show that (P,,..., Pd) x(Pj) then Pi (Ai, v(Ai)). Clearly, if x(Pi) that need to Pi from we Pj cannot occur for i :/= j if Pi is a DFd only prove stays away F (x (Pi), y) (as a polynomial in y) of is root if a multiple ramification point for x, i.e., y (Pi) As Pi Pj (i : :4 j) implies that u(A) has a multiple root x(Pi), in such a case F(x, y(Pi))
u(A)
determines
a
=
=
=
-
=
would have
a
multiple
point of singular point of 17F, ramification
0
DFd is
a
divisor
root
x
=
x(Pj), again "
(x (Pi), y (Pi))
x
then also
a
contradiction.
on
ey
because =
[
F(,\,v(,\)) U(A)
I
=
0. If
moreover
Pi is
a
-
0 and it follows that
(x (Pi), y (Pi))
is
a
SyMd IPF
holomorphic function. If U Sn, such d} be decomposed as S, U (PI Pg) E VFd let the set of indices 11, that all points Pi, with i running through one of the subsets Sj, have the same x-coordinate, which is disjoint from the x-coordinates of the points which correspond to the other subsets. For each Pi (i d) let xi denote the lifting of x to a small neighborhood of (P1,...' Pd) (corresponding to the factor Pi). Then a local defining equation Of VFd is given by This
means
that DFd is
given locally
as
the
zero
locus of'a
...
.
.
.
I... I
,
=
n
H J1
(Xj
-
i=1 J,kESi j
87
Xk)
=
0-
Chapter
OFd
0
is
a
111.
Integrable Hamiltonian systems and symmetric products
(Pi
curves
biholomorphism
We first construct the inverse Of Let
of
Pd)
E
OFd, which is closely related SYmd ]PP \ DPd- Clearly u(A) is taken as
to the map
S,
as
given by (2.3).
d
U(A)
MA
=
-
x(pi)).
(3.5)
i=1
If all
x(Pi)
axe
different then
whose value at A
=
x(Pi)
is
v(A) is uniquely determined y(Pj), i.e., v(A) is given by
as
the
polynomial
of
degree
d
-
I
d
A
1=1
and is
holomorphic there.
kol
If two values
a ramification point (since be solved uniquely as y
-
1: Y(PI) 11 X(pl)
V(,
coincide,
X(Pk) X(Pk)
(3.6)
-
x(PI)
P2 is not x(P2), then P, stay away from DFd), hence the equation F(x,y) 0 can f (x) in a neighborhood of P, P2. For Pl' and P2' in this say
=
=
we
=
=
=
neighborhood, substitute
f(X(Pl))
d =
f(x( P1 ))
+
(X(Pj)
x
(pl))
(X( P1 ))
dx
+ 0
(X( P11)
X(PI ))2,
(i
=
1, 2)
for yj and Y2 in (3.6), to obtain that v(A) has no poles as Pl', P2' -+ P3., hence extends to a holomorphic function on the larger subset where at most two points coincide. Since the complement of this larger subset in SymdrF \DFd is of codimension at least two, v(A) extends
holomorphic function on SyMd ]pF \ DFd- It also follows that this holomorphic function OFd on all Of SYM'1 ]PP \ I)Fd: if the point Pi has multiplicity 3j, then the first si I derivatives of v(A) at x(Pi) coincide with those of f (A) at x(Pi), hence F(A, y(Pi)) has a zero of order si at A x(Pi). Finally, the inverse of a holomorphic bijection between complex manifolds is always holomorphic (see [GH] Ch. 0.2), hence OFd is a biholomorphism. Since the symmetric product of any non-singular curve is irreducible, the same holds true to
a
is the inverse Of -
=
for -FPd-
0
Having a biholomorphism between affine varieties does not imply that this is an isomorphism. However this is so in the present case, as can be proven by using Newton's interpolation Theorem. This was done by Mumford for a special case (where the curve is hyperelliptic of genus d); since his (quite long) proof applies verbatim to the general case it is not repeated here (see [Mum5] pp. 3.23 3.25). -
Lemma 3.4
Even
if -FPd
is
singular
its dimension
equals
d.
Proof We may still associate to each singular curve IPF defined by F(x, y)
point =
in
Fd
a
0. The d-fold
divisor
consisting of d points on the symmetric product of such a curve is
but is still of dimension d. Therefore it suffices to show that the constructed map is finite on a Zaxiski open subset. In fact, if we stay away from the singular points in both TFd
singular and in
on
Sym drp then
the
proof of Proposition
these subsets. Thus all fibers
Proposition
(over
closed
3.3
applies
points)
3.3 and Lemma 3.4 finish the
to show that this map is
have dimension d.
proof of Proposition 2.3
sition 11.3.7.
88
upon
bijective I
applying Propo-
geometry of the level manifolds
3. The
3.3. The structure of the real level manifolds
Fd
SinceY
is
given
whose coefficients in
are
TFd
as
n
Fix(-r),
all real. We
it Consists of those out what this
figure
R" polynomials (u(X), v(,X)) E J: F'd for the corresponding point
means
SymdrF-
Under the biholomorphism OFd, the real level manifolds TFd correspond Proposition 3.5 Pd) on I7p, consisting only of real points to the set of all unordered d-tuples of points (Pi, P each ramification point (of x) occurring Pi E R2nr .F and complex conjugated pairs Pi and at most once, Pj Moreover its manifold structure derives x(Pi) x(Pj) only if Pi from the structure of the d-fold symmetric product of rp. =
=
-
Proof
only if its roots consist only of real roots and roots which occur in complex conjugate pairs. Obviously, if v (A) is real, v (A) and its derivatives take complex conjugate values when evaluated at complex conjugate points (and real values at real points). Also, if a polynomial of degree smaller than d is specified in d points that are real or occur in complex conjugated pairs then that polynomial is real (it is sufficient to prove this in case yi, this in which all points are distinct, which is easily done by using (3.6)). Since v(xi) those to Pd) on points (PI, TFd correspond means that the real polynomials (u(A), v(,\)) in Sym drF consisting of real points Pi (x(Pi), y(Pi)) E R2 and complex conjugated pairs
u(A)
is real if and
=
...
5
=
Pj
=
Wpj)' Y(pj))
(xX(Pk), i(A))= Pk, but
=
(of x)
of each ramification point
Proposition level manifolds
3.5
is at most one, and
be used to obtain
can
-FF d, as we show now for d
x(Pi)
=
to
DFd, i.e., the multiplicity
x(Pj) only
if
Pi
=
Pp
I
precise description of the topology of the real 2 2 (for d 1, -TF d is just IPFn R the real part
a =
belonging
not
=
,
2 of 17F). For a fixed F such that rp is smooth, let the connected components of rpn R and define for 1 < i, j, k < s and i < any) be denoted by r,, , r, .
roo
=
I7ij
=
]Pkk
-
.
(P, P) I P E 1Pp, x(P) 0 RJ, (PI P2) E ri x rj I X(PI) X(P2) =
P1
=*
i
(pl, P2)
rk
rk
x
.
S2
Then the union of
(if
.
I x(PI)
=
]POO with all the
x(P2)
sets
l7ij
=
(Pi
=>
P21i =
P2 and is
not
a
ramification
point)
I
-
and ]Pkk is easy identified with 0F,2(AF,2), the paths in it which are not contained in R2,
surface to be described. One observes that the only are
in
roo,
in fact
roo
exactly the surfaces rlk. Therefore, .rkk, nor to ]POO.
connects
connected to any other IF
....
if i
then
rij
is not
I7ij, say on r12- If the intervals x(rl) is either homeomorphic to a torus, a 1712 r, r2, and X(I72) are disjoint, the components r, and 172 are closed or open. If cylinder or a disc, depending on whether x(ri) and x(r2) have a point P in common, then one finds again these surfaces, but with a Therefore
we
first concentrate
then 1712
number of punctures
on
such
=
(holes), equal
x
a
subset SO
to
2
#IQ
E
ri I X(Q)
89
=
X(P)J.
Chapter if
and
x(ri)
Ill.
x(r2)
Integrable have
an
Hamiltonian systems and
symmetric products
of
curves
interval in common, r12 may even disconnect in different pieces. easily determined from a picture of the real part of the curve.
The structure of these pieces is
x (P2)1 is Namely, on a square representing rl X 172 the divisor I (PI, P2) E ri X r, I x (PI) drawn by counting points on the vertical lines x the needs to take care one constant, only is that if 171 (or r2) is closed, then an origin should be marked on it, and if one passes this origin, one needs to pass over the corresponding edge of the rectangle. The following table shows some examples (all possibilities for which 1P, and r2 are closed, and for which x is 2: 1 =
,
=
r2)-
when restricted to 171 and
Divisor
r, and r2
CD CD
Component 1712
1-1
torus
torus minus
C
Picture
(torus
minus
point
disc)
+ disc
0
C )
CD
C)
two
CD
FV1 I A I
cylinder
VN
C)
cylinders
+ disc
two discs
00 00
Table 2 In the
I
same
way
(PI, P2)
E
]Pkk is investigated by drawing the divisor
ri
x
ri I
x(PI)
r,' rectangle representi M, diagonal then represents S2 on a
For a
removed. in
a
The
Figure I.d.
x(P2)
and
(
P,
P2
P,
P2 is
or a
ramification point of
x
ri. Either triangle cut off from the rectangle by its main and I7jj is the complement of the divisor in the triangle.
x
as in Figure La below. Then Figure Lb shows of the rectangle), which is the divisor D to be anti-diagonal (the is drawn in Figure Lc and is redrawn in a simpler way resulting piece r1l For every I7j such a piece is found and will be glued to ]POO precisely along
example, consider
torus with
=
circle
on
a
component ri
it
90
3. The
the part of its
boundary which
geometry of the level manifolds
comes
from the
diagonal
in the
rectangle (the solid
lines in
Figure Ld). D b
_D
a
A a
A
1
b
a
D
A
I
2
D
a
(d)
(c)
(b)
(a)
---
---
A22
F,
1
Figure
explain how roo is described, we recall the classical picture of a (smooth, complete) algebraic curve P. An equation F(x, y) 0 of such a curve defines a m : I ramified covering of F(x, y) in y. This may be visualized by map to P' by (x, y) -+ x, when m is the degree drawing concentric spheres (called sheets), on which there are marked some non-intersecting intervals (called cuts, every cut is equally present on all sheets). The topology is such that if you are walking on a sheet i and pass a cut j (from a fixed side) then you move to a sheet of 11, ml. It is clear that the datum of cuts and their pj (i), each pj being a permutation corresponding permutations determines the topology of the curve completely. Since each cut connects two ramification points (of x), these cuts may, for a real curve, be taken on the real axis and orthogonal to it. roo is now given as follows. Consider the described picture for the smooth completion PF Of 17F. Clearly the conjugation map interchanges the upper and lower hemispheres and In order to
=
.
is fixed
on
the
equator(s) IP
E
PF I x(P)
.
.
,
E R U
ool.
A convenient way to
It follows that the open upper (lower) represent them is by drawing a disc for
hemispheres give precisely ]POO. each upper hemisphere and labeling the different parts of the boundary which correspond to the horizontal and vertical cuts. A moment's thought reveals that the different sheets are to be connected along those lines which correspond to the vertical cuts, while the pieces I7kk This gives a topological model of are to be connected to the corresponding horizontal cuts. rOO U U8k=l rkk as a disc with holes. The following example may highlight the different steps.
yl f (x) 0, where f has five hyperelliptic curve F(x, y) three and its graph zeros and one pair of complex conjugate zeros. The curve has genus P1 of are related representation as a cover given by
Example real and
3.6
We consider
X2
=
a
X3
CD
`4
=
-
"6
X7
X7
x
x,
Figure
imaginary ramification points (of x) upper hemispheres
where the two
91
2
are
not
seen
from the
graph.
For roo
we
get
Chapter
III.
Integrable Hamiltonian systems
and
symmetric products
of
curves
H21
DV DVI H,
H,3
H2
H23
V,
V,
V2'
V2'
Figure which become
r22 given
as
disc after
one
in
Figure
Ld
gluing the by
3
V11, V21-
vertical cut
We also get two subsets 1713. and
---------------
H,'
H,'
H11
H,1
---------------
Figure and
one
disconnected piece 1733 H
(since
00
4
13)-
IE
3
3
H
Figure 5 Now
glue Figures 3,
4 and 5
the other components Of
AF,2 It is shown in the
AP,2
one
same
Hj
according are
to their labels to find a disc with two holes. Since direct products of the real components, we find that
torus + two
way
that,
if
cylinders
F(x, y)
+
is of the form
n
F(x, y)
=
y2
+
m
]I (X
_
Ce,)
i=1
with aj,
#j
E R
(all
ai
being different,
AP,2
((n-l)/2)
tori +
AF,2
(n) 2
tori +
2
where g is the genus
[n+] 2
+
V 2
m
-
as
I of the
11 (X2 +
3
j=j
well
cylinders
disc with two holes.
one
+
as
all
pi2)'
then
one
disc with g
one
disc with g holes
curve
92
F(x, y)
=
0.
-
1 holes
if
n
if
n
is
odd,
is even,
3. The
Compactification
3.4.
We
discuss the
now
geometry of the level manifolds
of the
complex level manifolds
(smooth) compactification
of the manifolds TPd.
There is
one
a obvious and natural compactification, namely the compact manifold Synid PF, d similar way as Sym rp; as above rF denotes the smooth compactification of rp. However ,FP,d has the disadvantage that none of the vector fields XH, extends holomorphically to it in a holomorphic way a compactification such that at least one of these vector fields extends
defined in
-
it, will simply be called good. The interest in good compactifications is that it allows one to integrate the corresponding vector fields in terms of theta functions, or degenerations of theta functions. The purpose of this paragraph is to show that even for very simple choices of F(x, y), a good compactification Of -FFd does not exist. We believe that this is true for almost all choices of F(x, y). A special class of examples for which a good compactification does exist is considered in Chapter V. to
compute, for fixed F(x, y) how the vector fields XH. behave on the compact d manifold Symd PF, which relates to TFd (which we identified with Sym IFP\DPd) as follows: At first
we
d Sym PP
Here
E)Fd
to
TFd
U
DFd
U
EFd-
d
Sym rF andEpd is a divisor whose irreducible components the points ooi in PF \ rF, namely
is the closure of E)Fd in
SF,d(00i) correspond
=
9F,d(00k)
"
f(00ki&
A) I Pk
...
Vp
E
for 2 < k <
dj
XH, being a polynomial vector field on CId' it is holomorphic on TPd- We determine its behavior along the irreducible components of f)Fd and 9Fd, which may be done by computing the order of vanishing Of XH, at a generic point of each component, which in turn is done by using local coordinates at such a point. Each vector field
Proposition 3.7 Every vector field XHj has a simple pole along all irreducible components of tpd. It has a zero of order pk along Gd(00k) (i.e., a pole of order -pk if pk < 0), where pk
Pk
-
vk + d + I
Pk
Vk + I
-
Of X
-
< >
0, 0;
(3.7)
at 00k, and vk is the order
(of vanishing) of Pj'vi9v (x, y) at is 000X(000 if x finite
pk is the order
order
if Vk if Vk
=
of x
at 00k
(resp.
the
Proof
We first write down the vector field XH, at a generic point (u(A), v (A)) E F.Fd; the gener((XIIYI)I. I (XdYd)) all xi arediffericity condition taken here is that for 0Fd(u(A),vGX)) :--:
--
Varying the point (u (A), v (M points (xi, yi) is a ramification point in a small neighborhood, each xi gives a local coordinate on a neighborhood Uj C rF Of (Xd Yd)) (xi, yi) as well as a local coordinate on a neighborhood U C 37Fd Of ((XI Y0 ent and
none
of
of the
x.
I
Since
on
the
(xj
-
one
xj)
hand the derivative of
XH, xj, while
on
u(A)
the other
=
rld=l(,\ k
_
Xk)
d
o9F
XHAXj)
=
at A
hand, direct substitution
i(Xj'Yj) F,
k=i+l
93
UkXjk-i
in
=
I
xj is
...
I
I
XH,u(xj)
(2.12) gives
Chapter
we
Integrable Hamiltonian systems
111.
and
symmetric products
of
curves
find that
jj(Xj Xj)-j
X.U,xj
OF
19Y
10i
where
is the i-th
o-j(.,Bj)
(3.8)
_
symmetric
function in x,.... ) Xd, evaluated at xj
=
0.
right hand side of (3.8) has at a generic point Of DF,d a simple pole, hence each vector field XH, has a simple pole on (every component of) DPd. The behavior of XH, along ISF,d is slightly more complicated since it depends on F(x, y), and may even behave differently on each component 9Pd(00k). For a generic point in a neighborhood of a point of 9Fd(ook), let us introduce coordinates xi as above. If we denote by Mk and vk the integers introduced in the statement of the proposition, then clearly x, is given in a neighborhood of 00k in terms t'Ik (vk < 0), or as x, of a local parameter tj at 00k as x, cl + t' Ik (vk > 0), depending of on whether x is infinite in a neighborhood ook or has a finite value cl E C at 00k; also " 0. We define for 2 < j :! , d local parameters tj t"k (f, + 0 (t)) with (t)) (t) (xl yj &Y (centered at Pj, which may be assumed to be generic) by xj x(Pj) + tj. Direct substitution in (3.8) yields t'j' (Ci + 0 (4)), XHj tl The
=
=
=
I
=
=
XHjtj
=
Cj +
(j
0(tl)5
where Pk is defined in (3.7). We conclude that 46F,d(00k) if pl, ! 0 (resp. Pk < 0)-
XH,
=
has
2,..., d);
a zero
(resp. pole)
of order
jPkj along 1
Sym drF is not a good compactification, since 0 vector fields pole along T)Pd. This divisor can be contracted in some cases, as we XH, will show in Chapter V. The following example shows that a good compactification does not exist in general. Thus
we
have shown that
have at least
Example3.8 To show that
which then
can
a
Let
J7F 2
has
Sym2 PF
=
no
be found in
Y and
yl+f (x), where thedegree off is at least three, andlet d= 2. good compactification we use some results about algebraic surfaces [Har] Ch. V. Suppose that Y is a good compactification of YC F,2
F(x,y)
are
birational; for surfaces
of monoidal transformations
(also
known
as
this
means
blow-up's)
that there exists
a
finite series
which transforms Y into
SYM2 P J'.
2
(Zariski) open subsets U C Y and V C SYM PF to which all these monoidal transformations restrict as isomorphisms and the vector fields on U and V correspond exactly
Then there exist
isomorphism. In particular DP,2 is entirely contained in the complement of V and by one of the monoidal transformations, so at least we know that the Of DF,2 must be 0 (only P"s can be contracted).
under this
must be contracted
genus
We may however compute the genus Of DP,2 directly. points (PI, P2) with x(P.1) = x(P2) for which P, =A P2 or P, its smoothness is easy checked. ramified at the n deg f points =
by 3). So
it has the
same
Recall that it consists of the =
P2 is
a
ramification point,
so
However the map x expresses DF,2 as a 3 : I cover of P1, (xi, yi) for which f (xi) 0 (and at infinity if n is not divisible =
ramification divisor
as
PF,
hence
genus(f)F,2)
=
genus(rF)
>
0,
a
contradiction.
Bxample mann
3.9
In the one-dimensional
1) the level manifolds are punctured Rieunique compactification. If the genus of such a Riemann surface I good compactifications supports no holomorphic vector fields, so for d
surfaces and have
exceeds one, then it Of -TFd rarely exist.
case
(d
=
a
=
94
3. The
geometry of the level manifolds
application of (3.8), let us show how the vector fields can be integrated on these real level sets. Summing up (3.8) over all j (and for any W) we find that for any fixed integers i,r < d, As
an
d
E j=1
d
XW HjXj
X 3
d
UkX;
ap
W(xj,yj)-E'-'(Xj,yj)
-i+k-I
xj
j=1 k=d-j+l
-
XI
Therefore the d functions d-I =
Xr
X,r XH O
Xj
E W(xj'YjA-'(xj'Yj)' i
r
=
0,..., d
-
(3.9)
1,
Oy
have linear
dynamics
in time and lead to the
of
(connected components 3.5. The
significance
the)
the vector fields
integration of
real manifolds
XH, along
the
7:F d*
of the Poisson structures
d
We have constructed for any positive integer d and for any F (x, y) E C [x, y] \ C [x] a which is indexed by family of compatible integrable Hamiltonian systems (C2d, 1.,.}W d i AFd)
p(x, y)
E
C[x, y] \ 10}.
For
a
fixed
F(x, y)
all vector fields in
C2d
Ham(f-, -I'd AFd) ,
vector fields of the different
by
the
(compatible) integrable
-+
Spec AFj.
Since for
are
tangent
fixed W these vector fields generate at a general point the tangent space, the vector fields obtained for one choice of W can be written down in terms of another choice of W. This relation between the to the d-dimensional fibers of the momentum map
Hamiltonian systems is
a
given explicitly
following proposition.
0,..., d 1) denote the polynomials Proposition 3.10 For fixed F(x, y) and d let Hi (i on C2d, defined in (2.10), and let Xj' and Xf denote their Hamiltonian vector fields with respect to 1', Jd and I-, j1d. Then the transfer matrix TI'P, which is defined by =
(X p
Hd-11
is
X. F'J") (X1 =
...
I
Hd-11-"
-
X,HO
) 7-1 0'
given by
(-Ud-1 w(M, v(M)),
general transfer cocycle identities The
where
M
matrices 7- 2 -4 7W 2 7W21
are
-UO
-Ud-2
-Ud-3
1
0
0
...
0
0
1
0
...
0
0
...
0
1
0
immediately computed from (3.10) and
95
-"P X' 7WV2 7W2371
(3.10)
upon
using the
Chapter
Ill.
Integrable Hamiltonian systems
and
symmetric products
of
curves
Proof It suffices to express
711
in the
neighborhood
of
generic point (u(A), v(,\))
a
any local coordinates. At such a generic point all roots xi of coordinates; we denote v(xi) = yj as before. Then by (3.8),
XH o XH',o
=
A P
with
WOXH
denotes the matrix with entries
=
(XH")ij
=
u(A)
axe
in terms of
different and
serve as
diag(V(xi, yj),
W(Xd, Yd));
XH;,xi
X1. The above formula H
and XH
implies that V, is given by
TjW
=
W =
=
(XH)-l,&WXH VAWV-1
VW(AX, V(A'))V-l W
(VA'V-', v(VA'V-'))
WA V(M)), where M
requires
==
VA'V-1 is easily checked
some
extra work
(one
to have the form announced in
(3.8));
uses
also
we
(3.10).
Step (i)
have introduced the notation V for the
Vandermonde matrix d-1
(Xd-I Id
d-i
X2
...
Xd
...
Xd
d-2
2
X1
d-2
X2
V=
Remark 3.11
In the
special
case
where d2 and
W(x, y)
x
the transfer matrix has the
simple form -
U1
-UO
1
and
we
obtain what Caboz et al. call
a
(p, s)
0
bi-Hamiltonian structure
compatible integrable Hamiltonian systems clarify this concept.
inition of and
96
(see [CGR]).
and Proposition 3.10
Our def-
largely generalize
IV
Chapter
Interludium: the geometry of Abelian varieties
1. Introduction
For the convenience of the reader who wishes to go on reading the rest of the text we There is nothing new in this chapter, our a chapter about Abelian varieties.
include here
give a compact and coherent presentation of the theory of Abelian varieties in applications to integrable Hamiltonian systems. Our exposition is paxtly algebraic partly analytic, we think that both approaches highlight different aspects of the theory of Abelian varieties, see for example the theorems of Abel, Jacobi and Pdemann in Paragraph 4.3. Moreover, for applications to the theory of integrable systems a reasonable amount of understanding of both aspects and their interplay is needed. The main references for the theory of Abelian varieties are [LB], [Kem], [Mum2] and [Mum3]. However the relevant chapters in [ACGH], [GH] and [Mum4] are also highly recommended to learn this subject. We start by recalling the basic definitions of divisors and line bundles (on a complex manifold) and recall how they are related. The sections of an ample line bundle are used to construct embeddings in projective space and the dimension of this space is computed from
intention a
was
to
form suitable for
the Riemann-Roch Theorem. As
an
illustration we show that every compact Riemann surface
projective space, hence it is an algebraic curve. It is therefore common not to distinguish between compact Riemann surfaces and algebraic curves, the latter of course being assumed complete, non-singular, irreducible, reduced; we will also use these terms interchangeably, choosing the term compact Riemann surface or algebraic curve according to whether the analytic or the algebraic structure is more relevant. can
be embedded in
some
In Section 3 we give the Pdemann conditions which tell which complex tori are Abelian, i.e., can be embedded in projective space. The sections of an embedding line bundle are explicitly described by theta functions and their number is easily computed. A lot is simplified
97
P. Vanhaecke: LNM 1638, pp. 97 - 125, 1996, 2001 © Springer-Verlag Berlin Heidelberg 1996, 2001
Chapter
IV. The geometry of Abelian varieties
here because every effective divisor on an (irreducible) Abelian variety defines an ample line bundle and the third power of an ample line bundle on an Abelian variety always provides an
embedding. In particular in the case of Abelian surfaces everything can be computed very explicitly (except the embedding itself but that is where we win use the theory of integrable systems for!), an ample divisor is then nothing but an embedded curve and its genus relates to the number of sections of the line bundle defined by this divisor. We also treat in detail the Jacobian of
exaxnple of there is
an
Abelian
variety. There
axe
algebraic curve since it is the most important basically two (very different) ways to define it,
an
algebraic definition (using divisors on the curve) and an analytic/transcendental (using integration of differential forms over cycles). The fact that these correspond to the same object is a deep theorem, due to Abel and Jacobi. We could not resist to reproduce a proof of it here. We close Section 3 by discussing the Kummer surface of a Jacobi surface and its 16r, configuration, which will be used in Section VIIA. an
definition
We
use
aside from
the
0,
following
also the constant sheaves set U
terminology.
functions',
Z, Q, R and
and
On
complex manifold
a
M,
C and the sheaves
M
we
its sheaf of meromorphic
0*, M*, QP
will use,
functions,
defined for each open
by 0* (U)
=
the
multiplicative
group of
non-zero
(U)
=
the
multiplicative
group of
non-zero
9F(U)
=
the vector spare of
M
*
We add M
talking
9
notation and
its sheaf of holomorphic
as a
about
In this
notation
a
subscript sheaf
chapter
O(M)
on
functions
meromorphic
holomorphic p-forms
on
functions
on on
U,
U,
U.
in the notation when it is not clear from the context that
we are
M.
0 will
for the
holomorphic
only be used in this sense, so no confusion can arise with the ring of regulax functions on an affine variety, which is used in other
chapters. 98
2. Divisors and line bundles
2. Divisors and line bundles
we discuss divisors, line bundles and the way in which they are related. We explain how the holomorphic sections of a (very ample) line bundle lead to embeddings projective space. We fix a compact complex manifold M of dimension n, the dimensions 1 and n 2 being the most important for us.
In this section
also in n
=
=
2.1. Divisors
A subset V of M which is
locally given by the zero locus of a holomorphic: function analytic hypersurface. V is called reducible if it is the union of at least two hypersurfaces; otherwise it is called irreducible. A divisor on M is a finite (formal) sum is called
an
D
=
1: aiVi
(ai
(E
Z)
analytic hypersurfaces of M. Two divisors are added in the obvious way and resulting group, which is called the group of divisors on M, is denoted by Div(M). If all 1), the integer E ai is called ai are positive the divisor is called effective. For curves (n the degree of D and is noted by deg D. of irreducible the
==
If V is an irreducible hypersurface then to any meromorphic function f there can be assigned an integer, "the order of vanishing of f along V" as follows. By definition, V has a local defining function for some neighborhood around any point P of V; if g is such a local defining function then this integer is the largest integer n for which gn divides f in the ring of germs of holomorphic functions at P. This is well-defined because it turns out that this integer is independent of the point P and the local defining function g. We denote this integer -a < 0). a > 0 (ordV f by ordV f and say that f has a zero (pole) of order a if ordV f Then a divisor (f) can be assigned to every meromorphic function f by =
=
(f)
=
1: ordv f
-
V,
V
the
(finite)
sum
running over all irreducible analytic hypersurfaces V for which ordV f :74- 0; 0 for any meromorphic function f on a compact Riem deg(f
it
is not hard to show that
surface.
f,,,dzi A holomorpbic top-form w E Qn (M), ordv w is defined as follows. If w coordinate neighborhood U,,, then we define ordv w ordv f,, for any irreducible analytic hypersurfarce V intersecting U,, (again this is well-defined because the For
A
dzn
=
a
=
on some
transition functions lie in
0*).
Then the divisor of
E ordv
is the effective divisor
w
w
-
V,
V
the
(finite) sum running over all irreducible analytic hypersurfaces V for which ordv W 7 0. Div(M) admits a sheaf-theoretic interpretation, namely there is a natural isomorphisin Div(M)
---
HO (M, M* /0*).
99
Chapter IV. The geometry of Abelian varieties To
see
this let
f
=
ffj
be
a
global
section of
f..
E
b
ow.,
and the divisor associated to this section is
D
n
f,,
=
ordv fa because
up)
naturally taken
1: ordv f,,
=
then ordv
M*10*,
-
to be
V,
sum is taken over all irreducible analytic hypersurfaces V for which v n u,, = A o. Conversely, given a divisor, a global section of M*/O* is defined over small opens by taking the product of the local d,efining functions of all irreducible analytic hypersurfaces appearing in the divisor (with the coefficients as exponents). Also Div(M) and HO(M,M*10*) are easily seen to have the same group structure.
where the
2.2. Line bundles
By
a
one over
U,
n
Up
line bundle f-
on
M
we
will
9.,&-) (L.,
always
holomorphic
mean a
vector bundle of rank
M. For any two overlapping open sets U,, and Ujq the transition Anctions gq I -+ C* of L axe defined in terms of the trivializations 0,, : -7r- (U,,) -+ U,, x C by
is the fiber
over z
E
u,,
(0.
=
E
-
C*'
n U,a) and take values in
C*; one has that satisfy the so-called cocycle identities
and these transition functions
g,, 'q E 0 *(u,, n q 3
909fla
ga'3gj3'Yg-Ya hence the transition functions represent
-1
(61g.'a 1) (6 is the coboundary operator).
It is
Oech 1-cochain which is actually a 1-cocycle because
a
g6 9 PE 9-ya a
=
g6eg-!qfg-Y6
=
1,
standard result that to any set of functions
1gq I
satis-
cocycle identities there corresponds a line bundle with these functions as transition functions. Thus, the tensor product of two line bundles is again a line bundle and the set of all line bundles on M up to isomorphism becomes a commutative group, the Picard group of M, Pic(M). The inverse of a line bundleC is nothing but its dual and will be denoted byC*. fying
the
If other transition functions
there
are
functions
f,,
E
0*(U,,),
jh,61
given for L, coming from triviahzations such that by X,,
are
defined
hag i.e., have the
is same
a
Nch coboundary.
The
X,,, then
f. =
fo
upshot
group structure.
100
ag,
is that
Pic(M)
=
HI(M, 0*)
and both sets
2. Divisors and line bundles
To every divisor there corresponds a line bundle in the following way. Let V E -Div(M) locally defined by functions f E M* (U,,), then the corresponding line bundle [V] is defined by the transition functions
be
gap
and the definition is
clearly independent
461
of the choice of the local
defining
functions. The
map
Div(M) is
a
homomorphism of
groups, which is
Ker[-] because,
on
the
one
IV
=
hand, the
Pic(M)
surjective if M is algebraic. Its kernel
Div(M) 13f
E
-+
E
M(M)
transition functions for
90
=
for which
[(f)]
are
(f)
=
is
given by
D},
given by
f 1U.
Alb
=
f lu",
hand, if the line bundle [D] is trivial, then the local defining functions fc, give fc, on U, which is well-defined D, by defining f I u. f for which (f ) when the local defining functions axe chosen in such a way that the transition functions 1. If two divisors V and D' are defined to be linearly equivalent when V -1 V, if g,,,6 [D] [D], then we see that if M is (in addition) an algebraic variety, then and
on
rise to
the other
function
a
=
=
Div(M)
-
Pic(M)
-1
in
[-] corresponds under the identifications Pic(M)
natural way. The map
a
Div(M)
-5--
H'(M,.A4*10*) -
-
-
-+
connecting homomorphism. P
to the
HO(M, M*)
-+
HO(M, M*/O*) f+ H'(M, 0*)
exact sequence 0 -+ 0* -+ M* -+
coming from the short
in the
M*10*
-+
!-2-'
long
-+
-
-
HI (M, 0*) and exact sequence
-
0.
2.3. Sections of line bundles
0,,
Let L be
a
-7r-'(U,,)
-+
:
(holomorphic)
line bundle
U,,
map
s :
a
U,3 76 0,
M with
projection
0 s
I U. : U"
-+
7r-, (U")
sections
C
-+
M,
vector bundle charts a
-+
U"
X
C
map s,, : U,, -+ C, for each a. Two of these, say s,, and s,8, with U, n related by s,, g,,9 s,3. Conversely, a set of holomorphic maps Is,, I satisfying s,, each non-emptyu, nU,9 determines a global holomorphic section of the line bundle =
=
of holomorphic sections of a line bundle L will be denoted by O(L) and the sheaf by QP(L), the sheaf of holomorphic differentials with values in L. define the sheaf of meromorphic sections of L as O(L) M; then the meromorphic are given by meromorphic functions IsJ satisfying s,, g,6so on each non-empty
(9 QP will be denoted
We also
:
(holomorphic)
axe
g,,,,6s,6 on L. The sheaf
O(L)
7r
=
0" defines
over
C and transition functions g,q. A (holomorphic) section of L is M -+ L for which -7r o s 1m. The composition
x
101
Chapter IV. The geometry of Abelian varieties
u,, n U,6. Given
satisfy s,,It,, and t) by f,,
two
meromorphic sections s, t of L, the local defining functions Is,, I and It,, I s,61t,9. Hence we can define a meromorphic function f (the "quotient" of s s,,It,,. For example, for D E Div(M), defined locally by f,, E M(U,,), we have
=
=
taken
[D] to be the line bundle with transition functions gq Therefore every divisor D determines a line bundle [D] and
fJil,
=
so
any set of local
of D define
meromorphic
a
section sf of
[D].
It is customary to write
that f,, g,,,6f,6. defining functions
O(D)
=
for the sheaf
0([D])Just as we did for meromorphic functions, we can associate a divisor to every meromorphic section of a line bundle by first defining, for any irreducible hypersurface V c ordV s,, for any a for which u,, n v =A 0, and then setting M, ordV s =
(s)
=
E ordv
s
-
V
V
where the
sum runs over all irreducible hypersurfaces V C M. As before, ordV is well-defined because the transition functions gq s,,Isg are in 0*(U,, n U,3). Taking up the previous example again, the divisor of sf can be read off from the local defining functions f,,, giving =
(sf)
D. More
generally,
if
is any
meromorphic section of [D], then (s) -1 D because meromorphic function whose divisor is exactly E) (s). It follows that [(s)] [D] and that there is for any divisor V -1 D, a section s of [D] for V. Notice that line bundles which come from divisors are exactly those which which (s) have (non-zero) meromorphic sections. Since a section s is holomorphic if and only if (s) is effective, it follows that a line bundle is the line bundle of an effective divisor if and only if it has a non-trivial global holomorphic section. the
=
of sf and
"quotient"
s
s
defines
a
-
=
=
Actually, line bundle
instead of
[D],
one can
working
with the space HO (M, 0 (D)) of holomorphic sections of a L(D) of meromorphic functions on M defined
work with the space
by
L(D) To
see
this,
section
f,
=
s
E
s1so,
that
canonical
E
M(M) I (f)
+D >
0}-
fix wiy meromorphic section so of [V] for which (so) D. Then every holomorphic Ho (M, 0 (D)) corresponds to a meromorphic function f, by taking the "quotient", =
and
(h) so
If
=
Ei
L(V)
and the
+D
proof
=
(8)
of the
(80)
-
+ D
converse
=
(s)
>
0,
is similar.
We have established
a non-
isomorphism
L(D) Denoting by ID1 correspondence
++
HI (M, O(D)).
the set of all effective divisors
ID1
5---
P (L (D))
linearly equivalent
(2.1) to
D,
we
have the additional
P (Ho (M, 0 (D)))
for compact manifolds M, because V E ID1 implies the existence of a function f E D + (f) and f is uniquely determined up to a constant. Therefore, such that V =
a
L(D) IVI is
projective space; any subspaces of it axe classically called linear systems and ID1 itself is a complete linear system. The common intersection of the divisors in a linear system
called
is called the base locus of the system.
102
2. Divisors and line bundles
2.4. The Riemann-Roch Theorem The calculation of the dimension of HO (M,
O(V)) is done by using the Riemann-Roch give this theorem here only for (algebraic) curves, postponing the analogous theorem for surfaces to a later paragraph. To give three equivalent versions of this theorem, The holomorphic Euler characteristic X(L) of a line bundle L we need some terminology. over a compact complex manifold M is the integer Theorem. We will
X(L)
E(-1)-"dimH-(M, O(L)),
=
P>O
topological invariant of L and M.
paxticular the holomorphic Euler characteristic by X(Om) and is a topological invariant of M. Also, every compact complex manifold M has another distinguished line bundle, its canonical line bundle KM, defined as the line bundle corresponding to the divisor of any holomorphic top-form on M; this line bundle is well-defined because the "quotient" of two holomorphic top-forms defines a meromorphic function on M which establishes the linear equivalence of their divisors. The divisor of any holomorphic top-form is called a canonical divisor and will be denoted in the same way as its corresponding line bundle (although this divisor is only uniquely determined up to linear equivalence). Notice that the holomorphic sections of KM correspond to the holomorphic top-forms on M, 0 (Km) c ' QM (isomorphism of sheaves). which is
a
of the trivial line bundle
over
In
M is denoted
n
-
Then the Riemann-Roch Theorem
for
curves can
be stated in the
following equivalent
forms. LetI7 be
Theorem2.1 a
canonical divisor
on
compact Riemann surface ofgenus
a
1P. Then the
g, V be
following equivalent formulas
(i) XQD1) X(Or) +deg V, (ii) dim HO (r, o (D)) dim HO (r, n, (-v)) + deg D g (iii) dim HO (r, O(D)) dim HO (r, O(K. D)) + deg D
a
divisor
on
rand Kr
hold:
=
=
-
=
-
+ -
g +
Proof Instead of proving the Riemann-Roch three formulas and to
use
Theorem,
we
this theorem to compute the
prefer to show the equivalence of the degree of the canonical bundle on a
curve.
First, the equivalence of (ii) and (iii) is obvious from the isomorphism O(Kr) equivalence between (i) and (ii) we need another fundamental theorem, the Kodaira-Serre duality Theorem, which we will formulate in a more general form, since we will also use it for studying sections of line bundles on surfaces.
To show the
Let M be
Theorem 2.2
Hq(M, W(L)) For the
curve
and
r,
n
a
equals
that
Hq(r, o(,c))
=
are
1 in this theorem.
H q(r, so
compact complex manifold of dimension
Hn-q(M, gn-p(,C*))*
o(,c))
0 for q > 1.
=
n, then the vector spaces
isomorphic.
Taking p
Hq(r, flo(L))
-5--
=
0 and
a
line bundle L
Hl-q(]p, 01 (L*))*,
=
r,
we
find
(2.2)
Therefore, the holomorphic Euler chaxacteristic of C, X(,C),
equals
X(,C)
on
dim HO (r,
o(r_)) 103
-
dim H1 (r, o (,q).
Chapter IV. The geometry of Abelian varieties On the other
hand, for
q
=
X(L)
(2.2) gives H1 (r, o(,c)) - --'HO (r, n, (,c*))*
1
=
dimHo(]P, O(L))
-
The
it,
=
dim Ho (r, o (D))
X(00
=
dim Ho (r,
equivalence of (i) and (ii)
we
Or)
17, this gives respectively
over
dim H'(r, Q1 (-v)),
-
dim Ho (r,
is established if we
Q1)
=
I
-
dim Ho (1P,
Q').
show that dim Ho (r,
can
Q')
=
g. To show
consider the short exact sequence of sheaves 0
which
-
that
dimHO(r, n, (,c*)).
Taking for C the line bundle [D] and the trivial line bundle
X(['D])
so
gives
a
long
exact sequence in
0
-+
because H2 (r,
0)
-+
=
C
-+
0
(r, c)
-4
Q,
0,
-+
cohomology, namely
HO (r, C)
Ho (r, o)
Ho (r, n')
H' (r, q
H1 (r, o)
H' (r, P3.)
H2 (]p, C)
_+
0 from KodairaSerre
H
0
5---
H
0
(r, 0)
0,
duality.
Also I
we
H
1
c--'
Ho (17, Q').
5---
(r, n )
5---
know that
H2(r, q
c,
c
and
H' (1P, 0)
Therefore, counting dimensions
in the above exact sequence,
dim Ho (r,
proving We D
=
dim H' (r,
r2l)
q
find
we
=
g,
the claim.
proceed
to calculate
deg Kr, the degree of the canonical bundle on
Kr in the Riemann-Roch Theorem, dim Ho (r,
O(Kr))
dim Ho (17, 2
g +
-
0)
+
deg Kr
g + I
-
degKr.
Using Kodaira-Serre duality again,
deg Kr
=
=
=
=
which shows that the
degree
dim HO(r,
O(Kr))
+g
-
dimHO(r, Q'(Kr Kr)) dimHO(r, f2l) + g 2 -
2
+g
-
2
-
2g
-
2,
of the canonical bundle
104
equals 2(g
-
1).
a curve
r.
Taking
2. Divisors and line bundles
2.5. Line bundles and
embeddings
in
projective
space
One useful
application of linear systems is to construct embeddings of compact complex projective space, thereby realizing them as algebraic varieties. Let M be a compact complex manifold and IVI a complete linear system (one proceeds in the same way for general linear systems). If the base locus of JDJ is empty then, for every p, E M, the set of sections vanishing at p forms a hyperplane Hp in P(HO(M, O(D))), so we get a map
manifolds in
t
[,D1
:
M
-+
p F-+
P (Ho (M, 0 (D))) *
H,.
If z[-D] is an embedding then [D] is called a very ample line bundle. If f- is a line bundle and Ck is very ample for some k > 0 then f- is called ample. A necesSaXy condition for %['D] to be an embedding is expressed in the following theorem (Kodaira's Theorem). Let M be
Theorem 2.3 there exists
an
a
compact complex manifold andC
integer ko such
that
for
k >
ko the M
_+
a
positive line bundle. Then
map
pN
a well-defined embedding of M in pN P(H0(M,0(&)))*. In the language of divisors, if [D] is positive, then for some k E N, the functions with a pole of order at most k along D will provide an embedding of M into projective space. I
is
It is plain from the theorem that the positivity of L is crucial. A line bundle L is called is a positive positive if there exists a metric on L with curvature form 0 such that (1, I)-form (see [GH] Ch. 1.2). Actually, the condition that the line bundle is positive turns out to be
a
topological
condition and Kodaira's Theorem
can
be reformulated
as
follows.
A compact complex manifold M is an algebraic variety if and only if it has closed, positive (1, 1) -form w whose cohomology class [W] is rational, i.e., [W] E H2(M, Q). Such a (1, 1) -form is called a Hodge form.
Theorem 2.4 a
application of Kodaira's embedding Theorem, we show that every compact Riean algebraic curve, thereby justifying our terminology. Let g be any Hermitian metric on the compact Riemann surface IP and w the associated (1, I)-forrn Q g. Multiplying w by a constant if necessary, we may suppose that fr W 1, i.e., [W] is an integral cohomology class. It follows that r can be embedded in projective space by using the meromorphic functions on IP with poles at one point only. By Chow's Theorem, the embedded curve P is given by the zero locus of a set of homogeneous polynomials. Since the variety of chords of V has dimension three, we can project P to a hyperplane in this projective space and we can repeat this process, until we finally obtain an embedding of r in p3. In general r cannot be embedded in p2 ; however, since the set of tangents to the embedded curve has only dimension two, we can project the curve birationally to a curve f whose only singularities, are isolated (ordinary) double points, which means that the map IP -+ f is I : 1, except in a finite number of points. Conversely, it can be shown that every irreducible algebraic curve P has a normalization, i.e., there exists a compact Riemann surface r (which is unique up to As
mann
an
surface is
=
biholomorphism)
and
a
map
v :
IP
which is 1: 1 except at isolated points. 105
IV. The geometry of Abelian varieties
Chapter
Hyperelliptic
2.6.
curves
Most compact Riemann surface of genus g
! 2
be embedded in projective space in
can
canonical way, namely by using the canonical bundle of the Riemann surface. Suppose r has genus g ! 2 and let Jwi, .,wgl be a basis of W(r). In local coordinates, we can write
a
-
wi
=
-
and the canonical
fi(z)dz,
mapping
ZK is
given by
Pg_1
IP
(f, (P)
P
(P)).
The
complete linear system has no base points by the Riemann-Roch Theorem, hence SK is an embedding when it is injective and immersive. The compact Rlemann surfaces for which the above map is not an embedding axe called hyperelliptic (compact Riemann surfaces of
genus I being called elliptic). of the following theorem.
Theorem 2.5
Let r be
a
Hyperelliptic curves will be the
most
compact Riemann surface of genus g
interesting
for
us
because
> 2.
hyperelliptic if and only if it has a non-constant meromorphic function f for (f) + P + Q : 0, for some points P, Q E IP; (2) r is hyperelliptic if and only if IP is the normalization of an algebraic curve, given by an affine equation of the form y2 f(X), where f is a monic polynomial of degree 2g + 1 or 2g + 2 without multiple roots; (3) If g 2 then r is hyperelliptic.
(1)
r is
which
=
=
Proof The map W is I : 1 and immersive if and only if for any two points P, Q G 1P, there is holomorphic differential w for which w(P) :A w(Q) and there is an w' vanishing once at P exactly. These conditions are equivalent to dim HO (r, 0 (K P Q)) < dim HO (r, 0 (K P)) for any P, Q E 1P. This reduces to dim HO (r, o (P + Q)) < 2 upon using dim HO (r, 0 (K a
-
-
-
-
P))
=
g
-
1 and
dimHO(r, O(K
-
P
-
Q))
=
g
obviously belong to L(P hyperelliptic if and only if it admits proves (i).
-
Since the constants
+
surface is
a
This
As for
by
an
map
by
Q)
dimHO(r, O(P + Q)).
we
conclude that
a
compact Riemann
meromorphic function with only
two
poles.
Consider the map from IF to P' constructed in (:L). It is a 2 : 1 map and count (the Riemann-Hurwitz formula) the number of branch points of this
elementary
(i.e.,
xi,
(2),
3 +
...
the number of points where the map is I let I X2,+2
r,
=
J(x,y)
E
C2 1
Y2
=
:
1) equals 2g
]J(X
_
-
2.
Denoting these points
X,)J,
all i for which xi :A oo. The projection map x : r, -+ C is seen holomorphic map r -+ P1, when TP is obtained from r, by adding one or two points "at infinity". Finally r is shown to be a compact Riemann surface isomorphic to 1P.
where the
sum runs over
to extend to
a
To show
Since
deg K
(3),
=
2g
take IP of genus 2 and substitute V 2 2, we find
-
dimHO(r, O(K)) We conclude the
=
K in the Pdemann-Roch Theorem.
=
=
1 + 2
-
2 +1
proof by applying part (i) of the theorem
106
=
2.
to K
=
P +
Q.
2. Divisors and line bundles
general terms, a point P on a compact Riemann surface of genus g is called a point if there exists a function which has a pole of order at most 9 at P and which is holomorphic elsewhere. It follows that a hyperelliptic Riemann surface of genus g has 2g + 2 Weierstrass points which are the points for which there is a function with a double pole in one point only. Notice that these 2g + 2 points are intrinsically defined. In
more
Weierstrass
FinaJly, it is easy to check that on a compact hyperelliptic f (X), corresponding algebraic curve is given by an equation y2
Riemann surface 1P, whose the g differentials
xi-ldx Wi
=
Since they are independent, they form a basis of the vector space of all are holomorphic. holomorphic differentials on the Riemann surface. A hyperelliptic Riemann surface has a (holomorphic) involution (which is unique), the hyperelliptic involution, which is given by (x, y) -+ (x, -y) when an equation for the corresponding algebraic curve is written as y2 f (x). Hyperelliptic Riemann surfaces and their Jacobians will be dealt with in detail in Chapter VI. They will also appear frequently in Chapter VIL =
107
Chapter
IV. The geometry of Abelian varieties
3. Abelian varieties
3.1.
Complex
tori and Abelian varieties
If A is any lattice
(i.e.,
a
discrete
subgroup of maximal rank)
in
a
complex
vector spare V
of dimension g, then the quotient
T9
=
V/A
compact complex manifold, called
It is a commutative complex Lie a complex torus. algebraic variety; a complex torus which is at the swne i.e., which can be embedded in projective space, is called an Abelian variety. The conditions on A for the derived torus T9 to be an algebraic variety, which are computed from Theorem 2.4, are expressed by the famous Riemann conditions: is
a
group, but in general it is not an time a projective algebraic variety,
T-9
Theorem 3.1
I-X17 X2g} to fel'..., eg} ...
=
7
V/A
is
an
Abelian variety if and only if there exists an integral basis eg I of V such that the matrix of A with respect
complex basis Jel, given by
of A and is
a
A
=
(A,5 Z),
A6 diag(61,..., 6,) a diagonal matrix whose diagonal elements are positive integers satisfying dil&+, and Z a symmetric matrix whose imaginary part, Q (Z), is positive definite. In terms of coordinates xi, X2g dual to this basis of A, the Hodge form w is given by with
=
...
I
9 w
6idxi
A
dxi+,.
The Hodge form w and its cohomology class [w] carry extra information about particular embeddings of V/A in projective space. Namely, up to an integral multiple, w is obtained from an embedding z: V/A -+ pN as
t* 91%(V/A) i where Q is the associated
of pN. Different
embeddings may lead
(1, I)-form associated to the standard KRhler structure [w] which are different, even up to a multiple;
to classes
cohomology class [w] of w on an Abelian variety is called a polarization and the pair is called a polarized Abelian variety. When the embedding is done by using the sections of a (very ample) line bundle L, as explained in Paragraph 2.5, then the polarization is precisely the Chern class of C and any polarization is the Chern class of an embedding line the
(V/A, [w])
bundle
(Theorem
3.3
below)
-
importance of considering polarized Abelian varieties, rather than Abelian varieties, stems also from the fact that their moduli spaces are simpler. They break up in components, E 6i dxi A dxi+g are invariants given by the polarization type as follows. The integers 6i in w of the cohomology class of w and are called the elementary divisors of the polarization and 6g); if the elementary divisors (V/A, [w]) is said'O to have polarization type or type of w are all one then w is said to define a principal polarization and (V/A, [w]) (or just V/A) is called a principally polarized Abelian variety. Thus the moduli space of all polarized Abelian The
=
10
of
We will often
use
the
common
"polarized Abelian variety
abbreviation "Abelian
of type
6g)" 108
variety
of type
instead
3. Abelian varieties
varieties breaks up in a natural way in components indexed thus studied separately for each polarization type.
A
holomorphic
translation. If such an
by the polarization type and
are
map between Abelian varieties is a
group
homomorphism
isogeny. Surjectivity is almost reducibility Theorem.
a group homomorphism followed by a surjective with finite kernel then it is called view of the following theorem, known as the
is
automatic in
Poincar6
variety which contains a non-trivial Abelian subvariety 8ubvariety B of T9 such that A n B is a finite subgroup and such that there exists a suijective homomorphism A ED B -+ T9 whose kernel equals A n B: up to an isogeny such an Abelian variety is a product of Abelian varieties.
If Tq
Theorem 3.2
A then there exists
an
is
an
Abelian
Abelian
Abelian varieties which contain
a
non-triviaJ subtorus
are
called reducible Abelian
va-
rieties and it follows e.g. from an easy dimension count that a general Abelian irreducible. Interesting isogenies, used below, are obtained by the following: any
variety is polarized Abelian variety is isogenous to a principally polarized Abelian variety (but not in a unique way). Another interesting isogeny is an isogeny between a polarized Abelian variety and its dual which will be described in the next paragraph. 3.2. Line bundles
on
Abelian varieties
The Riemann conditions
give
necessary and sufficient conditions for
us
a
complex
torus
Abelian variety and Kodaira's Theorem says that the embedding can be done using the sections of a positive line bundle. The positive line bundles on an Abelian variety can to be
an
be described very explicitly and a basis of the space of holomorphic sections can be written down. To show this, let T9 V/A be a complex torus andC a line bundle on T-9. Then 7r*,C =
(ir
is trivial
:
V
-4
T9)
because V is contractible.
0: lr*L Then
(ir*L),
(-7r*L),+,x
=
and since
C
giving
a
linear
functions
1eA
0*(V)I.\EA e,\,
(z
are
+
X
(ir*L)-.
maps
(7r*L)-,
=
Line bundles
exponential
can
global trivialization
C.
to C
(ir*L),+,\
we
-+
get
C,
called the
A) e,\ (z)
=
e,\ (z +
A') e,\, (z)
=
Conversely, multipliers which satisfy these relations define multipliers. the
a
C, i.e., multiplication by a non-zero number e,&). The multipliers of L and satisfy
automorphism C
E
0.,
V
-+
Hence, there exists
be constructed
e,\+,\,
a
(--)
-
unique line bundle with these
using multipliers je,\(z)j
of
a
simple character.
sequence e
0 we
-4
Z
-+
0
!!T 0*
-+
0
get
H1 (7'9, 0)
-+
H1 ('T9, 0*) 54 H2 (T9' Z)
109
-+
H2(Tg, 0)
From
Chapter
IV. The geometry of Abelian varieties
where cl (L) is called the Chem class of the line bundle L. If type (1, 1) then
6,,dx,,,
A
is
W
a
positive integral form of
dx,,+,,,,
with respect to the basis JxI.... I X2gJ dual to some basis JAI.... .X2.1 of A. Setting e,\ ,\,, 16,, and letting zi, we get the following z, be linear coordinates dual to eI, , e, , theorem. .
The line bundle C
Theorem 3.3
(for
a
1
=
Up
to
....
a
g)
has Chem class
-+
.
.
7-9 with multipliers e,\.
=
1,
e,\,,,.
=
e
cl(,C)
==
-21riz ,
[w].
translation in T9 every line bundle is uniquely determined by its Chern class = so it is a positive integral form of type (1, 1). 'j
and this Chern class is c, (L)
[ ' 7-r E)]
The fact that the line bundle is
given by simple multipliers allows us to construct explicitly its holomorphic sections; they can be seen as functions on C9 which are periodic in g directions and "quasi-periodic" in g other directions. The number of independent holomorphic sections is given by dim HO (T9, 0 (,C)) 6,,, (3.1) where
6,) axe the elementary divisors of the polarization cl(f-). For a line bundle principal polarization, for example, there is only one section which, as a quasifunction on C9, is given by Riemann's theta function
(61
defining periodic
.....
a
ew'(',Zl)e21ri(l,z)
(A
=
(I Z)).
(3.2)
IEZn
Its divisor of zeros, E), is determined is called the Riemann theta divisor. The group of all line bundles of
uniquely by L, hence degree
0
on a
up to
a
translation
by [c, (,C)] and
polarized Abelian variety 7-9 is a complex a period matrix (A8, Z) then defining the lattice defining'fg is given by
torus, called its dual and denoted by t9. If T9 corresponds to a
"dual" basis
can
be
picked such that
the matrix
K A6 1, bn A6 IZA6 1)
-
(3.3)
representation it is easy to check the Riemann conditions, which show that the dual an Abelian variety. For L a fixed positive line bundle on 7-9 one defines an isogeny between T9 and its dual by v -+,C-I (& Tv*,C. The degree of this isogeny is 11 6il.
In this
is indeed
If T9 is irreducible then the line bundle of any effective divisor is ample; moreover the ample line bundle on an Abelian variety is very ample, hence gives an
third power of any
embedding in projective space (these two properties axe particular for Abelian varieties, for general algebraic varieties both are false; the last property is due to Lefschetz). For example, if L defines a principal polarization on ail irreducible Abelian variety T9, then C3 induces a polarization of type (3,3,...,3), and hence every irreducible principally polarized Abelian variety can be embedded in PHO(T-9, 0(,C3)) which is by (3.1) isomorphic to p311-I. 110
3. Abelian varieties
3-3. Abelian surfaces Since which
we are
techniques As
mainly
interested in two dimensional a.c.i. systems, the Abelian varieties often Abelian surfaces. In what follows we give some useful these surfaces.
will encounter
we
to
study
have
are
in the
of curves, varieties
are often studied by examining the (merospecifically by examining the divisors of these functions; we will call an effective divisor C on an algebraic surface S an (embedded) curve on S. The curve C is said to be smooth if it is a submanifold of S (taken with multiplicity 1) and we
morphic)
seen
functions
on
case
them,
more
irreducible if it is not the union of two effective divisors.
A fundamental result here is that the canonical bundle of canonical bundle of the surface itself
are
intimately related,
a curve on a
as
surface and the
is expressed in the
following
adjunction formula. Theorem 3.4
If S
bundles KS and KC
is an algebraic surface and of S and C are related by
Since
we
have shown that
deg(Kc)
a
smooth
0
2g
2 if C has genus g,
-
S, then the canonical
curve on
(KS
KC
of C
C
[C]) I C.
we can
calculate the genus
by I =
g
Now
on
S there is
a
natural
2
deg (Ks
[C]) I c
0
+ 1.
non-degenerate intersection pairing -
:
H2(Sj Z)
x
H2(S, Z)
-+
Z7
which counts the
(signed) number of intersection points of arbitrary transversely meeting 2-cycles representing the homology classes. The pairing also gives a natural definition of the intersection of divisors by taking the intersection of their fundamental classes in H2(S, Z). Under the natural isomorphism H2 (S, R) -+ H 2(S, R) -- HD2R which derives from it, each divisor D corresponds to a two-form qD E HD2R(S), its Poincar6 dual, and the intersection of cycles can be shown to be Poincar6 dual to the wedge product of forms, i.e., if Q denotes the top-form corresponding to the natural orientation, then " A ?7v, (V D')Q. This suggests to define the intersection C -,C' for two line bundles L and V as the cup product of their first Chem classes, thought of as an element of Z, =
(L)
C1
U C2
(,Cl)
-
(,C V) Q
=
Using the fact that the first Chem class of [D] is Poincar6 dual to D (which computing the curvature form of a metric connection on [D]) we see that
([E)] [D']) Q *
Hence, [D] [D'] -
=
D
-
D',
==
cl
([D])
U cl
([D'])
and the two definitions
between line bundles and divisors. It follows that
bundle f- and
a
divisor D
-
77v A 77D,
we can
-
D
=
L
-
[D] ill
=
be shown
*
the basic correspondence
also define the intersection of
cl(,C)(D).
by
(D D') Q
correspond under
by putting 'C
=
can
a
line
Chapter IV. The geometry of Abelian varieties
Using the intersection pairing explicitly as
we can
calculate the genus of the smooth
curve
C
on
S
more
Ks C+ C C
I g
=
2
arbitrary
For
-
-
deg(Ks
curves on
0
S
[C])Ic
+ I 2
(Ks
C)
+
C+ I
-
+
=
2
define the virtual genus -7r(C) by this formula. Hence, 7r(C) is homologous to C. Let o : r -+ C be a normalization of C
we
the genus of any smooth curve denote the and let JP1, , P, I
singularities of C, i.e. the points where C is not smooth. If by ki the multiplicity of C in Pi (i.e., the number of sheets in the projection, in a small coordinate disc in S around Pi, of C onto a generic disc), then -
we
-
-
denote
Ks-C+C-C
9 Or) <
and the at
Pi
equality holds
are
We
if and
ki(ki
+1-
if all Pi
only
-
1)
(3.4)
2
2
ordinary singularities (i.e., the ki tangents
axe
different).
are now
ready
to
prove the Riemann-Roch Theorem
give and
line bundles
for
on a
surface. Theorem 3.5
LetC be
a
line bundle
on a
(smooth) surface
S with canonical bundle KS.
Then L
X(L)
=
X(OS)
L
-
-
L
KS
-
+
2
Proof We
give the proof for line bundles of the form L S. Staxting from the short exact sequence
=
[D],
where D is
a
smooth, irreducible
curve on
0 -+
OS
-+
OS(,C)
-+
OD(,C)
-+
01
expressing the fact that the alternating sum of the dimensions of the appearing in the associated long exact sequence is zero, we obtain
and
X(,C) Now
X(OD (,C))
=
X(OS)
+
X(OV (,C)).
is the Euler characteristic of C restricted to the
calculated using the Rlemann-Roch Theorem for curves,
which leads to the
=
X(Ov)
proposed
+
deg(,CI-D)
formula for
=
X(,C).
112
I
-
D, and
curve
giving C
x(O-D(L))
vector spaces
g +,C -,C
-
L -,C
=
2
-
KS
it
can
be
3. Abelian varieties
It remains to compute the
This result is
holomorphic Euler characteristic of the trivial bundle given by Noether's formula:
Theorem 3.6
Let S be
complex surface and KS
a
over
S.
its canonical bundle. Then
1
X(OS) where
X(S)
In the
1-2 (KS Ks
=
+
-
denotes the Euler-Poincar6 characteristic
case
of
Let
of S.
Abelian surface 7-2 containing
an
an axbitrary extremely simple:
formula and the Riemann-Roch formula become Theorem 3.7
X(S)),
T2 be
an
Abelian
surface and C
a curve on
curve
C, the adjunction
T. Then
C-C
7r(C) If [C]
is
positive and induces
+ 1
=
=
XQCD
+ L
polarization of type (91, 2)
a
on
S, then
C-C
-7r(C)
=
+ 1
=
dimL(C)
+ I
=
6162
+ 1-
(3.5)
Moreover, for a general Abelian surface the intersection of two line bundles [C] and [D] is deduced from these formulas by replacing C and D by linear equivalent multiples of one divisor. Proof Let
(ZI, Z2)
be the coordinates
on
7' coming from C'. Then the two-form w
has
no zeros.
Hence its canonical bundle
decomposition with equals I 4 + 6 4 -
We
=
-
dzj
A
dz2
K 0. Because r has a natural cell0, 4), its Euler-Poincax6 characteristic string of formulas.
vanishes,
(4) i
cells of dimension i, (i + I 0, leading to the first
=
=
=
show that X ([D])
.
.
.
,
dim L (D) for positive line bundles, the equality dim L (D) 6162 being given by (3. 1). Clearly, it suffices to show that dim Hi (T2, 0 (D)) 0 for i > 1. For this purpose, we use the Kodaira vanishing Theorem, which states that HP (MI Qq (f-)) 0 for any positive line bundle L over a compact complex n-dimensional manifold M, if p + q > n. For Abelian surfaces, Q2(,C) O(K)(L) O(C), because K 0 and the Kodaira vanishing now
=
=
=
=
=
=
Theorem reduces for Abelian surfaces to
HP (7, 0 (L))
=
0
for p > 0.
The last claim follows from the fact that the N6rou-Sevieri group of a is isomorphic to Z.
113
general Abelian surface
Chapter IV. The geometry of Abelian varieties
4. Jacobi varieties
There
two very different ways to define the Jacobian of
a non-singular curve, the being given by two fundamental theorems, Abel's Theorem and the Jacobi inversion Theorem. We prefer to give both definitions here because of the importance of both of them in application to integrable Hamiltonian systems. We start by giving the algebraic definition, then we give the analytic/transcendental definition and finally prove their equivalence. It is also shown that the Jacobian of a curve of genus g is a principally polarized Abelian variety of dimension g.
are
equivalence of the
4.1. The
two definitions
algebraic Jacobian
We fix
a curve
(compact
Pdemann
surface)
17 of genus g. For the
algebraic definition,
constructed the group Pic(]P) of all line bundles on r, and showed that this to Div(r)/ -1, the group of all divisors modulo linear equivalence. Since is isomorphic group deg(f) 0 for any meromorphic function f , it follows that D -1 D' implies deg E) deg V,
recall that
we
=
hence
=
deg induces
a
homomorphism deg_
Div(r) -+
:
Z.
-1
We define the Jacobian
of 1P, Jac(r),
to be
Kerdeg_. Said differently, Jac(]P)
the group of divisors of divisors of degree zero modulo linear equivalence. allows us to define the degree of a line bundle C [D] by deg L = deg_ D.
=
Divo(r)/
The map
deg_ Defining Pic'(r) we have, with the
=
the group of (isomorphism classes of) line bundles of degree i on r above definition of the Jacobian, that Jac(r) is canonically isomorphic to the group PicO(r). Of course, PicO(r) is isomorphic to Pid'(r) for any other integer i but the isomorphism is
as
not canonical. For reasons that will become cleax later as
some
authors prefer to define
Jac(r)
Picg-'(r).
4.2. The
analytic/transcendental
Jacobian
t (wi, analytic definition, choose any basis 10 w,) of the space of holomorphic differentials on IP and let A denote the discrete subgroup of C9 consisting of all vectors in C9 of the form c , with -y running through HI(r, z). Since 1w,.... 1Wg' j'...' gj generate 'Y
For the
=
H1,0 E) H0,1 (the first the Rham group of IP and its splitting in the holomorphic and anti-holomorphic paxt), A is actually a lattice (called the period lattice of Jac(I7)). This shows that Cg/A is a complex torus, the analytic Jacoblan of r. More intrinsically, the analytic Jacobian of r is defined by
HD'R (r)
=
Jac(I') where H, (r,
phism
Z)
is viewed
T which maps -y E
HO(r, Q1)* =
H, (r, Z)
'
as a subgroup of HO (1P, Q')* via the natural injective homomorHI (1P, Z) to the linear map
,P(-y) : Ho (r, Q1) W
114
c
fy
4. Jacobi varieties
Before
showing how the algebraic and the analytic Jacobian are related we want to show analytic Jacobian is a (principally polarized) Abelian variety. Choosing a basis of H, (r, Z), for example a symplectic basis JAI A,, Bl,...,B,}, i.e., a basis for which 0 and Ai Bj Ai Aj Bi Bj Jjj, the lattice A is then conveniently represented as the that the
......
=
-
=
-
column space
=
-
(over Z)
of the matrix
fAj WI
...
fAj W9 The first g
x
B-periods.
For
fAg W1 fB, WI
fBg W,
fA,, W9 fB, W9
fBg W9
g block is called the matrix
of A-periods and the last g x g block the matrix of single 1-form its i-th period (I < i < 2g) is its integral over the cycle Ai if i < g, otherwise it is its integral over the cycle Bi-,. The following theorem states that the (analytic) Jacobian of IP is a principally polarized Abelian variety. Theorem 4.1
a
Let
JAI,_, Ag BI,
be a symplectic basis of HI (r, Z). For any ..' Bg} of A-periods is non-singular and hence a basis of the latter space can be chosen such that the matrix ofA-periods is the identity matrix. In this basis the matrix of B-periods is symmetric and its imaginary part is positive definite. Thus Jac(r) is a principally polarized Abelian variety. basis
the
I
of H'(r, QI) the
-
matrix
The main ingredient in the proof (given below) is the following proposition (known reciprocity law for differentials of the first and third kind).
as
Proposition 4.2 Let IP be a compact Riemann surface of genus g, equipped with a symplectic basis JAI,_, Ag, BI,-, B,} of HI(r, Z). Let w be a holomorphic 1-form and let 77 be a meromorphic 1-form whose poles are simple; we call these poles S,.... 8n. Also so E r denotes an arbirary fixed point. If IIi (resp. M) denotes the i-th period of W (resp. ofy) then 7
9
n
1101g+k
-
llg+k Wk
27ri
k=I
81
Res,,
q
W.
(4.1)
3=1
Proof
Representatives for the Ai and Bi may be chosen such that none of them pass throug poles of 77 and by cutting r along the cycles the surface may be represented by a polygon IF (with 4g sides) as in the following figure.
the
115
Chapter
IV. The geometry of Abelian varieties
Figure
6
boundary pieces which correspond to the cycles Ai and Bi will simply be called accordingly the cycles are sometimes denoted by the uniform notation 2g, (thus with this notation yj is just the boundary piece corresponding to the 1,
The
-(i resp. -yi+, and
Ji, i cycle Ji). =
.
.
.
,
Let C:
7r:
8
W, 80
then these
is holomorphic and -7r?7 is a meromorphic 1-form with poles poles are simple we have by the residue theorem
7r
f
?r,q=2-7riERes,,iri7=27ril:Res,,,q
o f;
right hand
which gives the
i
i
side of
(4.1).
I.
in the
points
si.
Since
W,
Also 2g
29
r
-7r?7
Fn
=
E
(7r(p+)
7r(P
_
77(p).
r
Here P+ and P-
that
77(P+)
=
are
?I(P ).
the two points To
finish, look
corresponding to P which lie on -yj figure and compute
resp.
yj-';
we
used
at the above P+
ir(P+) if P G yj, I <
j
< g.
-
Similarly
7r(P-)
=
fp,
IBI
W -
for P E -tj+,, g + I < j <
W
2g
=
-Hj+g
one
finds
P+
7r(p+)
-
7r(P-)
=
f
W,
'P
From this
we
find the left hand side of
fat
'7r?J
E j=I
-IIj+gq(P)
=
fA
W
=
11j.
(4.1):
Ilin(p)
+ +9
116
E IIJI'9+k k=1
IIg+kIIk-
(4.2)
4. Jacobi varieties
Proof of Theorem 4.1 For
a
holomorphic 1-form.
77 the
right hand side
(4.1)
in
vanishes and
gives
the
simple
relation 9
IIkIII,+k
-
IIg+kI-Ik
--`
0-
(4.3)
k=1
Denoting the complex conjugate of 77 by
A, which,
upon
using (4.2)
wA
=
A;
one
dir A
has
A d(7r
=
-
q)
=
fe
7r
-
'
to 9
A; wAq=EIlkfg+k -Hg+k
k
(4.4)
-
k=1
For
a
holomorphic
Morm
w
=
f (z)dz w
which leads for
=A
w
A Co
=
-2ilf (z) 12 dx A dy,
0 in combination with
0 > Q
ffr
(4.4)
wA
=
to
2!aEIIkfIg+k-
(4.5)
k=1
A first conclusion is that if all This
implies
some
to
A-periods
of
a
holomorphic 1-form
that the matrix of A-periods of the basis
JW1,
w are zero
then
w
=
0.
wg I is non-singular: otherwise non-trivial combination of the elements of this basis would have all its A-periods equal .
.
.
,
zero.
By changing
our
basis of the space of holomorphic differential$ are normalized in the sense that
(if necessary)
we
may
periods now takes the form (I Z), where Z is the matrix of B-periods. B-periods of two I-forms wi and wj one finds from (4.3) that IIg'+i lIg+j, i.e.,
For
therefore
assume
that the wi
iAj
Wi
=
6ij.
The matrix of the
=
fBi and Z is
a
symmetric matrix.
FinaJly!a,Z
is
positive definite.
Let 0
=4
ckwk with Ck E R. Then
w
9
Ilk k
CjWj
=
cjwj
=
k
Hk+g
Ck
j=1 k
4k E i=1
117
E CjZjki j=1
Chapter IV. The geometry of Abelian varieties which leads
by
substitution in
(4.5)
to
9
1: Ck (O Zkj) Ej
<
0-
k=1
period matrix has the desired form and we conclude from Theorem principally polarized Abelian variety whose dimension is the genus of 1P.
3.1
This shows that the that
Jac(r)
is
a
Thus the Jacobian of
a curve
of genus g is
principally polarized Abelian variety
a
whose
g. The following converse also holds: every irreducible principally polarized Abelian variety of dimension 2 or 3 is the Jar-obian of a curve of genus 2 or 3. In higher dimensions this is no longer true (as can be checked by an easy dimension count), and there
dimension
equals
is the famous
which asks for
Schotky problem
a
characterization of those matrices Z for which
(see [Mull], [Mum3] and [Shi]). The Riemann theta function, defined in (3.2), can be used to construct (all) meromorphic functions on Jac(17). Namely from the definition one checks the following quasi-periodicity
(Idg Z)
is
a
Jacobi variety
of this entire function
on
C9: if
E Z9 then
m
V(Z + M) ='O(Z), V(z + Zm) e`ri(,rn,Zm)e -27ri(m,z)V(_,).
(4.6)
=
From this it is clear that for all 1 <
i, j
u
(Z)
the
< g
=
meromorphic
function
a2 -log 19 (Z)
k9Zj'9Zj
u(z+Zm) u(z), hence it descends to a meromorphic function on the periodic, u(z+m) quotient with a double pole on the theta divisor. Another way to construct such functions, n which will also be used later, is by using theta functions with characteristics. For a, b E Q is
=
=
(called characteristics)
defines
one
,,
[ab ] (Z)
,i(1+a,Z(1+a)) e21ri(l+a,z+b)
1:
=
(4.7)
IEZ9
Formulas
(4.6)
become
now
[a]b (.v [a]b (z
+
=
e2ri(a,m)V
Zm)
=
e-"(7n zm)e -27ri(m,z),-27ri(b,m),9
From these formulas it is easy to such that E(aj a'j) (E Z9 and -
S
see
that if aj, bi and a'j, bil) (E Z9 then
E(bi
'=I
meromorphic
function
on
b'i
are
characteristics
(for i
=
1,
.
.
.
,
n)
S
n
a
(4.8)
[a]b
-
H is
[a]b
M)
+
[ai ] (_,) V (Z) [ail b,
V
bi
Jac(r). 118
(4.9)
4. Jacobi varieties
4.3. Abel's Theorem and Jacobi inversion We to the
show that the
now
object, namely
same
algebraic and the analytic/transcendental Jacobians correspond we
show that the Abel-Jacobi map 9
9
Ab
:
Divo (r)
Cg/A
-+
E Pi
:
-
Q
+
i=1
with c
and A
Pi
EI i=1
c
(mod A),
Qi
defined above, is surJective with kernel consisting of the
as
principal divisors.
We start with the latter which is the content of Abel's Theorem.
a
of degree 0, Ab(V)
For any divisor D
Theorem 4.3
=
0
if and only if D
is the
divisor of
meromorphic function.
Proof then d log f is a well-defined meromorphic 1-form with simple poles on the V of Qk) and residue I (resp. 1) at PI, (resp. at Qk). Denote by Wpk Q,, rn=,(pk support 'k the unique meromorphic 1-form with the same poles and residues as d log f but with its Aperiods zero. Then their difference is a holomorphic I-form If V
=
-
-
9
n
dlog f
C
=
tion
(cl, 4.2,
.
.
.
cg)
,
now
is
with
just the w
-
vector of
us
to
E CjWjl
k=1
j=I
A-periods
of d log f.
Another
application of Proposi-
wj and 17 =WPkQk leads to
=
fBj This allows
E WPkQk
wpk Q,
2-7ri
=
fPk
Wi-
Qkh
compute explicitly the integrals appearing in the Abel map: n
k=1
n
P
fQ kk
Wj
=
1
Iri
Ei
WpkQk
13
k=1
9
,1,J i7ri
dlogf-
CI
BFj
fB
WI
n
nj
-
EmIZjI, 1=1
in which the
integers mj and nj
This shows is
tempted
one
to define
are
up to
a
factor 27ri the A- resp.
direction of Abel's Theorem. For the other a
function
on
r
by P
f (P)
=
exp
(27ri 77) PO
119
B-periods
of d log
direction, if Ab(D)
=
f
.
0
one
Chapter IV.
The geometry of Abelian varieties
where 71 is a 1-form with residue in the points Pk and in Qk. Indeed, if f is well27ri 27ri defined then (f) D and we are done. The problem is thus to find an 71 with all its periods integral. Any such 77 can however be written as
;'
j:7r-'j
=
9
,q
770 +
=
E c1W1 1=1
where 970 has its
A-periods for
A-periods equal to zero. B-periods,
We must take ck
P-
n
fB where
we =
axe
h=1
0 the first term has the form nj +
To show
integral
9
cl Z1j'
Wj + k
computed the B-periods of qo from Proposition 4.2 (with w
Thus it suffices to choose cj
which
fQ
17 I
Ab(D)
in order to have
integral
77. As for the
-mj to obtain
a
the Jacobi inversion
surjectivity,
quite similar
=
71o). Since wj and 71 where ni, mi E Z (and m (m, .... M9))Morm 77 which has all its periods integral. n =
=
(Zm)j
=
Theorem,
it is better to define other maps we fix any PO E 17 and we define
to the Abel-Jacobi map. To do this
for any d E N d
Abd
:
SYM
P!
dr-+C9/A:(Pj,...'Pd) -+Ef. j=1
Theorem 4.4
The map
Abg
is
c
(modA).
(4.10)
PT,
surjective.
Proof
symg r and Cg/A are both compact connected complex manifolds and Abg is a holomorphic map. Hence the image of Ab. is a compact subvariety of Cg/A. Its differential in a general point (PI, P,) is given by .
.
.
,
Wi(PI)
(Pg)
W,
where
By
...
...
wg(pl) Wg Wg)
we have written wj locally as fj(z)dz and wj(P) is a (bad) notation for fj(P). wj Riemann-Roch there exists no holomorphic differential with its zeros on a general set =
fpj'...' Pgj
of
points, hence this matrix Ab. is surjective.
is invertible. This shows that the
image
of
Abg
has
dimension g, hence
The link between the
further in the
algebraic and the analytic/transcendental following theorem, attributed to Riemann.
Theorem 4.5
There is
a
constant,&
E
C9
(called
Riemann's
Jacobian is pursued
constant)
such that
g-1
V(Z)
==
0 4=*
3PI,
-
..'
P,
E r
:
Z
=
Ab
120
E(Pi
-
P)
A
(mod A).
(4.11)
4. Jacobi varieties
The
important condition
in the
side is that the
right-hand
sum runs over
g
I
-
points
I Because of this theorem mwiy people prefer to define the Jacobian by Pic-q- (r) (rather than Pico (17)) since with the former definition the theta divisor is canonical determined (while
only.
otherwise it is defined
only
up to
a
translation which depends
on
the choice of
a
point
on
r).
4.4. Jacobi and Kummer surfaces
Everything discussed above is easily specialized to Jacobi surfaces, i.e., Jacobians of curves. They are especially important when working with Abelian surfaces since, as we explained, every Abelian surface is isogenous to a principally polarized Abelian surface and these are (under the asswnption of irreducibility) Jacobians of genus two curves (recall also that every Pdemann surface of genus two is hyperelliptic, hence can it can easily be described explicitly). The theta divisor of a Jacobi surface is by Riemann's Theorem nothing but a copy of the curve, embedded in its Jacobian; conversely if in some Abelian surface a (non-singular) curve of genus two is found, then it defines by (3.7) a principal polarization genus two
on
the surface which necessitates it to be the Jacobian of this
curve.
Since the third power
of any ample line bundle on an Abelian variety is very ample, we may take the third power of the line bundle corresponding to the theta divisor to construct an embedding of a Jacobi surface in P8.
Associated to
an
Abelian variety T9 is
a
singulax variety,
its Kummer
variety, which is de-
the quotient surface K T9 / (- 1), where I is the involution given by (ZI, 1-+ , zg) (-zl,..., -z,) in linear coordinates coming from the universal covering space C9 of T9. Clearly the Kummer variety has 229 singular points which correspond to the two-torsion
fined
as
=
-
.
.
.
points (also called half-periods) of T9. We will consider in this text only the case g 2, the case of Kummer surfaces. Let A be an Abelian surface of type (611 62) Its Kummer surface K Al(-I) has sixteen singular points ej,...' e16. The desingularization of K can be described as follows. Let p : A -* A be the blow-up of A at all its half periods and denote the corresponding exceptional divisors by Ej. (-I)A extends to an involution (-l)A On A =
-
=
quotient ff Al(-l)A is a K-3 surface (see [Beall, Proposition VIII.11). ff is the desingularisation (minimal resolution) of K and we have the following commutative diagram.
and the
=
P
A
i
A
I
IrI k Associated to A there
-
K
surfaces which are desingularizasingular points. A similar construction can be performed when the Abelian surface admits an automorphism, different from the (-1)-involution, leading to a generalized Kummer surface. See Paragraph VII.3 for an example.
tions of K at
some
are
also several intermediate Kummer
but not all
of the Kummer surface of
2
surface, taking C [6], all sections Of raxe even and the singular surface can be embedded in P1 by using these sections. Being twodimensional the image is given by a single equation. To compute the degree of this equation, which is the degree of the hypersurface, we use the fact that this degree is given by fK Q, where Q is associated (1, I)-form of the standard Mililer structure on P1. Clearly this is* twice the volume of K, which itself is half the volume of the Jacobi surface (embedded with In the
case
a
Jacobi
121
=
Chapter IV. The geometry of Abelian varieties the
polarization of type (2, 2)). For w 2dxj A dX2 + 2dx3 A dX4 we get f W2 8, hence the degree 8, its volume is 4, the volume of K is 2 and the degree of K is 4. Explicit equations for this quartic polynomial, in terms of an equation for the underlying algebraic curve, will be given in Section VIIA. =
=
Jacobi surface has
Another classical result about the Kummer surface of
Jacobi surface is that it has
a
166 configuration. Namely for IP a curve of genus two, the (-I)-involution t on Jac(]P) has, apart from 16 fixed points aJso 16 invariant theta curves (i.e., translates of the theta divisor);
a
each of these 16 curves contains 6 of the fixed points and through any of the 16 fixed points pass 6 of the invariant theta curves. The configuration can be described completely as follows (see [Hud]). Let W1,...' W6 be the Weierstrass points on r, then the points W
Wij
=
fW",
0'
(mod A)
Jac(]P) since 2Wj -1 2Wj. There are sixteen half-periods in total since the Wji and Wii 1, Wjj hold for all i, j 6; apart from these they are all different. The sixteen invariant theta curves rij in Jac(IF) are the translates wij + rkk of the single curve ]PIi IP66, which can e.g. be taken as jAb(P WI) I P E rj. aoin this it is clear that IPII, hence aJI ]Pij, pass through six points Wk, and also that each point belongs to six lines ]Pij. Since the whole configuration is invariant under z it goes down to the are half-periods equalities Wij
of
=
=
=
.
=
*
*
*
.
.
,
=
-
Kummer surface in P3 and
gives there a 166 configuration of points and planes, classicaJly configuration. The sixteen points are nodes (singular points) and the sixteen planes the lines belong to are tropes (singular planes) of the Kummer surface.
called Kummer's
The 166
pair
configuration is diagrams, such
of square
W11 W45 W46 W56
best visuahzed as
W12 W36 W35 W34
the
by following.
W23 W16 W15 W14
W13 W26 W25 W24
the incidence
ri, r4, IP46 r.6
r.2 1 3, IP3. r34
diagram, which
P23 P,6 r,, r14
consists of
a
IP13 r2, r25 r24
Namely, the points incident with a line at position (m, n) in the second square diagram are those six points in the m-th row and n-th column, but not in both, of the first square diagram. Dua,lly, the same applies for the lines incident with a point. The 242 incidence diagrams obtained by permuting the rows or columns of both square diagrams in an incidence diagram (in the same way) are defined to be the same as the original incidence diagram (we will see in Section VIIA that there are 20 incidence diagrams which are different in this sense). These incidence diagrams will be used in Section VIIA to describe Abelian surfaces of type (1,4).
122
5. Abelian surfaces of type
5. Abelian surfaces of
Closely related
(IL 14)
type (1,4)
to Jacobi surfaces are Abelian surfaces of
is discussed here in detail because
they
ter VII. The results in this section
axe
type (1, 4). Their geometry
will appear in the two examples discussed in Chaptaken from [BLS]. As in that paper we will without
always restrict ourselves to those Abelian surfaces of type (1, 4) which are not isomorphic to a product of elliptic curves as polarized Abelian surfaces. Let PC be a line bundle of type (1, 4) on an Abelian surface 7-2. It follows from (3.7) that dim HO (7-2, 0 (,C)) 4 and C induces a rational map Oc : 7-2 _+ p3. further mention
=
5.1. The
generic
In the
birational
case
Oc(T2) generic case, the image of this map 0 its image. Let K(L) be the kernel of the isogeny =
C
p3 is
an
octic and
O'C
is
on
1'C :r a
-+
-+
r
taiC 0,C_1
r and its dual 'P (defined as the set of all line bundles on T2 of degree 0; t,, is the by a Ei T), then K(L) is a group of translations, isomorphic to Z/4Z ED Z/4Z. Picking any such isomorphism, let a and -r be generators of the subgroups corresponding to this decomposition. Then homogeneous coordinates (VO : yj : Y2 : Y3) for P3 can be picked, such that o-, -r and the (-I)-involution % on 7' (defined as %(Z1, Z2) (-Z11 -Z2) for between
translation
=
(zl, z2)
E
C2/A)
act
as
follows
(see [LB]):
0'(YO
Y1
Y2
Y3)
=
(Y2
Y3
YO
T(YO
Y1
Y2
Y3)
=
(Y1
YO
iY3
Y2
Y3)
=
(YO
t(YO (strictly speaking coordinates exist octic 0 is
given
Y1
Y1
it may be necessary to replace for (a, -r) and (3a, 3-r) or for
only
in these coordinates
2 2 2 2 ,\2OYOYIY2Y3
r
-YI),
*
:
Y2:
by 3-r; it
(a, 3,r)
(5-1)
iY2)i -Y3)i
and
is easily checked that these (3a, -r)). [BLS] show that the
by
2 + A2(YO4Y4 Y4Y34) +A3 (Y04Y343 + Y4IY24) + 2 3 2 2 + Y14Y4) + 2A,1\3 (Y2Y2 2A,1\2 (Y2Y2 Y2Y2) Y2Y2) Y2Y2) 2 3 + 0 2 1 3 1 2 (Y2Y2 0 1 3 0 1 + Y2Y2)(Y2Y2 0 3 2 2 0, 21\2X3(YlY2+YOY32)( Y12Y23 + Y2Y22) 0
+
'\2I ( YO4)Y41
+
_
_
_
2
(5.2)
2
=
(,\o : \, : 1\2 : X3) E p3 \ S where S is some divisor of p3 which we will determine 1, the coordinates (IEOYO : 'ElYl : IE2Y2 : (Paragraph VII.4.4). Notice that for any ej 1EO611E2Y3) will also satisfy (5.1) and these are the only coordinates with this property. It is also seen that, if (a,,r) is replaced by (3a, 3,r), then the coordinates (yo : yj : Y2 : Y3) are replaced by (YO : Y1 : Y2 : -Y3)- Since the equation of 0 depends only on yj2 these choices do not affect the equation (5.2), so there is associated to a decomposition K(,C) K, (D K2 (where K, and K2 are cyclic of order 4) an equation for 0. [BLS] also show that the polarized Abelian surface as well as the decomposition of IC(L) can be recovered from (5.2) and that every octic
for
some
later
=
=
123
Chapter IV. The geometry of the
type
(5.2) (with (AO : 1\1 (7-2, L)
Abelian surface If
:
)k3)
S)
is the
image Oc (T2)
of
some
(1, 4)-polarized
-
denote
we
/\2
:
of Abelian varieties
by
140(1,4)
Abelian surfaces for which
the moduli space of
OC
is
(isomorphism
classes
of) (1, 4)-polarized
birational, equipped with a decomposition of K(L)
as
above,
then it follows that
'40
_
(1,4)
if
Moreover,
we
denote
K the
by
subgroup K
=
-",::
p3\S ToZ -AO K(,C)
of
10, 2o-, 2T, 2T
(5.3)
of two-torsion
+
elements,
2o-J,
71/K is a principal polarized Abelian surface, which is the Jacobian of a curve of genus call T2/K the canonical Jacobian associated to 7-2. Recall that for a two-dimensional Jacobiau J its Kummer surface is the image of 0[28] C p3' where 0 is the theta divisor of J.
then
two;
we
Then it is
seen
from
(5.1)
that
an
equation for the Kummer surface of 7/K is given by yi2 by zi in the equation (5.2) for 0 and there choosing the origin of 7' such that L becomes line bundle Ar on T11K of type (1,I) via the canonical
the quartic Q in p3' obtained by replacing is a natural projection fi : 0 -+ Q. In fact,
symmetric, L is the pull-back of projection
a
p
and
OAr2 induces the
Kummer
T2
:
-*
mapping; [BLS] -2
7-2 1K, prove that the
following diagram
commutes
O.C
7
0
P1
If
(5.4)
Q O,vn
5.2. The If
Oc
non-generic
is not
case
birational,
then it is 2
:
I and
Oc (T2)
is
a
quaxtic
in
p3' given by
one
of the
equations Y6 Y21 +
_
=
0,
_
_
=
07
=
07
+ A2 (Y2Y2 Y2Y2) Y2Y2) 2 3 1 3 0 2 A 1 (Y2Y2 + Y2Y2 A3 (Y2Y2 1) Y2Y32) 2 3 0 1 2 0 2 2 + \3(y2y22 + Y02y32) /N2(Y1Y3 + y2y2) 0 2 1
A
the choice of the
depending on the decomposition
of
K(L)
can
case the Abelian surface as well as decomposition; only partly be recovered from these equations and r1K is
in this
124
5. Abelian surfaces of
type
(1,4)
product of elliptic curves (in particular T' is isogeneous to a product of elliptic curves). Squaring each of these equations we find equation (5.2) respectively with
a
A20
2 ('X22 +
A2) 3
-
'X23'A 0,
'X1A3 : 07 \21
_
'X237 0,
\IX2 7 07 A21
+
'\22A 0.
1\2,X3
0, A22
0
A20 X2
-2 (,\21 + =
2
f
A2) 3
(5.5)
=
0
1\ 0
=
2 (,\
X3
=
0
21
\2) 2
case (the generic case), OC(V) is an octic, 7-2/K is a Jacobian decomposition of K(L) can be reconstructed from the octic; in the other and T2 cannot be reconstructed case OC (7-2) is a quartic, 71 IK is a product of elliptic curves natural with a us rational surjective map The from the quaxtic. map OC provides
Summarizing,
and 7-2
,00
as
well
AO(I 4)
where
-+
AO(,,4)
as
in the first
the
((p3 \ S) U (three rational
denotes the moduli space of with
surfaces
together jection (5.3) defined complement of
.40(1,4)
boundaxy of 00 to Abelian
surfaces
in S
missing eight
(isomorphism
classes
the dense subset
'4 0(1,4)
of
AO
(1,4)
to the three rational curves, which
(A(,,4)), 0
surfaces, but
points)) /(Ao
of) (1, 4)-polarized
0), Abelian
The map 00 extends the biand maps the (two-dimensional)
decomposition of K(L) (as above).
a
on
curves
axe
thought
of
as
i.e., in S; the generic point of S however does to surfaces which
can
(see P3LS]).
125
be
interpreted
as
lying inside the not
degenerations
correspond of Abelian
V
Chapter
Algebraic completely integrable Hamiltonian systems
1. Introduction
integrable Hamiltonian systems of interest the general level set of the momentum to an affine paxt of an Abelian vaxiety and the flow of the integrable isomorphic map vector fields is linearized by this isomorphism. These two properties lead to the definition of an algebraic completely integrable Hamiltonian system (a.c.i. system). We will discuss three quite different definitions of an a.c.i. system, which have been proposed by different authors, and we extract from it a definition which is consistent with our approach to integrable Hamiltonian systems. All constructions of integrable Hamiltonian systems easily speciahze to the case of a.c.i. systems, except in the case of the quotient, which requires a real proof. The definitions and these properties will be considered in Section 2. In many axe
Painlev6
important tool for studying an (irreducible) a.c.i. system, since general level sets of its momentum map and to construct an explicit embedding of the completed general level set (which is isomorphic to an Abelian variety) in projective space. Although the nature of these Abelian varieties can often also be deduced from a Lax representation of the a.c.i. system (see e.g. [Aud3% Painlev6 analysis is at present the only available method to construct an embedding of these Abelian varieties in projective space. Since several such embeddings will be constructed and used, we will explain what Painlev6 analysis is about in Section 3. We wish to point out that Painlev6 analysis can, by a theorem of Adler and van Moerbeke (see [AM7]), in principle be used to prove that a certain integrable Hamiltonian system is a.c.i., but applying this theorem requires even for the simplest systems a considerable ainount of work. In Section 4 we recall from [Van2] our algorithm which allows one to lineaxize explicitly two-dimensional a.c.i. systems starting from the differential equations for one of the integrable vector fields. It is used in this text to construct morphisms of two-dimensional a.c.i. systems. it allows
us
analysis
is
an
to determine the nature of the
127
P. Vanhaecke: LNM 1638, pp. 127 - 142, 1996, 2001 © Springer-Verlag Berlin Heidelberg 1996, 2001
Chapter In Section 5
we
will
V. A.c.l. systems
explain briefly how Lax equations
appeax and what is their relevance
for a.c.i. systems, in paxticular when dealing with Lax equations with a spectral parameter. Lax equations have by now been obtained for all integrable Hamiltonian systems which were classically known, however some of these have been obtained after a lot of effort, reflecting the fact that it is not clear how to write down Lax equations for a given integrable Hamiltonian system, even if the underlying geometry has been completely revealed. For the general
integrable
Hamiltonian systems introduced by us in Chapter III for example it is not clear equations for the integrable vector fields (apaxt from the very special case
how to write Lax
Paragraph 111.2.4). Lax equations constitute a complete chapter in the theory integrable systems, however we will not treat them as being basic in their study since very often they only appear at the end. Of course, once obtained, Lax equations beautifully exhibit many aspects of the integrable Hamiltonian system in a unifying way; moreover, a careful analysis of the underlying algebra leads in many cases to generalizations and to a quantification of the integrable system. treated in of
128
2. Ax.i.
2. Ax.i.
systems
systems
by discussing three definitions of algebraic complete integrability. Often in the being used meaning that the general fiber of the integrable Hamiltonian system under consideration is an Abelian variety, an affine part of an Abelian variety or just a dense subset of an Abelian variety, the flow of the integrable vector fields being in any case We start
literature this term is
linear
on
it. Indeed that is the main idea and will be present in the different definitions of
a.c.i. systems Let
given below.
start with the Adler-van Moerbeke definition
us
(see [AM7]).
They
consider
a
Hamiltonian system (i.e., the Poisson bracket is polynomial as well the functions in involution) on IV' (fixing a basis of the integrable algebra, but that is
polynomial integrable as
here)
not relevant
they complexify in the natural general level sets of its momentum
which
conditions about the
way. They suppose the following map and the flows on them to be
verified: fiber of the momentum map is
1)
The
2)
Abelian variety; The flow of the integrable vector fields is
general
on
(isomorphic to)
the
an
affine part of
general fiber of the
an
momentum map
linear;
on C' restrict to the general level sets as regular functions meromorphic functions on the corresponding Abelian varieties; 4) The divisor to be adjoined to the general level set is minimal in the sense that for each irreducible component of this divisor there is at least one of these meromorphic functions which has a pole on it.
3)
The coordinate functions
which extend to
Next we give Mumford's definition (see [Mum5J pp. 3.51 3.54). He starts from a real symplectic manifold (M, w) of dimension 2n and a smooth function H on it. His requirement for algebraic complete integrability is the existence of a smooth real algebraic variety (MC, T) -
such that:
1) M is a component of the real part of (MC, -r); 2) MC is endowed with a (co-)symplectic structure (D which restricts to w along M; 3) There exists on MC a proper map which is submersive onto a Zariski open subset of Cn and whose
component functions The conditions these
by
are
components
(restricted
sufficient to
imply
C*1 and that the flow of the let
Third,
us
function H which define
on
a
M and
submersive map onto
A
a
bundle
a
divisor E) C
an a
7r :
an
-+
vector field Y
A
\D
are
Abelian varieties
He considers
-+
an
on
these
or
extensions of
(and
open dense subset B of
Cn,
c M -+ B c
(irreducible) complex
structure w,
functions in involution
B of Abelian
on
being dependent
vector fields is linear.
(see [Kn6]).
F-I(B)
M; isomorphism 0 : F-1 (B)
H
holomorphic symplectic
algebraic complete integrability he demands
1) 2) 3) 4)
involution,
that the fibers
holomorphic
n
F:
For
a
in
M).
integrable
recall Kn6rrer's definition
manifold M of dimension 2n with
are
to
a
holomorphic
in involution with
H)
Cn.
the existence of
varieties; A
\ D;
which restricts to
129
a
linear vector field
on
the fibers of ir;
Chapter so
that the
V. A.c.i. systems
diagram F
-1(B) P
A\D
I Z11
B
is commutative and such that the vector field Let
us
ate in the
case
XH is 95-related
to Y.
these definitions and deduce from it the definition which is appropriof integrable HandItonian systems on affine Poisson spaces. We don't discuss
comment
on
here the fact that all three choose
a specific basis of the integrable algebra since we discussed this aspect already in Chapter II (and it is not really relevant here). The last two definitions are situated on symplectic manifolds, while the first is on Poisson manifolds, a difference
which is easy to overcome. The first two definitions start from a real system and complexify it, the last one is completely situated in the complex. Also here, there is nothing to be worried
about; from the point of view of physics and when one wants to study the real topology of integrable Hamiltonian systems, one may want to impose a reality condition, but it is clear that the condition is not essential in the definition. So
the definition but put We
of the
now come
as a
to the
general level
restriction when the
more
think it should not be encoded in one
has in mind ask
so.
essential points in the definitions which all relate to the nature
sets of the momentum map.
The Adler-van Moerbeke definition
can
fiber of the momentum map (on from the ambient space C', is isomorphic to the
we
applications which
general
in this respect be reformulated saying that with its algebraic structure which comes
Cn),
an affine part of an Abelian variety; the added missing divisor is minimal relates to the possibility to perform the compactification of this general fiber into an Abelian variety by using the Laurent solutions to the differential equations describing one of the vector fields (see Section 3 below). We think that one should not insist on putting this condition in the general definition.
condition that the
D. Mumford's definition does not demand
about the nature of the
;ber of the momentum map but his assumptions anything imply what it is-
it is
an
general
Abelian variety
or an extensions of an Abelian variety by C*11. The key is that he asks properness of the morphism (which plays the role of the momentum map). As such, this is very strict: in the examples we know of one rarely finds complete Abelian varieties (but C*In are found in several examples). After completion however, as is discussed below, a proper map and hence an a.c.i. system in this sense" is found. An advantage of his definition is that he allows fibers which are more general than Abelian varieties.
H. Kn6rrer's definition asks that the level manifolds be
isomorphic to affine parts of family. Whereas the Adler-van Moerbeke definition asks that every generic level manifold can be compactified into an Abelian variety, his definition asks for a so-called partial compactification of the momentum map (at least over a Zariski open subset); this would then lead to a compactification of each general fiber by restriction. With these remarks in mind we will now define two notions of a.c.i. systems, one notion being
lbelian varieties, but
stronger
(but
even as a
harder to
verify)
than the other.
Strictly speaking the symplectic
or
Poisson structure may not extend to the
130
completion.
2. A.c.i.
Definition 2.1
Let
variety and denote an
(M,
A)
be
integrable Hamiltonian system on an affine Poisson by -7rA : M -+ Spec(A). Then (M, 1-, -1, A) is called
an
its momentum map
algebraic completely integrable
systems
Hamiltonian system
or an
a.c.i. system if there exists a A -+ B such that for
Zariski open subset B C Spec(A) and a bundle of Abelian groups 7r each b E B there exists a divisor Vb C 7r- I (b) and an isomorphism Ob such that the restriction of each vector field in vector field
on
Ham(A)
to
ir. '(b)
is
7r. ' (b)
7r-'(b) \ Z)b
-+
Ob-related
to
i
linear"
a
7r-'(b).
If there exists instead
Zaxiski open subset,
a
a
bundle of Abelian groups 7r : A -+ B, where B C Spec A is on A and an isomorphism 0 such that
a
divisor D
,ir. ' (B) VA
0 - -------
P-
A\V
I/ ir
B
is
a
commutative
diagram and
such that each vector field in
\ D which restricts to a linear vector (M, 1-, .1, A) is a completable a.c.i. system.
field
on
A
Here
are some
a.c.i. system:
one
field
remarks about this definition.
defines
Ob
=
01-7rAl(b)
and
on
Ham(A)
is
0-related
each level set of ir, then
Clearly
a
1)J-7r-1(b)-
completable
to
we
a
vector
say that
a.c.i. system is
an
Whether every a.c.i. system is
completable is a question which has been studied in more general terms in algebraic geometry. 13 In simple terms the way they put this question is demanding for a partial compactification of a morphism whose fibers over closed points are (isomorphic to) affine parts of algebraic groups; this partial compactification should be such that the compactified fibers are isomorphic to the corresponding algebraic groups. Their solution is to construct the points to be added as a SUM14 of two points in the affine part; more precisely they construct the fibred product of -7r. ' (B) with itself and define an equivalence relation by which two pairs of points are equivalent if they belong to the same fiber and they have the same sum (the sum to be taken 15 in the completed fiber). The question is now if this leads to an algebraic quotient. In general the answer is no, it works only up to a base extension. The problem of completing an a.c.i. system comes from the monodromy of the base space; in a real setting the monodromy is an obstruction for the global existence of action-action variables, as was shown by Duistermaat (see [Dui]). As Michble Audin pointed out to me, these ideas are very close. It turns out that many a.c.i. systems are actually completable (more precisely: is not known). One way to verify this is to compute an embedding of the
terexample
fiber of the momentum map in "
By
13
In
some
projective
space
PI; this
can
be done
a coun-
general explicitly by using
linear vector field on an Abelian group G we mean a G-invariant vector field. partial compactification only the fibers are compactified, not the base space. 14 In our case the Abelian varieties will appear without origin, i.e., as homogeneous spaces and one cannot assume the existence of a section over B which picks an origin on each of these Abelian varieties. Then the construction is modified by picking the fibred cube instead of the fibred square, using a similar equivalence relation. It is given that we can complete each level separately. a
a
131
Chapter
V. A.c.i. systems
the Laurent solutions to the differential equations describing one of the integrable vector fields will explain in Section 3. In all examples we have seen the embedding of Fb
=,7r. '(b)
as we
(b
B, regular E
Zariski open subset of Spec A) in projective space is given by functions which are on M and independent of the chosen fiber, i.e., there exist fi,..., fN E O(M) such a
that the map
0:
M _+ pN
given by P -+
(1: h(p):
fN(P))
has the
an
closure of its
this
following property: for any b G B the restriction of 0 to -Fb is image is an Abelian group (isomorphic to ir-'(b)). In have an embedding -7rAl (B) _+ pN x B given by
((1 -, h (p)
P -+
:
*
*
*
:
fN (P))
,
embedding and the case we obviously
IrA (P))
(7r. ' (13)) is the desired paxtial compactification. The same construction applies when the functions fl,..., fN depend regularly on B, i.e., if fi G O(B x M) for i and the closure of
N.
aom
now on we
will almost
ducible a.c.i. systems, defined
an
Example
in Section
An a.c.i. system is called irreducible if the an irreducible Abelian variety.
Definition 2.2 map is
exclusively (i.e., except
VII.4.5)
deal with irre-
follows.
as
general fiber of the
momentum
affine part of 2.3
If one of the vector fields of
an a.c.i. system is super-integrable then the a.c.i. Namely suppose that its general level sets are Abelian varieties, then the flow of the super-integrable vector field, being lineax on the Abelian variety on the one hand and being contained in a subvariety of lower dimension on the other hand, must evolve on an Abelian subvariety, hence the general fiber of the momentum map would not be an irreducible Abelian variety. Whether an a.c.i. system which is not irreducible admits a super-integrable vector field is not known.
not irreducible.
system is
Example In this
2.4
case
vector field
product of two a.c.i. systems is an a.c.i. system which is never irreducible. integrable vector field of the original systems leads to a super-integrable the product. The
every on
On each fiber
ir. '(b)
of
an
irreducible a.c.i. system the divisor A induces
a
polariza-
tiOn C-1 ([Dbl) since any effective divisor on an irreducible Abelian variety is ample. Thus one may think of the general fiber of the momentum map of an irreducible a.c.i. system as being
polarized Abelian variety. If the a.c.i. system is moreover completable then the polarization type of this general level set is constant since it is discrete; probably the assumption that the system is completable is superfluous here but we don't have a proof of this. In any case, if the polarization type of the general level sets of an irreducible a.c.i. system is constant then we call it the (polarization) type of the a-c-i. system. a
If a divisor is removed from a (completable) a.c.i. system as in Proposition 11.2.35 then resulting integrable Hamiltonian system is also a (completable) a.c.i. system. There are two very distinct possibilities.
the
-
If the function
f
whose
zero
divisor is removed
left out, the others remain intact. Of general, so we still have an a.c.i. system; are
132
course
belongs
to A then
some
the
which
left out
ones
are
level sets are
not
2. A.c.i.
systems
not belong to A its zero divisor cuts off from every level set a divisor (a change), of course the general level set is still an affine part of an Abelian variety (or Abelian group in general).
If
-
f does
dramatic
a.c.i. system which depends on parameters as in Proposition 11.3.24 we easily see big integrable Hamiltonian system which is given by the latter proposition is also a.c.i.; clearly it contains as its level sets aJ1 level sets of all the a.c.i. systems obtained by freezing the parameters. For
an
that the
Some
care
has to be taken when
restricting
a
(completable)
By Proposition
11.3.19
Less obvious is the useful
systems
as
given
in the
one
gets
on a
general level
a.c.i. system to
a
level set of
integrable Hamiltonian system; in order for it to be a (completable) a.c.i. system one has to verify in addition that the general level set of this restriction is contained in the collection of general fibers (which are known to be affine parts of Abelian groups) of the original system. For special level sets of the Casimirs we may not even get an integrable Hamiltonian system. the Casimirs.
set
an
property that the quotient construction also leads
to a.c.i.
following proposition 16.
Proposition 2.5 Let (M, 1-, -1) be a Poisson variety with a Poisson action by a finite group G. If (M, 1-, -1, A) is a (completable) a.c.i. system with an affine part of an Abelian variety as its general level set and for each g E G one has g A C A then (MIG, J.'.10' AG) is a (completable) a.c.i. system (with an affine part of an Abelian variety as its general level set) and the quotient map 7r is a morphism (I., jo is the quotient bracket on MIG). -
Proof
Proposition 11.3.25 it suffices to identify the general fiber of the momentum being isomorphic to an affine part of an Abelian variety. Clearly the action descends to an action on Spec A, (denoted in the same way) namely for each g Ei G one has the following commutative diagram. In view of
map
as
M
M
I
I
WA
Spec A
Since
A)
(M,
whose fibers
finite,
are
is a-c-i. there is
a
-
WA
Spec A
91
Zaxiski open subset B C Spec A and a bundle -7r : T -+ B compactify the fibers of 7rA over B. Since G is
Abelian varieties which
B may be assumed stable for the action of G; also we may assume that the action is effective.
by passing
to
a
smaller group if
necessary
Since each g- maps level sets (of 7rA) to level sets and a general point. invertible, each g- restricts to an isomorphism 7r '(c) -+ ir' '(g b) which after composing with Ob and 09-b leads to an isomorphism ir-'(b) \Vb -+ *7r-' (9 b) \Vg-b of the affine parts of the corresponding Abelian vaxieties; since g- is a Poisson map and Ob and 09-b linearize Let b E B be
since all g-
-
are
,
16
generated by a quasi-automorphism. (of filinite action the Z2 generated by a (-1)-involution then the example order). general level set of the quotient is an affine part of the Kummer variety of the Abelian variety; obviously also the vector fields do not descend to the quotient. The
proposition is
If
one
not valid if the action is
considers for
133
Chapter V. A.c.i. systems
the Hamiltonian vector fields, each g. extends to an isomorphism g. : ir-1(b) -+ -7r-'(g b) upshot is that the action of G on M induces an action on T (such -
hence extends to T. The
that g.
-
0
T
is
equivariant).
As
we
have
T restricts to each fiber
-+
as a
seen
in
group
Paragraph IV.3.1 this implies that each homomorphism followed by a translation.
map
At first suppose that for any g E G the level set of ir over any closed point is mapped ir then we easily pass to the quotient, giving a bundle TIG -+ BIG
to another level set of
whose fibers
are isomorphic to the original fibers of 7r : T -+ B, hence are Abelian varieties done. If for any g E G the level set of 7r over a general point is mapped to another level set of ir then we may replace B and T by Zariski open subsets which put us in the
and
we axe
previous situation leading again to an a.c.i. system. In this case, is completable then obviously the quotient is also completable.
if the
original
a.c.i. system
The situation is very different when the general fiber is stable for the action of G, because case we get non-trivial quotients of these level sets. In this case all points of SpecA
in this axe
fixed
(for
the induced
action),
hence all fibers
are
stable and
4
C
O(M)G.
Since for
morphism g. is Poisson, all vector fields Xf, f E A are preserved by the action; by linearity of these vector fields the ar-tion of ear-h g E G on the level sets of ir is by translation over an integral part of a period. The quotient of an Abelian variety by a Enite group of translations is again an Abelian variety and we may form the quotient of T by the action of G, obtaining a new bundle TIG of Abelian varieties over B. Thus in this case the quotient is also a.c.i., but the Abelian varieties which appeax in this quotient system are not isomorphic but merely isogeneous to the Abelian varieties which appear in the original a.c.i. system (in particular the quotient system will most often have a different polarization type). Again completability of the a.c.i. system (if present) is not lost. Notice also that in this last case the group G acts as a group of translations, hence G is commutative if its action is any g E G the
effective. We
are
left with the
case
in which
some
elements of G map the level sets
over
every
(or
closed point to another level set, while some other elements fix all these level sets. In this case one may take the quotient in two steps, since the subgroup of G which 0 corresponds to the latter elements is a normal subgroup of G. the
generic)
The following converse of the above proposition is not true. Let (M, be a Poisvariety with a Poisson action by a finite group G and suppose that (M, -1,A) is an integrable Hamiltonian system such that for each g E G one has g -A C A. If the quotient (MIG, 1.'.10' AG) (which we know to be an integrable Hamiltonian system) happens to be an a.c.i. system, then it does not follow that (M, 1-, -1, A) is itself a.c.i. See Paragraph VII.6.2 for a counterexample. son
Notice that in the first
case
treated in the
proof of the proposition the
map g-
:
B -+ B
the moduli space of Abelian varieties which appear as level sets in the a.c.i. system. The quotient BIG is an intermediate object between B and this moduli space. is
a
covering
map
over
134
3. Painlev,6
3. PalnleW
analysis for a.c.i. systems
analysis for ax.i. systems
The differential equations describing an integrable vector field of an irreducible a.c.i. possess families of Laurent solutions (see [AM7]). In slightly different terms this was
system
already known
to
Kowalevski;
her
original idea
was
taken up and extended
by Adler and
van
Moerbeke to give necessary and sufficient conditions for algebraic complete integrability in terms of Laurent solutions (see [AM7]). We restrict ourselves here to part of their result17.
Proposition 3.1 vector field XH, H
Let E
(M, 1-, -1, A)
be
irreducible a.c.i. system. Then
an
for
.4 the space of Laurent solutions has dimension dim M
any 1.
integrable
-
Proof Pick a general fiber F of the momentum map -7r : M -+ Spec.4. By assumption F isomorphic to T \ D where T is an Abelian variety and D is a divisor on T; we denote this isomorphism by 0. Let Z be any point on 7- and let us choose a system of generators zm of 0 (M); upon restriction to F they lead to a system of generators of 0 (.F), which zi, we still denote by zi. The functions zi o 0-' provide a system of generators of 0 (T\ V) and in terms of these the differential equations for the vector fields 0,,XH (H E A) axe linear, hence if Z E 7- \ D then one finds a solution for zi o 0', hence also for zi, which is holomorphic in t. It is just the description of the integral curve of the vector field XH starting from the point 0-'(Z) E M. Of course the dimension of the space of such solutions is dim M, since we have precisely one solution for every point of M.
is
.
.
.
,
Next, suppose that Z E D is such that one or several of the functions zi o 0-' have a pole at Z; notice that the functions zi o 0-' which are regular on T \ D uniquely extend to meromorphic functions which have their poles on at least one irreducible component of V. The divisor of e.g. z,
o
0-'
can
be written
(uniquely)
k
I
niVi
Di'
one or more
we
to
axe
miEX
-
(mi, ni
E
N
\ 10}),
different and irreducible. If z, o 0-' is not holomorphic around Z Di, but it may belong as well to some of the Di. In any case, if
where all Di and then Z
belongs
as
pick for each irreducible component of D
a
local
function around Z, say
defining
fi
for Di
and gi for Di' (if Z does not belong to some divisor then the local defining function may be taken as the constant function 1), then z, o 0-1 is written around Z as
Z'
0
fnl fn2 1 2
f
M1
91
with
f holomorphic
around Z and
f (Z) 9
...
M2
92
...
fknk gMJ 1
I
0.
for the torus, and think of the local We may take linear coordinates x, ti X21 7 Xn defining functions as being expressed in terms of these. The t-axis cannot be contained in ...
any of the divisors 17
Di
or
Di'
since otherwise the
general fiber would contain
a
subtorus, i.e.,
talking in this section about the solutions to some differential equations (in integral curves of a vector field) we will talk here about holomorphic and meromorphic functions rather than regular and rational maps. Since
we are
other words the
135
Chapter
V. A.c.i. systems
reducible, contrary to our assumptions. It follows that all these functions can (again up nonvanishing holomorphic function) be written as a (Weierstrass) polynomial in t (by the Weierstrass Preparation Theorem) and we see that the zero or pole z, has in Z (as a function of t) depends on the components of the divisor of Z, to which Z belongs but also on the singularity these divisors have in Z (since then the first few terms in the series vanish) and on the contact the vector field XH has with these divisors (for the same reason). be to
a
Proceeding in this way for all functions zi we find a Laurent solution to the differential equations, which staxts from Z. Since Z is an arbitrary point of a divisor of the general fiber, the space of Z from which there staxts
a
Laurent solution is of dimension dim M
-
1.
Notice that this divisor is contained in but is not
necessarily equal to the divisor which needs to be adjoined to the level in order to complete it into an Abelian variety. If there is for every component of D at least one of the functions zi o 0-' which has a pole on this component, then a Laurent solution stafts from an arbitraxy point of D and we have an exact bijection of the space of Laurent solutions and the points to be added to the general level sets in order to complete them into Abelian varieties. This is the case in all examples that we
will consider.
an
intersection of
0
organize themselves naturally
The Laurent series some
divisors
(contained
in families
in the divisor of
as
poles
follows- for every zi, fix of (zi)), fix an order of
singularity and an order of tangency of the vector field. On this set all zi are written as Laurent series depending on a number of free paxameters, equal to the dimension of this set (corresponding to the staxting point of the series which can be chosen in it) and in a dense subset the order of pole each expansion experiences is fixed. The pole may however become less severe in an analytic subset, obtained from the intersection with one of the divisors on which zi has a zero; in such a case the leading coefficient of the Laurent series must be (dependent on) a free parameter, so that it can in paxticular take the value 0. Thus, there is always at least one family of Laurent solutions which depends on dim M 1 free parameters (called a principal balance). If there is for every component of D at least one of the functions zi o 0-1 which has a pole on this component, then there are as many principal balances as irreducible components in D. There axe also always families of Laurent solutions which depend on fewer free parameters, known as lower balances. -
The geometric study of the Laurent solutions of one of the integrable vector integrable Hamiltonian system is called the Painlev6 analysis of this system. The different sets which
Remark 3.2
correspond
fields of
to the different balances do not
an
give
a
stratification of the Abelian variety in general; indeed, if, for example, z, and Z2 both have a pole on some smooth divisor and the intersection of these divisors is singulax, then the
singular locus of this intersection will in general not show up as a sepaxate family of Laurent solutions, leading to a singulax stratum. An example where the Laurent solutions do lead to a (family of) stratification(s) of all hyperelliptic Jacobians will be given in Paragraph VI.4.2.
the
Finding all Laurent following problems.
1)
It is not clear
solutions in
by looking
in order to find
2)
For
a
braic
a
a
direct way is in
at the differential
general a difficult task. One encounters
equations with which exponents
to start
(all) solution(s);
given choice of exponents one needs to solve a nonlinear system of algeequations (called the indicial equations) for the leading term, which may be
136
3. Painlev6
difficult, especially graph VI.4.2);
very
3)
when the number of variables is indefinite
a
matrix, depending
the number of
leading terms, but this is again very matrix, is indefinite; convergence of all Laurent solutions;
on
these
variables, hence
One also has to show the
e.g., Para-
difficult when
the size of the
figure out how the different sets the different families of Laurent correspond to are related (see [AM7]).
It is not obvious to
solutions The main
use
that
as
analysis in the remaining chapters is not to algebraic complete integrability (see [AM7]), embeddings of the Abelian varieties whose affine parts
will make of Painlev6
we
detect a.c.i. systems (see e.g. [Hai2]) but to construct explicit projective appear
(see
The presence of free parameters (giving information about the dimension of the corresponding subset) can in favourable cases be detected by computing the eigenvalues
of
4) 5)
for a.c.1. systems
analysis
or
to prove
the fibers of the momentum map.
Proposition3.3 Let T bean Abelian variety and letV some divisor such thatAr= T\V is an affine variety; consider also a linear vector field Y on T. If for every irreducible component of D the space of Laurent solutions, which corresponds to a general point of it, is explicitly known (in the form of the first few terms), then an explicit embedding of T in projective space can be computed concretely from it. Proof
over
ing
We know that any irreducible component Di of D is ample since T is irreducible; more3Di is very ample and we may construct an embedding of the Abelian variety by us3Di. For any irreducible component of D these Laurent solutions express a system of
generators
zi,
.
.
.
,
(restricted to Ar) as Laurent series in t, hence can be used O(M) (restricted to JV) in terms of t. What is important here is
z,, of 0 (M)
to express any element of
pole which the Laurent solution of a function zi on JV has in t coincides with the pole the extension of f to T has on the divisor which corresponds to the Laurent solution; this follows as above by writing zi (or zi o 0-1) as a quotient of holomorphic functions which in turn are expressed in terms of Weierstrass polynomials, but now this is done at a general point of the divisor. This allows to look for elements of O(M) which lead to independent meromorphic functions on T which have a certain pole at D' but axe holomorphic on the other divisors in D. This is a finite proces: we can first look for functions of degree one, then of degree two and so on and by Formula (IV-3.1) we know when we have found a complete 0 set of (independent) functions and these provide the embedding. that the
Amplification
3.4
One
can
also construct
an
embedding by using
some
combination of
the different components in the divisor, taking these components only with multiplicity one It is easy to figure out which set of multiplicities suffices when the N6ron-Severi or two. group of the Abelian variety is trivial (i.e., is equal to Z). Notice that since this is the for a generic Abelian variety this is a mild assumption. Then every component Vi is algebraically equivalent to a multiple of some fixed divisor and it suffices that, under this algebraic equivalence, the divisor which one picks to construct the embedding is equivalent to (at least) three times this divisor. One fixes such a choice of embedding divisor D' and determines as before the Laurent solutions which correspond to the irreducible components which belong to this divisor as well as the Laurent solutions which correspond to the remaining components in D; from it a concrete embedding can be constructed.
case
137
Chapter
V. Ax.i. systems
4. The linearization of two-dimensional a.c.i.
systems
Let (M, 1-, -1, A) be a two-dimensional a.c.i. system and let us denote for any c E Spec A the fiber of the momentum map over c by F,. If c is general then T, completes into an Abelian surface by adding one or several (possibly singular) curves. The following algorithm,
proposed in [Van2] leads to an explicit linearization (i.e., integration) of any integrable field H, H E .4. (steps (1) and (2) are due to Adler and van Moerbeke, see [AM9]).
(1) Compute and
the first few terms of the Laurent solutions to the differential
these to construct
use
vector
an
embedding of the general fiber.F,
in
equations, projective space
(Proposition 3.3).
(2)
Deduce from the
embedding the structure of the divisors V, to be adjoined to T, complete F,, into an Abelian surface 7,,. At this point the type of polarization induced by each irreducible component of D, can also be determined in order to
(see Section IV.3.3). (3) a) If one of the components image of the rational
of D, is
smooth
a
2
0[2r,. which is
a
curve
r, of genus two, compute the
map
singular surface
in
P3,
p3
_+
the Kummer surface IC, of
Jac(r,).
b) Otherwise, if one of the components of D, is a d : I unramified cover Cr. of a smooth curve IP, of genus two, p : C,, -+ ]U, the map p extends to a map jac(r. In this case, let 8r, denote the (non-complete) linear system p*12]P,l C 12C.1 which corresponds to the complete linear system 12r,J and compute now the Kummer surface Kc of Jac(]P,) as the image of oe'c c) Otherwise, change always be done for
(4)
Choose
a
case
infinity so
0[2r,,] (W)
=
on
(0
:
0
the :
to arrive in
as
case
a)
or
IP, and coordinates (zo
curve
(3) a) and Oe.JW) a singular point (node) for IC,
0
Then this point will be
(3) b).
p3.
_+
b).
This
can
any irreducible Abelian surface.
Weierstrass point W
for p3 such that in
the divisor at
V
:
1)
:
in
case
: =
z,
(0:
:
Z2
:
Z3) 1)
0: 0:
and IC, has
an
equation 2
P2(Z0iZ15Z2)Z3
P&O,ZI Z2)
+ P3(Z07ZI7Z2)Z3 +
of
where the pi are polynomials degree (0 : 0 : 0 : 1) we may assume that
i. After
a
=
0)
projective transformation which
fixes
P 2 ( Zo 'ZI, Z2 )
(5) Finally,
2 =
let x, and X2 be the roots of the
ZI
-
4zoz2.
quadratic equation
OX2
_,
+ _,IX + Z2
=
0,
whose discriminant is P2 (ZO ZI i Z2) , with the zi expressed in terms of the original Then the differential equations describing the vector field XH variables qI, , q4. .
are
rewritten
.
.
by direct computation
in the classical Weierstrass form
dX2
dxl -
+
A/f(XI) xldxl +
VY XIJ 138
= -
Vff (X2) -X2dX2
*V7f(X2)
aldt, a2dt,
4. The linearization of two-dimensional a.c.i.
systems
where a, and a2 depend on c (i.e., on the torus) only. From it, the symmetric functions X1 + X2 (--'z -zl/zo) and X1X2 (:-- Z2/ZO) and hence also all functions in O(M) can be written in terms of the Riemann theta function associated to the curve
y2
=
f(X).
(in Section VII.5) on a non-trivial example that this algorithm is very effective and easy to apply. Other worked-out examples can be found in [Van2]. We wish to make the remark that it is also shown in [Van2] how a Lax representation and action-angle We will show below
variables for the system derive from the above linearization.
139
V. Ax.i. systems
Chapter
5. Lax
equations
An interesting way to construct integrable Hamiltonian systems is by means of Lax equations. In many cases they turn out to be even a.c.i. We give a sketch of how this works in the
case
of finite dimensional
integrable
Hamiltonian systems.
The literature
on
equations is immense, original references are [Lax], [Flal], [Fla2], [AM2], [AM3], [RSI] and [RS2]; the approach we describe here is due to Semenov-Tian-Shansky (see [Sem]). We also wish to note here Garnier's paper [Gar], since it is the first paper (as early as 1919) in which Lax equations (with spectral parameter!) were written down. Lax
Let g be
a
Lie
algebra with
Lie bracket
[-, -]
and R
an
endomorphism
of Z.
The
new
bracket
1QRX, Y] + [X, RYI)
[Xi YJR satisfies the Jacobi
identity (hence
[BR(X, Y), Z]
2
defines another Lie bracket +
[Bp,(Y, Z), X]
g)
on
[BR(Z, X), Y1
+
if and
only
if
(5.1)
0,
=
with
BR (X, Y)
=
[RX, RY]
-
R
([RX, YJ
+
[X, RY])
If so, then R is said to define a structure of a double Lie algebra solutions to (5. 1) is found by looking for solutions to BR (X7 Y)
on
=
[RX, RYJ the so-called classical to the
modified
=
R
QRX, Y]
.
Yang-Baxter equation; more general solutions Yang-Baxter equation
is
a
constant. =
are
obtained via solutions
-c[X, Y],
=
(5.2)
If for
+ g- then the
=
R
as
paxticular class of
example 9 is a direct sum (as a vector space) of two Lie corresponding projection operators P+ : 0 -+ 0+ and ig+ 1), by taking -+,g- lead to a solution of this modified Yang-Baxter equation (for c c
subalgebras, ig P_
[X, RY])
g. A
0, i.e.,
classical
BR(X, Y) where
+
.
is easy to form
verify.
Notice that in this
P+
=
case
-
P_,
the formula for the R-bracket takes the
(5.3) following
simple
[X1 YIR
=
1P+X1 P+y]
-
1P_X' P_Y].
(5.4)
Having two Lie structures on g we also have two Lie-Poisson structures on g*, denoted by 1., -1 and I-, -JR. Then the relevance of double Lie algebras for integrable Hamiltonian systems comes from the following proposition: Proposition
5.1
Cas({-, -1)
is involutive
for
the R-bracket
Proof
Recall from
Example
11.2.8 the
explicit formula
If, gl(O
df-( ) Wg co ,
140
5. Lax
for the Lie-Poisson bracket
f Therefore,
if
f
,
E
(f
,
g E 0 (9*),
Cas 0 (g*)
V
= -
equations
E
E
g*).
9*,
From
it, it follows that
( 1&R(), x] )
Vx E 9
=
0-
g E Cas 0 (g*) then
If, 91R( ) showing that they
are
=
6
( [RdfT6), dg- 6)1)
+
6
( [jfT6), RcFg 6)1 )
=
0,
in involution with respect to the R-bracket.
algebra generated by Cas(j-, J) and Cas(j-, JR) is a good candidate of being in order for Cas(I., J) to be big enough to imply integrability one often has to restrict the phase space to a Poisson subvariety. For many choices of g and R one finds indeed interesting integrable Hamiltonian systems in this way; for example if Z is a simple Lie algebra then the root space decomposition of g leads to a natural choice for R similar to (5.3) and one finds a large family of integrable Hamiltonian systems, the generalized Thus the
integrable
on
(ig*, I-, JR);
Toda lattices. If g admits a nondegenerate, invariant bilinear form then the identified with 9 and the differential equations which describe the
Xf
=
I-, f}R
on
g
(for f
E
Cas(j-, -1))
have the
following peculiar form 1
Xf L
=
[L, Mf]
where
phase space Z* can be integrable vector fields
Mj
-
2R(df (L)).
(5.5)
Such an equation is called a Lax equation and an integrable Hamiltonian system on a Poisson subspace of the dual of a double Lie algebra will be said to be in Lax form or to be of Lax type (to be distinguished from Lax representations, defined below). The determination of the integral curves of (5.5) can be reduced to the Riemann problem (see [RS1]). One may however also consider the loop algebra jg[X, A-'] of a Lie algebra g, which inherits a Lie structure from g and has a natural decomposition =
9[/\]
ED
A-I
a nondegenerate, invariant bilinequations (5.5) retain their form but L and Mf are now dependent on A; this A is called a spectral parameter and the Lax equations Of course the loop algebra is are also said to be Lax equations with a spectral parameter. infinite dimensional but in concrete examples a finite-dimensional Poisson subspace is usually considered. An example will be discussed in detail in Section VI.3.
with ear
corresponding R-bracket given by (5-3). It
form if g has
one.
In this
case
also has
the above Lax
Integrable Hamiltonian systems of Lax type for which the Lie algebra is a loop algebra Still assuming that g has a nondegenerate, invariant bilinear -1 when form, we may identify Z* with g and the spectral invariants of 0 are Casimirs of and the viewed as functions on g*. Applied to the case of a loop algebra, L(,\) E characteristic polynomial det(L(,X) zId) defines the affine part of an algebraic curve rL(,\) in C x C*, namely the curve
often turn out to be a.c.1.
-
rL(A) : det(L(A) 141
-
zId)
=
0
Chapter V. A.c.i. systems which is called the curve
and
to it is
L(A)
spectral
moves
constant; what
curve.
Thus there is associated to each matrix
under the flow of
Xf
in such
a
way that the
curve
L(A)
an
algebraic
which is associated
under this flow is
a line bundle of degree 0 on this spectral curve, eigenvectors of L(A) (for the precise construction, see [Gri]) and thus the flow of L(,\) can be seen as a flow on Pico (rL(A)) i.e., on the Jacobian of rL(,\) (more precisely on the Jacobian of the Riemann surface obtained by compactifying L(,) over 0 and oo). Linearity of this flow is not guaxanteed, although for many examples of interest this vector field is linear and the above eigenvector map is said to linearize the integrable Hamiltonian system; the Lax equations constructed in Paragraph 111.2.4 for example do not have this property except in the special case considered in Paragraph VI.4.2. This can for example be checked by the necessary and sufficient conditions for the eigenvector map of a vector field (which is defined by a Lax equation) to linearize, which is given in [Gri].
moves
which is constructed from the
,
The in this
name
way).
be another
above).
Lax
Let
equation is also often used
(M, 1-, .11, A,)
integrable
Then
a
be
in the
following
sense
(but
it is not
explained
integrable
Hamiltonian system and (N, 1-, *12, A2) Hamiltonian system which is assumed to be of Lax type (as defined an
finite
resentation of the
morphism 0 : (M, 1-, -11, A,) -+ (N, J* ij2i A2) is called a Lax repintegrable Hamiltonian system (M, 1., -11, A,.). Notice that even if the
integrable Hamiltonian system on N is a.c.i. the one on M needs not be, moreover the Lax equations (5.5) lead to similar equations for (some of) the integrable vector fields on M, namely we may compose the Lax equation (5.5) with 0 to obtain
Xf L 0 o
=
[L
o
which represents the fields Xg with g E
0, Mj
o
0],
(5.6)
integrable vector field X4,.f on M. Notice that not all integrable vector A, axe of this form since O*A2 = 6 A, (in general). Also the vector fields (5.6) are not vector fields on an affine subspace of the dual of a Lie algebra. Special caxe has to be taken when determining information about the level sets of A,, which axe covers of (Zariski open subsets of) the level sets of A2; even when the degree is one, these need not be isomorphic.
142
P. Vanhaecke: LNM 1638, pp. 143 - 173, 1996, 2001 © Springer-Verlag Berlin Heidelberg 1996, 2001
Chapter In
Paragraph
4.3
we
VI. The Mumford systems
will exhibit
some
interesting features of the
even
and the odd
Mumford systems. Namely we will show how Painlev6 analysis leads to a stratification of the Jacobians which appear as the fibers of the momentum map; notice that the stratification which is obtained in the even case is very different from the one which is induced in the odd
case.
These stratifications
Grassmannian via
(an
axe
extension
also obtained from the natural stratification of the Sato the Krichever map, but this will not be discussed here
of)
(see [VanQ(Section 5) we show how to construct for any smooth curve in C2 Hamiltonian system which has a level set (of the momentwn map) which is an integrable isomorphic, to an affine part of the Jacobian of this curve. Moreover, the restriction of the integrable vector fields to this level are linearized by this isomorphism. For a large and In the final section
a.c.i. system. The construction which we Chapter III, in particular the construction is also completely explicit. A generalization, which is not a.c.i. was given in Chapter III and is further generalized in [Van5] to arbitrary families of curves. For a generalization of the construction in Section 2 to matrix differential operators, see [KV]. A variant of the (even) Mumford system in which the polynomials have parities (odd/even) has recently be most
give
important class of
here is
a
constructed in
curves
this leads to
modification of the
one
given
an
in
[FV].
144
2. Genesis
2. Genesis
2.1. The
algebra
of
pseudo-differential operators
We start by reviewing the basic definitions and properties of (formal) pseudo-differential operators. Let D denote the non-commutative algebra C[[x]][i9] of differential operators, the
multiplication being given by juxtaposition and applying the commutation rule [19, a(x)] a'(x); here a(x) is any formal power series, a(X) E C[[x]], and a'(x) its (formal) derivative. D has as a distinguished maximal commutative subalgebra the algebra D' C [a] of constant coefficient differential operators. D is contained in the larger algebra T C[[x]](((9-1)) of pseudo-differential operators, the (associative) rule of multiplication being formally derived from the commutation rule [,9, a(x)] a'(x), i.e., =
=
=
=
00
,9-'a(x)
=
i=O
An element
order
(of
q)
Q
C-
EiL
T, Q
z_=-.
aii9',
is said to have order q if aq
when aq
=A
0 and is said to be monic
0 then Q is called normalized (of order q). 1; if, in addition, ag-1 The subalgebra C((a-')) of T consisting of constant coefficient pseudo-differential operators is denoted by T1. The following properties of differential operators are easily established. =
Proposition 2.1 (1) Every monic pseudo- differential operator Q of order q has a unique inverse Q` in T. In particular the monic pseudo- differential operators of order zero form a group, called the Volterra group and denoted by Volt. Its subgroup of constant coefficient operators is denoted by Volt'. (2) Every normalized differential operator Q of order q > 0 has a unique monic q-th root
Q11q
(3) Every Q
=
in T. This root
normalized
T` 9qT for
element
Q11q
is normalized and has order 1.
differential operator Q of some
T E Volt which is
order q > 0 is
unique
up to
conjugated to a q, i. e left multiplication by an .,
of Volt'.
Finally we recall the definition of the Sato Grassmannian Gr. Let us denote by 6 Dirac's delta function, thought of as a zeroth order differential operator. It has the fundamental property that for any Q E T there exists a unique Q' E T' such that QJ Q'6. The left =
coset
C((,9-1))d
V:= is P
a -
left T-module in
(Q6)
=
(PQ)6.
A
a
natural way:
The
multiplication
allows
us
'F-6
for P E T and for
distinguished subspace H
=
=
of V is defined
C[a]6
=
Q by
C((,9-1))
E
c
T
we
define
D6.
to associate to each element in XF
a
subspace of
C((,9-'))6
in
a
natural way, namely given Q E IF define WQ C C ((a-')) 6 by WQ Q H. We call the set of all WT which correspond to elements T in the Volterra group the Sato Grassmannian, =
Gr:=
JWT I
T E
145
VOltJ
-
Chapter V1. The Mumford systems If T E Volt the linear space WT has a basis, similar to the standard basis of has a basis whose elements have the form T
a'
-
=
(0"
0(&-1))6,
+
(i
>
H, namely
it
(2.1)
0).
It follows that the map Volt -+ Gr given by P i-+ Wp is injective hence bijective. The following proposition gives a useful characterization of differential operators in terms of this
(see Nul3]).
map
Since
no
Let
2.2
Proposition
Q
confusion is
Then
E T.
possible
Q
E V
we remove
if and only if WQ
C H.
6 from the elements of
WQ, i.e.,
we
identify V
C((a-1)).
with
2.2. The matrix associated to two In this section
[P, Q]
=
we assume
0. We will
Q are monic differential operators which commute, them, say Q, is normalized. Finally we assume that positive and coprime. In this paragraph we show how to
that P and
that
assume
one
of
the orders p and q of P and Q axe associate a unique element M E ]g I, [A] to reconstruct
(P, Q)
commuting operators
(P, Q);
it will be shown in the next section how to
from M.
associate to the pair (P, Q) a pair (P, W) of elements P E Te and unique up to an equivalence specified below). Since Q is normalized T-1,917T. Choosing such there exists by Proposition (2.1) an element T E Volt such that Q P TPT-1 then 15 E T is monic let If H Gr. T we G W define T element WT we an 0. We claim that the latter implies that P E Tc. To show this it is of order p and [&, P] a
first step
we
W E Gr
(this pair
is
In
=
=
=
=
-
=
sufficient to show that if
a(x)
E
C[[xl]
is such that
[aq a(x)]
=
,
0 for
some
q > 0 then
a(x)
is
constant. Since
0
we
=
find indeed that
[Igq a(x)]
=
,
a(x)
=
aq a(x)
T'
-
a(x) aq
qa'(x)a7-'
=
+
Opq-2),
0.
pair (P, W) cleaxly depends Volt' then (P, W) is replaced by
The Te E
-
(15, W)
=
on
the choice of T:
(TP(T')-1, T' W) -
if T is
T'
-
replaced by
T'T where
W),
since constant coefficient differential operators commute. The above equation defines an action of Volt' on T' x Gr; we say that two elements in T1 x Gr are equivalent when they
correspond under this we use
action. We
it to show that the
Proposition
2.3
now
pair (P, Q)
give can
a
characterizing property of the pair
(P, W)
and
be reconstructed from it.
The element W E Gr is stable under the action
of & and
W,
P-WCW.
146
(2.2)
2. Genesis
Conversely, and
Q
(P, W) E T- x Gr satisfies (2.2), P being monic. The pair (P, W) is unique pair (P, Q) of commuting differential operators, such that P is monic
suppose that
associated to
a
is normalized.
Proof
(2.2)
The verification of
is easy, for
aq in which the inclusion
(P, Q)
from
(P, W).
Q
W
-
=
T
example -
(Q H)
C T
-
H C H holds because
-
Q
-
H
E V.
Since W E Gr it is of the form W
=
W,
=
Let
T
-
show how to reconstruct
us
H for
a
unique
T E Volt. If
we
define
Q
=
T-1 aqT)
P
=
T-'PT,
0. The crucial Q is normalized of order q while P is monic of order p; also [P, Q] property is that (2.2) implies that P and Q are differential operators. We have that WQ c H and Wp C H because e.g. for P one computes
then
=
Wp
=
P
H
-
=
T-1
-
(P W)
c
-
T-1 W -
H.
=
Using this, the fact that P and Q are differential operators follows from Lemma (2.2). Clearly, if we replace (P, W) by an equivalent pair then the same pair (P, Q) is obtained. 0 Notice that in the above
proposition the orders of P and Q need
The next step is to associate to the pair from which
(j5, W)
(-P, W) (up
be reconstructed. Since
can
P
to
not be
equivalence)
E Te and
P
-
coprime. a
matrix M E 0
[q [A]
W c W it follows that
P
endomorphism of W, hence can be represented by a semi-infinite matrix (with entries in C) by choosing any basis of W. This matrix becomes a periodic matrix (with period q) when a periodic basis (EO, El, ) is chosen, i.e., a basis such that for any basis element Ej the element &E, is also a basis element. Note that the existence of a periodic basis follows is
an
.
.
.
,
from the inclusion &W C W. We choose q vectors in W of the form
Ej and extend them to Periodic matrices are
polynomials
a
04
basis of W
can
in 'X
=
+
aq
.
as
Explicitly,
(0 for
< i < q
-
1),
arbitrary i
> 0, the vectors Ei+q &Eiprice of allowing entries which define the matrix'8 M [A] by (mij) Ei
by introducing,
be rewritten
=
0(ai-1),
=
square matrices at the
if
we
=
q-1 -
P
-
Ej
=
E MijEj,
(0
< i < q
-
1),
(2.3)
j=0
The
rows
and columns of the matrix M and of the matrices eij, introduced
labelled from 0 to q
-
1.
147
below,
are
VI. The Mumford systems
Chapter then the elements of M have the
:5
<
<
[p1q]
I
(2.4)
[p1q]
<
+ 1
-
[p1q]
[p1q]
+ I
[p1q]
[p1q]
[p1q]
[p1q]
[p1q]
constraints:
following degree
[p1q]
+ I
since
P
-
+ lower order in 19
Ev+i
Ei
-[
P+i Q
'E(p+i) mod
+ lower order in a.
q
implicit in (2.4) that when the degree is exact (no inequality signs) then the corresponding polynomial is monic. We denote the affine space of matrices of the form (2.4) by MV,q Let N - c GL(q) denote the subgroup of lower triangular matrices, which acts on Mp'q by conjugation. It is
.
Proposition of MMIN -.
The above
2.4
procedure associates
to the
pair (P, Q)
a
well-defined
element
Proof Two bases of the form
(2.1)
related
axe
by
an
element of
hence their matrices
Nq-
axe
replaced by any other representative T' (fl, W) then N --conjugated. precisely the same matrix M is obtained when using the basis of T' W which is obtained by multiplying all elements of the basis of W by T'. Notice that such a basis is aJways of the When
(P, W)
is
-
-
form
(2.1).
1
As it turns out the quotient space
subspace f4P,q of MPq. To by (2.4). For 0 < i, j :! q and
show this
is in
Mp,qlN -
we
-
generated by the matrices eij for which j i The projection 0 Iq -+ &, will be denoted by IIy. be the elements Of 91q, given by -
S=
Let d
( It7-Od 0)'
matrices
deg S
=
F[Plql+l M.Ai _i=O
%
-d and that
=
R=
Z E
=
q
M[plql+l
E
S+
R
+ED ,y
148
&Y
we
also write
deg Z
0
d. With this notation Mp,q consists of all
-
-y<-d E
affine
(0 1d). 0
deg R
an
notation which is motivated
p mod q, 0 < d < q, and let S and R
for which
M[,I,]
some
,
Wien
0
We have that
natural way isomorphic to
(q x q)-matrix with a I at position (i, j) q'7=11 q &Y where &Y is the subspace of 9 Iq
1 let eij denote the elsewhere and decompose g Iq as 9 Iq 0 -
zeros
a
need to introduce
jgy.
2. Genesis
The affine
subspace M-
of M- consists of those matrices in MP," for which
ML./"]+'
=
0
S,
ML./ql
where the stars denote
arbitraxy matrices of the intermediate space Mp"', which is also an affine RS consists of all matrices in Mp,q for which
M[plq]+l
=
S,
MLIl
=
Ad
(* *),
appropriate size. We also introduce an subspace of Mp,q. The subspace MM Rs
R
(E
+ED
9,Y.
,y
Let
us
denote the Lie
algebra
of
N - by n
OS which is the Lie
=
fX
and let gs denote the
E
nq I [X, S]
isotropy algebra of S,
01,
=
algebra of Gs exp9s, the stabilizer of S. We have that the (adjoint) invariant. Notice that 9 s is given explicitly as the algebra Gs on Mp,q leaves MPIq RS strictly lower triangular matrices M for which Mi+d,j+d Mij for all 0 < j < i < q =
action of of
=
Proposition
q(p +
q
-
2.5
Mpq A- is
2d) -dimensional affine
-
isomorphic Mp,q RS IGs which,
in
turn, is isomorphic
to the
space Mp,q.
Proof We show that every element in Mp,q is The
N --conjugated
proof then follows from the Let M M.Ai C Mp,q. E[p/qj+l i=O
fact that the
any element
exp
%
E
g- 1 and let g
Ad, MLI,]+, which after
projection
=
=
corresponding
We first show that
=
E
N -
to
a
unique element of 'A' p,q.
map Mp,q -+
M[plq]+l
is
f4p,q
N --conjugated
is
regular.
to S. Take
Then
.
expadC M[Plq]+,
=
M(Plqj+l
+
[ , M[Plq]+,]
10-d-I becomes
on
H-d-1 Since the linear map ads
(Ad, M[plq]+I)
=
II-d-1
(M[pl,]+I)
+
[ , S1-
9-1 _+ 9-d-I is surJective we can pick g (i.e. 0. Repeating this procedure with 6 E 0-,y where y :
6)
such that
II-d-I (AdgML,,1q]+I) 2,3.... and using the surJectivity of ads : 19-y _+ 9-d-,y we find that ML"1qI+I is N --conjugated to S. Since, by definition, Gs c N - is the stabilizer of S and since the map Mp,q _+ Mp,q can be RS picked regulax it follows that MP,qlNq- is isomorphic to MP,qlGs. RS =
We next show that
ML,,Iq]
is
Gs-conjugated
to
a
=
unique element of the form
0
Id
(* )
-
*
Notice that if M E MP' q then M[plq E R + RS I consider the space of matrices of the form *
...
*
*1
0
G-j
0
1
0
...
Fix any 6 E
11.... 7q
11
a
0)
0
1
(2.5)
149
Chapter V1. The Mumford systems
diagonal with the I,, axe precisely a by s minjd, q 61. It is easy to see exp(o-6 no,) on 0 Iq leaves the space (2.5) invariant. Also, at level q-d-6 (the diagonal where the *i in (2.5) are) the adjoint action induces for any element E G6 the affine map (translation) l9q-d-6 -- 19q-d-6 : Z -+ Z + [R, ]. In turn this exp g the linear map O[q -+ C' which map induces on C' the translation by II[R, ] where Il is this We denote z + II[R, ] for to matrix the by *,). map (*,, (2.5) X , thus X&) maps E 9-, n gs such that z G C8. We wish to show that there exists for any z C: C' a unique -+ Xc (z) 0. Equivalently, that for any z E C' the affine map X , : g-, n gs -+ Ca : XC (z) is a bijection. Taking the corresponding linear map this means that we need to show that
where the
and the
the stars
diagonal with
distance 6 apart. The number that the adjoint action of G6
is
s
in terms of 6
given
=
-
=
=
=
-
..'
=
X
is
an
9-,
:
n
-+
9.
C8
II[R, 6]
-+
:
spaces have dimension
isomorphism. Since both
s
=
minjd, q
it suffices to show
61
-
be in the kernel of X. If 6 : q
that X is injective. Let 6 and d
d then
-
s
=
q
-
6 >
-
X( ) that X is
so
p and q
for
are
some
(01 01
=
...
10,61,
16d-6)
...
-
(611
...
168)
injective. When 6 < q d the proof is more delicate and depends on the fact that relatively prime (indeed if p and q have a common divisor then X is not injective -
values of
Then
6). X( )
=
(01
...
1
01 C17 C2,
...
7
Cd-6)
-
(Cq-d-5+1
7... I
Cq-6)i
is in the d. If 1, the length of these vectors being 3 6. But d k for and 0 G Cq-d+k q-d remember that G Ck+d because C E gs. This means that the indices of C may be thought of as lying in Zd. Now the fact that p and q are relatively prime implies that d and q are 6 then implies d Cq-d+k for k 1, coprime, hence also q d and q. The fact that G that all .y are equal, hence they are all equal to 0. Thus X is injective in all cases and we can make all *j in (2.5) equal to zero by using a unique element of G6; doing this consecutively I leads to the desired result. for 6 1, 2,.. -, q
appears at position 8 + kernel of x then q-d-6+1
where
j
==
...
=
=
=
=
=
-
=
=
=
=
-
-
.
.
.
,
-
We call M E
Mp,q
the matrix of
(P, Q)
or
of
W).
2.3. The inverse construction show that every element M E Mp,q (with p and q coprime) is the matrix of commuting differential operators such that P is monic of order p and Q is pair (P, Q) normalized of order q. By Proposition 2.3 it suffices to show that M is the matrix of a pair W C W and P monic. Equivalently we need to (P, W) E Tc x Gr satisfying aq. W C W, We will
of
now
a
show that there exist such that ord Ej
=
a
monic element P
T' of order p and q vectors
E0,
.
.
.
,
Eq- I
in W
i, and such that q-1
Ej
1: mijEj. j=0
150
(2.6)
2. Genesis
In order to do this
and
we
define for
E
r
and Ej in terms of
expand
we
Z
an
P
=
qP 0
Ej
=
a+ g,
iF-1 ,,gi-I
+ plo
element MH
E
MH
0,
+P20W-2 +
(2.7)
g2iai-2 +...'
of, by where
(2.8)
mij r
Lemma 2.6
(3.) M(r)
0
for 1
O
(2) M 27)
r < 0; if p+i-j=Omodq,
if P+i-j:AOmodq;
(3) M(O) is the matrix of
a
cyclic permutation
or
of 10,
q
-
11.
Proof If
r
< 0
then p + i
and
-
j
r> q
-
P+i-i
I
mi(jr), which is the coefficient of ap+i-i-r, is
so
ord mij
q zero.
The
inequality holds for
same
7,3
when q does not divide p + i
j
-
that for such values of i and
so
j
m 9)
we
also have
m 9)
=
0
r
0. If q
is monic of order p + i 1. It implies that M(O) is j hence a permutation matrix. The fact that this permutation is cyclic follows from the fact that p and q are coprime; indeed, this permutation corresponds to the translation over p in Z.. I
divides p + i
-
j then mij
plug (2.7)
If we
and
-
(2.8)
into
(2.6)
then
can
also be written
1: 1: MlVg j-r,
r=O
r=O
E r=O
for any -y > 0. We show that .
.
.
,
=
i E
10,
q
-
1}.
Let
Then
0
Prg,qy M(O)
is
I
q-1
r
9j-r
be solved
recursively for pj
and
can
be written
as
90
(M(O)
g, .
Since po
gi0
=
1
permutation matrix, the equation (2. 10) is satisfied that we have constructed Pr and gr' for all r < -y and a
Ay
(Ay
(2.10)
r=O -
can
us assume
9 -r
LM(r)
equation (2. 10)
Ry
(2.9)
!4-r
Prgi-r
(2.10)
and since
1, 0, q 0. identically for y -
I
as 0
=
-
j=O
Pr! -r
for i
0 and for 0 < i < q
q- I
y
which
find that for any -Y
we
Eprg')
Y-r
=
2
-
1q)
gq
g'q-1 Y
151
+
(known stuff)
VI. The Mumford systems
Chapter where
"(known stuff)"
equation of the To
rows
only
the previous
done,
rewrite
(2.10)
gi
and p,.
Summing
up the q
rows
of this
of any permutation matrix equals the sum q-1 I and we use (2.10) to solve for I 7
rows ' =
9
how this is
see
involves
find p, because the sum of the of the identity matrix. We take
we
9
...
194
follows,
as
0
9
(MO)
I
9, -
1,)
(known stuff). g,qY
Recall that M(I) is
10, 1,
.
.
.
,
q
-
a
matrix which corresponds to cyclic permutation of order q we
permutation
Since
11.
is
o-
a
a
permutation
can
a
solve for the
of the set
g'Y
in the
0.2 (0) and so on. Notice that the last following order: solve first for i or(O), then for i equation is precisely the equation defining p, showing that the solution exists and is unique once the vector g.0, has been chosen. Clearly the freedom in choice for gOY corresponds to the left action of Volt'. Notice that it is only in the very last step that we used that p and q are coprime. Summarizing we have shown the following proposition. =
Let ME
=
Mp,q, where
p and q
Proposition
2.7
(P, W)
Gr such that P W C W and aq_W C W and such that M is the matrix
E T' x
are
coprime.
Then there exists
-
pair of P.
a
pair (P, W) is unique up to the left action of Volt'. The correspondence which associates pair (P, Q) its matrix is a bijection between mp,q and the space of all pairs (P, Q) of commuting differential operators with P monic of order p and Q normalized of order q. The
to the
2.4. The KP vector fields In this section
commuting fields
will realize the KP vector
we
vector fields
on
the Sato Grassmannian
fields, which are a naturaJ collection of Gr, as a collection of commuting vector
the affine spare 1CIp, q. In order to write down the KP vector fields on the Grassus first show that the tangent space at a point W of the finite-dimensional
on
mannian, let
Grassmannian G
this,
we
=
consider G
Stab(W)
C
of k planes in Cn is naturaJly given by Hom(W, Cn/W). To see homogeneous space GLn1 Stab(W), where GLn GL(n, C) and stabilizer of W, with Lie aJgebra
G(k, n) as
GLn is the
the
=
stab(W) Then one
TWG
=
TW(GLnl Stab(W))
associates to
a
=
=
10
E
g1n I O(W)
gIn/stab (W)
representative 0 of
an
c
Wj.
Hom(W, Cn/W); for the last equality gIn/stab(W) the composite map
=
element of
0
W
--,
Cn
,
Cn
,
V/W
of the Sato Grassmannian (which is infinite-dimensional) we define the tangent point W E Gr to be given by Hom(W, TI1W), where we consider W in the last equation as a subspace of V. In this language the i-th KP vector field is given by Vi : W -4 TI1W : w i-+ a'w mod W. In the
case
space at
a
152
2. Genesis
We first transfer these vector fields to the space of pairs (P, Q) of commuting scalar differential operators with P monic of order p and Q normalized of order q. To do this we fields use the bijection Volt -+ Gr : T i-+ WT, which gives the following commuting vector on
Volt dT
T(T-la'T)-,
dti
(see [1\4ul3]). If we have a U(t) T-I&T then
U
dU
Tti this for
Applying
KP vector fields
[T-lft (T-'O'T)-]
& given by
we
define
on
dP
i
[Q+/q7 Q17
=
=
[Q'Il, U]. + for the
a
[Q+ p]. ,
MP'q. More
on
precisely,
we
will write down the
the constant coefficient scalar differential operators
correspond to
each KP vector field is then
i/q
=
Tti
to write these vector fields down
vector fields that
[U, Q"q]
by 16 we find the following Lax representation commuting operators (P, Q),
lgq and
t-i proceed
=
the above space of
Q
We
pseudo-differential operator & and
constant coefficient
=
=
linear combination of these vector fields. We fix
the derivative in the direction of the vector field a periodic basis of W and by e. By the above interpretation
choose
corresponding
to
[P'/Ak]
[,5i/),k]
;
i, k and denote
by
a
dot. We
+
containing its first q elements Grassmannian, we can
denote the column vector
of the tangent space at W to the
write
e
[P'/Ak]
=
polynomial matrix (in A right hand side of (2.11) is smaller
where A is the of the
equation
6
(2.11)
A(aq) e,
such that the order of the i-th component equation (2.3), which is the defining
aq)
=
-
+
than i. Also
of the matrix M E Mp,q with respect to the chosen
periodic basis,
can
be rewritten
as
Pe If
differentiate
we
(2.12)
then
we
find P6
k,5
=
(MA
=
=
(2.12)
Me.
Me+ Me, which is easily rewritten
-
PA) 6
=
as
[M, A] 6,
that elements of D'- commute among themselves and with matrices which are [M, A], independent of x, such as M and A). Rom the last equality we can conclude that k & (rather than in 8). We claim that A can be because k, M and A are polynomials in A
(one
uses
=
=
taken
as
(MiIAI)+.
To prove this
must show that the i-th
we
(jii/Ak) has order smaller than i
(for
i
=
1,
.
.
.
,
e
-
+
q).
(Mi/Ak 415
Since
(pi/,\k) e
component of
=
153
P
commutes with M
(Mi/Ak) 15
we
have that
Chapter
VI. The Mumford systems
and it suffices to show that the order of the i-th component of
(.Pi/Ak)- 6_ (Mi/,Xk)_,5 is smaller than i. Since the i-th component of 6 has order i this is clearly the case for the first term; in the second term every component has negative order because M depends on A =,9q only. Thus we have shown that the KP vector fields lead to the following
commuting
vector fields
on
Mp,q: dM
dtij= IM, [Mi/A!"]+]Given M E MP,q
we
have shown in Section 2.2 that there exists
gMg-1 E kPA. Differentiating Y vector fields on kP,9:
Y
(2.13)
=
=
gMg-'
we
find for any
a unique g G N - such that i, j the following commuting
dY =
dtij In the next section
family
of
we
ly, lyi/Aj+lj+
-
w1l.
will show that these vector fields
compatible Poisson
structures
on
.1 jp,q.
154
are
(2.14) Hamiltonian with respect to
a
3. Multi-Hamiltonian structure and
3. Multi-Hamiltonian structure and
In the
section
previous
affine space
Mp,q
In this section
Hamiltonian. We will do this
3.1. The
The
j_Iq
iE
we
write X
inner
an
=
==
131JAI
ED
X+
=
product
on
+ X_
to the above
according
91q by (x, y)
product (-, -)
on
i1q
right
hand side is
a
a
element X
=
X(A)
=
(vector space) decomposition. non-degenerate and ad-
0(91q) carries a natural Poisson leads to
an
ad-invariant,
non-
Res,\=o (X (,X), Y (A))
=
shorthand for
Ei+j=-, (xi, yj)
=
,
EiEz Trace xjy_j_j.
an
Poisson bracket
the type
X(A)
-+
on
0(b
the
multi-
algebra 0(j_Q of functions on j-1. for which we can define a gradient as in Example 11.2.8, but which is large enough to contain functions of Res H(X'(X)) (for H E 0(g[,)), which will be important later. We define an algebra of functions by
We introduce
and
on
via
(X (,X), Y (A)) where the
an
T ace xy, which is
=
=
degenerate
are
A-I
invariant, (x, [y, z]) ([x, y], z). According to Example 11.2.8, bracket, which we will denote by I., J. The inner product inner
vector fields
i(q Of 91q-
loop algebra
loop algebra will be denoted by capital letters; for
Elements of the
E xi Ai
family of commuting
will show that these vector fields
we
j_1q of qIq is defined by 91q
We define
the
by using
a
symmetries
i-t,,
loop algebra
loop algebra
have constructed
we
q[q [A].
C
symmetries
=
C
IV] qf*
C' denotes, for C E q(q and for .5 E Z, the following algebra of functions:
linear map
FxjAj
i-+
C(x.).
On
it,
we
consider the
0(q[q)= F:j 1-+CjVnEZ:Fjb E O&Jjq
Thus, elements of the
O(ZI.)
gradient VF(X)
of
a
restrict to
polynomials
0(j_[,)
function F E
on
all
at X E
-
subspaces B
it,
is defined
d
(VF (X), Y)
Proposition
belongs
to
4
3.1 -
For any X E
Moreover, for
any
=
dt lt=o
91, F,
and F E
G E 0
IF, Gj(X) belongs
to
0(i-io,
making
0(j[q)
into
F(X
=
a
+
tY)
VY E
0(gl,), VF(X)
61q)
Poisson
155
algebra.
Example
11.2.8
by
i1q.
is
well-defined by (3.1)
the Poisson bracket
(X, [VF(X), VG(X)])
in
IF, Gj, defined by
(3.1)
and
Chapter For I c Z let
VI. The Mumford
R, denote the endomorphism of of, defined by R:
glq
RI:
DIq
-+
Oil
:
of,:
X
+
X -+
X+
IF, G11 (X)
guish
I E
Z,
a
2
family of compatible
X-,
RI satisfies the modified classical Yang-
(X, [RIVF(X), VG(X)]
=
-
R(A'X).
Since any linear combination of the endomorphisms Baxter equation (V.5.2) the brackets
form, for
systems
+
[VF(X), RIVG(X)]),
Poisson brackets. We call them R-brackets to distin-
them from the above-defined canonical Lie-Poisson bracket
the Ad-invariant functions Ki j E Z, the function
'
T ace
t-+
: x
(3.2)
on
0( i-iq)
-
Consider
i+1
i+1
,
(i
=
0'..., q
-
1)
on
jg[q
and
define, for
(X (A)) Hjj:j(q-+C:X=X(A) -+Res Ki),j+l Clearly Hij E O(j(q) and VHij(X) Proposition V.5.1 implies that
=
and
(3.3)
XiA-i-1. Therefore each Hij is
all functions
Hij
any
a
Casimir for
1-, -1
in involution with respect to all
are
R-brackets
1-, -11, which can also be deduced immediately from (3.2). As we have seen in Section V.5 the Hamiltonian vector fields that correspond to such functions can be written in Lax form; taking for example the 0-th R-bracket the function Hij leads to the vector field I , Hij 10, which takes the Lax form Xjj =
-
-
dX
1 =
dtij Two alternative ways to write this
,
2
RVHij (X)]
(3.4)
.
are
dX
[X, (VHij (X)) j
=
dtij Notice that the
equations (3.5) and (2.13)
spaces. The vector field
(3.5)
are
can
dX =
now
Xjj
axe
show that
on
linked
a
by
in
(3.5)
_
formally
the
same
but
axe
defined
=
on
different
I-, J,
I -, Hij+j 11.
.1 [X, R, VHi,j+l (X)] 2
since
(3.6) endomorphisms Rj,
.
it-, all the R-brackets I-, JI, the Hamiltonians Hij and the vector fields
vector field
these R-brackets. V is defined
by "shift
[X, (VHij (X)) ].
be written in Lax form with respect to all
dtij We
-
is in fact Hamiltonian with respect to all brackets
I -, Hij 10 Therefore, Xjj
=
V, which has the deformation property with respect
as
the infinitesimal generator of the action of C
A", 8,
E XiXi)
Xi
156
(A + S)i;
on
91,
to all
given
3. Multi-Hamiltonian structure and
here
we use
for
powers of A the formal
negative
+
s)-1
symmetries
expansion
=
i>O
which is
for small s, in particular it is the right definition if one wants fundamental vector field V of this action: the latter is easily computed as
actually convergent
to consider the
'9 The properties of V are given by the (i + 1)&' for any E gl*. q '6-1N X(A), i.e., V ' following proposition, whose proof is an easy consequence of the definitions.
Vx(,\)
=
=
(i.)
i, 1
Let
3.2
Proposition
The Lie derivative
E Z
-
of the 1-th R- bracket is
,CVIF, G11 for
any
(2) CVHjj (3) CVXij
F, G
=
=
E 0
-
1,CVF, G11
(up
-
to
a
factor -1) the (I
IF, f-VGjj
-
1) -th R-bracket, (3.7)
-11F, G}1-1,
=
(it,);
(j + I)Hi,j+,; [V, Xii] U + 1)Xi,j+i =
The conclusion is that
we
E Z
1, m
have for every
hierarchy with respect to the R-brackets 1-, -11 and I-, m 0) depicted as follows (we omit the coefficients).
and for every Hi a bi-Hamiltonian A typical fragment of it is (for
jm-
=
'C"
'C
Hjj+j
Hij
Hjj+21
-
0
0
0
1
Xi,j+l
Xij
3.2.
Reducing We will
now
3.3
.
to
Let
of the brackets.
some
M'jq
mp" S
denote the
=
E I[Plql+l
affine subspace of it, defined by
MiA'
1E
i=O
is Mp" S
a
Poisson
subvariety of
Ofq I M[p1q] +1
=
S
-[plqj+l C11- I JI, with respect to the Poisson structure E1=0 hence it has the arbitrary. Moreover V is tangent to Mp,q S ,
91q
complex numbers ci are deformation property with respect
where the
Xi,j+21
apply Proposition 11.2.27 to obtain a family of Poisson brackets and a vector Before doing this we truncate g(q to a finite-dimensional Poisson on Mp,q
relating subspace with respect
Proposition
'ClV
the R-brackets and the vector field V
them
field
'ClV
to each
of
these restricted Poisson structures
157
on
M11". S
V1. The Mumford
Chapter
systems
Proof
By Proposition 11.2.18, M jq Poisson ideal of 0
(i-Q.
linear function
on
QIj
For 0 <
is
Poisson
a
< q let
i, j
jljl
-
generated by
where
jj(S),
s
6ij
=
(-
eij A'
,
-
1)
as
+ 1. Since
gl,
E
,
0
if and
[,*
only
if its ideal I is
and consider for
8
E Z
a
the
follows
-
the functions
[p1q]
=
==
of
(- eij)
i-tq; it is convenient to write QIj illj
The ideal I is
subvariety
[p1q] + I I and by the functions jfj where s f 10, for any I E Z the following find we V j8j eijA-1-1 .
.
=
formula for the RZ-bracket.
Cj', di I I where
eft
eft
I if s, t < I and
=
1 (jkA81+t+1
68
=
=
-I if s, t
: 1; otherwise
above generators of I in (3.8), together with an where w linear function contains only terms
eft
0.
=
arbitrary ktj,
one
0 10,..., fp/q]
MP'q follows
Poisson ideal. The fact that V is tangent to for V. formula V jfj (s +
(3.8)
-Ji4aj
in
a
+
Substituting any of the finds that the resulting
11, showing
that -E is
similar way from the
1) j'j+'
=
a
explicit N
and the restriction of V to Mp1q The restriction of the above Poisson structures to M"' S S will be denoted in the We
are now
same
precisely
where 0
(w.r.t.
< I <
which subvariety of Mp,q S
Proposition bracket
son
3.4
The
E v/ql+' 1=0
is
as
way
in the
the
corresponding
structures
on
of Proposition 11.2.27: M"' is S
case
jp1q]
1)
+
if,an
affine Poisson
variety
which the stabilizer GS of S acts; MP1q is RS
on
a
Gs-stable.
GS, M,,q triple (Mp,q S RS )
&, -11,
,
is Poisson-reducible with respect to the Pois-
where the complex numbers
ci are
arbitrary.
Proof We first show that the action of
Gs
is Poisson (where Gs is given the trivial Mp,q S
on
the
action of
(diagonal) structure). Actually GL(q) on gfq is Poisson with respect 1-, -11 for any I r= Z. To see this, take any 1 E Z and g E GL q). It is sufficient to show that (Ad,) I fl, f2 11 I (Ad,) fl, (Ad,) f2 11 for any fi, f2 E 0 (jTq) that are linear. For fi linear one has that (Ad,)*fi is also lineax hence Vfj(X) is independent of X G gl, and we Poisson to
*
*
*
=
can
omit the argument X. Since d
Wt- jt=O f, (Ad, (X we
find that
(V(Ad,)*f,,Y)
=
+
tY))
=
f, (Ad, Y)
(Ad,--i Vfi,Y)
f (Ad,)*fl, (Ad,)*f2j, (X)
=
=
=
=
=
(Vfi, Ad, Y)
giving V(Ad,)*fj.
=
Ad,-i Vfi.
(X, [Adg-i Vfi, Adg-, Vf2].,,,) (Ad., X, [Vfi, Vf2]R,) ffi, f2 11 (Ad, X) (Adg)*jfj, f2j,(X).
158
Then
3. Multi-Hamiltonian structure and
Since
Ap,q
is
Gs-invariant
only left
we are
symmetries
with the verification of condition
ideal _T of MP,q in Mp" is generated by those elements of the form RS S i < d 0, q. For 1 [p1q] these elements are Casimirs of
andO<m,n
I andO< s<
'&[Plq], Cnn 1, (X) 'ij
'[Plq]'a I
.8
zero:
for
s
zA
=
1 this is
[plqj
Indeed,
if X E Mp'q S
then
(81111d/ ql+8+1-1(X) s
j6?J1q]i 6minjI(X) 6in6m[Pj/q]+1(X) =
(din emj
=
-
I
-1
P
in
at once, while for
seen
(11-2.17). The ij(R) for which
-
S3
=
-
which is
&"
one
6mjp
(X))
computes
[Vlq]+l in
(X)
6,njein, S) (femn eij] S) ([eij, S], emn)
=
=
-
i
=
and
finds again zero because j i + deg S j i d < -q. We have shown (11.2.17) s :A [p1q] + 1. Suppose now that I and notice [p1q] + 1. Take F E O(Mp,q, Mp,q)G S RS that this implies that F satisfies the infinitesimal condition one
=
-
when I
([VF(X), X], V) Verifying (11.2.17) for
-
-
=
amounts to
=
showing that
ij suchthatj-i
VX C
0,
f4p,q,
VV E OS.
(3.9)
101q] Fl,,I,,+,(X) =OforanyX Gf4P,q and wI
But
1 ?3 /"I, Fj [plq]+l (X) X, [eij, VF(X)] [A-[Vlql-leij,R (,X[p/ql+lVF(X))] %3
1
+
=
2 1
=
(X, [eij, VF(X)] [A-W'71-'eij,AWq1+'VF(X) (,\[Plql+'VF(X) (X, [eij, VF(X)]) (X, (AWq1+'VF(X)) j ) +
-
2
2
=
-
-
Both terms vanish: rewritten
the first
one
in view of
(3.9)
and the second
one
because it
can
be
as
(S, [eij, (AIP/ql+l VF(X))
([S, eijj, (A[Plql+-'VF(X))
_1
Proposition 2.5 we have shown that Mp,q carries a ([plq] + 2)-dimensional compatible Poisson structures. We will now show that the restriction of V to X4p,q
In view of
family
of
has the deformation property with respect to these Poisson structures.
Proposition
,CVj-, .1. (resp. the
3.5
Denote V
Let
by
is the
1-, -1
=
E Plql+' cjj-, J, 1=0
W the restriction
be any R-bracket
on
Mpj"
and let
I., JV
of V to M-14. Then Cwf- j'-j' I-, J'V where I-, -I' reduced brucket of I-, j (resp. I., JV) on MP,q. In particular W has
deformation property with respect
=
,
to
159
Chapter V1. The Mumford systems
Proof First notice that the
triple (M111, GS, M,,q S RS )
is Poisson-reducible because
[p/q]+l
I-, -IV
an
(3.7).
immediate consequence of
restriction of V to
W).
Cal-,
Let
MpWr7S by W (notice
(Mp,q)Gs RS
0
fl, f2
4'p,q and let us denote the 'Qp,q is also denoted by
that its further restricition to
We need to show that
Iflif211V =LWIflif2li-fcWfl)f2lt-ffl7,CWf2liTo
see
that this formula makes
to show this
sense one
(Mp,q)Gs RS
it suffices to prove that
C
q)Gs (.A4p, RS
the fact that the actions of LW and GS commute, a consequence of the fact that the flow of DW is given by 0, : X (A) --+ X (A + s), while GS acts by simultaneous
inclusion,
conjugation.
If
we
let
use
F1, F2
C
0
(MP,q) S
such that
f,
=
p(FI)
and
f2
=
p(F2)
then
Iflif211V =pfF1,F2JV =
pCV IF,, F21
=
pCV IF,, F21
-
-
pJCVF1, F21 JpCVF1, f2l
-
-
pJF1, LvF21 Ifl, pCvF21
Let us denote M_,,q pCv(F) =,Cwp(F) for any F E 0 (M,,q, S RS ). __+ 0 (Mpq) (which may be thought of as coming ftom. (Mp'q)'s RS RS " the quotient map Mp" and by Mp" -+ M' _+ Mp,q the inclusion map. Then RS RS IGs) RS S Therefore Mpq z*(F) ir*p(F) for any F E 0 (Mp,q, S RS ). so
it suffices to show that
by
ir* the inclusion map 0
=
7r*LWp(F) =,CW7r*p(F) and the result follows from
injectivity
=
of 7r*. In
Mp, 1CV 0 (Mp,q, D S RS an
_, easy consequence of f
(Mp,q)Gs RS
Mp,q)Gs Z.,CVO (Mp,q, S RS
=
=
There is
VHoj (X)
=
a
C
and the vector field W
are
we
used the fact that
COO4p,q S
7
Mp,q)Gs RS
'
LW%*O (Mp,q, MV,q)Gs ='CJ'V7r*O S RS
7r*LWO (MPi")
the coefficients of
space to traceless matrices.
GS
(i)
z*,CVF _Q 7r*pCVF
=
o,q Gs. WRSJ
final, innocent, reduction
Idq A_j_'
Lwz*(F)
that
can
Gs
c 0
be done
Trace(X)
axe
(Mpq)Gs S Mp,q)Gs. (Mp,q, S RS
on
these systems.
Casimirs and
The multi-Hamiltonian structure, the
just the restrictions of the original
space.
160
we can
Namely,
since
restrict
phase
commuting
ones
vector fields
to this smaller
phase
4. The odd and the
4. The odd and the
(odd)
4.1. The
even
Mumford systems
Mumford systems
even
Mumford system
preceeding section we have described a family of commuting Hamiltonian vector affine subspace of it,, which, as we said, can be taken as an affine subspace Of ;Iq 2 and p For q 2g + 1 we call the corresponding system the (genus g) Mumford system, because it was Mumford who first wrote down (in [Mum5]) explicitly these commuting vector fields and showed their algebraic complete integrability (the Hamiltonian structure is absent in [Mum5l). For reasons that will become clear in the next paragraph we will often refer to the Mumford system as the odd Mumford system. Since we are considering special values for p and q we will simplify the notation, used earlier in this chapter; the new notation will also take into account the fact that we only consider traceless matrices from now on. We will denote the phase spare of the genus g Mumford system, which consists of the traceless matrices in X4, 2g+1,2 by Mg. EXplicitlyj Mg In the
fields
on an
-
=
=
consist of all matrices
A(A) where
and R
u
and
monic, deg u
w axe
0
1
0
0
=
=
g
deg w
=
-
1 and
'
deg v
< g. We have that S
=
)
so
that the group
GS consists of all matrices of the form traceless matrices in
A(A)
element of either
03 (A)
-
one
'g M2g+I RS
resp.
0
with
a
=
(0 0) 0) 1
1
algebra gs g-,. The space of all denoted by M9RS resp. M9S. A general by Lie
U (A) ) -V(A) J
v(A) ( W(A)
-
I
M2g+l,g S
will be
of these spaces will be denoted
ii (A)
1, while both ii and 0 have degree at most g, with the extra Since GS acts by conjugation the quotient E Mg R s.
for which t7v is monic of degree g + condition that fi is monic when
A(,\)
map
Mgs
-+
Mg is
explicitly given by Ui
=
iii,
Vi
=
Oi
wi
=
iv-i
-
+
fiiOgI
i
20iO,
-
(for
0 < I < g +
1)
on
...
191
(4.1)
g
where ui, fii.... denotes the coefficient of Ai in u (A), ii (A), the reduced brackets on Mg, as given by (11.2.27). To do
brackets
07 11
fii F, 2,
MgS
ffiw jll
are
Using (4. 1) it is easy to compute this, notice first that the Poisson
given by ij
='51
i4+j+1_11
10i'l7vj}1 Jt7vi, fij I I t
1 if s, t < I and el (recall that e-I (between linear functions) are zero.
10i
-
fiiO, iv-j
+
NjO,
-
=
2e'jJ Oi+j+l -1
1; otherwise el t 0) and all other brackets For example if I =A n then Ivi, wj 11 is found from
=
-
I if s, t >
=
fii+j+1-IOg2 + 20i+j+,-,Og) + fiidj',
fijOg211 161
V1. The Mumford systems
Chapter
giving Ivi, wjjt 0, 1, given, for I =
ej'3'wj+j+j-j +%M. In this way the reduced brackets
=
lui, Vj 11
Ivi, WjIl jwj, uj 11 while the bracket
=
lui, Uj 11 Ivi, VjIl
61 Ui+j+1-11
ijwi+i+l-l + UX 2dj'j Vj+j+j-j'
=e
=
I-, -lg+l
is
quadratic and
lui, Vj 19+1
=
N, Wj Ig+1
=
ei"+l wi+j+-g
jwj, ujl,+,
=
2e'j 9+ lvi+j-g
?g,+, ui+j-g
-
jwj, wj 1,
is
Namely, let
be written.
I., JV
=
-
Eq
0
W
0,
=
26ivj
(4.2) -
261jvi,
0,
UiWj'
IN, Vi Jg+1
=
01
2ujvi,
jwj, wj Ig+1
=
2viwj
Ej9-0 cjA'
=
0,
=
=
-
-
=
lui, Uj Ig+1
UiUj'
be
i
=
(4.2)
that
(4.3)
0...
of
polynomial
a
Then it follows from
cif-, -Ii.
generating functions
found to be
given by
For linear combinations of the Poisson structures can
are
by
g,
g
degree
I-, -1 0
can
-
a
2vjwi. compact formula let'9
at most g and
be written in terms of
follows:
as
IUM, UMII
=
IUMMOI,
=
IVM' V(Y)I1 0, U(X)W(Y) U(Y) O(X) =
-
X-Y
ju(x),w(y)j'P
=
-2v(X)W(y)
-
V(Y)V(X)
(4.4)
7
X-Y
W(X)W(Y)
IV(X), WMY
-
.
W(Y)W(X)
U(X)W(Y),
=
X-Y
jw(x), w(y)j o The Hamiltonians in involution
=
are
2
(v(x)W(y)
-
v(y)V(x))
the coefficients of
.
TraceA(A)2/2, i.e.,
the coefficients
U(,\)W(,\) + V2(,\). Rom (4.4) we can compute the Hamiltonian vector fields H(A) where y E C as well as the vector fields Xi H(y)J1, I-, I-, Hil', where Hi is the Xy i-th coefficient of H(A) (notice that the function Hi, used here, coincides to the function Hji, defined by (3.3)). For example
of
=
=
=
X1,(U(A))
leading
=
IU(A), U(Y)W(y)
=
2u(A)v(y)
U(Y)V(A)
A-V
to
d-A(A)
A(A),
=
WKY Writing Hi
=
Resv=o H(y)/yi+l
we
d
The minus
A(A)
=
0
0)]
-
A
-
Y
U
(Y )
0
find
[A(A), (A(,\)/,\'+')+ ( 0)].
sign has been put
with the brackets
A(Y) -
0
Tti 19
V2(Y)j1
+
defined in
(4.5)
-
Ui
in to make these brackets
Chapter
III.
162
0
coincide
(when
q
=
2)
4. The odd and the
4.2. The
even
In the
even
Mumford systems
Mumford system
previous section
have constructed the (odd) Mumford system. Alternatively it hyperelliptic case considered in Paragraph 111-2.4. We show how this can be done be constructing an integrable Hamiltonian system that is very similar to the Mumford system. It is an open problem to obtain this variant of the Mumford system (which was first constructed in [Van2]) from the algebra of pseudo-differential operators and/or from a reduction on the loop algebra 4In the language of Paragraph 111.2.4 we take F of the form F(x, y) y2 f(X), where 1 and choose d deg f 2g + 2, we take W g, the genus of the hyperelliptic curve y2 f (X). Then the Lax equation (111.2.14) takes the form can
we
also be obtained from the
=
=
=
_
=
d
A(A)
dti
=
[A(,\), [Bi(,\)]+]
,
where
A(,\)
The
=
V(A) ( W(A)
U(A) -V(A)
A(,X)
highest
flow
A(A)
[A(,\), B(A)],
where
=
one
computes
=
b(A)
b(A)
=
Bi(,\)
=
Xa.,-,A(A) A(,X)
A(A) u(A)
[u(A)] Ai+1
is the most
=
V(,X) (W(A)
U(,X) -V(A )
and
W(A)
[ F(l\ (A)v(,\))] u
important for
M11)] ('X)
)
and is
us
B(A)
,
(4.6) +
simply given by
(bo) ('X 1),
(4.7)
]+'
(4.8)
=
0
fR (X' Y)
=
y2
_
(X2g+2
U(,\)
phase
space in the
+ a2g+1 X2g+l +
an affine variety isomorphic to C2g+l hyperelliptic curve of genus g. The g + 2
which is
V2(,\)
I
+
We want these systems to be extended to a larger tion 111.2.6. As deformation family M we take
every
w(,X)=_
from
U
M=
and +
.
-
-
-
+ aix +
Notice that M contains coefficients
a
.....
sense
Amplifica,
ao)j,
an
a2,+l
of
(odd) equation for Casimirs for this
are
larger system and the above Lax pair, supplemented with the equations coming from these Casi:rnirs, describes the integrable Hamiltonian system on the larger phase space Cg+2 X C2g C3g+2 (which is equipped with coordinates uo, agl. Ug-1 Vo vg-,, a2g+1 -
There is however which
we
W(A)
=
A9+2
+
W
-
,
g+,Ag+" + WgAg +
7
....
,
...
,
-
-
-
+
WJA
+ Wo
computed from (4.8); of course they will be polynomials in the ui, vi and ai. One sees by inverting these relations the ai can also be written as polynomials, but now terms of ui, vi and wi. The upshot is that we can look at this system as defined on C3g+2
are
however that in
-
better way to look at this larger system. Notice that in the way in have presented it, the g + I coefficients of a
163
Chapter with coordinates now an
juo,...' ug-li'vol
V1. The Mumford systems
...
7Vq-1iWO)Wli
...
,
w,+, I and the Lax operator
A(A)
is
arbitrary element
V(A) ( W(A)
U(A) -V(A)
where
deg u(A)
=
deg v(A)
< g
u(A) monic,
g,
1,
-
w(A)monic.
degw(A)=g+2, A
similar to
of Poisson
family polynomial
W of
brackets, degree at most
IUW' UM11
g
=
=
ju (x), w (y) 1W
=
given, for
any
by
NX), VM11 U(X)W(Y)
1U(X)'V(Y)l1
for the odd Mumford system, is
one
=
0,
U(Y)V(X)
-
X-Y V
-2
(T) W(Y)
-
X
1V(X)'W(Y)l1
W(X)V(Y)
V
(Y) W (X)
Y
WM O(X)
-
=
X
jw(x), w(y)J P
-
=
-
a(x
+
y)u(x)W(y),
Y
2a(x + y) (v(x)V(y)
-
v(y)W(x)).
u + w,+, a(u) u,-I. Rom these formulas one obtains, as in the case of the odd Mumford system, easily Lax equations for the integrable vector fields X, I-, H(y)ll, as well as the vector fields Xi I-, Hill, where Hi is the coefficient of Xi in H(,\) u(,\)W(,\)+V2 (X).
where
=
-
=
=
=
point out that when the odd Mumford system is constructed in this way precisely at the Poisson structures 1.,Jw, given in (4.4). This means that
It is worthwile to then we
one
arrives
have two different constructions for the multi-Hamiltonian structure of the odd Mumford
system. 4.3.
Algebraic complete integrability
and Laurent solutions
Having shown integrability of the odd and the even Mumford systems in Chapter III we comment on its algebraic complete integrability. It was shown by Mumford in [Mum5] that the general level set of the odd Mumford system is an affine part of a Jacobian (of the corresponding hyperelliptic curve) obtained by removing the theta divisor and that the flow of the integrable vector fields is linear on it, hence leading to algebraic complete integrability. In Section 5 we will modify our construction given in Chapter III in order to obtain similar systems which axe a.c.i. Since this new construction coincides with the original one in the hyperelliptic case it will give a proof that the odd and even Mumford systems are a-c-i- For the odd case explicit solutions in terms of theta functions are given as follows (see [Mum5] for details). Let D(t) Ab(D(t)) be an integral curve starting at D D(O) then D(t) At+D, where A is a fixed vector which depends on the chosen vector field (going with t) and the corresponding polynomial u(A) is computed from its values at the Weierstrass points ak of the curve using now
=
==
U(ak)
=
Ck
(
19k] (At + D) 0 [9] (At + D)
0 [J +
2
)
the vector 77k and 8 are characteristics which are described explicitly in of (IV.4.9) these define indeed meromorphic functions on the Jacobian. 164
=
(4.9)
[Mum5].
In view
4. The odd and the
For both
cases
Mumford systems
even
the smooth level sets of the momentum map appear as the first level of we describe next (for more details and proofs see [VanQ.
stratification which
a
(complete, irreducible) complex curve (say, a compact Riemann hyperelliptic. The hyperelliptic involution o- extends linearly We introduce a decomposition of Jac(IF) with -+ D'. an involution D arbitrary fixed point P on the (hyperelliptic) curve r. Let E, denote the set
Let r be
a
smooth
surface) of genus to Div(]P) giving respect to
an
g which is
-T, which we
we
define
order a
1(m, n)
=
by (m, n) :5 (m', n)
subset Divmn (17,
of
P)
10
E N x N
if and
if
only
<
m
n
<
m' and
<
m
+
gJ n
<
Then for
n.
(m, n)
E
Ig
Div(r) by
9-m-n
Pi +mP+nP' -gP I Pi
Divmn (17, P)
the term If
we
gP
E
r\JPPJ
is introduced here in order to make every element in
and
i:A:j
=*
Pi
=,I=Pj'
Divmn (r, P) of degree 0.
introduce20 g
Divo (r, P)
g-n
U U
=
Divmn (r, P)
n=O m=O
then the Abel map Ab : Div'(r) -+ jac(r) : D -+ Ab(D) restricts to -+ Jac (17). It is shown in [Van4] that the sets
a
bijection Ab
Divo (17, P)
Jm,n (r, P)t"T Ab(Divmn (r, P))
(or Jm (r, P)t'e-f Ab (Divmo (r, P)) P PI) define a stratification of Jac(r), meaning that they are disjoint differenmanifolds, whose boundary is a finite union of lower-dimensional sets i,,(r, P) (resp. J,,(I`,P)). In the case P = 4'PO* the stratification is completely described by the following proposition. in
case
=
tiable
Proposition 4.1 If P =A P' then Jac(r) is stratified by the (g submanifolds Jm,n Wi P), whose closure is given by the (finite) union
U
-7m,n (r, P)
-
m
-
n)-dimensional
(4.10)
ik,1 (r, P)
(k,l) !(m,n)
Each stratum
Jm,n (17, P)
has two
boundary components which P
P.,
,F
20
Divo (r, P) is
of which it is
a
=
Ab(Pa
-
P)
fp
not to be confused with
W11
...
Divo (r),
subset.
165
Ifp
W9)
are
translates
of each other by
(mod A).
the group of divisor of
degree
zero on
r,
Chapter V1. The Mumford systems
of dimension g i are translates of each other by n15 for some of the (g I)-dimensional strata ji, (r, P) and Jo,j (r, P) are 11,... i}. translates of the theta divisor and are tangent along their intersection jij (r, P).
More n
all i + 1 strata
generally,
E
-
The closures
,
-
Thus the different strata fit
represent the different spaces
together
as
horizontal line and depict inclusions resented by the following.
by
by the partial order
dictated
jm,n (171 P) by .7m,n
< on
put those of equal dimension
i
axrows, then the stratification is
1g:
on
the
if
we
same
schematically
rep-
JO, 0
40
jo, 1
-70,2
j, 0 We also
J019
give the corresponding proposition for
P
PO".
P' then Jac(r) is stratified by the (g Proposition 4.2 If P ifolds J,,, (r, P), whose closure is given by the (finite) union =
U
im (r, P)
-
m)-dimensional
subman-
jk (r, P)
k>m
and each stratum In this
case
j, (17, P)
has
the stratification is
Jg
jo
=
Jac(r), j,
just
is
a
-+
one
boundary component.
simply depicted
Jg-1
-+
Jg-2
as
-4
translate of the theta divisor and
The relevance of these stratifications for the
the fact that each stratum
J1
ig
A
is the
origin
in
Jac(r).
and odd Mumford systems resides on one family of Laurent solutions for one
even
corresponds to precisely special vector field of these systems (the most basic one). These Laurent solutions are given by the following two propositions (for the relation to the Sato Grassmannian, see [Van4]).
Proposition4.3 For the odd Mumford system there are g+1 families of Laurent solutions. The m-th family corresponds to the stratum J,(]P, P) and the functions ul,..., ug have the following Laurent expansions starting at points of the stratum J,,, (17, P) U
g-i
=
(- 1)
.(2i-l)!!(m+i)! 2"il (M i)! --
-
Ug-i
=
1
"
t2i
+O(t-2i+l)
(i
M), (4.11)
O(t -2i+l) 166
4. The odd and the
even
Mumford systems
Proof
Equations (4.7) to P
PO')
=
are
written out in the
fi(x) ,b(x) ib(x) or
just
of the odd Mumford system
case
(corresponding
as =
2v(x),
=
-w(x)
=
-2(x
(x
2u,-I)u(x), 2u.-I)v(x),
+
-
-
third order equation,
as a
; i (x)
=
4
(fii- I
2ug- I fii
-
-
(i
fig- I ui)
=
0,
-
-
-
,
g
-
1;
u
-
I
=
0).
(4.12)
The ansatZ21 00
Ui
uijti
t2i j=O
leads for the
leading coefficients
u9-i,O to the recursion relation
ai
2i + 1 ai+1
To solve this recursion one
i < g +
1,
we
j-+-1 +
relation, notice that
some m E
10,..., gJ
which leads
(2i aj
and a,,,+, from it by
+
[
2
if ai
1)
=
+ a,
1
(4.13)
ai.
0 then ai+1
=
0; since
0 for at least
ai
find that a,
for
[i(i
2
by -
m(m + 1)
induction
1)!! (m (M
2ii!
+ -
(4.14)
immediately
i)! i)!
to the formula
(i
M),
0, hence also to (4.11). The series for vi and wi follow immediately a. differentiation, in particular they do not give rise to separate families of Laurent =
solutions.
As fication
a
M
corollary of the proposition
on
In the see
we see
that the odd Mumford system induces by the subsets Jm (17, P).
a
strati-
Jac (17) which coincides with the stratification case
of the
even
Mumford system
we
have the
following result (for
a
proof
[Van4)).
Proposition 4.4 For the even Mumford system there are (g+l)(g+2) families of Laurent 2 solutions, one for each element of the set -Eq. The (m, n) -th family corresponds to the stratum Jm,n(r, P) and the functions uo,..., ug-1 have the following Laurent expansion in t: m-n
ug-1 ug-i In
t
(4.15)
0(0),
(i
particular, the even Mumford system induces stratification by the subsets J,,,,(I7, P).
a
=
2,
.
..'
g).
stratification
on
with the
It
can
be shown that this
gives all
Laurent
167
solutions,
see
[Van4].
Jac(r)
which coincides
Chapter V1. The Mumford systems
5. The
general
We will
case
modify
now
our
construction of
Chapter
III to construct for
a
large class of
a.c.i. system whose general fiber of the momentum map is the affine part of the Jacobian of a deformation of this curve. This new construction will coincide with the previous curves an
in
one
one case
(called
denote the genus
the
considered in Section 4.
hyperelliptic case),
0. Let g plane curve rp C C', defined by an equation F(x, y) of the smooth completion Pp of ]Pp, which we assume to be non-zero. Each
We start with
a
smooth
holomorphic differential
w on
Vp
be written
can
as
R(x,y) (X, Y)
!2y- UX) '9Y
for
some
polynomial R(x, y),
hence the choice of
a
basis of the space of
holomorphic differ-
entials leads to g polynomials Ro (x, y), Having fixed such a basis , R,- iL (x, y). for any c (co,...' c,-,) c C9 a polynomial F,. with corresponding curve rp, by -
-
-
we
define
=
g-1
F,(x, y)
=
F(x, y)
+
E ckRk (Xi Y)
(5.1)
-
k=O
The
following will
in
one
of the statements below be assumed
Assumption For a general point on r,, is given by
c
E
C9,
on
the
basis of the space of
a
Rk(XiY) dx 0 V (X, Y)
(k
I
=
0,..., g
-
curve
l7p:
holomorphic differentials
1).
This assumption, which is easily checked for any concrete curve at hand, is obviously valid hyperelliptic, trigonal, say n-gonal curves. For curves with a bad singularity at infinity however, the condition may fail; an example of such a curve was kindly communicated to us by H. Kn6rrer and will be given below (Example 5.2).
for
by constructing the affine Poisson vaxiety on which our a.c.i. system will live; closely related to the one in Paragraph 111.2.2 to which we will refer several times. We suppose that rp and the polynomials have been fixed (leaving aside the assumption at the moment). For simplicity we take the standard Poisson structure jyj, xj I Jjj on C9 (which corresponds to W I in Paragraph 111.2.2); also we take d equal to the genus g of the curve r, Let f h E O(Cg) be defined by We staxt
the construction is
=
=
,
f
=
JI(X, Xj)2, _
i<j
h
=
det
and let A' denote the union of their
depends
on
F
only (and
not
on
2
( R. (Xj
zero
,
Yj ))I
divisors, A'
the choice of
Rk)
168
=
so we
(f )
U
(h)
=
(f h).
will denote it
Notice that
(h)
by AF. Thus A'
5. The
general
case
A U AF. Proposition 11.2.35 leads to an affine Poisson variety (MI, morphism M, -+ 09 \ A'. Explicitly MI is given by
(XO' X0, (XI, YO'... (X q, Y9
MI
X0
,
II(X,
_
X
j)2
=
x
0
and
det' (R, (xj, yj))
a
=
Poisson
1
i<j
In view of the squares in the definition of
action
on
M, leading
to
M2
a
=
quotient (M2 I
f and h the symmetric *
*
i
i
I (t, t, u (A), v (A)) I
computed22 det2 (R, (xj, yj)) polynomial R(u, v). Finally we define Here
we
=
group
S,
defines
a
Poisson
1
t disc u (A)
=
t'R (u, v)
=
1}.
in terms of the ui and vi and called the
have
MF21
1 2)
I (t', u (A), v (A)) I t'R(u, v)
=
resulting
I
variety we were aiming at. This variety is determined up to isomorphism depend on the choice of basis Ri so we use a subscript F instead of R). As 111.2.2 we get a commutative diagram,
which is the affine
by in
(it does Paragraph F
not
MI
------------
0'
M2
IP2
PI1 (C2) \ A/ (a Paragraph
in which S is as
in
on
MF29,
restriction
of)
the map
,
S
(111.2.3) and all maps turn out to J* *12 on M2 descends to a
111.2.2 the Poisson structure
also denoted
by J* '12- Apart 1
M.F29
1
be Poisson maps: Poisson structure
from the verification that the brackets of the ui and
the vi do not depend on t, in the present case one also has to verify that the same is true for the brackets of these with t'. However, by Proposition 11.2.35 the latter are given by
so
they
are
independent of
jui't'll
=
IV,, till
=
t. For
present construction reduces
_t/21U, R(u, v)},, _t/2fV, R(u, v)jl,
hyperelliptic
curves
to the construction which
We now adapt our construction given replaced by the map23
=
was
h hence A'
given
in
=
A
=
Paragraph
in Section 111.2.3. The natural map
ftF : (C2) 2g 22
f
A.F and the 111.2.2.
fIFd
is
now
Cg
effectively compute R(u, v) for a concrete example one first replaces Vi by symmetric polynomial in xl,..., x.. This polynomial is easily rewritten in v(xi) terms of the elementary symmetric functions ul,. Ug. 23 We do not add the dimension as a subscript here, because it is implicit in F, namely In order to to obtain
a
..'
d
=
g is the genus Of
rF-
169
Chapter
A,-,
whose components
are
VI. The Mumford systems
defined by requiring that the
polynomial
g-1
j>1i(O11y1)7
...
,
(xg yg)) R., (x, y)
(5.2)
,
i=O
(x, y) (xj, Vj) the value F(xj, yj), where j 1, g. Solving for the f1i involves only the determinant of the matrix with elements Ri (xj, yj), hence we arrive at a regular morphism Hp : M -4 C9 which makes the following diagram commutative: has for
=
=
-
M,
M2
PiI
JP2
(C2)9 \ (A U A/)
M29 F
S
H1,
C9
Although the components of Hp depend on the choice of R, the algebra AF generated by these is clearly independent of it. It leads to an integrable Hamiltonian system (which is an a.c.i. system if the assumption is satified) by the following proposition (for a different proof of integrability see [Van5]). Proposition Curve
rF
C
5.1
Let
F(x,y)
C2 of positive
E
C[x,y]
genus g.
be such that
Then
(M.29, 1. F
F(x,y)
'12,
AF)
=
0
defines
defines
an
a non-singular integrable Hamil-
tonian system whose level set over 0 is isomorphic to an affine part of the Jacobian of 17,v and the flow of the restriction to this level of all integrable vector fields is linearized by this
isomorphism. When the above
assumption about I7p
is
satisfied then this integrable Hamiltonian system an affine part of the
a.c.i. system whose general level set over c E C9 is isomorphic to Jacobian of the deformation rF,, defined by (5.1) of the curve rFis
an
Proof Since S is
a
Poisson map it suffices to show that the components of ftF are in involution. 0 for all i, j. From. the definition of the components Ak of 7
Clearly IF(xi, yi), F(xj Yj)jh
fl.p
we
=
have
IfloRo (xi, yi) +... +A,-,R,-,(xi,yi),floRo(xj,yj)+---+A,-iLR,-I(xj,yj)},=O. (5-3) Since the second component depends only on (xj, yj) (although each individual Hi depends on all (xi, y,)) it is in involution with all coefficients of R1, (xi, yi) and (5.3) can be rewritten as
g-1
E IfIk, Al JRk (Xi, yi)RI (xj, yj) k,1=0
170
=
0.
5. The
general
case
0 for all k and 1. polynomials Rk are independent it follows that jAk, Al I algebra Ap generated by H1,...' H, is involutive. It has dimension g by the same argument as used in the proof of Proposition 111.2.3, hence Ap C[HI, H,]. As for completeness of A, notice that choosing a closed point in the spectrum Of AF Consists in fixing the values of the Hi, hence the level sets (in MF29) axe given by
Since the
=
Thus the
=
...'
g-1
F(A, v(A))
ciRi (A, v(A))
-
=
0 mod u(A),
i=O
where the ci are these fixed values. Denoting the polynomial on the left hand side by F, described by Proposition 111.3.3 and we see that they correspond to the level sets FF. ,, .
Lemma 111.3.4.
More
precisely they
particular the general level
In order to say curves
valid. If not to
Ho
=
...
by removing
a
divisor.
In
an
integrable algebra, hence that
(M2g, 1. F
12
1
AF) defines
Hamiltonian system.
integrable
those
obtained from these
set is still irreducible and every level set has dimension g. This
completes the proof that A.F is an
axe
more
about the
general level
sets
we
restrict ourselves from
now on
to
IPF for which the assumption (announced at the beginning of this paragraph) is then our argument is still valid for the fiber over 0 (i.e., the one which corresponds =
H,_1
=
0).
The main observation to be made here is that the restriction
Ap,
of the divisor 16kF Of
general fiber c of the momentum map (which is an (affine part of) a g-fold symmetric product of the curve l7pj is precisely the divisor which is blown down by the Abel-Jacobi map in order to construct the Jacobi variety of IPF,, from the g-fold symmetric product of rF,,. To see this, notice that the points of AF, correspond exactly to those divisors of degree g on IPF, for which the matrix R,(xj, yj) becomes singular, meaning that the space of holomorphic differentials with zeros at these points has positive dimension, hence (by Riemann-Roch) the dimension of the linear system defined by this divisor has positive dimension and this whole linear system is collapsed to a point by the Abel-Jacobi map. Thus, removing the zero divisor of (h) from C2g, which led precisely to our phase space MF29, has the effect of removing from each fiber the divisor which is blown down by the Abel-Jacobi map. h to
a
we had already removed a divisor (with several irreducible composymmetric product which included at least one component which maps to a translate of the theta divisor (in the notation of Paragraph 111.3.4 the latter components are the -6Fd(OOk))- Clearly each of the divisors in A. is linear equivalent to an effective divisor which contains ook, hence is collapsed to a point in the image of SFd(ook) (under the AbelJacobi map); since this is true for all k the image of A. is contained in the intersection of the translates of the theta divisor which were all missing already in our original affine part 2 of the syrmnetric product. It follows that our level set in MF'9 is now isomorphic to an affine the of If the of Jacobian the curve r F is not satisfied then we still assumption on part I7p.. have that the fiber over 0 is isomorphic to all affine part of the Jacobian of rp (this property
Recall however that
nents)
from the
may still hold for
some
We have checked
Abelian
variety;
and construct 7-
we as
Jacobian), giving
other
now
fibers). (under the assumption)
that the
general fiber is an affine part
of an
base N the space of all curves ]FF,, which axe non-singular the universal Jacobian over it (replacing every point by the corresponding may take
us
as a
the necessary
ingredients
for the
171
proof that
we
have
an
a.c.i. system.
Chapter VI. The Mumford systems
Only linearity of the a general point
flow
on
the Jacobians remains to be checked.
By
construction
we
have
at
g-1
1: AiRi (xj, yj),
F(xj, yj)
i=O
where
f1i
jyj, xj Ig
are =
the components of we have
Ap. Taking
the bracket
with xj, and
recalling that
Jij,
g-1
g-1
c9F
E Ri (xj, yj) X.#, xk + E Ai
57- (xj, yj) Jjk Y
i=O
Restricted to the invariant manifolds
"
(Xj, Yj)Jjk-
i=O
Aj
=
ci we have
9-1
-5Y (XjYj)Jjki
Rt(Xj) Yj)Xftj Xk i=O
g-1
i=0
which is
easily rewritten
Ri (xj, yj) 2F , ev (Xj,
Yj)
Xf-ii Xk
-Jjki
as
g
Rj (xi, yi) XX-6'kHk (X,, Y& ey
OF
assumption, we have on the left exactly a basis of the space of holorF. so we find that the vector fields X.#. (hence also the vector fields XHJ linearize under the Abel-Jacobi map, i.e., their flow is Unear on the Jacobian of 1P.F,. This shows that we have an a.c.i. system. Without the assumption we still have linearity of the flow on the Jacobian of IP.F (the invariant manifold over 0). Since rp satisfies the
morphic differentials
on
Let us also show that these a.c.i. systems satisfy the Adler-van Moerbeke condition algebraic complete integrability (which we did not impose in our definition of an a.c.i. system). That is, we show that along each of the components of the divisor which is missing from Jac(:PF,j at least one of the functions ui, vi, t' has a pole. Notice that ui and vi (when restricted to rF..) are given as symmetric functions on rF,, hence giving indeed meromorphic In the notation of 111-3.4 the missing components are the images functions on Jac(:rF.
for
DF, and gpg. As for the former, the functions latter, on a component 9F,,(OOk) where X(000 pole, and if Y(000 is infinite, then at least vg-1 has a pole along it.
under the Abel-Jacobi map of the divisors obviously all have a pole along it, for the
infinite, all
ui have
Example
5.2
valid for all
general
Here is Kn6rrer's
curves.
constant and
the
a
equal
curves
example
which shows that the above
assumption
vi
is m
is not
Notice that the assumption implies that the genus of the curves 17F. is to the genus Of IPF (for generic c); in the present example the genus of
will be
higher
than the genus of ]Pp
F(x,,U)
8 =
x
Y2+ Y 10
172
providing the counterexample.
Take
5. The
general
case
y) X6y. It is a curve of degree 10 whose closure X8y2 +y1O ZIO 0 has (I : 0 : 0) only singulax point. It is checked that this point is a cusp whose tangent has a contact of order 10 to the curve. It follows that the genus of the smooth completion of the curve (10-1)(10-2) 5 31. If however one adds a multiple of R, to the curve, which gives equals 2 and R, (x, as
=
_
=
its
-
=
RjX, Y) E C it still has
=
X8Y2 + y1O + CX6Y
singularity at infinity but it is now a cusp whose giving 33 as its genus. Thus for general c the genus of rp,, is bigger than the genus of 1F.F; if other holomorphic differentials are taken into account then it can of course only get worse. Thus I7p does not satisfy the assumption. then for
general
tangent has
a
c
only
one
contact of order 6 to the curve,
173
VII
Chapter
Two-dimensional a.c.i. systems and
applications
1. Introduction
The last
chapter is devoted to the study of several two-dimensional integrable Hainiltoare of interest, especially from the point of view of algebraic geometry.
nian system which
We specialize the odd and even Mumford systems to the case of genus 2 in Section 2. give explicit equations for these systems, which will be useful when studying the other examples, which will all be shown to be related to the genus 2 Mumford systems. An appli-
We
(genus two) even Mumford system was worked out in collaboration with JoS6 (see [BV2]) and is given in Section 3. We consider an arbitrary curve of genus two
cation of the
Bertin
which has
automorphism of order three. This automorphism can be extended to its Ja, a singular quotient, similax to the classical Kummer surface. We exhibit a 94 configuration on this quotient and show that it is a complete intersection of a quadric and cubic hypersurface in p4. By using the even Mumford system we explicitly compute equations for the quadric and cubic hypersurface, thereby giving a projective realization of these generalized Kummer surfaces. an
cobian and leads to
We study by Garnier as
in Section 4
an
integrable quartic potential
on
the plane, which
was
discovered
very special case of a laxge family of integrable Hamiltonian systems which he derived from the Schlesinger equations (see [Gar]). In Paragraph 4.1 we will show in two a
different ways that the general fiber of its momentum map is an affine paxt of an Abelian surface of type (1, 4). One method uses some of their specific geometry, as given in the beautiful paper
[BLS],
the other
It follows that the Gaxnier to the odd Mumford
Paragraph
4.5
we
one
potential
system leads
consider the
is based
is to
an
a
limiting
on a
morphism
to the odd Mumford
system.
a.c.i. system of type (1, 4) Moreover the morphism Lax representation of the Gamier potential. In
case
in which the
175
P. Vanhaecke: LNM 1638, pp. 175 - 241, 1996, 2001 © Springer-Verlag Berlin Heidelberg 1996, 2001
potential is
a
central
potential.
Chapter VIL Two-dimensional a.c.1. systems Then the is
a
general
C* bundle
fiber of the momentum map is not
over an
elliptic
aai
affine part of
an
Abelian variety but
curve.
a (1, 4)-polarization there is associated a rational map (we assume here that the Abelian surface is not a product of elliptic curves). The image of this map was shown in [BLS] to be an octic and its coefficients parametrize a 24: 1 cover of the moduli space A(1,4) of (1, 4)-polaxized Abelian surfaces. The octic is an unramified cover of the Kummer surface of a Jacobian which is canonically associated to the (1, 4)-polarized
To the line bundle which defines
to
p3
Abelian surface. One way to characterize this Jacobian is that it is the unique Jacobian J for which the map 2j (multiplication by 2) can be factorized via the Abelian surface. We will give in Paragraph 4.3 some other characterizations of this Jacobian. It should also be
remarked that the
morphism
to the odd Mumford
system precisely
maps the
general
level
affine part of an Abelian surface of type (1,4) to its canonical Jacobian, in particular we can use this morphism to explicitly describe the natural map between A(1,4) and the moduli space of Riemann surfaces of genus two. The result is surprisingly simple; looked at in another way it shows how to write down explicitly for a curve (given by its equation)
set, which is
the
an
equation of the
Kummer surface of its Jacobian in the classical Rosenhain form.
We also get an explicit description of the moduli space A(1,4) It is obtained as a quotient of the parameter space which is given by the coefficients of the octic. The group which is acting on it has order 24 and is isomorphic to Z/4Z x 63; the action was suggested by an automorphism of the system. The final result is that A(1,4) is birational to a cone M3 in -
weighted projective
space
p(1,2,2,3,4)
for which
we
give explicit equations.
Another a.c.i. system that is naturally related to Abelian surfaces of type (1, 4) is the in geodesic flow on SO(4) corresponding to some special metric, as was first pointed out [BV4]. In this case, studied in Section 5, the general fiber of the momentum map is an affine of a Jacobian, but there is a group of translations acting on this fiber, so
hyperelliptic. naturally associated
part
that there is
to this fiber
an
Abelian surface of type
the cone M3 in p(1,2,2,3,4) will be map from the base space to for its relation to the moduli problem for a.c.i. systems.
(1, 4).
The
explicitly computed.
resulting
See
[BV4]
Their a hierarchy of integrable potentials V,, on the plane. of the general level sets of the mothe but checked geometry revealing easily integrability mentum map requires more work. For the first non-trivial member V3 we define a morphism to the odd Mumford system and deduce from it that V3 is algebraic completely integrable and we identify the general fiber of the momentum map as an Abelian surface of polarization Painlev6 analysis by Adler and van type (1,2) (this result was previously established by using Moerbeke, see [AM9]). For V4 there is a similar morphism to the even Mumford system and of a hyperelliptic Jacobian, which is ramified we show that its general fiber is a 2 : 1 cover this result could also be obtained by using the theta of translates divisor; two touching along Moerbeke system, which is discussed in Paragraph 2.3. a morphism to the Ber-hlivanidis-van We look in Section 6 at is
For
V,,
we use a
morphism
to show that the level sets
axe
2
-
1 unrarnified
covers
n+L
of
a
certain
hyperelliptic curve of genus [ 2 1. We also exhibit Lax equations (with a spectral parameter) for the hierarchy. In the last section we reconsider the periodic three body Toda lattice. We use again The order three a morphism to give a new proof of its algebraic complete integrability. to an a.c.i. automorphism which comes from a cyclic permutation of the three particles leads this of realization concrete quotient system. system of type (1,3) and we give a
two-dimensional stratum in the Jacobian of
a
176
2. The genus two Mumford
2. The genus two Mumford
systems
2.1. The genus two odd Mumford For future
doing this, let
give explicit
use we now
us
restrict the odd and
systems
system
formulas for the genus 2 Mumford systems. Before Mumford systems to a Casimir which is common
even
to all the Poisson structures I., JW, where deg W :5 2, without restricting the class of (hyperelliptic) curves that appear as spectral curves for these systems. Namely, we consider for the odd (resp. even) Mumford system the hyperplane given by 0 (resp. H2,+I H2, 0). =
Since this
=
fixing a level of a Casimir all formulas are easily transcribed: for the odd (resp. even) case, simply substitute w. -+ ug-I (resp. wg+l -+ ug-1) in all formulas. We start with the odd case. Then the Lax operator is given by corresponds
to
A(A) and the Lax
(
=
0
A2 +
+ Vo
WI'X +
+
UI A + Uo
Vo
)
X, and X2
are
-VIA
Wo
vector fields
1
(A-2u,
=
-
Bo (A)
0
('\2
=
(2.1) given by XH,A(A)
,&(A)
=
2v(A),
b(A)
=
(A
zb(A)
=
-2v(A)(A
-
2u,)u(A) -
by
a
prime, these
w(A), 2u,),
In order to write down
-
are
\+Ul)
v1 u
1A+WI
denote derivation in the direction of the first vector field
of the zeroth vector field
even
UI'X2
where
BI (A)
we
_
equations for the Hamiltonian
[A (A), Bi (A)],
If
VIA
A3
written out
u'(A)
=
2v(,\) (A
+
V/ (A)
=
U
(,\) ('\2
_
w'(A)
=
2w(A)vI
a
-VI
dot and in the direction
as
ul)
-
2v,u(A),
U1 'X + W,
-
2v(A) (,\2
_
-
explicitly the compatible Poisson namely in the following form:
by
-Uo
Uo)
-
(A + UIMA),
UIA + W1
_
Uo).
structures it is useful to have these
further written out,
fil
2vI,
ul
2vo,
fio
2vo,
uO
2(uIvo
2u
I
=
uo
o
=
-2uuo
tbi
=
4uIvI
tbo
=
4uvo,
Recall also that the
-
2
-
_
-
WI,
wo,
2vo,
v'
-
uovj),
=
-2u,uo
vo
=
_UIwo + Uo (WI
wl
=
2(uIvo
+
=
2(uovo
+ vIwo
wl0
corresponding integrable algebra A U(A)W(,X), i.e., by
is
-
a
wo, -
UO),
uovI), -
vow,).
polynomial algebra generated by
the coefficients Of V2 (A) +
2
H3
=
uo + w,
H2
=
-uluo + uIwI + wo + v, ,
HI
=
ulwo + uow, +
H0
=
UoWo +
-
ul, 2
V201 177
2vIvo,
(2.2)
Chapter VIL Two-dimensional ax.i. systems
0). We now explicit for this system the formulas for the three 1, A"\2 Their Poisson matrices are easily I., -J'P, where W compatible computed from (VI-4-4) in terms of the system of generators ul, uo, vi, vo, wi, wo (in that order). Notice that since S4 is not a Casimir of I-, -IA' the latter Poisson structure does not
(recall
that
we
have put H4
=
Poisson structures
restrict to the level S4
It is easy to
=
.
-I'
0. The Poisson structure
=
is
given by
0
0
0
1
0
0
0
0
1
U1
0
0
-1
0
0
1
-2vi -2u,
-1
-U,
0
0
-Ul
0
0
-1
0
2vi
2%
compute that H2 and H3
U1
uO
w,
-
W1
-
0
2v,
-2vj
0
(2.3)
UO
Casimirs and that H, and HO give the first and (i.e., it is a modified Lie-Poisson
are
zeroth vector fields. Notice also that this structure is affine structure, see Example 11.2.14). The Poisson matrix of the
(affine)
structure
is
given
by
In this
case
HO and H3
0
0
1
0
0
0
0
0
0
-UO
0
2vo
-1
0
0
0
-2ui
0
0
UO
0
0
-UO
-WO
0
0
2u,
UO
0
-2vo
0
-2vo
0
WO
2vo
0
are
\
In this
case
HO and H,
that this structure is
polynomials HO,
.
.
.
,
the Casimirs and H, and H2 %2
the Poisson matrix of
is
(2.4)
give the
0
-U,
-UO
2vi
0
0
-UO
0
2vo
0
U1
UO
0
0
-WI
-WO
UO
0
0
0
-WO
0
-2v, -2vo
-2vo
W,
WO
0
0
0
WO
0
0
0
are
Finally
2vo
0
even
two vector fields.
given by
(2.5)
)
Casimirs and H2 and H3 give the two vector fields. Notice The Hamiltonian vector fields which correspond to the
linear.
H3 using these
I., Hjw
1.
j'\2
structures
are
summarized in the
H3
H2
Hi
HO
0
0
X,
X0
0
X,
X0
0
XI
X0
0
0
Table 3
178
following table.
2. The genus two Mumford
2.2. The genus two We
systems
Mumford system
even
give the corresponding formulas for the genus 2 even Mumford system. The I., .1w, which we computed from Proposition 111.2.1. These structures are linear nor affine. The Lax operator is given by now
Poisson structures neither
A(A) and the Lax
(
=
VIA A4
UIA3
_
+
equations for the Hamiltonian
[A(A), Bi (A)],
A2
+ UIA + UO -VIA VO
+ VO
W2A2
+
WIA +
Wo
vector fields
-
X, and Xo
)
given by XHA(A)
are
where
Bi (A)
0
1
2 A2 -2uA+2u,-UO+W2
0
=
)
,
and
They
are
A3
=
2vI,
ito
=
2vo,
i),
=
2u3+ UIW2 I
,bo
=
?b2
=
2(2uIvI
?bI
=
2(-2u2v, I
=
2 1
-
UIA2
+
vector fields
as
fil
?bo The
written out
A + U,
V1
Bo(A)
(W2
-
C7
on
uo)A + 2u,uo
uO
2u2UO 1
3uuo
2+ UOW2
-U 0 -
-2vo(2u
-
V,I
wi,
V
vo), + _
2uIvo UO +
-
=
2vo,
=
2(uIvo
=
2u 12UO
uovI),
-
-
U2+ 0
I
WOi
-
-VI
as
U1I
-
+ w,
W,2
=
W,I
=
W,0
=
VIUO)i
VIW2 +
W2)5
corresponding integrable algebra A
is
now
the
0
=uow, + u,
2(uIvo
UOW2
(2U20
_
WO),
uovI), -2(2uuov, + VO(W2
2(vlwo
WOi
-
+
-
vow,
-
-
UO)),
2u,uovo).
polynomial algebra generated by
2
H4
=
W2
H3
=
WI
H2
=
UOW2 + UIWI + WO + V17
H,
=
ulwo + uow, +
Ho
=
uowo + vo,
-
-
U1 + UO, U1UO + UIW27 2
(2.6)
2vIvo,
2
We also write down the three compatible Poisson structures for the genus 2 even Mumford system (for W 1, A, A2); their Poisson matrices are now written in terms of the system of generators U1, U07 VIi VOi W21 Wli WO (in that order). Here is the Poisson matrix of 1.,jI =
0
0
0
1
0
0
0
0
0
1
U1
0
0
-2v,
-2u,
0 -
0
1
-1
0
0
1
-Ul.
0
0
-UI
0
-1
U1
0
0
0
0
2vI
uo
-
2u, 2U21
UO -W2 -W2
-2uuo
-
w,
179
W2
-
UO
0
0
0
-2v,
4uIvj
W2
-
uo +
2u,uo 2v, -4uiv,
w, +
0
2
2u,
(2.7)
Chapter VIL Two-dimensional a.c.i. systems It has
H2, H3 and H4 0
1
0
0
0
0
0
0
0
-UO
0
0
2vo
-1
2U21
0
0
0
0
-2u,
0
UO
0
0
-UO
2ujuO
-WO
0
0
2ul
UO
0
0
0
0
one
vector fields. The
given by
is
0
\ 0
This
give the first and zeroth
Casimirs and H, and HO
as
Poisson matrix of
UO
-
W2
has HO, H3 and H4
structure is
-
2u 2I
as
-
UO +
-2uuo
0
0
-2vo 4u,vo
WO
2vo
-4u,vo
0
0
-2vo
W2
Casimirs and H, and H2
(2.8)
-
the two vector fields. A third
give
given by
0
0
-Ul
-UO
0
0
0
-UO
0
0
U1
UO
0
0
UO
0
0
0
-2v, -2vo
-2vo 0
UO
_
2vo
-WI
-WO
0
0
2uuo
-WO
0
-2u,uo
0
-2vo + 4u,vl
4uvo
W1
WO
0
0
4u,vl 2vo -4u,vo
0
WO
0
0
0 uO
2U2I
2v, 2vo
-
2U21
-
(2.9)
case HO, H, and H4 axe Casimirs and H2 and H3 give the two vector fields. Finally, a fourth structe, with H2, H3 and H4 as Casimirs, H, and H2 givng the two commuting vector fields, is given by the following (in order to make this anti-symmetric fit on this page we only
In this
print its
upper
0
0
0
triangular part) 2
U,
-
2v, 2vo 0UIW2 0UOW2
2(vo
UOU1
UO
U20
UOU1
0
-
-
ulvi)
-
-2u,vo -2uovo
-2uovl U1W1
W1 WO
0
-
UIWO
WO
UOWj
UOWO
2vjW2
2vOW2
2(vow,
0
-
(2.10)
vlwo)
0
As above
we
using these
summarize the Hamiltonian vector fields which correspond to HO, a table.
H4
upon
structures in
I., Hjjw
H4
H3
H2
H,
HO
I ,11
0
0
0
X,
X0
1.'.Y\
0
0
X,
X0
0
\2
0
X,
X0
0
0
X,
X0
0
0
0
j.'j
1-'-11'3
Table 4 It
can
be checked
by direct computation that
these brackets also
verify (VI.3.7),
conceptual proof (given for the odd Mumford systems through the loop algebra missing. a
180
but
g1q)
is
2. The genus two Mumford
2.3. The Bechlivanidis-van Moerbeke
systems
system
In this
paragraph we wish to consider an integrable Hamiltonian system on C' which was by Bechlivanidis and van Moerbeke (see [BNq) in order to understand the geometry of the Goryachev-Chaplygin top. This system is sometimes called the s even- dimensional system, but since the dimension of an integrable Hamiltonian system was defined as the dimension of the general fiber of the momentum map, which is two in *this case, we prefer not to call it this way; we prefer to call it the Bechlivanidis-van Moerbeke system. It was constructed as a pair of commuting vector fields together with five independent functions whose derivatives in the direction of these vector fields was zero; this was done without making reference to any Poisson structure for which the equations of motion were Hamiltonian. Several such Poisson structures (which are compatible) were constructed for it by us in [Vanl]. Not only will these be reproduced here but we will show that the Bechlivanidis-van Moerbeke system is isomorphic to the genus 2 even Mumford system, thereby proving its algebraic complete integrability (a proof of this has been given nowhere) and "explaining" where the bi-Hamiltonian constructed
structure of this
system
which is found in
comes
from. There is for this system
a
third
(compatible)
structure
natural way; it is closely related to the Poisson structure of the Toda lattice discussed in the last section of this chapter; of course this structure can also be transported to the even Mumford system, thereby leading to another compatible structure for that a
system. We
pick coordinates
sl,
C7 and consider the following algebra of functions,
87 on
A
=
C[Sli S2, S3i S4, 651,
where 2
S,
=
S2
=
si
S25
_
-
8S41
4s6,
5132 +
S3 =,92 4
Also consider the Poisson matrix
432
-
+ 531
S4
S486 + 3537 + 52831
S5
-86 +
2
82-SIS37
(18il 8jD,
0
0
-16s5
0
-8
0
0
0
0
0
0
0
4
16s5
0
0
2s5
2S4
-4S2S5
0
0
0
-1
0
8
0
-2S5 -2S4 4S2S5 -4S2S4
4S2S4 -2S2
1
0
-2S2
0
0
2S2
2S2
0
0
0
-16S2
-4
16S2
2
0
4S22 +
(2.12)
-4S2 S,
_
S,
0
we do not suggest to the reader to do this (because it is long), it can be checked by direct computation that this matrix defines indeed a Poisson structure on C7; we will check it in another way. What is however checked at once is that S1, S2 and S3 are Casimirs for
Although
181
Chapter this structure and that S4 and
VIL Two-dimensional a.c.i. systems
S.5
are
to the vector fields
involution; they lead
in
(up
to
a
constant) bi
=
-8S7,
si
s'2
b2
=
4s5,
h
=
2(S4S7
h
=
-4S2S5
b5
=
S6
=
-,5ls5 +
b6
b7
=
+
506)1
S3
S14 S15
S77
-
48234,
-
2S2S6
S154 +
S16
2S2S77 483.
-
=
8(sIS5
=
4s7,
=
4(S2S586
=
s1s5
+
-
=
S134 +
=
-SIS7
817
=
2S2S7)j + 82S4S7
-
2s3s5)7
2SO7, 232S6 -
-
4S3,
2slS2S5
882S3 + U01284
2
-
452S77 2 4s 2 S6
-
-
8186-
writing down other Poisson structures, let us prove that the Bechlivanidis-van Moerbeke system is isomorphic to the even Mumford system by giving explicitly the isomorphism; this map was constructed in [Van2] as an illustration of the algorithm (recalled in Section VA) which was developed there. The isomorphism is given by the map Before
0: C7
C7
_+
:
S7)
(Sl
'-+
(U (A)
,
V
(A)
2
W
(A))
where
U(,X) v(A) W(A)
=
=
=
A2
+
2S2A +
Sli
8S7,
8s5A
-
A4
2821\3
+
(SI
4(s, '922
+
85256
-
+
-
16S4 -
-
4S2),\2 2
-
4(-92(Sl
-
8S4
-
2S2) 2
+
4s6)A
1653)-
Of course this map is regular, but it is even biregular: S1, 82, s5 and S7 are of course regular in terms of the coefficients of u(A) and v(A); given this S4 is regular in terms of these and
the coefficient of 'X2 in
w(A); the same w(X). Next it
coefficients in A and XO in
0*(W2 0* (WI
follows for S6 and S3 upon
using respectively
the
is easy to check that
884), uo) 2(s, 4s 22 -4(.5182 + 4s6), UIUO + UlW2) 2 U1
-
+
=
-
=
-
0*(UIWI
+ UOW2 + Wo +
0*(ulwo
+ uow, +
0* (UOWO
+
V2) 0
V2) 1
2vivo) 2
=64 (8 7
_
=
'926
=
-64(324
_
-128(54,56 _
$183)
+
'925 +S3)+(SI
+ S587 +
4(s182
52S3)
-4 S2 2-8S4 )2, -
4(s,
-
4 S22
-
84)(SIS2
+
4S6)7
+ 486 )2.
(2.13) A, where A! denotes here the integrable aJgebra of the (genus 2) even Mumford Since 0 is a biregular map the fact that A! is integrable for some Poisson structure, system. O*A is integrable for the corresponding Poisson structure; moreover since A! is a.c.i. the same holds true for A. The Poisson structure which corresponds to (2.9) is given (up to a constant) by (2.12) hence (2.12) defines a Poisson structure and 0 is a Poisson map with respect to the Poisson structures (2.9) and (2.12). Since 0 is biregular we can also transport the other
Thus
0* A!
=
182
2. The genus two Mumford
Poisson structures. If
constant)
In this
the
case
we
following
systems
transport the Poisson structure given by
(2.8)
then
we
get (up
to
a
Poisson matrix:
0
0
-1657
0
0
0
0
0
0
0
-4
0
1637
0
0
2S7
-432-97
0
0
0
0
4
-432
0
-Sl
0
0
0
0
31
0
2s,82
8s,
0
-2S7 286 48237 8S3 4S286
-2,36 4S2
81
0
-2s,S2
0
S1, S2 and S5
-
are
-8s, 0
8S3
-
0
48256
-SI
Casimirs and S3 and S4 generate
(up
to
a
constant)
the vector
fields X, and X2; this can be checked directly or by using (2.13). One might think now that the structure (2.7) will give a Poisson bracket for which S1, S4 and S5 are Casimirs but S2 and S3 generate the two vector as follows:
fields, but this
is not the
O*H4
=
O*H3
=
-4S2,
O*H2
=
S12
0*H1
=
-128S4
0*H0
=
6485
case.
To
see
this, let
us
rewrite
(2.13)
2S1,
-
64S3,
+
-
4SIS2,
4S22.
isomorphism it maps Casimirs to Casimirs. Thus, if H2, H3 and H4 S.1, S2 and S3 are Casimirs for the corresponding structure. Similarly, if H0, H3 and H4 are Casimirs then S1, S2 and S5 are Casimirs for the corresponding structure. However, if H0, H, and H4 are Casimirs then S1, 32S4 + S, S2 and 16S5 + S21 are Casimirs, so that (up to a multiple) S2 and S3 generate X, but S4 and S5 give respectively SiX2 and S2X2Since
are
0
is
It follows
has one
a
Poisson
Casimirs then
that there is
even
no
linear combination of these three Poisson structures which
S1, S4 and S,5 as Casimirs, while S2 and S3 give the vector fields X, and X2. finds by trial and error easily that the following structure does this job:
4(s586
S487)
452S6
0
0
287
-2S6
2s,85 + 482s7
0
0
0
-S5
-54
37
36
+
54,57)
-2s7 286 2s,85 -4S237 2sS4 48286 -
-
-4(s586
+
-
However
2s, 84
0
0
0
0
-28385
2S3,94
85
0
0
0
0
-S3
94
0
0
0
-S3
0
-S7
2S385 -2S384
0
-53
0
-2-92,53
-3
0
2S283
0
-S6
(2.14) important because it will appeax later when studying the Toda lattice. It is compatible with the previous ones in view of Theorem 14 in [Van2] which says that if in a two-dimensional integrable Hamiltonian system all integrable vector fields axe Hamiltonian with respect to two different Poisson brackets, then these brackets are compatible. Of course This
one
is
it leads also to
generalization
a
fifth Hamiltonian structure for the genus 2 even Mumford system. For even Mumford systems, see [FVJ.
a
of this special structure to all
Using the isomorphism everything can be transported from the genus 2 even Mumford system to the Bechlivanidis-van Moerbeke system. Since we have Lax equations for the even Mumford system we also have them for this system. Similaxly we obtain all possible Laurent 183
Chapter VIL Two-dimensional ax.i. systems
solutions from those of the real
even
Mumford system, and since all coefficients of the map
are
knowledge about the topology of the real level sets of the even Mumford system to describe the topology of these level sets for the BechlivaiAdis-van Moerbeke system. Finally, in the next section we will use the even Mumford system to study a certain problem in algebraic geometry; clearly we could use the isomorphic Bechlivanidis-van Moerbeke system we can use our
instead.
184
3.
3.
Application: generalized
Kummer surfaces
Application: generalized We
now
which
was
which
use
give
an
Kummer surfaces
application of integrable
Hamiltonian systems to algebraic geometry, (see [BV2]). For other applications
worked out in collaboration with Jos6 Bertin the
same
3.1. Genus two
technique,
see
with
curves
[PV2]. an
of order three
automorphism
equipped with an automorphism of order three, quotient I'l-r has genus zero and r has By four fixed points. Since r has genus two it is also hyperelliptic; as before the hyperelliptic involution will be denoted by a. We have the following diagram We consider
denoted by
a curve
r of genus two,
the Riemann-Hurwitz formula the
7%
-ITL11- P1
IF
7r,I P
1
points to Weierstrass points, hence the commutator IT, a] o--r since the only automorphisms which fix all points and we see that ra Weierstrass points are o- and identity. It follows on the one hand that -r induces on P' hand that the four a fractional linear transformation -7 of order three, and on the other therefore We o--orbits. of two consist of fixed points -r suppose that - is given by may coordinate x on P' such that these two orbits a c choosing by exp(!'-), ex, -T(x) 3 0 and x oo. The images of the Weierstrass points form two orbits of correspond to x A3 and X3 A-3, three points under f, which correspond to the roots of the equation X3 A-3, possibly after a rescaling of x. Obviously A :;:4 0; since both orbits are different, A3 i.e., A6 =? 1. This shows that IP has an equation -r
necessarily
maps Weierstrass
fixes all these
=
=
=
=
=
=
Y2
=
(X X6
with
r.:;4
3 _
X3) (X 3
+ 2nx
3
+
_
A-3)
(3.1)
1,
1.
of genus every equation of the form (3.1) with K: 1, defines a smooth curve automorphism. (x, y) -+ (ex, y) of order three; also, if K in (3. 1) is replaced by -K then an isomorphic curve is obtained. Conversely, let there be given two isomorphic curves r and 17' with respective automorphisms -T and r' of order three. We may suppose that the r'o. We claim that if r, isomorphism. 0 : IP -+ r' respects the automorphism, i.e., Or
Clearly,
two with
an
=
and 172
are
written
as
above
as
Y2 2 then nj2 K2. To see this, notice that 0 linear transformation 0 which satisfies =
O(x, y)
=
(px, y), iving pl
=
=
X6
+
2nix3
+
1,
i bviously pmmutes with a hence there is an induced O(ex)
1. It follows that
=
eo(x),
X2 A 1 "
185
=
for all
can
x
E
be taken
P'. Thus as
O(x)
=
px and
modular parameter.
Chapter
VIL Two-dimensional a.c.i. systems
The
automorphism group of r contains a subgroup which is isomorphic to S3 x Z/2Z, as immediately from (3.1); it actually coincides with this group, unless K 0 (in which case the group of automorphisms jumps to D6 x Z/2Z). Namely, there is, apaxt from the hyperelliptic involution a, an action of S3 by means of which the Weierstrass points belonging is
seen
=
to oner-orbit
can be at random permuted. For future use we choose an element p of order two symmetry group S3 corresponding to a transposition in S3, say P(X, V) (X-1, yX-3) and notice that it commutes with a but not with r. Its fixed points axe the two points in
in this
=
7r-1111,
hence
IP/p
is
an
elliptic
curve.
We will find it convenient to denote the fixed
of -r, which
points
are
mapped by
7r, to 0
(resp. oo) by ol and 02 (resp. oo, and 002). Then o" 027 000 002 and we may suppose oo, giving also /L(02) p(oi) 002- In the same way we denote the Weierstrass points Akorbit by Ai, -r(Ai) corresponding to the x3 Ai+1 (indices are taken modulo 3) and the 3 A-3 -orbit by i, p(Ai) ones corresponding to the x i. Then the action of S3 x Z/2Z =
=
=
=
=
=
on
these
=
points is contained
=
in the
following table.
order
01
02
001
002
Ai
i
3
01
02
001
002
Ai+1
i_l
0'
2
02
01
002
001
Ai
i
A
2
001
002
01
02
Xi
Ai
Table 5
3.2. The 94 Let
J(r)
configuration
on
the Jacobian of 1P
denote the Jacobian of r and for in
corresponding point
J(r)
as
a
divisor D of
degree 0, let Ab(D) denote the Q1, Q2 E r, every element
before. Recall that for any fixed
r. J(r) can be written as w Q1 Ab(Pi + P2 Q2); moreover this representation is unique if and only if P, =7 P21, all P + PO' and Q + Q11 (P, Q E 1P) being linearly equivalent, P + P" -1 Q + Q1. In the present case of curves (3. 1) which have an automorphism T of order three, the cover -7r, associated to -r provides in addition (using the notations of the previous section for the fixed points of -r) the following linear equivalences
w
=
-
3o, The
by
302
automorphism -r extends in a natural It is given and well-defined for w
-r.
Ab(-r(PI)
+
-r(P2) -,r(Ql)
Proposition3.1 on
-1
The
-
-
3ool
-1
way to
an
-1
Ab(PI
=
3002-
(3.2)
automorphism on J(I`), also denoted P2 Q, Q2) as follows: -r(w)
+
-
-
7(Q2))-
automorphism
-r
has nine fixed points and nine invariant theta
curves
J(r).
Proof The
J(r)
-+
principal polarisation
i(r)
from
follows from the first
J(r)
on
J(]P)
to its dual
is invariant under
j(]P)
is
Aut(r), hence the isomorphism Aut(r)-invariant and the second statement
one.
186
Kummer surfaces
Application: generalized
3.
We count the number of fixed points holomorphic Lefsehetz fixed point formula
E (- I)-" Trace f
of
in two different ways.
r
HP,o (m)
we use
the
(3.3)
B,,)'
det (I
f (P.) P.
P
At first
holomorphic map f : M -+ M, where B,, is the linear part of f at the fixed point p,,. -r and M i(r); in this case HP,O(j(r)) may be identified with the apply it for f p-th skew-symmetric power of the cotangent bundle at any point of J(r). For the left-hand side in (3.3), the basis of HPO (j(r)) may thus be taken in a point Ab(PI + P2 Q, Q2) as jQIi n2l JWI (PI) +WI(P2)i W2(PI) + W2(p2)}, where wi x'-'dxly and Q1 A Q2 generates H2,0 (j(r)). Since -r* Qj e'Qi, (i 1, 2), the left hand side in (3.3) gives for
a
We
=
=
-
-
=
=
=
=
2
1: (- 1)" Trace -r* I H-0 (.T(r))
1
=
-
Trace
( 0' 02)
P=O
As for the
all B,,
right hand side, obviously
equal,
are
in fact
0
B,,
and the number of fixed
points of
B,,)
-
axe
(1
=
(3.4)
IE2
0
when local coordinates dual to Q, and Q2 det (I
(5 )
=
picked around the point P,,. Therefore _
6)(1
_
62)
=
3,
is indeed nine.
-r
A second way to determine the number of fixed points of -r is by writing down an explicit list: if we write every point W E J(r) as w Ab(PI + P2 2ool) then rw w if and only if =
7*(Pl) +r(P2) we
arrive at
10,
01
-
P2, i.e., P, the following list -1 P, +
021 02
The nine invariant of IP in
i(r) by
four fixed
-
011 001
M21 002
-
curves are
the map
then
-+
x
P20'
=
or
-
0011 01
given by
Ab(x
-
are
both fixed points for
001) 02
-
0021 01
the nine translates
ool).
-
=
-
P, and P2
Since this
curve
over
-
0021 02
these
obviously
-r.
-
Using (3.2)
0011-
(3.5)
points of the image exactly the
contains
points
10, each of the nine invariant
point belongs
curves
to four invariant
Ix
002
-+
0011 01
-
will contain
curves
Ab(x
-
-
0011 02
-
0011i
exactly four fixed points. Dually, every origin 0 belongs to the four curves
fixed
since the
ooi),
x
-+
Ab(x
-
oi),
i
=
1, 21.
Notice that the fixed points form a group F (isomorphic to Z/3Z ED Z/3Z) which is a subgroup of J3 (1P), the three-torsion subgroup of J(]P). On J3 (1P) there is a non-degenerated alternating form (-, -) induced by the Riemann form corresponding to the principal polarisation. The subgroup F C J3(r) has the following property. 187
Chapter
VIL Two-dimensional a.c.i. systems
The group F of fixed points of Proposition 3.2 of J3 (IF) with respect to the Riemann form
-r
on
J(17)
is
a
totally isotropic subgroup
Proof -r
is
where
x
of j3(r) (Z/3Z)4, which satisfies 1 + T +,r 2 0; also 2. It follows that F consists exactly of the elements of the form 1) -r(x) x J3(r). Finally, if y E F, then obviously (y,,r(x) x) 0. 0
symplectic automorphism
a
dim ker(T
=
=
-
-
E
=
-
Apart
from the Riemann
form, which coincides on J3 (1P)
with Weil's
pairing e3 (see [LB]) analogous to Mumford's quadratic form (theta chaxacteristic) on the two-torsion subgroup J2 (17) of J(r) and can be defined in complete generality. It measures the obstruction for a line bundle L to descend to the quotient J(]P)Ir. One can define it as follows. Choose a lineaxisation of L with respect to the cyclic group Z/3Z generated by r, i.e., an isomorphism 0 : C4,r*(L) with 0(0) Idco. When x is a fixed point of T, then 0 induces an isomorphism, of C., which is multiplication by a root of unity e(x), and e : x i-+ e(x) is the desired function. It depends on the choice of C itself and not only on the polaxisation. If E) is the (theta) divisor which corresponds to L, i.e., C [E)J, then the corresponding e ee may be computed as follows. 0 be a local defining function for 19 in x. Since the divisor E) is non-singular, the Let f leading part h of f is linear and we have r*(h) e(x)h. Since the singular points are of type A2, as is seen from (3.4), there exist local coordinates ju, vJ at x such that r* (u) eu a
function
can
be defined
on
F with values in the group of cubic roots of unity. It is
=
=
=
=
=
=
and if
x
or
e 2v. Therefore we have either h u and e(x) T*(v) 1. It follows that ee is explicitly given 0 19 then e(x) =
=
=
=
e,
or
for all
h x
=
v
EE F
and
e(x) by ee(x)
=
e2 Also .
=
-r,,ITe,
equivalently
ee(x)v The
p and
automorphisms
It is desirable to have
a
ain observation of this
a
act
=,r*v
on
F
for all well
as
G
v
TxE).
as on
(3.6)
the set of invariant theta
curves.
"totally symmetric" theta curve, i.e., invariant by -r, o- and p. The paragraph, from which the 94-configuration is a consequence, is the
following. There is a unique totally symmetric theta curve among the nine invariant function ee associated to this curve E) is a quadratic form on F; it is given suitable basis of F and upon identification of the group of cubic roots of I with F3 by
Proposition
3.3
theta
The
in
a
curves.
ee(r, s)
2 =
r
_
S2
(mod 3).
Proof The existence of the
Aut(r),
may find invariant divisor. It is we
an
curve
is clear:
since the
polarisation is invariant by the group gives this polarisation, hence also an
invariant line bundle which
unique since if there are two Aut(]P)-invariant curves, then their (two) points must be invariant under Aut(r) which is impossible (see Table 5). It is easy to identify 0: it is given by the image of P -+ Ab(P + oojL 2002). To see this, notice that this image can be written as intersection
-
P -4
Ab(P + S,
+
S:OL
-
3S2),
independent of the choice of Si, S2 E 1011 02 0017 002}. From this representation clear that 19 contains the four points Ab(Sf SI)7 S, E 1011 021 0011 00211
-
188
it is also
3.
Let
Ab(02 v
E
-
Application: generalized
Kummer surfaces
determine ee in terms of the basis 1 1, 21 where (I = Ab(002 ool) and (2 2 1 it follows using the chain rule that if r(x) x and Sincer mra and a
us
-
00-
=
=
=
7:x., E) then
ee(xo")v hence
ee(x")
Therefore, given by
if
=
we
=
-rv
=
o-.,ro-,,v
=
ee(x)o,.o-.v fromr
ee(x). In the same way it follows identify the group of cubic roots
ee(rC,
+
s 2)
r2
=
=
=
ee(x)v,
Mr-'11
that
of unity with F3
_
82
by
ee(p(x))
ee ((I)
=
(mod 3).
=
ee(x)-l.
I then ee is
(3.7) 1
The 94
configuration
is
now
W
E
described E) + W,
as
follows. if
ee (W
- =
-
w
and w'
are
two fixed
points, then
W') :A 0.
It follows that every invariant theta curve passes through four fixed points and that every point belongs to four invariant theta curves. Moreover we have seen that the function ee
fixed
determines the direction of the tangent to 19 in the fixed points of r. w and E) + w' are tangent in a common point x E F if and
then 0 +
ee+,, (x)
Since ee+,,, (x)
=
ee
(x
-
w),
=
-
w)
=
if w, w' E F
ee+w, (x).
this condition is rewritten
ee(x
Therefore, only if
ee(x
-
as
w')
2x w (only). We conclude that the four invariant curves running through one fixed point come in two pairs: since any two theta curves always intersect in two points (which may coincide), the curves of one pair are tangent in their unique intersection point and the curves of opposite pairs intersect in two different points (see Figure 7, which also contains the dual picture, equally present in the 94 configuration).
which -is satisfied for w'
=
-
dual
Figure
189
7
Chapter
V11. Two-dimensional a.c.i. systems
,
There is also
a neat way to display the incidence relations of the points and the curves of configuration by an incidence diagrain analogous to the one for the 166 configuration on the Kummer surface, as we explained in Paragraph IVAA Let us define W,,, r ,. + 8 2 and r,,, 0 + W, where (I Ab(002 ool) and (2 Ab(02 01) as in the proof of Proposition 3.3; also E) is the totally symmetric theta curve given by Proposition 3.3. It follows from (3.7) that one can use for the 94 Configuration the following incidence diagrams.
the 94
=
=
=
WOO WIO W20
3.3. A
W01 W11 W21
projective embedding
In this section
algebraic
we
=
-
W02 W12 W22
of the
roo rio r2o
-
ro, ril r2,
1'02 r12 IF22
generalised
Kummer surface
will compute explicit equations for the quotient s i(r)/,r as an T has nine fixed points, S has nine singular points and we have =
surface in p4. Since
that
they are of type A2. The minimal resolution of these singularities of S leads to a (a generalized Kummer surface), which we will denote by X (see [Beal]). Let J(17) -+ S be the quotient and denote by E) the unique divisor given by Proposition 3.3.
seen
K-3 surface ir :
Proposition 3.4 ample, leading to threefold in p4.
Let M be the divisor an
embedding of S
on
the
as
S for which ir*(M) [30]. Then M complete intersection of a quadric and =
is very cubic
a
Proof
Using the quadratic form i.e.,
lr* (M)
=
L03 Let .
us
we see
ee
denote
by
the canonical map from X to S. Then 18
so
that M
-
M
=
usingC -,C
='C(&3.'C(&3
6, which
Riemann-Roch Theorem
(for
that 'C03
[36]
=
N the line bundle
(deg 7r)M
=
2
=
-
descends to
we
M
line bundle M
on S, pull-back of M by
find 3M
=
-
M,
is also the self-intersection of N.
K-3
a
X which is the
on
Therefore,
we
find
by
the
surfaces), N-N
X(N) It follows
i >
0,
so
moreover
that
from Serre
=
X(Ox)
duality
dimHO(X, O(N))
=
+
=
and Kodaira
X(N)
=
2 + 3
=
5.
2
vanishing
that dim Hi(X,
O(N))
=
0 for
5.
The
morphism ON corresponding to N can be factorized via the blow-up p : X -+ S and provide an injective morphisin 0 : S _+ p4. More precisely, it can be seen by analyzing theta curves on J that ON is one to one away from the exceptional curves. If we consider now the surJective map is shown to
Sym HO (X, N) whose kernel leads to the count
as
-+
defining equations
above that the kernel contains
a
EDt>oHO (X, M t),
for the
quadratic
image as
Since the degree of N equals six, we see that the image quadric and a cubic hypersurface in p4.
190
of S in
well
as an
is the
p4' we see by a dimension (independent) cubic form. complete
intersection of
a
3.
Consider
now
Application: generalized Kummer surfaces
the genus 2
even
Mumford system;
our
system of generators
(2.6)
of
integrable algebra have the property of being weight homogeneous when the variables uo'...'wo are assigned the weights given by the weight table below. We call a Poisson bracket weight homogeneous of degree i when the bracket of any two weight homogeneous elements is weight homogeneous and its
I_XkXk7 AIXII
jXk
X1 I
,
weight homogeneous elements xk and x, (of weight k resp. 1), where s denotes the weight Of fXk, x1j. Then the Poisson brackets f-, J1, I-, -J-X and f.'.j,\2 have weights 2, 3 and 4 respectively. Everything is contained in the following table. for all
1
2
3
4
Ul
UO
VO
W2
V1
W1
weight
5
6
H,
HO
WO
H3
H2
1-74
1.1-1
H4 2
1.1-1 It follows from the table that the genus 2
even
Mumford system has
an
automorphism defined
by
(U1iUO V15VOiW1jW2)
(Mlif 2UO, f 2Vi, VO, E2WO
'-+
i
W1 i
EW2)
(3.8) allows
taking the second structure as Poisson structure. Having a (finite) automorphism Here a quotient system (which is also a.c.i. in view of Proposition V.2.5). however we are only interested in the level sets which are invariant for this automorphism. Again by the weight table these are the subsets of C7 given by
upon us
to construct
W2 W1
-U
-
2 I
+UO
0,
=
UIUO + UIW2 2
UOW2 + UIWI + WO + V1 ulwo + u0w, +
2v,vo
=
=
(3.9)
0,
0,
2
UOWO + VO
for
and p
n
arbitrary. This
is
an
affine part of the Jacobian of the
Y2
=
X6
+
2NX3
curve
+ /'t'
These axe presicely the curves of genus two curve is smooth, i.e., p =A 0. automorphism of order 3; since we have normalized the equation of our curve as 1. x6 + 2=3 + 1 we will focus in the sequel only at the level sets for which p
whenever this with 2
an
Y
=
=
description of the level sets as affine parts of the symmetric square of the automorphism on these level sets comes from the order three automorphisms on the curves. Explicitly our map S (defined in general by (111.2.3)) is given by X2- (XI + X2)1\ + XIX27 U (/\) From
curve
our
it is also clear that the
=
((XI YI) (X2 Y2)) i
i
i
1-+
V(,\)
Y1 X1
191
-
-
Y2 X2
A
+XIY2 X1
-
-
X2YI X2
(3.10)
Chapter V11. Two-dimensional a.c.i. systems and
w(,\)
by the fundamental relation u(X)w(X)
our
2
equation
x6 + 2r=3
=
y
v'(X)
+
+ 1 for the
=
f (X), where f (x)
r. Since the action
curve
is the on
the
given (x, y) -+ (ex, y) we see from (3.10) that it corresponds indeed with (3.9), hence automorphism extends the automorphism on the special Jacobians.
curve
our
is defined
hand side of
right
is
Now that
have
explicit equations for an affine part of the Jacobian of ]Pr (i.e., (3.9) regular functions on this affine part which extend to meromorphic functions on the Jacobian of rr. having a pole of order < 3 along a fixed recall that two translates of the theta component of the divisor to be adjoined at infinity divisor need to be adjoined to each affine part in order to complete them into Jacobians. As we explained in Section V.3 this is done by using the Laurent solution (the principal balances) e.g. to the vector field X, of which we computed the first terms in Paragraph VI.4.2. From these first terms one finds easily (i.e., by solving linear equations) a few extra terms, to wit with p
1),
=
we
want to construct the
we
-
1 UO
(2a
=
t
-
2a 2t + +
U1
(I
=
t
+
dt2
+
-
ad+ 2a 2b)t3
+ 2ab 2+ 2bc
(2ae
at+bt2
(2ac
+
Ct3
et4
+
+
ft5
-
bd
-
2a3 b+a 2 d
-
2f)t4 +
+
denote the free parameters; the Laurent solutions for v, and vo follow from it Here a, , f by differentiation and those for w(A) upon using the invariants. The distinguishes between .
the
.
.
principal balances which correspond quotient
to the two different divisors. In order to have the
functions which descend to the
look at those invariant
we
polynomials (invariant
with respect to the action of -r) which have no poles upon substituting the Laurent solutions when picking the sign and have a pole of order at most 3 when picking the + sign. Notice -
that invariance
just
means
We arrive at the =
1,
ZI
=
UJUO
Z2
=
2u,(uo
Z3
=
2uOv 12+ 2V20
z4
=
2v
3
0 mod 3.
VO, + v,
-2 +
being weight homogeneous of weight
list of functions.
following
zo
-
here
-
+
2),
ul
2uovl(2uo
(U2I + 4UO)V21
+
2vo(2r. + 15uluo
U2) 1
_
2
2uvo(u 1
+
IOVO(UIVJ -
5u 3) + I
-
vo)
2(U21
+
_
vi
_
-
UO)
2v, (7UOU21
UO)3
_
1OU30
_
-
U41
_
I
image of J(r) in p4 it suffices to eliminate the variables ui, vi In fact, from the first three equations of (3.9) the variables linearly and the other equations reduce to
(3.11).
2n(uo
_
U2) 1
+
3ulU20
-2nuluo so
_
UIV21 4-
+ uOu I
-
4uOU31
+
2v,vo + u51
3U2U2 UOV2I 1 0
+
U30
+
1U2) 0
4nuluo.
To find the and
(3.11)
3
+ 2u 01
V20
=
and wi from wi
are
(3.9)
eliminated
0,
(3.12)
it suffices to eliminate ul, uo, v, and vo from (3.12) and (3.11) (we have already eliminated (3.11)). In the latter zi and z2 are solved linearly for v, and vo,
the wi-variables in
VO
=
V1
=
U1UO 2 U1
-
Zi, Z2
-
UO +
192
2u,
(3-13)
and the for uo
new
Kummer surfaces
Application: generalized
3.
equation for
z3, obtained
by substituting (3.13)
in
is then solved
(3.11)
linearly
as
2 U!2
2-1 (Z3
UO
-
ZIZ2
-
2Z2).
(3.14)
1
Zi
After substitution of of
(3.12),
be
a
3: 1
we are cover
-
4 8z I
-
+
in the last
(3.14)
2nz2z3
4
(2nz2
4
(2
+
6Z3 + -4)
+
ZI +
14Z3
2z4
+
4mz3
Z4
(1
-
and in the
(3.11)
of
3z2z3
-
) Z2I
equations
-
2rz22
-
Z2Z4 -
:--
4Z2
01
-
3z2z3
-
z2z4) zi
=
-
Using the first equation, the second equation 8
2r,,z22
-
+(8r. + 4NZ3 2z22+ z3z4 + 542 0.
2KZ2 + 6Z3 +
+
equation
3 which reflects the fact that i(r) will in ul, eliminate U3I we arrive at the following two equations:
equations
of its image in p4. If we
24NZ2I 8z3I
_16r.Z3I
and
(3.13)
left with three linear
be replaced
can
by
"2) Z21+4(-2r.+(1-2n"2) Z2-6KZ3-nz4)z1+2(1-n"2) Z2 0, 5Z3 r.Z2 (5Z3 + z4) + 2z3 (2m"2 7) Z3Z4 2z4
3n
2
2
-
or
equivalently by tZAZ
-
_
-8r,.
-8K
16(l
0
4(1
2(2is
=
-
0, where
=
0
A
-
2 -
7)
-2
3 M2 )
-
-
r2 -7) 2(2"
-2
2" r2)
-24n
-4r,.
r2)
-5n
-K
0
4(1
2X2 )
-
4(1
t'
_
,
-24n
-5n
-10
-1
-4n
-K
-1
0
equations for the quadric and cubic hypersurfaces which define S identify in the sequel with S) may not seem very attractive, we will see that natural coordinates can be picked for p4 in Which these equations take a very symmetric form. Indeed, the projective basis of p4 that we used is rather arbitraxy: for example, the coordinates of the nine fixed points for r do not possess special coordinates in terms of the present basis. One observes that if the five fixed points for r which do not lie on E) are taken as base points for p4 then the coordinates of the four fixed points on 0 take a simple form and are independent of n. It is read off from the incidence diagram that
Although at this point as a
subset of p4
these
(which we
will
1WO07 W11i W12) W21, W221 are
the five points which do not lie
and take
x
=
on
E). To find their
coordinates,
use a
t, Y
picking either sign around
ol
Y
or
=
:L
(I + Mt3)
02, and in the
(t-3
+ 0
same
r2 1-2" +
r
+ 2
193
(t6),
way,
t3)
x
==
+ 0
t-' and
(t4)
local parameter t
VIL Two-dimensional a.c.i. systems
Chapter for oo, and 002. Then
careful
a
computation yields the following coordinates:
W117 W12
:(0:0:0:0:1), : (0: 0: 1 : I: T-3
W21 W22
:
Woo
i
(I
:
1
::F2
:
-
2r.
-
2r.),
T-2r.
:
:
4N2
14n +
4).
We take the points
JW00i W121 W21i W12) W11 as
base
for
points
coordinates yo,
that
we see
.
.
.
p4 (in that order), i.e., 0 (I : 0 : 0 y4. Then the four fixed points on 0 have
lie
they
W10
=
(1:
1: 1: 0:
0),
W20
=
WO,
=
(1:
1: 0: 1:
0),
W02
=
on
the
(1: (1:
0
:
=
,
as
0), etc.,
:
with associated
coordinates
0: 0: 1:
1)1
0: 1: 0:
1)7
(2-dimensional!) plane YO
-:=
Y2 + Y3
==
Y1 + Y4i
see that in fact 19 is contained in this plane. The translations twO and two, correspond to projective transformations of the surface and take in terms of these coordinates the simple form
and it is easy to
twl"
(-I
I
1
0
0)
-1
1
0
1
-1
0
1
1
0
-1
0
1
1
0
-1
1
0
0
1
0
0
0
1
0
0
0
1
0
0
-1
1
0
0
1
0
1
0
0
0)
0
1
0
0
0)
and
tw.,
0)
equations for the planes to which the other invariant curves belong, axe This configuration of nine points in p4 is characterized by the fact that there exist nine planes with the property that each of these planes contains four of the nine points and every point belongs to four of the planes. Thus we have recovered in a direct way a configuration that has been studied in the work of Segre and Castelnuovo on nets of cubic hypersurfaces in p4 (see [Cas] and [Seg]). from which the
obtained at
once.
The equations of the
quadric and cubic hypersurfarces Q following symmetric form:
coordinates the
Q C(YI + 3 3 C
where
c
adding
=
2 Y2
Y4) (Y2
(YIL
+ Y3 + Y4
+ Y3 + Y4
2
YJ (Y2 + Y3 + Y4
I +
-
n
to it the
-
-
YO)
+
YO)
-
YO)
Z (Y2
+
Y3) (YI
eCY2 ((YI
+
Y4)(Y2
EC2 Y1 ((Y2
+
Y3) (Y1
+
and C take in terms of the
+ Y3 + V4 -
-
-
YO)
CY42+Z Y32,
YO)
+ YOY3 +
YIY4)
YO)
+ YON +
Y2V3)
0)
I and Z n. The cubic equation can be simplified in a significant E2 Y2. The result is equation for Q multiplied with c2y, =
-
_
C2YIY4(Y2 + Y3
-
YO)
-
52 Y2Y3(YI
194
+ Y4
-
YO)
=
0-
new
way
by
3.
If
we
Application: generalized Kummer surfaces
define
then S is
given
as an
C
X1
"
X2
"
X3
-::::::
X4
YI + Y4
-Y1)
X5
Y2,
-Y4,
X6
Y3;
YO
-
Y2
-
algebraic variety
c2XIX2X3
+
Y3,
Y07
P5 by
in
F-2X4X5X6
--":
01
Q C(XIX2 + X2X3 + XIX3) + E(X4X5 + W
-
XI + X2 + X3 + X4 + X5 + X6
:::::::
X5X6 +
X4X6)
--":
(3.15)
0:
07
and the 94
configuration is presented in the form used by Segre and Castelnuovo. In fact, singular points W,, have a I on position r + s mod 3, a I on position 3 + (r s mod 3) and zeros elsewhere; the nine planes they belong to are given by W n (xi 1,...,3. Moreover the theta curves are mapped to the nine conics 0) for i,j Xj+3 Cij, (I < i, j < 3), given by Wij 3Cij. For example, C16 is given as
the coordinates of the
-
-
=
=
=
=
CX2X3 +
eX4X5
=
0,
X2 + X3 + X4 + X5
Notice that if is obtained
one
changes the sign of
(interchange
c
++ E
and xi
n
in the
++
195
0-
equations (3.15) then an isomorphic surface 1, 3), in agreement with the fact
Xi+3 for i
that n2 is the modular parameter.
=
=
V11. Two-dimensional a.c.i. systems
Chapter
4. The Garnier
4.1. The Garnier
It is shown in
potential
potential and its integrability
[CC]
that for any A
=(AI)
iAn)i
...
n
V\
1: A,qi2,
+
q,
i=1
defines
an
taken
as
(4.1)
j=1
integrable Hamiltonian system on R2n symplectic structure W
with the standard
equipped
potential
n
( 2)2
=
the
=
J(qj.... dqj
=
A
iqniPli
.
.
.
7
Pn) I
qi I pi E
Rj,
dpi, when the Hamiltonian
is
the total energy
H,\
==
T+
Pi2
T
V,\,
2
is the kinetic
(T
energy). It was pointed out to me by A. Perelomov that the integrability of potentials was already known to Garnier; therefore we will call V,\ the Garnier potential. 2 (two degrees of freedom) writing We study here the case n these
=
V.,3 It would be
of
=
interesting to study also algebraic geometry.
(q21
the
2)2
+ q2
2
+ aq,
+,8q2'2.
higher-dimensional potentials
from the point of view
only interested here in the complex geometry we will from now consider being defined on the affine Poisson variety C4 with the standard Poisson structure. However it is sometimes useful to extend it by the parameters a and 3, i.e., we consider C' as the phase space with coordinates qj, pi, a and 8 and with jqj,pi I I and all other brackets between these coordinates are zero (see Proposition 11.3.24). The Poisson structure on both C4 and C6 will be denoted by 1-, -1. Since
the
we are
potential
as
=
For
a: -#
we
define
A,9
=
F
=
(q,P2
_
G
=
(qIP2
_
C[F, G]
P1)2 q2 P1)2 q2
where F and G 2
(a
defined
-
+
Notice that the Hamiltonian 1 H
=
2
which
corresponds
to the
(P21 + P2) 2
+
2
(ql
-
G
=
2)2
+ q2
potential V,,,3 belongs F
to
2(,6
-
+ aql2+,3 q2
A,,,3
2,
since
a)H.
We also define
A
C [F,
G, H, a, 0]
==
idl(F
-
by
2q4l + 2ql2 q22 + 2aq,2), 2 2 #)(P2 + 2q24 + 2ql q22 + 2,6q22).
a) (Pi
+ +
are
G+
196
2(a -,3)H)
4. The Garnier
It is easy to check that IF, GI of the vector field XH which is
=
=
42
for
=
be done
example
-2q,(2q 1
Pi,
2
-2q2(2q,
fi2
P21
When this vector field is taken and A
can
by using
the
explicit form
given by 2
41
that both
0. This
potential
on
C' then of
course
2
ce),
+
2q2
+
2q52+,0).
&
+
=
=
(4.2)
0
are
added.
It follows
involutive and
clearly they have maximal dimension. All level sets of the momentum map associated to Aq are two-dimensional and the general fiber is irreducible hence A,6 is complete. It follows that A is also complete. Thus we have verified that (C4, 1. J, Ao) and (C6, I-, J, A) are integrable Hamiltonian systems (for all a : - 0).
A,6
axe
,
We will be interested in the level sets of the momentum map over closed points. Apart from Paragraph 4.5 we will only be interested in those for which a =1-,6. We will denote these
by by
or
the
Ff,
when
a
8 have been fixed.
7
Explicitly Ff, is given
as a
subset of C4
following equations:
In order to
(qIP2
-
q2PI
(W2
-
q2PI
)2 )2
a)(,92I 0)(P22
+ +
(a
-
simplify some of the formulas by f g 2(,6 a) h.
h be determined
-
=
+
+
in the
+ 2aq 12) 2q,4 + 2q2q22 1 2q 4 + 2q,2 q22 + 2,6qi2)
sequel
=
=
g.
let, for given f and
we
g, the constant
-
lot of automorphisms. For future reference we list them in the quasi-automorphisms which are of interest (i.e., 31 and 32)- r.2 is an automorphism of order three, e being a primitive cubic root of unity and 32 is not of finite order, A being an arbitrary non-zero complex number.
Clearly this system has
a
table below. We also add two
21
q,
q2
Pi
P2
a
16
F
G
H
-qi
q2
-Pi
P2
a
0
F
G
H
G
H
72
q1
-q2
Pi
-P2
a
0
F
31
qJ
q2
-Pi
-P2
a
0
F
G
H
32
Aq,
Aq2
A2p,
IN2P2
A2Ce
A2#
A6F
A6G
X4H
KI
q2
q,
P2
Pi
0
a
G
F
H
K2
62 qi
62q2
epi
1EP2
ea
60
f
9
62h
Table 6 The involutions z1, 22 and 3 restrict to all level surfaces Ff, and their restriction will be by the same letter. Notice that the automorphisms in the table lead to many different
denoted
quotients and they will all play a special role in this text; they are not all the automorphism they are the ones which are seen "at once". In view of the automorphism 32 it is natural to consider (a,fl, f g) as belonging to the weighted projective space 24 p(1,1,3,3).
but
,
24
A quick introduction to
weighted projective
197
spaces is
given
in
an
appendix
to
[AM7].
Chapter V11. Two-dimensional ax.i. systems Notice that if
a
=,6 then F(= G) is just the =
q
which is
obviously
qIP2
-
(4.3)
q2PI 1
in involution with the energy
corresponding to a central potential. What equations defining Ff, can be rewritten and the momentum q, giving precisely the equations (IV.5.2)
is remarkable however is that if
(birationally)
square of the momentum
in terms of q1, q2
6 then the
=A
a
of the octic 0 with
4(a _,6)2 (a +,6)
,XO
=
A21
=
A22
=
2(a
'\23
=
f,
-
2(f
+
g),
91 -
0)3,
Yo
=
yi
=
Y2
=
Y3
=
/2 (a q1 Iffg, q2 /2(a -,3)lg. q,
(4.4)
-
It follows that for
general f,g the surface Yf, is birationally equivalent to the affine part o n Jyo -:A 01 of the octic 0 which is itself birationally equivalent to an Abelian surface oo of type (1, 4). We show in the following proposition that Yf, actually is (isomorphic to) an affine part of an Abelian surface of type (1, 4). =
Proposition
is
4.1
Fixing _
(qIP2
_
for general25 f, g
E
C a
a
54,8
q2p,)2 + (,6 q2 pl)2 + (a
(qIP2
obtained by removing
any
isomorphic smooth
E
C, the affine surface.Ff,
-
a) (P21
an
field XH extends
to
+
=
=
g,
affine part of an Abelian surface
=
Tf2g,
of type (1, 4),
r f"g \ Dfg,
linear vector
a
C4 defined by
Dfg of genus 5,
curve
-Ffg and the vector
2q,4 + 2q 21 q22 + 2aq 12) 2 2 P2 + 2qi4 + 2ql q22 + 2,6q 2)
-
to
C
field
on
T2 fg.
Proof
(i) Let G be the group generated by the involutions *11, Z2, and 3. Our first aim is to show that .Ff,IG is (isomorphic to) an affine part of a Kummer surface. Since f and g are general, we (1\0 : Al : 1\2 : A3) given by (4.4) do quadric (Kummer surface)
may suppose that
the
not
belong
to S. For these
2 2 X2ZO_,, + X2(Z2Z2 Z2Z3 + \2I (Z2 OZI + Z2Z32) + X2(Z2Z2 0 2 0 2 + Z2Z2) 1 3 3 0 3 + Z2Z2)+ 1 2 2,\1'\2(Z0Z1 + Z2Z3)(ZlZ3 ZOZ2) + 2/\11\3(ZO-'3 -'IZ2)(ZO-'l Z2-3)+ -
2A2A3(ZIZ2 which is obtained from 0
-+
25
Q;
+
ZOZ3)(Zl-'3
+
-
ZOZ2)
(IV.5.2) by setting
this map restricts to
a
Precise conditions will be
map
po
:
00
--`
zi
-+
-
\i, let Q
be
(4.5)
07
yi2, i.e., there is an unramified 8 : I cover Q0, where Q0 Q n Izo =A 01. Also the rational =
given later (Proposition 4.6).
198
=
4. The Garnier
Yf,
map
rise to
a
-+
potential
00 given by (4.3) and (4.4) induces diagram
birational map
a
77f.q
I
and q
7r) (ql ql'p'2
I
=
E
q1P2
q2 7 PI 7
-
,
P2)
(ql, q2, PI P2)
q2pl. Then
(
=
o
q2PI',) forfl , E21e
i
ir) (q, q211 p, p'2) E 1-1, 11. Then
then q,
7
I
one sees
(ql, q2, PI P2)
=
,
where i'll k
means %k
-7r(q]'L,q2'7p'17p'2), I isomorphic
in
and
to the
I and
case Ek
is
injective.
(affine)
Kummer
(XI
X2 i
i
61 qJ q2
=
=
62q2'
7
if
(4.4). (ql,q2,Pl P2) E -Ffg i
On the other
X3).
surjective.-
determined from
P1,P2 exist such that
=
is
Obviously
xi and let q1, q2, q be
I
-
bijective.
is
is
X3)
(4.6)
Q0
normal, it suffices to show that QO let (Y1 Y21 Y3) be such that yi2 Then these satisfy the condition under which
Q0
X21
Qo, giving
PO
.Ff,q I G
(XI
--+
00
ir
Since
0 : Yf, / G
commutative
q
=
hand,
(
if
cq', (where
o
q
that
ZC2 SCI Zf (q', 1 q2, p 2 1
identity for
Ck
=
This shows that
,, p'), 2
1. It follows that
0 is surface defined by Q0.
an
7r(ql, q2, PI P2) 7
isomorphism, hence TfglG
is
(ii) We proceed to show that Tf, is isomorphic to an affine part of an Abelian surface, more precisely to the normalization A of 00 (the octic is singular along the coordinate planes). This normalization can be obtained via the birational map 0'C : 7' -+ 0. In particulax, by restriction of (IV.5.4) to an affine piece we get a commutative diagram Oo
A
POI
IPO
(4.7)
Qo
Ko 010 where
0Ar2
is
an
isomorphism.
If
we
combine both
Ff 9
diagrams (4.6) and (4.7)
get
A
18:1
8:11 .Ff., / G
K0
and 0 the isomorphism with p the birational map 0-10 1C and A -+ Ko axe only ramified in discrete points; the
.Ff, IG are
we
replaced by their closures: the closure of A
is
199
Now the two same
covers
Ffg Ff,
holds true if A and
just 7-2 and the closure
of
Ffg
is obtained
V11. Two-dimensional a.c.i.
Chapter
systems
explicit embedding which will be given in 4.4-1. By Zaxiski's main Theorem the normality of T' implies that the lifting W of 0 must also be an isomorphism and we get from the
'Ffg for
some
divisor
1 unramified
hence
Dfg
Df.,
on a
of
cover
7-2 f9 \Dfg
=
,,
Abelian surface
(1, 4)-polarized
It is
f9
seen
that
E)fg
is
a
4
translate of the Riemann theta divisor of the canonical
a
is smooth and has genus
5;
equation for Dfg will be given
an
in
Jacobian, Paragraph 4.4.
2 1, 01 (iii) Finally we show that XH extends to a linear vector field on V qI fg Letting 00 q2, we have shown that an equation for the Kummer surface of the canonical Jacobian associated to Ffg is a quartic in these variables. From (4.4) and (IV.5.2) the 2 leading term in 023 is given by ((a + MOO + 01 + 02) 4(a,800 +,301 + a02) or, in terms of =
.
and 03
=
=
-
,
the
original variables, 2
( q, We let x,
2
+ q2 +
C,
and X2 be the roots of the
+
2
p)2
-
4(a,6 + aq2
as
#ql2).
(4.8)
polynomial 2
X
+
2+(q2+ q2+a+#) ,
suggested by the algorithm recalled
+
X
ap
in Section VA
+
aq22 +,8ql2 because
("suggested"
we
did not prove
yet that the system is a.c.i.). Explicitly, let 2
2
X1 + X2
-':--
-(ql
+ q2 +
XIX2
=
aO +
aq22
+
+
l + -' 2
Oq12,
Xl-' 2 + -' IX2
a
then it is not hard to rewrite the X1 7 X2;
G
F
-2(qlpl
+
q2P2)i
2(oqlpl
+
aW2),
defining Ff,
g,
(4.9) in terms of
6bl &2. This gives
8(xi so
equations
=
+
a) (xi +,8) (0
(a + #)x? (XI
+
+ -
(a,8 )2
-
h)xi
+
(,3f
-
ag)/2(a -,8))
X2
that
dxj
dX2 +
-
A/f (XI)
=
0,
=
2v/2dt,
-
NRf (X ) 2
xldxl
(4.10)
X2dX2
-
+
-
Nrf (X2)
NR (X1) where
f (x)
=
Integrating (4.10) to
a
(k
=
(x
+
we see
linear vector field
1, 2)
axe
a) (X + 16)
( 3+ (a X
+
(afl
-
h)x
ag
+
2(a -,3)
)
-
Ff,, which obviously extends (VI.4.9) explicit solutions for qk
on
Vfg.
given by
where 77, and 772 Notice that
axe as a
2
Y
curve
#)X2
that XH is a lineax vector field From this expression and
on
qk
for the
+
=
-
O[J + qk] (At + D) 0[ (At + D)
halfperiods which correspond by-product
we
(X + a)(X +,a)
find
X3
+
an
to
-a
and -,0.
equation
(a + O)x2
+
(a#
-
h)x +
Pf
whose Jacobian is the canonical Jacobian associated to
200
-ag
2(a -,6) fg.
(4.11)
4. The Garnier
potential
Alternatively, the linearizing variables (4.9) suggest a morphism 0 from the potential V,,,5 on C4 to the genus 2 odd Mumford system, namely one defines O(qj,q2,P1,P2) (u (A), v (X), w (A)) where
U(A)
=
A2
+
(q,2
+
q22
2
+
a
+,6)A + afl + aq2
flq217
+
1
v(A)
=
w(A)
=
[(qlPl
T2
A3 +
+
W2))k
q21
(a +
-
+
(NIPI
aq2P2)]
+
q22)A2
2
-
The genus 2
odd Mumford system
2
(a+#) (ql
+
2
a#
(2a
: !-
e2 + q
+
2,6
2 1
2
+ q2
which is considered here is the
2)
a'O
+ q2
)
(4.12)
A
-
C7 and the Poisson
one on
before26, namely 2+ (a +,6)x-a,3. This morphism 0 is neither in ective nor surjective, however the latter is easily cured since the image is a level set of the Casimirs of the odd Mumford system which carries an induced a.c.i. system. As for the injectivity of 0, clearly it can be factorized through the quotient system obtained by dividing C' out by the group (of order four) generated by zi and 22- It follows from Example 11.2.31 that the regular functions on this quotient are generated by structure is
it
a
linear combination of the Poisson structures
corresponds to
=
W
only
have considered
-x
Qj with
we
Rj2
two relations:
%2,
=
Pi
=
QjPj, (i
=
pj2,
R,
=
1, 2). Then
=
(i
PjQj, the
1, 2)
=
quotient Poisson
structure is
given
by
( T2)7
where
Ti
0
and the map
U(A)
via which
A2
+
(Q1
0
4Rj
-4R,
0
2Qj -2Pj
-2Qi
2Pi
0
0
T,
factors is
0
Q2
+
+
a
+,6),\ + a,6 + aQ2
+
#Q17
1
V(A) W(A)
T2 A3
[(Ri
+
+
R2)A + (#Rl
(a +,6
-
R,
-
+
aR2)], P1
R2),\2
+
P2
2
P,
-
0
( Ta
+
+(a+#)(Q,+Q2)-a,6
L2 + Q1
+
20
A
Q2
is linear, this map is a biregular map. Thus the quotient system is everything in isomorphic to a trivial subsystem of the genus 2 odd Mumford system and is a.c.i. We may conclude as in Paragraph 6.1 that (for a 54 j8) (C4, 1-, -1, Aq) is an a.c.i. system of type (1, 4). The same is of course true for (C', 1-, .1, A). As a by-product we get also a Lax representation for the potentials V,8. For the vector field XH for example it is given by
Since
d dt
where
26
(X) ( w(A) v
u(A), v(A)
These
were
and
given
U(A) -v(.X) w(A) in
vl'2-
are
2.3,
V(A) [( W(A)
U(A) -V(A)
0 A
-
2(q,2
+
q22
0
given by (4.12).
2.4 and 2.5
on
C6 and 201
are
easily
rewritten
on
C7.
Chapter
V11. Two-dimensional a.c.i. systems
4.2. Some moduli spaces of Abelian surfaces of
(1,4)
type
paragraph we describe a map,0 from the moduli space A(1,4) Of Polarized Abelian cone M3 in some weighted projective space. To be (1, 4)-polarized Abelian surfaces which are products of elliptic curves (with the product polarization) are excluded from A(1,4). The map will be In this
surfaces of type (1, 4) into an algebraic precise we recall from Section IV.5 that
bijective
the dense subset
on
A(1,4)
which is the moduli space of
polarized Abelian surfaces
for which the rational map Oc : r _+ p3 is birational. Thus we construct a projective model for the moduli space A(1,4). The main idea in this construction is to see how the Galois
(T',,C)
group of the
AO(,,4)
cover
-+
A(1,4)
acts
on
M3
P and define
will be easy to calculate since it is a quotient of which acts linearly. The fact that this action is
(a
to be the
quotient. This quotient of) p3 by a group
Zariski open subset
simple is surprising and was suggested by the formulas (4.4) which show that the sign of the Xi does not matter and on the other hand by the automorphism r,., which shows that the Abelian surface which corresponds to a, #, f g is isomorphic to the one which corresponds to 6, a, g, f which indicates that the modular parameters A, and X3 can be permuted (upon adding the proper i's or signs for A0 and X2). A posteriori we can forget about the integrable Hamiltonian system and proceed as follows. to us
on
the
so
hand
one
,
Recall from
Paragraph
\
P
P
S
=
AO
-
_A0
IV.5 that
U
(three
A'(,,4)
maps onto
rational
in
curves
each
S,
missing eight points),
bijectively on the first component (which is dense); the three rational curves are thought of as lying in Pl/(Ao -,Xo) at the boundary of this component. AO(,, 4) is a 24 : 1 (ramified) -
cover
of
let
A(1,4): K, K2
K3
or
and
r
be elements of order 4 such that
10, a, 2a, 3al, 10,,T, 2T, 3,rl, 10, o- +,r, 2o- + 2-r, 3a
+
3,rl,
K4
=
K5
=
K6
=
K(L)
=
(a)
E)
(-r),
and define
10, a + 2T, 2o-, 3a + 2,rl, 10, 2a + -r, 2T, 2o- + 3-rl, 10, or + 3-r, 2a + 2T, 3o- + -rJ.
These are the only cyclic subgroups of order 4 of K(L). It is easy to see that taking possible isomorphisms K(L) 5--- Z/4Z ED Z/4Z we find exactly the 24 decompositions
K(L) We describe the
=
Ki
ED
Kj, (I
<
A
24:1
i, j :5 6, Ji
we
0
:
construct
A(,,4)
p(1,2,2,3,4))7
jJ :A 0, 3).
cover
A(1,4)
(1,4) and
-
all
_+
a
24: 1
M3,
cover
p
such that there results
A0(1,4) 00
M3 and
_+
where M3 is
a
map
algebraic variety (lying commutative diagram
an
a
24:1 ,,
weighted projective
space
A(1,4)
1 -p
in
(4.13)
_
24:1
M3 202
M3
E)
4. The Garnier
in which the restriction
determined
OfO
A(1,4)
to
a
bijection (D
is
a
divisor
on
MI which will be
explicitly).
The group G
GL(2, Z/4Z)
=
acts
K(L)
(a)
=
(,r)
E)
and
transitively
on
(ordered!)
bases
as
follows: if u,,r
are
b
(c d) a
such that
is
potential
E G then
b
( d) a
-
(a, T)
(ao- + bT, ca + dr),
=
c
giving a new decomposition K(f-) (ao- + br) E) (co- + dr). We denote by H the normal subgroup of G which consists of those elements of G which are congruent to the identity matrix, modulo 2. Then H acts on the set of decompositions of K(L), thus H acts on AO( 1,4)7; to determine the corresponding action on the isomorphic space P, it is sufficient to take any element of H, act to obtain a new basis and determine the new coordinates (YO YI Y2 Y3) according to (IV.5.1). Substituting these in (IV.5.2) the new parameters (AO Al A2 A3) are found immediately. The result is contained in the following table (since diagonal matrices act trivially, only one representative of each coset modulo diagonal matrices is shown): =
basis
K(,C)
(o-, -r)
K, ED K2
(o- + 2,r, -r)
K4E)K2
for p3
coo.
(YO (YO
:
YI
:
Y2
Y3)
(AO : 1\1 : A2 :A3)
iY3)
iY2
Y1
moduli in P
.
(a, (a
+
2a +
2-r,
7-)
(YO iY1
K, E)K5
-r)
2o- +
Y2:
(YO iY1 iY2
K4EDK5
:
(AO -AI : A2 A3)
iY3)
(AO
Al
-A2
-Y3)
WO
1\1
A2
:
A3) -A3)
Table 7 The
upshot of the table
is that all
Abelian surface. The quotient space is
P/
(AO
:
Aj
:
A3) correspond
to the
same
P =
(AO
:
1\1
:
1\2
:
A3)
-
(AO : A1 : A2 : A3)
(P3 \ S') U (three rational curves in Y, upon
A2
given by
defining
pi
=
A? %
for the three rational fact that there
each
(4.14) missing three points),
coordinates for the quotient p3, from which in particular equations well as for the three points are immediately obtained (the
as
curves as
three
missing points instead of
two is due to ramification of the quotient The divisors S and ffl will be calculated later. We will also interpret this "intermediate" moduli space P. are
map at two of the three
Notice that
GIH
on
P (which extends
we
find
as
above the
is
points).
to the permutation group S3, so we have an action of S3 p3 since it is linear). Choosing six representatives for GIH
isomorphic
to all of
following
table:
203
Chapter
V11. Two-dimensional a.c.i. systems
for P3
basis
K(L)
(U"r)
KIEDK2
(YO:YI:Y2:Y3)
(ILO: AI: A2: A3)
(,r, 3o-)
K2EDKI
(YO:Y2:Yl:iY3)
(-AO: P2: AI: A3)
(o-, o-+-r)
K, EDK3
(VITY2: YI: N/Z7YO: Y3)
(AO: A3:
(o-+-r, -0
K3 E) K2
(YI: YO: N/ZY2: NrZY3)
(AO: -ILI: /-13: A2)
(3-r, o-+,r)
K2 ED K3
( rVl: iY2: 07YO: YO
(AO: -P3: ILI: -P2)
coo.
moduli in P'
7
(o-+,r, 3o-)
(NrZY2: VZYO: -Yl: -iY3)
K3EDKI
-/42:
ILI)
(/ZO: A2: -A3: -Al)
Table 8
The Tables 7 and 8
together show how to reconstruct explicitly the decomposition of equation of the octic. More important, it allows us to construct the quotient is shown in the following proposition.
K(L)
from the
space
M3
as
Proposition defined by
There is
4.2
bijective
a
f42 weighted projective
in
space
=
f1 (4f23
p(1,2,2,3,4) (with
f, (f,
f4
D2
512f4
=
=
-
(16f22 + 72f, f2
16
particular the moduli
in
a
a
space
A(1,4)
-
cone
(fo
f4))
off from.A43 by
and D
the
=
D, + D2 is
hypersurfaces
27f2 1
-
48fo f3)
has the structure
+
3f02 (fO2 + 24f,
-
32f2)
of an affine variety. The
(4.15) .
map
extends
map
0 the
is the
3f2),
-
In
natural way to
cut
are
1), where M3
27f32)
-
coordinates
the divisor whose two irreducible components
DI
A43
map
:
4(1,4)
_+
m3,
image of the (two-dimensional) boundary -4(1,4) \ curve (inside V) given by
A(1,4)
being
C
\ fP, QJ,
where C is the
rational
C
and P, Q E C
from
are
its vertex
for
(I
given by :
some
0
:
0
:
P
0
:
(4:
=
0),
a,fl, f and
all
:
V02
=
0: 3: 2:
points
g, with
4(4f2 0),
-
and
in the
fl), Q
=
(2:
1: 1: 0:
M3 correspond
cone
-2). Moreover, apart to
some
level
surface
a
Proof we describe the quotient of p3 by the action of S3, and show that it is (isomorphic f, (4f23 27f32) in weighted projective algebraic variety M3 given by an equation f42 space p(1,2,2,3,4). To do this we use the (induced) action of S3 on C3 which is given in terms of affine coordinates xi pi/po for C3 by
First
to)
the
=
-
=
(1, 2) (XI
X2
(1,2,3) (XI
X2
X3) X3)
=
(-X2; -XI -X3) (-X3 X1; -X2) i
=
204
7
4. The Garnier
Since the action is
it must be
orthogonal,
then u, is anti-invariant for to ul. Then invaxiants
for the action of S3
are
invariant line and
invariant
an
X1 + X2
U2
X1
U3
X1 + X31
-
-
X31
(4.16)
X21
(1, 2,3);
U2
and U3
are
chosen
orthogonal
2
f2
=
U22
f3
77--
U2U3 (U2
U2U3 + U3;
U3)
-
found. Also there is
(1, 2)-anti-invariant
f3 generate the f2 and f3,
=
U22(2U2
-
3U3)
U23 (2U3
+
3U2)
-
and
invariants
and
U1
and is invariant for
(1, 2)
A
of
reducible, having an
to it. Indeed let
plane orthogonal
which is
potential
(1, 2, 3)-invaxiant, giving a new invariant f4 ul A. depending on U2 U3 the invariant A2 is expressible =
i
A2
=
4f23
-
Since
f2
in terms
27f32,
i.e., A2 is nothing else than the discriminant of the cubic polynomial X3
-
It follows
f2X + f3.
that
f42
=
fl(4f23
-
27f32),
(4.17)
U2. Notice that (fl, f2, f3, f4) have degree (2, 2, 3, 4) so that the quotient of p 3 1 by the action of S3 is given by (4.17) viewed as an equation in weighted projective space p(1,2,2,3,4) with respect to coordinates (fo : f, : f2 f3 : f4). In conclusion we have established where
the
f,
=
cover
-p
_+
M 3 and there is
an
induced map
A(1,4)
24:1
AO
_+
M3 which makes
A(1,4)
(1,4)
1PO
P
M
24:1
into
a
diagram (since the
commutative
actions
on
AO(14)
the
are
by construction).
same
,
The reducible divisor D is known. Since
of S to
we
know of
Paragraph 4.4,
no
easily computed
explicit equations for S (or S) are S, we postpone the computation
once
easy direct way to determine
where the
potentials will be used
to
compute S in
will show there that Y breaks up in four irreducible pieces 11, way; and disc(P3/(x)) 0 where P3 is the polynomial we
-=
a
01
straightforward M2
01
/Z3
0
x,
0
=
P3 and
=
4142X
disc(P3"(x))
=
3 _
(/to
+
2M,
+
6M2
+
2113 )X2
0 denotes its discriminant
and eliminate X2 and X3 from
fl, f2
and
f4
+
(in x).
(/to
f, (f, 205
-
21t,
Granted
f4. Then the =
-
3f2)
+
2A2
this,
relation
61L3)X
-
we
take pi
-
4P3i
=
0, let
Chapter V11. Two-dimensional a.c.i. systems is found at once;
0. The computation obviously the same equation is found for A2 01 P3 0 is longer but also straightforward. Namely, by a simple translation in x the monic polynomial P3'(x)/(4/12) can be written as X3 -ax+b, with discriminant 4a 3- 27b2. When this discriminant (depending on pi) is written in terms of ui using the inverse of (4.16), the equation (4.15) for V2 is read off immediately.
f6i
disc(P3"(x))
As for the
==
curve
=
0 and po
=
2(/-12
(A(1,4))
to be added to
S3 identifies the three rational equation (as a subvariety of VI) of
pi
+
to obtain
(A(1,4)),
to
A
=
fi
=
=
we
notice that the action
single curve. To compute its fi, let, according to (IV.5.5),
a
in terms of the coordinates
Then in terms of po and /-Z2
M3).
10
(4.14), leading
in
curves
f2
leading
=
=
get
mo,
(2/12
-
po/2)2, M20
POIL2
1122
+
_
2
4
to
3f02
=
4(4f2
-
fi),
by elimination of po and /12. As for the two special points P and Q on this curve, it is easy to check that picking Mi 0, P2 2(P2 + P3) leads to the point (4 : 0 : 3 : 2 : 0) A3 and po and alternatively taking p, 09 AO 2P3 leads to the point (2 : 1 : I : 0 : -2). This /-12 gives explicit equations for all these spaces and proves the announced result in (4.13). =
=
=
Finally, let (fo of this
cone.
point. Define
:
...
:
:4 0 a, 0, f, g by
Then JL2
f4)
=
=
=
E
M3 be
for at least
a
f
any point different from the vertex (I : 0 : 0 : 0 : 0) of the six points (po : p, : p2 : jL3) lying over this
one
=
po +
2p,
+
=
po +
2p,
-
=
128 112P3, 2
2A2
+
2/431
2/-12
+
2/13,
(4.18)
2
g
=
128P2/-111
54,3 and a, 0, f and g satisfy (4.4). M3 correspond to some level concludes the proof of the proposition. then
a
in the
cone
4.3. The
This shows surface
that, apart from the vertex, all points for some a =7 #, f aald g. This
precise relation with the canonical Jacobian
In this paragraph we want to show that a (1, 4)-polarized Abelian surface T2 E A(1,4) is intimately related to its canonical Jacobian J(T2) (introduced in Paragraph IV.5), hence also to some curve of genus two, denoted IP(T2). In fact there is more: at the level of the Jacobian, let J(T2) be represented as C2/A, then T2 induces a non-degenerate decomposition of the lattice A and at the level of the curve, 7-2 induces a decomposition of the set of Weierstrass points of r(T2) which in turn corresponds to an incidence diagram for the 16r, configuration on its Kummer surface; moreover, the Abelian surface can be reconstructed from either of
these data
(Proposition 4.3).
206
4. The Garnier
potential
Recall that the canonical Jacobian of
A(1,4)
where K is the an
a (1, 4)-polarized Abelian surface (T2, L) E (irreducible principally polaxized) Abelian surface J(TI) TI1K, (unique) subgroup of two-torsion elements of K(,C). We have seen that such
is defined
as
the
=
Abelian surface is the Jacobian of a smooth
where A is the
r of genus two,
curve
i.e., it is given
as
C2/A,
lattice
period
0
A
H,.(r, z)
t (W1 W2), the wi being (independent) holomorphic differentials consisting of all periods of 0 and HI (17, Z) on r. The Abelian group HI (r, Z) has an (alternating) intersection form can be decomposed into non-degenerate planes (in many different ways), =
i
HI (17, Z) Such
a
=
H1(17, Z)
=
HI (D H2 and A
==
fy
=
A,
E)
c
I
0(-)H2 non-degenerate.
and
decomposition A
a
Ai both
0(-)H,
HI (D H2,
decomposition leads to
A,
=
E)
HiI
y E
A2
upon
defining
(4.19)
;
A2 will be called non-degenerate decompositions. come from simple
called in addition simple if each Hi is generated by cycles which closed curves (Jordan curves) in P I under some (hence any) double cover
They
are
simple, non-degenerate decompositions Paragraph IV.4.4) for (1, 4)-polaxized Abelian surfaces proposition. called in
Proposition 4.3 There is phism, classes of) data:
(1) (2)
7r :
and incidence
The relevance of
a
(1, 4) -polarized
a
Jacobi
surface
a
natural
Abelian j
=
is
seen
r -* P 1.
diagrams (refollowing
from the
correspondence between the following four (isomor-
surface r E + a simple, non-degenerate decomposition
C2/A
A
=
=
#W2
A,
A2
ED
of A,
(3)
(4)
smooth genus two curve r + of its Weierstrass Points. a
a
a
decomposition )IV
smooth genus two curve r + an incidence corresponding Kummer surface.
=
W,
U
diagram for
W2, #WI
the
=
16r, configuration
3, on
its
The
correspondence (i)
++
(2)
is established in two ways,
namely
J may be taken
as
the
quotient of T2 using A2 or as a cover of T' using A, (or WI). Moreover, interchanging the components of the decomposition in (2) amounts to taking the dual P of T2 in (:L). J is the Jacobian of the curve r which appears in (3) and (4) and interchanging A, and A2 in (2) amounts to interchanging W, and W2 in (3) and taking the transpose of both square diagrams in the incidence diagram in (4).
Summarizing
we
have the
following
commutative
diagram, determined by T2 (only),
A2
_P
J A,
1 \2, 1
T2
A,
(4.20)
J A2
where 2j denotes multiplication by 2 in J and a Ai is considered on the quotient torus that is obtained
207
labeling an arrow means that a projection by doubling the sublattice Ai.
Chapter
V11. Two-dimensional a.c.i.
systems
Proof
WUI/V2 of its Weierstrass genus two curve rand a decomposition W 3, let -7r : r -+ P1 be any two-sheeted cover of P'. It is well known has branch points exactly at W; the points in W as well as their projections under 7r Given
(3)-+(2)
points, with #Wi that
ir
will be denoted
=
a
=
by Wl,...' Wr,,
also
7r(Wi)
will
just be
written
as
I/Vi. If P1 is covered with 0 then H, (r, Z)
connected open subsets U, and U2 for which I/Vi C Ui and U, n U2 n W decomposes as H, ED H2 where H, and H2 are defined as
Hi
=
1-y
(=-
H, (17,
Z) I
7r.-( E
H, (Ui
\ I/Vi, Z) 1.
Among the cycles in Hi there are those which come from simple closed curves in Ui \ I/Vi encircling two points in Wi and these generate Hi. Since any (different) of these intersect (once) the restriction 0(.)H, is non-degenerate, hence leads (upon using (4.19)) to a nonAi ED A2 for the period lattice. Thus C2 /A and A degenerate simple decomposition A A, ED A2 provide the corresponding data. =
only depend (up to isomorphism) on the isoW, U W2. Let o- : r -+ IP be an automorphism which r, W permutes the Weierstrass points (such an automorphism only exists for special curves r). Then o- extends lineaxly to Jac(r) e_- C2 /A, hence also to the lattice A, giving a new decomWe
now
morphism
show that the constructed data
class of the data
=
o-Al ED o-A2. The lattice aAi contains the periods corresponding to the points position A aWi (with respect to the same basis of the space of holomorphic differential forms), hence o-W, U o-W2. A o-Ai ED oA2 corresponds to the decomposition W =
=
=
By the classical Torelli Theorem, r can be reconstructed from its Jacobian, (2) -* (3) actually in dimension two, r is isomorphic to the theta divisor of Jac(r). The lattice A c C2 is the period lattice of IP with respect to some basis 0 JW1 W21 of the space of holomorphic A -+ H, (17, Z), which in turn leads differentials on 1P, which determines an isomorphism to a decomposition H, (IF, Z) O(Ai). H, ED H2 upon defining Hi I
=
=
by 7r r --+ P, any two-sheeted points by above, then Hi has a system of generators Xil AQ Where 7r* Aij is a simple closed curve in P1 \ W, encircling an even number of branch points Wi, which reduces to two in this case (there axe only six points Wi and encircling four points amounts to the same as encircling the other two points). Since the decomposition is non-degenerate, ?r.Ail and 7r*,\i2 encircle a common point, so we may take of r and
1/V the set of Weierstrass
If -we denote
cover as
i
Wi Then
#W1
=
#W2
=
=
ir-11points
in W encircled
3 and it is easy to
see
,
by ir.Ail
that w, n w2
-
=
or
7r*,Xi2J.
0.
again that the constructed data are independent of the choice of the basis JW1 W21 and are well-defined up to isomorphism. To do this notice first that when the choice I AOY for of basis 0 (wi, w2) is not unique, say V is another basis producing A, then c We show
i
=
=
some
A E
GL(2, C),
hence
i
O=A
it
0
for any y G H, (r, Z). We find that A AA, i.e., A has a non-trivial symmetry group. Then A, ED A2) C2 /A has a non-trivial automorphism group and the data (C2 /A, A Jac(]P) and (C2 /A, A AA, ED AA2) are isomorphic. Thus it suffices to show that the constructed =
=
=
=
data
are
well-defined up to isomorphism. This follows 208
(as
in the first
part of the proof)
at
4. The Garnier
potential
from the property that if Jac(]P) has a non-trivial automorphism a, then it is induced automorphism on r. To see this property (which is paxticular for the case in which the genus of r is 2) let 19 be a generic translate of the Riemann theta divisor passing through the once
by
an
origin 0 of Jac(]P). Then a(()) is another translate passing through 0 (since every curve in Jac(]P) which is isomorphic to IP is a translate of 0) hence composing a with this translate determines an automorphism of r. This shows the constructed data are well-defined.
(2)
-+
(1)
Given J
=
C2/A
and A
A,
=
C2/Al
A2
ED
we
A!
with
form the
=
complex
torus
IA, E) A21
2
the first lattice is doubled in both
(i.e.,
directions) and equip this torus with the polarization by the principal polaxization on J. We claim that r is a (1, 4)-polaxized Abelian surface which belongs to A(1,4). To show this, first notice that the cycles J/\11 7'X211 A121 A22} introduced above, form a symplectic basis of H, (1P, Z), i.e., 0 (Ali AN) 07 0 (Ail Ai2) 17 hence these cycles lead to a period matrix of the form (see [GH]) induced
*
1
(0 satisfying the period
which leads
=
=
'
b
c)spanned by All and A12 (which correspond
A' has in terms of slightly different coordinates
matrix 1
j
b
matrix)
(0 neous
a
1
Riemann conditions. Since H, is
to the first and third columns of this
the
0
=
to J
C2 /A
the result that
immediately to
(notice
that the
0
a
4
2b
7-2 is
2b
4c) (1, 4)-polarized Abelian surface,
a
4: 1
isoge-
right block
of this matrix is positive definite). Since the original is the canonical Jacobian of T2, we axe in the generic case of Paxagraph IV.5
implies 7' E -4(1,4)Dually the surface is (up
which
r but this
to
=
isomorphism) C2 /A"
also constructed
with
A"
=
A,
ED
by taking
2A2i
decomposition induces a 4: 1 isogeny from J to (this) 71. correspondence is well-defined, observe that
To show that the
(C2 /A,
A
=
A,
ED
A2)
t-
(C2 /A,
A
=
A,
E)
A!2)
implies
0
/(21A,
(D
A2
C2
/(2IA' A2) 1
and
ED
the last two
C2/ (A,
E)
2A2)
0
/ (Ai
(D
2A!2),
isomorphisms being isomorphism of polarized Abelian surfaces. For given r E AO let J be its canonical Jacobian J(7-2). Then 7-2 -+ J -"(14), is part of the isogeny 2j : J -+ J hence there is a unique complementaxy isogeny J -+ 71 with kernel Z/2Z ED Z/2Z. Writing J as j C2/A, the latter isogeny induces an injective lattice homomorphism 0 : A -+ A whose cokernel is isomorphic to Z/2Z E) Z/2Z. Then 0 determines a unique decomposition A, E) A2 of A for which OJA2 is an isomorphism and OIA,
(3.)
-+
(2)
,
=
209
Chapter V11. Two-dimensional a.c.i. systems
multiplication by 2. We have seen that such a decomposition is simple. It is also nondegenerate, since otherwise 7-2 would not have an induced (1, 4)-polarization (see below). Observe that in the exceptional case that 71 -+ J is another part of the isogeny 2j, the two isogenies combine to an automorphism of J, leading to isomorphic data in W-
is
(3)
++
)/V, say W
(see [Hud]); we prove it as follows. Given a decomposition JW4, W5, W61 the corresponding incidence diagram is taken
This is classical
(4)
JW1, W2, W31
=
W11 W45 W46 W56
U
W12 W36 W35 W34
r12 IP36 IP35 1734
ip,, IP45 IP46 IP56
W13 W26 W25 W24
W23 W16 W15 W14
of as
IP13 1726 1725 1724
IP23 1P.6 171, IP14
obviously the decomposition of W is reconstructed from it at once. To show that every diagram is of this form, notice at first that we have the freedom to permute the W66 in the upper left corner. well as the columns, so that we can put W11 rows as The curves ]Fij this point W11 belongs to are the entries in the first row and the first column (except r1l) of the square diagram on the right. If the origin belongs to rij n ipjk, (j =,A k), then it also belongs to rik. Then ril is easily identified as the image of the map P -* Jac(]P) defined by and
incidence
=
=
...
P
P
-+
Wh
fW" f
4
4 +
(mod A),
V W
and the other three the incidence
and rl,, with Ji, j, k, 1, m, n} 11, 2,3,4,5,61. Hence from which the decomposition of W can be read =
curves axe
diagram takes the above form
Off. If the curve has non-trivial automorphisms, we define diagrams which correspond to such automorphisms as being isomorphic, so as to obtain the equivalence (3) ++ (4) at the level of isomorphism classes. Finally we concentrate on the dual 1' of T' and its relation with the canonical Jacobian of T. At first recall from [GH] that the period matrices of 71 and f' relate as
(1
0
0
a
4
2b
2b)
4c
(4
0
4a
1
2b
2b) (1
0
c
2b)
4
2b
4a J
_
0
0
c
instead of taking !A, ED A2 when showing that 'P is constructed from J by taking A, ED IA2 2 2 constructing 7-2 from J. It follows that the isogeny 21 can be factorized via 11 as well and that taking the dual of 7-2 corresponds to interchanging the components of the decomposition N of A. This finishes the proof of the proposition.
Remarks 4.4 one considers simple degenerate decompositions (instead of non2 and in (3) is altered into W IN, U W2 U W3, #I/Vi decomposition degenerate) the order of the components in the decomposition of IN is now irrelevant. The corresponding
1.
(2)
If in
above
then the
object
in
(3.)
is then
Jacobi surface
(or
=
=
a
the
Jacobi surface
curve)
(different
from the
cannot be reconstructed.
210
one
in
(2))
from which the
original
4. The Garnier
2.
potential
20, there axe 20 different incidence diagrams (6) 3 isogeny 2j : J -4 J, some of which are isomorphic non-trivial automorphism group (i.e., different from Z2). Since
=
and 20
possible decompo-
sitions of the
if and
has
It follows from the above
a
proposition that the 20 intermediate Abelian surfaces 3. Let
the
C(2)
only
if J
appear in 10 groups of dual
denote the moduli space of all smooth
curves
(hence r) pairs.
of genus two. Then
we
have
following isomorphisms W1 W2 W3
A(I,4)
i
C(2)
JJW1
i
and both spaces
A(1,4)
has
WiEP
W4, W5 W6
i
7
w2i W37 W47 W5, W61 I Wi
are
related
by
an
natural structure of
E
P17 i 34 i
obvious unrainified affine
jJ1 mod PGL(2, C), Wi 0 WjJ / mod PGL(2, C);
ii 6i=*WiA =
cover
variety which
A(1,4)
W
-+
C(2).
We have
seen
that
is
compactified in a natural way into its projective closure, which is the (singular) algebraic variety M3. On the other hand, C (2) also has a natural compactification (the Mumford-Deligne compactification). It would be interesting to figure out how both compactifications are related. 4.
of
T',
a
Among here is
a
an
the different ways to define final one: J J(T2) is the =
(and chaxacterize) only
the canonical Jacobian
Jacobian. for which the
J(TI)
diagram
r 4:1
1 \\ 77-2
7
4:1
commutes (2T is multiplication by 2 on 71). The proof is easy using the ideas of the above proof. Observe that this diagram is (4.20) with 7-2 and J interchanged; we could drop a superfluous triangle since i J. =
4.4. The relation with the canonical Jacobian made We have shown in
Paragraph
type (1, 4) the Jacobi surface of have
a
explicit
4.3 that there is associated to genus two
curve
IP and
some
an
Abelian surface of
additional data.
Also
we
(in Paxagraph 4.1)
that these Abelian surfaces appear as level of the integrable Hamiltonian system defined by one of the potentials V,8. This allows us to make this relation very explicit (in two different ways) and to calculate precisely the locus S in p3 for which the seen
associated quartic fails to be
a
Kummer surface
Abelian surface fails to be birational to
using the theory of integrable systems)
an
octic).
(and
hence the associated
(1, 4)-polarized (i.e., without
We know of no direct method
to do this.
Our calculations
rely on the explicit construction of an embedding for 7-2 in projective by using the Laurent solutions to the differential equations (4.2). Since we know that the potential V,,,6 is a.c.i. (for a the vector field XH has a coherent tree of Laurent solutions (see Section V.3), in particular it has Laurent solutions depending on 1 dim C4 3 free parameters (principal balances). Moreover, since the divisor Df, to be adjoined to a (general) fiber.Ff, of the momentum map is irreducible, there is only one such family. Also q1, q2 and q qIP2 q2p, have a simple pole along Df, since their squares space, which is found
_
=
=
-
211
Chapter VIL Two-dimensional a.c.i. systems
descend to
Jac(r)
information the
with
q1
=
q2
=
double
a
principal
pole along (some translate of) its theta divisor. With this given by
balance is
1[a +2((, I[b 2((,
+a
+
a
-b2)a + 2aV O)t2
0; the
=
b2
-
a
3
t
where 2a 2 + 2b 2 + I
+
2)b + 2ba2a)t2
series for p., and P2
Laurent solutions it is easy to find
induces
2
t
of
-
ad
found
are
embedding
an
+ bd
Tf2,g
polarization of type (2,8), it is very ample and a double pole along Dfg, to wit,
in
this
3
3
+
+
1 (9(t4) 1 O(t4)
(4.21) 1
by differentiation. Using the projective space: since 2Dfg can be done using the sixteen
functions with
2
1,
z8
=
%,
=
q1,
z9
=
q1q,
z2
=
q27
Z10
=
q2q,
Z3
=
zil
=
Z12
=
Z13
=
ZO
=
z,
2
q
=
q1P2
Wli
Z4 =P1,
If,, f2l
depend regularly completable.
+q22)q + aqIP2
=
Z6
=
q17
Z14=2qlq2(ql+q22)
Z7
=
q1q2,
Z15
P2j
jJ2 on
-
2
f1j2,
-
&2PIi
(4.22)
Jq1, qJ, Jq2,qJ,
Z5
2
where
(q 1
+PIP2i
2 =
q
7
f2 and fl. Since the embedding variables fl, f and g) it follows that this a.c.i. system is
the Wronskian of
the base space
(i.e.,
on
a,
We compute the correspondence between the data by using the cover J -+ T2; this can also be done using the cover T2 -+ J (see [Van3]). RecaJ1 from Paragraph 4.3 that given there is a unique Jacobian J J(7-2) such that =
J P1
I\ J
T
J P2
yields a factorization of the map 2j (multiplication by 2). This implies the existence of a r(7-2) as is shown singular divisor in T2 whose components are birational equivalent to r in the following proposition. =
Proposition 4.5 The image pl(k) of Kummer's 16r, configuration IC consists of four the sixcurves, all passing through the half periods of 7-2; these points are the images of teen points in the configuration and each of the four image curves has an ordinary three-fold point at one of these points, with tangents at this point, which are different from the tangents to the other curves. Each curve is birational equivalent to r and induces a (1, 4)-polarization r with an ordinary on r. The image P2 (PI (IQ is one single curve, birational equivalent to six-fold point.
212
4. The Garnier
potential
Proof The map pi identifies all half-periods which appear in a row in the first square diagram diagram which corresponds to 71. Therefore p, also identifies the curves
of the incidence
which appear in a row in the second square diagram of this incidence diagram and we obtain four curves passing through the four image points, every curve having a three-fold point at
image of the three points in the same row (but not the same column) of the first square diagram. Since IC induces a (16,16)-polarization on J, pi(IC) induces a (4,16)-polarization on 7, hence each component induces a (1, 4)-polarization. The virtual genus of each component is thus five, and since each is obviously birational to r via pl, the threefold point must be ordinary and there axe no other singular points. the
The intersection of two of these components is the self-intersection of one of them (since are translates of each other), hence is by Theorem IV.3.7 equal to 2(5 8; on the 1)
they
-
=
hand, since each passes through the three-fold point of the other and since they have 8 so all tangents must be two simple points in common, this gives already 3 + 3 + 1 + I different and there are no other intersection points. The fact that P2 (PI (IC)) has an ordinary six-fold point and is birational equivalent to r is shown in a similar way. 0 other
=
The image 2 j (0) is an
essential
divisor A with
six-fold point, first studied in [Van2] (where it was linearizing variables for integrable Hamiltonian
a
nothing but p*2 A. We have also shown there that this divisor is the leading term in the equation of the Kummer surface of J (when normalized) the algorithm in Section VA
systems)
and p, (K) is
locus of the
zero as
a
in the construction of
ingredient
in
To
apply this
in the
the Kummer surface of J
investigate
its
zero
present
(rf*g,)
can
is
use
the
expressed
leading
term
(4.8)
in terms of the
of the
equation of
original variables), and
locus, i.e.,
(q2I This
we
case,
(which
be factorized
2
+ q2 +
completely
+p)2
Ce
2
-
4(a,6 + Pq 1
+
aq2) 2
=
0.
as
[q2
-
el
V4a
-
13
-
62iql]
=
0.
fi=l
reflecting 61
A/a
the fact that +
p*A 2
is reducible. In order to find
CAI in the equations for Ffg. Eliminating P2
equation for
an
one
finds
an
r(Tf2g),
let q2
equation for the
curve
-
2
ACI 62 : pjQ(qj)(qj
-
6162i ,Fa
-
#)ql +p2( qI )=O,
where
Q(X) P is
some
=
IE162i(a
-
polynomial
6)312X3 + (a -#)(2a of
degree
3. This 2
Z
==
X(X
_
curve
-
#)X2 +6162iVfa is
clearly isomorphic
iftf2A/a
213
j8(h+a(,3-a))x-2'
6)Q(X)-
to the
curve
(4.23)
Chapter VIL Two-dimensional a.c.i. systems In order to decide to which decomposition of the Pl'...' P4 be the following points in P15
-iVa-_-O VGa--# +i-.,fa---,3 +V/-a--,3 -Vfa---# +iV a -iVa-_-O +V/a---#
0
P,
=
(0
:
...
P2
=
(0:
...
0
P3
=
(0:
...
0
P4
=
(0:
...
0
:
points this corresponds, let
Weierstrass
-
-
1
+i(a -#)),
1
+i (a
1
-i(a -,3)),
1
-i(a -,8)),
-
and let q6 denote the three roots of Q(x). Then it is easily checked by picking local parameters around the points at infinity of A,1,2 that the incidence relation of the Pi on the A,,,, is
given by the following table: q, -+ 0
q, --+
q1 -+ q6
oo
A+I,+l
P,
P4
3P3
P2
A-1'+1
P2
P3
3P4
P,
A+I,-l
P3
P2
3P,
P4
P,
3 P2
P3
P4
A_1'_1
P
6162iVa
q1
Table 9
agreement with the fact that each
The table is in
through
has
curve
a
three-fold
Moreover it shows that the three
singulaxities. when going from
the other
under the map pi and W2 10,00,611E2iNFa
711-
If
J to we
7-2,
points
point and
q6
hence these form the subset I/V1 in
were
passes
identified
Proposition 4.3
substitute x+a X 1-+
in the
equation (4.23) for the
Y2 Then the
(a#
-
(x + a)(x
=
curves
+
decomposition of W
h)x
+
(,8f
-
ag)/(2a
-
0)
is
6,1,2
X3
given
20),
a-
then
we
(a + O)X2
+
as
7'
find the
+
(aO
equation (4.11),
-
h)x
+
Of 2(a
-,8))
-
ag
(4.24)
'
(a + t3)X2
follows: W, contains the roots of x3 +
and W2
=
+
foo, -a, -,31.
Suppose that (T2, f-) E A(1,4) and let the surface be represented by a surface JF(,p,f,,), for some a :A # (using (4.18)). Then the curve IP(7) corresponding to it under the basic bijection explained in Paragraph 4.3 must be smooth. Since we know from (4.24) that an equation for r(T2) is given by 2
Y
we
=
(X
+
Cl)(X
conclude that
just
that
f :
+
p)p3(X),
p3(X)
=
X3
+
=,A
0.
+
(aO
-
h)x +
ag
2(a
-
0)
,
(4.25)
0 and P3 (-a) : k 0, P3 (-0) = - 0, the last condition meaning Conversely, both conditions together are sufficient to guaranty
disc(P3 (x)) 7
0 and g
(a + O)X2
214
4. The Garnier
that the
curve
is smooth and the
corresponding Abelian surface
p3,
this result in terms of the coordinates yj for form y2 where X(X I)pl'(X) 3 =
and y
use
(4.18)
is in
A(1,4). In order to state (4.25) in the simple
to rewrite
-
P3"(x) (x
potential
=
41Z2X3
-
(po
2p,
+
slightly resealed);
+
6/-42
+
2/t3 )X2
(po -,2p,
+
in this
+
2p2
+
61-13)X
-
4131
representation W2 10, 1, ool and W, contains the P1 : P2 : P3) to correspond to a surface in 4-(1,4) is 0 and disc(P3"(x)) 0. It shows that the locus S' is given by the four now that /11A2/-t3 0 and disc(P31'(x)) divisors MIA2,A3 0 and the exceptional locus S is found immediately from it by substituting A? for pi in these equationS27. Combining this with Proposition 4.1 we have shown the following proposition. roots of
are
P31'(x).
The condition for
=
(po
=
=
The surface F(, 4.6 is (isomorphic to) an affine part T2 D of an surface (T2, [D]) E A (1,4) if and only if a 6, f 54 0, g : : 0 and disc(P3 (x)) -A 0. Equivalently (MO : p, :/12 : A3) E p3 are moduli coming from the birational Map28 O'C : r2 _+ p3 With (7-2, L) E A(1,4) if and only if Al/Z2/13 74- 0 and disc(P3" (x)) :A 0. The curve ]P(r) corresponding to the canonical Jacobian of T2 is then written as
Proposition Abelian
y2 =X(X
-
1) (4P2X3
when the coordinates
-
(MO
for P'
+
2p,
+
6/42
+
2P3 )X2
is taken such that W2
+
(po
-
2pi
+
2A2
+
6P3)X
-
4/-t3)
10, 1, ool. Conversely
the equation of of the genus two curve and a decomposition W W1 U W2 of its set of Weierstrass points: the coefficients of the octic are Aj Vp-i where pi are essentially the symmetric functions of W2 when the coordinate x for P1 is taken such that W2 10, 1, ool. Taking also the non-generic case into account, there is an Abelian surface _77(Q,#j,g) (M3 \ D) U (C \ fp7Qj). corresponding to each point in the image 0 (A(1,4) )
the octic
(IV.5.2)
x
is written down at
once
when
=
giving
the equation
=
=
=
=
following corollary follows
The
Corollary
once
from this
proposition.
A(1,4)
4.7
(isomorphic to)
at
a
For any Abelian surface (T2, [D]) E complete intersection of two quartics in
the
offine variety 7-2 \
E) is
C4.
Remarks 4.8
Recalling the description Of A(1,4) from Remark 4.3.2
1.
of the moduli space
A(, ,4)
-
=
27 are
in
jjW4j W57 W61 I Wi
These equations for S
[BLS]
213
I(IWI, W2, W317 JW41 W5, W61) I
that the
has the
following description
E
C
\ 10, 1}7
Wi i
E
6j
P11 i 76 j = ,
Wj i4
= ,
Wi 7 Wj
Wjj
mod
PGL(2, C)
14
principle be found purely algebraic, but the calculations easily overlooked. In fact it is claimed (without proof) only condition is lLlA2lJ,3 : 0, the more subtle condition disc(p3Nx)) 5A 0
very tedious and
being
one
A(1,4):
can
in
some cases are
overlooked.
Recall that pi
=
X?,
where
Ai
are
taken from
215
(IV.5.2).
Chapter V11. Two-dimensional a.c.i. systems where the action of S3 consists of permuting 0, 1 and oo in the equation W4) (x W5) (x W6), i.e., it is generated by replacing x by 11x and I
y2
=
X(X
_
1)(X
in this
equation. Obviously the ring of invariants of the symmetric functions of W4, W5 and W6 is just the cone M3, which explains why A(1,4) has such a nice structure. Using Tables 7 and 8, this leads to a geometric interpretation of the "intermediate" moduli space p3 \ S, namely -
-
-
P3 \ 9
5--'
JJW41 W5, W61 I Wi
E C
\ 10, 11, i:74- i
= ,
x
Wi 7 Wj I
-
explain this, notice that taking the basis vectors mod 2 in the first column of Table 8 an ordering for the 4 half-periods on the canonical Jacobian which correspond to the lattice A2, which in turn induce an ordering in the points in W2; on the other hand, all
To
determines
elements in the first column of Table 7 2. In the classical literature
are
the
defines
one
same
mod 2.
Rosenhain tetrahedron for
a
a
Kummer surface
tetrahedron in p3 with
singular planes of the surface as faces and singular points of it as vertices. In [Hud] it is shown that the equation for the Kummer surface with respect to a Rosenhain tetrahedron is written as the quartic (4.5). It then follows from Proposition 4.6 how to read off from the equation of a Kummer surface with respect to a Rosenhain tetrahedron, an equation for the curve corresponding to this Kummer surface and vice versa. It seems as a
that this result is not known in the classical
4.5. The central Garnier
recent literature.
or
potentials
In this final
paxagraph we concentrate on the potentials V,,,,, which were excluded up interesting to compare the classical linearization of the central potential V,,,, which uses polar coordinates with the a 0 limit of the linearization of the perturbed potential V,,,3 (a :A 3): they will be seen to coincide. We will also construct a Lax pair for this limiting case and discuss the geometry of its level manifolds. to
now.
It is
=
At
first, consider for general values
h, k the level surface Yhk defined by
of
1
h
=
2
Yhk k
=
(p21 + p2)
qIP2
which in terms of polax coordinates
leading
=
(qJ2 + q22)2 +
p
a
(ql2+ q22),
Wli
becomes
(A2 P2 2) +
=
2
k
-
(p, 0) 1
h
+
2
+
#4 + ap 2,
26,
to
1 2
This suggests
setting
0.
P2 2
=
p6
+
04
-
hp2 +
k 2
p2, yielding 6,2 --=a
3
+au
2
8
k2
-ha+ 2
216
(4.26)
4. The Garnier
reduces for
(4.9)
the transformation
Secondly
+X2
X1
=
and
(4.10)
=
=,3
a
to
(q 21 + q22 + 2a)
-
2
XIX2
potential
(4.27)
2
2
+ aq, + aqi
a
becomes
8(xi
&?%
+
(a2- h)xi (XI X2 )2
a)2 (2S_ + 2ax?
+
%
-
(ha + f/2))
-
(4.28)
-
The on
becomes clear after the
(4.26) and (4.28) (4.27) becomes
of
equivalence
8182
a
xi +
a
equality
is
xi
that
only one of the
definition),
1
=A
(q21 + q2) 2
s,
linearizing variable
or
=
_s,
(the
last
introduced above. In terms of
s
(4.28)
equation which reads
one
h2
3
+as
--=s
2
8
which is
2
01
=-
si differs from zero, say 0
which matches the
is reduced to
(q 2+ q2),
+52
51
so
simple translation
the curve; indeed
exactly (4.26)
f
since
(qlP2
=
-
q2pj
)2
f
-hs+2'
=
k2.
a Lax pair for the interesting that the Lax pair gives in the limit a The polynomials u(A), v(A) and w(A) axe now all divisible by (A + a),
It is also
potential V,,,,.
u(A)
=
q22 (A + a) (A + q2+ I
+
a),
I
V(A)
=
w(A)
=
which leads to
Finally
a
simpler
over
the
(a
+
2) A_'2 (pi +p2)
q,
elliptic
q2
-
(4.26),
curves
k, h
E
=
2
k 0 then
2
as
Yhk
is
a
=
(P21 + p2) 2
q1P2
C* -bundle
-
2)
+ q2
a).
potentials V,,". These turn out described in the following proposition.
the
over
+
q2), (qJ2 + q22) 2+ a (q2+ 1 2
elliptic
Moreover the C*-action
being the
on
:
to
Yhk is
3 =
-
or
+
V
2 aor
2 a
(4.29)
curve
2
to it
(q,
q2PI-
T
Ehk
sponding
+
-a
C, let J7hk denote the affine surface defined by
1
h T,hk
If k - 4
2
2
2 -
pair by canceling the factor (A
Lax
For any
4.9
Proposition
V
(A+ a)
q2P2)
+
describe the level surfaces for the central
we
be C*-bundles
(A + a) (qlpl
,/2-
-
hor +
Hamiltonian action, the Hamiltonian
momentum qIP2
-
Wl-
217
(4.30)
_Y* function
corre-
Chapter
V11. Two-dimensional a.c.i. systems
Proof The
linearizing variables, calculated above suggest
C4
to consider the map
-+
C2
given by
(q,
I
Our first claim is that the one
easily
(q2I + q22,qlpl + q2P2)-
q2 i P1 i P2)
image (,Fhk) : L 07
is
given by the plane elliptic
curve
(4.30). Indeed,
obtains for q,2 + q22
q2k Pi
-
qiT
-
q]2L
qik P2
7
2
+ qi
+ q2*T
-
2
2
q, + q2
which leads
by
direct substitution in the first equation of T
T For
q2
+
q22
=
0, i.e.,
q2
=
+iql
2
deduce
we
r
=
ha +
-
T*
gets
one
h
from which
to
k
or3+ aor2
==
-
(4.29) immediately
(p21
+
p2), 2
k
q, (P2 T-
iPI);
T
qi(p,
iP2)i
ik, giving the point (o-, -r)
=
(0, :Lik)
on
Ehk, proving the first
claim.
Secondly, we determine the fiber -l (a, -r) over each point multiplicative group of non-zero complex numbers,
on
Ehk. To do this, observe
that the
C*
acts
on
-5--
SO (2,
a
C)
b
a) I
a2+ b2
=
I
Fhk by a
b
q,
pi
aq, +
-b
a
q2
P2
aq2
-
bq2 bql
aP1 + aP2
-
bP2 bP1
surjective map is C*-invariant. It is proved by direct calculation that the action free, hence each fiber of consists of one or more C's. If (a,,r) E Ehk then p, and P2 are determined from q, and q2 (at least if q 21 + q22 4 0), which themselves are determined (up to the action of C*) by q2+ q22 p, so exactly one C* lies over each point (qi,q2iP11P2) for 1 which q12 + q22 :? 0; in the special case that q12 + q22 0, the same is true, since pi and P2 are determined (up to the action of C*) by p2 2h, and qj, q2 are uniquely determined 2 1 +p2 from p, and P2 It follows that Fhk is a C*-bundle over the elliptic curve Ehk. and the
is
=
=
=
-
q2PI
Finally, observe is given by
that the Hamiltonian vector field
41
=
42
=
-q2i
P1
=
qj,
P2
=
218
corresponding to -P21 Pli
the momentum qIP2
-
4. The Garnier
from which it is
Let
us
define a
(and calculate)
central
vector field is
complex flow of this proposition.
that the
seen
proving the last claim in the
face Thk of
potential
potential
the moduli
for k
0
given by the C*-action, 0
(in p(1,2,2,3,4)) corresponding
to
a
level
sur-
the limit29
as
-2
2
f=k
computation shows that this limit exists, is independent of f :A 0, h and a is exactly equal to the special point P at the boundary of '0 (A(i,4)) defined Proposition 4.2. Namely for f -+ g and a -+,6 one finds
Then
and
an
easy
moreover
(PO so
:
PI
:
A2
:
/13)
=
(-4
:
I
:
0
:
in
1)
that
(fo, fl, f2, f3, f4)
=
(-4:
0: 3: -2:
0)
by weight homogeneity the associated moduli correspond to P. Notice that the point is independent of a # as well as of f g, so the map 0 does not distinguish between any of the level surfaces of any central potential V,,,,,.
hence
=
=
21
Recall that
f
-
g
=
2(j6
-
a)h. 219
Chapter V11. Two-dimensional a.c.i. systems
5. An
integrable geodesic flow
5.1. The It
geodesic shown
was
flow
by Adler
SO(4)
on
and
classes of left-invariant metrics
for metric II
Moerbeke
van
SO(4)
on
SO(4) for completely integrable system (a.c.i. system)
(unpublished proof)
that there exist three
which the
geodesic flow reduces to an algebraic on its Lie algebra 50(4). In the sequel, we will consider the second case, known as the case of metric II. In suitable coordinates, the first vector field X1 of this a.c.i. system is given by the differential equations il
on
2Z5Zr,,
:--
i2
The second vector field
z4'
=
Z24i
z2(2z3
-
z6),
the vector fields X, and X2 admit four ing functions:
F,
=
F4
2 3
_
i3
=
Z5'
=
(ZI
+
Z4),
i6=2zlz5_
given by the differential equations
z4(2Z3
-
Z6),
Z3,Z47
ZI3
=
Z4Z5,
Z6'
=
ZIZ2;
(5.2)
independent quadratic invariants, given by the follow-
-'i2,
(5.3)
2
zi
z =
z/2
is
Z5
z251
2
F3
verify that there
z
_,2I
F2
It is easy to
2z3Z4,
X2, commuting with X1,
=
ZI
==
i5=z3(zl+z4),
i4=2Z2Z3,
(z,
exist
+ Z4
)2
4(_,23
+
precisely
-
Z2Z5
-
Z3Z6)-
three
linearly independent linear Poisson structures on C6 with respect to which X, and X2 axe Hamiltonian; moreover, these Poisson structures are compatible, implying that the integrable system admits a tri-Hamiltonian structure. Explicitly, for any (a,,6, -y) E C3, the matrix 0
aZ6
-aZ6
0
-OZ5 2,yz4
a(Z6
OZ5
-2-yz4
0
-aZ5
a(2Z3
0
#Z3
+
\,6(2Z5
-
Z2)
-
ZO
0
2^(Z6 azi
az5 +
'Y(Z1
+J6Z4
is the Poisson matrix of
a
0
-
-,6Z3
2,fz6
-
2z3)
0
2,yz2
-Y(ZI
+
Z4)
aZ3
0
-J6Z2 2,yzl
6Z2
-2-yzi
0
P,6,
on
X, and X2
C'. If (a,,6,'y) 5A (0, 0, 0) then P"'6,
as
described in the
P,3,
table.
F,
F2
F3
F4
0
2 X2
-2XI
0
2 X,
0
8 X2
0
Pool
2XI
PZ4
0
0
system of generators of the algebra of Casimirs of these structures
0
2z5)
-
-az3
vector fields
Ploo
-
Z4)
0
Polo
6(Z2 -aZl
2^(Z2
+
Poisson structure
generates the Hamiltonian
-
2(XI
-
X2)
0
Table 10
220
following table;
a
also follow from the
5. Geodesic flow
It
which
on
SO(4)
by Adler and van Moerbeke in [AM7] that, for any f some3o Zariski open subset -H of C4, the affine surface
shown
was
belongs
to
Af
=
Jz
C6 I Fi(z)
E
=
gi, i
=
(fl, f2, f3, A)
4}
=
isomorphic to an affine part of the Jacobian of a compact Riemann surface rf of genus (which depends on f e R), Af 5--- Jac(Pf) \ Df and that the vector fields X, and X2 are linear, thereby proving that the above system is algebraic completely integrable. The affine is
two
part
Af,
the divisor
and the Rlemann surface
Df
that the group T of involutions, 011
(Z1
Z6)
=
I... I
U2
(Z1
Z6)
=
7... ,
commutes with the vector fields
fact
they generate,
tori
Jac(:Pf).
For
labeled
(or by
el
the free
are
a
can
be described
follows. First notice
as
(-Zli -Z21 Z3, -Z4i -Z5, -6), (-ZI Z2 -Z3 -Z4 Z5 -4), i
7
(5.4)
1
i
7
Af invariant;
X, and X2 and leaves the affine surfaces of translations
Tf
f E W, a group consequence, the divisors for any
half
over
periods
in
in the
Df are also stable under these translations. Df one applies Painlev6 analysis to the vector
precise description of the divisors
a more
field X,
As
rf
generated by
four principal balances, any combination of X, and X2). It has has precisely e = + 1, 162 = 1 , whose first few terms are explicitly given as follows (a, b, .
parameters)(a
1)el
-
(1
4
'El 'E2 Z2
=
Z3
=
Z4
=
Z5
=
Z6
=
(a
t 62
2t
(1 + bt
bt +
abt +
-
((a
((a
(b
2
-d-
1)(ae
-
1)e + ad
-
e
-
-
)t2
c
-
c
-
+
O(t3)) a2d)t2 + 0 (t3))
ab2) +
ab 2)t2 +
O(t3)) (5.5)
61
(-a + abt + Ct2 + O(t3))
T
IE162
(I + bt + dt2 + O(t3))
2t
(a
-
-
-,.
1)62
(-1 + bt
-
et2
+
O(t3))
fi, When any of these families of Laurent solutions is substituted in the equations Fi(z) i 1,...,4, the resulting expressions are independent of t. This leads to four algebraic equations in the five free parameters, giving explicit equations for an affine part rf of Pf =
=
Each of these
equations
Y2 In what recover
follows,
=
we
X(I
_
easily
X)
Vf(1E1i62)i
axe
621
=
rewritten
[4X3f,
will refer to the
the Riemann surface
Since there
Pf
-
(4f,
curve
from it
as
+
in
one
f4 )X2
62 :2
=
1, of the =
curve
Pf (1, 1)
Explicit equations for R will
+
Pf,
y4
f3
_
_
adjoin
(5.5),
one
f2 )X as
+
the
f31
the divisor
Df
given
+
rf (-1, 1)
in the next
221
+
]Pf.
In order to
by oof. consists of four copies
TPf (-1, -1).
paragraph.
(5.6)
.
curve
point which
i.e.,
Pf (1, -1)
be
+
C2, given by (5.6),
has to
four families of Laurent solutions
E)f 30
is
we
denote
Chapter V11. Two-dimensional a.c.i. systems The Laurent solutions in
can also be used to compute an explicit embedding of the tori Jac(rf P'5: the sections of the line bundle on Jac (r- f ), defined by Df correspond to the meromor,
phic functions on Jac(Pf) with a simple pole (at worst) at the divisor Df and, in turn, these are found by constructing those polynomials on C6 which have a simple pole in t (at worst) when any of the four families of Laurent solutions are substituted in them (see Chapter V). I and the functions zi, i Apart from the constant function zo 1, 6, one easily finds the following independent functions with this property: =
=
.
The
embedding of
Z7
=
z5(2z3
-
Z6)
-
z8
=
zi
(2z3
-
z6)
-
z9
=
z4(2z5
z1o
=
(2z5
--1.1
=
(2z3
Jac(Pf)
0: Af
__
-
Z2)
Z2)2 Z6 )2
_
_
p
Z2-'3, =
ZIZ2Z3
Z13
=
Z2Z3Z6
Z14
=
Z2Z5Z6
Z15
=
ZIZ2Z5
4Z64,
-
ZIZ4Z5i
-
(5.7)
ZIZ27
Z62'
_
Z227
=
Z12
z4z6,
_
in P15 is
p15:
-
.
given
(_,l
on
'... ,
the affine part
ZO
4
(I
:
ZI
Z1Z3Z41
-
Z3Z4Z6-
-
the map
Af by
(P)
:
...
These functions will be used later to construct two maps which quotients of Af birationally into p3.
:
Z15(P))-
are
similar to
0 and which
map two different
5.2.
Linearizing variables
paragraph we show that from the point of view of moduli, the family of affine E W, can be replaced by a family of polarized Abelian surfaces of type (1, 4). order to do this we will first construct an explicit map from the affine surface Af (f e W) an affine part of Jac(:Pf). Following [BV4] we do this by following the algorithm which In this
surfaces In to
was
Af, f
outlined in Section VA. We define W to be the set of those
f
=
(fl, f2, f3, f4)
C4 for which the
E
curve
(5.6)
is
a
of genus two, i.e., that its right hand side is of degree 5 and has no multiple 0 for all f E W. It will follow from roots; notice that this entails in partic-ular that flf2f3 our construction that, for every f E W, Af is indeed an affine part of the Jacobian, thereby
non-singular
curve
justifying the notation W. In order to apply the procedure described in SectionVA, we fix an arbitrary element f E W and we choose one component, say C Pf (1, -1), of the divisor Df on Jac(ff). The meromorphic functions on Jac(ff) which have at worst a double pole along the divisor C can be obtained by constructing those polynomials on C' which have at worst -1 are a double pole in t when the Laurent solutions (5.5) corresponding to el 1, 62 substituted into them (and no poles when the other solutions are substituted). It is easily computed that the space of such polynomials is spanned by =
::--
=
X0
=
where
11 we
X1
=
(-2
+
-4) (Z3
think of these
Jac(ff) \ C
+
Z5),
X2
polynomials
=
as
(Z3
+
being
Z5) (ZI
+
Z6)
X3
i
restricted to
Af.
=
(ZI
The
+
Z6) (Z2 +,4), (5.8)
mapping 0, given
by
0: Jac(rf) \ c
-+
P3
'
P
=
(ZliZ2i
...
I
Z6)
222
'-+
(XO (P)
:
X1 (P)
:
X2 (P)
:
X3 (P))
on
5. Geodesic flow
SO(4)
on
Jac(:rf) to its Kummer surface, which is a singular quartic in P3. An equation for this quartic surface can be computed by eliminating the variables zl,...,ZC, from the equations (5.3) and (5.8): solving the equations (5.8) and the first three equations in (5.3) for the variables zi, z2, z6 and substituting these values in the remaining equation, the equation for the Kummer surface of Jac(:Pf) can be written in the form maps the surface
..
X32((XI where
+X2-2f, )2 +8flXl)+f3(XIIX2)X3+f4(XliX2):--Oi
f3 (respectively f4)
It follows from
.'
(5.9)
is
(5-9)
polynomial of degree three (respectively four) in X, and X2system of linearizing variables (xi, X2) is given by the equa-
a
that
a
tions
-2f,(x,+X2):--':XI+X2-2f,, This is checked in the present tions (5.10) as
(Z3 Since
f
C W the
divide
can
Z5)(Z2
+
the vector
+
Z4)
=
variables
and X2
x,
xi and
of
we can
(5.12)
+
ZO (ZI
ZO
+
1)-1
-
I
Lb2X2
=
+
=
the
-
+
use
of
(5.8),
2f, (xi
-
to rewrite the equa-
1) (X2
both different from I and from 0
are
1 as necessary. Deriving by xi field X, given by (5.1) we find that
by
Lb&l
of the two
W
-2fjxjX2i
JxiI
Then
follows. First make
case as
(5.10)
-2flxIX2:--Xl-
z, + z4 +
&2(X2
-
-
so
1)
-
(5.11)
that below
we
equations (5.11) with respect
to
2Z37
1)-1
(5.12) z, + z4 +
=
2z5.
solve the first three equations of (5.3), together with (5.11) and the difference in (5.12) for zj, Substituting these values in the second equation I Z6
equations
-
...
find that
we
2
&1
&2
(xl(xl 1))
X2(X2
-
-
1)
)2
1 =
-
X1
-
[4f,
f2
f3
+
-
(X1
-
WX2
-
1)
+
-
(5.13)
X1X2
equation is linear in &21 and &22 Finally we substitute the values for zi, Z6 equation of (5.3) to find another equation which is linear in &21 and &2, 2 leading
Notice that this in the fourth
X2
.
.
.
.
,
to
Xxi) (XI X2) -21
&?
(i
=
1, 2),
-
where
g(X) (We
note that the
=
X(I
curve
X)[4fI X3
_
y2
=
g(X)
in terms of the coordinates X1 7 X2
-
(4f, +f)X2+(f 4 4 _f2
is precisely the given by (5. 10),
curve
_
f3 )X+fl. 3 It follows that, equations (5. 1) reduce to
]Pf given by (5.6).)
the differential
the Jacobi form
&I
X2 -
-
+
Nrg(xl) so
that x, and X2
are
indeed
-
Vrg(X2)
X2X2
XIXI =
0,
-
Nrg(xl)
linearizing vaxiables.
223
+
= -
Nrg(X2)
1,
Chapter V11. Two-dimensional a.c.i. systems The construction of these
Jac(Pf)
linearizing vaxiables leads
U(X)
=
X
an
map into the Jacobian
explicit
by defining
a
polynomial
v
of
2 v
-
at most 1
degree
as
-
(5.14)
z,5)
follows
v(1)=u(1)(z1+z4+2Z5)-
(5.15)
by u(x), as can be checked by a direct computation, so that a point of Jac(:Pf) \ Of, where Of is (a translate of) the
is divisible
(X)
2(z3
z,,
v(0)=u(0)(z1+z4+2z3), Indeed, g(x)
Z2 + Z4
(ZI+Z2+Z4+Z6 2(z3
2+
-
and
to
by defining
the above formulas indeed define theta divisor of
Jac(Pf).
system because g is above map is
Notice that
regular;
moreover
f
E
-
z2 + z4 and z, + z6
can
Z2 + Z4
::
+Z6
=
Z,
I (V(O) 2 i(O)
-
Z5
a
0 and hence Z3
R, fl
it is birational because
Z3
while, using (5.14),
such this does not define
as
not monic. Since
be rewritten
V(1) Nl) V(O) NO)
V(O) U(O) V(1) U(1)
Z5
:A 0, showing
that the
(5.15) gives
V(1) U(1)
-
map to the odd Mumford
-
)
as
(5.16)
I
follows:
.) U(0)' ') U(1).
(5.17)
Using the invariants F1, F2 and F3 one easily finds formulas for Z3 + Z5, z2 z4 and z, Z6 showing that the map is birational. On the one hand this proves that when f E 71, i.e., when l7f is a non-singular curve of genus two, then Af is isomorphic to an affine part of Jac(rf). On the other hand it leads to explicit solutions for (5.1) with respect to initial conditions which correspond to a point f E W, in terms of theta functions, -
U(O) as
follows from
from it because We
see
-
'0
(
(VI.4.9). they
are
V[60](At + B) 0[ (At + B)
u(O)
roots of
u so
=
that the
)
U(1)
,
=
C,
(
V[61](At + B) 0[ (At + B)
2
)
'
The expressions for v(O) and v(1) in terms of theta functions follow the derivatives of u(O) and u(1) with respect to t.
that the inverse map, given 0 and u(l)v(O) 0, u(1)
divisors
2
-
=
corresponding
by (5.16) -
and
u(O)v(l)
divisors
are
=
(5.17),
is
0. When
away from the 0 then 0 is one of the
holomorphic
u(O)
=
of the form Wo + P, where Wo stands for
0 and P E Pf Similarly, u(1) 0 corresponds to the the Weierstrass point over 0, x(WO) divisors W, + P, where W, stands for the Weierstrass point over 1. In order to avoid a rather involved explicit computation for the third divisor we appeal to the fact that the divisor at =
=
-
is invariant for the group Tf. Knowing that Df consists of the theta divisor of (consisting divisors oof + P) besides the two divisors that we have just determined we can identify the elements of Tf as translations over [W1 Wo], [oof Wil and [Wo oof ]. Thus,
infinity Df
-
-
-
corresponds to the effective divisors in [Wo+W1+P-oof]. It is now easy to see that the four points 2oof, oof + Wo, oof + W, and Wo + W, (which constitute a single Tf orbit) each belong to exactly three of the four curves and that these the divisor
u(1)v(0)-u(0)v(1)
=
0
224
5. Geodesic flow
four
curves
following
have
no
other intersection points.
Thus,
SO(4)
as a
byproduct, Df.
we
have recovered3l the
intersection pattern of the components of the divisor
Figure We will
now use
the above results to
M where
on
=
8
study the moduli
JAf I f
E
space M defined
by
WI/isomorphism,
isomorphism. means isomorphism of affine algebraic surfaces. We will relate this moduli cone M', introduced in Section 4. In the following two propositions we show
space to the
how M and
M(1,4)
are
related.
Proposition 5.1 For any f surface Tf The line bundle Cf induced map
OCf : Tf
-+
quotient Af ITf is an affine part of an Abelian induces a polarization of type (1, 4) on Tf and the to its image.
R the
E =
.
[Df ITf I
P3 is birational
Proof
We have shown in Proposition 4.3 that the translations of the form
10, [W2
-
by a group of WO, W, and W2 are is an Abelian surface of type (1, 4), more precisely
Wj], [Wi
-
quotient of
Wo], [Wo
-
a
Jacobi surface
W2]1,
where
points on the underlying curve, belongs to -4(1,4). The divisor Df descends to the irreducible divisor Df ITf which has a triple point which corresponds to the singular points of Df. Since Df induces a polarization of type (4,4) on Jac(Pf), DfITf induces a polarization of type (1,4) on 7-f. In order to see that the induced map Ocf is birational onto its image one considers TflKf where Kf is the group of two-torsion elements inside the kernel of the natural isogeny from Tf to its dual Abelian surface 'ff. Since Tf Jac(:Pf)/Tf the map Jac(rf) -+ TflKf is an isogeny whose kernel consists of the sixteen half periods of Jac(Pf). This means that this isogeny is multiplication by 2 in Jac(:Pf) and hence that 7f' lKf is a Jacobi surface. This implies that 0 the map OCf : 7-f _+ p3 is birational to its image. Weierstrass it
=
31
This intersection pattern
to the vector field
was
first determined in
Xi.
225
[AM9] by using
the Laurent solutions
Chapter
V11. Two-dimensional a.c.i. systems
The above correspondence between affine surfaces Af and Abelian Proposition 5.2 faces T induces a bijection X : M
sur-
Proof For f E ?i we know that,Tf is a group of four translations of Jac(rf) over half periods leaving Df invariant. Since the group of translations over half periods acts transitively on the set of theta curves this property characterizes Tf. It follows that isomorphic surfaces Af and Ak lead to isomorphic quotients Af ITf and AkITk and hence to isomorphic polarized Abelian surfaces (7-f ,Cf ) and (Tk, Lk). This shows that the given correspondence between
affine surfaces
Af
Starting from
and Abelian surfaces T induces
a
map X
:
M
-+
A(1,4).
polarized Abelian surface (T, Oc) of type (1, 4) for which the induced map is birational there exists a Riemann surface V and a partition W-= W1 U W2 JWo, W1, W21 U JW3, W4, W51 of its Weierstrass points such that T Jac(]P)/T, where 'I is the group of translations, given by T 10, [Wo Wj], [WI W2], [W2 WO] I. Moreover the triple (f W1, W2) is uniquely determined up to isomorphism (see Proposition 4.3). Let us pick one particular triple (P, W1, W2) and let us choose coordinates for PI such that the P1 is given by 0, 1 and oo (in some order). .mage of W, under the natural double cover Then we find an equation of the form any
=
=
=
-
-
-
,
Y2 in which the
right
x(I
=
hand side has
-
x)(AX3
+
BX2
+ Cx +
D)
double roots.
Obviously we can find then at least one rf, given by (5.6). By construction (the isomorphism class of) the affine surface Af is contained in the fiber X- 1 (TC), showing the surjectivity of X. Finally, a triple (V, Wl, W2) which is isomorphic to (TP, W1, W2) leads to an isomorphic surface Ak because Af is intrinsically described in terms of the triple (r, W1, W2) as being the affine part of the Jacobian of I-', obtained by removing the translates of the theta divisor, corresponding to the half periods 10, [Wo Wj], [Wi W2], [W2 WO] f
E W such that this above curve
no
corresponds
to the
curve
-
where W,
:--:
-
-
IWO W1 W2 I 7
5.3. The map M
,
-+
M1
It follows from Section 5.2 that for any f E W the line bundle Lf which corresponds to defines a birational map 0,cf from Tf to an octic surface in P3. We will compute an
Df ITf
equation of this octic because the coefficients of this equation, which depend us to construct explicitly the map M _+ M3. Since Tf Jac(ff)/Tf the =
on
f,
will allow
vector space of
provide this map consists of the Tf-invariant functions on Jac(rf) with a simple pole along Df (at worst), i.e., the T-invariant functions in the span of jz07 Z151Using (5.4) and (5.7) one finds the following four independent invariant functions: functions which
...
00
=
Zo
01
=
02
03
=
1
1,
z1o
=
(z2
=
Z11
=
(2Z3
=
Z12
=
-
2z,5 )2
_
Z1Z2Z3
Z6 -
Z62' )2_Z221 _
Z4Z5Z6-
In order to compute an equation for the octic it suffices variables z, Z6 from the equations (5.3) and (5.18). In
in principle to eliminate the practice, doing the calculation in a
-
226
(5.18)
-
5. Geodesic flow
straightforward such can
MuPad
as
or
Maple. Therefore
results,
we
when
even
will describe in
variety
using
a
want to
we
h
X3
-
X5,
0
X1X4
_
f2
XI
-
x6i
0
X2X5
-
f3
X2
-
x4i
0
X3X6
-
f4
=
X, + X4 + 2ZI + 4X3
01
=
4X5
-
4Z2 + X2
-
X6,
02
=
4X3
-
4Z3
X6
-
X2i
02= XIX2X3 3
+
X4X5X6
+
-
computation
compute is isomorphic
Z2, 1 2
4Z2
-
computer algebra package
a
detail how this
some
first step we notice that the octic which defined by the following equations:
be done. As
to the
way leads to disastrous
SO (4)
on
-
Z2
i
Z2, 3 (5-19)
4Z3,
2ZIZ2Z3.
see this, we consider a regular map V from the variety given by (5.3) and (5.18) to the 6 zj2 and Zj Zi Zj+3, where i 1, variety given by (5.19). The map V is given by Xj all and T because the orbits of is are constant hand the On on one and j Xi 2,3. Zj 1, V T-invariant; on the other hand it is easy to check that every fiber of W contains precisely four points, hence the degree of W is four. This shows that (5.19) represents the image of Af /Tf in projective space, obtained by using the sections of the line bundle associated to Df ITf Six of the equations in (5.19) are linear and we can use these equations to eliminate Z12, this X21 X3) X5, X6, Z2 and Z3 from the four non-linear equations. Apart from X,X4 leaves us with the following three equations (we have used X,X4 Z12 to simplify them)
To
=
=
=
.
..
,
=
-
=
=
2
2
2
[f3XI -
-
2(f2 (h
4f3X,
4(f3
+
f2X4
-
-
+
f3
f3
4(f2
02)(XI
(f2
-
-
-
A
f4
+
+
01) (f3 f2)
-
-
f3
-
01
-
02)XIZ1
02)(4f,
-
02)ZI21
+
f2
-
2f3(4f, -
f3
-
2(4f, -
f4
X4)
+
4XI X4
-
(f2
4X4f2
+
4XIX4
-
(4f,
+
A +
-
f2
-
01)Z1 -
f2
f4
-
-
8 023
-
+
f3
f3
02
+
+
2f2(f4
-
=
+
f4
0I)X4ZI
f4
02)XI
+ 01 +
f3
+
+
01
-
02)X4
0,
2ZI )2
+
-
=
0,
01 + 2ZI )2
=
0.
(5.20) computation feasible stems from the following observation. If we multiply the second equation by X, and the third equation by X4 to remove from the first equation in (5.20) those terms which contain X21 and X '2, then the resulting equation is a linear equation in XI, X4 and ZI (the relation XJX4 Z12 is again used to simplify this expression) so that (5.20) is equivalent to a linear system of equations in XI, X4 and ZI, which is solved at once. An equation for the octic is then given by substituting the expressions for XI, X4 and ZI in the only remaining equation X,X4 Z12. The resulting equation is monstrous (it has 2441 terms), in contrast with the equation (IV.5.2) for the octic, corresponding to an Abelian surface of type (1, 4). By a recaling of some of the coordinates by roots of -1 we write the octic. in the following more symmetric The first trick that
we use
to make the rest of the
=
=
form: 2
+ P2( 41,43 + VI4V4)+ + t42(y4V4 4Y4) 2V32 + P2(V4Y4 2 1 3 + Yo 2 3 Yo 2 2 3 0 1 + Y4Y4) 1 2 2 + Y2Y2) + Y2y2)(Y2Y32 (Y2Y2 2t,2) 2Z/32)(y2V2 2/12P3 0 2 21A, 02 (Yo YI 1 2 Y3 1 V2 0 3 1 0 2 0. + Y2y22)(y2Y2 y2y2) 2P3P1 (y2y2 2 3 1 0 1 0 3 2
2
IL YO YI Y2 -
-
_
_
-
=
227
_
(5.21)
Chapter
V11. Two-dimensional ax.i.
systems
The difference between these two equations lies of course in the choice of coordinates. In order to compute the coordinate transformation which reduces our equation to the symmetric form
(5.21)
we use
A043
B023
the
following geometric farct. Since the octic that we obtained has the form 0 the octic has a singular point of order four at (0 0 0 : 1) and such a singular point necessarily comes from four of the sixteen half periods on Jac(rf). Clearly, (5.2 1) also has a singular point of order four at (0 : 0 : 0 : 1). On the other hand, we see that the tangent cone to (5.2 1) at (0 : 0 : 0 : 1), is the union of four hyperplanes because the zero +
+
C2
-
=
locus of the coefficient of
(YO Where Yo
in
Y1 + Y2) (YO
+
has the form
(5.21)
Y1 + Y2) (YO + Y1
-
V//_13YOi Y1 V/P-2YI and Y2 irrelevant). The coefficient =
=
square root
y34
-
Y2) (YO
-
Y1
-
i,//-IIY2 (the particular
=
A of 04 in 3
are
-
our
Y2)
choices made for each
equation for the
octic must also be
product of four linear factors, but these are harder to determine because this can only be done by passing to an extension field of the field C[f3., f2, f3, f4]. However, if one uses the sections of a symmetric line bundle to map a Jacobian in projective space, then symmetric equations for the image are usually obtained by explicitly introducing the Weierstrass points on the curve, rather than working with the coefficients of a polynomial that defines the underlying curve (see [PV2]). In view of the equation (5.6) for ]Pf we are therefore led to 32 defining the
4x 3f,
Indeed,
-
(4f,
+
in terms of the
f4)X2 + y4
Ai
_
f3
finds the
one
f2)X + f3
_
(X
:--
Al) (X
-
following factorization
for
-
A2) (X
-
A3)
A,
3
A
00
=
JJ[AAi(Ai
1)00 -Ai01
-
-
(Ai
-
1)021-
i=1
In order to find the
required coordinate transformation, YO
+
Y1 + Y2
=
00i
-YO
-
Y1 + Y2
=
K1
-YO + Y1
-
Y2
=
Y1
-
Y2
=
YO
-
(AINI (AI
r1,2(AA2(A2 r-3(A/\3(,X3
we can now use
-
1)00 -A101
-
(Al
-
-
1)00 -A201
-
(A2
-
-
1)00
-
(A3
-
-
A301
the
following
1)02)1
ansatz-
(5.22)
1)02)1
1)02)-
The coefficients ni are uniquely determined by the compatibility equations, which stem from the vanishing of the sum of the left hand sides of these four equations. If we denote A(x) =
A(X Al)(X A2)(X A3) then the solution to the compatibility equations 1,...,3). Substituting these values for r,,i in (5.22) we -1/A'(Ai), (i -
-
-
is
can
=
equation for the octic
in terms of the coordinates
determine the pi such that
we
obtain
given by
rvi
rewrite
=
our
Yo,...'Y3. Putting Yi piyj we can precisely (5.21). It gives the following values for =
Ai /Ili /12 7 P3:
32
N?
=
Ai(I
P2
=
12(or22
Ai)(Ai+1
-
_
0.20.3) 1
The final result will be symmetric in
-
Ai+2)3i (5.23)
+
2(c2
-
al)(U10'2
+
Al, A21 A3, hence does
these parameters.
228
9(73)1 not
depend
on
the order of
5. Geodesic flow
on
SO(4)
where ai is the i-th symmetric function Of /\I i 1\2 7 1\3 and A4 1\1 i \5 the parameters pi explicitly in terms of the Weierstrass points of the
1\2 This determines -
curve
ff.
The
sign of
the parameters p and pi is not important. Indeed, the coefficients (A,A1,/-12,A3) are only intermediate moduli for Abelian surfaces of type (1, 4), the moduli themselves being given by the following expressions which realize the moduli space as the cone C : f42 f, (4f23 27f32) =
in
weighted projective f0
=
f1
=
f2
=
f3
=
f4
=
space
-
p(1,2,2,3,4):
P2, (/'121 + /122 + tZ2)2' 3 1141 + P42 + /.143 P2P2 P2P2 /12t,2 1 2 2 3 3 11 _,2 (P22 P2)(/,2 /,2)(/ 112), 2 1 3 3 1 (/t21 + t,22 + /'Z2)(P2 2-2 P2) 1 + 142 2 + P2 3 3 (/12 3 _
_
_
_
_
-
2/.z2)(/.A2 1 1 3 + P2
-2 A2). 2
2 2 2 symmetric group S3 Oil C [1\1 A2 1\31 induces on C [/.tl 'P2 /.13] an and action which is determined by (1, 2) (j12' 2, /-, 32) (-M2, (1, 2, 3) -(/42, 1 /,42, 1 -/.12) 3 2 p2) 2 -p2, 1 it 2 3 is either invariant or antiTherefore, every symmetric fimetion in C[p2' (112, 2 p2] 1 p2' 3 1 2 /,12, 3 p2) invariant with respect to this induced action. It follows that the above polynomials fo, f4 axe symmetric in Al /\2 A3. They axe easily expressed in terms of f (fl f2 f3 f4) giving an explicit formula for the map M -+ M3 C p(1,2,2,3,4).
The standaxd action of the
i
.
i
=
.
.
=
7
i
i
229
i
i
7
.
.
,
V11. Two-dimensional ax.i. systems
Chapter
6. The H,6non-Heiles
hierarchy
6.1. The cubic H6non-Heiles On C'
by jqi, q2J
we :--
potential
take qj, q2, pi and P2
JPI) P2}
=:
0; jqi, pj I
coordinates and the standard Poisson bracket
as
1
K3
2
L3
They
are
_
(P21 + P2) 2 q2
42 j
qI
P27
-8qlq2, 2 I
-
3
8q2
+
24 q22'
=
4q2q2' 1 2
2
+ q, (q, +
following
Pi,
-4q
2
+
P2I + qlpIP2
in involution and determine the
41
given by A3 the algebra C[K3, L,3] where
We denote
Jij.
=
(commuting)
two
-2q2pi
q'2
=
qipj,
P11
=
-PIP2
4q22).
+ q021
2
P12 ==P21
vector fields:
-
4qj (q1
+
2q22)7
2
8qlq2.
-
Spec A3 its corresponding level set has dimension two and is (by Proposition 11.3.7). It follows that A3 is integrable. We now define a regular map 03 : C4 _+ C6, where C6 is the phase space of the genus 2 odd Mumford system, which we view again as the affine space of Lax operators (2.1). The map 03 is given by
Clearly, for
any closed
point
in
irreducible hence A3 is complete
1
(qj, q2
It is verified
i
U(A)
=
(A)
=
w(A)
=
V
P1 i P2)
'\2
2q2A
vf2A'
2
q,
-
(P2 A + q1PI)
+
2q2 A2
+
(q2
+4q 2),X
p2 --i _
2
by direct substitution that
0*3 (_U21
+ U0 +
0*3 (_UIUO
WI)
==
0,
+ U, W, + Wo +
0*(ulwo 3
+ uow, +
0*3 (UOWO
+
V2) 0
=
2vivo)
V2) 1 =
=
_K3,
-L31
0.
H3 of the integrable algebra A for the genus 2 odd -L3- It follows that -K3, OW, 0, O*H2 3 3 03*A A3 and the Poisson structure on C6 which makes 0,3 into a morphism. (of integrable Hamiltonian systems) is the one given by (2.4), denoted here by 1'7 *12-
In terms of the system of generators Mumford system, this means 0*H0 3
Ho, =
.
.
.
,
O*H3 3
=
=
=
=
This morphism is neither injective nor surjective. It is however finite and its image is 0. Thus we know from Corollary 11.2.16 that 0 is the composition H3 given by Ho of a surjective and an injective morphism. Indeed, since the Poisson structure 1, *12 has Ho and H3 as Casimirs, it restricts to a Poisson structure on the level over any point of 0. Since .97 is irreducible H3 CaS(I','12), in particular to the level Y defined by Ho =
=
1
=
230
=
6. The Mnon-Heiles
it is
affine Poisson
an
Hamiltonian system
variety and since
(here 1* 12
it is four-dimensional
(37,
and A denote their restrictions to
A) F).
is
Thus
an
integrable 03
we can see
surjective morphism,
as a
03 to
hierarchy
(C4, 1,
:
A3)
(-T-77 J* *}27 A)
-+
i
trivial
a
We
subsystem of the genus 2 odd Mumford system. decompose further this surjective morphism. 03- Consider the
action
T
Of Z2
on
C4
given by T
(ql, q2) P1 P2)
(-ql
I
7
q2 j -PI 7 P2)
an automorphism of the 116non-Heiles system since it is a Poisson map and 7-*A3 A3; in fact even more is true: r*f f for all f E A3. By Proposition 11.3.25 we have a quotient
It is
=
=
integrable Hamiltonian system which we denote as (C4/T, f_, '10 A3) (Since all elements of A3 go to the quotient). Notice that in this case the level sets are quotients of the original level sets. Moreover the action leaves 03 invariant and 03 can be factorized via the quotient C4/7-, i.e., we have a morphism (between affine varieties) 3 : C41-r -+.F which makes the following diagram commutative. c4 ,
ir
I \\
C4/,r
jr
0 -
03
Since
03 and
7r axe
Poisson the
same
3
is true for
3 : (C4/Tj 1, 1, 10 A3) 5
is
a
surjective morphism
of
*3A
and
-+
(-177 1
A3 by surjectivity of 7r. Thus
=
'12 A)
*
1
1
Hamiltonian systems.
integrable
11.2.31 that Q, p2 p2' Q) P1 qlpl q,2, Q2 P2, R 1 generate the algebra of invariant functions on C4 for the action of r; they generate the algebra of regular fimetions on the quotient, which therefore is a cone with equation R2 QJPJ. The map 3 is then given by It follows from
Example
=
U(,X)
=
(A)
=
W(A)
=
v
(Q1 Q2, P1 P2 R)
-
-vF2
i
,
A2
A3
=
2Q2/\
-
=
=
Q1;
(P2 A+ R),
+
2Q2 A2
+
(Q1
+4 Q2)'X 2
P1 _
2
important to note here that this map is biregular. The first conclusion is that the 116non-Heiles potential admits an automorphism of order two whose quotient is isomorphic to a trivial subsystem of the genus 2 even Mumford system. It is easy to see that the quotient and it is
map is
even
unramified.
3
is
an
general fiber
is
an
Since
a.c.i.
we use
seen that (C4/,r, A3) is an a-c.i. system (whose Jacobian). To conclude that the H6non-Heiles system is the explicit expressions (VI.4.9) of the variables ui in terms of theta functions. It
isomorphism we have affine part of
a
follows from that formula that uo is the square of a quotient of two theta functions, hence qj, which is its square root, is a quotient of two theta functions. It follows that the generators of the ideal which define the level sets
define
an
affine part of
an
Abelian
axe expressible variety.
231
in terms of theta
functions,
hence
they
Chapter VIL Two-dimensional a.c.i. systems
quartic Mnon-Heiles potential
6.2. The We
A4
=
turn to the H6non-Heiles
now
structure
the cubic H6non-Helles
as
C[K4, L4]
potential V4. It is defined on C4 with the same Poisson potential, the integrable algebra now being given by
where 1
K4 L4
=
2
(p21 + P2) 2
+
q41
+
2
=
2+ 16q42, 12q2q 1 2 2
q2P1 + qIPIP2 + q,
3
(8q2
+
2
4q, q2
A4 is also contained in the algebra of invariant functions (for T) and leads also to a quotient system (C4/,r, I., .1o, A4). The analog of the map 03 is in this case the map 04 : C4 _+ C7 defined by
(ql, q2, PI P2)
U(A)
=
V(A)
=
(A)
=
A2
N12-
i
2
2q2A
(P2A
-
+
qII
qlpl), 2
W
C7
'X4
+
2q2 A3
+
2
(q1
+
4q22) A2
+
2
4q2 (q 1
+
2q22) -Pi 2
interpreted as the phase space of the genus 2 even Mumford system. The map 04 morphism when the Poisson structure (2.8) is chosen for it. As in the previous paragraph it follows that the quotient system is isomorphic to a trivial subsystem of the 0. In H3 Ho genus 2 even Mumford system, the level now being defined by H4 particular the quotient system is an a.c.i. system (with an affine part of a two-dimensional Jacobian as the general fiber of its momentum map).
Here is
a
is
Poisson
=
case.
=
=
Our argument which showed that the H6non-Heiles potential is a.c.i. is not valid in this To see this let us fix one Jacobian from the trivial subsystem of the genus 2 even
Mumford system (or from the isomorphic quotient system) and recall from Section VIA that the divisor to be adjoined to the general fiber of the momentum map consists of two translates of the theta divisor and that uo has on each of these a simple pole. If the quotient map,
general level set, extends to an unramified map 7r over the whole Jacobian simple pole on the inverse images of the two translates of the theta divisor; since 7r is unramified these inverse images are reduced. But then 7r*uo cannot be the square of a meromorphic function on the inverse image of the level set (which is a level set of the quartic potential). It follows that -7r must be ramified and since it is unramified on the affine part it must be ramified at the divisor at infinity. Thus the completed general fiber of the momentum map of the quaxtic potential is a ramified cover of a two-dimensional Jacobians. restricted to
a
then ir*uo has
A first
a
example of
Bechlivanidis and appeax
as
ramified
computed by
van
an
covers
L. Piovan
integrable
Moerbeke in
Hamiltonian system with this property was given by The aJgebraic invariants of the surfaces which
[BM].
of Jacobians in the context of
(see [Pio2]).
232
integrable
Hamiltonian systems
were
6. The Mnon-Heiles
6.3. The H6non-Heiles
hierarchy
hierarchy
higher potentials the situation is even worse. We can still construct a quotient these quotient systems are not a.c.i. They are however still trivial subsystems of the (two-dimensional) hyperelliptic systems which we considered in Paragraph 111.2.4; these hyperelliptic. systems correspond to higher genus curves and are not a.c.i. (recall that only if 2 and g > 3). d they are a.c.i.; here d g For the
system, but
even
=
=
still considering the standard Poisson structure on C4 and consider now the A,, algebra integrable [Kn, Ln] which generalizes the algebras A3 and A4. The polynomials Kn and Ln are given by We
axe
=
where Vn is defined
Kn
=
Ln
=
(P21 +P2) 2
2
+ Vn,
2
-q2p, + qipIP2 +qI2Vn-17
by [n/21 2
Vn
n
n-2k
-
k
k
) 2kqn-2k. q,
9, 2
k=O
and satisfies the recursion relation
Vi+2 We still have
7-
defining
2q2Vi+l
=
automorphism of
an
+
q211
order two and
system. The analog of the morphisms 03 and 04 is the
(qj, q2
i
U(,\)
=
V(A)
=
w(A)
=
A2
_
V2-
P1 i P2)
we can
consider the
quotient
On defined by
map
2q2A
2
qj I
-
(P2A + qlPI)7
n-1
1: VjAn-i
2 1 -
2
i=O
Using the recursion relation it is
compute that
easy to
U(A)W(A)
V2 (A)
+
=
An+2
_
A2Kn
-
ALn.
can be seen as in the previous cases as a morphism. to a trivial subsystem of one of hyperelliptic systems discussed in Paragraph 111.2.4 (namely the one associated to the y2 Xn+2 + aX2 with parameter a). These are also multifamily of polynomials F(x, y)
This the
=
-
Hamiltonian and the Poisson structure which has to be taken
morphism)
is
again the
w(A) in terms of the luo, vo I Jul, vi I
determines the brackets of the coefficients of of
and
u(A)
to W
=
which
are
more
order to have
Chapter III)
surfaces
are
a
Poisson
Casimirs; this
brackets of the coefficients =
-uo
(which corresponds
-
in the level sets of the momentum map of the potential unramified 2 : I covers of the level set FF,2 (defined
are
Xn+2 A2Kn ALn- We determine these level sets for F of the of any dimension lower than the genus of the curve rp and f (x) y assumed non-singular) in the following proposition. =
general is
except
zero
merely interested here
>
(which
0
are
4). Clearly these (111.3.1)) where F y2
Vn (n in
(A),
in the notation of
x
We
v
(in
2 for which all but the coefficients of A and A
one
form F
-
_
-
2
=
-
233
Chapter V11. Two-dimensional a.c.i. systems
yl f (x), the level set Fpd is for Proposition 6.1 In the hyperelliptic case F(x, y) d < g biholomorphic to a (smooth) affine part of a distinguished d-dimen8ional subvariety Wd of Jac(Pp), namely =
where 6 E
Jac(P.F)
TF, d
Wd
Wd -I
TF,d
Wd
(Wd-1
is
given by
6
deg f (x) odd, U
Ab(ool
=
-
(e + -
Wd- 1))
deg f (x)
even,
10 mod Ap,. Also
002)
Jac(fp),
W,
=
W,-i
=
theta divisor E) C
Wi
=
curve
Wo
=
origin of
PF
Jac(f.F),
embedded in
Jac(PF),
Jac(rF).
Proof We prove the proposition only for the case in which deg f (x) is odd. In this case IFF compactified by adding one point which we call oo. We choose this point as the base point for the Abel-Jacobi map (on the symmetric product) and define Wk for k 1,...,g as Wk Abk (Sym" VF). By Jacobi's Theorem Wg Jac(:PF) and by Riemann's Theorem, W,_1 is (a translate of) the Pdemann theta divisor. Clearly for each k < g, Wk-1 is a divisor in Wk and, Wk \ Wk-1 is smooth. We claim that is
=
=
=
d
Abd(SYM IPF \ DFd) more
precisely Abd realizes Pd) C
(P
a
=
Wd \ Wd-1,
holomorphic bijection between
these smooth varieties.
SyMd ]pp \ DFd Pi
Vi Pi
:,4-
oo
and 3i
: jx (Pi)
=
x
Abd(Pli... Pd) 0 Wd 7
we
used Abel's Theorem in the last step. It follows that
biholomorphic, hence by Proposition 111.3.3, Abd OOFd manifolds -7Fd and Wj \ Wd- I are biholomorphic. to the
case
of the H6non-Heiles
hierarchy
the
fiber of the momentum map of the n-th potential of this cover of an affine part of the W2 stratum of a hyperelliptic
part is completed into the W2
Pj
and Pi is not
ramification
point of
x
Wd-li
are
Applied
=
(Pj) a
where
Namely,
SYMdr. \ DPd and Wd \ Wd-1 is a biholomorphism and the 0
proposition says that the general hierarchy is an unramified 2 : 1 curve
of genus
["+']; 2
this affine
by adding one copy of the curve if n is odd, otherwise two copies need to be added. As before, the morphism. on to the Mumford systems lead to a Lax representation (with spectral parameter) for the H6non-Heiles hierarchy. Notice that although we have a Lax equation with spectral parameter the system is not a.c.i.: its general level surface corresponds to a non-linear subvariety of a higher-dimensional hyperelliptic stratum
Jacobian.
234
7. The Tbda lattice
7. The Toda lattice In this paragraph we look at the st(3) periodic Toda lattice and some of its variants. For generalization to other Lie algebras, Lax equations and a physical interpretation see [OP2] and [AM8]. For the non-periodic case, see [FH]. a
7.1. Different forms of the Toda lattice Consider the
Poisson matrix
following 0
T)
-tT
we
denote the
where
are
0
-t1L
t2
0
tI -t2
-t3
t3
0
T
0
C6 by 1-, -1. One easily find that t1t2t3 algebra of Casimirs is generated by these polynomial algebra.
corresponding Poisson
and t4 + t5 + t6 two elements, in
C:
on
structure
on
Casimirs. We show that the
particular
it is
a
Cas(C6,1.,.I)=C[tlt2t3,t4+t5+t6l-
Lemma7.1
Proof denote a t1t2t3 and b t4 + t5 + t6. Then F E Cas(M, 1., -1) can be written in tj I... I t5 and the Casimir b by replacing t6 with b t4 t5- We call the resulting polynomial Fl. Since F is a Casimir the same holds true for F, and we find ftom It,, F, I It2: F1 I 0 that Let
us
=
=
terms of
-
-
=
oViL
OF, =
=
ji5
54
0'
depends on t1, t2 and t3 only. If F, is symmetric in these variables it is a polynomial t1t2 +t2t3 +t3t1 and V3 tl +t2 +t3 and V2 t1t2t3, so FI(tl,t2, W F2(vl, V2, V3)0 that Since F2 and v3 are Casimirs we find from 14, F21
hence F, in VI
=
=
Since both derivatives
actually zero, symmetrizes it, are
are
=
as
above
one
t1t2t3 only and Recall that
ple
9F2 (9V2
polynomials in vi only it follows from this that these derivatives a polynomial in t1t2t3 only. If F, is not symmetric then one
hence F, is
F2(Vli V27 V3) and
OF2
Yv l
=
finds that F2
we are
we
Fl(tli t21 t3)
depends
On
+
Fl(t21 t31 t1)
+
Fl(tO17 Wi
only. This implies that F, is
V3
done.
a
polynomial
in M
the invariant
computed
polynomial
of this Poisson structure in Exam-
11.2.56. We define
T1
=
t1t2t3i
T2
=
t4 + t5 + t61
T3
=
T4
=
1 2
(t24 + t25
t4t5t6
-
+
t2) 6 + t,
t1t4
-
235
t2t5
+ t2 + t31 -
W6
Chapter V11. Two-dimensional a.c.i. systems
and A
=
C [TI,
T2, T3, T4]. Since T1 and T2 completeness
Paragraph Let
us
Casimirs and
axe
is involutive. We do not show
JT3, T41
here because it will follow
0 the algebra A automatically from =
7.2.
look at the level sets of the Casimirs
closed points.
over
t1t2t3
=
t4 + t5 + t6
=
They
are
given by
a,
b,
where a, b E C are arbitrary; we denote this level set by ab If a :A 0 then four-dimensional and the Poisson structure has rank four (even at every -
1,r_7 ab
is
irreducible,
point).
If a 0 then Yab has three irreducible components, each of which is a four-dimensional plane and the Poisson structure has rank four. It follows as in Proposition 11.2.42 that if we restrict =
the Poisson structure to any of these levels then its algebra of Casimirs is still maximal. Similarly the integrable algebra leads to an integrable Hamiltonian system on (the irreducible
components) to
a
=
of these level sets, as in 0, but often it is
I and b
=
Proposition 11.3.19. The original Toda lattice corrsponds just as easy to work on the larger space C6, or on the
0. One may however also want to consider W C C' defined by t4 + t,9 + t6 levels of the Casimirs at once, except the reducible ones, i.e., the ones for which 0. This is easily done by using Proposition 11.3.29. Then the phase space is taken as the
hyperplane all possible a
=
affine
=
variety defined by t0t1t2t3 algebra of Casimirs is
for to. The
come
a
Casimir the
same
holds true
given by
now
CaS(C7) and the
C7. Since t1t2t3 is
I in
=
C[TI0, T1, T2]/idl(TOTI
=
-
1)
is the tensor
corresponding integrable algebra
product Paragraph 7.2.
back to this form of the Toda lattice in
of it with C [T3, T4]. We will
As is even apparent from the physical origin of the problem, the Toda lattice has automorphism of order three, which is given by the map -T : C6 _+ C6, 7' :
(t1i t21 t31 t41 t5i t6)
an
(t21 t31 t1i t57 t67 t4)-
=
preparation of the computation of the quotient by this automorphism (in Paragraph 7.3) a simple linear transformation giving coordinates which are diagonal for the action of r. Let c denote a fixed cubic root of unity and define As
a
we
do
X1
Then
-T
is in
=
X2
=
X3
=
tl + t2 + t37
Y1
=
t4 + t5 + t67
t31
Y2
=
t4 + d5 +
t2 + 'Et37
Y3
=
t4 +
t1 + d2 + 2
tl +
diagonal form and
6
is
2 'E
integrable algebra A
is
now
62t,5 + ft6-
given by
'r(XI, X2, X31 Y11 Y2, Y3) The
1E2t6l
given by A
X3I
(XI, 62X20EX3 Yl,62Y21fY3)-
=
=
C [XI I
3
X,
=
X2
=
X3
=
3xi
X4
=
Y23
3
+ X2 + X3
-
X2, X3 X4] where 7
3xlX2X3,
Yli + YMi
+
Y33
+
9(X2Y3
236
+
X3Y2)-
T. The Tbda lattice
The Poisson structure is
(up
to
vr--3) given in these
factor
a
0
X
-tx
0
)
and the
algebra of Casimirs reduces
7.2. A
morphism
where
to
-X2
(0
X=
coordinates
by
X3)
0
-X3
X1
0
-XI
X2
C[XI, X2].
to the genus 2
even
Mumford system
morphism from the sl(3) periodic Toda lattice to the Bechlivanidiswe have shown to be isomorphic to the genus 2 even Mumford system (on C7). For a generalization, giving a morphism. from the sr(g + 1) periodic Toda lattice to the genus g even Mumford system, see [FV]. For explicitness we recall that we 0 consider the Toda lattice as being defined on the hyperplane W defined by 4 + t5 + t6 in C6. If we do not suppose this then we do not have a morphism to the Bechlivanidis-van Moerbeke system, but still to the genus 2 even Mumford system, which we now have to take as being defined on C', namely we cannot assume anymore that the coefficient of X5 in the We
van
now
describe
a
Moerbeke system which
=
f (x) vanishes. Since the formulas are simpler in this case and since there equation y2 no phenomenologic difference we will not consider this more general case. Consider the map 0: -H C C6 -+ C7 defined by =
81
=
W6
82
=
-4/2,
83
=
is
t1i
-
-t2t3/16,
+
S4,5
=
(t2
86,7
=
(tA
W/8, t2t5)/8.
Then
0*(s,
-
4s 22
(3182
+
(,q24
'925
_
8S4)
436) +
+ 84S6 +
0*(827
826
_
-
(7-1)
07
=
0*(S2-53
-T3i
-T4/2,
=
SO
=
307)
3183)
=
=
01
-T1116.
A where X denotes the Bechlivanidis-van Moerbeke Thus 0 is a regular map and O*A! system. Moreover it is easy to check that the Poisson structure which has to be taken on the latter system in order for 0 to be a Poisson map is the one given by (2.14) and we have obtained a morphism of integrable Hamiltonian systems. =
As in the H6non-Heiles
example this morphism
is
again neither injective
nor
surjective.
first attempt to cure the non-surjectivity we restrict the Bechlivanidis-van Moerbeke system to the level set Y (of the Casimirs) given by
As
a
824
_
82+ 53 5
=
07
3233 + -9486 + 85-97
=
01
which is irreducible and of codimension two, hence it is it contains the
bracket
on
image of 0. Let
us
an
affine Poisson
compute the invariant polynomial
F. The matrix 2.14 has rank 4 for X3 237
p(Y)
variety; clearly of the Poisson
:A 0, hence by irreducibility of
F and since
Chapter VII. Two-dimensional a.c.i. systems
dim.F
=
5, the leading
points where
83
term of
0 and
=
-4(s586
(at
+
is less than up in the
2, i.e., all
cases
S5
S6
0,
85
=
time)
(2.14)
2s7
-286
2sis5 + 482S7
-S5
-S4
S7
4S2S6
that
54,
(
-2s6
-
(7.2)
_
S25
+ 83
=
0 it
Splits
left with
287
-2s6
T84
-S4
:F2(2S236
4S2S7
+
2s,S4
432S6
S7
SIS4)
-
T86
-84
2sIS4
86
2 2 determinants must vanish. Since on.F S4 ,
we are
go down at those
only
caai
the rank of the matrix
)
-
2s,S4
S6
0 then the rank of this matrix is smaller than 2 if and
=
means
x
0
det
This
2
=LS4 and
4S4(87 =L S6)
( If S7
=
same
507)
0
is RIS5. The rank of
p(.F)
the
2(S26
S4(2S2S6
-
only
-
if
SIS4))
=
0-
have found two irreducible components which are defined (in C7) by 53 2 -54(2S286 SIS4); each of these components has dimension 3. 0 then the rank of (7.2) can only be smaller than 2 if S4 = S5 0, a 86 = s7 s7
we
=
zFs6 and 36
_
-
If S7 :L S6 :A locus which is contained in both irreducible components, hence this one does not contribute to p(,F). Finally the rank is zero if and only if S3 = S4 S5 = S6 S7 0, a two-dimensional =
=
plane inside.F. The upshot
is that
p(.F)
p(.F)
=
=
=
=
given by
is
R 2S5 +
2RS3
+
S2.
This should be
compared with the invariant polynomial for the Poisson structure of the Toda W, which we denote by p(W). The invariant polynomial for the Poisson structure of the Toda lattice on C6 was found in Example 11.2-56 and computed as R2S6 + 3RS4 + S3; it follows at once (devide by S) that its restriction to the hyperplane W (given by t4 + t5 +t6 0) equals R2S5 + 3RS3 + S2. lattice
on
=
The small difference between
p(Y) and p(W) suffices to conclude that 0 is not a biregular (although it is regular and dominant). The polynomials actually indicate that something goes wrong at the rank 2 level. Let us denote the subsets of.F (resp. W) were the rank is at most 2 by.Fj (resp. WI). Thus 0 restrictS33 to a regular map 01 : W1 -+.Tj and W., has three irreducible components whileY, has two. Thus either (at least) two irreducible components are identified or (at least) one of them is mapped completely inside 770, the locus inside.F of
map
points where the rank is
hence the latter 33
zero.
case occurs
Recall that the rank at
34
On the component t2 81
=
W6
S2
=
-4/2,
33
=
and
an
we
-94
-
=
=
0 Of W1 the map
0
is
given by
t1i
85
=
86
=
conclude that
image point
t3
=
is
never
57
0
=
07
is not
larger
a
finite map.
than the rank at the
point,
hence
restricts indeed. 34
The other two components of W,
are
mapped neatly 238
to the two
components of F1.
7. The Tbda lattice
Our way to deal with this is to cut away the bad piece: from the Toda side we remove the divisor of T, tIt2t3 and from the Bechlivanidis-van Moerbeke side we remove
(reducible) the
zero
:
locus of S5
both T, and
=
M2
axe
(MI, 1-, .11, AI)
systems M,
S,5
2_ 87 862_813,3: it follows from (7. 1) that these correspond under 0. Since Casimirs we obtain from Proposition 11.3.29 two integrable Hamiltonian
=
=
J(toltll
...
J(50i 811
(M2 J* C12 A2) i
W I t0t1t2t3
...
7
=
Ii t4
2_
2_
and Ai
I-, .1i
I
and
787) 1 50(87
86
+
where MI and M2
t5 + t6
SISO
=
=
11 824
are
given by
017 _
'925 +S3
=
8253 + 8486 + 3587
=
017
obtained
accordingly. The morphism MI -- M2 which corresponds to morphism of integrable Hamiltonian systems) will be denoted by the same letter 0. Now both invariant polynomials are the same (being given by R 2S5), however 0 : MI -+ M2 is not surJective, since it is obviously missing the points of M2 where s3 vanishes. Since 83 does not belong to A2 and (M2 I' *12) A2) satisfies the conditions of Proposition 11.3.7, removing the zero locus of s3 leads toanother integrable Hamiltonian system(M3 1" 1'13 A3) where M3 is given by and
0 (and
are
which is also
a
i
1
,
I(SO
M3 and
f
13
7
1
50 1 SI 7
...
ISO I SO(S 2_s2_ SIS3) 7 6
and A3 derive at
Finally
we
have
a
from
biregular map!
f' 12
It is
SI0 S3
7--
and A2
17 s24
_
S25 +S3
82S3+S4S6+S5S7
-16to,
to
=
-so116,
s'0
=
-16toti,
t,
=
s'0 (s 2_ 7
81
=
t5t6
t2
=
82
=
t3
=
t4
=
-2S2)
t5
=
-80(S4
-
35)(86
-
S7)7
t6
=
-SO(S4
+
35)(86
+
87)-
t1i
S4,5
-4/2, -t2t3/16, (t2 t3)/8,
85,6
(t3t6
once
0},
explicitly given by
=
-
=
-
so
33
Now it follows at
once
--:::
t2t5)/8
that the Toda lattice
82) 6
4(84
+
807
4(84
-
85)7
(on W)
-81,
is a.c.i.:
since the Bechlivianidis-van
Moerbeke system is a.c.i. the same is true for the system we constructed on M3 since just removed a divisor (with two components), however the level sets have changed since
we we
belong to the integrable algebra. Thus our system on MI is a.c.i. and since it contains the general level set of the original Toda lattice (on W) as its general level set, the Toda lattice is a.c.i. and its general level set is an affine part of a Jacobian; using the order three automorphism one easily recovers the well-known fact that this affine part is obtained from the Jacobian by removing three translates of a genus two curve, earch pair of which is tangent at their intersection point. Each of these curves induces a principal polarisation on its Jacobian, hence the Toda lattice is an a.c.i. system of polarization type (3,3). removed the divisor of
a
function which does not
239
Chapter
VIL Two-dimensional a.c.i. systems
7.3. Toda and Abelian surfaces of
type
(1,3)
Having shown that the Toda lattice is a.c.i. of polarization type (3,3) we are now ready a new a.c.i. system of polarization type (1, 3). Using Proposition V.2.5 it is obtained as follows: the order three automorphism r fixes all fibers of the momentum map f for all f E A). Therefore the general level set of the quotient a.c.i. system (since r*f is isogeneous to a Jacobian and the isogeny is a 3 : 1 (unramified) map, showing that this level set carries a polarization of type (1,3). It is also easy to determine the divisor which is to be adjoined to the general fiber (of the momentum map) in the quotient system: it is the quotient of the Toda divisor by T which acts as a translation permuting the three components of this divisor (and the three intersection points). Hence the quotient is an irreducible divisor, birationally equivalent to the genus two curve and having one singular point which is a tacnode. It follows from IV.3.4 that its virtual genus equals 4 which is by Theorem IV.3.7 consistent with the fact that it induces a polarization of type (1,3) on the to construct
=
surface. an explicit realization of the quotient system, then the main object C6 1-r (or Wft): the Poisson structure and the involutive algebra are obtained at once. Thus we need to construct a system of generators of for O(C61-r) (or O(Rl-r)) as well as a generating set of relations between these generators. To do this, we use the coordinates xi, yi, constructed in Paragraph 7.1, which are diagonal with respect to the action of -r. Obviously the following elements are invariant with respect to r.
If
to be
one
wants to have
computed
is
X1
X1,
X2
Y1
X2X3i
=
Y1,
Y2
X4
=
3, y4 X2
X6
=
X22Y2i Y6
-=
M3i
Y23,
=
=
X5
X3 =
=
X3, 3
X021
X2Y22, X7
=
y5
=
Y3
=
X2Y37
=
X3Y3
Y33,
X23Y3 Y7
2
,
We claim that these fourteen elements generate O(C61-r). To show this, let F be an invariant polynomial. Writing it as a polynomial in x, and y, it suffices to show that an invariant poly1101nial in X27 X31 y2 and Y3 can be written as a polynomial in the above candidate generators. Because of the elements X4, Y4, X5 and Y,5 it suffices to check this for a polynomial of degree less than three in X2, X31 Y2 and Y3- Since the action is diagonal it suffices to check it for
monomials; there
are
27 of
these, namely the
XilXi2Yi3 2 Yi4 3 2 3 It is easy to check that
with
monomials
ii + N2 + i3 + 24
they all depend
on
the fourteen
=
ones
0
(mod 3).
above.
Finding all relations requires more work: there are quite a lot of them. We give half of the explicit list, the other half is found by interchanging X and Y in our list; our list is ordered by the degree of the monomials (in the xi, yi) they come from. In order four there is just
one,
X2Y2 in order five there
are
=
X3y3i
(two times) six, =
X2X6i
X2Y4
=
X3y6i
X2X7
=
x5y3i
X6Y2
=
Y3Y61
X2Y5
=
Y3Y77
X7Y2
=
X3Y71
X3X4
240
7. The Toda lattice
finally,
in order six there
are
(ten plus) twelve, 3
X32
X4X51
X20Y3
X33
X5Y41
X2Y
X4Y4:-` X6Y6; X5Y5
2
X4X7i --X4y7i
X A2X3
X6 X6
X7y7i
X2X3Y3
X6X71
X6'
Y4 Y6)
X2Y2Y3
X6Y7)
X27
X5y7,
2
X2 X2=yy 5 '63
More important than this list is how to prove that it is complete; we do this as before by reducing it to a finite list. Every relation must come from all identity in the variables xi, yi, since x, and yj are invariant themselves we may forget about these as before. Second, since the variables xi and yj are diagonal for the action these identities come from identities between monomials, i.e., from identities which are written in terms of the xi, yj as il
ii
ki
11
X2 X3 Y2 Y3
.
j2 k2 12 Xi2 2 X3 Y2 Y3
:==
i3
j3
k3
4
13
X2 X3 Y2 Y3
*
j4
k4
14
X2 X3 Y2 Y3
(7.3)
with i, + 2j, + k, + 21, 0 All powers in this equation (mod 3) by invariance (s 1, , 4). may be supposed smaller than three; to check this for the powers Of X2, suppose that X2 appears on the left (hence also on the right) with a power at least three, then this must come =
=
.
from
one
of the
following
terms
(or
Now check that the
.
multiple of it):
a
2 2 x3' X2'Y35 X2'y6i X2Y 2, X2y3y6i 2
reducing
.
by using the relations order, for example ',6 X22y
one can
=
2 X2 y2' Yj3, Y Y6 Y3 Yd2, y3. r 6 i
always
X2X3X6
factorize X4
(on
both
sides) thereby
2
==
X3X4-
Since now all exponents in (7.3) may be supposed smaller than three we have a finite list to check. In this way it is verified that all relations are a consequence of the relations which we have given and we have an explicit description of the ax.i. system.
241
Index Abel's Theorem, 119
Chern
class,
110
Abelian variety, 108
closed
point,
37
-
-
-
dual,
110
compatible
(principally) polarized, reducible,
108
-
109
-
Poisson
brackets,
integrable
25
Hamiltonian systems, 62
Abel-Jacobi map, 119
completable a.c.i. system, 131
adjunction formula,
complete algebra,
Ill
affine Lie-Poisson group, 30 a.c.i. -
complex torus, 108
algebraic completely integrable system,
-
completable,
-
irreducible,
-
completion,
system, 131 131
131
Darboux
132
(polarization) type of,
132
analytic Poisson manifold,
65
-
coordinates,
-
Theorem,
66
66
deformation property, 25
ample line bundle, 105
degrees of freedom, dimension, 37,
base space, 49
bi-Hamiltonian 63
-
hierarchy,
-
integrable system,
-
vector
62
field, 62
branch point, 106
-
canonical,
-
elementary,
-
degree of,
-
group, 99
-
pole,
-
zero, 99
canonical
coordinates,
66
Garnier 66
algebra,
divisor,
-
coordinates,
66
line
103
bundle,
140
variety,
potential,
-
fiber,
-
level set, 37
-
point, 37
37
Casimir -
decomposition,
-
function, 19,
-
level set
38
65
of, 38
253
Hamiltonian,
19
-
derivation,
19
-
vector
field, 19,
110
196
general
103
-
108
99
dual Abelian
-
-
103
99
double Lie
brackets,
49
49
divisor
Ber-hlivanidis-van Moerbeke system, 181
-
47
48
65
Index
H6non-Heiles -
-
K-3
233
hierarchy,
potential, 230, 232, 233
-
hyperelliptic -
case, 83
-
curve, 106
-
-
-
Serre
duality,
-
107
involution,
-
bi-Hamiltonian, 63
H6non-Heiles,
Hodge form,
vanishing Theorem,
233
generalized surface, intermediate
-
surface, 121
-
variety,
Krull
equations,
algebra, 49,
136
69
Hamiltonian system,
49, 69
multi-Hamiltonian system, 62
field, 49,
69
curve, 68
first,
-
-
representation,
-
type, 141
general,
37
Casimirs,
of the
integrable system, 50
variety
41
41
derivative, 20
-
affine group, 30
-
structure, 23
-
modified structure, 25
line bundle
involutive 47
-
ample, 105
-
canonical,
irreducible a.c.i. system, 132
-
holomorphic,
isogeny, 109
-
-
algebra,
-
Hamiltonian system, 47
very
lower
inversion
100
ample, 105
balances,
136
maximal
Theorem,
120
-
Mumford system, 143
-
surface, 121
-
103
114
Jacobi -
38
Lie-Poisson
polynomial,
Jacobian,
142
of the
Lie
1
matrix,
141
equation,
equation with spectral parameter,
-
invariant of affine Poisson -
solutions, 136
-
-
47
-
37
-
-
-
121
level set
integral closure,
121
Lax
bi-Hamiltonian system, 62
-
121
105
integrable
vector
113
surface,
dimension,
Laurent indicial
103
Kummer
hierarchy -
surface, 121
Kodaira
of
-
algebra
-
spectrum, 37
Casimirs,
momentum map, 50
variety, 114 254
39
141
Index
morphism -
-
-
of affine Poisson of
varieties,
26
Hamiltonian systems, 54, 70
integrable
of Poisson spaces, 68
multipliers, 109 multi-Hamiltonian -
integrable system,
-
vector
field,
62
62
Mumford. system -
-
163, 179
even,
odd, 161, 177
node,
122
Noether's
Painlev6
formula,
113
-
ideal, 29
-
isomorphism, 27,
-
manifold,
-
matrix,
-
morphism, 26, 68
-
product bracket,
-
reducible,
-
space, 65
-
standard structure, 23
structure, 19, 65
-
subalgebra,
-
trivial structure, 22
-
vector
analysis, 136
reducibility theorem,
-
space, 49
-
period
-
-
121
algebra of, monic,
periods, 115
matrix of B
periods,
i-th
115
period,
145
145
normalized, 145 order
of,
polarized
matrix of A
lattice,
145
Abelian vaxiety, 108
polarization
115
-
114
principal,
108
-
type, 108
-
type of a.c.i. system, 132
Picard group, 100
principal balance, 136
Poisson
principally polarized
-
action, 31
-
algebra,
-
-
109
pseudo-differential operator -
-
111
dual,
space, 38
-
field, 20
-
-
half,
31
-
map, 38
-
30
32
-
-
-
65
21
Poincax6
parameter
phase
68
19
Abelian variety, 108
quasi-automorphism,
affine
brackets,
affine
subvariety,
61
25
27
ramification point, 87
19
-
algebra,
-
analytic manifold,
-
bracket, 19,
-
canonical structure, 23
reciprocity laws, 115
-
cohomology,
real level sets, 85
-
constant
rank
65
-
65
-
20
structure, 23
of Poisson structure,
22,
rank
41
decomposition,
69
reducible Abelian variety, 109 255
Index
regular Poisson structure,
22
Toda, lattice
Pdemann -
-
-
-
-
-
-
conditions,
108
-
constant, 120 Hurwitz
generafized,
141
three
235
body,
trope, 122
formula,
106
Roch
Theorem, 103,
theta
divisor,
theta
function, 110
trivial 112
110
-
Poisson structure, 22
-
subsystem,
58
vector field
Sato
Grassmannian,
Schouten
bracket,
Schotky problem,
145
-
20
-
118
bi-Hamiltonian,
62
Hamiltonian, 19,
65
-
integrable, 49,
seven-dimensionaJ system, 181
-
KP,
spectral
-
linear,
-
multi-Hamiltoinian,
-
Poisson,
-
super-integrable,
-
parameter,
-
curve, 142
141
symplectic 115
-
basis,
-
manifold, 65
-
two-form,
-
decomposition,
-
foliation,
69
152 131 62
20
49
virtual genus, 112 Volterra group, 145
65 67
Weierstrass point, 107
67
super-integrable
weight homogeneous, vector
field,
191
49
Yang-Baxter equation type of a.c.i. system, 132
-
theta
-
-
curve, 122
-
function,
-
-
110
function with
divisor,
characteristics, 118
110
256
classical,
140
modified
classical,
140
