CONTEMPORARY TRENDS IN ALGEBRAIC GEOMETRY AND
ALGEBRAIC TOPOLOGY
NANKAI TRACTS IN MATHEMATICS Series Editors: Shiing-shen Chern, Yiming Long, and Weiping Zhang Nankai institute of Mathematics
Published Vol. 1
Scissors Congruences, Group Homology and Characteristic Classes by J. L Dupont
Vol. 2
The Index Theorem and the Heat Equation Method
byY.LYu Vol. 3
Least Action Principle of Crystal Formation of Dense Packing Type and Kepler's Conjecture by W. Y. Hsiang
Vol. 4
Lectures on Chern-Weil Theory and Witten Deformations by W. P. Zhang
Nankai Tracts in Mathematics - Vol. 5
CONTEMPORARY TRENDS IN ALGEBRAIC GEOMETRY AND
ALGEBRAIC TOPOLOGY
Editors
Shiing-Shen Chern Lei Fu Nankai Institute of Mathematics P. R. China
Richard Hain Duke University, USA
fe World Scientific III
New Jersey • London • Singapore • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
CONTEMPORARY TRENDS IN ALGEBRAIC GEOMETRY AND ALGEBRAIC TOPOLOGY Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-02-4954-3
Printed in Singapore by Mainland Press
PREFACE
Wei-Liang Chow and Kuo-Tsai Chen are two of the most distinguished Chinese mathematicians of the 20th century. Chen's collected papers have been published in 2001 by Birkhauser, while Chow's collected papers have been edited by V. Shokurov and myself and are awaiting publication by World Scientific. A memorial conference organized by the Nankai Institute of Mathematics, took place in Tianjin in Oct. 9-Oct. 13, 2000. This volume contains only a small sample of the papers presented at the conference. We are glad that it covers broad areas.
S. S. Chern
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CONTENTS
Preface
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Mathematics in the 20th Century Sir M. Atiyah
1
The $4 of Minimal Gorenstein 3-Folds of General Type M. Chen
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Morphisms of Curves and the Fundamental Group M. Cushman
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Iterated Integrals and Algebraic Cycles: Examples and Prospects R. Hain
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Chen's Iterated Integrals and Algebraic Cycles B. Harris
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On Algebraic Fiber Spaces Y. Kawamata
135
Local Holomorphic Isometric Embeddings Arising From Correspondences in The Rank-1 Case N. Mok
155
Multiple Polylogarithms: Analytic Continuation, Monodromy, and Variations of Mixed Hodge Structures J. Zhao
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Deformation Types of Real and Complex Manifolds F. Catanese
195
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Contents
Wei-Liang Chow, 1911-1995 S. S. Chern
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Comments on Chow's Work S. Lang
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The Life and Work of Kuo-Tsai Chen R. Hain and Ph. Tondeur
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MATHEMATICS I N T H E 20TH C E N T U R Y
Sir Michael Atiyah
Thank you for inviting me here to take part in this program. Of course, if you talk about the end of one century and the beginning of the next you have two choices, both of them equally difficult. One is to survey the mathematics over the past hundred years; the other is to predict the mathematics of the next hundred years. I have chosen the more difficult task. Everybody can predict and we will not be around to find out whether we were wrong. But giving an impression of the past is something that everybody can disagree with. All I can do is give you a personal view. It is impossible to cover everything, and in particular I will leave out significant parts of the story, partly because I am not an expert, partly because it is covered elsewhere. I will say nothing, for example, about the great events in the area between logic and computing associated with the names of people like Hilbert, Godel, and Turing. Nor will I say much about the applications of mathematics, except in fundamental physics, because they are so numerous and they need such special treatment. Each would require a lecture to itself. But perhaps you will hear more about those in some of the other lectures taking place during this meeting. Moreover there is no point in trying to give just a list of theorems or even a list of famous mathematicians over the last hundred years. That would be rather a dull exercise. So instead I am going to try and pick out some themes that I think run across the board in many ways and underline what has happened. Let me first make a general remark. Centuries are crude numbers. We do not really believe that, after a hundred years something suddenly stops and starts again. So when I describe the mathematics of the 20th century, 1
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I am going to be rather cavalier about dates. If something started in the 1890's and moved into the 1900's — I shall ignore such detail. I will behave like an astronomer and work in rather approximate numbers. In fact many things started in the 19th century and only came to fruition in the 20th century. One of the difficulties of this exercise is that it is very hard to put oneself back in the position of what it was like in 1900 to be a mathematician, because so much of the mathematics of the last century has been absorbed by our culture, by us. It is very hard to imagine a time when people did not think in those terms. In fact, if you make a really important discovery in mathematics you will then get omitted altogether! You simply get absorbed into the background. So going back, you have got to try to imagine what it was like in a different era when people did not think the same way. 1. Local to Global I am going to start off by listing some themes and talking around them. My first theme is broadly under what you might call the passage from the local to the global. In the classical period, people on the whole would have studied things on a small scale, in local co-ordinates and so on. In this century, the emphasis has shifted to try and understand the global, large-scale behaviour. And because global behaviour is more difficult to understand, much of it is done qualitatively, and topological ideas become very important. It was Poincare who both made the pioneering steps in topology and who forecast that topology would be an important ingredient in 20th century mathematics. Incidentally, Hilbert, who made his famous list of problems, did not. Topology hardly figured in his list of problems. But for Poincare it was quite clear it would be an important factor. Let me try to list a few of the areas and you can see what I have in mind. Consider, for example, complex analysis ("function theory", as it was called), which was at the centre of mathematics in the 19th century, the work of great figures like Weierstrass. For them, a function was a function of one complex variable and for Weierstrass a function was a power series. Something you could lay your hands on, write down, and describe explicitly; or a formula. Functions were formulas: they were explicit things. But then the work of Abel, Riemann, and subsequent people, moves us away, so that functions became denned not just by explicit formulae but more by their global properties: by where their singularities were, where their domains of
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definition were, where they took their values. These global properties were the distinguishing characteristic feature of the function. The local expansion was only one way of looking at it. A similar sort of story occurs with differential equations. Originally, to solve a differential equation people would have looked for an explicit local solution: something you could write down and lay your hands on. As things evolved, solutions became implicit. You could not necessarily describe them in nice formulae. The singularities of the solution were the things that really determined its global properties. Very much a similar spirit, but different in detail, to what happened in complex analysis. In differential geometry, the classical work of Gauss and others would have described small pieces of space, small bits of curvature and the local equations that describe local geometry. The shift from there to the large scale is a rather natural one, where you want to understand the global overall picture of curved surfaces and the topology that goes with them. When you move from the small to the large, the topological features become the ones that are most significant. Although it does not apparently fit into the same framework, number theory shared a similar development. Number theorists distinguish what they call the "local theory", where they talk about a single prime, one prime at a time, or a finite set of primes, and the "global theory", where you consider all primes simultaneously. This analogy between primes and points, between the local and global, has had an important effect in the development of number theory, and the ideas that have taken place in topology have had their impact on number theory. In physics, of course, classical physics is concerned with the local story, where you write down the differential equation that governs the smallscale behaviour; and then you have to study the large-scale behaviour of a physical system. All physics is concerned really with predicting what will happen when you go from a small scale, where you understand what is happening, to a large scale, and follow through the conclusions. 2. Increase in Dimensions My second theme is different. It is what I call the increase in dimensions. Again, starting off with the classical theory of complex variables: classical complex variable theory was primarily the theory of one complex variable studied in detail, with great refinement. The shift to two or more variables
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fundamentally took place in this century, and in that area new phenomena appear. Not everything is just the same as in one variable. There are quite new features, and the theory of n variables has become more and more dominant, one of the major success stories of this century. Again, differential geometers in the past would primarily have studied curves and surfaces. We now study the geometry of n-dimensional manifolds, and you have to think carefully to realise that this was a major shift. In the early days, curves and surfaces were things you could really see in space. Higher dimensions were slightly fictitious, things which you could imagine mathematically, but perhaps you did not take them seriously. The idea that you took these things seriously and studied them to an equal degree is really a product of the 20th century. Also it would not have been nearly so obvious to our 19th century predecessors to think of increasing the number of functions, to study not only one function but several functions, or vector-valued functions. So we have seen an increase both in the number of independent and dependent variables. Linear algebra was always concerned with more variables, but there the increase in dimension was going to be more drastic. It went from finite dimensions to infinite dimensions, from linear space to Hilbert space, with an infinite number of variables. There was of course analysis involved. After functions of many variables, you can have functions of functions, functionals. These are functions on the space of functions. They have all essentially infinitely many variables, and that is what we call the calculus of variations. A similar story was developing with general (non-linear) functions. An old subject, but really coming into prominence in the 20th century. So that is my second theme.
3. Commutative to Non-Commutative A third theme is the shift from commutative to non-commutative. This is perhaps one of the most characteristic features of mathematics, particularly algebra, in the 20th century. The non-commutative aspect of algebra has been extremely prominent, and of course its roots are in the 19th century. It has diverse roots. Hamilton's work on quaternions was probably the single biggest surprise and had a major impact, motivated in fact by ideas to do with physics. There was the work of Grassmann on exterior algebras — another algebraic system which has now been absorbed in our theory
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of differential forms. Of course the work of Cayley on matrices, based on linear algebra, and of Galois on group theory were other highlights. All these are different ways or strands that form the basis of the introduction of non-commutative multiplication into algebra, which is as I said the bread and butter of 20th century algebraic machinery. We do not think anything of it, but in the 19th century all these examples above were, in their different ways, tremendous breakthroughs. Of course the applications of these ideas came quite surprisingly in different directions. The applications of matrices and non-commutative multiplication in physics came with quantum theory. The Heisenberg commutation relations are a most important example of a significant application of non-commutative algebra in physics. That was subsequently extended by von Neumann into his theory of algebras of operators. Group theory has also been a dominant feature of the 20th century and I shall return to this later. 4. Linear t o Non-Linear My next theme is the passage from the linear to the non-linear. Large parts of classical mathematics are either fundamentally linear or, if not exactly linear, approximately linear, studied by some sort of perturbation expansion. The really non-linear phenomena are much harder and have only been tackled to a great extent, seriously, in this century. The story starts off with geometry: Euclidean geometry, geometry of the plane, of space, of straight lines, everything linear; and then through various stages of non-Euclidean geometry to Riemann's more general geometry, where things are fundamentally non-linear. In differential equations, the serious study of non-linear phenomena has thrown up a whole range of new phenomena that you do not see in the classical treatments. I might just pick out two here, solitons and chaos, two very different aspects of the theory of differential equations which have become extremely prominent and popular in this century. They represent alternative extremes. Solitons represent unexpectedly organised behaviour of non-linear differential equations, and chaos represents unexpectedly disorganised behaviour. Both of them are present in different regimes, are interesting and important but they are fundamentally non-linear phenomena. Again, you can trace back the early history of some of the work on solitons into the last part of the 19th century, but only very slightly.
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In physics, of course, Maxwell's equations (the fundamental equations of electromagnetism) are linear partial differential equations. Their counterparts, the famous Yang-Mills equations, are non-linear equations which are supposed to govern the forces involved in the structure of matter. The equations are non-linear, because the Yang-Mills equations are essentially matrix versions of Maxwell's equations, and the fact that matrices do not commute is what produces the non-linear term in the equations. So here we see an interesting link between non-linearity and non-commutativity. Non-comrnutativity does produce non-linearity of a particular kind, and this is particularly interesting and important. 5. Geometry versus Algebra So far I picked out a few general themes. I want now to talk about a dichotomy in mathematics, which has been with us all the time, oscillating backwards and forwards, and gives me a chance to make some philosophical speculations or remarks. I refer to the dichotomy between geometry and algebra. Geometry and algebra are the two formal pillars of mathematics, they both are very ancient. Geometry goes back to the Greeks and before; algebra goes back to the Arabs and the Indians, so they have both been fundamental to mathematics, but they have had an uneasy relationship. Let me start with the history of the subject. Euclidean geometry is the prime example of a mathematical theory and it was firmly geometrical, until the introduction by Descartes of algebraic co-ordinates, in what we now call the Cartesian plane. That was an attempt to reduce geometrical thinking to algebraic manipulation. This was of course a big breakthrough or a big attack on geometry from the side of the algebraists. If you compare in analysis the work of Newton and Leibniz, they belong to different traditions, Newton was fundamentally a geometer Leibniz was fundamentally an algebraist and there were good, profound reasons for that. For Newton, geometry, or the calculus as he developed it, was the mathematical attempt to describe the laws of nature. He was concerned with physics in a broad sense, and physics took place in the world of geometry. If you wanted to understand how things worked, you thought in terms of the physical world, you thought in terms of geometrical pictures. When he developed the calculus, he wanted to develop a form of it which would be as close as possible to the physical context behind it. He therefore used geometrical arguments, because that was keeping close to the meaning. Leibniz, on the other hand, had the aim,
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the ambitious aim, of formalising the whole of mathematics, turning it into a big algebraic machine, It was totally opposed to the Newtonian approach and there were very different notations. As we know, in the big controversy between Newton and Leibniz, Leibniz's notation won out. We have followed his way of writing partial derivatives. Newton's spirit is still there, but it got buried for a long time. By the end of the 19th century, a hundred years ago, the two major figures were Poincare and Hilbert. I have mentioned them already, and they are, very crudely speaking, disciples of Newton and Leibniz respectively. Poincare's thought was more in the spirit of geometry, topology, using those ideas as a fundamental insight. Hilbert was more a formalist, he wanted to axiomatize, formalise, and give rigorous, formal, presentations. They clearly belong to different traditions, though any great mathematician cannot easily be categorised. When preparing this talk, I thought I should put down some further names from our present generation who represent the continuation of these traditions. It is very difficult to talk about living people — who to put on the list? I then thought to myself: who would mind being put on either side of such a famous list? I have, therefore, chosen two names: Arnold as the inheritor of the Poincare-Newton tradition, and Bourbaki as, I think, the most famous disciple of David Hilbert. Arnold makes no bones about the fact that his view of mechanics, physics, is that it is fundamentally geometrical going back to Newton; everything in between, with the exception of a few people like Riemann, who was a bit of a digression, was a mistake. Bourbaki tried to carry on the formal program of Hilbert of axiomatizing and formalising mathematics to a remarkable extent with some success. Each point of view has its merits, but there is tension between them. Let me try to explain my own view of the difference between geometry and algebra. Geometry is, of course, about space, of that there is no question. If I look out at the audience in this room I can see a lot, in one single second or microsecond I can take in a vast amount of information and that is of course not an accident. Our brains have been constructed in such a way that they are extremely concerned with vision. Vision, I understand from friends who work in neurophysiology, uses up something like 80 or 90 percent of the cortex of the brain. There are about 17 different centres in the brain, each of which is specialised in a different part of the process of vision: some parts are concerned with vertical, some parts with horizontal,
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some parts with colour, perspective, finally some parts are concerned with meaning and interpretation. Understanding, and making sense of, the world that we see is a very important part of our evolution. Therefore spatial intuition or spatial perception is an enormously powerful tool and that is why geometry is actually such a powerful part of mathematics — not only for things that are obviously geometrical, but even for things that are not. We try to put them into geometrical form because that enables us to use our intuition. Our intuition is our most powerful tool. That is quite clear if you try to explain a piece of mathematics to a student or a colleague. You have a long difficult argument and finally the student understands. What does the student say? The student says, "I see!" Seeing is synonymous with understanding, and we use the word "perception" to mean both things as well. At least this is true of the English language. It would be interesting to compare this with other languages. I think it is very fundamental that the human mind has evolved with this enormous capacity to absorb a vast amount of information, by instantaneous visual action, and mathematics takes that and perfects it. Algebra, on the other hand (and you may not have thought about it like this), is concerned essentially with time. Whatever kind of algebra you are doing, a sequence of operations is performed one after the other and "one after the other" means you have got to have time. In a static universe you cannot imagine algebra, but geometry is essentially static. I can just sit here and see, and nothing may change, but I can still see. Algebra, however, is concerned with time, because you have operations which are performed sequentially and, when I say "algebra", I do not just mean modern algebra. Any algorithm, any process for calculation, is a sequence of steps performed one after the other, the modern computer makes that quite clear. The modern computer takes its information in a stream of zeros and ones, and it gives the answer. Algebra is concerned with manipulation in time, and geometry is concerned with space. These are two orthogonal, aspects of the world, and they represent two different points of view in mathematics. Thus the argument or dialogue between mathematicians in the past about the relative importance of geometry and algebra represents something very, very fundamental. Of course it does not pay to think of this as an argument, in which one side loses and the other side wins. I like to think of this in the form of an
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analogy: "Should you just be an algebraist or a geometer?" is like saying "Would you rather be deaf or blind?" If you are blind, you do not see space, if you are deaf, you do not hear, and hearing takes place in time. On the whole, we prefer to have both faculties. In physics, there is an analogous, roughly parallel, division between the concepts of physics and the experiments. Physics has two parts to it: theory — concepts, ideas, words, laws — and experimental apparatus. I think that concepts are in some broad sense geometrical, since they are concerned with things taking place in the real world. An experiment, on the other hand, is more like an algebraic computation. You do something in time; you measure some numbers; you insert them into formulae, but the basic concepts behind the experiments are a part of the geometrical tradition. One way to put the dichotomy in a more philosophical or literary framework is to say that algebra is to the geometer what you might call the "Faustian Offer". As you know, Faust in Goethe's story was offered whatever he wanted (in his case the love of a beautiful woman) by the devil in return for selling his soul. Algebra is the offer made by the devil to the mathematician. The devil says: "I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine". [Nowadays you can think of it as a computer!]. Of course we like to have things both ways, we would probably cheat on the devil, pretend we are selling our soul, and not give it away. Nevertheless the danger to our soul is there, because when you pass over into algebraic calculation, essentially you stop thinking; you stop thinking geometrically, you stop thinking about the meaning. I am a bit hard on the algebraists here, but fundamentally the purpose of algebra always was to produce a formula which one could put into a machine, turn a handle and get the answer. You took something that had a meaning; you converted it into a formula; and you got out the answer. In that process you do not need to think any more about what the different stages in the algebra correspond to in the geometry. You lose the insights and this can be important at different stages. You must not give up the insight altogether! You might want to come back to it later on. That is what I mean by the Faustian Offer. I am sure it is provocative. This choice between geometry and algebra has led to hybrids which confuse the two, and the division between algebra and geometry is not as straightforward and naive as I just said. For example, algebraists frequently
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will use diagrams. What is a diagram except a concession to geometrical intuition? 6. Techniques in Common Let me go back now to talk not so much about themes in terms of content, but perhaps in terms of techniques and common methods that have been used. I want to describe a number of common methods which have been applied in a whole range of fields. The first is: 6.1. Homology
theory
Homology theory starts off traditionally as a branch of topology. It is concerned with the following situation. You have a complicated topological space and you want to extract from that some simple information which involves counting holes or something similar, some additive linear invariants you can associate to a complicated space. It is a construction, if you like, of linear invariants in a non-linear situation. Geometrically you think of cycles which you can add and subtract and then you get what is called the homology group of a space. Homology is a fundamental algebraic tool which was invented in the first half of the century as a way of getting some information about topological spaces. Some algebra extracted out of the geometry. Homology also appears in other contexts. Another source of homology theory goes back to Hilbert and the study of polynomials. Polynomials are functions which are not linear, and you can multiply them to get higher degrees. It was Hilbert's great insight to consider "ideals", linear combinations of polynomials, with common zeros. He looked for generators of these ideals. Those generators might be redundant. He looked at the relations and then for relations between the relations. He got a hierarchy of such relations, which were called "Hilbert's syzygies" and this theory of Hilbert was a very sophisticated way of trying to reduce a non-linear situation, the study of polynomials, to a linear situation. Essentially Hilbert produced a complicated system of linear relations which encapsulates some of the information about non-linear objects, the polynomials. This algebraic theory is in fact very parallel to the topological theory, and they have now got fused together into what is called "homological algebra". In algebraic geometry one of the great triumphs of the 1950's
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was the development of the cohomology theory of sheafs and its extension to analytic geometry by the French school of Leray, Cartan, Serre, and Grothendieck, where you have a combination of the topological ideas of Riemann-Poincare, the algebraic ideas of Hilbert, and some analysis thrown in for good measure. It turns out that homology theory has wider applications still, in other branches of algebra. You can introduce homology groups, which are always linear objects associated to non-linear objects. You can take groups, finite groups, for example, or Lie algebras: both of these have homology groups associated to them. In number theory there are very important applications of homology theory, through the Galois group. So homology theory has turned out to be one of the powerful tools to analyse a whole range of situations, a typical characteristic of 20th century mathematics. 6.2.
K-theory
Another technique, which is in many ways very similar to homology theory, has had wide applications, and permeates many parts of mathematics, was of later origin. It did not emerge until the middle of the 20th century, although it is something that had its roots much further back as well. This is what is called "K-theory", and it is actually closely related to representation theory. Representation theory of, say, finite groups goes back to the last century but its modern form, K-theory, is of more recent origin. Ktheory can also be thought of in the following way, it can be thought of as the attempt to take matrix theory, where matrices do not commute under multiplication, and try to construct Abelian or linear invariants of matrices. Traces and dimensions and determinants are Abelian invariants of matrix theory and K-theory is a systematic way of trying to deal with them and sometimes is called "stable linear algebra". The idea is that if you have large matrices, then a matrix A and a matrix B which do not commute will commute if you put them in orthogonal positions in different blocks. Since in a big space you can move things around, then in some approximate way you might think this is going to be good enough to give you some information and that is the basis of K-theory as a technique. It is analogous to homology theory, in that both try to extract linear information out of complicated non-linear situations. In algebraic geometry, this was first introduced with remarkable success by Grothendieck, closely related to the story we just discussed a moment
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ago involving sheaf theory, and in connection with his work on the RiemannRoch theorem. In topology, Hirzebruch and I copied these ideas and applied them in a purely topological context. In a sense while Grothendieck's work is related to Hilbert's work on syzygies our work was more related to the Riemann-Poincare work on homology, using continuous functions as opposed to polynomials. It also played a role in the index theory of elliptic operators, in linear analysis. In a different direction, the algebraic side of the story, with potential application to number theory was then developed by Milnor, Quillen and others. It has led to many interesting questions in that direction. In functional analysis, the work of many people, including Kasparov, extended the continuous K-theory to the situation of non-commutative C*algebras. The continuous functions on a space form a commutative algebra under multiplication, but non-commutative analogues of those arise in other situations, and functional analysis turns out to be a very natural home for these kinds of questions. So K-theory is another area where a whole range of different parts of mathematics lend themselves to this rather simple formalism, although in each case there are quite difficult technical questions specific to that area, which connect up with other parts of the subject. It is not a uniform tool; it is more a uniform framework, with analogies and similarities between one part and the other. Much of this work has also been extended by Alain Connes to "noncommutative differential geometry". Interestingly enough, very recently, Witten in working on string theory (the latest ideas in fundamental physics) has identified very interesting ways in which K-theory appears to provide a natural home for what are called "conserved quantities". Whereas in the past it was thought that homology theory was the natural framework for these, it now seems that K-theory provides a better answer. 6.3. Lie
groups
Another unifying concept which is not just a technique is that of Lie groups. Now Lie groups, by which we mean fundamentally the orthogonal, unitary and symplectic group, together with some exceptional groups, have played a very important part in the history of 20th century mathematics.
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Again, they date from the 19th century. Sophus Lie was of course a 19th century Norwegian mathematician, and he, Felix Klein and others pushed "the theory of continuous groups", as it was called. Originally, for Klein, this was a way of trying to unify the different kinds of geometry: Euclidean geometry, non-Euclidean geometry. Although this subject started in the 19th century, it really took off in the 20th century. The 20th century has been very heavily dominated by the theory of Lie groups as a sort of unifying framework in which to study many different questions. I did mention the role in geometry of the ideas of Klein. For Klein, geometries were spaces which were homogeneous, where you could move things around without distortion, and so they were determined by an associated isometry group. The Euclidean group gave you Euclidean geometry; hyperbolic geometry came from another Lie group. So each homogeneous geometry corresponded to a different Lie group. But later on, following up on Riemann's work on geometry, people were more concerned with geometries which were not homogeneous, where the curvature varied from place to place and there were no global symmetries of space. Nevertheless, Lie groups still played an important role because they come in at the infinitesimal level, since in the tangent space we have Euclidean co-ordinates. Therefore in the tangent space, infinitesimally, Lie group theory reappears, but because you have got to compare different points in different places, you have got to move things around in some way to handle the different Lie groups. That was the theory really developed by Elie Cartan, the basis of modern differential geometry, and it was also the framework which was essential to Einstein's theory of relativity. Einstein's theory of course gave a big boost to the whole development of differential geometry. Moving on into the 20th century, the global aspect, which I mentioned before, involved Lie groups and differential geometry at the global level. A major development characterised by the work of Borel and Hirzebruch gave information about what are called "characteristic classes". These are topological invariants combining together the three key parts: the Lie groups, the differential geometry and the topology, and of course, the algebra associated with the group itself. In a more analytical direction, we get what is now called noncommutative harmonic analysis. This is the generalisation of Fourier theory, where the Fourier series or Fourier integrals correspond essentially to the commutative Lie groups of the circle and the straight line. When you
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replace these by more complicated Lie groups, then we get a very beautiful, elaborate theory which combines representation theory of Lie groups and analysis. This was essentially the life's work of Harish-Chandra. In number theory the whole "Langlands program", as it is called, which is closely related also to Harish-Chandra theory, takes place within the theory of Lie groups. For every Lie group, you have the associated number theory and the Langlands program, which has been carried out to some extent. It has influenced a large part of the work in algebraic number theory in the second half of this century. The study of modular forms fits into this part of the story including Andrew Wiles' work on Fermat's Last Theorem. One might think that Lie groups are particularly significant only in geometrical contexts, because of the need for continuous variation, but the analogues of Lie groups over finite fields give finite groups, and most finite groups arise that way. Therefore the techniques of some parts of Lie theory apply even in a discrete situation for finite fields or for local fields. There is a lot of work which is pure algebra; work, for example, which George Lusztig's name is associated with, where representation theory of such finite groups is studied and where many of the techniques which I have mentioned before have their counterparts. 7. Finite Groups This brings us to finite groups and that reminds me: the classification of finite simple groups is something where I have to make an admission. Some years ago I was interviewed, when the finite simple group story was just about finished, and I was asked what I thought about it. I was rash enough to say I did not think it was so important. My reason was that the classification of finite simple groups told us that most simple groups were the ones we knew, and there was a list of a few exceptions. In some sense that closed the field, it did not open things up. When things get closed down instead of getting opened up, I do not get so excited but of course a lot of my friends who work in this area were very, very cross. I had to wear a sort of bulletproof vest after that! There is one saving grace. I did actually make the point that in the list of the so-called "sporadic groups", the biggest was given the name of the "Monster". I think the discovery of this Monster alone is the most exciting output of the classification. It turns out the Monster is an extremely interesting animal and is still being understood now. It has unexpected
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connections, with large parts of other parts of mathematics, with elliptic modular functions, and even with theoretical physics and quantum field theory. This was an interesting by-product of the classification. Classifications by themselves, as I say, close the door; but the Monster opened up a door. 8. Impact of Physics Let me move on now to a different theme, which is the impact of physics. Throughout history, physics has had a long association with mathematics, and large parts of mathematics, calculus, for example, were developed in order to solve problems in physics. In the middle of the 2 0 t h century, this perhaps had become less evident, with most of pure mathematics progressing very well independent of physics but in the last quarter of this century things have changed dramatically. Let me try to review briefly the interaction of physics with mathematics, and in particular with geometry. In the 19th century, Hamilton developed classical mechanics, introducing what is now called the Hamiltonian formalism. Classical mechanics has led to what we call "symplectic geometry". It is a branch of geometry which could have been studied much earlier, but in fact has not been studied seriously until the last two decades. It turns out to be a very rich part of geometry. Geometry, in the sense I am using the word here, has three branches: Riemannian geometry, complex geometry and symplectic geometry, corresponding to the three types of Lie groups. Symplectic geometry is the most recent of these and in some ways possibly the most interesting, and certainly one with extremely close relations to physics, because of its historical origins in connection with Hamiltonian mechanics and more recently with quantum mechanics. Now, Maxwell's equations, which I mentioned before, the fundamental linear equations of electromagnetism, were the motivation for Hodge's work on harmonic forms, and the application to algebraic geometry. This turned out to be an enormously fruitful theory, which has underpinned much of the work in geometry since the 1930's. I have already mentioned general relativity and Einstein's work. Quantum mechanics, of course, provided an enormous input. Not only in the commutation relations but more significantly in the emphasis on Hilbert space and spectral theory. In a more concrete and obvious way, crystallography in its classical form was concerned with the symmetries of crystal structures. The finite
16
Sir M.
Atiyah
symmetry groups that can take place around points were studied in the first instance because of their applications to crystallography. In this century the deeper applications of group theory have turned out to have relations to physics. The elementary particles of which matter is supposed to be built appear to have hidden symmetries at the very smallest level, where there are some Lie groups lurking around, which you cannot see, but the symmetries of these become manifest when you study the actual behaviour of the particles. So you postulate a model in which symmetry is an essential ingredient and the different theories which are now prevalent have certain basic Lie groups like SU(2) and SU(3) built into them as primordial symmetry groups. So these Lie groups appear as building blocks of matter. Nor are compact Lie groups the only ones that appear. Certain noncompact Lie groups appear in physics, like the Lorentz group. It was physicists who first started the study of the representation theory of noncompact Lie groups. These are representations which have to take place in Hilbert space because, for compact groups, the irreducible representations are finite dimensional but non-compact groups require infinite dimensions, and it was physicists who first realised this. In the last quarter of the 20th century, the one we have just been finishing, there has been a tremendous incursion of new ideas from physics into mathematics. This is perhaps one of the most remarkable stories of the whole century. It requires perhaps a whole lecture on its own but basically, quantum field theory and string theory, have been applied in remarkable ways to get new results, ideas, and techniques in many parts of mathematics. By this I mean that the physicists have been able to predict that certain things will be true in mathematics based on their understanding of the physical theory. Of course, that is not a rigorous proof, but it is backed by a very powerful amount of intuition, special cases, and analogies. These results predicted by the physicists have time and again been checked by the mathematicians and found to be fundamentally correct, even though it is quite hard to produce proofs and many of them have not yet been fully proved. So there has been a tremendous input over the last 25 years in this direction. The results are extremely detailed. It is not just that the physicists said, "this is the sort of thing that should be true". They said, "here is the precise formula and here are the first ten cases (involving numbers with more than 12 digits)". They give you exact answers to complicated
Mathematics
in the 20th Century
17
problems, not the kind of thing you can guess; things you need to have machinery to calculate. Quantum field theory has provided a remarkable tool, which is very difficult to understand mathematically but has had an unexpected bonus in terms of applications. This has really been the exciting story of the last 25 years. Here are some of the ingredients: Simon Donaldson's work on 4dimensional manifolds; Vaughan-Jones' work on knot invariants; mirror symmetry, quantum groups; and I mentioned the Monster just for good measure. What is this subject all about? As I mentioned before, the 20th century saw a shift in the number of dimensions ending up with an infinite number. Physicists have gone beyond that. In quantum field theory they are really trying to make a very detailed study of infinite-dimensional space in depth. The infinite-dimensional spaces they deal with are typically function spaces of various kinds. They are very complicated, not only because they are infinite-dimensional, but they have complicated algebra and geometry and topology as well and there are large Lie groups around, infinite-dimensional Lie groups. So, just as large parts of 20th century mathematics were concerned with the development of geometry, topology, algebra, and analysis on finite-dimensional Lie groups and manifolds, this part of physics is concerned with the analogous treatments in infinite dimensions and of course it is a vastly different story, but it has enormous pay-offs. Let me explain this in a bit more detail. Quantum field theories take place in space and time; and space is really meant to be three-dimensional but there are simplified models where you take one dimension. In onedimensional space and one-dimensional time, typically the things that physicists meet are, mathematically speaking, groups such as the diffeomorphisms of the circle or the group of differentiable maps from the circle into a compact Lie group. These are two, very fundamental examples, of infinite-dimensional Lie groups which turn up in quantum field theories in these dimensions, and they are quite reasonable mathematical objects which have been studied by mathematicians for some time. In such 1 + 1 dimensional theories one can take space-time to be a Riemann surface and this leads to new results. For example, the moduli space of Riemann surfaces of a given genus is a classical object going back to the last century. Quantum field theory has led to new results about the cohomology of these moduli spaces. Another, rather similar moduli space is
18
Sir M.
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the moduli space of flat G-bundles over a Riemann surface of genus g. These spaces are very interesting and quantum field theory gives precise results about them. In particular, there are beautiful formulae for the volumes, which involve values of zeta functions. Another application is concerned with counting curves. If you look at algebraic curves in the plane of a given degree, of a given type, and you want to know how many of them, for example, pass through so many points, you get into enumerative problems of algebraic geometry, problems that would have been classical in the last century. These are very hard. They have been solved by modern machinery called "quantum cohomology", which is all part of the story coming from quantum field theory, or you can look at more difficult questions about curves not in the plane, but curves lying on curved varieties. One gets another beautiful story with explicit results going by the name of mirror symmetry. All this comes from quantum field theory in 1 + 1 dimensions. If we move up one dimension, where we have 2-space and 1-time, this is where the theory of Vaughan-Jones of invariants of knots comes in. This has had an elegant explanation or interpretation, in quantum field theory terms. Also coming out of this is what are called, "quantum groups". Now the nicest thing about quantum groups is their name. They are definitely not groups! If you were to ask me for the definition of a quantum group, I would need another half hour. They are complicated objects, but there is no question that they have a deep relationship with quantum theory. They emerged out of the physics, and they are being applied by hard-nosed algebraists who actually use them for definite computations. If we move up one step further, to fully four-dimensional theory, (threeplus-one dimension), that is where Donaldson's theory of four-dimensional manifolds fits in and where quantum field theory has had a major impact. In particular it led Seiberg and Witten to produce their alternative theory, which is based on physical intuition and gives marvellous results mathematically as well. All of these are particular examples. There are many more. Then there is string theory and this is already passe! M-theory is what we should talk about now, and that is a rich theory, again with a large number of mathematical aspects to it. Results coming out of it are still being digested and will keep mathematicians busy for a long time to come.
Mathematics
in the 20th Century
19
9. Historical S u m m a r y Let me just try to make a quick summary. Let me look at the history in a nutshell: what has happened to mathematics? I will rather glibly just put the 18th and 19th centuries together, as the era of what you might call classical mathematics, the era we associate with Euler and Gauss, where all the great classical mathematics was worked out and developed. You might have thought that would almost be the end of mathematics, but the 20th century has, on the contrary, been very productive indeed and this is what I have been talking about. The 20th century can be divided roughly into two halves. I would think the first half has been dominated by what I call the "era of specialisation", the era in which Hilbert's approach, of trying to formalise things and define them carefully and then follow through on what you can do in each field, was very influential. As I said, Bourbaki's name is associated with this trend, where people focused attention on what you could get within particular algebraic or other systems at a given time. The second half of the 20th century has been much more what I would call the "era of unification", where borders are crossed over, techniques have been moved from one field into the other, and things have become hybridised to an enormous extent. I think this is an oversimplification, but I think it does briefly summarise some of the aspects that you can see in 20th century mathematics. What about the 21st century? I have said the 21st century might be the era of quantum mathematics or, if you like, of infinite-dimensional mathematics. What could this mean? Quantum mathematics could mean, if we get that far, understanding properly the analysis, geometry, topology, algebra of various non-linear function spaces and by "understanding properly" I mean understanding it in such a way as to get quite rigorous proofs of all the beautiful things the physicists have been speculating about. One should say that, if you go at infinite dimensions in a naive way and ask naive questions, you usually get the wrong answers, or the answers are dull. Physical application, insight, and motivation has enabled physicists to ask intelligent questions about infinite dimensions and to do very subtle things where sensible answers do come out and therefore doing infinite-dimensional analysis in this way is by no means a simple task. You have to go about it in the right way. We have got a lot of clues. The map is laid out: this is what should be done, but it is long way to go yet.
20
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What else might happen in the 21st century? I would like to emphasise Connes' non-commutative differential geometry. Alain Connes has this rather magnificent unified theory. Again it combines everything. It combines analysis, algebra, geometry, topology, physics, number theory, all of which contribute to parts of it. It is a framework which enables us to do what differential geometers normally do, including its relationship with topology, in the context of non-commutative analysis. There are good reasons for wanting to do this, applications (potential or otherwise) in number theory, geometry, discrete groups, and so on, and in physics. An interesting link with physics is just being worked out. How far this will go, what it will achieve, remains to be seen. It certainly is something which I expect will be significantly developed in the first decade at least of the next century, and it is possible it could have a link with the as-yet-undeveloped (rigorous) quantum field theory. Moving in another direction, there is, what is called "arithmetic geometry" or Arakelov geometry, which tries to unify as much as possible algebraic geometry and parts of number theory. It is a very successful theory. It has made a nice start but has a long way to go. Who knows? Of course, all of these have strands in common. I expect physics to have its impact spread all the way through, even to number theory: Andrew Wiles disagrees and only time will tell. These are the strands that I can see emerging over the next decade, but there is what I call a joker in the pack: going down to lower-dimensional geometry. Alongside all the infinite-dimensional fancy stuff low-dimensional geometry is an embarrassment. In many ways the dimensions where we started, where our ancestors started, remain something of an enigma. Dimensions 2, 3, and 4 are what we call "low". The work for example of Thurston in three-dimensional geometry, aims at a classification of geometries one can put on three-dimensional manifolds. This is much deeper than the two-dimensional theory. The Thurston program is by no means completed yet, and completing that program certainly should be a major challenge. The other remarkable story in three dimensions is the work of VaughanJones with ideas essentially coming from physics. This gives us more information about three dimensions, which is almost orthogonal to the information contained in the Thurston program. How to link those two sides of the story together remains an enormous challenge but there are recent
Mathematics in the 20th Century
21
hints of a possible bridge. So, this whole area, still in low dimensions, has its links to physics, but remains very mysterious indeed. Finally I should like to mention that in the physics what emerges very prominently are "dualities". These dualities, broadly speaking, arise when a quantum theory has two different realisations as a classical theory. A simple example is the duality between position and momentum in classical mechanics. This replaces a space by its dual space and in linear theories that duality is just the Fourier transform. But in non-linear theories, how you replace a Fourier transform is one of the big challenges. Large parts of mathematics are concerned with how to generalise dualities in non-linear situations. Physicists seem to be able to do so in a remarkable way in their string theories and in M-theory. They produce example after example of marvellous dualities which in some broad sense are infinite-dimensional non-linear versions of Fourier transforms and they seem to work. But understanding those non-linear dualities does seem to be one of the big challenges of the next century as well. I think I will stop there. There is plenty of work, and it is very nice for an old man like me to talk to a lot of young people like you; to be able to say to you: there is plenty of work for you in the next century.
T H E 04 OF MINIMAL G O R E N S T E I N 3-FOLDS OF GENERAL TYPE Dedicated t o the memory of Professor Wei-Liang Chow and Professor Kuo-Tsai Chen
Meng Chen Department of Applied Mathematics, Tongji University Shanghai, 200092, P. R. China mchen@mail. tongji. edu. en
1. Introduction This paper is devoted to the birational classification of minimal algebraic Gorenstein 3-folds. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. According to [2,5,6,9,15,17] and [21], we have the following theorem. Theorem 1.1. Suppose X is a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. Then the following holds. (i) The m-canonical map <j>m is a birational morphism for all m > 6. (ii) 05 is birational with possible exception for pg(X) < 2.
24
M. Chen
curious about some birational properties of ^4. Our approach is different and elementary. In order to make our statement simpler, let us first fix the terminologies. X is refered to as $4-standard if there exists a fibration / : X' —> C onto a projective curve C, where X' is birationally equivalent to X and the general fiber of / is a smooth projective surface of general type with invariants K2 = 1 and pg = 2. X is called ^4-semi-standard if X is fibred by curves of genus two, i.e. there is a fibration g : X' —> W onto a normal projective surface W where X' is birationally equivalent to X and the general fiber of g is a smooth projective curve of genus two. If X is ^-standard, one can easily see that X is (^4-semi-standard by taking the relatively canonical map of / . It is well known that the 4-canonical map of a smooth projective surface of general type is birational if and only if (K2,pg) ^ (1>2). This leads to the trivial fact that, if X is <j>4-standard, the 4-canonical map 4 of X fails to be birational. A very natural question is whether the converse is true. In this paper, we would like to study this problem and to show that the converse is true under some reasonable conditions. Since \4K\ is base point free, 0 4 is automatically generically finite. Here we shall give an alternative and simpler proof. Our results are as follows. T h e o r e m 1.2. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. The following holds. (i) Suppose pg(X) > 41 and dim
i(X) ^ 2. Then (j>A is birational if and only if X is not ^^-standard. (ii) Suppose pg(X) > 41 and X is not 4-semi-standard. Then <j>i is birational. (iii) 4>4 is generically finite. Throughout the ground field is assumed to be algebraically closed of characteristic 0. For a Q-divisor D on a smooth variety V, we denote by rD~[ the round-up of D, which is the minimum integral divisor such that r £ P - D > 0. rN-'lin H16cLIlS HllGctr GQU.i.VEll.GIlCC. '"SJnum means numerical equivalence. We would like to thank Prof. Catanese for stimulating discussions and helps during the writing of this paper.
The 4 of Minimal
Gorenstein
3-Folds of General Type
2. Proof of Theorem 1.2 Definition 2.1. A normal variety X is called Gorenstein if the dualizing sheaf ux is invertible and X is Cohen-Macaulay. We refer to [18] for the definitions of canonical, terminal singularities. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. It is well known that Kx is a positive even integer, xiPx) < 0 and that Pm(X)
:=
h°(X,Ox(mKx))
= (2m - 1)
m(m 12
-V-x(o x )
(2.1)
Suppose pg(X) > 2. We can define the canonical map fa. Set KX ~lin
Mi+Zi,
where Mi is the movable part of \Kx\ and Zi the fixed one. Taking the birational modification 7r: X' —> X, according to Hironaka, such that (1) X' is smooth; (2) the movable part of \iv*(Kx)\ is base point free; (3)
TT*(KX)
has supports with only normal crossings.
Denote by g the composition fa o n. So g:X'
—^'CPP'W-
1
is a morphism. Let g : X' - A W -?-> W be the Stein factorization of g. We can write 7r*(Mi)~un5i+£7i, where Si is the movable part. Then we have *'{Kx)~ilaS1+E', where E' = Ei + ir*(Zi) is the fixed part of \n*(Kx)\- We note that 1 < dim (W) < 3. We shall formulate our proof according to dim (W). Remark 2.2. Although [5,6] and [17] only treated smooth minimal 3folds, the method is still effective for Gorenstein minimal 3-folds. In order
25
26
M. Chen
to avoid unnecessary redundancy, we would like to cite several basic facts from there without giving the proof. Theorem 2.3. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. Suppose pg(X) > 5. //dim4>\{X) = 3, then fa is birational. Proof. It's obvious that a general member S\ is a smooth projective surface of general type. Because pg(X) > 0, it is sufficient to verify the birationality for fa\si by virtue of the Tankeev principle. We consider the system \KX,+2**{KX)
+ S1\.
The vanishing theorem gives \KX, + 2ir*{Kx) + S1\ \Si = \KSl + 2L\, where L := w*(Kx)\si is a nef and big divisor on Si. If \L\ gives a birational map, then so does \Kst + 2L\. Otherwise, \L\ gives a generically finite map of degree > 2. Noting that h°(Si,L) > pg{X) — 1 > 4, we have L 2 > 2(h°{Si,L)-2) > 4 . If \KSl+2L\ doesn't give a birational map, then there is a free pencil of curves on Si with a general irreducible element C such that 2L • C < 2 according to Reider's result [18, Corollary 2]. The only possibility is L • C — 1. On the other hand, L • C > 2 since \L\ gives a generically finite map on C and C is a curve of genus > 2. The contradiction shows that $\KSl+2L\
is birational. Therefore fa is birational.
D
Theorem 2.4. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. Suppose pg(X) > 41 and X is not fa-standard. If dimi(X) = 1, then fa is birational. Proof. In this case, W is a smooth projective curve. We have a fibration f : X' —>W. Denote b := g(W). Let F be a general fiber of / . Then F is a smooth projective surface of general type. In general position, Si can split into a sum of different fibers, i.e. a
Si ~lin 2 J F* ' i=l
The <j>i of Minimal Gorenstein
3-Folds of General Type
27
where a > pg(X) — 1. The vanishing theorem gives the surjective map H° (X\ KX> + 2n*(Kx)
+ £ # )
a
—•
QH°{Fi,KFi+2n*(Kx)\Ft)-+0. i=l
This means that 4>i can distinguish general different fbers of / . In order to prove the theorem, it is sufficient to verify the birationality of ^p for a general fiber F. Denote F :— n(F). Then Mi ~ „ u m aF. Noting that F2 is a quasi effective 1-cycle on X, we have Kx • F 2 > 0. Let a : F —> Fo be the contraction onto the minimal model Fo of F. Suppose KXF2 = 0. Then we have 0F(ir*(Kx)\F)
= OF(cr*(KF0))
(2.2)
according to [6, Lemma 2.3]. We have Tr*(Kx)~n»maF
+ E'.
Thus ir*(Kx)
- F - \ E '
~ n u m (l - - \
-K*{K.x)
is a nef and big Q-divisor, since a > 1 under the condition of the theorem. Denote G:=
n*(Kx)--E'
.
a
The Kawamata-Viehweg vanishing theorem yields \KX.
+ 2TT*(KX)
where L := TT*{KX)\F
+G\\F=
\KF
+ 2L + G\F\,
(2.3)
~Hn <**(KFo). We can see that
is an effective divisor. Because X is not (^-standard, F can't be a surface with (K2Fo,Pg(F)) = (1,2). If (K2Fo,Pg(F)) ± (2,3), then $, 3 i 0 p , is birational. Since $\KF+2L\
= $|3K>| >
28
M. Chen
we see that i\F is birational and so is ^4. If (KF ,pg(F)) — (2,3), we can show that $ \KP+2L+G\F\ is birational. In fact, we have KF + 2L + G\F >KF
+ 2L +
1 - - \E'\ a,
where
1-i a
E'\
n*(Kx)\F
is a nef and big Q-divisor. It is well known that \a*(KFo)\ gives a generically finite map [1]. For simplicity, we can suppose the movable part of |cr*(.Ki?0)| is base point free and C is a general member in the movable part of this system. It's sufficient to prove the birationality of ^\KF+2L+r(l-^)E'\F^\
1C-
We study the system KF + L +
1--)
a.
E'\F
C
The vanishing theorem gives KF + L +
E'\F
1
+C
= \Kc + L\c + D\
where deg(L|c) > 2 and D is a divisor of degree > 0. Obviously, \Kc + L\c + D\ gives an embedding. Thus ASuppose Kx • F2 > 0. We want to show that (p4\F is also birational. In this case, (2.2) doesn't hold. However, we still have (2.3). First we have to study \2L\. We claim that \2L\ gives a generically finite map whenever pg{X) > 41. Suppose Mi is the movable part of |2.RTx'|- Then M2 < 2n*(Kx). It's obvious that Kx> + G < 2KX> • Denote by M£ the movable part of \KX> + G\. Then M'2 < M2. The Kawamata- Viehweg vanishing theorem gives the surjective map H°{X', KX' +G)^
H°{F, KF + G\F) —>• 0.
The (p^ of Minimal Gorenstein 3-Folds of General Type
29
We also have a natural map H°{X',M'2)^H°{F,M'2\F). When pg(X) > 41, we have pg(F) > q(F) > 5 by [7, Theorem 2(3)]. Thus \KF\ can't be composed of a pencil of curves according to [23]. Denote by H the movable part of \KF\. We have h°(F, M'2\F) > dime imifi) = dime tm(a) = h°(F, KF + G\F). Whereas, M^p < KF + G\F. We see that H < M^p. Thus \M2\F\ is not composed of a pencil of curves and neither is \2L\. We have H < 2L. If \H\ already gives a birational map, so does \Kp + 2L + G\p\- Otherwise, 2L-H>H2>
2(Pg(F) - 2) > 6 .
Thus L • H > 3. For simplicity, we can suppose \H\ is base point free. This means that we can take H be a smooth curve. Using the vanishing theorem again, we have r
KF+
/
-i
1\
[l-^\E'\F
+H
= \KH + D0\, H
where Do is a divisor on the curve H with deg(D0)>
(I-^\E'\F-H=(I-^)L-H>2.
So KH + Do is very ample. Noting that
KF+
+H
(I-^\E'\F
+ 2L,
we see that $\KF+G\F+2L\\H is birational and so is $\KF+G\F+2L\shows that n is birational.
This
Theorem 2.5. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. Suppose pg(X) > 5 and X is not 4-semi-standard. If dim i(X) = 2, then $4 is birational. Proof. In this case, we have a fibration / : X' —> W onto a normal projective surface W. Let C be a general fiber of / . Because X is not ^4-semi-standard, C is a smooth curve of genus > 3. We can see that a2
-Sl|Si ~lin 2__,C« ~n«m a 2C, i=l
•
30
M. Chen
where a-i > pg(X) — 2 > 3 and we take C be a smooth fiber contained in Si. Note that a general member Si is a smooth projective surface of general type. The vanishing theorem gives \KX> + 2**(KX) + Si| | S l = \KSl + 2L\, where L := n*(Kx)\si
is nef and big and
/i°(Si,L) > ^ ( S ^ S i l s J >
Pg(X)
- 1.
Suppose \L\ is not composed of a pencil of curves. If \L\ gives a birational map, so does \Ksx + 2L\. Otherwise, L2>2(h°(S1,L)-2)>4. If \Ks1 + 2L\ doesn't give a birational map, according to Reider, there is a free pencil on Si with a general irreducible member C such that 2LC < 2. This means L • C = 1. This is impossible, because \L\ gives a finite map on C and C is a curve of genus > 2. Thus $\KS +2L| i s birational. So ^4 is birational. Suppose \L\ is composed of a pencil of curves. Since
we can see that a generic irreducible element of the movable part of |L| is a smooth fiber C contained in Si. We have L2 >L-Si\Sl
>a2>3.
If lA'si + 2L\ doesn't give a birational map, according to Reider, there is a free pencil on Si with a general irreducible element C such that 2L • C < 2. The only possibility is L • C = 1. Obviously, C should be algebraically equivalent to C. Otherwise, dim$|x,|(Cr) = 1. Which means L • C > 2, since C is a curve of genus > 2. Therefore we have seen that C is actually a fiber of / . So we should have L • C = 1. We want to derive a contradiction by proving that L • C > 2. We can write L ~Hn S i | s ! + J ~ n u m CL2C + J ,
where J is an effective divisor on Si and C is contained in Si. So L-C
J ~num I 1
)L
The 4 of Minimal Gorenstein 3-Folds of General Type
is a nef and big Q-divisor. Considering the system r
n
1
KSl+L+
L-—J a2
we get from the Kawamata-Viehweg vanishing theorem that
KSl+L+
L--J a2
Kc + L\c+
L
J fl2
We shall use a parallel analysis to the one in the proof of Theorem 2.4. Denote by M 4 the movable part of |42Cx'|- Then M 4 < 4n*(Kx)- Denote by M 4 the movable part of \KX'+2ir*(Kx)
+ S1\.
Then M 4 < M 4 . Denote by N the movable part of \KSl +2L\. We have the exact sequence H°(X',KX,
+ 2^{KX)
+ 5i) - ^ H°{SuKSl
+ 2L) — • 0
and the natural map H°(X',M'4)
Atf°(Si>M4k).
Since M 4 | S l < KSl + 2L and h°(Si,M'4\Sl)
> dim c im(/?i) = dim c im{ax) = h°{Su KSl + 2L),
we see that M^ls, > N. Denote by JV' the movable part of
KSl+L+
L-—J a2
Then N > N'. We have the surjective map
H°(s1,KSl+L+ -^H°(c,Kc
L-^-J^j + L\c+
and the natural map &>&,&)-UH°(C,N'\C).
L-—J
31
32
M. Chen
So h°(C,N'\c)
> d i m c i m ^ ) = dimcim(p)
h°(Kc + L\c+
L-—J
V
«2
Y
c)
Since L - —j]
a2 J
• C = ( \ - — \ L - C > 0 ,
\
a2J
we see that r
. /
h°[Kc+L\c+
1
L
n
\
J
> 9(C) + 1.
Thus h°(C, N'\c) > g{C) + 1. The R-R on C shows at once that N' • C > 2<7(C) > 6, because g(C) > 3. Thus we have
^*(Kx)\Sl-C>N'-C>6. We get L • C > 2, a contradiction. Thus |isTs,1 + 2 i | gives a birational map and so 4 is birational. D Theorems 2.3-2.5 directly imply (i) and (ii) of Theorem 1.2. Theorem 2.6. Let X be a minimal projective Gorenstein 3-fold of general type with only locally factorial terminal singularities. Then $4 is generically finite. Proof. Since we are treating the general case without any assumption on pg(X), we can't consider the canonical map. However we have pz(X) > 4 according to (2.1). So we can study fa. Set 2KX ~iin M2 + Z2, where M2 is the movable part of |2.Kx | and Z2 the fixed one. Taking the birational modification -K2 : X' —> X, according to Hironaka, such that (1) X' is smooth; (2) the movable part of |27T2 (/iTjsf) | is base point free; (3) both 7r|(2.Kx) a n < i ^(^Kx) have supports with only normal crossings.
The 4>4 of Minimal Gorenstein
3-Folds of General Type
33
Denote by g2 the composition <j)2oTr2. SO g2 : X' —> W'2 C P ^ ( ^ ) - i is a morphism. Let 52
. x' A ^ 2 - ^ W^
be the Stein factorization of 52- We can write n$(M2) ~un S2 + E2 , where S2 is the movable part. Then we have n2{2Kx)
~iin S2 + E2,
where E2 = E2 + ir2{Z2) is the fixed part of \n2(2Kx)\- We only have to consider the case when dim<j>2(X) < 3. Suppose dirci(f)2(X) — 1. We have a fibration f2 : X' —> W2 onto a smooth curve W2. A general fiber F of f2 is a smooth projective surface of general type. Because 2Kx> < 4Kx', 4>i can distinguish different fibers of g2. In order to prove the generic finiteness of $4, it is sufficient to show that 4>
S2 ~lin 2 J "^ ' where a2 > P2{X) - 1 and the F(s are fibers of f2. The vanishing theorem gives \KX> + *i{Kx) where L2 := -n^Kx)^-
+ S2\ \Sa = \KS2 + L2\,
According to [16, Claim 9.1], we have
Os2(^(Kx)\S2)
^
Os2(a*2(KSo)),
where cr2 : S2 —> SQ is the contraction onto the minimal model So of S2. Thus Ks2 + L2 ~, i n Ks, +
34
M. Chen
and so ®\KS2+L2\
=$\2KSa\-
From [6, Theorem 3.1], we know that 52 can't be a surface with pg = q = 0. By [21, Theorem 1], $|2iCS2| *s generically finite. Thus 4 is generically finite. Suppose dim^>2(-^0 = 2. We want to derive a contradiction assuming that 4>i is not generically finite. We consider the following two natural maps H°(X',4n*2(Kx))
^
A4 C
tf°(52,4L),
H0(X>,2TT*2(KX))
- ^ A2 C
H°(S2,2L),
(2-4)
where Aj is the image of Oj for i = 2,4. By our assumption, A4 should be composed of a pencil of curves on the surface 52. On the other hand, it's obvious that A2 C A4 and A2 = P2|s 2 | •
Noting that |52|s 2 | is a free pencil, we can see that, in this situation, the movable part of A4 is also base point free and that both A2 and A4 have the same generic irreducible element. Because the movable part of A4 is base point free, there is a divisor H4 (movable part of A4) in S2 such that \Hn\ C A4 and h°{S2,Hi)
=dimcA4.
(One should note that A4 is not a complete linear system in general.) We can write 62
S2IS2 ~lin 2 ^ Ci ~ nU m b2C , i=l
where b2 > P-z{X) — 2, the C[s are fibers of f2 and C is a smooth fiber of f2 contained in 52. Then we have H4 ~ nU m b±C, where 64 > dime A4 — 1 and we think of A4 as a C-vector space. The vanishing theorem gives that
\KX> + ^(Kx) + S2\ L = \KS2 + L 2 |,
The 4>i of Minimal Gorenstein 3-Folds of General Type 35
where L2 := ir2(Kx)\s2-
I* is obvious that
Ks2+L2>2L2>S2\s2. can This means that $\KS2+L2\ distinguish different fibers of $%• For a generic C contained in 52, we want to study $\KS +L2\\C in order to derive a contradiction. If dime A4 > 6, i.e. h°(S2, H4) > 6, then we can see that 64 > 5. Noting that Hi < AL2, we have
4L2 ~num biC + Z4 , where Z± is an effective divisor. Thus L2 ~num —C + -Z4 4
4
and L2 — C — T-Z4 ~num I 1 — 7— I L2 64 \ 64) is a nef and big Q-divisor on 52. Thus the vanishing theorem yields r
1
Ks2+
"
L2--Z4
= \KC + D\,
04
c
where
deg(D) > U2 - ^Z*\
•C=(l-y)L2-C>0.
So \Kc + D\ gives a finite map on C. Noting that r
Ks2+
1 L-i-j-Zi o4
n
we see that $|/c s +L 2 |lc is finite and so &\KS +L2\ is generically finite. This means fa is generically finite, a contradiction. If dime A4 < 5, because P±{X) > 21 by (2.1), we see from the map (2.4) that |4TT2*(KX)-452|^0.
36
M. Chen
So we can write 47r;CK'A-)~iin4S'2+G4, where G4 is an effective divisor. Thus n
2^x)
L2 = 1t2{Kx)
~nnm $2 + 4G4
~num S2\s2 + 7 ^ 4 | s 2 ~num b2C + -G4\s2
,
where b2 > P2{X) - 2 > 2. We have that La
" C " 4 ^ G 4 | S 2 ~num V1 " b~2) L2
is a nef and big Q-divisor. The vanishing theorem gives Ks2
+
L2 —
-TT-G4\S2
w2
= \KC + D'\
where deg(D') > (L2 - ±-G4\s^j
• C = ( l - i ) L2 • C > 0.
This means that \Kc + D'\ gives a finite map on C. Noting that r
Ks2 + L2-
1 -^-G4\s2
Q\KS2+L2\,
we see that \Ks2 + L2\ gives a generically finite map and so that <j>4 is also generically finite, a contradiction. In a word, <j>4 is generically finite. • Example 2.7. The assumption pg(X) > 5 is sharp in Theorem 2.3. There is a trivial example with pg(X) = 4 and K\ = 2 on which dim<^i(X) = 3 and 4 is a finite map of degree 2. On P^, take a smooth hypersurface S of degree 10. S ~u n 10H. Let X be a double cover over P 3 with branch locus along S. Then X is a nonsingular canonical model, K\ = 2 and pg(X) = 4 and fa is a finite morphism onto P 3 of degree 2. One can easily check that <j>4 is also a finite morphism of degree 2.
The <j>i of Minimal Gorenstein 3-Folds of General Type 37 Acknowledgments This project is partially supported by the National N a t u r a l Science Foundation of China (10131010), the post-doc fellowship of the Georg-AugustUniversitat Gottingen, Shanghai Scientific & Technical Commission (Grant 01QA14042) and S R F for R O C S , SEM.
References [1] W. Barth, C. Peter and A. Van de Ven, Compact complex surface, SpringerVerlag (1984). [2] X. Benveniste, Sur les applications pluricanoniques des varietes de type tres gegeral en dimension 3, Amer. J. Math. 108 (1986) 433-449. [3] E. Bombieri, Canonical models of surfaces of general type, Publications I.H.E.S. 42 (1973) 171-219. [4] F. Catanese, Canonical rings and special surfaces of general type, Proc. Symp. Pure Math. 46(1987) 175-194. [5] M. Chen, On pluricanonical maps for threefolds of general type, J. Math. Soc. Japan 50 (1998) 615-621. [6] M. Chen, Kawamata-Viehweg vanishing and quint-canonical maps for threefolds of general type, Comm. in Algebra 27 (1999) 5471-5486. [7] M. Chen, Complex varieties of general type whose canonical systems are composed with pencils, J. Math. Soc. Japan 51 (1999) 331-335. [8] C. Ciliberto, The bicanonical map for surfaces of general type, Proc. Symposia in Pure Math. 62 (1997) 57-83. [9] L. Ein and R. Lazarsfeld, Global generation of pluricanonical and adjoint linear systems on smooth projective threefolds, J. Amer. Math. Soc. 6 (1993) 875-903. [10] R. Hartshorne, Algebraic Geometry, GTM 52, Springer-Verlag (1977). [11] Y. Kawamata, A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann. 261 (1982) 43-46. [12] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem, Adv. Stud. Pure Math. 10 (1987) 283-360. [13] J. Kollar, Higher direct images of dualizing sheaves, I, Ann. of Math. 123 (1986) 11-42. [14] J. Kollar and S. Mori, Birational geometry of algebraic varieties, Cambridge Univ. Press (1998). [15] S. Lee, Remarks on the pluricanonical and adjoint linear series on projective threefolds, Comm. in Algebra 27 (1999) 4459-4476. [16] S. Lee, Quartic-canonical systems on canonical threefolds of index 1, Comm. in Algebra 28 (2000) 5517-5530. [17] K. Matsuki, On pluricanonical maps for 3-folds of general type, J. Math. Soc. Japan 38 (1986) 339-359.
38 M. Chen [18] M. Reid, Young person's guide to canonical singularities, Proc. Symposia in Pure Math. 46 (1987) 345-414. [19] I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. 127(1988) 309-316. [20] E. Viehweg, Vanishing theorems, J. Reine Angew. Math. 335 (1982) 1-8. [21] P. M. H. Wilson, The pluricanonical map on varieties of general type, Bull. Lond. Math. Soc. 12 (1980) 103-107. [22] G. Xiao, Finitude de I'application bicanonique des surfaces de type general, Bull. Soc. Math. France 113 (1985) 23-51. [23] G. Xiao, L'irregularite des surfaces de type general dont le systeme canonique est compose d'un pinceau, Compositio Math. 56 (1985) 251-257.
M O R P H I S M S OF CURVES A N D T H E FUNDAMENTAL GROUP
Matthew Cushman Deephaven Capital Management, 4699 Old Ironsides Dr. Santa Clara, CA 95054, USA mcushman@alumni. uchicago. edu
We study the question of when a map between the fundamental groups of algebraic curves is induced by an underlying geometric map on the curves themselves. The quotients of the group ring by powers of the augmentation ideal J of the fundamental group of algebraic varieties are known to have additional structures (mixed Hodge structures, Galois module structures or motivic structures). These structures must be respected by any map induced by geometry. We describe how this question can be reduced to the calculation of an invariant lying in a certain Ext group, and calculate it explicitly for the mixed Hodge fundamental group module I . We show that this invariant has a if-theoretic interpretation, and give an example of a map between fundamental groups of curves module I3 which is not geometric.
1.
Introduction
Homotopy groups of algebraic varieties, and in particular the fundamental group, have long been known t o have special properties. Some of these come from comparison with the homology and cohomology groups, which are easier t o u n d e r s t a n d a n d handle. These simpler groups c a r r y various additional structures, which depend u p o n t h e field of definition k of t h e variety X a n d t h e particular definition of t h e homology or cohomology theory. For example, if X is defined over C, the simplicial cohomology groups of X(C) have a functorial mixed Hodge structure (MHS) as shown by Deligne in [3] a n d [4]. 39
40
M.
Cushman
In [6], Hain gives a method for getting an analogous structure on the fundamental group ni(X(C),x) where x £ X(C). This structure is not a MHS on the fundamental group exactly, since the fundamental group isn't generally abelian. The MHS is defined on the prounipotent completion of the group ring of 7Ti(X(C), X), defined as
]imZ[ni(X(C),x)]/r where / C Z[ni(X(C), x)] is the augmentation ideal, i.e. the kernel of the augmentation map Z[ni(X(Z),x)] -> Z. There is an analogue (which predates this theorem) of this result in Galois theory: the profinite completion of the fundamental group of a variety defined over any field (a notion which itself needs to be defined) carries an action of the absolute Galois group of the field. There is a philosophy, going back to Grothendieck, that these various homology and cohomology theories (Hodge and Galois are the examples we use here) are reflections of an underlying motivic theory which would subsume them both. There has been a lot of exciting developments in this direction recently, and several good definitions of a category of motives are now known, for example Voevodsky [13]. In [9,10], Nori defines a homology theory with values in a category £HAi(k) for k C C with many good properties. These include a universality property, which states that any reasonable homology theory on the category of varieties defined over k factors through £/HM(k), and a map from pairs of varieties to complexes in £/HM(k) which compute the homology of the pair. In [2], we show that fundamental groups of varieties defined over C also carry a motivic structure as defined in [9]. This theorem is similar to Hain's, in that the structure in not on the fundamental group itself, but on the prounipotent completion. Our proof only uses general properties of homology theory defined by Nori, and so can be translated into other theories as well. In this paper, we consider a specific problem to which this theory can be applied: when is there a morphism between two pointed curves preserving the basepoints? If there is such a geometric map (X,x) —> (Y,y), functoriality dictates that there is a map on the fundamental groups (and pro-unipotent group rings). This map will be motivic, and will preserve the Hopf algebra structures. We consider whether or not a map on fundamental groups necessarily comes from an underlying geometric map. The classical Torelli theorem states that two smooth curves with identical Hodge
Morphisms
of Curves and the Fundamental
Group
41
structures on H 1 (including polarizations) are isomorphic. Hain and Pulte [7,11] show that smooth pointed curves are determined up to abstract isomorphism by the Hodge structure on Z[TTI(X,X)]/73, with ambiguity up to two points in the case where X is hyperelliptic. We apply some homological algebra to this problem to show an equivalence between the presence of a map on fundamental groups and the vanishing of certain obstructions in Ext groups (see Theorem 4.2). Our main specific result is negative: we show that there is a map on fundamental groups modulo I3 from the punctured projective line to itself which does not lift to a geometric map. This map was suggested to us by Nori. Even this result is really a weaker statement about the Hodge theoretic fundamental group, not the motivic fundamental group itself, since morphisms and extensions in Nori's category are not well understood. The author would like to thank Madhav Nori for inspiring the problem we consider and for many useful conversations. Thanks are also due to Richard Hain for sharing his ideas and encouraging this article. Several inaccuracies in the original draft were noticed by the referee, who also pointed out that Proposition 4.1 is standard.
2. Motives in
£7iM
In [9], Nori defines a category of mixed motives for a variety defined over a field of characteristic 0. It is a "universal homology theory" for varieties defined over k, in a sense we will make precise below. One remarkable property of this category which we exploit is the existence of a well-defined functor from pairs of varieties to complexes in £'HM.{k). Definition 2.1. Fix a field k. We define the following categories: (1) V(k) is the category of pairs of algebraic varieties over k. (2) V*(fc) is the category whose objects (X, x) are varieties X defined over k together with a A;-rational base point x, and morphisms defined over k which preserve the base points. (3) V**{k) is the category whose objects are triples (X,x\,X2) where X is a variety defined over k, and x\ and X2 are basepoints. The first category is relevant to the definition of £HA/i(k), characterized by the following theorem:
which is
42
M. Cushman
Theorem 2.2 [9]. Let k be a subfield of C. There is a tensor abelian category £'HM.(k), which is determined up to equivalence by the following: (1) There is a functor H n : V(k) ->• £rlM(k) for every n € N. (2) For every triple of varieties (X, Y, Z) defined over k there are maps d : H n +i(X, Y) —> Hn(Y,Z) which induce a long exact sequence •••-»• Un+1(X,Y)
-» Hn(Y,Z)
->• H n ( J t , Z ) -»• H n ( X , y ) -+ • • •
(3) There is a faithful exact functor T : £rlM.(k) —¥ AB. The composite functor (X, Y) i->- T(Hn(X, Y)) is naturally isomorphic to the singular homology group functor (X, Y) \-+ Hn(X(C),Y(C); Z). (4) £H/A(k) (together with the data of the functors, maps and natural isomorphism in (l)-(3)) is initial with respect these properties. Prom the universal property of this category it is easy to construct a functor £rlM(k) —¥ MriS, which corresponds to sending the homology group of a variety to its mixed Hodge structure as denned in [3] and [4], as well as a functor from £7iAi(k) to the category of Gal(fc/A;)-representations on %t-modules, which corresponds to etale homology. 3. The Motivic Fundamental Group The structures which naturally appear on the homology and cohomology groups of algebraic varieties, mixed Hodge structures and Galois actions, have been revealed on suitably completed versions on the fundamental group of the variety as well. As mentioned earlier, Grotendieck [5] showed that there is an algebraic theory of the fundamental group which allows one to define a fundamental group 7r" 9 (X) for any variety X defined over an arbitrary base field. For X a variety defined over C, 7r™ 3(X) is isomorphic to the profinite completion of the topological fundamental group of X(C). More generally, if X is defined over k C C there is an exact sequence: 1 -> 7fT(X(C)) -»• Trf9(X) -». Gk -»• 1 where Gk is the absolute Galois group of fc, and TFJ" indicates the profinite completion of TTI. An element x S X(k) induces a splitting of this exact sequence, giving an action of Gk on 7rT(X(C)). Hain showed in [6] that the pro-unipotent completion of the integral group ring of the fundamental group has a Hodge structure. These results show that suitable completions
Morphisms
of Curves and the Fundamental
Group
43
of the topological fundamental group of varieties carry structures like those on the homology and cohomology groups. In [2] we show that there is a corresponding motivic theory of the fundamental group. The precise statement is Theorem 3.1. In the statement of the theorem, we use the notation 11* (X; 0:1,3:2) as shorthand notation for the "motivic version" of Z[ni(X;xi,X2)]/IkThis is to avoid confusion in the statement of the theorem itself, but in the future we will write out the whole expression "L\KI(X; XI, X2)]/Ik, which is more expressive. We use iri(X; xi, X2) to denote the space of based paths up to homotopy from x\ to X2, and / is the augmentation ideal in TJ[K\(X, X2)] which acts on the right on TTI(X; xi, ^2). Equivalently, one can take Jk\Z[jri(X; x\, X2)} where J is the augmentation ideal in TL\K\{X,XI)], which acts on this group the left. Theorem 3.1. For every k G N there is afunctorUk These functors have the following properties: (1) For every k there is a natural
: V**(k) -»•
SHMik).
transformation
Uk+1(X;x1,x2)^Uk(X;x1,x2). (2) For every embedding a : k —> C, Fa(Hk(X; x\, x2)) is naturally isomorphic to Z[ni(Xa(C);xip,X2(T)]/Ik as abelian groups. Here, Fa is as in Theorem 2.2. (3) For every integer k > 0 there are natural transformations: Uk(X;xux2)
® Uk{X;x2,x3)
->
Uk(X;xux3)}.
(4) These composition transformations are compatible with the composition of paths, the comultiplication and the inversion operator in the path space via the natural isomorphisms given in (2). This data is equivalent to giving a pro-SHM. structure on the inverse limit limZ[7Ti(X; x\, X2)]/I£1 such that all of the obvious maps are motivic, and the completed ideal I is a sub-motive. The main idea behind the proof is to apply Nori's motivic theory to a particular cosimplicial variety. This cosimplicial variety, known as the geometric cobar construction, is constructed from X in the following manner (see [12,14]):
44
M.
Cushman
XxX
m XxX
xX
inn X
xXxXxX
mnu X
xXxXxXxX
In this diagram, the up arrows are projections which omit a coordinate other than the first or last coordinate. The down arrows are hyperdiagonals, which duplicate a coordinate. A very natural way to define this cosimplicial space is as X1', where 7, is the model of the unit interval as a simplicial set (precisely, it's the functor on A* represented by the object [1] = {0,1}). This cosimplicial space has a natural map to the constant cosimplicial space XxX, which corresponds to mapping a path to its endpoints (this is the path space fibration from topology). The zeroth homology groups (regarded as a motive) of fiber of this map over a point (zi, £2) £ X x X is the motivic fundamental group. 4. Some Homological Algebra The motivic structure on Z[iri(X;xi,X2)]/Ik is functorial, so a morphism of pointed varieties (X, x) —> (Y, y) induces a morphism of motives Z[-Ki(X,x)}/Ik —>• Z[7Ti(y,y)]/Ik. This map preserves the Hopf algebra structure as well. We want to investigate the converse problem: when does such data on the fundamental group imply the existence of a morphism between the underlying varieties? We give a criterion which yields a more manageable description of the fundamental group data in terms of extension classes when the map I®k -» J fc /J fc + 1 i s surjective. The motivic fundamental group ring described above is filtered by powers of the augmentation ideal Ik. A morphism between the underlying algebraic varieties induces a morphism between the motivic group rings which respects this nitration, as well as the multiplication, comultiplication and inversion operator. Our approach is to view maps on motivic fundamental group rings as a map between the associated graded rings which
Morphisms
of Curves and the Fundamental
Group
45
lift to maps on the filtered group rings, and then study when such a lifting exists. Of course, a map between the associated graded rings does not always yield a map on the underlying filtered rings. The obstructions lie in certain Ext groups, which we now discuss. The following lemma is a standard result in homological algebra. We include an intrinsic "element-free" proof as it is important for our approach to the problem. Proposition 4.1. Let 0 ^
A
—> B —•
C
—>0
0 —» A' —+ B' —> C
—• 0
be a diagram of objects in an abelian category with exact rows. Let r\ 6 Ext (C, A) and rj' £ E x t ^ C " , ^ ' ) be the corresponding extension classes. Then there is a morphism <J>B : B —>• B' which makes the diagram commute if and only if the class 8 = <$>*cr( — 4>A*I) S Ext 1 (C, A') vanishes. Proof. The short exact sequences corresponding to the extension classes 4>QT]' and 4>A*TI are the obvious pull-back and push-forward exact sequences, respectively. We fit these into the following diagram: 0—s-
A —• lA
B I
—> C ||
0 —•
A'
— • A'
0 —•
A1
II 0 —> A'
—>0
C
—>- 0
—>. B' xc> C — *
C
—^0
I
I <£c
—>
J5'
—>• C" —>• 0
If 0 = 0, then the middle two exact sequences are isomorphic. Thus, there is a morphism A' ®A B —> B' f j c , C; composing this with the obvious maps yields a map B -* B'. Conversely, if there is a map B —> B' we can use the universal property of puUbacks to get a map B —> B' f j c , C. Then the universal property for pushforwards yields a map A' ®A B —> B' \\C, C. This is necessarily an isomorphism from the five-lemma. • We are interested in the case where we are given a morphism <)>: Z{n1(X,x)}/Ik
->
Z[iri(Y,y)]/Jk
46
M.
Cushman
which is a map of filtered Hopf algebras in the category of motives. We use J to denote the augmentation ideal of the fundamental group ring of Y to avoid confusion. We break this into a sequence of maps <j>k : / f c //* ; + 1 -4
Jk/Jk+l.
In the case oik = 0, it's easy to see that there is a canonical isomorphism J 0 / / 1 = J°/J1 = Z, and (f>o is necessarily the identity. Thus, the morphism yields no information in this case. The case of k = 1 is only moderately more interesting. Simple algebra shows that for any integral group ring Z[G], I/I2 is the abelianization of G. Thus, 1112 = 7ri(X, x) a b = H\{X, Z). The exact sequence 0 —> Hi(X,x) —> Z ^ X , ^ ] / / 2 —• Z —* 0 naturally splits (the inclusion of the basepoint x ^-> X yields a splitting), so a map on the associated graded is equivalent to a map on the filtered group ring. Thus, a map modulo I2 is equivalent to a map Hi(X,Z) —> Hi(Y,Z), which is equivalent to a map between generalized Jacobians. The case of k = 2 is more interesting. In this case we have a map of exact sequences 0 —• J 2 / / 3 —• I/I3
4-
4-
—» Hi(X,Z) —• 0
I i
0 —> J 2 / J 3 —• J / J 3 - 4 Hi(Y,Z) — • 0 Using Proposition 4.1 we see that the existence of the middle vertical morphism is equivalent to the vanishing of a certain invariant in Ext1(Hi(.X'1a:), J2/J3). We describe this in more detail in a special case in Sec. 5. If one extends this to the next stage the invariant becomes complicated. The obvious exact sequence to consider is 0 —• I3/I4
- ^ Z/7 4 —> I/13
—> 0
o —• J3 / J 4 —> JI J 4 — • J / J 3 —> o which gives an invariant in E x t 1 ( / / J 3 , J 3 / J 4 ) . Continuing on in this manner, we obtain a sequence of invariants in Ext 1 (//7 f c , Jk/Jk+1). Since k I/I gets increasingly complicated as k grows, these groups are difficult to understand. Under certain circumstances these invariants are equivalent to a invariants in the simpler groups Ext 1 ( / / J 2 , Jk/Jk+1).
Morphisms
of Curves and the Fundamental
Group
47
Proposition 4.2. Let X, Y are varieties defined over k, and let x £ X(k), y € Y(k). Let I be the augmentation ideal of R = 7,[TTI(X, X)] and J the augmentation ideal of S = Z[7ri(F, y)]. Fix n G N. Assume (1) Kom£HM(I2/In,Jn/Jn+1)=0(2) The kernel of the multiplication map <2>z i/in
i/r
i2/r+1
—-•
is generated by elements of the form aj3 <8> 7 — a ® f3j where a, 7 € J together with the elements in the subgroup J2d+e=n+i Id/In+1 <8> (3) The same holds for J C S. Then given maps R/In ->• S/Jn and / " / / " + 1 ->• Jn/Jn+^ invariant defined above lies in the image of the map Ext 1 (I/I2, Jn/Jn+1)
-> Ext 1 ( / / / " ,
as above, the
Jn/Jn+1).
Furthermore, this map is an injection. Proof. The injectivity statement follows from the long exact sequence for ExtsuM in the first argument: -s- Kom(I2/In, 1
2
n
n+1
Ext (I/I , J /J
1
)
1
n
n+1
-> Ext {I/r , J /J
2 n
)
-> Ex\}{I /I ,
Jn/Jn+1) n
-»•
J /J
n+1
)
together with the assumption that YLOTO.£-HM{I2IIn•, Jn/Jn+1) = 0. We want to show that the invariant 9 as defined in Proposition 4.1 is in the image of the bottom left map in the above exact sequence. It suffices to show the 9 maps to zero in the next term of the sequence, which is E x t f H ^ ( / 2 / / n , Jn/Jn+1). Using Proposition 4.1 again, this is equivalent to a map ip : p/In+1 —>. J2fJn+1 making the following diagram commute: 0
>• / " / / " + !
iV
0 —• Jn/Jn+!
>. / 2 / / n
¥
pjjn
li>
4^"
_ + J^/jn
_ > J2/JU
>
Q
>Q
There is a similar diagram which maps to this in which the center map does exist: 0 -> / " " I / / " ® 11 p
4 p'
_> IJ jn Q / / / n _^ J/jn-l
4p
® J / j n _> Q
4 p"
0 -> J " - 1 / . / " (g) J / J " -». J / J " ® J/J™ _• J / J " - i ® J / J " -> 0
48
M.
Cushman
This diagram is obtained by tensoring the exact sequences at the previous stage by 7/7™ and J/Jn. These exact sequences map to the ones in the previous diagram by multiplication. We want to use the center horizontal map between these two short exact sequences to construct the desired map. We can do this by assumption (2) in the statement of the proposition. D 5. The Case of P 1 \ S In this section we discuss the fundamental groups of punctured projective lines. We show that there is a map between two such lines truncated at the first nontrivial stage, 7 3 , which is not induced from an underlying geometric map. This counterexample was suggested to us by Nori, based upon heuristics from if-theory; we discuss that calculation as well and relate it to extension classes of mixed Hodge structures. Throughout this section, we fix two finite sets S,T C P1(/e), both including co. Set S = S\{oo} and T = T\{co}. We will analyze the obstructions defined above for the existence of a map X —> Y where X = P J \ 5 and Y — P 1 \ T . The most interesting case is where T has three points; an elementary argument shows that the general case can be built up from this inductively. By applying an automorphism of P 1 it suffices to consider T = {0,1, co}. We fix basepoints x 6 X(k) and y £ Y(k). We denote the kernel of the augmentation ideals in 7Ti (X, x) and it\ (Y, y) by 7 and J respectively. It is easy to see that, as abelian groups, Hi(X(C),Z) can be naturally identified with Z[S] (choose an oriented 1-cycle around each element of S). As a motive, Hi(X) is naturally isomorphic to [Z(l)] s , where Z(l) = H2(P) is the Tate motive of weight 2. Topologically, 7Ti(X(C),a;) is the free group generated by small loops around each point a £ S (which are only defined up to conjugacy). This implies that, under the isomorphism 7/7 2 = Hi(X, Z), multiplication in the fundamental group ring induces isomorphisms H1(X)®"_^7n/7n+1. Since group ring multiplication is motivic, these isomorphisms are motivic as well. In this case, a graded ring homomorphism (of abelian groups or motives) from gr7Z[7ri(X,x)} to gijZ[-Ki(Y,y)] is equivalent to a map
Morphisms of Curves and the Fundamental Group 49
Hi(X) -4 Hi(Y) (of abelian groups or motives respectively). We are interested in the question of which maps lift to map TL\K\(X,x)]/In —> Z[TTI(Y,y)}/Jn in the category of motives. By the results in the previous section, we can attack this question by calculating certain extension classes. At this point, we encounter some difficulties with the category SUM. itself. The Ext groups in the category £%A4 are not known. Even worse, there are only conjectures for these groups after tensoring with . Thus, we must retreat from out goal at this point and work within a category which we understand better. We use the category of mixed Hodge structures J^VriS. This has the advantage that the Ext groups in this category are explicitly known (see [1]). Even better, the extension groups of the Hodge theoretic fundamental group of punctured projective lines are known in terms of poly logarithms (see [7] and [8]). We begin with a review of extension classes of mixed Hodge structures Proposition 5.1 [1], Let A = (A%, F\, W^) and B be mixed Hodge structures such that, for some n, W„ = A and W^ = 0. Let C = Hom(A, B) be the mixed Hodge structure Horn, so that C% = Hom^s(Az,-Bz)- The /titrations are defined by WnC = {4>\ for all i, <j>(WiA) C Wi + „A} (similar for F). Then Ext V s ( A B) = CC/(F°C + CZ). A straightforward application of this result shows Extj^ W(S (Z(i), Z(j)) = C* if i < j . Otherwise, this group is 0. In the construction of the obstructions in Theorem 4.2, we took the difference between a pull-back extension class, and a push-forward extension class. In the category of mixed Hodge structures it is very easy to calculate the push-forward or pull-back of an extension class, as is shown by the following proposition, which we state without proof. Proposition 5.2. Let 0 -> Ai -*• Bi -> Ci -> 0 be a short exact sequence in MM.S for i = 1,2. Let &i : (Cj)c —> (J4»)C represent the corresponding extension classes ej as in Proposition 5.1. Let f : A\ —> Ai and g : C\ —> Ci be morphisms of mixed Hodge structures. Then the extension class f»e\ is represented by / c o e±, and the extension class f*e2 is represented by e% o gc.
50 M. Cushman In [7], Hain calculated the extension class of the 7 / J 3 part of the fundamental group of P 1 \ { 0 , 1 , oo}. Restating this for P 1 minus an arbitrary number of points yields the following proposition: Proposition 5.3. Let x € X = P 1 \ 5 , and let I be the augmentation ideal in the group ring Z[iri (X, x)]. The extension class of the sequence of mixed Hodge structures 0 —>• I2/I3 -¥ I/I3 -¥ III2 -¥ 0 is given by the element 0:3-2;
a3 — a2 OL2 — OJ3
(oti,a2,a3)
i-> < a - x 2 1, -1,
if ct\ = a2^
a3 ;
if cti = a3 j= a2 ; if a i ^ a2 and <*i ^ <*3 ; if a-i = a2 — a3 ,
1
under the identification of Ext (Ki(X),H.i(X)®2) with [C*]SxSxS (the first copy of S corresponds to the decompositon Hi(X) = [Z(l)] s , the second two copies correspond to the analogous decomposition Hi(X)® 2 = [Z(2)] S x S ). We can now explicitly perform the calculation to see when a map Hi(X,Z) -> Hi(Y,Z) lifts to the next level in the filtration of the Hodge theoretic fundamental group. Proposition 5.4. Let X,Y, S,T be as above. Let 4>: Hi(X,Z) -»• Hi(Y,Z) be a map of motives. Let (\jja^\a £ S,f3 s T) be the integer matrix representation of <j> regarded as a linear map [Z(l)] s —> [Z(1)] T . The invariant described in 4.2 is given by the collection of elements A^ 7//4ji 7 for a S S and /3,7 £ T, /8 7^ 7, where X and fi are given by the following equations:
A?I7=(-1)***- n
\j^~)
C€S\a
Ufa ^ (-l)* 1 " 3 . under the identification Ext 1 (Hi (X), Hi (Y)®2) = [ C * ] S x T x T .
Morphisms
of Curves and the Fundamental
Group
51
Proof. The elements A^7 (resp. /u^7) are the classes in Ext 1 (if^-X-), H2(Y)) of the push-forward (resp. pull-back) exact sequence which occurs in the proof of Proposition 4.1. These can be calculated in a straightforward manner using Propositions 5.2 and 5.3. • Corollary 5.5. The obstructions to mapping X=Fl\S are
X5M = ( - l ^ - V - H l - y)-*-o Yl
C-a
to
Y=F1\{0,1,00}
VaOVCl-^nl^O
C65\a
Proof. Just take T = {0,1} in the previous proposition.
•
This data is not enough to guarantee the existence of a geometric map (X, x) —• (Y, y), however. We give a counterexample to this below. Set Y = P 1 \ { 0 , 1 , oo}. The generalized Jacobian Jac(y) is then isomorphic to G m x G m . For any X = P i \ 5 , a map Ki(X, Z) to Hi(y, Z) induces a morphism X —>• Jac(y) = G m x G m , which may be regarded as two elements ofKi(X) = K(X)*, say / and g. These are uniquely defined if one stipulates a point x € X{k) to map to (1,1) e G m ® G m . Specifying a point y S Y(k) to map to the identity in Jac(F) yields an inclusion r\ : Y <-+ G m x G m (the Abel-Jacobi map). Explicitly, r)(z) = (z/y, (1 — z ) / ( l — y)). The pair / , g corresponds to a geometric map X —>• Y if and only if, when regarded together as a map to Jac(y), their image is contained in the image of Y. In other words, they must satisfy the equation yf + (i - y)g - 1 •
(5.1)
In particular, the iiT-theoretic product of these two elements {yf, (l — y)g} € K2(X) must vanish. However, by the bilinearity of the product, there will be many pairs / and g for which this product vanishes but which don't satisfy (5.1). In general, the tame symbol of these two elements {/, g}a at a point a G S is an element of k* = Ki(k). Taken together, these yield an element of (k*)s. Since E x t ^ ^ ( Z , Z(l)) is conjecturally k*, this is isomorphic to Extg W A ^(/ 2 /i 3 , J/J2). The obvious guess is that the element given by the tame symbol is the same as the obstruction to lifting the map to the Z[7ri(X,a;)]/7 3 stage. First, suppose that these elements do agree. By the above argument, any pair of maps f,g : X ->• G m satisfying {yf, (l-y)g} = 0 e Ki(X) will yield
52
M.
Cushman
a motivic Hopf algebra map Z[TVI(X,X)]/I3 ->• Z[iri(Y,y)]/J3. This will be induced by a geometric map if and only if yf + (1 — y)g = 1. If we start with / , g so that yf + (1 — y)g = 1, then any pair formed of powers of these goes to zero in K2(X): {ypfp, (1 - y)qgq} = 0. If we choose f,g,x,y,p,q carefully we can arrange for this second pair to satisfy {y'f, (1 — y')g) = 0 for some other point y'. As an example, take X = P 1 \ { 0 , 1 , oo}, with f,g induced by the identity map X —>• Y with the basepoints x = y = 2. In this case, f(z) = z / ( - 2 ) , g(z) = (1 - z)(l + 2) = (1 - z)/3. Taking p = 2, g = 1 and y' — 4 we find { 4 2 ( - z / 2 ) 2 , (1 - 4)(1 - z ) / ( l - (-2))} = {z 2 , - ( 1 - z)} =
{z,(-(l-z))
2
}
= {z,(l-z)2} = 2{z,l-z} = 0. These do not correspond to any geometric map, since 4(z/2) 2 + (1 — 4)(1 - z)(l - 2) = z2 + 3z + 1 ^ 1. This calculation assumes that the obstruction to lifting a map on homology to the next stage of the fundamental group in E x t g W ^ , ( / 2 / 7 3 , J / J 2 ) is the image of this element of ^ ( X ) . Since we can't describe Ext in the category £%Ai we cannot make this completely precise. We can do the calculation in Hodge theory, and compare the formula which arises with the tame symbol. We know that Ext,M?is(Z(l),Z) = C* = Ki(C), so these are indeed the same group. Proposition 5.6. Let X = P 1 \(S'Uoo) with oo £ S and Y = P 1 \ { 0 , 1 , oo}. Identify Hi(Y, Z) = Z(l) © Z(l) by choosing two small 1-cycles around 0 and 1; similarly, identify Hi(X, Z) = [Z(l)] s . Let f,g G K\(X) correspond to the map : Hi(X, Z) —>• Hi(Y,Z) as described above with respect to base points x,y. Then the obstruction given in Proposition 4.1 for lifting <j> to a map on fundamental groups based at x and y modulo I3 is given by {{yf,(i-y)9}a)aesProof. Let <j> be given by the matrix (ipap\a £ S,(3 e {0,1}) with respect to the usual basis of Hi(X, Z) and Hi(Y, Z). It is then easy to calculate the
Morphisms of Curves and the Fundamental Group 53
induced maps f,g:X^>-
Gm: z
Ces
x
- ( \ * < °
^
•«-n(Hf)
V>a
Let a g 5 . The tame symbol at a is given by (f,9)a = (-lf°°^(yff^((l
-y)g)-+<«\z=a
(
n
_ f\ V'olV'CO-V'aoV'Cl -)
This agrees with, the expression we obtained for the obstruction /-^oi/'^oi ^ Eq. (5.1). •
References [1] J. Carlson, Extensions of mixed Hodge structures, Journees Geometrie Algebrique d'Angers (1980) 107-128. [2] M. Cushman, The motivic fundamental group, Ph.D. thesis, Department of Mathematics, University of Chicago (2000). [3] P. Deligne, Theorie de Hodge. II., Inst. Hautes Etudes Sci. Publ. Math. 40 (1971) 5-57. [4] P. Deligne, Theorie de Hodge. III., Inst. Hautes Etudes Sci. Publ. Math. 44 (1974) 5-77. [5] A. Grothendieck, Seminaire de geometrie algebraique, 1960. [6] R. Hain, The de Rham homotopy theory of complex algebraic varieties. I, K-Theory 1 (1987) 271-324. [7] R. Hain, The geometry of the mixed Hodge structure on the fundamental group, Algebraic Geometry, Bowdoin (1985), (Proc, Symp. Pure Math. 46 (1987) 247-281). [8] R. Hain, Classical polylogarithms, Motives Part 2 (Proc. Symp. Pure Math. 55 (1994) 271-324). [9] M. Nori, Motives, notes, 2000. [10] M. Nori, Construcible sheaves, in Proceedings of the International Colloquium on Algebra, Arithmetic and Geometry, Tata Institute for Fundamental Research (2001). [11] M. Pulte, The fundamental group of a Riemann surface: mixed Hodge structures and algebraic cycles, Duke Math. J. 57 (1988) 721-760.
54 M. Cushman [12] J. Stallings, Quotients of the powers of the augmentation ideal in a group ring, in Knots, groups and 3-manifolds, Ann. of Math. Studies, vol. 84, Princeton University Press (1979). [13] A. Suslin V. Voevodsky and E. Priedlander, Cycles, transfers and motivic homology theories, Ann. of Math. Studies, vol. 1, Princeton University Press (2000). [14] Z. Wojtkowiak, Cosimplicial objects in algebraic geometry, in Algebraic K-theory and Algebraic Topology, Kluwer Academic Publishers (1993).
I T E R A T E D I N T E G R A L S A N D A L G E B R A I C CYCLES: EXAMPLES A N D PROSPECTS
Richard Hain* Department of Mathematics, Duke University, Durham, NC 27708-0320, USA hain@math. duke, edu
The goal of this paper is to produce evidence for a connection between the work of Kuo-Tsai Chen on iterated integrals and de Rham homotopy theory on the one hand, and the work of Wei-Liang Chow on algebraic cycles on the other. Evidence for such a profound link has been emerging steadily since the early 1980s when Carlson, Clemens and Morgan [13] and Bruno Harris [40] gave examples where the periods of non-abelian iterated integrals coincide with the periods of homologically trivial algebraic cycles. Algebraic cycles and the classical Chow groups are nowadays considered in the broader arena of motives, algebraic if-theory and higher Chow groups. This putative connection is best viewed in this larger context. Examples relating iterated integrals and motives go back to Bloch's work on the dilogarithm and regulators [9] in the mid 1970s, which was developed further by Beilinson [7] and Deligne (unpublished). Further evidence to support a connection between de Rham homotopy theory and iterated integrals includes [48,4,30,57,50,22,37,59,58,25,38,51,55,19,60,20]. Chen would have been delighted by these developments, as he believed iterated integrals and loopspaces contain non-trivial geometric information and would one day become a useful mathematical tool outside topology.
'Supported in part by grants from the National Science Foundation. 55
56
R. Hain
The paper is largely expository, beginning with an introduction to iterated integrals and Chen's de Rham theorems for loop spaces and fundamental groups. It does contain some novelties, such as the de Rham theorem for fundamental groups of smooth algebraic curves in terms of "meromorphic iterated integrals of the second kind," and the treatment of the Hodge and weight nitrations of the algebraic de Rham cohomology of loop spaces of algebraic varieties in characteristic zero. A generalization of the theorem of Carlson-Clemens-Morgan in Sec. 10 is presented, although the proof is not complete for lack of a rigorous theory of iterated integrals of currents. Even though there is no rigorous theory, iterated integrals of currents are a useful heuristic tool which illuminate the combinatorial and geometric content of iterated integrals. The development of this theory should be extremely useful for applications of de Rham theory to the study of algebraic cycles. The heuristic theory is discussed in Sec. 6. A major limitation of iterated integrals and rational homotopy theory of non-simply connected spaces is that they usually only give information about nilpotent completions of topological invariants. This is particularly limiting in many cases, such as when studying knots and moduli spaces of curves. By using iterated integrals of twisted differential forms or certain convergent infinite sums of iterated integrals, one may get beyond nilpotence. Non-nilpotent iterated integrals and their Hodge theory should emerge when studying the periods of extensions of variations of Hodge structure associated to algebraic cycles in complex algebraic manifolds, when one spreads the variety and the cycles. Some developments in the de Rham theory, which originate with a suggestion of Deligne, are surveyed in Sec. 12. Iterated integrals are the "de Rham realization" of the cosimplicial version of the cobar construction, a construction which goes back to Adams [2]. The paper ends with an exposition of the cobar construction. Logically, the paper could have begun with it, and some readers may prefer to start there. I hope that the examples in the paper will lead the reader to the conclusion, first suggested by Wojtkowiak [57], that the cosimplicial version of the cobar construction is important in algebraic geometry, and that the numerous occurrences of iterated integrals as periods of cycles and motives are not unrelated, but are the de Rham manifestation of a deeper connection between motives and the cobar construction.
Iterated Integrals and Algebraic Cycles
57
This paper complements the survey article [33], which emphasizes the fundamental group. I highly recommend Chen's Bulletin article [15]; it surveys most of his work, and contains complete proofs of many of his important theorems; it also contains a useful account of the cobar construction. Polylogarithms are discussed from the point of view of iterated integrals in [34]. There is much beautiful mathematics that connects iterated integrals to motives which is not covered in this paper. Most notable are Drinfeld's work [23], in particular his associator, which appears in the study of the motivic fundamental group of P 1 - {0,1, oo}, and the Kontsevich integral [45], which appears in the construction of Vassiliev invariants. 1. Differential Forms on Path Spaces Denote the space of piece wise smooth paths 7 : [0,1] -» X in a smooth manifold X by PX. Chen's iterated integrals can be defined using any reasonable definition of differential form on PX, such at the one used by Chen (see [15], for example). We shall denote the de Rham complex of X, PX, etc by E'{X), E*(PX), etc. We will say that a function a : N —> PX from a smooth manifold into PX is smooth if the mapping 4>a : [0,1] x N -+ X defined by (i,x) H-> a(x)(t) is piecewise smooth in the sense that there is a partition 0 = to < ti < • • • < tn-i < t„ = 1 of [0,1] such that the restriction of <j)a to each [tj_i,tj] x N is smooth.* They key features of the de Rham complex should satisfy are: (i) E'(PX) is a differential graded algebra; (ii) if N is a smooth manifold and a : N —> PX is smooth, then there is an induced homomorphism a* : E'(PX)
->
E'(N)
of differential graded algebras; a Recall that a function / : K —> K from a subset K of R ^ is smooth if there exists an open neighbourhood U of K in K.N and a smooth function g : U —> R whose restriction to K is / .
58
R. Hain
(iii) if D and Q are manifolds and D x PX —¥ Q is smooth (that is, Dx N -> Dx PX ->• Q is smooth for all smooth iV -> P X , where iV is a manifold), then there is an induced dga homomorphism
E%Q)^E'{DxPX). (iv) If D is compact oriented (possibly with boundary) of dimension n and p : Dx PX —• PX is the projection, then one has the integration over the fiber mapping p . : Ek+n(D
x PX) ->
Ek(PX)
which satisfies ptd±dpt
= (p\dD)*-
Chen's approach is particularly elementary and direct. For him, a smooth fc-form on PX is a collection w = (wa) of smooth fc-forms, indexed by the smooth mappings a : Na -» PX, where wa S Ek(Na). These are required to satisfy the following compatibility condition: if / : Na —> Np is smooth, then Wa = f*W/3 .
Exterior derivatives are defined by setting d(wa) = (dwa). Exterior products are defined similarly. The de Rham complex of PX is a differential graded algebra. This definition generalizes easily to other natural subspaces W of PX, such as loop spaces and fixed end point path spaces. Just replace PX by W and consider only those a : Na —> PX that factor through the inclusion W "-4 PX. It also generalizes to products of such W with a smooth manifold Q. To define a smooth form w on Q x W, one need specify only the wa for those smooth mappings a of the form id xa : Q x N -> Q X W. Lest this seem ad hoc, I should mention that Chen developed an elementary and efficient theory of "differentiable spaces", the category of which contains the category of smooth manifolds and smooth maps, which is closed under taking path spaces and subspaces. Each differentiable space has a natural de Rham complex which is functorial under smooth maps. The details can be found in his Bulletin article [15].
Iterated Integrals and Algebraic Cycles
59
2. Iterated Integrals This is a brief sketch of iterated integrals. I have been deliberately vague about the signs as they depend on choices of conventions which do not play a crucial role in the theory. Another reason I have omitted them in this discussion is that, by using different sign conventions from those of Chen, I believe one should be able to make the signs in many formulas conform more to standard homological conventions. Chen's sign conventions are given in Theorem 7.2 and will be used in all computations in this paper. Suppose that w-i,... twr are differential forms on X, all of positive degree. The iterated integral / W\W2 • • • Wr
is a differential form on PX of degree - r + deg w\ + deg wi -\ Up to a sign (which depends on one's conventions) / W1W2 • • • wr = 7T»0*(piu;i A p%W2 A • • • A
\- deg wr.
p*wr)
where (i) pj : Xr —> X is projection onto the jth factor, (ii) A r = {(ti,... ,tr) : 0 < ti < t2 < • • • < tr < 1} \s the time ordered form of the standard r-simplex, (iii) <$> : Ar x PX —>• Xr is the sampling map 4>(tu... , t r , 7 ) = (7(*i),7(*2),-.- ,7(M)> (iv) 7T» denotes integration over the fiber of the projection TV
: A r x PX -»• PX.
When each Wj is a 1-form, f u>i • • • wr is a function PX —> R. Its value on the path 7 : [0,1] —>• X is the time ordered integral wi---wr:= fi(ti)---fr(tr)dti---dtT, (1) Jf Jo
u>i---wr
(2)
60
R. Hain
where for a € [0,1], pa : PX -^ X is the evaluation at time a mapping 7 >->• 7(a)If W is a subspace of PX (such as a fixed end point path space, the free loop space, a pointed loop space), we shall denote the subspace of its de Rham complex generated by the restrictions of iterated integrals to it by Ch'(W) and call it the Chen complex of W. It is naturally filtered by length: Ch'0{W) C Ch\{W) C Ch\(W) C • • • C
Ch'(W),
where Ch's (W) consists of all iterated integrals that are sums of terms (2) where r < s. The standard formula it*d±dttlt
— (nld&r-)*
implies that iterated integrals are closed under exterior differentiation and that, with suitable signs (depending on one's conventions), d I w\ • • • wr = V J ±
w\ • • • dwj • • • wr
j=l rr -- ll ^ ± J.7=1 '=I
/ Wi • • • Wj-.\(Wj
A Wj+i)lVj+2
•••Wr
^
± I I w\ • • • wr_i I A p\wr ± P o ^ i A / u>2 • • • wr . This implies that each Ch's(PX), and thus each Ch*(W), is closed under exterior differentiation. The standard triangulation b of A r x A s gives the shuffle product formula Wi---Wr/\
b
/ W r + i • • • Wr+S =
^2 a£sh(r,s)
±
/ W
T h i s is
ArxAs=
(J
{(t CT (i),t ff (2),... ,t ff (r+«)):0
(3)
Iterated
Integrals
and Algebraic
Cycles
61
where sh(r, s) denotes the set of shuffles of type (r,s) — that is, those permutations a of {1, 2 , . . . ,r + s} such that a-\l)
<•••
<<j-l{r)
and cr _1 (s + 1) < <7_1(s + 2) < • • • < cr _1 (s + r). With this product, Ch'(W) is a differential graded algebra (dga). In many applications, one considers the restrictions of iterated integrals to the fixed end-point path spaces PX,VX := { 7 G PX : 7 (0) = x,j(l)
= y} .
Multiplication of paths A* • Px,yX X Py Z A —> PXZJ(. induces a map of the complex of iterated integrals0: fl*
I W\ • • • Wr
= 2_j TTl / W\ • • • Wj A 7I"2 / Wj+\
• • • Wr
where -K\ and TT2 denote the projections onto the first and second factors of Px,yX x Py,zX. The inverse mapping PXlyX -> Py,xX ,
7 h-> 7 _ 1 ,
induces the "antipode" / IDl Wl • • • W Wr r H> ±
I/ Wr • • • Wi .
The closed iterated line integrals H°(Ch°(PXtVX)) are precisely those iterated line integrals that are constant on homotopy classes of paths relative to their endpoints. When x — y, the Chen complex of PX,XX is a differential graded Hopf algebra with diagonal / W\ • • • Wr M- \ .
J c
j=1
I U>1 • • • Wj ®
J
/ W j + l • • • Wr .
J
H e r e we u s e t h e c o n v e n t i o n t h a t w h e n s = 0, J s = 1.
62 R. Hain
Its cohomology H*(Ch*(PXtXX)) is a graded Hopf algebra with antipode. Each element of H'(Ch'(PXtXX)) defines a function iti(X, x) -> R. Restricting elements of Ch'(Px,xX) to the constant loop cx at x defines a natural augmentation Ch*(Px,xX)^R. Denote its kernel by IChm(Px,xX). These are the iterated integrals on the loop space PX,XX "with trivial constant term." Of course, if one takes iterated integrals of complex-valued forms E'(X)c, then one obtains complex-valued iterated integrals. We shall denote the Chen complex of complex-valued iterated integrals by Ch*(PX)c, Ch'(PXtyX)c, etc. 3. Loop Space de Rham Theorems Chen proved many useful de Rham type theorems, (A comprehensive list can be found in Sec. 2 of [30].) In this section we present those of most immediate interest. Theorem 3.1. If X is a simply connected manifold, then integration induces a natural Hopf algebra isomorphism H'(Ch'(PXtXX))
s
H'(PX,XX;R).
This, combined with standard algebraic topology, gives a de Rham theorem for homotopy groups of simply connected manifolds. First a review of the topology: Suppose that (Z, x) is a connected, pointed topological space and that A is any coefficient ring. Consider the adjoint h* : H*(Z;A)
-> Eomz{n.(Z,z),
A)
of the Hurewicz homomorphism. An element of H' {Z\ A) is decomposable if it is in the image of the cup product mapping H>0{Z; A) ® H>0{Z; A) ->• H'(Z; A). The set of indecomposable elements of the ring H'(Z; A) is defined by QH'(Z; A) := H'(Z; A)/{the
decomposable elements} .
Iterated Integrals and Algebraic Cycles
63
Since the cohomology ring of a sphere has no decomposables, the kernel of hl contains the decomposable elements of H'(Z; A), and therefore induces a mapping e : QH*(Z;A)
->• Hom z (7r.(Z,z), A).
Typically, this mapping is far from being an isomorphism. However, if A is a field of characteristic zero and Z is a connected H-sp&ce, then e is an isomorphism. (This is a Theorem of Cartan and Serre — cf. [47].) When X is simply connected, PX>XX is a connected H-space. Chen's de Rham theorem and the Cartan-Serre Theorem imply that integration induces an isomorphism QHj{Px,xX;R)
-=> Hom(7T j (P IiX ,c x ),R) S Hom(7r i+1 (X,:r) I R)
for each j . There is a canonical subcomplex QCh'(Px>xX) of the Chen complex of PXiXX, which is isomorphic to the indecomposable iterated integrals (see [29]) and whose cohomology is QH'(Ch'(Px<xX)). This and Chen's de Rham Theorem above then yield the following de Rham Theorem for homotopy groups of simply connected spaces: Theorem 3.2. Integration induces an isomorphism H'{QCh\Px>xX))
^ > Hom(7r.(X,a;),R)
of degree +1 of graded vector spaces. Both sides of the display in this theorem are naturally "Lie coalgebras." It is not difficult to show that the integration isomorphism respects this structure. We now turn our attention to non-simply connected spaces. The augmentation e : Z^i (X, x) —> Z of the integral group ring of 7i"i (X, x) is defined by taking each element of the fundamental group to 1. The augmentation ideal J is the kernel of the augmentation e. The diagonal mapping 7Ti {X, x) —>• 7i"i (X, x) x 7Ti (X, x) induces a coproduct A : ZTTJ (X, x) -» ZTT! (X, x) ZT^ (X,
x).
The most direct statement of Chen's de Rham theorem for the fundamental group is:
64
R. Hain
T h e o r e m 3.3 ( C h e n [14]). The integration pairing H°(Ch'(PXiXX))®Ziri(X,x)
-> C
is a pairing of Hopf algebras under which H°(Ch°(PXtXX)) js+i y^g induced -mapping
annihilates
-»• Hom z (Z7r 1 (X, a ; )/J s + 1 ,C)
H\Ch's{PX)XX)) is an isomorphism.
An elementary proof is given in [33]. An equivalent statement, more amenable to generalization, will be given in Sec. 12. A second version uses the J-adic completion RTTI(X,
xy := limR7Ti(X, x)/Js s
of M.7Ti(X,x). It is a complete Hopf algebra (cf. [52, Appendix A]), with diagonal
A:
Rni(X,xy^Riri(X,x)~®Rni(X,x)~
induced by that of Rni(X, x). Corollary 3.4. If H1(X;R) a natural homomorphism
is finite dimensional, then integration induces
RTr1(X,xr^Rom(H°(Ch's(PXiXX)),R) of complete Hopf algebras. Of course, each of these theorems holds with complex coefficients if we begin with complex-valued forms. R e m a r k 3.5. It is tempting to think that one can extend Chen's loop space de Rham theorem, or its homotopy version, to a de Rham theorem for higher homotopy groups of non-simply connected spaces. While this is true for "nilpotent spaces" (which include Lie groups), it is most definitely not true for most non-simply connected spaces that one meets in day-to-day life. In fact, Example 7.5 shows that there is unlikely to be any reasonable statement. One point we wish to make in this paper, however, is that in arithmetic and algebraic geometry, the cohomology of iterated integrals is intrinsic and may be a more interesting and geometric invariant of a complex algebraic variety than its higher homotopy groups or loop space cohomology.
Iterated Integrals and Algebraic Cycles
65
4. Multi-Valued Functions In this section, we give several examples of interesting multi-valued functions that can be obtained by integrating closed iterated line integrals. Although elementary, these examples are reflections of the relationship between iterated integrals and periods of certain canonical variations of mixed Hodge structure. The following result is easily proved by pulling back to the universal covering of X and using the definition (1) of iterated line integrals. Proposition 4.1. All closed iterated line integrals of length < 2 on PX,VX are of the form ^
ajk
jk+ / £ + a constant,
j,k
where each (f>j is a closed 1-form on X, the ajk are scalars, and £ is a 1-form on X satisfying d£ + ^2 ajk <j)j A k = 0 . j,k
A relatively closed iterated integral is an element of Ch'(PX) that is closed on Px,yX for all x, y £ X. The iterated line integrals given by the previous result are relatively closed. Multi-valued functions can be constructed by integrating relatively closed iterated integrals. For example, suppose that X is a Riemann surface and that wi and w^ are holomorphic differentials on X. Then /
W1W2
is closed on each PXtVX. This means that for any fixed point x0 £ the function
r
X H-> /
W\W2
Jx0 is a multi-valued function on X. It is easily seen to be holomorphic. Example 4.2. If X = P ^ C ) - {0,1,00}, then dz dz 0 1- z z
I
X,
66
R.
Hain
is a multi-valued holomorphic function on X. In fact, it is Euler's dilogarithm, whose principal branch in the unit disk is denned by
ln2(x) = Y, ^2 • n>l
More generally, the fc-logarithm
n>l
can be expressed as the length k iterated integral fc-i
L 0
dz dz 1— z z
dz z
From this integral expression, it is clear that lnj; can be analytically continued to a multi-valued function on C — {0,1}. Note that CC0> the v a l u e of the Riemann zeta function at an integer k > 1, is lnfc(l). More information about iterated integrals and polylogarithms can be found in [34]. More generally, the multiple polylogarithms v-^ • • • ixn)
i->mi,...,mn\pli
'•=
x k l x1*2 • • • x k n
£__/ J , " n J,m 2 tjn^ 0
>
\xj\
< *• '
and their special values, Zagier's multiple zeta values C( n ii • • • ,nm), be expressed as iterated integrals. For example, r
/ M
\_ '
/ ,(x,!/) f dy dx 7(o,o) \ l - y l - x
d(xy) / dy l-xy\l-y
dx 1-x
can
dx\\ x))'
This expression defines a well defined multi-valued function on C 2 - {(x, y) : xy(l - x)(l - y)(l - xy) ± 0} as the relation dy 1- y
dx 1— x
dixy) f dy dx dx\ 1 — xy \ 1 — 2/ 1 — a; x )
holds in the rational 2-forms on C 2 . Formulas for all multiple polylogarithms and other properties can be found in Zhao's paper [60].
Iterated Integrals and Algebraic Cycles 67
Closed iterated integrals that involve antiholomorphic 1-forms can also yield multi-valued holomorphic functions. Proposition 4.3. Suppose that X is a complex manifold and w\,... are holomorphic 1-forms. If £ is a (1,0) form such that d£ + J ! a i k W j
,wn
/\wk=0,
then the multi-valued function fX
F :xt-¥'^2ajk
px
/ WjWk + / £ Jx0
AU
Jx„
is well defined and holomorphic. Proof. Since each Wj is holomorphic,
dF(x) = Y^ajk(
wA Wk + £ •
Since £ has type (1, 0), dF also has type (1, 0), which implies that F is holomorphic. • Example 4.4. Take X to be a punctured elliptic curve E — C / A - { 0 } . The point of this example is to show that the logarithm of the associated theta function 6(z) is a twice iterated integral. We may assume that A = Z + Z T where r has positive imaginary part. Denote the homology classes of the images of the intervals [0, 1] and [0, r] by a and /3, respectively. These form a symplectic basis of H\(E,Z). The normalized abelian differential is dz and dz = a* + r/3* from which it follows that
dz/\dz = 2ilmr £ H2(E,C) = C . The multi-valued differential /x = (z — z)dz satisfies /J,(Z + 1) = fj.(z) and fj,(z + T) = fi(z) + 1i \va.rdz . On the other hand, the corresponding theta function 9(z) := 9(z, T) satisfies 0(z + 1) = 6(z) and 6(z + T)= exp(-i7rr -
2niz)6(z).
Thus
y( z
+ 1 =
)
y( z )
and
-j(z + T) = -g(z)-2™dz-
68
R. Hain
It follows t h a t
£:=
Im Td9
•fi(z)
is a single-valued differential on E — {0} of type (1, 0) having a logarithmic singularity at 0 which satisfies dz A dz + d£ = 0 in E2(E - {0}). (In fact, these properties characterize it up to a multiple of dz.) It follows that f dzdz + £ is relatively closed, so that x i-»
/
dzdz + £
J xa
is a multi-valued holomorphic function on E — {0}. Applying the definition of iterated integrals yields log0(z) = log0(:zo) +
Imr
/ (dzdz + £) -(z(x) - z{x0))2 + -(z(x0) - z{x0))< Jx0 where z{x) = JQ dz. This example generalizes easily to theta functions of several variables by replacing E by a principally polarized abelian variety A and E — {0} by A — 0 , where 0 is its theta divisor. Remark 4.5. This example can be developed further along the lines of Beilinson [7] and Deligne's approach to the dilogarithm. (An exposition of this, from the point of view of iterated integrals, can be found in [34].) Set 1 G
C
C\ C
/l and
F°G =
0
C
0\ 0
1/ 1/ The Lie algebra g of G is the Lie algebra of nilpotent upper triangular matrices. Let
£E\E-{0})®S.
d
I n the language of Beilinson and Levin [8], / dz is the elliptic logarithm for E, and log 6 is the elliptic dilogarithm.
Iterated Integrals and Algebraic Cycles
69
This form is integrable: dw + wAw — 0. It follows that the iterated integral
(I T =
0
Jdz 1
Jdzdz + J^\ Jdz
0
1
/
is relatively closed. (See [15] or [33].) It therefore defines a homomorphism e:m(E-{0},xo)^G, 7->, which is the monodromy representation of the flat connection on (E— {0}) x G defined by w. There is thus a generalized Abel-Jacobi mapping v : E - {0} ->•
T\G/F°G
that takes x to (T, 7), where 7 is any path in E — {0} from x0 to x. It is holomorphic as can be seen directly using the formulas above. Denote the center
/1
0
cr
0 1 0 \o of G by Z. The quotient T\G/(F°G and the corresponding projection
0
1
• Z) is naturally isomorphic to E itself
T\G/F°G
-> £;
is a holomorphic C*-bundle. The formulas in the previous example show that the section u above is a non-zero multiple of the section 6 of the line bundle L —> E associated to the divisor 0 g E:
L - (0-section) —=—*-
E - {0}
^E
T\G/F°G
— » - T\G/(F°G
• Z)
This has an interpretation in terms of variations of MHS, which can be extracted from [39]. The construction given here of the second albanese mapping is special case of the direct construction given in [32]. There is an analogous construction with a similar interpretation where the pair (E, 0) is replaced by an abelian variety and its theta divisor (A, 9 ) .
70
R. Hain
5. Harmonic Volume Bruno Harris [40] was the first to explicitly combine Hodge theory and and (non-abelian) iterated integrals to obtain periods of algebraic cycles. Suppose that C is a compact Riemann surface of genus 3 or more. Choose any base point x0 £ C. Suppose that L\, L2, L3 are three disjoint, nonseparating simple closed curves on C. Let fa be the harmonic representative of the Poincare dual of Lj,j = l, 2, 3. Since the curves are pairwise disjoint, the product of any two of the fas vanishes in cohomology. Thus there are 1-forms <j>jk such that d<j>jk + fa A k = 0
and 4>jk is orthogonal to the d-closed forms. These two conditions characterize the fak. By Proposition 4.1, the iterated line integral /
fak + fak
is closed in Ch'(PXo<XoC). Choose loops 7,- (j = 1, 2, 3), based at x0 and that are freely homotopic to the Lj. Harris sets I{Li,L2,L3)=
fafa
+ fa2-
J13
He shows that this integral is independent of the choice of the base point x0 and that -T(£<7(i), L„(2), £
modi?
Jx„
If the coordinates in R 3 are (z\,Z2,zz), then $*dzj = <j)j for j = 1, 2, 3. Since H2(T, Z) is spanned by the dzj A dzk and since / dzj Adzk=
i*
/ fa A fa = 0,
Jc
Iterated Integrals and Algebraic Cycles
71
the image of C is homologous to zero in T. One can therefore find a 3-chain r in T such that dT = $»C. Since V is only well denned up to a 3-cycle, the volume of T is only well denned mod Z. Harris's first main result is: Theorem 5.1. The volume ofT is congruent to I{Li,L2,L^)
mod Z.
By an elementary computation, the span in A3H\(C, Z) of classes L\ A Li A L3 is the kernel K of the mapping A3#i(C,Z)->tfi(C,Z) defined by a A b A c i-> (a • b)c + (b • c)a + (c • a)b. The harmonic volume / thus determines a point in the compact torus Hom(ff,R/Z). There is a lot more to this story — it has a deep relationship to the algebraic cycle Cx — C~ in the jacobian of C. This is best explained in terms of the Hodge theory of the operator 8 rather than d. This shall be sketched in Sec. 9. 6. Iterated Integrals of Currents There is no rigorous theory of iterated integrals of currents, although such a theory would be useful provided it is not too technical. The theory of iterated integrals makes essential use of the algebra structure of the de Rham complex. The problem one encounters when trying to develop a theory of iterated integrals of currents is that products of currents are only denned when the currents being multiplied (intersected) are sufficiently smooth (or sufficiently transverse). Nonetheless, this point of view is useful, even it if is not rigorous. The paper [28] was an attempt at making these ideas rigorous and using them to study links. E x a m p l e 6 . 1 . In this example X is the unit interval. Suppose that 0,1,0,2,-•• ,ar are distinct points in the interior of the unit interval. Set Wj = 6(t — a.j)dt, where 6(t) denotes the Dirac delta function supported at * = 0. Let 7 : [0,1] ->• X = [0,1] be the identity path. Recall that A r is the time ordered simplex A r = {(ti, *2, • • • , U) : 0 < tt < • • • < tr < 1} .
72
R. Hain
By definition, / wiw2 • • -wr = /
JAr
J-t =
/
S(ti - ai)8(t2 - a 2 ) • • • 6(tr — ar) dtidt2 •• dtr
<5(oi,...,a P )(*li-" ,U) dtidt2
• • • dtr
Since the a.,- are distinct numbers satisfying 0 < a,- < 1, /•
| 1,
if ai < a2 < • • • < ar
/ Wllt)2 • ••Wr = J-f
E x a m p l e 6.2. More generally, suppose that Hi,... ,Hr are real hypersurfaces in a manifold X, each with oriented (and thus trivial) normal bundle. Suppose that 7 e PX is transverse to the union of the Hj — that is, the endpoints of 7 do not lie in the union of the Hj and 7 does not pass through any singularity of their union. We can regard each Hj as a current, which we shall denote by Wj. For such a path 7 which is transverse to Hj, Wj = (Hj • 7) := the intersection number of Hj with 7 . /' 7 For simplicity, suppose that 7 passes through each Hj at most once, at time t — CLJ, say. Then
i'wj
=ej6j{t-aj)
where tj is 1 if 7 passes through Hj positively at time a,j, and —1 if it passes through negatively. By the previous example, i"1 I eie 2 • • • e r , if 0,1 < 0,2 < • • • < ar / w\W2 • • • wr = J 7*u>i 7*u>i• • -7*w • • 7*uv r = \ h Jo 10, otherwise. This formula can be used to give heuristic proofs of many basic properties of iterated line integrals, such as the shuffle product formula, the antipode, the coproduct and the differential. For example, suppose that u i i , . . . , wr are 1-currents corresponding to oriented lines in the plane and that a and /3 are composable paths that are transverse to the union of the supports of the Wj. (See Fig. 1.) Note that J wi- • -wr is non-zero on a/3 if and only if there is an i such that a passes through wi,... ,Wi in order and (3 passes through
Iterated Integrals and Algebraic Cycles
/•j
W£
W:
Fig. 1.
Wi+i,...
W,'1+1
W,r-1
73
Wr
Pointwise product of iterated integrals.
, wr in order. In this case
/
Ja0 a/3
Wi • • • W Wrr == WI
Wi • •• -Wi • Wi / j t f j + l • • • Wr I/ Wi-
Ja Ja
JB Jp
= 53 I wWjr--••w-Wi i \I A-n J a
Wj+l
••
-Wr
JB
as all the terms in the sum are zero except when j = i. Examples using higher iterated integrals also exist. The simplest I know of is a proof of the formula for the Hopf invariant of a mapping / : g 4 n _ 1 —> S2n. It is a nice exercise, using the definition of iterated integrals, to show directly that if / : S'4"'-1 —>. S2n is smooth and p and q are distinct regular values of / , then
/
6P6q,f)
is the linking number of f~1{p) and / _ 1 ( ? ) m SAn~l. Here 6X denotes the 2n-current associated to x G S2n. This formula is equivalent to J.H.C. Whitehead's integral formula for the Hopf invariant [56] and Chen's version of it [15, p. 848].
74
R. Hain
6.1. First
steps
Suppose that T\,... ,Tr are closed submanifolds of X (possibly with boundary), where Tj has codimension dj. Denote the d,-current determined by Vj by Sj. Suppose that N is a compact manifold and that a : N —> PX is smooth. We shall say that a is transverse to J Si • • • Sr if the mapping a:ArxN^Xr,
((*i,... ,tr), u) n- ( a ( u ) ( t i ) , . . . ,a(t.)(t r ))
is transverse to the submanifold T := Ti x • • • x r r of Xr. That is, the restriction of a to each stratum of A r x N is transverse to each boundary stratum of T. If N is has dimension — r + d\ -\ +dr and a is transverse to T, then we can evaluate the iterated integral /
<$i • • • <* r
on a. This transversality condition is satisfied in each of the examples above. 7. The Reduced Bar Construction Chen discovered that the iterated integrals on a smooth manifold have a purely algebraic description [15,16]. This algebraic description is an important technical tool as it allows the computation of various spectral sequences one obtains from iterated integrals, applications to Hodge theory, and it facilitates the algebraic de Rham theory of iterated integrals for varieties over arbitrary algebraically closed fields, (cf. Sec. 13.) This algebraic description is expressed in terms of the reduced bar construction, a variant of the more standard bar construction [24], which is dual to Adam's cobar construction [2]. Chen's version has the useful property that it generates no elements of negative degree when applied to a non-negatively graded dga with elements of degree zero, unlike the standard version of the bar construction. In this section, we use Chen's conventions for iterated integrals. In particular, our description of the reduced bar construction gives a precise formula for the exterior derivative of iterated integrals. Suppose that A' is a differential graded algebra (hereafter denoted dga) and that M' and N* are complexes which are modules over A*. That is, the structure maps A'<sM'-¥
M* and A' N' -)• N'
Iterated Integrals and Algebraic Cycles
75
are chain maps. We shall suppose that A', M' and N* are all non-negatively graded. Denote the subcomplex of A' consisting of elements of positive degree by A>0. The (reduced) bar construction B(M,A',N) is denned as follows. The underlying graded vector space is a quotient of the graded vector space : = 0 M ' ® (A>0[l]®r) ® N* .
T(M',A',N')
s
Following convention m®a\ • • • ar ®n G T(M', A', N') will be denoted by m[ai\ • • • \ar]n. To obtain the vector space underlying the bar construction, we impose the relations m[dg|ai| • • • \ar]n = m[gai\ • • • \aT]n — m • g[a\\ • • • \ar]n; m[ai\ • • • \ai\dg\ai+i\ • • • \ar]n = m[ai\ • • • \ai\gai+\\ • • • \ar]n — m[ai\ • • • |ajsr|a; + i| • • • |a r ]«,
1 < i < s;
m[ai| • • • |ar| g • n — m • g ®1. Here each a* G A>0, g G A0, m G M', n G JV*, and r is a positive integer. Define an endomorphism J of each graded vector space by J : v i-> (-l)desvv T h e differential is denned as d = dM ® I T ® lw +
JM
® ^B <8>
^-N
+
JM
® JT ® &N + dc •
Here T denotes the tensor algebra on A >0 [1], ds is denned by dB[ai\ • • • \ar] = ^T (-lYlJai\
• • • \Jai-i\da.i\ai+i\
• • • \ar]
l
+ ^2
(-l)i+1[Ja1\---\Jai-i\JaiAai+1\ai+2\---\ar}
(4)
l
and dc is denned by d c ^ [ a i | • • • \cLr]n = (—l) r Jm[Jai\... \Jar-\]ar The reduced bar construction B(M*,A',N*) AT ® Nm = BQ(M', A', Nm) C BX(M\
A',N')
• n — Jm- aifal • • • \ar]n. has a standard nitration c B2(M', A\ N') C • • •
by subcomplexes, which is called the bar filtration. The subspace B3(M',A',N')
76
R. Hain
is defined to be the span of those m[ai\ • • • \ar]n with r < s. When A' has connected homology (i.e., H°(A') = R), the corresponding (second quadrant) spectral sequence, which is called the Eilenberg-Moore spectral sequence (EMss), has E\ term E'3'* = [M* H>0(A')®S
® N']*.
A proof can be found in [16]. This computation has the following useful consequence: L e m m a 7 . 1 . Suppose that A' is a dga and M' and N' are right and left A*-modules, where j = 1,2. Suppose that }A '• A\» —¥ A*, is a dga homomorphism and fM : Ml -¥ M'2 a n d fN
: TV* -»• JV2«
are chain maps compatible with the the actions of A* and A\ and f. If fA, fM and fN induce isomorphisms on homology, then so do the induced mappings Bs{Ml,A\,N'x)
-j- B,{M5, Al,Ntf
and B{M{, A\,N{)
-»• B(M^,A'2,Nfi
.
When A', M' and N* are commutative dgas (in the graded sense), and when the A'-module structure on M* and N* is determined by dga homomorphism A' ->• M* and A' -¥ N', B(M',A',N') is also a commutative dga. The multiplication is given by the shuffle product: (m'[ai| • • • |a r ]n') A (m"[a r + i| • • • \ar+s]n") = ^2
±(m'Am")K(i)|---|aCT(r+s)](n'An").
s/i(r,s)
It is important to note that the shuffle product does not commute with the differential when A* is not commutative. Many complexes of iterated integrals may be described in terms of reduced bar constructions of suitable triples. Here we give just one example — the iterated integrals on PX:VX. A more complete list of such descriptions can be found in [15] and [30, Sec. 2]. Suppose that X is a manifold and that xo and Xi are points of X. Evaluating at Xj, we obtain an augmentation Cj : E*(X) —¥ R for j = 0, 1. Suppose that A' is a sub dga of E'(M) and that both augmentations restrict to non-trivial homomorphisms 6j : A' -¥ R. We can take M' and
Iterated Integrals and Algebraic Cycles
N' both to be R, where the action is given form the corresponding bar construction Define Ch'(PXo>XlX;A') to be the spanned by those iterated integrals J w\...
77
by eo and ei, respectively. Now B(R,A',W). subcomplex of Ch'(PXo>XlX) wr where each Wj € A*.
Theorem 7.2. Suppose that X is connected. If H°(A*) = E and the natural map Hl{A') —> H1(X;M) induced by the inclusion of A' into E'(X) is injective, then the natural mapping B[K,A
, K) —> Ch (PXOtXlX; A'),
[wi\---\wr]
^•Jw1---wr
is a well defined isomorphism of differential graded algebras. This and Adams' work [2] are the basic ingredients in the proof of the loop space de Rham theorem, Theorem 3.1. The previous result has many useful consequences, such as: Corollary 7.3. If X is connected and A' is a sub dga of E'(X) for which the inclusion A' <->• E'(X) induces an isomorphism on homology, then the inclusion Ch'(Px>yX;A')
->
Ch*(Px,yX)
induces an isomorphism on homology. This is proved using the previous two results. It has many uses, such as in the next example, where it simplifies computations, and in Hodge theory, where one takes A' to be the subcomplex of C°° logarithmic forms when X is the complement of a normal crossings divisor in a complex projective algebraic manifold. Example 7.4. A nice application of the results so far is to compute the loop space cohomology H'(Px<xSn;M.) and real homotopy groups n •jrn(S , x) <8> K of the n-sphere (n > 2). This computation is classical. The first thing to do is to replace the de Rham complex of Sn by a sub dga A' which is as small as possible, but which computes the cohomology of the sphere. To do this, choose an n-form w whose integral over Sn is 1 and take A' to be the dga consisting of the constant functions and the constant multiplies of w. By Corollary 7.3, the iterated integrals constructed from elements of A' compute the cohomology of Sn. But these are all linear
78 R. Hain combinations of 6m : = I w • -w,
m > 0.
E a c h of these is closed, a n d no linear combination of t h e m is exact. It follows t h a t
fR0 mi j = m ( n - l ) , I0,
otherwise.
The ring structure is also easily determined using the shuffle product formula (3). When n is odd, we have 9™ = mWm; when n is even #1 A 0 2 m = 0 2 m + l ,
#1 A 0 2 m + l = °> a
n d
#2 A 0 2 m = ( m + l ) # 2 m + 2 •
Applying Theorem 3.2 we have: (R, 7Tj(5 n ,a;)(8)R= < R , [o,
j=n, j = 2n-
1 and n even,
otherwise.
Example 7.5. This example illustrates the limits of the ability of iterated integrals to compute homotopy groups.*5 The main point is that there are continuous maps / : X —> Y between spaces that induce an isomorphism on cohomology, but not on homotopy. Properties of the bar construction (cf. Lemma 7.1) imply that for such / the mapping / * = H'{Ch'(Pf(x)t/(x)Y))
-»•
H'(Ch'(Px,xX))
is an isomorphism. The prototype of such continuous functions is the mapping X —> X+ from a connected topological space X, with the property that ni(X,x) is perfect, to X+, Quillen's plus construction. By a standard trick, one can extend de Rham theory (and hence iterated integrals) to arbitrary topological spaces.f In this setting, one can take a "Minimal models do no better or worse. If (X, x) is a pointed topological space with minimal model M'x, there is a canonical Lie coalgebra isomorphism QM^ Si H*(Ch,(Px,xX)). This follows from [17, Sec. 3]. 'Basically, one replaces a space by the simplicial set consisting of its singular chains. This is canonically weak homotopy equivalent to the original space. One then can work with the Thorn-Whitney de Rham complex of this simplicial set. It computes the cohomology of the space and is functorial under continuous maps.
Iterated Integrals and Algebraic Cycles
79
perfect group T and consider the mapping cj>:BY-> BT+ from the classifying space of T to its plus construction. This mapping induces an isomorphism on homology, and therefore a quasi-isomorphism :E'(Br+)^E,(BT).
cj>*
This induces, by Lemma 7.1, an isomorphism H'(Ch*{PXiXBT+))
->
H'{Ch'{Px<xBT)).
Since the universal covering of B r is contractible, PXiXBT is a disjoint union of contractible sets indexed by the elements of I\ On the other hand, BT+ is a simply connected H-spa.ce, the loop space de Rham theorem holds for it. It follows that QHj(Ch'{Px,xBT))
£* H o m ( 7 r J + 1 ( B r + , x ) , E ) .
In particular, take T = SL{%), a perfect group. Prom Borel's work [10], we know that R,
j = 3 mod 4 ,
0, otherwise. ! For those who would prefer an example with manifolds, one can approximate BSL(Z) by a finite skeleton of BSLn(Z) for some n > 3 or take T to be a mapping class group in genus g > 3. 7.1. An integral
version
Suppose that X is a topological space and that R is a ring. Each point x of X induces an augmentation ex : S'(X;R) —> R on the R-valued singular chain complex of X. If x, y € X, we have augmentations ex : S%X;R)
-»• R and ey :
S'{X;R)^R,
which give R two structures as a module over the singular cochains. We can thus form the reduced bar construction B(R, S'(X; R), R). The following result, which will be further elaborated in Section 14 and is proved using Adams cobar construction, is needed to put an integral structure on the cohomology of Ch*s{Px,yX), regardless of whether X is simply connected or not.
80
R. Hain
Proposition 7.6 (Chen [15]). For all s > 0, there are canonical isomorphisms H'{BS(Z,
S'{X, Z), Z)) ® z R = # ' ( ^ ( R , S'{X, R),R))
It is very important to note that the naive mapping B{I) : B{%E'(X),W)
-*
B(R,S'{X;W),R),
\wi\ • • • \wr] !->• [J(wi)| • • • \I(wr)] induced by the integration mapping / : E*(X) —> S'(X;W), is no* a chain mapping. This is because / is not an algebra homomorphism (except in trivial cases), which implies that B(I) is not, in general, a chain mapping. 8. Exact Sequences The algebraic description of iterated integrals gives rise to several exact sequences useful in topology and Hodge theory. We shall concentrate on iterated integrals of length < 2 as this is the first interesting case — Hk{ICh\{Px>xX)) is just Hk+1(X;R). Lemma 8.1. If X is a connected manifold, then the sequence 0 -4 QH2d-\X;R)
H2d-2{ICh'2(Px>xX))
->
-» \H>0{X;R)®2]2d
^
H2d(X;R)
-»• QH2d(X;M)
-»• 0
is exact. This sequence has a natural Z-form and exactness holds over Z as well. Sketch of Proof. By the algebraic description of iterated integrals given in the previous section, the sequence 0 -»• ICh\{PXtXX)
-»• (E>0{X)/dE\X))®2
-> IChl(PXtXX)
-»• 0
is exact. This gives rise to a long exact sequence. The formula for the differential and the identification of ICh\(Px>xX) with E>°(X)/dE°(X) imply that the connecting homomorphism is the cup product [H>0(X;R)®2)k
^
Hk(X;R).
The integrality statement follows from Proposition 7.6 using the integral version of the reduced bar construction. •
Iterated Integrals and Algebraic Cycles
81
Combining it with the de Rham Theorems yields the following two results. For the first, note that the function 7Ti(X, X) -> J/J2 ,
7
^ ( 7 - 1) + J2
is a homomorphism and induces an isomorphism
H1(X,Z)^J/J2. Here J denotes the augmentation ideal of Zni(X, x). Corollary 8.2. For all connected manifolds X, the sequence 0 -> H\X;
Z) -> H o m ( J / J 3 , Z) ^ H^X;
Z)®2 ^
tf2(X;
Z)
is exact. TTie mapping tp is dual to the multiplication mapping # i ( X ; Z ) ® 2 S (J/J2)®2
-i- J / J 3 .
The analogue of this in the simply connected case is: Corollary 8.3. If X is simply connected, then the sequences 0 -> H3(X; Q) -»• Hom(7r3(X, a;), Q) -> S 2 tf 2 (X; Q) ^
#4(X)
and 0 -> tf3(X; Z) -»• H2{PX,XX; Z) -> tf2(X; Z)®2 ^
tf4(X;
Z)
are exact. 9. Hodge Theory Just as in the case of ordinary cohomology, Chen's de Rham theory is much more powerful when combined with Hodge theory, and is especially fertile when applied to problems in algebraic geometry. The Hodge theory of iterated integrals is best formalized in terms of Deligne's mixed Hodge theory. I will not review Deligne's theory here, but (at the peril of satisfying nobody) will attempt to present the ideas in a way that will make sense both the novice and the expert. More details can be found in [30,31,33,39].
82
R. Hain
9.1. The Riemannian
case
In the classical case, the Hodge theorem asserts that every de Rham cohomology class on a compact riemannian manifold has a unique harmonic representative which depends, in general, on the metric. If X is a compact riemannian manifold, then every element of H'(Ch's(PXiVX)) has a natural representative, which I shall call "harmonic" even though I do not know if it is annihilated by any kind of laplacian on PXtVX. This is illustrated in the case s = 2. Every closed iterated integral of length < 2 is of the form WjWk + £
(5)
where j,k
The iterated integral (5) is defined to be harmonic if each Wj is harmonic and £ is co-closed (i.e., orthogonal to the closed forms). This definition generalizes to iterated integrals of arbitrary length. Classical harmonic theory on X and the Eilenberg-Moore spectral sequence imply that every element of H'(Ch'(PX!yX)) has a unique harmonic representative. Harris's work on harmonic volume (cf. Sec. 5) is a particularly nice application of harmonic iterated integrals. 9.2. The Kdhler
case
This naive picture generalizes to the case when X is compact Kahler. In classical Hodge theory, certain aspects of the Hodge Theorem, such as the Hodge decomposition of the cohomology, are independent of the metric. Similar statements hold for iterated integrals: specifically, H'(Ch'(Px,yX)) has a natural mixed Hodge structure, the key ingredient of which is the Hodge filtration, whose definition we now recall. The Hodge filtration E'{X)c
= F°E'{X)
D F^'iX)
D E'(X)
D•••
of the de Rham complex of a complex manifold is defined by .
££. ^^
_ Jdifferential forms for which each term ofl \each local expression has at least p dz's J
Iterated Integrals and Algebraic Cycles 83
where Ep'q(X) denotes the differential forms of type (p, q) on X. Each FPE* (X) is closed under exterior differentiation. A fundamental consequence of the Hodge theorem for compact Kahler manifolds is the following: Proposition 9.1. If X is a compact Kahler manifold, then the mapping H'(FPE'(X))^H'(X;C) is injective and has image FpH'{X):=Q)Hs't(X). In other words, every class in FPH'(X) is represented by a class in P p F E'(X), and if w 6 F E'(X) is exact in E'{X)C, one can find ip £ FpEm(X) such that dip = w. The Hodge filtration extends naturally to complex-valued iterated integrals: FpCh*s{PXiyX) is the span of
where r < s and uij £ FPiE'{X), where pi + • • • + pr > p. The weight filtration is simply the filtration by length: WmCh\Px,yX)
=
Ch'm(PXtVX).
The Hodge theory of iterated integrals for compact Kahler manifolds is summarized in the following result. A sketch of a proof can be found in [31] and a complete proof in [30]. Theorem 9.2. If X is a compact Kahler manifold, then Ch'(PXtVX), endowed with the Hodge and weight filtrations above, is a mixed Hodge complex. In particular. (i) H*{FpCh'(PXtVX))^H*{Ch'{Px,yX)c) is injective; (ii) H*(Ch'(PXtVX)) has a natural mixed Hodge structure with Hodge and weight filtrations defined by FpH'(Ch*(Px,yX))
=
H9(FpCh'(Px,yX))
and WmHk{Ch'{Px,yX))
= im{Hk{Ch'm_k{Px,vX)
-»•
Hk(Ch'(Px,yX))}
84
R. Hain
If H1(X;Q) point x.
= 0, this mixed Hodge structure is independent of the base
This theorem generalizes to all complex algebraic manifolds (using logarithmic forms) and to singular complex algebraic varieties (using simplicial methods). Details can be found in [30]. Corollary 9.3. If X is a complex algebraic manifold, then H2d~2 {ICh\(PXiXXy) has a canonical mixed Hodge structure defined over Z and the sequence 0 ->• QH2d-\X)
->
-»• [H>0(X)®2]2d
^
H2d~2(ICh'2(Px,xX)) H2d(X)
-> QH2d(X)
-> 0.
is exact in the category of TL mixed Hodge structures. The minimal model approach to the Hodge theory for complex algebraic manifolds was developed by Morgan in [48]. From the point of view of Hodge theory, iterated integrals have the advantage that they provide a rigid invariant on which to do Hodge theory, whereas the minimal model of a manifold is unique only up to a homotopy class of isomorphisms, which makes the task of putting a mixed Hodge structure on a minimal model more difficult. Chen's theory is also better suited to studying the non-trivial role of the base point x € X in the theory, which is particularly important when studying the Hodge theory of the fundamental group. On the other hand, minimal models (and other non-rigid models) are an essential tool in understanding how Hodge theory restricts fundamental groups and homotopy types of complex algebraic varieties, as is illustrated by Morgan's remarkable examples in [48]. 10. Applications to Algebraic Cycles Recall that a Hodge structure H of weight m consists of a finitely generated abelian group Hi and a bigrading
Hc= ® fl™ p+q=m
of He = Hz®€- by complex subspaces satisfying Hv,q = HI
Iterated Integrals and Algebraic Cycles 85
compact Kahler manifold. Its dual, Hm(X), is a Hodge structure of weight —m.g For and integer d, the Tate twist H(d) of H is denned to be the Hodge structure with the same underlying lattice Hz but whose bigrading has been reindexed: H(d)p'q
= Hp+d'q+d
.
Equivalently, H(d) is the tensor product of H with the 1-dimensional Hodge structure Z(d) of weight —2d. The category of Hodge structures is abelian, and closed under tensor products and taking duals. 10.1. Intermediate
jacobians
and Griffiths'
construction
The dth intermediate jacobian of a compact Kahler manifold X is defined by Jd{X) := J(H2d+1(X)(-d))
S
Eom(Fd+1H2d+1(X),C)/H2d+1(X-Z).
It is a compact, complex torus. For example, Jo(X) is the albanese of X and JdimX-i(X) is Pic X, the group of isomorphism classes of topologically trivial holomorphic line bundles over X. Suppose Z is an algebraic rf-cycle in X, that is trivial in homology. We can write Z as the boundary of a (2d+l)-chain T, which determines a point /rin Hom(Fd+1H2d+1(X),C)/H2d+1(X;Z)
S
Jd(X)
by integration: / : [w] M- I w where w <E Fd+1E2d+1(X). This mapping is well defined by Stokes' Theorem, Proposition 9.1, and because Fd+1E2d(Z) = 0. It is also convenient to define Jd(X) = Jn-d(X), where n is the complex dimension of X. sJust define Hm(X)-P<~q to be the dual of Hp'i(X).
86
R. Hain
10.2. Extensions
of mixed Hodge
structures
In this paragraph, we review some elementary facts about extensions of mixed Hodge structures (MHS). Complete details can be found in [12]. Suppose that A and B are Hodge structures of weights n and m, respectively, and that O-^B-iBA^^O
(6)
is an exact sequence of mixed Hodge structures. In concrete terms, this means: (i) there is an exact sequence 0 -> Bz -+ Ei 4 A z -» 0
(7)
of finitely generated abelian groups; (ii) Ec := Ez ® C has a filtration • • • D FPE D Fp+lE Bc n FpE = 0
B*'m— and
D • • • satisfying
7r(F p £) = 0
As'n~s.
When Az is torsion free, the extension (6) determines an element tp of the complex torus J(Hom(A, B)). This is done as follows: by the property of 7r, there is a section SF : Ac —> Ec that preservers the Hodge filtration; since Az is torsion free, there is an integral section sg : A% —>• Fz of 7r. The coset ip of sp — s% in J(Hom(A, B)) is independent of the choices S F and si. 10.3. The Theorem of
Carlson-Clemens-Morgan
This is the first example in which periods of (non-abelian) homotopy groups were related to algebraic cycles. Here X is a simply connected projective manifold. By Corollaries 8.3 and 9.3, the sequence 0 ->• H3{X; Z)/(torsion) -»• Hom(7r 3 (X), Z) -> K - • 0
(8)
is an extension of Z-mixed Hodge structures,11 where where K is the kernel of the cup product S2H2(X;Z)^H\X;Z). h T h e integral statement is proved in [13] — however, H3(X;Z) be torsion free.
is implicitly assumed to
Iterated Integrals and Algebraic Cycles
87
Denote the class of a divisor D in the Neron-Severi group NS(X)
:= {group of divisors in X}/(homological equivalence)
of X by [D]. If the codimension 2 cycle
Z:=J2n3kDjnDk is homologically trivial, where the rijk are integers and the Dj divisors, then
is an integral Hodge class of type (2, 2) in K. Pulling back the extension (8) along the mapping Z(—2) —> K that takes 1 to Z, we obtain an extension 0 -> H3(X; Z(2)) -»• Ez -> Z -»• 0 of mixed Hodge structures. This determines a point
=
J2(X).
On the other hand, the homologically trivial cycle Z determines a point $(2)eJ2(I). Theorem 10.1 (Carlson-Clemens-Morgan). The points z and$z J2(X) are equal. 10.4. The Harris-Pulte
of
Theorem
Pulte [50] reworked Harris' work on harmonic volume using the Hodge theory of 8 and the language of mixed Hodge theory. Suppose that C is a compact Riemann surface and that x € X. Corollaries 8.2 and 9.3 imply that the sequence 0 ->. H\C)
-> H°(ICh'2(Px,xC))
-» K -> 0
is exact in the category of Z-mixed Hodge structures, where K is the kernel of the cup product Hl{C) i? 1 (C) -> H2(C). It therefore determines an element mx of
JQiomiK,!!1^))).
88 R. Hain An element of H o m ^ , Hl{C)) can be computed using the recipe in the previous paragraph. For example, if u := ^2iaik
[w-?'l ® [*fc]
e
^
j,k
where each Wj is holomorphic, then, by Proposition 9.1, there is £ G F 1 ^ 1 ^ ) such that d£ + ^2aikWj
AiBk=0.
j,k
Thus
/
J2*ik Wjwk + £ ) G FlH°{ICh°2{Px,xC)). J.k
(sp can be chosen so that this is sp(u).) The value of the extension class ^ on u is represented by the homomorphism H\(C) —> C obtained by evaluating this integral on loops based at x representing a basis of H\(C; Z). (Full details can be found in [33].) These integrals are examples of the 8 analogues of those considered by Harris. On the other hand, one has the algebraic 1-cycles Cx := {[z] -[x]:zeC}
and C~ := {[a:] - [z] : z G C}
in the jacobian J a c C of C. These share the same homology class, so the algebraic cycle Zx :— Cx — Cx is homologically trivial and determines a point vx G J i ( J a c C ) = J ( A 3 F i ( C ) ( - l ) ) . The linear mapping A 3 # i ( C ) -» K* Hi(C) defined by a Ab /\c>-^ -> I
Jaxb
u> ® c + | u 4 / J
K
Jbxc
u> ® a+
I
u> ®b JcXa
J
is an injective morphism of Hodge structures, and induces an injection A : J^JacC) ^
J(Hom(tf,#1(C))).
Theorem 10.2 (Harris-Pulte [40,50]). With notation as above, vx = 2A(mx).
Iterated Integrals and Algebraic Cycles
89
Remark 10.3. If C is hyperelliptic and x and y are two distinct Weierstrass points, the mixed Hodge structure on J(C — {y},x)/J3 is of order 2. In this case Colombo [19] constructs an extension of Z by the primitive part PH2(Ja.cC; Z) of H2(3&cC) from the MHS on J(C - {y}, x)/J4 and shows that it is the class of the Collino cycle [18], an element of the Bloch higher Chow group CH9(Ja,cC, 1). This example shows that the MHS on 7Ti(C — {y}, x) of a hyperelliptic curve contains information about the extensions associated to elements of higher if-groups, {K\ in this case), not just .Rol l . Green's Observation and Conjecture Mark Green (unpublished) has given an interpretation of the CarlsonClemens-Morgan Theorem. He also suggested a general picture relating the Hodge theory of homotopy groups to intersections of cycles. In this section, we briefly describe Green's ideas, then state and sketch a proof of a modified version. 11.1. Green's
interpretation
If one wants to understand the product CHa{X)
® CH\X)
->
CHa+b(X)
the first thing one may look at is: CHa{X)
® CH\X)
-> TH2a+2b(X; Z(o + b))
After this, one may consider: ker{CHa(X)
CHb{X)
- • Ja+b(X)
-> r # 2 a + 2 6 ( X ; Z(o + 6))}
= E x t ^ Z , H2a+™-\X;
Z(o + b))).
(9)
What Green observed is that when X is a simply connected projective manifold and a = b = 1, the result of Carlson-Clemens-Morgan implies this mapping is determined by the class e(X)
eExt1Hodee(K,H3(X;Z(2)))
of the extension 0 - • H3{X, Z(2)) -> Hom(7r3(X), Z(2)) -»• K -»• 0, where K is the kernel of the cup product S2H2(X,Z(1)) This works as follows: since the diagram
->•
H4(X,Z(2)).
90
R. Hain
CH1{X)^CHl{X)
H2{X;Z{1))®H2{X;Z{1))
•
I
2
CH {X)
4
•
I
H (X;Z(2))
commutes, there is a natural mapping 1ax{CH\X)
® CH\X)
-> TH4(X,Z(2))}
-> n f ( 2 ) .
The result of Carlson-Clemens-Morgan implies that cupping this homomorphism with e(X) gives the mapping (9). He went on to conjecture that all the "crossover mappings" (9) — more generally, all crossover mappings associated to the standard conjectured filtration of the Chow groups of X — are similarly described by cupping with extensions one obtains from the mixed Hodge structure on homotopy groups of X. In his thesis [5], Archava proves that a conjecture of Green and Griffiths implies the analogue of Green's conjecture in the case where the category of mixed Hodge structures is replaced by the category of arithmetic Hodge structures of Green and Griffiths [26]. 11.2. Iterated
integrals
and crossover
mappings
This section proposes a generalization of the theorem of Carlson-ClemensMorgan to cycles of all codimensions and also to algebraic manifolds which may be neither compact nor simply connected. Suppose that X is a complex algebraic manifold. By Corollary 9.3, the sequence 0 -> QH2d-\X) -> [H>0(X)®2}2d
->• ^
H2d-2(IChl(PXiXX)) H2d(X)
-> QH2d(X)
->• 0.
(10) ev
is exact in the category of Z-mixed Hodge structures. Denote by H (X; Z) the sum of the even integral cohomology groups of X of positive degree. Let Kev = kev{Hev(X; Z)®2 -4 Hev{X; Z)} . This underlies a graded Z-Hodge structure. We can pull the extension (10) back along Kev —• K to obtain a new extension 0 -> QH2d~\X;
Z) -> E -> Kev ->• 0
(11)
Iterated Integrals and Algebraic Cycles
91
which underlies an extension of MHS, which can be seen to be independent of the choice of the basepoint x. There is a natural mapping
ker < ^
CHa{X)
® CH\X)
-s- TH2d(X, Z(d)) l ^
VKM(Q
.
a-\-b—d a,b>0
This, the quotient mapping H*(X) determine a homomorphism
$:ker^ J ]
—> QH'(X),
and the extension (11)
CHa(X)®CHb(X)^H2d(X,Z(d))
a,b>0
^E^odz^QH2d-\X-1{d))). The following, if proven, will generalize the theorem of Carlson, Clemens and Morgan. Conjecture 11.1. If X is a quasi-projective complex algebraic manifold, the mapping $ equals the composition of the crossover mapping (9) with the quotient mapping Jd{X) -» J(QH2d+1(X)(d)). Heuristic Argument. By resolution of singularities, we may suppose that the quasi-projective algebraic manifold X is of the form X — D, where X is a complex projective manifold and D is a normal crossings divisor. Suppose that Z\,... , Zm are proper algebraic subvarieties of X of positive codimensions c\,... ,cm, respectively. By the moving lemma, we may move them within their rational equivalence classes so that they all meet properly. Suppose that the rijk are integers and that the cycle
j,k
is homologically trivial in X of pure codimension d. The basic idea of the argument is easy. The extension class associated to W is the difference sp(W) — sj,(W) mod Fd of Hodge and integral lifts of the class W := 5 > , f c [ Z , ] ® [Zk] G K2d to W2dH1d-1 (ICh'2(Px,xX)).
92 R. Hain Suppose that Wj € Eci'C:i{X) is a smooth form representing the Poincare dual of the closure of Zj in X. Since W is homologically trivial, there is a form £ e FdW1E2d-1{X\ogD) satisfying d£ = ^njkWj Awk (cf. [30, 1.3.2.8]).' It follows that Y^rijk 3,k
f WjWk + inti e
FdW2dH2d-2(ICh'2(PXtXX)),
J
which we take as the Hodge lift sp{W) of W. In the integral version, we shall use King's theory of logarithmic currents [43,44]. We would like to take the integral lift of W to be
*z(W) •= [J2nik
6 5k
i
- { r £ W2dH2d-2(ICh'2(Px,xX))z ,
(12)
where T is a chain of codimension 2d — 1 whose boundary is W, and dj is the integration current defined by Zj. To make this argument precise, one has to show that si(W) makes sense. Assume this. The final task is to compute the extension data. Denote the complex of currents on X by D*(X), and King's complex of log currents for (X, D) by D'(XlogD). These have natural Hodge and weight nitrations. There is a log current i>j G Fc> WiD^iXlog
D)
such that dipj = Wj — Sj. Using the formula for the differential, we have: / (ujjWk - SjSk) — -d / (tpjdk - Sjtpk + i>jdipk) - j (ipj A 5k + 6j Atpk+ tpj A dipk) = — I (V'j A 6k + Sj A ipk + tpj A dipk) mod exact forms Combing this with the relations d£ = 2_]njk Wj A Wk and ddr = — VJrtjk Sj A 8k 3,k j,k 'The < there should be an equals.
Iterated Integrals and Algebraic Cycles
93
we have, modulo exact forms, sF(W) - sz(W)
= f{i + <5r) - J2nik
I®!
A J
fc + SJ A ^ + 4>j A d^fc)
mod i7"* + exact forms, • / which is the desired result. The deficiency in this argument is that the theory of iterated integrals of currents is not rigorous. To make this argument rigorous, it would be sufficient to show that there is a complex of chains whose elements are transverse to s%(W), on which sz(W) takes integral values, and that computes the integral structure on H'(ICh\(Px,xX)). One possible way to approach this is to triangulate X so that D, each Zj and T are subcomplexes, and then to obtain the cycles that give the integral structure from some analogue of Adams-Hilton construction [3] associated to the dual cell decomposition. So far, I have not been able to make this work. This argument suggests that it is the Hodge theory of iterated integrals (or more generally, the cosimplicial cobar construction) rather than homotopy groups which determines periods associated to algebraic cycles, as this result holds even when the loop space de Rham theorem and rational homotopy theory fail. It would be interesting to have an example of an acyclic complex projective manifold where $ is non-trivial to illustrate this point. This argument also applies in the relative case where the variety X and the cycles are defined and flat over a smooth base S. In this case, the map <3? will take values in EXt^(s)(Zs>.RM-1/.Z;C(d)) where f : X —¥ S and Hodge(S) denotes the category of admissible variations of mixed Hodge structure over S. This can be seen using results from [30, Part II] and [39]. By combining this with the standard technique of spreading a variety defined over a subfield of C, one should get elements of the Hodge realization of motivic cohomology as considered in [6], for example. 12. Beyond N i l p o t e n c e The applicability of Chen's de Rham theory (equivalently, rational homotopy theory) is limited by nilpotence. Using ordinary iterated line integrals,
94
R. Hain
one can only separate those elements of -K\ (X, x) that can be separated by homomorphisms from TT\ (X, x) to a group of unipotent upper triangular matrices. If the first Betti number 61 (X) of X is zero, all such homomorphisms are trivial, and iterated line integrals cannot separate any elements of Ki{X, x) from the identity. If bi(X) = 1, then the image of all such homomorphisms is abelian, and iterated line integrals can separate only those elements that are distinct in Hi(X;W). Thus, in order to apply de Rham theory to the study of moduli spaces of curves and mapping class groups (61 (X) — 0) or knot groups (&i(-X") = 1), for example, iterated integrals need to be generalized. Before explaining two ways of doing this we shall restate Chen's de Rham theorem for the fundamental group in a form suitable for generalization. First recall the definition of unipotent (also known as Malcev) completion. A unipotent group is a Lie group that can be realized as a closed subgroup of the group of a group of unipotent upper triangular matrices. (That is, upper triangular matrices with l's on the diagonal.) Unipotent groups are necessarily algebraic groups as the exponential map from the Lie algebra of strictly upper triangular matrices to the group of unipotent upper triangular matrices is a polynomial bijection.-" Suppose that V is a discrete group. A homomorphism p from T to a unipotent group U is Zariski dense if there is no proper unipotent subgroup of U that contains the image of p. The set of Zariski dense unipotent representations p : r —> Up forms an inverse system. The unipotent completion of r is the inverse limit of all such representations; it is a homomorphism from T into the prounipotent group U{Y)
:=]imUp.
Every homomorphism T —>• U from T to a unipotent group factors through the natural homomorpism T —>• U(T). The coordinate ring of U(T) is, by definition, the direct limit of the coordinate rings of the Up: 0{U(Y))=YimO{Up).
JHere and below, I shall be vague about the field F of definition of the group. It will always be either R or C. Also, I will not distinguish between the algebraic group and its group of F-rational points.
Iterated Integrals and Algebraic Cycles
95
It is isomorphic to the Hopf algebra of matrix entries / : T —> R of all unipotent representations of I\ The following statement is equivalent to the statement of Chen's de Rham theorem for the fundamental group given in Sec. 3. Theorem 12.1. If X is a connected manifold, then integration induces a Hopf algebra isomorphism 0(U{m{X,x)))
S
H°{Ch'{Px,xX)).
One recovers the unipotent completion of iri(X,x) as Specif 0 (Ch' (PXtXX)). The homomorphism ni(X,x) —*• U(TTI(X,X)) takes the homotopy class of the loop 7 to the maximal ideal of iterated integrals that vanish on it. 12.1. Relative
unipotent
completion
Deligne suggested the following generalization of unipotent completion, which is itself a generalization of the idea of the algebraic envelope of a discrete group defined by Hochschild and Mostow [41, Sec. 4]. Suppose that S is a reductive algebraic group. (That is, an affine algebraic group, all of whose finite dimensional representations are completely reducible, such as SLn, GLn, 0(n), G m , ) Suppose that T is a discrete group as above and that p : Y —> S is a Zariski dense homomorphism. Similar to the construction of the unipotent completion of T, one can construct a proalgebraic group £ ( I \ p), which is an extension i4U(r,p)4g(r,/,)As4i of S by a prounipotent group, and a homomorphism T —>• Q(Y,p) whose composition with p is p. Every homomorphism from T into an algebraic group G that is an extension of 5 by a unipotent group, and for which the composite r —> G —> S is p, factors through the natural homomorphism T^G(T,p). The homomorphism T —> Q(T,p) is called the completion ofT relative to p. When S is trivial, the relative completion reduces to classical unipotent completion described above. The definition of iterated integrals can be generalized to more general forms to compute the coordinate rings of relative completions of fundamental groups. Suppose now that T — TCI(X,X), where X is a connected manifold. The representation p determines a flat principal 5-bundle, P —> X,
96
R. Hain
together with an identification of the fiber over x with S. One can then consider the corresponding (infinite dimensional) bundle 0(P) —> X whose fiber over y G X is the coordinate ring of the fiber of P over y. This is a flat bundle of R-algebras. One can, consider the dga E'(X,0(P)) of 5-finite differential forms on X with coefficients in O(P). In [35], Chen's definition of iterated integrals is extended to such forms. The iterated integrals of degree 0 are, as before, functions PX,XX —> M. Two augmentations 6:E*(X,0{P))^0(S)
and
e : E'(X,0(P))
-» R
are obtained by restricting forms to the fiber S over x and to the identity 1 S S in this fiber. These, give 0{S) and R structures of modules over E'(X, 0(P)). One can then form the bar construction B(R,E*(X,0(P)),0(S)). This maps to the complex of iterated integrals of elements of E'(X,
0(P)).
Theorem 12.2. Integration of iterated integrals induces a natural isomorphism H°(B(R, E'(X, 0(P)), 0(S))) <* OiGfa(X,
x),p)).
The corresponding Hodge theory is developed in [35]. It is used in [36] to give an explicit presentation of the completion of mapping class groups Tg with respect to the standard homomorphism Tg —> Spg to the symplectic group given by the action of Tg on the first homology of a genus g surface when g > 6. One disadvantage of the generalization sketched above is that these generalized iterated integrals, being constructed from differential forms with values in a flat vector bundle, are not so easy to work with. A more direct and concrete approach is possible in the solvable case. 12.2. Solvable iterated
integrals
In his senior thesis, Carl Miller [46] considers the solvable case. Here it is best to take the ground field to be C. The reductive group is a diagonalizable algebraic group: S = (C*)fc x M d l x • • • x fidm .
Iterated Integrals and Algebraic Cycles
97
He defines exponential iterated line integrals, which are certain convergent infinite sums of standard iterated line integrals of the type that occur as matrix entries of solvable representations of fundamental groups. Exponential iterated line integrals are linear combinations of iterated line integrals of the form /
e5°wieSlw2eS3
• • • e6"-1
wneSn
• So u>\ d\ • • • di w2 S2 • • • S2 • • • <S„_i • • • 6n-i
wn5n---Sn
kj>0J
where <5o, • • • ,Sn,wi,... , wn are all 1-forms. This sum converges absolutely when evaluated on any path. The terminology and notation derive from the easily verified fact that exp I w — > k>0
/ w...
w .
J
"l
Theorem 12.3 (Miller). Suppose that X is a connected manifold and p : TTI(X,X)
—> S
is a Zariski dense representation to a diagonalizable S is the Zariski closure of the "semisimplification" of the Alexander module of K. 13. Algebraic Iterated Integrals A standard tool in the study of algebraic varieties over any field is algebraic de Rham theory, which originates in the theory of Riemann surfaces and
98 R. Hain
was generalized by Grothendieck [27] among others. This algebraic de Rham theory extends to iterated integrals and several approaches will be presented in this section. I will begin with the most elementary and progress to the abstract, but powerful, approach of Wojtkowiak [57]. 13.1. Iterated
integrals
of the second
kind
The historical roots of algebraic de Rham cohomology lie in the classical result regarding differentials of the second kind on a compact Riemann surface. Recall that a meromorphic 1-form w o n a compact Riemann surface X is of the second kind if it has zero residue at each point. Alternatively, w is of the second kind if the value of fw on each loop in X- {singularities of w} depends only on the class of the loop in H\(X). A classical result asserts that there is a natural isomorphism j
^ {meromorphic differentials of the second kind aa.X} {differentials of meromorphic functions}
This can be generalized to iterated integrals. Suppose that X is a compact Riemann surface and that S is a finite subset. An iterated line integral of the second kind on X — S is an iterated integral
HYjaiJ r<s\I\ =
-
j Wi< "
u ' Wlr l
)
where ai £ C and each Wj is a meromorphic differential on X, with the property that its value on each path in X that avoids the singularities of all Wj depends only on its homotopy class (relative to its endpoints) in X — S. Example 13.1 (cf. [33, p. 260]). We will assume that S is empty (the case where S is non empty is simpler). Suppose that w\,... ,wn are differentials of the second kind on X and that ajk 6 C. Since differentials of the second kind are locally (in the complex topology) the exterior derivative of a meromorphic function, for each point x € U we can find a function fj, meromorphic at x, such that dfj = Wj about x. Define rjk(x)
=
Resz=x[fj(z)wk(z)}.
Since Wk is of the second kind, changing fj by a constant will not change rjk(x). If
^2^2ajkrjk(x) = 0, x€X
j,k
Iterated Integrals and Algebraic Cycles 99
then there is a meromorphic differential u on X (which can be taken to be of the third kind) such that Res 2=a; u(z) =
-22ajkrjk(x).
The iterated integral ^2ajk j,k
/ WjWk+ J
/U J
is of the second kind. This can be seen by noting that the integrand of this integral near x is Y^ajkfj(z)wk(z)
+ u{z),
j,k
which has zero residue at x. Equivalently, the pullback of the integrand of this iterated integral to the universal covering of X — S is of the second kind. Theorem 13.2. If X is a compact Riemann surface, S a finite subset of X, and 1 < s < oo, then, for all x, y € X — S, integration induces a natural isomorphism TjOf r
QW \ ~ J ^ e
se
* °f iterated integrals of the\
H {Chs(^Xty{X - b))c) = {second kind
ofkngth
< s 0nx-s] •
Proof. This is just an algebraic version of the proof of Chen's -K\ de Rham theorem given in [33, Sec. 4]. Familiarity with that proof will be assumed. I will just make those additional points necessary to prove this variant. Set U = X — S. Suppose that s < oo. We consider the truncated group ring CTTI(U,x)/J3+1 to be a 7i"i(U,x)-module via right multiplication. Let Es —• U be the corresponding flat bundle. This is a holomorphic vector bundle with a flat holomorphic connection. It is filtered by the flat subbundles corresponding to the filtration C-KX{U,x)IJs+l
D J/Js+1
D • • • D J3/Js+1
D0
of Ciri(U, x)/Js+1 by right -K\{U, a:)-submodules. Denote the corresponding filtration of E by Es = E°s D El D • • O Ess D 0 .
100
R. Hain
By the calculation in [33, Prop. 4.2], each of the bundles El/El+1 has trivial monodromy, so that each El has unipotent monodromy. By the results of [21], each of the bundles El has a canonical extension El to X. These satisfy: (i) each El is a subbundle of Es := E°; (ii) the connection on Es extends to a meromorphic connection on Es which restricts to a meromorphic connection on each of the El; (iii) the connection on each of the bundles El/El+1 is trivial over X. (Take El = El when S is empty.) The following lemma implies that there are meromorphic trivializations of each Es compatible with all of the projections • Es -> E.-i
->
>E0=Ox
and where the induced trivializations of each graded quotient of each Es is flat. Moreover, we can arrange for all of the singularities of the trivialization k to lie in any prescribed non-empty finite subset T of X. The connection form u>3 of Es with respect to this trivialization thus satisfies us G {meromorphic 1-forms on X} ® J - 1 End(C7r1(C7, with values in the linear endomorphisms of Cni(U, the filtration Cm(U, x)/Js+1
D J/Js+1
X)/JS+1
D • • • 3 J"/Js+1
x)/Js+1) that preserve
2 0
and act trivially on its graded quotients. This connection is thus nilpotent. Note that, even though u>3 may have poles in U, the connection given by u)s has trivial monodromy about each point of U. This is the key point in the proof; it implies that the transport [33, Sec. 2] 1+
us+
usvs H
1-
us---us
is an EndC7ri(C/, a;)/J s+1 -valued iterated integral of the second kind on U. Its matrix entries are iterated integrals of the second kind. k
A meromorphic trivialization <j> : E —> C>J)J is singular at x if either cf> has a pole at x or if the determinant of <j> vanishes at x.
Iterated Integrals and Algebraic Cycles
101
The result when x = y now follows as in the proof of [33, Sec. 4]. The case when x ^ y is easily deduced from the case x = y. The result for s = oo is obtained by taking direct limits over s using the fact that OJS is the image of us+i under the projection {meromorphic 1-forms on X} J - 1 End(Ciri(U,
x)/Js+2)
-> {meromorphic 1-forms on X} J - 1 End(C?ri (U, x)/Jsa+1).
•
L e m m a 13.3. Suppose that
is an extension of holomorphic vector bundles over a compact Riemann surface X. IfT is a non-empty subset ofX, there is a meromorphic splitting of p which is holomorphic outside T. Proof. Set f = Hom(^",C»x)- Riemann-Roch implies that HX{X,T{*T)) = 0, where f{*T) is defined to be the sheaf of meromorphic sections of J7 that are holomorphic outside T. It follows from obstruction theory for extensions of vector bundles that the sequence has a meromorphic splitting that is holomorphic on X — T. • Remark 13.4. Note that if S is non-empty, the proof shows that the algebraic iterated line integrals built out of meromorphic forms that are holomorphic on X - S equals H°(Ch'(Px,y(X - S))c). Since X - S is affine, this is a very special case of Theorem 13.5 in the next paragraph, a consequence of Grothendieck's algebraic de Rham Theorem. The result above can also be used to show that if X is a smooth curve defined over a subfield F of C, then H°(Chm{Px,v{X - S))c) has a canonical F-form — namely that consisting of those meromorphic differentials of the second kind on X — S that are defined over F. It would be interesting and useful to have a description of the Hodge and weight filtrations on H°(Ch*(Px^y(X — S))c), possibly in terms of some kind of pole filtration, as one has for cohomology. 13.2. Grothendieck's integrals
theorem
and its analogues
for
iterated
Suppose that X is a variety over a field F of characteristic zero. Denote the sheaf of Kahler differentials of X over F by Cl'x/F. Denote its global sections
102
R. Hain
over X by H°(QX,F). It is a commutative differential graded algebra over F. When F = C and X is smooth, every algebraic differential w € H°(ilx,c) *s a holomorphic differential on X. The corresponding mapping H°(QX,c) —> E' (X)c is a dga homomorphism. Theorem 13.5. If X is a complex affine manifold, then natural homomorphism H-(H°(WX/C))^H'(X;C) is a ring isomorphism. Note that Theorem 13.2 is a consequence of this when 5 is non-empty and S = T. If F C C and X is denned over F, then H°(QX/F) ®F C 9* fi"°(n^/c). One important consequence of Grothendieck's theorem is that if F is a subfield of C, then H'(X(C);
C) S
ff'(fl°(n£/F))
®F C.
That is, the de Rham cohomology of the complex manifold -X"(C) has a natural F structure which is functorial with respect to morphisms of affine manifolds over F. This can be generalized to arbitrary smooth varieties over F by taking hypercohomology. Define the algebraic de Rham cohomology of X by
HhR(X) = m'(x,crx/F). As above, if F is a subfield of C, then the ordinary de Rham cohomology of X(C) has a natural .F-structure: H'(X(C);C)^H'DR(X)®FC. Using the classical Hodge theorem, one can show that if X is also projective, the Hodge filtration
FpHm(X(C)) := 0ff s ' m - s (X(Q) is obtained from a natural Hodge nitration H'DR{X) = F0H'DR(X)
D FlH'DR{X)
D F2H'DR(X)
of the algebraic de Rham cohomology by tensoring by C.
D•••
Iterated Integrals and Algebraic Cycles
103
This can be extended to iterated integrals on affine manifolds in the obvious way. For an affine manifold X over F and F-rational points x, y € X(F), define the algebraic iterated integrals on Px,yX by HbR(P*,yX)
=
H'(B(F,H°{tTx/F),F))
where F is viewed as a module over H°(il^,F) via the two augmentations induced by x and y.1 It follows from Corollary 7.3 and Grothendieck's theorem above that if F is a subfield of C, there is a canonical isomorphism H'DR{Px,yX)
®F C £* H'(Ch'(PXtVX(C))).
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When X is not affine, one can replace X by a smooth affine hypercovering U, -¥ X and apply the methods of [30, Sec. 5] or Navarro [49] (see below) to construct a commutative dga A'(U9) over F with the property that when tensored with C over F, it is naturally quasi-isomorphic to E'(X)c- One can then define HjjR(Px,yX) to be the cohomology of the corresponding bar construction as above. It will give a natural F-form of H'(Ch'(PXiyX(C))). However, in this general case, it is better to use Wojtkowiak's approach, which is explained in the next paragraph. 13.3. Wojtkowiak's
approach
The most functorial way to approach algebraic de Rham theory of iterated integrals on varieties is via the works of Navarro [49] and Wojtkowiak [57]. This approach has been used in the works of Shiho [51] and Kim-Hain [42] on the crystalline version of unipotent completion. Suppose that D is a normal crossing divisor in a smooth complete variety X, both defined over a field F of characteristic zero. Set X = X — D and denote the inclusion X «-• X by j . One then has the sheaf of logarithmic differentials Cl*y.(\ogD) on X, which is quasi-isomorphic to j*F. For a continuous map / : U —> V between topological spaces, Navarro [49] has constructed a functor M^wf* from the category of complexes of sheaves on U to the category of complexes of sheaves on V with many wonderful properties. Among them: 'This is not an unreasonable definition, but one should recall that that when X is not simply connected and F = C, the de Rham theorem may not hold as we have seen in Example 7.5.
104
(i)
R. Hain
takes sheaves of commutative dgas on U to sheaves of commutative dgas on V; (ii) if V is a point, then the global sections TR^w f*Qu of R^vy f*Qu is Sullivan's rational de Rham complex of U; (iii) My^/* takes quasi-isomorphisms to quasi-isomorphisms; (iv) it induces the usual i?/» from the derived category of bounded complex of sheaves on U to the bounded derived category of sheaves on V. RJW/*
For convenience, we denote the global sections TlRyjy of R^w an arbitrary topological space Z, define A'(Z)
Dv
^TW- ^ o r
= RfoyQz •
This is the Thorn-Whitney-Sullivan de Rham complex of Z. Its cohomology is naturally isomorphic to H'(Z;Q). In the present situation, we can assign the commutative differential graded algebra L'{X,D):=WrwQ.'x{\ogD) to (X,D), where we are viewing X as a topological space in the Zariski topology. This dga is natural in the pair (X,D). If x, y are ^-rational points of X, there are natural augmentations L'(X,D) -4 F. We can therefore use them to form the bar construction B(F,L'(X,D),F). Following Wojtkowiak"1 [57], we define H'DR(Px,yX)
= H'(B(F,
L'(X, D), F)).
This definition agrees with the ones above. If F is a subfield of C, then the naturality of Navarro's functor implies that there is a natural dga quasi-isomorphism A'{X{C))
<8>Q C «H> L'(X,
D) ®F C
where we regard X(C) as a topological space in the complex topology. This quasi-isomorphism respects the augmentations induced by x and y. Thus we have: ""Actually, he does not use logarithmic forms, just algebraic forms on X. However, it is necessary to use logarithmic forms in order to compute the Hodge and weight nitrations.
Iterated Integrals and Algebraic Cycles
105
Theorem 13.6 (Wojtkowiak). If F is a subfield ofC, there is a natural isomorphism H'DR(Px,yX)
% C ^ H'(Ch'(Px,v(X(C)))).
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This result can be extended to the Hodge and weight nitrations. The Hodge filtration of L'(X,D) is defined by = i ^ p l ^ l o g D ) -> ^ + 1 ( l o g D ) - • . . • ] ,
F?L'(X,D)
where fl^ (log D) is placed in degree p. This extends to a Hodge filtration on B{F,L*{X,D),F) as described in [30, Sec. 3.2]. The Hodge filtration of Hj)R(PXtyX) is defined by FPH'DR(Px,yX)
= im{H'{F*B{F,L'{X,D),F)) S H'(FPB(F,
-»•
H'DR(Px,yX)}
L'(X, D), F)).
Similarly, the weight filtration of L'(X, D) is defined by WmL*{X,D)
=
P^wr<mn'x(logD).
Like the Hodge filtration, this extends to a weight filtration of B(F, Lm(X, D), F) as in [30, Sec. 3.2]. The weight filtration of H*DR{Px,yX) is defined by WmH2>R{PXlVX)
= im{Hn(Wm-.nB(F,L'(X,D),F))
-»• H'DR(Px,yX)}
.
Theorem 13.7. Suppose, as above, that X is a smooth complete variety and D a normal crossings divisor in X, both defined over F. If X = X — D, then there is a Hodge filtration Hbii(Px,yX)
= F H})R(PXtyX)
D H})R(PXtyX)
D H^R(PXtyX)
D •••
and a weight filtration • • • C WmH'DR(Px,„X)
C Wm+1H'DR(Px,yX)
C • • •C
HDR'(Px,yX)
which are functorial with respect to morphisms of smooth F-varieties and are compatible with the product and, when x = y, the coproduct and antipode. These filtrations behave well under extension of scalars; that is, if K is an extension field of F, then there are natural isomorphisms F"H'DR(Px,yX
® F K) = (F"H'DR(Px,yX))
®F K
106
R. Haiti
and WmHhR(Px,yX
® F K) s (WmH'DR(Px,yX))
® F K.
When F = C, these filtrations agree with those defined in [30]. Proof. The first point is that there is a natural filtered quasi-isomorphism
(E'(XlogD),F')
o
(L'(X,D),F').
The second is that there are natural quasi-isomorphisms j.Fx
- > n > ( l o g D)<-+j*Slx.
D
14. The Cobar Construction In this section, we review the cobar construction (a cosimplicial models of loop and path spaces) and explain how iterated integrals are the "de Rham realization" of it. The applications of iterated integrals in earlier sections, and their role in the algebraic de Rham theorems for varieties over arbitrary fields, suggest that the cosimplicial version of the cobar construction plays a direct and deep role in the theory of motives and that the examples presented in this paper are just the Hodge-de Rham realizations of such motivic phenomena. Additional evidence for this view comes from the works of Colombo [19], Cushman [20], Shiho [51] and Terasoma [55]. The original version of the cobar construction, due to Prank Adams [1,2], grew out of earlier work [3] with Peter Hilton. Adams' cobar construction can be viewed as a functorial construction which associates to a certain singular chain complex S, \X,x) of a pointed space (X, x), a complex Ad(S, '(X, x)) that maps to the reduced cubical chains on the loopspace PX,XX and which is dual, in some sense, to the bar construction on thedualofS< 0 ) (X,a;). The map from Adams' cobar construction to the reduced cubical chains is a quasi-isomorphism when X is simply connected. In the non-simply connected case, a result of Stallings [53] implies that Ho(Ad(Si '(X,x))) is naturally isomorphic to Ho(Px,xX;Z) = "LTfi(X,x). We begin with the abstract cobar construction and work back towards the classical one. The abstract approach appears to originate with the book of Bousfield and Kan [11]. Much of what we write here is an elaboration of the first section of Wojtkowiak's paper [57], Chen has given a nice exposition of the classical cobar construction in the appendix of [15].
Iterated Integrals and Algebraic Cycles
14.1. Simplicial
and cosimplicial
107
objects
Denote the category of finite ordinals by A; its objects are the finite ordinals \n] :— { 0 , 1 , . . . , n} and the morphisms are order preserving functions. Among these, the face maps d? : [n - 1] -»• [n],
0 < j < n
play a special role; d? is the unique order preserving injection that omits the value j . A contravariant functor A —> C is called a simplicial object in the category C. A cosimplicial object of C is a covariant functor A -> C. Example 14.1. Denote the standard n-simplex by A n . We can regard its vertices as being the ordinal [n]. Each order preserving mapping / : [n] —¥ [m] induces a linear mapping | / | : A " —• Am. These assemble to give the cosimplicial space A*
A0 = = £ A1 ^ ^
A2 ^ ^
A3
whose value on [n] is A n . Example 14.2. Suppose that K is an ordered finite simplicial complex (that is, there is a total order on the vertices of each simplex). Then one has the simplicial set K, whose set of n-simplices Kn is the set of order preserving mappings <j> : [n] —> K (not necessarily injective) such that the images of the 4>{j) span a simplex of K. In particular, we have the simplicial set A" whose set of m-simplices is the set of all order preserving mappings from [m] to [n]. If one has a simplicial or cosimplicial abelian group, one obtains a chain complex simply by defining the differential to be the alternating sum of the (co)face maps. Likewise, if one has a simplicial or cosimplicial chain complex, one obtains a double complex.
108
R. Hain
14.2. Cosimplicial
models of path and loop
spaces
Suppose that X is a topological space. Denote the simplicial model of the unit interval A\ by I,. Let
X1' =Hom(/.J). This is a cosimplicial space which models the full path space PX. Its space of n-cosimplices is Hom(J n , X). Since there are n + 2 order preserving mappings [n] -> {0,1}, this is just Xn+2. The j t h coface mapping dj
. JJfJn-l _ j . X1"
iS n
3
~3
id x • • • x id x(diagonal) x id x • • • x id : Xn+1 -> Xn+2 . We shall denote it by P'X and its set of n-cosimplices by PnX. The simplicial set dlm is the simplicial set associated to the discrete set {0, 1}. Since (dl)n consists of just the two constant maps [n] —>• {0,1}, the cosimplicial space X91' consists of X x X in each degree. The mapping X1' —>• X91' corresponds to the projection PX - > I x I that takes a path 7 to its endpoints (7(0),7(1)). One obtains a cosimplicial model P£yX f° r Px,yX by taking the fiber of X1' ->• X9Im over (x,y). Specifically, P£yX — Xn, with coface maps d? "-tX - • PlyX given by '(x,xi, dP(xx,...
, a ; „ _ i ) = < (xi,... ^Xi,...
14.3. Geometric
• • :Xn—i)
3=0]
,
, X j , Xj , . . .
i 2-n—lj )
, xn_i,y),
0 <j
j = n.
realization
As is well known, each simplicial topological space X, has a geometric realization \X,\, which is a quotient space | X . | = m X \n>0
n
x A M / ~ , J
where ~ is a natural equivalence relation generated by identifications for each morphisms / : [n] —>• [m] of A. If if" is an ordered simplicial complex and K, the associated simplicial set, then \K,\ is homeomorphic to the topological space associated to K.
Iterated Integrals and Algebraic Cycles
109
Dually, each cosimplicial space X[»] has a kind of geometric realization ||X[»]||, which is called the total space associated to X' (cf. [11]). This is exactly the categorical dual of the geometric realization of a simplicial space. It is simply the subspace of
nxwA" n>0
consisting of all sequences compatible with all morphisms / : [n] —>• [m] in A, where X[n] A " denotes the set of continuous mappings from A„ to X[n] endowed with the compact-open topology. Continuous mappings from a topological space Z to \\X[»] || correspond naturally to continuous mappings A* x Z -»• X{»] of cosimplicial spaces. As in Sec. 1, we regard A" as the time ordered simplex An = {(h,...
,tn) : 0 < t i < • • • < * „ < 1 } .
There are continuous mappings PX -+ \\P'X\\ and PXiVX -+ ||P;,yA"|| defined by 7
M- {(«!, . . . , t„) M- ( 7 (0), 7 (*l), • • • ,7(*n),7(l))}
and 7 h4 { ( t i , . . . , t „ ) i - 4 ( 7 ( * i ) , . . . ,7(*n))} These correspond to the adjoint mappings A* xPX^
P'X and A* x PXiVX ->
P'tVX,
which are the continuous mappings of cosimplicial spaces used when defining iterated integrals in Sec. 1. 14.4.
Cochains
Applying the singular chain functor to a cosimplicial space X[ ] yields a simplicial chain complex. Taking alternating sums of the face maps, we get a double complex 5*(X[»]; R) where
Ss+t(X[s];R)
110 R. Hain
sits in bidegree (—s,t) and total degree t — s. n The associated second quadrant spectral sequence is the Eilenberg-Moore spectral sequence. Elements of the corresponding total complex can be evaluated on singular chains a : A* -» ||X[»]|| by replacing a by its adjoint ffiA'xA'^
X[»].
To evaluate c £ S'pffs];/?) on a, first subdivide A3 x A* into simplices in the standard way and then evaluate c on this subdivision of A s x A* —> X[s\. When X is a manifold we can apply the de Rham complex, as above, to obtain a double complex Et(X[»]), where Es(X[t\) is placed in bidegree (—s,t). Integration induces a map of double complexes
£•(*[•])->S'(X[.];R). This is a quasi-isomorphism as is easily seen using the Eilenberg-Moore spectral sequence. When X[»] is a cosimplicial model of a path space, we can say more. I will treat the case of P'X; the case of P',yX being obtained from it by restriction. The first thing to observe is that E'(X)®(-S+2^ can be used in place of E\Xs+2)
=
Em{PaX).
The corresponding double complex has [E*
(X)®^2^]^1
in bidegree (—s,t). The associated total complex is (essentially by definition) the unreduced bar construction B(E*(X),E'(X),E"{X)) on E*(X). Here E'(X) is considered as a module over itself by multiplication. The chain maps B(E,(X),E'(X),E'(X))
-»• E'(P'X)
-»•
S*(P'X;R)
are quasi-isomorphisms (use the Eilenberg-Moore spectral sequence). Similarly, in the case of P'yX, B(H,E'(X),R)
-> E'{P^yX)
-> S'(PlyX;R)
are quasi-isomorphisms. "Note that this has many elements of negative total degree.
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Iterated Integrals and Algebraic Cycles
111
We can get cochains on PX by pulling back these along the inclusion PX <-»• ||P*X||, which allows us to evaluate elements of S'(P'X;R) on singular simplices a : A* -»• PM as above. In particular, if a is smooth and w' ® u/i ® • • • ® tu, ® u/' 6 £ * p 0 ® ( s + 2 ) , then (a, u/ x wi x w 2 x • • • x ws x w") =
(w' x wi x w2 x • • • x ws x w")
Jo
= ± (<J,Pow' A I / w\ • • • wr J A P i i o " )
where the sign depends on one's conventions. Thus the cosimplicial constructions naturally lead to Chen's iterated integrals. In the case of Px,yX, the chain mapping B(R,E'(X),M) —> B(R, E'(X),R) is a quasi-isomorphisms. The cohomology of Sm(P£yX; Z) then provides the cohomology of iterated integrals with the integral structure described in Paragraph 7.1 via (15). 14.5. Back to
Adams
What is missing from the story so far is chains, which are useful, if not essential, for computing periods of iterated integrals and mixed Hodge structures. They are especially useful in situations where the de Rham theorem is not true for loop spaces, but where the cohomology of iterated integrals has geometric meaning. Adams' original work constructs cubical chains on Px,xX from from certain singular chains on X. Denote the unit interval by / and let e°, e] : Z™-1 —• In be the j t h bottom and top face maps of the unit n-cube: e
j
:
(*li
• • • >'n—l)
l—
^ (ill
j
• • • > tj-l)
e
n
i tj>
• • • > Wi-l)
j
•
For 0 < j < n, let fj : A ->• A and r, : A -*• A " denote the front and rear j'-faces of A n . These correspond to the order preserving injections [i] —* [n] uniquely determined by fj(J) = j and rj(0) = n — j . The starting point is to construct continuous maps 0
en : r-1
-> P0,nAn
°With care, these can be made smooth — details can be found in [15].
112
R. Hain
with the property that when 0 < j < n, p 6n o e° = P{dj) o 0„_i : In~2 and 6n o e) = {P{fj)°Oj)*{P{rn-j)°en-i).
(16)
These are easily constructed by induction on n using the elementary fact that Po,nA n is contractible. When n = 1, the unique point of 7° goes to any path from 0 to 1 in A 1 . The cases n = 2 and 3 are illustrated in Fig. 2.
Fig. 2.
02 and 0 3 .
For a pointed topological space (X, x), let S, (X, x) be the subcomplex of the singular chain complex generated by those singular simplices a : A™ —> X that map all vertices of An to x. If X is path connected, this computes the integral homology of X. For each such singular simplex a : A" ->• X, we have the singular cube P(a)o6n:In-1^PXtXX. Set P(cr) o 0„ - cx , n = 1; cr
=
P(p)oOn,
(17)
n>\,
where cx denotes the constant loop at x. Set q [<7i|<7 2 | • • • K ]
=
[<7i] * [a2] *•••*
[crn].
PFor a : U -v P ^ X and /3 : V -> P ^ ^ X , define a * /3 : [/ x V -> P X , Z X by (u,u) Ma(u)/3(^). q Strictly speaking, we need to use Moore paths as we need path multiplication to be associative.
Iterated Integrals and Algebraic Cycles
113
This extends to an algebra mapping 0 ( 4 ° o ) ( X , a ; ) ) ® s -»• {reduced cubical chains on PX<XX} , s>0
o\ • • • as
H->
[ci| • • • \
which is easily seen be to injective. T h e formula (16) implies t h a t
d[cr} = -[da}+ Y, (-l) J 'K)k ( "- J ) ],
(18)
l<j
where a^ denotes the front j face and &(n~i) the rear (n — j ) t h face of a. Adams' cobar construction is, by definition, the free associative algebra
Ad(Sl°\x,x))
=
@{sW(X,x))*> s>0
on 5 ; (X, x) with the differential (18), where a\ ® • • • (g> as has degree — s + ^degCTj. This is an augmented, associative, dga, where the augmentation ideal is generated by the [a]. Adams' main result may be stated by saying that the chain mapping Ad(Si '(X, x)) —» {reduced cubical chains on
PX,XX}
is a quasi-isomorphism when X is simply connected. Stallings' result [53] for Ho is more elementary. Proposition 14.3. If X is path connected, then there are natural augmentation preserving algebra isomorphisms H0(Ad(si°\x,x)))
£* H0(PX,XX;Z)
*
Zm(X,x).
Sketch of proof. The second isomorphism follows directly from the definitions. We will show that Ho(Ad(Si '(X,x))) is isomorphic to ZTTI(X, x). Let Simp, (X, x) denote the simplicial set whose fc-simplices consist of all singular simplices a : Afc —> X that map all vertices of A™ to x. After unraveling the definitions (17) and (18), we see that Ho(Ad(Sl '(X,x))) is the algebra generated by the 1-simplices Sinu^ (X, x) (augmented by taking the generator corresponding to each l-simplex to (1) divided out by the ideal generated by Ooi — C02 + C12, where a G Simp 2 (X, x) and <jjk is the singular l-simplex obtained by restricting a to the edge jk of A 2 . It follows from van Kampen's Theorem that Ho(Ad(Si '(X,x))) is naturally isomorphic to the integral group ring of the fundamental group of the geometric
114
R. Hain
realization of Simp,(X, x). The result follows as the tautological mapping |Simp.(X, x)\ —¥ X is a weak homotopy equivalence. • With the standard diagonal mapping A:Si0)(X,x)^si0)(X,x)®si°\X,x),
a M-
^
a ( n - J >
0<j<degcr
5 ; ' (X, x) is a coassociative differential graded coalgebra. The cobar construction can be defined for any connected, coassociative dg coalgebra C,. The homology analogue of the Eilenberg-Moore spectral sequence implies that if C, —> Si ' (X, x) is a dg coalgebra quasi-isomorphism, then the induced mapping H.(Ad(C.))
->•
H.{Ad(Si0)(X,x)))
is an isomorphism provided H\(X; Q) = 0, and that H0(Ad(C.))/r
->•
H0(Ad{Si°\X,x)))/r
is an isomorphism for all s in general. Elements of S'(P'XX) can be evaluated on elements of Ad(S± '(X, x)) to obtain a chain mapping S'(P;iXX; R) - • Hom(MS( 0 ) (X, x)), R). which is a quasi-isomorphism for all coefficients R as can be seen using the Eilenberg-Moore spectral sequence. Consequently, the integral structures on H'(Ch'{Px,yX)) one obtains from from S'{X*;Z) and Ad(Si0){X,x)) agree.
Acknowledgements It is a great pleasure to acknowledge all those who have inspired and contributed to my understanding of iterated integrals, most notably Kuo-Tsai Chen, my thesis adviser, who introduced me to them; Pierre Cartier, who influenced the way I think about them; and, Dennis Sullivan who influenced me and many others through his seminal paper [54] which still contains many paths yet to be explored.
Iterated Integrals and Algebraic Cycles 115
References J. F. Adams, On the cobax construction, Proc. Nat. Acad. Sci. U.S.A. 42 (1956) 409-412. J. F. Adams, On the cobar construction, Colloque de topologie algebrique, Louvain, 1956, Georges Thone, Liege; Masson &: Cie, Paris (1957) 81-87. J. F. Adams and P. Hilton, On the chain algebra of a loop space, Comment. Math. Helv. 30 (1956) 305-330. K. Aomoto, Addition theorem of Abel type for hyper-logarithms, Nagoya Math. J. 88 (1982) 55-71. S. Archava, Arithmetic Hodge structures on homotopy groups and intersection of algebraic cycles, Thesis, UCLA (1999). M. Asakura, Motives and algebraic de Rharn cohomology, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), CRM Proc. Lecture Notes 24 (2000) 133-154. A. Beilinson, Higher regulators and values of L-functions of curves, Funktsional. Anal, i Prilozhen 14 (1980) 46-47. A. Beilinson and A. Levin, The elliptic polylogarithm, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Part 2, pp. 123-190, Amer. Math. Soc, Providence, RI (1994). S. Bloch, Applications of the dilogarithm function in algebraic JiT-theory and algebraic geometry, Proceedings of the International Symposium on Algebraic Geometry, pp. 103-114, Kyoto Univ., Kyoto, 1977. A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. Ecole Norm. Sup. 7 (1974) 235-272. A. Bousfield and D. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag (1972). J. Carlson, Extensions of mixed Hodge structures, Journes de Gometrie Algbrique d'Angers, Juillet 1979 Sijthoff & Noordhoff, Alphen aan den Rijn (1980) 107-127. J. Carlson, C. H. Clemmens andd J. Morgan, On the mixed Hodge structure associated to 7T3 of a simply connected complex projective manifold, Ann. Sci. Ecole Norm. Sup. 14(4) (1981) 323-338. K.-T. Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975) 83-98. K.-T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 8 3 (1977) 831-879. K.-T. Chen, Reduced bar constructions on de Rham complexes, Algebra, topology, and category theory, pp. 19-32, a collection of papers in honor of Samuel Eilenberg (1976). K. T. Chen, Circular bar construction, J. Algebra 57 (1979) 446-483. A. Collino, Griffiths' infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. (1997) 393-415. [19] E. Colombo, The mixed Hodge structure on the fundamental group of a hyperelliptic curve and higher cycles, preprint (2000).
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[20' M. Cushman, Morphisms of curves and the fundamental group, this volume. [21 P. Deligne, Equations diffrentielles a points singuliers reguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag (1970). [22; P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groups over Q (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ. 16, pp. 79-297, Springer, New York (1989). [23 V. Drinfeld, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q), Algebra i Analiz 2 (1990) 149-181; translation in Leningrad Math. J. 2 (1991) 829-860. [24 S. Eilenberg and J. Moore, Homology and fibrations, I: Coalgebras, cotensor product and its derived functors, Comment. Math. Helv. 40 (1966) 199-236. [25; A. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995) 197-318. [26 M. Green and P. Griffiths, Unpublished manuscripts. [27 A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Etudes Sci. Publ. Math. 29 (1966) 95-103. [28; R. Hain, Iterated integrals, intersection theory and link groups, Topology 24 (1985) 45-66; Erratum, Topology 25 (1986) 585-586. [29; R. Hain, On the indecomposable elements of the bar construction, Proc. Amer. Math. Soc. 98 (1986) 312-316. [30 R. Hain, The de Rham homotopy theory of complex algebraic varieties, I and II, K-Theory 1 (1987) 271-324 and 481-497. [31; R. Hain, Iterated integrals and mixed Hodge structures on homotopy groups, Hodge theory (Sant Cugat, 1985), Lecture Notes in Math., Vol. 1246, pp. 75-83, Springer, Berlin (1987). [32 R. Hain, Higher albanese manifolds, Hodge theory (Sant Cugat, 1985), Lecture Notes in Math., Vol. 1246, pp. 84-91, Springer, Berlin (1987). [33 R. Hain, The geometry of the mixed Hodge structure on the fundamental group, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, Part 2, pp. 247-282, Amer. Math. Soc, Providence, RI (1987). [34 R. Hain, Classical polylogarithms, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, Part 2, pp. 3-42, Amer. Math. Soc, Providence, RI (1994). [3& R. Hain, The Hodge de Rham theory of relative Malcev completion, Ann. Sci. Ecole Norm. Sup. 31(4) (1998) 47-92. [36; R. Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997) 597-651. [37] R. Hain and R. MacPherson, Higher logarithms, Illinois J. Math. 34 (1990) 392-475. [38; R. Hain and J. Yang, Real Grassmann polylogarithms and Chern classes, Math. Ann. 304 (1996) 157-201. [39; R. Hain and S. Zucker, Unipotent variations of mixed Hodge structure, Invent. Math. 88 (1987) 83-124.
Iterated Integrals and Algebraic Cycles 117 B. Harris, Harmonic volumes, Acta Math. 150 (1983) 91-123. Hochschild and G. Mostow, Pro-affine algebraic groups, Amer. J. Math. 91 (1969) 1127-1140. H. Kim and R. Hain, A De Rham-Witt approach to crystalline rational homotopy theory, math.AG/0105008. J. King, The currents defined by analytic varieties, Acta Math. 127 (1971) 185-220. J. King, Log complexes of currents and functorial properties of the AbelJacobi map, Duke Math. J. 50 (1983) 1-53. M. Kontsevich, Vassiliev's knot invariants, I. M. Gel'fand Seminar, Adv. Soviet Math., Vol. 16, Part 2, pp. 137-150, Amer. Math. Soc, Providence, RI (1993). C. Miller, Exponential iterated integrals and the solvable completion of fundamental groups, Senior Thesis, Duke University (2001), math. GT/0202237. J. Milnor and J. Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965) 211-264. J. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Etudes Sci. Publ. Math. 48 (1978) 137-204; Correction: Inst. Hautes Etudes Sci. Publ. Math. 64 (1986) 185. V. Navarro Aznar, Sur la thorie de Hodge-Deligne, Invent. Math. 90 (1987) 11-76. M. Pulte, The fundamental group of a Riemann surface: mixed Hodge structures and algebraic cycles, Duke Math. J. 57 (1988) 721-760. A. Shiho, Crystalline fundamental groups and p-adic Hodge theory, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), CRM Proc. Lecture Notes, 24, pp. 381-398, Amer. Math. Soc, Providence, RI (2000). D. Quillen, Rational homotopy theory, Ann. of Math. 90 (1969) 205-295. J. Stallings, Quotients of the powers of the augmentation ideal in a group ring, in Knots, groups, and 3-manifolds (Papers dedicated to the memory of R.H. Fox), Ann. of Math. Studies, No. 84, pp. 101-118, Princeton Univ. Press (1975). D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Etudes Sci. Publ. Math. 4 7 (1977) 269-331 T. Terasoma, Mixed Tate motives and multiple zeta values, math.AG/0104231. J. H. C. Whitehead, An expression of Hopf's invariant as an integral, Proc. Nat. Acad. Sci U.S.A. 33 (1947) 117-123. Z. Wojtkowiak, Cosimplicial objects in algebraic geometry, Algebraic Ktheory and algebraic topology (Lake Louise, AB, 1991) pp. 287-327, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 407, Kluwer Acad. Publ., Dordrecht (1993). [58] J. Yang, Algebraic K-groups of number fields and the Hain-MacPherson trilogarithm, Ph.D. thesis, University of Washington (1991).
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[59] D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992) pp. 497-512, Progr. Math. 120, Birkhauser, Basel (1994). [60] J. Zhao, Multiple polylogarithms: analytic continuation, monodromy, and variations of mixed Hodge structure, this volume.
CHEN'S ITERATED I N T E G R A L S A N D A L G E B R A I C CYCLES
Bruno Harris Mathematics Department, Brown University Providence, RI 02912, USA bruno@math. brown, edu
1. Introduction Chen's iterated integrals on compact Kahler manifolds X can be used to study complex cycles. We will begin by recalling the results of [5] and [6] where iterated integrals of real harmonic 1-forms on a compact Riemann surface S were used to give a formula for the Abel-Jacobi image of the cycle S — S~ on the Jacobian J(S) (S~ denoting the image of S under the —1 involution of J(S)). This formula was used to give non-triviality results for this Abel-Jacobi image, in particular the first specific example (as opposed to an existence result) of an algebraic cycle homologous to 0 but not algebraically equivalent to 0, and in fact a cycle defined over Z. Our aim in this paper is to explain this result in a wider context, which we do in two steps. The first is to use the real Archimedean height pairing of cycles in a real Riemannian manifold as an improvement of the AbelJacobi map and to show that the same iterated integral formula calculates it for both the embedding of S in J(S) and the diagonal embedding of S in S x S x S. The second aim is to introduce a holomorphic analogue of iterated integrals: instead of regarding an indefinite integral as inverting exterior differentiation d, we invert ddc. As application of these holomorphic iterated integrals, we show that they assign to a compact Kahler X and k complex subvarieties (or cycles) C\,... , Cfc a real number depending (multilinearly) only on the homology classes of the C< (and the Kahler structure of X), provided the Q satisfy 119
120
B. Harris
two conditions: the sum of their codimensions is (dimX) + 1, and the intersection of any k — 1 of them is empty (so fc > 3). If the Cj are all divisors, then this real number is the same as that denned by Deligne in [3], Determinant of Cohornology (he assumed only that the intersection of all k = n + 1 of the Cj is empty, but in this more general case the number does not depend only on the homology classes and does not have the same iterated integral expression). Finally, we note that the complex iterated integral expression is developed from the ordinary Archimedean height pairing involving ddc on complex Kahler manifolds and has exactly the same form as the real iterated integral expression derived from the real pairing involving d on real Riemannian manifolds. 2. Chen's Iterated Integrals on Compact Riemann Surfaces and Generalizations 2.1. Iterated
integrals
involving
two differential
forms
If X is a differentiable manifold, p : [0,1] —> X a differentiable path and « i , 0:2 1-forms on X, we define a function A\ on [0,1] by Ai=
p*a>i Jo
and the iterated integral J (0:1,0:2) by / (01,0:2) = / JP Jo
Aip*a2
where at\, a2 will be assumed always to be closed forms, but even then the iterated integral is not a homotopy invariant of the path. In order to have integrals which have all the usual properties of integrals of single closed forms over closed paths, we make several assumptions: (a) X is a compact Riemann surface S. All results will be independent of the additional choice of a metric. (b) a i , a2 are harmonic 1-forms (real) representing integral cohornology classes and a i Aa 2 is exact; we let 7712 be the unique 1-form orthogonal to closed forms such that dr)i2 = « i A 0 2 •
Chen's Iterated Integrals and Algebraic Cycles
121
(c) The path p is a closed path and
L
a i = 0 = / Q2 . JP
Under these assumptions (a), (b) and (c), the integral I(ai,a2;p)=
{ai,a2)-T]i2 (2.1) JP is a (skew-symmetric) function of the 3 cohomology classes [ai], [02], [0:3] = Poincare dual to p (03 again harmonic). As shown in [5], writing ^([ai],N,N)
=I(ai,a2,p)
gives an abelian group homomorphism 2 7 : /\3{H\S;Z)y
-+R/Z,
where the ' superscript denotes the kernel of A3H1{S,Z)
-+Hl(S\Z),
a A b A c —> (a • b)c + (b • c)a + (c • a)b (a • b = intersection pairing), i.e. the kernel of interior product with the symplectic form. This kernel, the domain of 27, can be identified with the primitive part PH3(J; Z) of H3{J; Z), J = J(S). Thus 27 is an element of the corresponding intermediate Jacobian of J(S). In J(S) we have the cycles S, S~ (see introduction) and S — S~ which is homologous to 0 and so has an Abel-Jacobi image i/(S — S~) in this intermediate Jacobian. Theorem 2.1. [5] The iterated integral homomorphism 27 and the AbelJacobi image v = v(S — S~), both in the primitive intermediate Jacobian of J(S), are related by i/ = 27.
This theorem say that if we choose three mutually disjoint closed curves Ci, C2, Cz in S and denote by a\, a2, 0:3 their Poincare dual harmonic 1-forms, so that for i ^ j ai A otj = drjij
(77^ coexact)
122
B.
Harris
and if in J we choose any real 3-chain D so that
dD =
S-S~
then / a i A a2 A 03 = 2 / 7D
(0:1,0:2) - 7712
modZ.
JC3
We will sketch a proof shortly using a height pairing in real Riemannian manifolds. But first we want to mention an example studied in [6] (with further information in [7]) and subsequently developed much further by Spencer Bloch in [1] using etale cohomology. (Further work was done by Chad Schoen and David Zelinsky). This is the Fermat curve x 4 + yA = 1 where all the iterated integrals reduce to
(/o(l-*4)-1/2^)(/o(l-^)-3/4^)' It is very easy to estimate that this real number is not an integer and consequently the Abel-Jacobi image of 5 — S~ is not zero, and so this cycle is homologous but not algebraically equivalent to 0. In fact Bloch showed using etale cohomology that this cycle (and some other similar ones) is of infinite order modulo algebraic equivalence, but we don't know if the above number is irrational. See R. Hain [4] for geometric applications of S — S~. 2.2. Real Archimedean height pairing on a compact oriented Riemannian manifold X of real dimension
m
This pairing assigns to a pair of cycles (integral linear combinations of differentiable simplices) A, B of real dimensions p, q with p + q = m— 1 and disjoint supports a real number denoted (A, B)x or just (A, B). (See [9]). We can do this (as in [9]) by using the Laplacian L = dd* + d*d on forms. We denote Lp this Laplacian on p-forms. Let exp(—tL) be the heat
Chen's Iterated Integrals and Algebraic Cycles 123
operator, t > 0. It has a C°° kernel which is an m-form k(x, y, t) on X x X (and a function of t) satisfying: for each form a = a(x) on X, exp(-tL)a(y)
= / a(x) A k(x, y, t). Jx€X If we choose for each p and eigenvalue A of L an orthonormal basis of eigenforms a\ [p = degree), then the kernel is k(x, y,t) = Y^
*<*A(Z)
A a%(y)
exp(-Xt).
p,X
The terms with A = 0 give h(x,y) the kernel for projection P onto harmonic forms. We modify these operators by ± signs for use when the dimension m of X is allowed to be either even or odd: for exp(-tL) acting on p-forms, namely exp(—tLp) we introduce the modified operator (_1)"»(™-P)
exp(-tLp).
The corresponding heat kernel is K(x, y, t) = J2(-l)m(m-p)
* «l(x) A apx(y) exp(-tA).
p,X
Let H(x, y) be the sum of terms with A = 0. It is easy to check that (for each fixed t) K(x, y.t) is a closed m-form on X x X representing the Poincare dual to the diagonal A, i.e. for any closed form <j>{x, y,) on X x X, / (x,y)AK(x,y,t)= JXxX
/ <j>{x,y) = / (x,x) JA JX
(take (f> = a%(x) A */3%(y) with A = 0). H(x, y) is of course closed, and is cohomologous to K(x,y,t). These closedness statements would not be true for m odd if we had omitted the ± signs. On X x X we take the product metric. Then H is the harmonic Poincare dual to the diagonal. As t —¥ oo, K(x,y,t) —> H(x,y) and as t —• 0, K(x, y, t) —> the Dirac current 5A of integration over the diagonal. Most of the time we will assume m is even and all signs above will be + 1 . However, we must still be careful about orientations when we integrate. If X, Y are manifolds oriented by top degree forms vx, vy then we orient X x Y by vx A vy = pr\(vx) Apr^ivy). Then pru commutes with d. As
124 B. Harris
an example, if X = S x S x S, Y = Ti x T 2 x T 3 and a point in I x 7 is denoted by (x,y) = (zi, £2,2:3,2/1,2/2,2/3) then to calculate pri* (integration over y) of a form involving the Xi and 2/j, we must first write it as a sum of forms a(x) A f3(y) and then integrate j3. If S is a submanifold of X or more generally a cycle on X given by smooth maps of simplices to X, then we can pull back K(x,y,t) or H(x,y) to X x S and then integrate over y £ S obtaining Ss(x,t) = Js^, a smooth form representing the Poincare dual to 5, or as the harmonic Poincare dual to S. As t —> 0, 5s,t —> 8s, the Dirac J-current of the cycle S. On X x X, K(x,y,t) and H(x,y) are cohomologous, so there exist (smooth) forms 7(2;, y, t) satisfying (for each fixed t) d{l{x, y, t)) = K(x, y, t) - H(x, y) Suppose now A, B are disjoint cycles on X satisfying: dim A + dim B = dimX - 1. We define (A, B)X £ R by (A,B)x=}im[
fr{x,y,t)
- P(r,)]
(where P = projection to harmonic forms; usually we will choose 7 with P(7) = 0). This definition is independent of the choice of 7. We define lB,t =lB(x,t) as fyeB-y{x,y,t). The form jB,t on X satisfies djB(x,t)
= 8B(x,t)
-
aB(x)
and (A,B)X
= lim I 7 B ( M ) P{lB{x,t)). *->-0 J A We will always choose 7 and 73 with harmonic part 0. (See the eigenform expansion of K). The limit of jB,t for t —> 0 is called the Green's form for B and is singular on B. If A, B are both homologous to 0 (with integer coefficients), then (A, B)x is an integer, the topological linking number. We assume now that only A is homologous to 0. Then the map B -»• (A, B)x
mod Z
for fixed A depends only on the homology class of B in Hq (X, Z) mod torsion [q = dimB = codimA — 1) and gives a homomorphism of this
Chen's Iterated Integrals and Algebraic Cycles
125
group into R/Z, i.e. a point on an intermediate Jacobian of X. To see all this in terms of the harmonic form as representing the Poincare dual cohomology class of B (with integer coefficients, modulo torsion classes) we chose a chain D with dD = A and meeting B transversely at smooth points. Then since Q5 = <5B(:E,£) — d'ygj, QR
/.-
= lim
/ 6B(x,t) - / drfB,t
t-vO .JD
JD
= integer - / 75 JA
= -(A,B)x
modZ.
Thus for fixed A homologous to 0, the Abel-Jacobi image of A as represented by the map OLB
I
—• / Q-B
mod Z
is represented mod Z by the real number —(A, B)x if we choose a collection of cycles B representing a basis of the cohomology group, and chosen to be disjoint from A. We can say then: the Abel-Jacobi map into a real torus is lifted to a map into a real vector space. In the situation of Sec. 2.1, where X = J(S) and o^ A 0:2 A 0:3 is a primitive harmonic form on J [on being forms on S with integer periods), we take A = S and B a suitably chosen real (2g — 3) dimensional subtorus of J (g = genus of 5) which does not intersect S and has a\ A 0:2 A a.$ as Poincare dual form. B is chosen as follows: restricting the integrals on S defining the Abel-Jacobi map S —>• J(S) to the three integrals of the on gives a map S ->• T 3 =
R3/Z3
and a homomorphism h : J ^ T3 which is a fiber bundle over T3. The metrics on J and on T 3 are the usual Euclidean metrics and the harmonic forms are the translation invariant ones. Thus h* and h* preserve harmonic forms. (We assume a\ A a^ A 0:3 generates a direct summand in PH3(J; Z) and a i , «2, ^3 generate a direct summand in i/ 1 (J r ; £)). Let now zo be any point in T 3 not in h(S) and T 2 s " 3 = /i _1 (a; 0 ). The height pairings in J and in T 3 are related by (S,T2°-3)J{s)=(h(S),x0)T3.
126
B. Harris
To see this recall that the left hand side is fs^yT2g-3, the right side is Ih(S) 7x» — Is ^*(^o)- C1* °f a current is the transpose of /i, of a form). It is easy to check that in this situation (fibration, in fact a trivial one with preservation of harmonic forms) we have h*iXo
= Jh-i(x0)
~ 7T2»-3 •
We then have to discuss the height pairing in T3. Since h(S) is homologous to 0 in T 3 (because J a; A ctj = 0 by assumption), this height pairing reduced mod Z is the negative of the Abel-Jacobi map, and this last was calculated in [5] and shown to be given by the iterated integral (2.1). We can also calculate the height pairing directly in T 3 with product metric form T1 x T1 xT1, but we won't do this here since this is an odd dimensional manifold and there are ± signs to keep track of. However, the method will become clear from calculations we will now do in other product manifolds with product metric. 2.3. Iterated integrals for the diagonal in S x S x S
embedding
of S
Let now S be a Riemann surface, with any metric, X — S x S x S with product metric, A = A s , the diagonal in X, and B = C\ x Ci x C3 where C\, C2, C3 are mutually disjoint 1-manifolds on S. Let cti be the harmonic Poincare dual form to Cj in S. Choose any 1-form 1712 on S with d-qi2 = OLI A a.2- Then we have: Theorem 2.2. The pairing onX = SxSxS is expressed as follows: {As,C1xC2xC3,)sxSxs
= / Jc3
given by (As, C\xC?,x
C3)
[ ( a i , a 2 ) - 7 7 i 2 ] + / »7i2Aa 3 Js
for any choice 0/7712 on S satisfying drju = a i A 02. //7712 is chosen to be orthogonal to the harmonic 1-forms then the last integral is 0. The pairing depends only on the homology classes of the Ci in S and on the complex structure of S (but not on the choice of metric on S). It is skew-symmetric in the 3 homology classes. Proof. For x, y G X write x = (xi,x2,x3), y = (2/1,3/2,2/3)- The heat kernel Kx = K(x,y,t) of X can be written in terms of the heat kernel Ks
Chen's Iterated Integrals and Algebraic Cycles
127
of 5 as Kx(x,
y, *) = Ksfa,
3/1, t)Ks{x2,y2,
t)Ks(x3,y3,
*),
(from the eigenform expansion, or just by writing the Laplacian on X as L = Li+L2 + L3). We abbreviate this as K =
KXK2K3.
Similarly the harmonic part is H = H(x1,y1)H{x2,y2)H{x3,V3)
=
H1H2Ha.
On 5 x 5 we choose a form 7s(si,S2,*) satisfying djs{si,s2,t) = Ks(si,s2,t) - Hs(si,s2) and write j i < t = 1s(xi,yi,t). We assume 7s is orthogonal to harmonic 1-forms on S x S. On X x X we can then choose the corresponding form 7x,t = 7(xi,X2,x3;2/1,2/2,3/3;*) as 7x,t = 7(^1 > 2/i. t)K(x2,y2,
t)K(x3,2/3, *)
+ #(&!, 2/1)7(2:2,2/2, *)-^(a:3,2/3, *) + i?(a:i, y1)H(x2, = H,tK2K3
+ Hii2tK3
y2)j(x3,y3,t) + HiH-z^j •
Then (A, B) = (A,s,Ci xC2x C3) is obtained by integrating each of these three terms over (2/1,2/2,2/3) & Ci X C2 x C3 and then the resulting three terms on X over the diagonal x\ = x2 = x3. However, we must first move all dxi to the left and dyi to the right which will introduce a sign in each term: thus in the first term this procedure introduces first a — sign, then turns the K(xi,yi,t) factors (i = 2,3) into Sci(xi,t) = 5citt on X and 75(11,2/1,*) into 7ci,t on S i.e. the first term after the 2/i integrations becomes — idixi, t)6c2{x2,t)8c3(x3,t) on S x S x S and restriction to the diagonal gives the product — 7Ci,t^C2,t 0: we obtain 0 because 5ct t approaches the Dirac Sct and C2, C3 are disjoint. The same procedure applied to the second term (with t > 0 still), gives
1 a
CilCz,t&C3,t
•
128
B. Harris
Now as t -> 0, the function 7c 2 ,t on X approaches an L1 function (in fact one having a simple jump across C2) satisfying Lebesgue dominated convergence as this is done. In fact we need to look only in a neighborhood of C3 disjoint from Ci as Sc3,t approaches the Dirac dc3- Furthermore, if we stay away from C\ as well, act is exact: a^ = dA\. So as t —> 0 the above integral approaches /
aci7c2 = /
J C3
dAi7c2 = - /
J 03
Axd^c2
J O3
= /
^10:2
-f Cz
since d*yc2 — &c2 — ac2 = -®c2 outside C2. The last term acquires a "—" sign, and gives - / aClac27c3,t
= -
driufca.t =~
m2^7c 3 , t
= / r]\iOLc3 - \ m2^c3,t • JS
JS
Letting M O w e get - / »?12 + / V12CHC3 Jc Js IC33 JS The sum of the three terms gives the formula in Theorem 2.2. This formula in fact does not involve C\ but only a\ and so shows dependence only on the homology class of C\. To prove the same thing for C 2 and C3 and the skew-symmetry we just have to calculate with modifications of 7 which have the Ki and Hi in a different order — see the proof of the more general Theorem 2.3. • We summarize now the results of this subsection. Let a\, 0:2, 0:3 be harmonic 1-forms on the Riemann surface S, Poincare dual to 3 disjoint circles (or 1-cycles) C\, C2, C$ and choose any 1-form 7712 on S such that drji2 = ct\ A a 2 - Use the notation: 7712 = {oc\ A 02,1) and 7712 A 0:3 = (ai A 0:2,0:3). Write I =
(ai, a 2 ) - (ai A a 2 ,1) + / ( " I A a 2 , a 3 ) . •/c 3 Js Then 7 expresses two height pairings: / = ( A 5 , d x C 2 x C7 3 )sxsxs = - ( 5 , T 2 f f - 3 ) J ( s )
mod Z
and 1/ = 27 mod Z gives the Abel-Jacobi image of the cycle S — S~ (on J(5)) in the intermediate Jacobian of J(S).
Chen's Iterated Integrals and Algebraic Cycles 129 2.4. Generalized
pairing in S X S X S
Let S be real Riemannian (compact oriented as usual) of real dimension m which is even and C i , . . . , Cfc, (fc > 3) be smooth submanifolds (or smooth cycles) of codimensions p\,... ,pk satisfying Pi H
1- pfc = m + 1.
We make general assumptions: (1) The intersection of any k — 1 of the Cj is empty. (2) The intersection of any set of | Q | (supports) occurs at smooth points and is transverse; thus this intersection is smooth and has the expected dimension, or is empty. Let X = Sk (product metric always), A = diagonal, denoted S, B = C\ x • • • x Cfc. We will show that (S, C\ x • • • x Ck)sk is given by a sum of iterated integrals generalizing the Riemann surface case, and is a function of the homology classes of the Cj satisfying a graded skew-symmetry condition. To state the result we use the following notation: C\,... , Cfc are the given cycles, their supports will also be denoted Cj, intersection of supports will be denoted by Cj D Cj, intersection of cycles (here meeting properly) will be denoted Cj • Cj. We write Ck+\ = S. Then by the assumption 1, we have for i = 1 , . . . , k (and even for i = k + 1),
Ci n • • • n Cj_i n Ci+1 n • • • n C fe+1 = 4> and so for i = 2 , . . . , k, the form a.\ A • • • aj_i, is exact when restricted to any neighborhood of Cj + i n • • • (~lCk+i which does not meet C\ Pi • • • DCj_i. For i = 2 , . . . , k we write r]y... j _ i for any form on (a neighborhood of) Cj + i n • • • n Cfc+i satisfying drji ... j _ i = a\ A • • • A « j _ i
and write 771 . . . & = 0 (noting a\ A • • • A a.k = 0 ) . We then write (a x A • • • A OJJ_I, aj) = 771...,_! A aj
and {ai A---Aaj-1,1) = 771... i _ i .
130
B. Harris
Generalizing Theorem 2.2 we then have Theorem 2.3. With the above notations and assumptions on the even dimensional Riemannian manifold S and the cycles Ci,...,Ck (of codimensions pi) on S we have (S,C,x---xCk)sk k
= ( - 1 ) 9 + 1 Y, where q = ^2r<sPrPs-
/
( a i A • • • A Oi-i,ai)
- ( « i A • • • A a ( > 1)
(The last term i = k has ( a i A • • • A a*, 1) = 0 by
assumption). Furthermore, this expression (i.e. either side of this equation) is unchanged if we replace any Ci by a homologous cycle C[ (satisfying the same intersection conditions), and is multiplied by (— l)?*?^ 1 if d, Cj+i are interchanged. Proof. As in the previous proof, we start with the choice of jx(x, X x X given by
2/> *)
on
k
7 x ( z i , . . . ,xk;yi,...
,yk;t) = J ^ f f i A • • • A # i _ i A 7* AKi+i
A • • • AKk
where all Hj,jj,Kj are forms in (xj,yj) on S x S (and t). We need only the terms of degree m — pj in yj, for integrations over Cj, and so of Xj degree pj (for Hj, Kj) or pj — 1 (for 7-,). Thus in the ith term moving all the dxj to the left introduces the sign (since m — pj = pj mod 2) (—1)9+*'' where q — Tir<sprps, Ki = S^ _ 1 pj. Now we can integrate over y in C\ x • • • x Ck and obtain from the ith term (-l)q+7Uai(xi)
• • •ai-i(xi-i)'yci(xi,t)5ci+1(xi+1,t)..
.5Ck(xk,t).
Restricting to the diagonal S : X\ = • • • = xk gives the product on S, which then has to be integrated over S. When t —¥ 0, -ya,t — 7Ci(^i,i) approaches the form 7 ^ on 5 which is not continuous on Ci but is L1 on S since near a smooth point of Ci, if r denotes distance in the normal direction to Ci, then the leading singularity is of order r 1 _ P i (as one can calculate from the heat kernel in Euclidean space, see [10]; conpare also the differential character discussion in [8, p. 72-74]). The transversality condition on the Cj implies that 7 ^
Chen's Iterated Integrals and Algebraic Cycles
131
when restricted to C;+i (!• • -Pi Cfc (transverse to d) is also L1. We also need to argue that the convergence of 7Cj,t to 7c 4 as t —• 0 satisfies Lebesgue dominated convergence and finally that the integral of the ith term as t —> 0 has limit (_!)«+* f
ai
A • • • A ai_l7ci
•'Ci.)_i-...Cjb+i
provided i > 2. Next, «i A • • • A ai_i here is dr]i„.i-i and cfycs = Sc€ — a* restricted to this integration domain. Applying Stokes gives now (-1)"
/
m...i-i -
JCi-...Ck+i •••Ck+i
/
Vi-.i-i
Aa;
JCi+i-...-i JCi+1-...-Ck+i
The term i = 1 has integrand 7ci<5c2*---<5cfe* which as t —> 0 approaches 0 since Ci n • • • n Cfc = ^>. Thus the sum of the A; terms gives the formula in Theorem 2.3. This formula shows that C\ itself does not enter into any of the integrals, but only the form a.\ does, so we can replace C\ by C[. Next we prove the skew-symmetry with respect to interchange of Ct, Ci+i of (S, C\ x • • • x Ck)sk i-e. that a factor (—I)P*P*+I is introduced. We will do the calculation for interchange of C\, C 2 , other i working out the same way. Since we can choose any 7' on X x X with dq' = K — H — K\... Kk — H\... Hk to calculate any (A, B)x = (A, A x B)xxX we choose 7' = ^172^3 ...Kk
+ 7 l t f 2 K 3 • • • Kk + EjLaiTi • • • ^ i - i 7 i ^ i + i • • • #fc •
Then (5,C 2 x d x C 3 x • • • x Cfc) = lim f
7'
and the same calculation as before gives this as (—l) P l P 2 (5, C\ x • • • x Cfc), concluding the proof. • As a last,but important, remark, we state that if S is compact Kahler the quantity in Theorem 2.3 is independent of the choice of Kahler metric and depends only on the complex structure. This follows from an easy calculation using the iterated integral expression and the ddc lemma.
132
B. Harris
3. The Complex Kahler Height Pairing Using ddc S will now be compact Kahler of complex dimension n and C\,... ,Ck will be complex cycles with (complex) codimensions p i , . . . ,Pk- The Poincare dual harmonic forms a\,... , a^ have types (j>i,Pi) and similarly all currents or forms will have even degrees and types (r, r). ddc is defined as ^dd. All maps will be complex analytic and so pull backs and push forwards by them commute with ddc. On S x S we have the same real forms Ks = K(x, y, t) and H as before, and the ddc lemma gives existence of real forms F(x, y, t) of type (n— 1, n—1) satisfying ddcT = K-H
for each t.
Integration of T in the second variable y as y runs over a subvariety Z of S (or over a desingularization Z) gives a form Yz,t on S such that ddcTz,t = dz,t - OLZ (where 8z,t, OLZ are as in the previous section) and as t —> 0, Tz,t —• Tz which is an L1 form on S (and the convergence is a dominated one). Details are given in [10]. We assume (1) Any k — 1 of the Cj have empty intersection. This implies that for i = 2 , . . . , k, Q I A • • • A OJ;_I restricted to (a neighborhood not meeting C\ fl • • • n Cj_i of) Cj + i Pi • • • n Ck+i (Ck+i = S) can be written as OJI A • • • A a,_i = rfdc^i...i_i (such forms (J,i...i-i exist e.g. — Tz where Z = Ci • ... • Cj_i). We set Mi...fe = 0 . We make any choice of such forms Hi...i-i, i = 2,...k and write (ai,a2)
= Hia2,
(a! . . . a j _ i , l ) = /ii...i_i,
( a i a 2 , 1 ) = M12 , («i ...ai_i,Q!i) =/xi...i_iai.
(2) We assume further that the (underlying subvarieties of the) cycles Cj meet tranversely at smooth points. Theorem 3.1. Let X = S with product metric. For any subvarieties A, B of X with dim A + dimB = (dimX) - 1 and disjoint support we define
Chen's Iterated Integrals and Algebraic Cycles 133 the complex Archimedean
height pairing,
[A,B\x (P = harmonic
= Yim[
as in [9], by
(T(x,y,t)
-
P(T))
projection).
Taking A — diagonal in Sk, denoted S, and B — C\ x • • • x Ck, we have I ^ i X - x C
t
]
S l
fc = ~YL\
[(ai A - - - A a i _ i , a i ) - ( a i A - - - A a i , l ) ] .
(3.1)
i = 2 •'Ci+i'-'-'Cfc+i
T h e real number given by either side of (3.1) depends only on t h e homology classes of the Q and is unchanged by permutations of these Cj. P r o o f . This is exactly the same proof as for T h e o r e m 2.3 b u t with t h e advantage t h a t all ± 1 signs there are + 1 here. • We remark t h a t in the case whre all Cj are divisors, we can choose as Green's currents Yct = log ||(jj|| 2 ,
134 B. Harris [9] B. Harris, Cycle Pairings and the Heat Equation, Topology 32 (1993) 225-238. [10] G. Hein, Computing Green Currents via the Heat Kernel, Preprint, Humboldt-Universitat zu Berlin (2000). [11] J. Jorgenson and S. Lang, The Ubiquitous Heat Kernel, pp. 655-682 in Mathematics Unlimited: 2001 and Beyond, B. Engquist, W. Schmid (eds.), Springer (2001).
O N A L G E B R A I C F I B E R SPACES
Yujiro Kawamata Department of Mathematical Sciences, University of Tokyo Komaba, Meguro, Tokyo, 153-8914, Japan kawamata@ms. u-tokyo. ac.jp
An algebraic fiber space is a relative version of an algebraic variety. We prove some basic topological and analytical properties of algebraic fiber spaces. We start with reviewing a result by Abramovich and Karu on the standard toroidal models of algebraic fiber spaces in Sec. 1. We can eliminate the singularities of the fibers in a topological sense by performing the real oriented blowing-up on the toroidal model (Sec. 2). The rest of the paper concerns the Hodge theory of the degenerate fibers. In Sec. 3, we define the weight filtration on the cohomology with coefficients in Z. We prove the logarithmic and relative version of the Poincare lemma in Sec. 4. Finally we prove that the logarithmic de Rham complex gives rise to a cohomological mixed Hodge complex in Sec. 5 (Theorem 5.2). As a corollary, we prove that certain spectral sequences degenerate. We consider only varieties and morphisms defined over C. The topology is the classical (or Euclidean) topology instead of the Zariski topology unless stated otherwise.
1. Weak Semistable Model An algebraic variety in this paper is a reduced and irreducible scheme of finite type over SpecC. An algebraic fiber space is a relative version of an algebraic variety; it is a morphism / : -X" —> Y of algebraic varieties which is generically surjective and such that the geometric generic fiber is reduced 135
136
Y.
Kawamata
and irreducible. We look for a standard model of an algebraic fiber space in the category of toroidal varieties. Definition 1.1. A toric variety (V,D) is a pair consisting of a normal algebraic variety and a Zariski closed subset such that an algebraic torus T = (C*) n acts on V with an open orbit V\D. A toroidal variety is a pair (X, B) consisting of an algebraic variety and a Zariski closed subset which is locally analytically isomorphic to toric varieties in the following sense: for each point x € X, there exists a toric variety (VX,DX) with a fixed point x', called a local model at x, and open neighborhoods Ux and U'x, of x S X and x' € Vx in the classical topology such that (UX,B C\UX) is isomorphic to (U'X,,DX n U'x,). A toroidal variety (X, B) is called strict if any irreducible component of B is normal. A strict toroidal variety (X,B) is called a smooth toroidal variety if X is smooth and B is a simple normal crossing divisor. Namely, a local model of a smooth toroidal variety has the form (C™, div(xi • • • xn>)), where (x\,... ,xn) are the coordinates. A strict toroidal variety (X, B) is called a quasi-smooth toroidal variety if its local model is a quotient of a smooth local model by a finite abelian group action which is fixed point free o n I ' \ 5 . Namely, a local model has the form (Cn/G, div(xi • • • xn>)/G), where G is a finite abelian subgroup of GL(n, C) which acts on C n diagonally on the coordinates (a;i,... , xn>). Let (V, D) and (W, E) be toric varieties with the actions of algebraic tori T and S. A toric morphism g : (V, D) —• (W, E) is a morphism g : V —> W of algebraic varieties which is compatible with a homomorphism go : T —> S of algebraic groups. Let {X,B) and (Y, C) be toroidal varieties. A toroidal morphism f : (X, B) —> (Y, C) is a morphism / : X —> Y of algebraic varieties such that f(X\B) C Y\C and that for any point x € X and any local model (Wy,Ey) at y = f(x) e Y, there exists a local model (VX,DX) at x and a toric morphism g : (VX,DX) -> (Wy,Ey) which is locally analytically isomorphic to / . The resolution theorem of singularities by Hironaka implies that there is always a smooth birational model for any algebraic variety; for any complete algebraic variety X, there exists a birational morphism /x : Y —> X from a smooth projective variety.
On Algebraic Fiber Spaces
137
As for the relative version of the resolution theorem, one cannot expect that we have a birational model which is a smooth morphism. Indeed, we cannot eliminate singular fibers. Instead of the smooth model, Abramovich and Karu obtained a toroidal model by using the method of de Jong on the moduli space of stable curves. Theorem 1.2 (Abramovich-Karu [1]). Let /o : Xo —*• YQ be a surjective morphism of complete algebraic varieties whose geometric generic fiber is irreducible. Let Bo and CQ be Zariski closed subsets of Xo and Yo such that fo(Xo\Bo) C YO\CQ. Then there exist a quasi-smooth projective toroidal variety (X,B), a smooth projective toroidal variety (Y,C), birational morphisms fi : X —> XQ and v : Y —> Yo, and a toroidal and equi-dimensional morphism f : (X, B) -» (Y, C) such that n(X\B) C X0\B0, v{Y\C) C Y0\C0 and v o f = f0 o fi. Let / : (X, B) —>• (Y, C) be a toroidal and equi-dimensional morphism of quasi-smooth toroidal varieties. We can describe / explicitly by using local coordinates as follows. Let us fix x G X and y = f(x) G Y. Let n = dimX and m = d i m y . We have local models of X and Y: there are integers 0 < n' < n and 0 < ml < m, finite abelian groups G and H which act diagonally on the first n' and m' coordinates of the polydisks A" = { ( z i , . . . ,xn)\\xi\ < 1} and A m = { ( y i , . . . ,ym)\\yj\ < 1}, and open neighborhoods U and V of x € X and y G Y in the classical topology such that {U,BC\U) = (A"/G,div(a;i • • -xn>)/G) and (V,C n V) ^ (Am/H,div(y1 • ••ym,)/H). We may assume that the fixed locus of each element of G and H except the identities has codimension at least 2. Then the morphism / induces a morphism / : A" —> A m . The fact that / is toroidal and equi-dimensional means the following: there are integers 0 = to < ii < ti < • • • < tmi < n' and 1 < li (i = 1 , . . . , tmi) such that
k=l
for j = 1 , . . . ,m' and f*(yj) = xn>-mi+j for j — m' + 1 , . . . ,m. Indeed, since / is toroidal, it is expressed by monomials by some local coordinates. The equi-dimensionality implies that the sets of indices i of the Xi on the right hand side for different j's are disjoint.
138
Y. Kawamata
By using [5], Abramovich and Karu obtained a model with reduced fibers: Corollary 1.3. Let (X,B) be a quasi-smooth projective toroidal variety, (Y, C) a smooth projective toroidal variety, and f : (X, B) —> (Y, C) a toroidal and equi-dimensional morphism as in Theorem 1.2. Then there exists a finite and surjective morphism ny '• (Y1, C) -¥ (Y, C) from a smooth projective toroidal variety such that 7r - 1 (C) = C and the following holds: if we set X' to be the normalization of the fiber product X Xy Y' and B' = nx1(B) for the induced morphism irx '• X' —> X, then (X',Br) is a quasi-smooth projective toroidal variety and the induced morphism f : (X', B') —¥ (Y1, C) is a toroidal and equi-dimensional morphism whose fibers are reduced. The morphism / ' is called a weak semistable reduction of /o- We note that the fibers of / ' are reduced if and only if all the exponents U are equal to 1 in the corresponding local description (1.1) for / ' .
2. Real Oriented Blowing-up It has been known that the general fiber of a semistable degeneration is topologically homeomorphic to the real oriented blowing-up of the singular fiber (e.g. [9]). In [6] this knowledge is used to put a Z-structure on a certain cohomological mixed Hodge complex. The real oriented blowingup is a special case of the associated logarithmic topological space to a logarithmic complex space defined by [4]. Definition 2.1. The real oriented blowing-up of a quasi-smooth toroidal variety (X,B) is a real analytic morphism from a real analytic manifold with boundary to a complex variety px : X# —>• X defined by the following recipe: (0) If there is no boundary, then px is the identity: if X = A = {z G C||z| < 1} and B =
On Algebraic Fiber Spaces
139
(3) The real oriented blowing-up of a quotient of a smooth toroidal variety by a diagonal action of a finite abelian group which is free on the complement of the boundary divisor is the quotient of the real oriented blowing-up: if X = Xl/G and B = Bx/G, then X* = Xf/G and px = PXi /G. We note that the action of G on Xf is free so that X* has no singularities. (4) We can glue together the real oriented blowing-ups of local models: if X = U, Xi and B = \J. Bh then X* = U Xf and px = \J. pXi. Proposition 2.2. Let {X,B) be a quasi-smooth toroidal variety. Then the real oriented blowing-up X& is homeomorphic to the complement of an e neighborhood of the boundary B in X with respect to some metric for sufficiently small e. The real oriented blowing-up is functorial: Proposition 2.3. Let f : (X, B) —¥ (Y, C) be a toroidal morphism between quasi-smooth toroidal varieties. Let px '• X& —¥ X and py '• Y^ —> Y be the real oriented blowing-ups. Then a morphism of real analytic varieties with boundaries / # : X * —• Y& is induced so that the following diagram is commutative:
X*
Px
> X Xy
Y*
pn
Vri
Y*
»
» X f
Y*
PY
» Y.
By using the real oriented blowing-up, we can eliminate the singularities of fibers topologically: Theorem 2.4. Let f : (X,B) —»• (Y, C) be a proper surjective toroidal and equi-dimensional morphism of quasi-smooth toroidal varieties. Then the induced morphism f& : X& —> Y# is locally topologically trivial in the following sense: each point y' € Y& has an open neighborhood V such that ( / # ) _ 1 ( 7 ' ) is homeomorphic to V x {f*)~l{y') over V. Proof. We use the local description explained after Theorem 1.2. We write p*^(xi) = rie"/z:i9i (1 < i < ri) and p^fa) = s^e^1^ (1 < i < m'). Then
140
Y.
Kawamata
the map / # is described by the following formulas:
(/*)*-j='lf fc=i
feS.
(2-D
and
(/#)Vi=
E
^-i+fc^-i+*
fc=i
for j = 1 , . . . ,m' and (/#)*(%•) = aj n /-m'+j for j = m' + 1 , . . . ,m. We note that the actions of G and H are rivial on the Vi and s^- while those on the 6i and 4>j are translations. Since
j ( n , • • • , rt)|t< G [0,1), J J t i = c I S [0, l ) * - 1 for any c e [0,1), the fibers of / # : ?7* —>• V# are homeomorphic to [ 0 , 1 ) " ' - ™ ' X (S1)*'-"1'
X (£)2)n-n'-m+m' _
The restriction of / # on l / # n ( / # ) _ 1 ( V ) for sufficiently small V is homeomorphic to the first projection of V x ( / * ) _ 1 ( y ' ) to V. By gluing together, we obtain our result. • Corollary 2.5. Rp{f^)tf'Lx* p>0.
i-s a locally constant sheaf on Y& for any
Proposition 2.6. Let (X,B) be a quasi-smooth toroidal variety and let B — X)i=i Bi be the irreducible decomposition. Let E be a connected component of the intersection P | i = 1 Bi of codimension I, and let G = Xw=i+i Gi for Gi = Bi fl E. Then (E,G) is a quasi-smooth toroidal variety. Let px '• X# —> X and PE '• E# —> E be the real oriented blowing-ups. Then px induces a map p'x : px (E) —* E& which is an l-times fiber product of oriented S1 -bundles. Proof. Since the boundary B has no self-intersection, E is normal, hence (E, G) is toroidal and quasi-smooth. E\G is smooth and the normal bundle N(E\G)/(X\G) is the direct product of line bundles. Since E# is homeomorphic to the complement of an e neighborhood of G in E, we obtain our assertion. •
On Algebraic Fiber Spaces
141
Corollary 2.7. P / N
\
Rp(px)*Zx#=f\{(&ZBi\
,
and v (
N
\
Rp{px)*%p-xHE)= f\ [llE® 0 Z G , where the exterior products are taken respectively as Zx-modules modules.
and Z#-
In the case p = 0, the above formula means that (px)*'^x* — %X (pjf)»Z„-!(£) — ^E- We note that Gi may be empty or reducible.
an
d
3. Weight Filtration We put a weight filtration on a complex on singular fibers. The definition of the filtration is natural thanks to the geometric construction of the real oriented blowing-up. One can compare with rather complicated earlier definitions in [10] and [3]. In this section, we denote by / : (X, B) —• (Y, C) a weak semistable reduction as in Corollary 1.3. We fix y € Y and let Ei,... ,Ei be irreducible components of the fiber E = f~1{y)- We have diml?fc = n — m for any k. For any combination of integers 1 < io < • • • < it < h we define Ei0t... tit = f)k=o Eik. We note that dim-Ei0i... ^ may be larger or smaller than n—m—t. For example, if a local model / : A 4 —> A 2 is given by f*(yi) = x\x2 and f*{y2) = X3X4 with y = (0,0), then the irreducible components of E are Ei = {zi =x3= 0}, E2 - {xi = x± = 0}, E$ = {x2 = x3 = 0} and £4 = {X2 = X4 = 0 } SO t h a t E14 = E23 = ^ 2 3 4 = •E'134 = -£7l24 = •£'123 = •E'1234-
We call each Ei0 it a stratum of E. Let Gi 0) ...,i t be the union of all the strata which are properly contained in -Ei0,...^t. Then the pair (.-E'io,... ,it > " , i o i . . .
Mi
is again a quasi-smooth toroidal variety. Let Pi0t...,it : E* i( —>• i?i 0 ,...,i t be the real oriented blowing-up. Let EW be the disjoint union of all the -E7i0]... j ( . Then we have an exact sequence 0 ->• ZE -> ZE[o) -> ZB[i] ->
>• Z £ [ t ] - > • • •
142
Y.
Kawamata
where we identified the sheaves ZE\t\ with their direct image sheaves on E and the differentials are given by the alternate sums of the restriction maps. If the number of irreducible components of C which contain y is m', then l pY {y) is homeomorphic to (S1)™'. Let D = p£(E) = ( / # ) - 1 P y 1 ( y ) - W e have natural maps px • D -> E and / # : D —• pZl{y). Corresponding to the irreducible decomposition of E, we have D = (Ji=i A for Di = px (Ei). We write Dio it = f l L o A fc and D® - U I < H , < . . . < W < I D i o , . . . M . Then we have similarly an exact sequence 0 -> ZD -*• ZD[0] -> Z^m ->
• Z£,(t] - > . . . .
We fix a point y £ Py^J/) a n d let -D = ( / # ) _ 1 ( y ) - We set A 0 ,...,i» — 5 n A 0 ,...,i, and SW = Uii0)...,jt is homeomorphic to the direct product 5i 0> ...,i t x pY1(y), and / # corresponds to the second projection. Moreover, Z)io,...,it is a t'-times fiber product of S1bundles over Ef •. Proof. We assume first that t = t' = 0. Then we have Dio = i ? ^ . By Proposition 2.6, px induces an (S 1 )" 1 -bundle p'x : Di0 —¥ JS*, and we have a map
p'xxf*:Di0^E*xpY\y). Since the fiber of / is reduced, we may take the homeomorphism to the e neighborhood as in Proposition 2.6 in a suitable way to conclude that P'x x / # *s bijective and a homeomorphism. In the general case, if we restrict p'x x / # to .Di0,...,it, we obtain a homeomorphism A 0 ,...,it -> P70\Eio,-,h) By applying Proposition 2.6 again to Ei0
x PyHv) • i, C Ei0, we conclude the proof.
• Definition 3.2. We define the weight filtration Wq on the complex of sheaves R(PX)*ZD on E x pYl{y) by Wq(R(px)tZD)
= (T
-»• r < , + 1 ( i J ( ^ ) , Z f l ( 1 1 )
On Algebraic Fiber Spaces 143 in the derived category of complexes of sheaves on E x pY (y), where r denotes the truncation (cf. 1.4.6 of [2]). If we take the successive quotients with respect to this nitration, then we can check that all the differentials of the resolution become trivial: Proposition 3.3. Gi™ (R(px).ZD)
[-q] ® Rq+\Px)^DW
S* R«{px)*ZDm q+t
e • • • e R (Px)*zDlt]
[-q - 2]
[-q - 2t] © • • •
Corollary 3.4. The local monodromies of Rp(f#)*Zx# tent for any p>0-
on Y* are unipo-
Proof. We consider the restriction of the locally constant sheaf i? p (/#)*Zx# on Pyl{y)- Since / * = pr 2 opx, the weight filtration induces a spectral sequence EP.9
=
RP+i(pr2)^G^p(R(Px)tZD)) p
By Proposition 3.1, R (px)*ZD[t] have our result.
=* R?+*{f*)*Zx#
.
is a constant sheaf for any p, t. Hence we •
4. Relative Log d e Rham Complex Let (X, B) be a quasi-smooth toroidal variety of dimension n. Then the sheaf of logarithmic 1-forms Clx(logB) is a locally free sheaf of rank n. Indeed, if the local model of (X,B) at x S X is the quotient of type (An/G,Bni/G) with Bn> = div(x 1 • • -xni), then the action of G on the basis dxi/xi (1 < i < n') and dxi (n' < i < n) of the sheaf Q,\n{logBni) is trivial. The sheaf of logarithmic p-forms ^ ( l o g B ) = f\p Q,x(logB) is a locally free sheaf of rank ("). Let / : (X, B) —> (Y, C) be a toroidal and equi-dimensional morphism of quasi-smooth toroidal varieties. Let dimX = n and dim Y = m. Then ^/y(log) =
fi3r(lagB)//*n^(l°gC)
is a locally free sheaf of rank n — m. Following [10] and [4], we define the structure sheaf Ox# of the real oriented blowing-up by adding the logarithms of the coordinates: Definition 4.1. (0) If there is no boundary, then Ox# = Ox-
144
Y.
Kawamoto.
(1) If (X,J3) = (A, {<)}), then k
Ox* = £ fcez>0
Px\Ox){\ogx)
where pxx denotes the inverse image for the sheaves of abelian groups, and x is the coordinate. The symbol log x is identified with a local section of px1(Ox) on X\B. We note that the multivalued function logx on X becomes locally single valued on X # . Therefore, the stalk of Ox# at (r, 9) is isomorphic to 0XiX if r ^ 0 and to OX,Q ® Z[t] if r = 0, where t is an independent variable corresponding to log x. We note that a section of Ox# is a finite polynomial on log x instead of an infinite power series. (2) If (X,B) = (Xi x X2,piBi +p*2B2), then Ox*
= (Ph\Oxf)
®prrip-i(0xi)
Px\Ox))
® P rr 1 p-l(o X a ) P ^ (
®P-x\ox)(v*2\°x*)
For example, if X = A n with 5 = div(:ri • • • xni), then we have n'
Ox*=
£ fci
PiHOxJlI^e^)*'fc„/€Z>0
i=l
In this case, the stalk of Ox* at any point over the origin of X is isomorphic to OX,Q ® Z [ t i , . . . ,tni], where the U are independent variables corresponding to the logo;*. (3) If (X,B) = (Xi/G^i/G), then Ox* = (Ox*)G. (4) We can glue together the structure sheaves of local models. Lemma 4.2. Let (X,B) be a quasi-smooth toroidal variety, x £ X a point, and let n' be the number of irreducible components Bi of B which contain x. Then Ox# <8> - I / Q \ px (C^) is a locally constant sheaf L on px(x) = (S1)™ whose fibers are isomorphic to C[t\,... ,tmi], where the tj are independent variables corresponding to the logarithms of the local equations of the Bi at x. The monodromies Mi of L around the loop in pxl{x) corresponding to the Bi are given by the following formula Mi(tj)
=ij+25ij7rv/zl.
Definition 4.3. We define the log de Rham complex on the real blowing-up by npx* = P~x ("x(log B)) ® „ - i ( 0 x ) Ox*
On Algebraic Fiber Spaces
145
for p > 0. The differential of the complex fi^-# is defined by the rule: dlogx = dx/x. The relative log de Rham complex is defined by
The differential of the complex $l'x# ,Y# is induced from that of fiV#. If B = 0, then the Poincare lemma says that A * ->•
•{!£_>....)
in the derived category of sheaves on X. We have the following logarithmic version: Theorem 4.4. (1) Cx#
=ftx#
in the derived category of sheaves on X#. (2) R(Px)Mx#^nx(logB) in the derived category of sheaves on X. (3) R(px)Xx*=nxQ°gB) in the derived category of sheaves on X. Proof. (1) This is a special case of Theorem 3.6 of [4]. For example, we check the case where (X,B) = ( A , { 0 » . Let h = £ f c hk(\ogx)k for hk e Oxja- If dh = dx/x^2(xtik
+ (k + l)hk+i)(\ogx)k
=0,
k
then we have xh'k + (k + l)hk+i = 0 for all k. Since hk — 0 for k 3> 0, we conclude that h 6 C by the descending induction on k. On the other hand, we can solve the equations x
9k + (k + l)gk+i = hk
by the descending induction on k to find g = Y^k9k(\°Zx)k hdx/x = dg.
such that
146
Y.
Kawamata
(2) We check the case where (X, B) = (A, {0}) and p = 0, that is, (px)*Ox* = Ox and R1{px)*Ox# = 0. The case p > 0 follows from the projection formula, and the general case is similarly proved. The first equality follows from the fact that log x is multi-valued on X near 0. We have an exact sequence 0 -»• Px\Ox)
-> Ox#
-> L ® Ox,o -* 0
where L = L/C is a locally constant sheaf on pxl (0) = S1 such that the stalk Lo at 0 £ S1 has a C-basis {ek}kez>0 corresponding to the (logx)fe with the monodromy action given by
M(efe)=£rj(27r>/=l)fc-^. Then H°(S\L) (px).Px\°x)
= Cex and H\Sl,L)
= 0. Hence
= (px)*Ox# ,H°{S\L)
® Ox,o =
R\px)*PxX{Ox)
and R1(Px)*Ox* =0. (3) is a consequence of (1) and (2). This follows also from Proposition 3.1.8 of [2], because we have
R{px)Xx* for the inclusion j : X\B
=
RjXx\B
-> X by Proposition 2.2.
•
In the relative setting, if B = C — 0, then we have / _ 1 ( O y ) = fi^c/y in the derived category of sheaves on X. The logarithmic version is the following: Theorem 4.5. (1)
(f*)-\oY#)*nx#/Y# in the derived category of sheaves on X*. (2) R{Px)*Wx*/Y*
- ( P r i r ^ ^ l o g ) ) ®(pr2)-v-1(CM
in the derived category of sheaves on X xY
Y*.
(pv2)-\0Y*)
On Algebraic Fiber Spaces
147
(3) - ( p r J - ^ V ^ ) ) ®(pra)-^l(Ov) (P'2)_1(CV#)
R^xUf^-HOy*)
in i/ie derived category of sheaves on X Xy Y # . Proof. (1) We check the case where (X,B) = (A 2 ,div(x 1 x 2 )) ) (Y,C) = (A, {0}), and f*(y) — xix 2 . The general case is similar. Let 5> f e l , f c 2 (log*i) f c l (logZ2) f c 2
h=
fci,fc2
for hkl,k2 € £>x,o- If d/l = dxi/xi
22 (^l^fci.fca.xi k\,k2
X2hkltk2,x2
+ (fci + l)hfcl+i,fca - (fc2 + l)^ 1 , f c 2 + i)(logx 1 ) f c l (logx 2 ) f c 2 = 0 in £lx# / y # , then we have Zl/lfci,fc 2 ,xi - X2hkuk2,x2
+ (h
+ l)/lfc 1 + l,fc2 - (fc2 + l)^fci,fc 2 + l — °
for any k\, k2- Since hkltk2 = 0 for fei S> 0 or A;2 3> 0, we prove by the descending induction on (fci,^) that there exist hk G Ox,o for k G Z> 0 such that hkltk2 = (fclfcffc2)/ifc1+fe2, and if we expand hk in a power series hk = 2 / i ak,ii,i2xix2' then we have aktilti2 = 0 unless li = £2- Therefore, /i G CV#]0On the other hand, we can solve the equations ^fci,*2 = XldktM.X!
- Z25fci,fc2,x2 + (^1 + l)3fci+l,fc 2 - (&2 + l)3fci,fc 2 +l
by the descending induction on (k\, fc2) to find 5 = J2
5fe1,fe2(logx1)fcl(logx2)fe2
fci,fc2
such that hdxi/xi = dg in f2^ # , y # . (2) We check the case where (X,B) (A, {0}), f*(y) — x\X2 and p = 0, that is (px)*Ox#
= (A 2 ,div(xix 2 )), (Y,C)
£* ( p r J - ^ O x ) ® ( p r a ) - w i ( o y ) ( P r 2 ) _ 1 ( ^ y # )
and i?1(;o3f )»Cx# = 0- The general case is similar.
=
148
Y.
Kawamata
We have p ^ ( 0 ) ^ (S 1 ) 2 and / ^ ( O ) * S 1 . The map px : p ^ ( 0 ) -» ^ ( 0 ) is given by p ^ (0i, 02) = 01 + fcLet /i = X)fe1,fc2'lfci,*!2(loga;i)*!l(loga;2)*!2 be a local section of O x * , where hkltk3 G Ox,o- The monodromies of the multi-valued functions log^i and log X2 along the fibers of px which are homeomorphic to S1 's are given by l o g X\ >-> log X\ + 27T\/—I,
l o g X2 l-> l o g X2 — 2ltV^l
.
The function h is single valued along these Sl,s if and only if there exist hk € Ox,o for fc G Z>o such that hkuk2 — {klkk2)^lk1+k2- Therefore we have the first equality. We have an exact sequence
® ( / # ) - v - 1 ( 0 y ) (/ # ) _1 (cv#)
o -> Px\ox) -> Ox#
- j . {Lx/(f#)-lLY)
® OXi0 -> 0
where L x and Ly are the locally constant sheaves on p ^ (0) == (511)2 and Py^O) = 5 1 such that the stalks Lx,(o,o) and Ly, 0 at (0,0) e (S1)2 and 0 £ S1 have C-bases {efclik2}fc1,fc2€Z>o anc ^ {efe}fe6Z>o whose the monodromies Mi, M2 around the first and second factors and M are given by
Mi(ekuk2)
= E
) (27rv / =I) fcl - J e Ji fe2,
[•
M 2 (e fcllfca ) = E j=o V
J
(27rV=l)fc3-;''efcl j , I
and the homomorphism ( / # ) _ 1 £ y —• Lx is given by
(/#)
l
ek ^ E
( • ) eJ.fc-J •
On Algebraic Fiber Spaces 149
In other words, we have
(Ml-M2)(ekuk2) = £ £(-!)<— f fcl l I"2) x(27rV=l)kl+k*-h-hejuJ2, (Mi - M 2 )((/#)" 1 e f c ) = ( / # ) - 1 e f c . We have (Cx).(/#)"1iy~(Px).^ = i r , ^(Px)^/*)"
1
^^^.
i? 1 (Px)*ix = 0.
Hence
(PxULx/f*)-1^)
* L y , R^xULx/f*)-^)
=0•
Since ( ^ ) * ^ 1 ( ^ ) = (pr1)-1(^x), ^ ( P x W ^ x ) = C ( p y ) -i(o) ® OXi0 and
we have {plx)*{Px\Ox)
®(/#)-v-(o,) (/#)_1(Oy#))
= (WirHOx)
®( P r a )-i / v 1 (o y ) ( P r 2 ) _ 1 ( ^ y # )
^ Ly ® CX,0 • Therefore, we have the desired result. (3) is obtained by combining (1) and (2).
•
We assume that / is weakly semistable in the rest of this section, and use the notation of Sec. 3. Let m! be the number of irreducible components of C which contain y e F,and let j / i , . . . ,ym' be the corresponding
150
Y. Kawamata
local coordinates. We fix y € Py1(y) restrictions of px to D and Di0t... tit.
anc
^ denote by p^, and p£,.
. the
Definition 4.6. We define the structure sheaves of D and -Dj0l...,», by the following formula: 0D = °Di0
{0x#®p-.{Ox)Px\OE))/I,
= {Ox* ®p-x\ox)P'xiPEi0
it
H))/Iio,..M
where I and /»„,... ,tt are ideals generated by logy.,- — c* (j = 1 , . . . , m') for some constants CjGC corresponding to the choice of y e pY (y)- We define
n| /c (io g ) = fi^/y(iog) ®0x oE and define the log de Rham complex on the fiber by
% = p~D&E/c(iog)) %i0
H
®P-.(OJ0
oD = npx#/Y# ®ox# oD,
= PiH^E/cOos)) ® P - ( 0 ,, <X,...,,t =
fi
x#/y#
® c x # OD.O
.t
for p > 0. We note that there is no ideal sheaf of D corresponding to / because of the monodromies. The following is easy: L e m m a 4.7. Let t' = dim.E — dim.Ej0i... tit as in Proposition 3.1. (1) Ift = t' =0, then
npE/c(hg)®oEoEio^tfEio(iogGio). (2) / / 1 ' > 0, then 0
-+ VEi0_M(hgGiQ -)• Of*'
it)
-> nE/c(\og)
®oE OEi0
-* 0
^ 0
it
where the arrow next to the last is the residue homomorphism. The following is similar to Theorem 4.5: Theorem 4.8. (1) C
D
-&•£>>
C
Di0
in the derived category of sheaves on D.
i t
-^D
i 0
it
it
On Algebraic Fiber Spaces 151 (2)
R{PD)tnpD = npE/c(iog), R(PDi0
l t ).n* i o
H
- ^ / c ( i o g ) ®OE oEio
it
in the derived category of sheaves on E. (3) = "fi/cOog)
R{PD)*^D
R(pDi0
it),cDio
it
s n^ /c (iog) ®OE oEio
it
in the derived category of sheaves on E. 5. Cohomological Mixed Hodge Complex We prove the main result (Theorem 5.2) of this paper. We assume that / is weakly semistable in this section, and use the notation of Sec. 3. We shall construct a cohomological mixed Hodge complex on E = f~1(y). On the Z-level, we put a weight filtration Wq on the complex of sheaves on R(PX)*%D -E a s m Definition 3.2 by Wq(R(Px),ZD)
= {T
~>
r
• T
-> • • •)
in the derived category of complexes of sheaves on E. We have an exact sequence 0 -> OE
-> OEV>\ -> OEW
->
• 0E[t]
-> • • •
as before, where we identified the sheaves 0E\t\ with their direct image sheaves on E. On the C-level, we define the weight filtration and the Hodge filtration on the complex £1%E ; c (log) = ClXiY(log) ®ox ®EDefinition 5.1. The weight filtration {Wq} on n^y C (log) is defined by
w,(n^ / c (io g )) = (w,(n^ /c (iDg) ® om)
-> wq+1(nE/c(iog)
® oEW)
-> — • w„ +t (n^ /c (iog) ® oEW) ->•••), where W on the right hand side is the filtration defined by the order of log poles at [2, 3.1.5].
152
Y.
Kawamata
The Hodge filtration {Fp} on £l'x,Y(\og) **(Ify y (log)) =
is defined by
a>p(n-x/Y(log))
where <7>p denotes the stupid filtration (cf. [2, 1.4.7]). We consider its restriction {Fp} on the fiber E: Fp{SlE/c{\og))
= <7>p(^/c(log)).
The following is the main result of this paper: Theorem 5.2. {(R(px)*CD,
W), (fi£ / c (log), W, F)}
is a cohomological mixed Hodge complex on E [2, 8.1.6]. The proof is reduced to Propositions 5.3 and 5.6. Proposition 5.3. (R{px),CD,W)
S
(QE/c(log),W)
in the derived category of filtered complexes of sheaves on E. Proof. Since the truncation is a canonical functor, this is a consequence of Theorem 4.8 (3) and [2, 3.1.8]. • The following is similar to Proposition 3.3: Lemma 5.4. G r ^ ( ^ / c ( l o g ) ) £* G r f (SlE/c(]og) ®
Om)
®GT%.1(n'E/cQog)®0EW)[-l] © • • • © Gr£ t (fi£ / c (log) ® Om)[-t]
©•••
Lemma 5.5. Let (X,B) and (E,G) be as in Proposition 2.6. Then the Poincare residue induces the following isomorphisms of filtered complexes: (G^(nx(logB)),F)
-
{n'm[-q],F[-q])
0
nBi^
iq[-q],F[-q]
^l
(Grf (fi^(logB)®0x
OE),F)
= \&{n'alt])(*-')[-q],F[-q}\
.
On Algebraic Fiber Spaces
153
We note that G^ = E by definition. Proof. The first formula is in [2, 3.1.5.2]. The second is similar.
•
Proposition 5.6. (Gr™(n-E/c(log)),F) is a cohomological Hodge complex of weight q on E [2, 8.1.2]. Proof. We have to prove that (#"(£, Gr£t(^/c(log) ®
Om)[-t]),F)
^ ( # " - < ( £ , G r £ t ( ^ / c ( l o g ) ® Om)),
F)
is a Hodge structure of weight n + q: Hn-t(E,G^+t(n-E/c(log)®0Elt]))
=
^ F n F " . p
By Lemmas 4.7 and 5.5, we can write ( G r £ t ( n j . / c ( l o g ) ® O e [ t,), F) - ( £
n £ [ - g - t],F[-q - t})
for some set of the strata G. Thus the problem is reduced to prove
Hn-g-2t{E,
fl'G) = ^2 F p _ 9 _ t n Fn~p-t. p
But this is the usual Hodge theorem on G.
O
By [2, Scholie 8.1.9], we obtain the following corollary (cf. [3]): Corollary 5.7. (1) The spectral sequence wE™ = Hr+i(E,Gr™p(R(px)*CD))
=>
H^{D,C)
degenerates at Ei. (2) The spectral sequence FE™
= H«(E,npE/c(\og))
=• H"+"(D,C)
degenerates at E\. Combining with Corollary 2.5, we obtain the following ([7] and [8]): Corollary 5.8. i?'/,,fij^,y(log) is a locally free sheaf for any p, q.
154
Y.
Kawamata
References [1] D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0, Invent. Math. 139 (2000) 241-273. [2] P. Deligne, Theorie de Hodge II, III, Publ. Math. I.H.E.S. 40 (1971) 5-57; 44 (1975) 5-77. [3] T. Fujisawa, Limits of Hodge structures in several variables, Compositio Math. 115 (1999) 129-183. [4] K. Kato and C. Nakayama, Log Betti cohomology, log etale cohomology and log de Rham cohomology of log schemes over C, Kodai Math. J. 22 (1999) 161-186. [5] Y. Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981) 253-276. [6] Y. Kawamata and Y. Namikawa, Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties, Invent. Math. 118 (1994) 395-409. [7] J. Kollar, Higher direct images of dualizing sheaves I, II, Ann. of Math. 123 (1986) 11-42; 124 (1986) 171-202. [8] N. Nakayama, Hodge filtrations and the higher direct images of canonical sheaves, Invent. Math. 85 (1986) 217-221. [9] U. Persson, On Degeneration of Surfaces, Mem. AMS. 189 (1977). [10] J. Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1976) 229-257.
LOCAL H O L O M O R P H I C ISOMETRIC E M B E D D I N G S ARISING FROM CORRESPONDENCES I N T H E R A N K - 1 CASE Ngaiming Mok Department of Mathematics, The University of Hong Kong Pokfulam Road, Hong Kong nmok@hkucc. hku. hk
Let U and fi' be bounded symmetric domains, equipped with Bergman metrics. We are interested in characterizing germs at a base point of holomorphic isometric embeddings / from fl into il'. The question is whether they are necessarily totally-geodesic embeddings. The essential case is when fl is irreducible. When fi is of rank > 2, the zeros of holomorphic bisectional curvatures are preserved, which is enough to imply that / is a totallygeodesic embedding as a consequence of the Gauss equation. When fi is of rank 1, i.e., biholomorphic to the unit ball Bn, holomorphic bisectional curvatures are strictly negative, and arguments of Hermitian metric rigidity do not apply. The problem is then completely different. A general tool for the study of holomorphic isometric embeddings for complex manifolds equipped with real-analytic Kahler metrics is the use of local Kahler potentials. It led to the powerful notion of the diastasis introduced by Calabi [2] in 1953, which was used in [2] to treat the problem of characterizing local holomorphic isometric embeddings in Euclidean spaces, among other things. The method was applied by Umehara [7] to study Kahler-Einstein local complex submanfolds of the complex hyperbolic space, proving that all such local submanifolds are necessarily totally geodesic. His result implies that every local holomorphic isometric embedding between two complex hyperbolic space forms is totally geodesic. In another direction, starting with the question of characterizing modular correspondences among correspondences K on compact quotients X of 155
156
N. Mok
bounded symmetric domains by torsion-free discrete groups of automorphisms, Clozel-Ullmo [3] were led to study among other things certain local holomorphic isometric embeddings of the n-dimensional complex hyperbolic space forms into finite products of n-dimensional complex hyperbolic space forms. They proved in the case of n = 1 that such local mappings are necessarily totally geodesic. They based their proof first of all on easy parts of the method of Calabi's to obtain functional equations on potentials for Bergman metrics, and relied also on the theory of normal forms for a real hypersurface containing a germ of holomorphic curve, due to d'Angelo [4], for the key step of obtaining an algebraic extension of a germ of holomorphic isometric embedding. A straightforward generalization of the same proof for general n appears difficult. In the current article we adopt the same approach as in [3], but introduce a technique for algebraic extension making it applicable to arbitrary dimensions. The technique consists of polarizing functional equations, as is commonly used for the Schwarz Reflection Principle, and the problem boils down to a question of verifying non-degeneracy. In place of one real-analytic functional equation arising from equality of Kahler potentials we have an infinite number of algebraic functional equations on the germ of holomorphic isometric embedding. The proof of algebraic extension amounts to showing that the infinite number of constraints on the extension of a germ of holomorphic isometric immersion are sufficiently algebraically independent to define a variety of the proper dimension containing the graph of the germ, forcing the germ to extend globally and algebraically. The non-degeneracy is checked using simple estimates involving elementary symmetric functions. 1. Statement of the Main Result and Preliminaries 1.1. Main
result n
For flCC C a bounded symmetric domain we denote by ds^ the Bergman metric on CI. Note that (Cl, ds^) is of constant Ricci curvature —1, by Mok [6, proof of Proposition 3 in Chapter 4, p. 59]. For r > 0 we write B™ = {z e C" : ||z|| < r}, Bn = Bf the unit ball. The main purpose of this article is to give a proof of Theorem 1.1. Theorem 1.1. Let e be such that 0 < e < 1; p, q be positive integers; and f : {B™,qds2Bn\B") -> {(Bn)p,ds?B„y) be a holomorphic isometric embedding. Writing f = ( / * , . . . , / p ) , assume that for each k, 1 < k < p,
Local Holomorphic
Isometric
Embeddings
157
/ : 5™ —> Bn is of maximal rank at some point. Then, q = p and f is the restriction to B™ of a holomorphic totally-geodesic embedding F : (Bn,pds%n) -> ((Bny,dsfBn)p). Theorem 1.1 for n = 1 is the same as Clozel-Ullmo [3, Theoreme 2.2]. The analogue of Theorem 1.1 for bounded symmetric domains 0 of rank > 2 in place of the unit ball Bn follows readily from arguments on Hermitian metric rigidity of Mok [5], as explained in the Introduction. From the proof, the non-degeneracy condition on individual direct factors fi of / is unnecessary. For our purpose we are contented with proving Theorem 1.1 under the non-degeneracy condition that fi : Bn —> Bn is of maximal rank at some point, as stated, since the problem originated from a question regarding isometric holomorphic correspondences, in which case the non-degeneracy condition is always satisfied.
1.2. Equality
of
potentials
Without loss of generality we may assume that /(0) = 0 and that fk : B™ —> Bn is unramified at 0. The Bergman metric on Bn is given by ds2B„ = 2Re Y/9ijdzi ® dzi, where gi:j = ^ j - - (n + l)log(l - ||z|| 2 ). For z G (Bn)v write in Euclidean coordinates z' = ( z 1 , . . . ,zp), zk = (zh,... ,z£) for 1 < k < p. Write ||zfc|| = £ L i \z\|2. We can take as Kahler potentials for dsB„ resp. ds^B„^p the real-analytic functions — (n + l ) l o g ( l - ||z|| 2 ) resp. - ( n + l ) ^ f c = i 1 ° g ( 1 - ll^ll 2 )- From the assumption that / : (B£,qdsBn\B?) —> ((Bn)p,ds?B„^p) is a holomorphic isometric embedding it follows that
-v^I<wf>g(i - ||/fc||2) = - g x^ia9io g (i - ||Z||2), fc=i
hence - f > g ( l - ||/ fc || 2 ) = - 9 l o g ( l - ||z|| 2 ) +Reh
(1.1)
fc=i
for some holomorphic function h on B™. Since f(0) — 0, comparing Taylor expansions we conclude as in Clozel-Ullmo [3] that h = 0.
158 N. Mok
1.3.
Polarization
We polarize Eq. (1.1), as follows. Write (z,w); z = (z\,... ,zn), w = (wi,... ,wn); for Euclidean coordinates on Bn x Bn. Then, on B™ x B™ write
- ^log (i - ^ / f M / f c ) ) = -glog (i - it****) + H^w"> • Expanding in power series we have H(z,w)=
£
Hu-z'w-1,
where I = (ii,... ,in) and J = ( j i , . . . , jn) are multi-indices of nonnegative integers z1 = z\,... , z£*, etc. Restricting to the diagonal {z = w} we had H(z; z) = 0, i.e., J2I,J^O Hijz'z3 = 0, hence Hjj = 0 for all I, J, so that in fact H(z; w) = 0 on B™ x B™. In other words, we have
-£tog(i-;£7fH/*(*)) = - g i o g ( i - £ « * * ) . fc=i
y
j=i
j
\
i=i
/
(i.2)
2. Algebraic Extension of a Germ of Holomorphic Isometric Embedding 2.1. Statement
of Proposition
2.1
We consider the germ of / at 0 as a germ of vector-valued holomorphic function, / = (f1,... , fp); fk = (fk,... , / * ) . As the main step towards establishing Theorem 1.1 we proceed to prove the following proposition. Proposition 2.1. Let VQ c C n x Cpn be the germ of graph offatO. there exists an affine algebraic variety V C C n x C p n extending VQ.
Then,
The starting point is the identity (*). For any w € B™, coordinates off / i ( z ) . • • • . /n(*). • • • . /n( z )> •••>/£) satisfy a system of algebraic relations. The germ of graph VQ of / at 0 lies in the common intersection of affine-algebraic varieties Sw denned by w £ B™. The question is one of non-degeneracy: viz., whether the algebraic relations are sufficiently algebraically independent so that the common intersection S of Sw reduces to a variety whose germ at 0 is of dimension n. We consider first of all the case where n = 1.
Local Holomorphic
2.2.
Prof of Proposition
Isometric
Embeddings
159
2.1 for n = 1
Write z = zx and / ( z ) = ( / ^ z ) , . . . , / p ( z ) ) . We have the identity - X > g ( l -JHw)fk(z)) fc=i
= - g l o g ( l - iDz)
for z,w E Bl- Let (z; £*,... , £p) be a point on the germ of graph Vb of / at 0. Then, for each w G B\, (z; £ x , . . . , £p) satisfies the identity - £
log(l - 7^HC f c ) = - 9 log(l - z),
(2.1)
n(i-7^Rcfc) = (i-^) 9 -
(2.2)
i.e.
k=\
Let S^ be the afnne-algebraic hypersurface in C 1 x C p , with coordinates (z;C), defined by Eq. (2.2) with w fixed, w € B\. We have Vo C P L e s 1 &*> : = *^- The S e r m °f ^o agrees with an irreducible branch of the germ of 5 at 0, if we can show that S is of dimension 1 at 0. This is the case if the fiber of the projection IT : S —> C 1 into the first factor of C 1 x C p has finite fibers over any z € B\. To prove the latter it suffices to find a uniform a-priori bound on ||C|| for all points {z;Q € S, \z\ < e. We will prove this starting with Eq. (2.2). Expand fk(w) in power series fk(w)
= a\w + akw2 + akw3 -\
.
As assumed af ^ 0 for 1 < k < p. We have
n(i-7^Hc fc )=n f1- f £ < > m )
fc=l
\
\m=l
/
/
+w> fe^RcV - £^< f c ) - • • • • (2-3) Generally, p
oo
fe
n(i-7^c )=i+E(- i ) m T -^ i ----^ p ) , z ) m k=l
m=l
(2-4)
160
N. Mok
where for C = ( C 1 , - - - , C p ) , r ? f c = 4 C f c , p
n(c) = E ° ^
v
fc
= E,?fe;
k=\
(2-5)
fc=l
7a(o = E ° f ° i c V - E ^k
= £ . y - E ( 4 V = E^v+£W°A = -4;(2-6) fc<€
75,(0= £
k
\
a
i J
k<e
a
k
i
fc
^K cV-EScV+E^c
k<e<s
k^e
fc
k
= E ^W+E c*^v+Edfer?fe for some c^'dfc • • • • ( 2 - 7 ) fc<^<s k
£
^•••r1k*.
(2.8)
ki<---
Writing 77 = (77 1 ,... , rf), of Xp - MvW-1
rj1,...
+ C72tq)Xr-2
,rf
-•••
are the roots, counting multiplicities,
+ ( - I f " V p _!(77)X + ( - 1 ) ^ ( 7 ? ) = 0 . (2.9)
For 1 < m < p we have generally rm(0 = M f ) +
ft.-i(l),
where Pm_i(r]) is a polynomial in 77 = (77 1 ,... ,rf) From (2.2) and (2.4) we have for (2; 0 £ Vb, r
m ( 0 = —77
^™,
of t o t a l degree < m— 1.
forl<m
m\{q-m)\ ,7-m(0=0,
(2-10)
^
^
form>g + l.
From (2.10) and (2.11), for 1 < m < p, we have either maxfc |7?fc| < 1, or \
(2.12)
Local Holomorphic Isometric Embeddings 161 for some positive constants C\, C^. Recall that 77fc = ak£k. For z £ B\{z,£) € S, write M0(() = max{|
(2.13)
In particular there exists C5 > 0 such that for each z S BE' we have
M0(cr
(2.14)
It follows that for S = f]weBi Sw, the fibers of the projection S c C ' x C - * C 1 are finite for z £ B\, so that the germ of S at 0 £ C 1 x C p is of dimension 1. The irreducible component V of S containing Vo, the germ of Graph (/) at 0, gives the desired extension of Vo as an affine-algebraic variety, as desired. 2.3.
Proof of Proposition
2.1 for general
n
As in the case of n = 1 let (z; C1, • • • , Cp) = (21 > • • • , zn\ Ci, • • • , <„> • • • » Cf,... , Cn) be a point on the germ Vo of the graph of / at 0. From (*) we have the following identity analogous to (2.2)
n(i-E^NcH = f i - i : ^ .
fc=i \ Expand fk(w)
j=i
1
in power series, fk = (fk,... # M = £ ahw' 7^0
fc=l \
j=l
I
fc=l
\
\
i=i
/
(2.i5)
, /*)
> ™J = <
w
2 • • • wn ;
\/^0
p n 1 — j y j Wi I Y j Y J ^iC? 1 + higher order terms i=\ \fc=ij=i J (2.16)
162
N. Mok
- l + ^(-l)l7lr7(CK.
(2-17)
Without loss of generality we may assume, by unitary changes of coordinates of each n-tuple « f , . . . ,<*), that df0{-£-) = a J i 5 ^ + • *" + a i i S ^ > axl ^ 0 for 1 < k < p. Then, the coefficient of it/T in (2.16) is given by Y^k=i a nCi : - Computing coefficients of w\, wf, etc., we have, analogous to (2.10), the formulae Tm,0
0(C) = ^mia^ClWlCl
• • • ,^Cl)
+ Qm-l(C) ,
(2.18)
1
where Qm-\ is a polynomial in the pn variables £ = (C ) of total degree < m— 1. There is one principal difference in the case n > 1, in that Qm-i(C) involves more than the n variables (Cf, • • • , C? )> so that we do not get an a-priori bound on (Cfi••• iCF) right away. For (z,C) G S write Mo(C) for max{|C*| : 1 < k < p, 1 < j < n } , and M{() = max{l, Af0(C)}. From the analogue of (2.13) we obtain only that, for any z G B™, and any k, l
(2.19)
for some constant Co. We conclude therefore ICil^CMtO'r1
(2.20)
for some constant C. Note here that (Cf) are p of the coordinates (Cf )J where we have chosen Euclidean coordinates on C p n = C n x • • • x C n (p times) under unitary changes of bases (so that the form of the potential for Bergman metrics do not change). We may put an index 1, writing C — (Cm,) to mean that the coordinates have been chosen to be adapted to g£-, so that 4fo(g§-) is of the special form
{b\ = a\x in previous notations.) For each £, 1 < £ < n, we may choose similarly Euclidean coordinates (Cm,) o n C p n so that
Since any two norms on a Euclidean space are equivalent, we may replace the sup-norm MQ(C) implicit in (2.20) by the Euclidean norm ||£||, which is
Local Holomorphic
Isometric
Embeddings
163
invariant under unitary changes of coordinates. We have thus proven that for z G B£, (z, C) G S, and for 1 < k < p, 1 < I < n, IC^I^maxCC'HCH^.l)
(2.21)
for some absolute constant C. To get an a-priori bound for ||£|| it remains to observe that m a x ^ KmJ is equivalent to ||£||. Consider £ = (C\ • • • , Cp) as a vector and write ™
(k
=
d
22tft)3~^7k~•
We have
K> where (•, •) denotes the Hermitian inner product on the Eucliean space C" (the fc-th direct summand of the image space Cpn). Now R = d/o (g§-), and the pn vectors df£ (g£-), • •. ,dffi (gg-); 1 < k < p; are linearly independent by assumption, so that A^e = 0 for all £, 1 < I < n if and only if C,k = 0. In other words, the linear mapping A : C n —> C n defined by •MCfe) = (Cmi> • • • >((n)n) i s a n isomorphism, so that max< \Cmi\ is equivalent to ||Cfe||- In particular, maxfc^ \Cmt\ i s equivalent to ||£||, so that for some absolute constant C"
||C||<max(C"||C||^\l), implying IICH < Constant, uniformly for (z, 0 S S, z G £ " .
(2.22)
The rest of the argument for n = 1 works verbatim for general n to complete the proof of Proposition 2.1, as desired. 3. Proof of Theorem 1.1 Theorem 1.1 for n = 1 was proven by Clozel-Ullmo [3]. For the concluding argument we restrict ourselves to the case n > 2. In this case it is easy to deduce Theorem 1.1 from the algebraic extension and the functional Eq. (2.2) by means of a classical result of H. Alexander [1] in relation to holomorphic mappings on the unit ball. We have
164
N. Mok
Theorem 3.1 (Alexander [1]). Let Bn denote the unit ball of the complex Euclidean space C™, and dBn denote its boundary. Assume n > 2. Let b € dBn and let U be a connected open neighborhood of b in C n . Let g : U —»• Cn be a nonconstant holomorphic mapping such that g(UC\dBn) C dBn, Then there exists a biholomorphic automorphism G : Bn —> Bn such that G\unBn = g\unBnWe continue with the proof of Theorem 1.1 for n > 2. Restricting the functional Eq. (2.2) to the diagonal, we have
n(i-n/T)=(i-Niy.
(3.i)
Recall that / was originally defined on B™, /(0) = 0. By Proposition 2.1 the germ of graph VQ of / at 0 can be extended to an algebraic variety V C C n x Cpn. In order to apply (3.1) we proceed to carefully choose some open set on which / can be defined as a 'univalent' holomorphic map, as follows. Denote by -K : V —¥ C™ the projection to the first factor C™. Then, there exists an algebraic hypersurface H C C™ such that, writing H = 7r - 1 (iJ), n\v_fI -> C™ — H is a finite covering map. Even though 7r|y0 is unramified, it may still happen that 0 e H. Pick a point a £ f?" — H and write a for f(a). Let 7 C Bn — H be a smooth path joining a to a point b on dBn — H. Let T C C" — H be a contractible open neighborhood of 7. There is an open subset T C V — H such that a, s T and such that ir\f is a biholomorphism of T onto T. Choosing 7 and T properly we can adjoin Vo to T so that n \vuf r e m a m s a biholomorphism onto its image 5™ U T := T". On T" we may thus regard / = (f1,... , fp) as being a univalent holomorphic map satisfying (3.1). The connected open set V C C" contains an open neighorhood U of b G dBn. Applying (3.1) to U, we conclude that for some index k, where 1 < k < p, 1 - ||/ fc || 2 vanishes on UC\dBn. In other words, ||/ fc || = 1 on Ur\dBn. Since fk is nonconstant, by Theorem 3.1 there exists Fk £ Aut(Bn) such that Fk\unBn = fk\unBnNote that Fk extends to a rational map on C n . We also denote by Fk the extended map. Since T" D U is connected it follows that actually Fk agrees with fk on T". In particular, Fk(0) = fk(0) = 0. This forces Fk to be a unitary mapping, so that ||/ fe || 2 = ||z|| 2 . We may take k to be p and reduce (3.1) to n £ j ( l - \\fk\\2) = ( 1 - ||z|| 2 )« - 1 . li q > p the same argument applies inductively to show that each fk is
Local Holomorphic Isometric Embeddings 165 the restriction of a unitary mapping, and t h a t in fact q = p. In this case / is a totally-geodesic embedding. If q < p then we will reach a point where, rearranging the indices if necessary, we have Il£~^(l — ||/ f e || 2 ) = 1. Since / f c (0) = 0 for 1 < k < p — q this would force each fk to vanish identically on a neighborhood of 0, contradicting the assumptions. T h e proof of T h e o r e m 1.1 is complete.
Acknowledgement T h e author would like to t h a n k the organizers of the Wei-Liang Chow and Kuo-Tsai Chen Memorial Conference held on October 9 - 1 3 , 2000 at Nankai I n s t i t u t e of Mathematics, especially Professor S. S. Chern, for their kind invitation. T h e author would also like to t h a n k Laurent Clozel for introducing him t o t h e circle of problems involving t h e differential geometry of holomorphic correspondences, especially the problem of characterizing m o d u l a r correspondences in t h e rank-1 case as t r e a t e d here, a n d for spotting a blunder in t h e concluding argument of the original version of t h e article.
References [1] E. Alexander, Holomorphic mappings from the ball and the polydisc, Math. Ann. 209 (1974) 249-256. [2] E. Calabi, Isometric imbedding of complex manifolds, Ann. Math. 58 (1953) 1-23. [3] L. Clozel and E. Ullmo, Correspondances modulaires et mesures invariantes, preprint. [4] A. d'Angelo, Real hypersurfaces, orders of contact, and applications, Ann. Math. 115 (1982) 615-637. [5] N. Mok, Uniqueness theorems of Hermitian metrics of seminegative curvature on quotients of bound symmetric domains, Ann. Math. 125 (1987) 105-152. [6] N. Mok, Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds, Vol. 6, Series in Pure Mathematics, World Scientific, Singapore-New Jersey-London-Hong Kong, 1989. [7] M. Umehara, Einstein Kahler submanifolds of a complex linear of hyperbolic space, Tohoku Math. J. 39 (1987) 385-389.
MULTIPLE POLYLOGARITHMS: A N A L Y T I C C O N T I N U A T I O N , M O N O D R O M Y , A N D VARIATIONS OF M I X E D H O D G E S T R U C T U R E S Jianqiang Zhao Department of Mathematics, University of Pennsylvania PA 19104, USA jqz@math. upenn. edu In this exposition we shall describe a new way to analytically continue the multiple polylogarithms by using Chen's theory of iterated path integrals. Then we explicitly determine the good unipotent variations of mixed Hodge-Tate structures (MHS) related to multiple logarithms and some other multiple polylogarithms of lower weights. Following Deligne and Beilinson we define the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. At the end, motivated by Zagier's conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms. The main results of this paper with complete proofs will appear elsewhere.
1. Introduction In recent years, there is a revival of interest in multi-valued classical polylogarithms and their single-valued cousins. In the mean time there have been a number of generalizations of these functions such as Grassmannian polylogs [21,22,25], Chow polylogs [16], elliptic polylogs [3,28,30,36], p-adic polylogs [9], infinitesimal (p-adic) polylogs [6,13], finite polylogs [5,13,29], and multiple polylogs [14,18,19,20]. For any positive integer m i , . . . ,m„, Goncharov [14] defines the multiple polylogs of complex variables as follows:
167
168
J. Zhao ™ fc l ~ f c 2 . . .
Li
(x,
x)=
V
2
*
0
*
„kn
1ZL_
b-|<1
^
(1.1) Conventionally one refers n as the depth and Jf := mi + • • • + m n as the weight. When the depth n = 1 the function is nothing but the classical polylogarithm. More than a century ago it was already known to H. Poincare [31] that hyperlogarithms
J
Jbn
Jb2 Jbi *1
_
a
l
dti
dtn
^2 — 0.2
tn — an
are important for solving differential equations. Notice that the multiple polylogarithm m i - l times
^•"-
(I1
"
,) =
mn — 1 times
/ f l i . O , . . . , 0 , . . . , a n , 0 , . . . ,0 \ ft (o.o,...1o o, oJ...,o1J'
(L2)
where aj = l/(xi • • • xn) for 1 < i < n. It is an iterated path integral in the sense of Chen [7] whose path lies in C. One thus can easily enlarge its domain of definition to some open subset of C n . However, it is not obvious that this actually gives a genuine analytic continuation in the usual sense. It is one of our goals in this paper to analytically continue the above function to Cn as a multi-valued meromorphic function using Chen's iterated path integrals with paths all lie in C n . Note that even the classical polylogarithm is not holomorphic on C, for example, Li\(x) = — log(l — x) is clearly multi-valued. In early 1980s Deligne [10] discovers that the dilogarithm gives rise to a good variation of mixed Hodge-Tate structures. This has been generalized to poly logarithms (cf. [23]) following Ramakrishnan's computation of the monodromy of the polylogarithms. The monodromy computation also yields the single-valued variant Cn{z) of the polylogarithms (cf. [1,37]). These functions in turn have significant applications in arithmetic such as Zagier's conjecture [37, p. 622]. On the other hand, as pointed out in [17,19], "higher cyclotomy theory" should study the multiple polylogarithm motives at roots of unity, not only those of the polylogarithms. For this reason we want to look at the variations of mixed
Multiple Polylogarithms:
Analytic
Continuation,
Monodromy
169
Hodge structures associated with the multiple polylogarithms and see how far we can generalize the classical results. According to the theory of framed mixed Hodge-Tate structures the multiple polylogarithms are period functions of some variations of mixed Hodge-Tate structures (see [2], [13, Sec. 12] and [14, Sec. 3.5]). Wojtkowiak [35] studies mixed Hodge structures of iterated integrals over CP 1 \{0,1,00} and investigates functional equations arising from there. In this paper we adopt a down to earth approach different from [35] and compute explicitly the variations of mixed Hodge-Tate structures related to the multiple logarithms £ n ( z i , . . . ,xn) := i i 1 ;
;
i (xi,...
, xn).
n times
The key step of this approach is our new definition of analytic continuation of the multiple polylogarithms using Chen's iterated path integrals over CPn\Dn with some non-normal crossing divisor Dn. In order to have "reasonable" variations we should be able to control their behavior at "infinity" Dn. This requires us to deal with the natural extension of the variations to the infinity using the classical result of Deligne [11, Proposition 5.2]. By the same idea we are able to treat all the weight three multiple polylogarithms and present a result for the double polylogarithms. We observe that the old form (1.2) of polylogarithms is not suitable for the investigation of the MHS at the infinity. As an important application of the above explicit computation, in the last section of this paper we describe an approach to computing the singlevalued real analytic version of the multiple polylogarithms following an idea of Beilinson and Deligne [1]. For example, we find the single-valued real analytic double logarithm £i,i{x, y) = lm(LiiA(x,
y)) - arg(l - y) log |1 - x\ x{l - y) - arg(l - xy) log x-1
^(fff)-^)-^> where Ci{z) is the famous single-valued dilogarithm. Similar identities in weight three case are also listed. It is a remarkable phenomenon that in all
170
J. Zhao
these identities no terms of lower weight occurs which is drastically different from the classical situation. The motivation of this paper comes from [15, Sec. 2.3] where the HodgeTate structures coming from the double logarithms are discussed, and from [1] where an elegant construction of the single-valued real analytic version of classical polylogarithms are given. The author wishes to thank his advisor Sasha Goncharov for his encouragement and generous help and R. Hain for answering some of my questions concerning the good unipotent variations of mixed Hodge structures. H. Gangl kindly informed the author of the preprint [35] of Wojtkowiak in which conjectures generalizing Zagier's are also considered. The author also thanks the referee whose comments and suggestions make the exposition clearer. I'm very glad to have this opportunity to thank all the faculty and staff in Nankai Institute of Mathematics and Mathematics Department of Nankai University for their kindness, help and encouragement while I was an undergraduate student there more than a decade ago. 2. Analytic Continuation of Multiple Polylogarithms In this section we define the analytic continuation of the multiple polylogarithms different from (1.2) by using Chen's theory of iterated path integrals. 2.1. Chen's theory of iterated
path
integrals
The primary references of this subsection are two of Chen's papers [7] and [8]. For a 1-form f(t)dt over R the integral J f(t)dt is understood in the usual way. For r > 1, define inductively
/ f1(t)dt---fr(t)dt=
Ja
f ( f
Ja
\Ja
h{T)dT.--fr^{T)dr\fT{t)dt. /
When r = 0, set the integral to be 1. For example, the classical polylogarithm dt dt dt
Lin(x) = I — Jo 1
t t
t
(n—1) times
More generally, let wi, u>2,... be 1-forms on a manifold M and let a [0,1] -> M be a piecewise smooth path. Write a*Wi — fi(t)dt
Multiple Polylogarithms:
Analytic
Continuation,
Monodromy
171
and define the iterated path integral [ Wi---Wr= Ja
f
f1{t)dt---fr{t)dt.
JO
Here we follow Chen's original convention which is different from the one adopted by Deligne [12] who writes Jawr---w\ to mean our Jaw\ • • • wr. The following results are crucial for the application of the Chen's theory of iterated path integrals. Lemma 2.1. Let Wi (i > 1) be C-valued 1-forms on a manifold M. (i) The value of J w\ • • • wr is independent of the parameterization of a. (ii) If a, /3 : [0,1] —> M are composable paths (i.e., a(l) = /3(0) or a followed by /3), then /
Wi • • • Wr — Y j
Ja3
/
U>i • • • Wi
„._„ J a
/ U>i+\
• • • Wr .
JB
3=0
Here, we set J <$>\- • • 4>m = 1 if m = 0. (iii) For every path a, /
J a-1 (iv) For every path a,
Wi • • • wr
= (—l)r
/ wr
• •• w\.
Ja
/ wi • • • wr / wr+i • • • wr+s = X ] / Ja
Ja
a
W
°W '''
w
v(r+»)'
Ja
where a ranges over all shuffles of type (r,s), i.e., permutations a of r + s letters with
172
J. Zhao
2.2. The index set &(m1,...
,mn)
Our analytic continuation of the multiple polylogarithms of depth n will be produced by some Chen's iterated path integrals over C n . To write down the formula explicitly we need to introduce an index set with two different kind of orderings.
2.2.1. Definition Define the index set 6 ( m i , . . . , m n ) — { i = ( i i , . . . ,in) : 0 < it < mt for t = 1 , . . . , n } and the weight function | • | on 6 ( m i , . . . , m n ) by | ( i i , . . . ,in)\ = ii H
+in.
For brevity, we write 0 = ( 0 , . . . ,0) € &(mi,... , m n ) which is the only index of weight 0 in 6 ( m i , . . . , m n ) and IK = ( m i , . . . , m n ) e © ( m i , . . . , m n ) which is the only index of the highest weight K := mi + • • • + m„ in 6 ( m i , . . . , m„). We also define the depth function of the index ( i i , . . . ,in) by fl{i : it ^ 0}, i.e., the number of nonzero components.
2.2.2. A complete ordering The complete ordering is defined as follows. Let i = ( i i , . . . ,in) and j = (ji, • • • ,jn)- If |i| < | j | then i < j (or, equivalently, j > i). If |i| = | j | then i > j if max{i t : 1 < t < n) > max{j t : 1 < t < n}. Otherwise, we compare the second largest components of i and j , and so on. If { i i , . . . , in} — {ji, • • • > jn} a s two sets then the usual lexicographic order from left to right is in force with 0 < 1 < • • •. For instance, (0,0,1) < (1,0,1) < (1,1,0) < (0,2,0) in 6(1,2,1). Remark 2.3. In the multiple logarithm case, namely, mn = 1, there is a one-to-one correspondence between non-negative integers less than 2™. Thus one is tempted tional order of positive integers in binary forms. However in our situation.
when mi = • • • = © n and the set of to use the conventhis is not suitable
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173
2.2.3. A partial ordering and the retraction map The partial ordering is defined as follows. Let i = (i\,... ,in) and j = O'l. • • • >in)- We set j ^ i (or, equivalently, i >- j) if j t < it for every 1 < t < n. For example (0,0,1,0) -< (0,1,1,0) in 6(1,1,1,1) but (1,0,0,0) ^ (0,1,1,0) and (1,0,0,0) )/- (0,1,1,0). Clearly j -< i implies j < i but not vice versa. Suppose i has depth k with iTs ^ 0 for 1 < s < k while j has depth I and j t r ^ 0 for 1 < r < I. If j -< i then we can write tr = rar for 1 < r < I. For such i and j we define the i-th retraction map p; from 6 ( m i , . . . , m„) to &(iTl,... ,iTk) as follows. The entry of pi(j) is j T a r if it is at the arth (1 < r < I) component and 0 at all other components. For instance P(020io)(01000) = (10) G 6(2,1). In particular, p\(i) = (in,... ,iTk) has highest weight in &(iTl,... ,iTk)2.2.4. Vector indices Let &K(mi,... ,m„) be the set of .ftT-tuples "j* = ( J i , . . . , j i f ) of 6 ( m i , . . . ,mn) such that |jt| = t and j i -< ••• < ]K = l x - One may think "j* as a length K queue of indices of 6 ( m i , . . . ,m„) in which each index is produced by increasing some component of the preceding index by 1. 2.2.5. Additional
notation
We fix u s := ( 0 , . . . , 0 , 1 , 0 , . . . , 0) G 6 ( m i , . . . , m„) of weight 1 throughout the exposition, where the entry 1 is at the s-th component. Whenever the s-th component is of i satisfies is < ms we can increase is by 1 to get a new index which is denoted by i + u s . If is > 0 we similarly define i — u s as the index with the s-th component of i decreased by 1. Fix v s = IK — ms\is G 6 ( m i , . . . , m n ) whose components are nonzero except at the s-th position. When mi = • • • = mn = 1 we write 6 ( 1 , . . . , 1) = 6 „ . 2.2.6. Transposition functions Let "j* = (jii • • • JAT) € &K(mi,... Jr
=
, m n ) . For 1 < r < K we write
u
J r - 1 + s — (*li • • • )£«>«,... , 0, ta, . . . , tn) ,
0 < s
+
l,tajt=0.
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J. Zhao
Here if a = n + 1 then the last nonzero component of j r is t3. We define the transposition functions on i = (i\,... ,in) 6 &(m,... , m) with m — m a x { m i , . . . ,mn} by T0r = id,
rr(i) = (v(i),...,V(n)))
where if ta > 1 or a = n + 1 then a = id whereas if fa = 1 and a < n then cr is the transposition in the symmetric group of n elements that exchanges s and a. 2.3. Analytic
continuation
of multiple
polylogarithms
In this section we define the analytic continuation of the multiple polylogarithms which will be used to calculate the monodromy of multiple polylogarithms. Let x — (xi,... , xn) be a variable over C n . Define Sn = Cn\Xn where the divisor Xn=lx£Cn: [
J ] Xiil-xj) l
J| (l-xr--xk) l<j
= o\
•
(2.1)
J
Set the domain Dn = Ux!,...
,xn) eCn
: Xj - - < -,j
= 1 , . . . ,n\
c
Sn.
Suppose the depth of i = ( i i , . . . ,in) is fc and iTl ^ 0 , . . . ,iTk ^ 0. We define at = a t (x) := (xt • • • z n ) - 1 ,
1 < t < n,
and Tm+l-l
x(i) = y = (yi,...
,3/it),
ym =
fj a=T„
/ ..
xa =
Tm+1 a
,
1 < m < k,
rm W (2.2)
with Tfe+1 = n + 1 and a n + i = 1. We also write a m (y) = (y m • • -j/fc) l = a T m (x). Note that x(i) e Cfc which is the reason why we call k the depth ofi. We begin with some 1-forms which will be used to express the multiple polylogarithms. Take "j* e &K(mi,... , m n ) and j r = j r _ i + us as given in
Multiple Polylogarithms:
Subsec. 2.2.6. For any (Si,... r = K and
Analytic
Continuation,
Monodromy
175
, 6K) € &K, namely, St = 0 or 1, let y = x if
y = (y1,...,y,)=X(r;r++1io...oTfK(jr))
if
l
where I is the depth of j r because the transposition functions do not change the depth of an index. We let tai ^ 0 , . . . , tai ^ 0 and s = a\ (because ts =£ 0) and set 0,
if ts > 1 and dr = 1,
dy\/y\ ,
if ts > 1 and Sr = 0 ,
rr,5r
f(y) ==
- yx),
if ts = 1 and Sr = 0 ,
dy\/y\(y\ - 1), if A < I, ts = 1 and Sr = 1, 0,
if A = /, t3 = 1 and J r = 1.
Obviously u>!i.*"(y) is always a closed 1-form whose singularities lie only along Xn. Proposition 2.4. Let 0n = 0 be the origin in Cn. Let L\_\r=iWr the iterated integral f wi • • • WK • Then for every x € Dn Lir, b m\
mn
denote
))) J0
JO
(5i « j c ) e 6 K ~f€&K(mi,...,m„)
r=l
where the paths of the iterated integral lie entirely in Dn Proof. This is proved by induction on K by using the power series expansion (1.1) to express the derivative of Li m i ] ... ] m i l (x) in terms of polylogarithms of weight K — 1. • By the above proposition we can now define the analytic continuation of Limi,... ,m„(x) to Sn as the iterated path integral Limi
m„(x)=
T "
Jo
2 (<5i sK)eeK
Jye6K(mi,...,m„)
\Jwf(x(T£\o---oTsKKQr))). r=\ (2.3)
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J. Zhao
where all the paths lie inside Sn. Note that all the 1-forms appearing in (2.3) are rational forms with logarithmic singularities along Xn. E x a m p l e 2.5. When n = 1, Lii(x) = / dlog I — — J = - l o g ( l - x ) . When n = 2, 6 2 = {(0,0), (0,1), (1,0), (1,1)} and there are two elements in 6 2 : ((0,1), (1,1)) and ((1,0), (1,1)). Let x = (a;, y) then x(0,l)=2/,
x(l,0) = a;y,
x(l, 1) =
(x,y).
Thus Liitl(x,y)
= / u;i(x(0, l))u;i(x)+wi(x(l,0))ii;2(x) Jo /• (x ' y) dy dx 7(0,0) l - y l - x
d{xy) ( dy l-xy\l-y
dx x(x-l)J
\ '
L e m m a 2.6. The iterated path integral of (2.3) depends only on the homotopy class of the path from 0 to x. Proof. By induction on the weight K and Lemma 2.2. 2.4. Multiple
•
logarithms
We now specialize to the multiple logarithm cases w h e r e m \ = ••• = mn = 1. Then we have 6 ( 1 , . . . , 1) = 6 n and K = n. Though we can get the analytic continuation of the multiple logarithms by (2.3) immediately, we want to derive a cleaner expression in this special case. For any i = {i\,... , in) € 6 „ with is = 0 we define pos(i, i + us) = s as the position where the component is increased by 1. For example pos((l,0), (1,1)) = 2. We define the position functions / ^ , . . . , / " on ~f € 6 £ as follows: ACf)
= 1,
fnCf)
= pos(jt-i,jt),for 2 < t < n .
These functions tell us the places where the increments occur in the queue ofl*.
Multiple Polylogarithms:
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177
The following closed 1-forms will play important roles in the rest of this note: Wl(x)
:= dlog (^—
J ;
wt(x) := dlog (
l
~ **~l 1 ,
2
Proposition 2.7. The multiple logarithm £„(x) is a multi-valued meromorphic function on C". For x £ Sn one has £nW=
5Z / ^=ai.-j-)G6»,/0
u,
/i(^)(X(il))«'/a(y)(x(i2))"-«'/»c/)(x(j„)).
(2.4) 3. Multiple Logarithm Variations of M H S In this section we will define the variation matrix M[n](x) coming from the multiple logarithms of depths up to n. We will show that it is a 2 n x 2 n multivalued matrix which defines a good variation of a MHS over Sn = Cn\Xn where Xn is the divisor denned by (2.1). Remark 3.1. In fact, the irreducible component xn = 0 in Xn is not needed in the case of multiple logarithms. But the variation matrix corresponding to general multiple polylogarithms may have singularities along this component, for example, ^1,2(^1,^2) of the double polylogarithm Liit2{xi,X2). See Sec. 5. 3.1. The variation
matrix
The double logarithm was treated in [15, Sec. 2] by Goncharov. We rewrite his A\ti(x,y) as Mi,i(x,y) and try to generalize this to arbitrary multiple logarithm variation matrix M.[n](x) for x € Sn. Observe that on the index set &n the depth and the weight functions coincide. Definition 3.2. Suppose |i| = k and in = • • • — iTk = 1. Suppose | j | — I and jh=...= j t i = l. (1) If j / i, we define the (i,j)-th entry of M[n](x) to be 0. (2) If j -< i then we let tr = rar for 1 < r < I. Set io = ^o = 0, ti+i = n + 1 ,
178 J. Zhao a/+i = fc + 1. Define the (i, j)-th entry of M[n](x) as (2ni)lEij(x)
where
£uW = 7*(j)(x(i)) := Xjai—i\xTl • • • xT2 — i, xT2 • • • xT3—i,...
, xTail
• • • Xtj_ij
XJ[ ^a>-+l-Q!r-l r=l
1 \ l
X
' • 'XTar
Xtr
• • •XTar
tr
-
+ + 1
I -
2-l
_! ' " ' ' 1 -
Xtr
Xtr
•• • • ••
Xtr+1-l Xr^^-i
(3-1)
Here £ 0 = 1. E x a m p l e 3.3. On the last row of M[n](x) one has ' 1 - xtrxtr+i r=0
^
Xtr
l-xtr ' " '
' 1 - X t r
•••xtr+1-i
(3.2)
•••Xtr+l-2
x
In particular, Elto = 7o ( ) = -Cn(x) and Eltl = 7J(x) = 1. Another interesting case is when |i| = I, | j | = I — 1 and j -< i. Suppose kx = • • • = it, = 1 and j t , = 0 then £ y ( x ) = - l o g / y ( x ) where 1 — x\ • • • x^ /i,j(x) = {
xt.-! • • • xt,-i(xt, l - » t . _ !
if s = 1, • • • xu+1-i
- 1) if s > 2 .
(3.3)
•••Xt3-1
We now fix a standard basis {a : i £ 6n} of C 2 consisting of column vectors. Suppose |i| = k. It follows from definition that the i-th row of •M[n](x) is R{ := X > i r i ) U I 7 £ a ) ( x ( i ) ) e f = (27ri)*ef + J ^ ^ H ^ W M
• (3.4)
where e j are now row vectors. Note that 7*,.% = 7J = 1 by definition. It is clear that the first entry (i.e., j = 0) of this row is £fc(x(i)). Let us call the minor of .M[n](x) consisting of rows beginning with ktuple logarithms the k-th block. It has (£) rows with row indices |i| = k.
Multiple Polylogarithms:
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179
Lemma 3.4. The matrix _M[nj(x) is a lower triangular matrix. Moreover, the columns with | j | = k of the k-th block of A^[„i(x) is (2ni)k times the identity matrix of rank (£). Proof. The lemma follows directly from Eq. (3.4) because if j ^ i then j
•
Example 3.6. Let £o = 1. By definition or the above proposition the first column C 0 (x) = [ £ | i | ( x ( i ) ) : i e 6 n ] T . Proposition 3.7. The columns o/.M[ n ](x) form the set of the fundamental solutions of the system of differential equations dXo = 0, (3.5) dXi = E|k|=|i|-i,k-;i-^'krflog/i,k(x)
for all 1 < |i| < n
where /i,k(x) are rational functions defined by (3.3). 3.2. Monodromy
A/l[n](x)
of
Fix an embedding C" *-> C P n . Let Xn = Xn U ( C P n \ C n ) . Let M P (C) be the set o f r x r matrices over C. Put <" = ( c u)i,j6e„ e H°(CPn,nhpn(log(Xn)))
® M 2 n(C),
where f-dlog/,j(x), C
if |j| = | i | - l , j ^ i ,
U = \
0,
otherwise.
(3.6)
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J. Zhao
All of the 1-forms in u have logarithmic singularity on Xn because j \ j (x) are all rational functions given by (3.3). E x a m p l e 3.8. When n = 2we have 0 -dlog(l-y) (jj
-cHog(l 0
0
-xy)
0
-d\og(l — x)
— dlog
V)
We have seen in Proposition 3.7 that ,M[„](x) is a fundamental solution of the first order linear partial differential equation dA = wA,
(3.7)
where A is a possibly multi-valued function Sn M2»(C). Moreover .M[n](x) is a unipotent matrix for very x. £ Sn. Applying d on Eq. (3.7) and plugging in A = M[n](x) we get 0 = dwM[„](x) - w A dM[n](x) = {du - o> A
u)M[n](x)
Because A/l[n](x) is invertible and u is closed we get dui = 0,
wAw=0.
(3.8)
This shows that u> is integrable. The main goal of this section is to show that if we analytically continue every integral entry of jVf[n](x) along a common loop q 6 7Ti(5 n ,x), the resulting matrix will still be a fundamental solution M.[n](x)M(q) of (3.7) where M(q) e GLi2n(Z). In what follows we also denote this action of q by &(q) operating on the left. We then define the monodromy representation Px : 7Ti(5 n ,x) —> GL 2 "(Z)
q^M(q)T. Here we take the transpose to ensure p x to be a homomorphism because M(pq) = M(q)M(p) by our convention. From the explicit computation in Theorem 3.12 we will see that p x is a unipotent representation. By the definition in (3.1) in order to determine the monodromy of . M r ^ x ) it is imperative that we resolve the monodromy of the multiple logarithms £„(x) first. The next three lemma can be found by induction using integral computations.
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Lemma 3.9. Let 1 < s < n. Let p be a path from 0 to x in Sn. Let Is 6 Ti(Sn, x) enclose the component T>sn — {xs • • • xn = 1} only once in Sn but no other irreducible components of Xn such that J dlog(l—xs • • • xn) = —2iri. Then (@{qs) - i d ) £ „ ( x ) = -27ri£ s _ 1 (a?!,... , x s _ i ) - £ „ _ s ( y ( s ) ) , where / \
/J-
y(s) =
^s^s+1
—rz \
1
•*•
^s ' ' ' xn
\
>• • •. i Xs
1
• 3*s ' ' ' ^n— 1 /
Lemma 3.10. The monodromy of £„(x) about T>u = {x^ — 1}, 1 < i < n, and T>ij = {xi • • • Xj = 1}, 1 < i < j < n, is trivial. Lemma 3.11. Let n > 1. For any 1 < a < b < n set Faa = 1 and I? /"„\ o I fiab(X) = £ b - a \
x
a.Xa+l
1 - Xa
, ••• ,
L — Xa • • • Xb 1 - Xa ••
-Xb-1,
Let 1 < j < n and QJQ € iri(Sn,x) (resp. 1 < j < n and qij, 2 < j < n and qjn) be a loop turning around the component T>JQ = {XJ = 0} (resp. T>ij = {xi • • • Xj = 1}, resp. T>jn = {XJ • • • xn = 1}), only once but no other irreducible components of Xn such that J dxj/xj = 2ni (resp. f dlog(l — xi • • • Xj) = 2-Tti, resp. J dlog(l — Xj • • • xn) = 2m). Then n-l
(@(qj0) - id)F l n (x) = -2ni Y,
Fls(x)Fs+hn(x),
s=j
(G(qij)
- id)Fi„(x) =
(G(qjn) - id)Fln(x)
2TriFhj(x)Fj+hn(x),
= -27rtFij_i(x)Fj n (x).
Proof. The lemma follows from the monodromy property of £ n ( x ) .
•
Combining Lemmas 3.9 to 3.11 we have Theorem 3.12. Let M.[n](x) = [•Bi,j(x)]ijee„ where -Eij(x) are defined by (3.1). Let 1 < i < j < n and q^ € ni(Sn,x) (resp. 1 < j < n and qjo) enclose T>ij = {xt • • • Xj = 1}, (resp. T>JO = {XJ = 0}) only once but no other irreducible component of Xn such that J dlog(l — X{ • • • Xj) = 2-7ri (resp. J dlogXj = 2ni). Then M(qj0) = I + [nij] i)je 6„ ,
M(qij) =1+ [ m i j ] i j e e „ ,
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J. Zhao
where I is the identity matrix of rank 2™,
{
-1,
if tr<j
0,
otherwise,
- 2 , r > l , i = j + u s + i and j
<s
and
(
1,
if tr=i<j
< tr+i - 2, r > 1, i = j + u J + i ,
-1, t/tP + l < i < j = t r + i - l , r > 0 , i = j + U i , (3-10) 0, otherwise. Here i and j satisfy the condition in Definition 3.2(2) in the cases o / m , j = ± 1 and n y = —1. It follows immediately that we have Corollary 3.13. The monodromy representation of .M[ n i(x) px:7r1(5n,x)^GL2"(Z) is unipotent. 3.3. Mixed Hodge structures
of multiple
logarithms
Having analyzed the monodromy properties of .M[n](x) associated with the n-tuple logarithm we now turn to its mixed Hodge structures. Define a meromorphic connection V on the trivial bundle C P n x C 2 " —• CPn
(3.11)
by
Vf = df-uf, where / : Sn —> C 2 is a continuous section. This connection has regular singularities along the divisor Xn = XnU(CPn\Cn) because u; is integrable by (3.8) and all the 1-forms in u are logarithmic in any compactification of Sn. Proposition 3.7 further implies that the columns (27ri)lJlCj(x) of Al[ n ](x) satisfy V / = 0 and are therefore flat sections of (3.11). Even though they are multi-valued, their Z-linear span is well defined thanks to Theorem 3.12. Hence Vjn](x) forms a local system over Sn.
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To define the MHS on Vjnj one need provide two compatible filtrations: an increasing weight filtration W, of Vjn](x) and a decreasing Hodge filtration T* of Vj n ] C = Vjn](x) (g) C. The weight filtration is defined by W2k+i = W2k and W_2feVjn](x) = ((27ri)l i IC i :|i|>fc)Q. In particular, W_2fc Vjn] (x) = 0 if A; > n and W_2jt Vjn] (x) = Vj„] (x) if A; < 0. By regarding ei's as column vectors one can define the Hodge filtration on V[n],c by ^ ~ % ] , c := ( ei : |i| < k)c • So in particular, T~kV^c = 0 for k < 0 and T~kV^tc = V[n],c f° r By induction on n and using Lemma 3.4 it is easy to show that ' 0,
k>n.
if p < k - 1, i
F-* n W_ 2 f c Vj n ] i C = < ((27ri)l lei : fc < |i| < p), if k < p < n, <(27ri)lilei : fc< |i| < n ) , if
p>n.
This implies that F-pEi%kV[n],c
0,
if p < k - 1, r
. ^_ 2fc V [n] ,c/W -2fc-i^„j lC ,
if P > k.
In other words, ^ g r ^ f c ^ H . c = 0 for q > -k + 1 and J F ^ g r ^ V j ^ c = Sr^2fe^in],C f° r 1 ^ — k. This means that the Hodge filtration induces a pure HS of weight — 2k on each weight graded piece. Furthermore, it is not hard to see by checking the powers of 2ni appearing on the diagonal of M[n](x) that this induced structure on gr^fc^H.c is isomorphic to the direct sum of (£) copies of the Tate structure Z(fc) by Lemma 3.4. 4. Limit M H S of Multiple Logarithms Let the monodromy of M[n](x) at any subvariety V of CPn be given by the matrix T-D and the local monodromy logarithm by Nv = logTv/lm. Note that T-p is unipotent so N-p is well-defined. Now let us recall the construction of the unipotent variations of limit MHS at the "infinity" with normal crossing. Let 5 be a complex manifold of dimension d. Suppose that S is embedded in S, via the mapping j , such that D = S — S is a divisor with normal crossings. Let V be any local
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J. Zhao
system of complex vector spaces on S, and V the corresponding vector bundle. According to Deligne there is a canonical extension V of V over S ([11, Proposition 5.2]). Moreover, when the local monodromy is nilpotent V is a subsheaf of j * V. The local picture of S C § is (A*) r x A d - r C Ad where A is the unit disk and A* is the punctured one. We let t i , . . . , tT denote the variables on (A*) r , and Ni,... ,Nr the (commuting) local nilpotent logarithms of the associated monodromy transformations of the fibre. For zi,... ,zrin the upper half-plane, the universal covering mapping for (A*) r is given by tj = exp(2nizj),
j = 1 , . . . , r.
Let vi,... ,vm be a basis of the multi-valued sections of V over (A*) r x Ad~r, the formula
(
r
\
r
- ^2-nizjNj J = [vi,... , vm] Y[ tJNj 3=1 J 3=1 determines a basis of the sections of V over Ad and these provide, by definition, the generators of V over Ad. In our situation, although the divisor Dn is not normal crossing the image of the global holomorphic logarithmic forms in the complex of smooth forms on Sn is independent of the normal crossings compactification Sn (cf. [22, Proposition 3.2]). In fact, the forms we are considering lie in the subcomplex generated by 1-forms of the type df/f where / is a rational function. Such forms are automatically logarithmic in any compactification and therefore our connection is automatically regular. Hence the admissibility and the existence of the limit MHS is an automatic consequence of the admissibility of our variations restricted to every curve in Sn. Moreover, the pullback of our trivial bundle (3.11) restricted to Sn to Sn is exactly Deligne's canonical extension of (3.11), and the pullbacks of the subbundles T% and Wm are the correct extended Hodge and weight subbundles. Therefore we have T h e o r e m 4 . 1 . The n-tuple logarithm underlies a good unipotent gradedpolarizable variation of mixed Hodge-Tate structures (Vjn], W,, T') over
sn=
l
J
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185
with the weight-graded quotients gr^2fc being given by (£) copies of the Tate structure Z(fc). Proof. It is clear that all the odd graded weight quotients are zero so that we can let the polarizations on the weight graded quotients gr^2fc D e the ones that give each vector 2iriej (|j| = k) length 1. Then everything is clear except the Griffiths transversality condition. But this condition is also satisfied because dCj = wCj for every j G 6n by Proposition 3.7. D If we want to determine the limit MHS of multiple logarithms explicitly we can still apply the techniques used in the normal crossing case. We will carry this out only for the double logarithm case. The general picture is similar but much more complicated. We have
Mi,i(x,y)
=
1
0
0
0
Mv)
1
0
0
Zi(xy)
0
1
0
£2(2,2/)
£1(2)
Tl,l(2-Ki).
H(x,y)
1
where H(x,y) = &i(y) — £i(x) — log a; and 7-1,1(27™) = diag[l, 2ni, 2ni, (27ri)2]. To save space we let Mij be the 4 by 4 matrix whose entries are all zero except that the (i,j)th entry is 1. (i) Take the divisor T>io = {x = 0} and the tangent vector d/dx. We have T{x=o} = - M 4 j 3 ,
^=°>
=
Define [so si s2 S3] = l i m ^ o Mi,i{t,y)
logT{x=0} 2ni
1 ~-^iM4'3-
• logt • M^zf2m
then it's easy to
see t h a t
so si s2 s3] =
1
0
0
0
My)
1
0
0
0
0
1
0
0
0
My)
1
Tl,l(27Ti).
Let Vqrx=0\ be the Q-linear span of SQ, S±, S2, S3, and Vc;{x=o} = C <8> 4 VQ,{X=O}- Let {eo,ei,e2,e3} be the standard basis of C where the only
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nonzero entry of ej is at the (j + l)st component. Then the limit MHS on {(a;, y) : x = 0, y ^ 1} along d/dx are given by ((VQ,{X=0}>W.),(K:>{X.O},J"))>
where for k = 0 , . . . , 3 , W-2kV®t{x=0}
= (Sk, • • • , S3), W-2k = W-2fc+l
(4.1)
and (4.2)
F *Vb,{x=o} = (eo, • • • , e f c ) .
(ii) A similar calculation shows that on the divisor V\\ = {(1,2/) : y 7^ 1} and along the tangent vector d/dx if one sets [so S\ s2 S3] = limt_>i Mi p i(t, J/) • log(l - t)(M 4)2 - M 4]3 )/27ri then one has the limit MHS
[s0 si s2 s3\ =
1
0
£1(2/)
0
£1(2/)
1
.£2(1, y)
0
T M (27ri).
^(y)
It is easy to see by differentiation that £2(1,2/) = (£i(y)) 2 /2. We can similarly deal with the next two cases: (iii) On D22 = {(x, 1) : x ^ 0,1} along the tangent vector
d/dy.
(iv) On D12 = {(1/y, y) '• y 7^ 0,1} along the tangent vector d/dx. limit is given by (a;, y) —> (1/y, 2/) f° r every fixed 2 / ^ 0 , 1.)
(The
For 0-dimensional singularities we only have the following two cases: (v) T>io n X>22 — (0, !)• From (i) we see that there are limit MHS on the open set £>io\{(0,1)} of T>i0. We now can easily extend these MHS to (0, 1) along the vector d/dy and find the limit MHS to be the Q-linear span of so, • • • ,S3 where [S0 Si S2 S3] = T i i l ( 2 7 T l ) .
If we start with (iii) and then extend the MHS to (0, 1) along tangent vector d/dx we will get the same limit MHS. (vi) Vn n X>i2 = X>i2 n D 2 2 = P11 n V22 = (1,1). We can start with either case (ii) or (iii) or (iv). Extending the limit MHS of case (ii) we see
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Monodromy
187
immediately that the along the tangent vector d/dy the limit MHS at (1, 1) is given by the Q-linear span of [s0 si s2 s3] = Ti,i(27rr).
(4.3)
If we start with case (iii) and then use tangent vector d/dx we find that only the lower left corner entry is different from the above. Instead of 0 it is •E4,i = lim Li2 ( —^— ) + - log 2 (l -x)x->l
\X — 1 /
logxlog(l - x)
I
since Li2(l - *) + ^ ( 1 - l/<) + log 2 1/2 = 0 for any t ^ 0. But if we take s'0 = So — S3/48 we get the same basis as in (4.3). The same phenomenon occurs if we start with case (iv) and then use tangent vector d/dy. Problem 4.2. We find that in higher weight cases the limit MHS of the multiple logarithm sometimes correspond to MHS of some multiple polylogarithm of the same weight. Can all the multiple polylogarithm variations of MHS be produced this way by multiple logarithms? 5. Some General Results and Problems One can similarly generalize the above theory to multiple polylogarithms. In single variable case, Deligne defined a variation of mixed Hodge-Tate structures related to the classical n-logarithm (cf. [23]). In the case of two variables, we have the next result for weight 3. Theorem 5.1. Each of the weight three depth two multiple polylogarithms underlies a good variation of mixed Hodge-Tate structures over S2 = C2\{xy(l — x)(l — y)(l — xy) = 0}. For Li2y\ the graded weight quotients are isomorphic ioZ(O), Z(1)©Z(1), Z(2)©Z(2), andZ(3), respectively. ForLi1}2 they are isomorphic to Z(0), Z(1)©Z(1), Z(2)©Z(2)©Z(2), and Z(3), respectively. By this theorem and previous results on Li$ and ^ i 1,1,1 we have now completely settled the cases of weight 3 multiple polylogarithms. We now can generalize Theorem 5.1 to Theorem 5.2. The double polylogarithm Lir,s underlies a good unipotent graded-polarizable variation of a mixed Hodge-Tate structure with the graded
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weight piece gr^ f c being direct sums of Ck copies of Z(fc) where (dk(r,s)
+ l,
ifr^k
= s,
Cfc = <
I dk(r, s),
otherwise ,
and
dk(r,s)
0,
ifk<0ork>r
k + 1,
if 0 < k < min{r, s} ,
min{r, s} + 1,
if min{r, s} < k < max{r, s} ,
r + s + 1— k ,
+ s,
if max{r, s} < k < r + s .
Among all the double polylogarithms Lir>r{x,y) is the most regular. It satisfies CQ = c^r = 1, ci = C2r_i = 2 , . . . , c r _i = c r + i = r, cr = r + 1. In general, we expect the multiple polylogarithm Limi m„(x) underlies a good variation of mixed Hodge-Tate structures over Sn with the graded weight piece gr^2fc being direct sums of Ck copies of Z(fc) for some positive integer Ck- It would be very interesting to solve Problem 5.3. (1) Find a closed formula of Ck only depending on m i , . . . ,mn and k. (2) Determine the variation matrix -M m i l ..., m n (x) explicitly. (3) Determine the connection matrix u: explicitly. (4) Determine the monodromy actions explicitly. 6. Single-Valued Version of Multiple Polylogarithms If part (2) of Problem is solved then following an idea of Beilinson and Deligne [1] as given in [4] one can easily discover the single-valued variant of Limi,...,mn{x\,... ,xn) which we denote by £ m i ,..., m „ {xx,... ,xn) which should be a real analytic function. In what follows we outline the procedure for multiple logarithms only. 6.1. General
procedures
For any n > 2 let L[n] = L[„](x) = [Co---Ci] be the matrix with 2 n columns Cj (j G <S„) as before and M[n] — M[n]{x.) = L[n](x)r[n](27ri) where r w ( A ) = diag[AUI]je5„.
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189
Define the matrix
where M[n] is the complex conjugation of M[ny From our calculation of the monodromy we see that B is a single-valued matrix function defined over S„. Moreover S
W =
B
H
since T[„](t) = T[ n ](i) _1 . Now that B[„] = I + N with I the identity matrix and N a nilpotent matrix we see that log B is well defined and satisfies \ogB[n] = - l o g 5 [ n ] , namely, logBr n i is a pure imaginary matrix. Then we define —l/(2i) times the lower left corner entry of log B to be £[„](x) which is a single-valued real analytic variant of the multiple logarithm £ n ( x ) . Remark 6.1. Our method is slightly different from that in [1]. In fact when we are in the polylogarithm case the matrix B constructed as above is the conjugate of the one in [1] by r(i). 6.2. Single-Valued Weights
Multiple
Polylogarithms
of Lower
By the above procedure we find the single-valued double logarithm Ci,i(x, y) = C2 ( f l y )
- £2 ( ^ y )
- C2(xy)
where the dilogarithm function C2{z) = Im(iz 2 (^)) + arg(l - z) log \z\. The function £iti(x,y)
satisfies the functional equations
Ci,i(x,y)
= -A,i ( l - z . ^ — [ J
by the functional equations C2(x) = —£2(1 — x) =
—C2(l/x).
(6.1)
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J. Zhao
Similarly, we compute the single-valued versions of Li2,i{x,y) Lii,2(%,y) as follows: £1,2(2, y) = ReLih2(x,
y) - a r g ( l - xy)[L2{x) +C2(y)] +log |1
— log |2/|ReLii,i(a;, y) — log |1 — x
and
-x\ReLi2(y)
\Re Li2{xy)
- -log|a;2/2|log|l-a;2/|log|l-a;_1| + 3 log |y|(2 log |1 - 2/| log |1 - x\ + log |1 - xy\ log |z(l -
y)\),
and £1,2(3,2/) = -^2:i{y,x)-C3(xy),
(6.2)
where £3 is the single-valued trilogarithm given by (cf. [37]) £ 3 (z) = Re(Li 3 (z)) - log \z\Re(Li2{z))
- |(log \z\f log |1 - z\.
One should compare (6.2) with the identity Lh,2(x, y) + Li2,i{y, x) + Li3(xy) = - log(l -
x)Li2(y).
Finally we find the interesting identity
\
l-x
)
\xy(l-z)J
\ y - l j
We remind the readers that such identities in higher weight cases do not exist in general. For example, £2,2(2,2/) cannot be expressed by £4 only (see the explanation on [15, pp. 244-245]). 6.3. A problem
of multiple
Dedekind
zeta
values
In general there exists single-valued real analytic version of the multiple polylogarithm Z 2 the value of this function is given by the power series expansion (1.1) when
Multiple Polylogarithms: Analytic Continuation, Monodromy 191 \xi\ < 1. We end our exposition by posing a problem generalizing Zagier conjecture about special values of Dedekind zeta function over number fields. Denote by Op t h e ring of integers of a number field F a n d Ip t h e set of integral ideals of Op. Let M b e t h e n o r m from F t o Q . T h e n we define the multiple Dedekind zeta function of depth d over F as CF(81,...,8d)=
M(m)-S1---M(nd)-Sd-
£ A/"(ni)<-<JV(n d )
This function is well defined for R e ( s i ) > 1 , . . . , R e ( s d - i ) > 1, Re(s<j) > 1. P r o p o s i t i o n 6 . 2 . For any integers m i , . . . an expression O / ( F ( ^ 1 V i " i d ) in terms evaluated at F rational points up to some number field F (such as the discriminant, embeddings, etc.)?
, m ^ - i > 1 and nxs. > 2, is there of a determinant o/£miv..]md factors determined only by the the number of real and complex
W h e n F = Q t h e problem has a n easy answer: C Q ( T O I , . . . ,md)
= £mi,...,m
References [1] A. A. Beilinson and P. Deligne, Interpretation motivique de la conjecture de Zagier reliant polylogarithmes et regulateurs, in: Proc. Sym. Pure Math. 55 Part 2, pp. 97-121, Amer. Math. Soc. (1994). [2] A. A. Beilinson, A. B. Goncharov, V. V. Schechtman and A. N. Varchenko, Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles in the plane, in: Grothendieck Festschrift II, Prog, in Math. 87, pp. 78-131, Birkhauser, Boston (1991). [3] A. A. Beilinson and A. Levin, The elliptic polylogarithm, in: Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Part 2, pp. 125-190, Amer. Math. S o c , Providence, RI (1994). [4] S. Bloch, Lectures on mixed motives given in Santa Cruz, 1995, available online http://www.math.uchicago.edu/bloch/publications.html. [5] A. Besser, Finite and p-adic polylogarithms, to appear in Comp. Math. Available online http://www.cs.bgu.ac.il/~bessera/polyfin/polyfin.html. [6] J.-L. Cathelineau, Remarques sur les differentielles des polylogarithmes uniformes, Ann. Inst. Fourier, Grenoble 4 6 (1996) 1327-1347. [7] K.-T. Chen, Algebras of iterated path integrals and fundamental groups, Trans. Amer. Math. Soc. 156 (1971) 359-379. [8] K.-T. Chen, Rerated path integrals, Bull. AMS 83 (1977) 831-879.
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[9] R. Coleman, Dilogarithms, regulators and p-adic L-functions, Inv. Math. 69 (1982) 171-208. [10] P. Deligne, Letter to Spencer Bloch, April 3, 1984. [11] P. Deligne, Equations differentielles a points singuliers reguliers, Lecture Notes in Math., Vol. 163 (Springer-Verlag, 1970). [12] P. Deligne, Lectures notes on multi-zeta values (IAS, 2001). [13] P. Elbaz-Vincent and H. Gangl, On poly(ana)logs I, available online http://www.math.uiuc.edu/Algebraic-Number-Theory/0248/index.html. [14] A. B. Goncharov, Polylogarithms in arithmetic and geometry, in: Proc. ICM, Zurich, pp. 374-387, Vol. I (Birkhauser, 1994). [15] A. B. Goncharov, Geometry of configurations, polylogarithms and motivic cohomology, Adv. in Math. 114 (1995) 179-319. [16] A. B. Goncharov, Chow polylogarithms and regulators, Math. Res. Letters 2 (1995) 95-112. [17] A. B. Goncharov, The double logarithm and Manin's complex for modular curves, Math. Res. Letters 4 (1997) 617-636. [18] A. B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Letters 5 (1998) 497-516. [19] A. B. Goncharov, Multiple £-values, Galois groups, and geometry of modular varieties, available online http://xxx.lanl.gov/abs/math.AG/0005069 [20] A. B. Goncharov, The dihedral Lie algebras and Galois symmetries —({0,oo} U HN))> available online http://xxx.lanl.gov/abs/math.AG/ 0009121. [21] A. B. Goncharov and J. Zhao, Grassmannian trilogarithms, Comp. Math. 127 (2001) 83-108. [22] R. Hain, The existence of higher logarithms, Comp. Math. 100 (1996) 247-276. [23] R. Hain, Classical polylogarithms, in: Proc. Sym. Pure Math. 55 part 2, pp. 3-42 (Amer. Math. Soc, 1994). [24] R. Hain and R. MacPherson, Higher logarithms, 111. J. Math. 34 (1990) 392-475. [25] R. Hain and J. Yang, Real Grassmann polylogarithms and Chern classes, Math. Ann. 304 (1996) 157-201. [26] R. Hain and S. Zucker, Unipotent variations of mixed Hodge structure, Inv. Math. 88 (1987) 83-124. [27] R. Hain and S. Zucker, A guide to unipotent variations of mixed Hodge structure, in: Hodge theory (Sant Cugat, 1985), Lecture Notes in Math. 1246 pp. 92-106 (Springer, 1987). [28] A. Huber and G. Kings, Degeneration of 1-adic Eisenstein classes of the elliptic polylog, available online http://www.math.uiuc.edu/AlgebraicNumber-Theory/0087/index.html. [29] M. Kontsevich, The 1 ^-logarithm, unpublished note (1995), available as the appendix to [13].
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[30] A. M. Levin, Elliptic polylogarithms: an analytic theory, Comp. Math. 106 (1997) 2678-282. [31] H. Poincare, Oevres, Vol. 2 (Paris, 1916). [32] D. Ramakrishnan, On the monodromy of higher logarithms, Proc. AMS 85 (1982) 596-599. [33] R. Ree, Lie elements and an algebra associated with shuffles, Ann. of Math. 68 (1958) 210-220. [34] J. Steenbrink and S. Zucker, Variation of mixed Hodge structure. I, Inv. Math. 80 (1985) 489-542. [35] Z. Wojtkowiak, Mixed Hodge structures and iterated integrals I, preprint. [36] J. Wildeshaus, On an elliptic analogue of Zagier's conjecture, Duke Math. J. 87 (1997) 355-407. [37] D. Zagier, The Block-Wigner-Ramakrishnan polylogarithm function, Math. Ann. 286 (1990) 613-624. [38] J. Zhao, Motivic complexes of weight three and pairs of simplices in projective 3-space, Adv. in Math. 161 (2001) 141-208. [39] J. Zhao, Variations of mixed Hodge structures of multiple polylogarithms, in preparation. [40] J. Zhao, Analytic continuation of multiple polylogarithms, Submitted.
D E F O R M A T I O N T Y P E S OF REAL A N D COMPLEX M A N I F O L D S
Fabrizio Catanese Lehrstuhl Mathematik VIII Universitat Bayreuth D-95440, BAYREUTH, Germany Fabrizio. Catanese@uni-bayreuth. de
The present article grew directly out of the lecture delivered at the ChenChow Symposium, entitled Algebraic surfaces: real structures, topological and differentiable types, and partly evolved through the transparencies I used later on to lecture at FSU Tallahassee, and at Conferences in Marburg and Napoli. The main reason however to change the title was that later on, motivated by some open problems I mentioned in the Conference, I started to include in the article new results in higher dimensions, and then the theme of deformations in the large of real and complex manifolds emerged as a central one. For instance, we could summarize the Leit-Faden of the article as a negative answer to the question whether, for a compact complex manifold which is a K (TT, 1) (we consider in some way these manifolds as being simple objects), the diffeomorphism type determines the deformation type. The first counterexamples to this question go essentially back to some old papers of Blanchard and of Calabi (cf. [10,11,13] and [77]), showing the existence of non Kahler complex structures on the product of a curve with a two dimensional complex torus (the curve has odd genus g > 3 for Calabi's examples which are, I believe, a special case of the ones obtained by Sommese via the Blanchard method). In the more technical Sec. 4 we show, among other things, that any limit of a product of a curve of genus g > 2 with a complex torus is again a variety of the same type. We show also 195
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Catanese
that the Sommese—Blanchard complex structures give rise, for fixed g, to infinitely many Kuranishi families whose dimension is unbounded, but we are yet unable to decide whether there are indeed infinitely many distinct deformation types. It is sufficient for our purposes to give a positive answer to a question raised by Kodaira and Spencer (cf. [59]), showing that any deformation in the large of a complex torus is again a complex torus. In the same section we also give a criterion for a complex manifold to be a complex torus, namely, to have the same integral cohomology algebra of a complex torus, and to possess n independent holomorphic and d-closed 1-forms. Whether complex tori admit other complex structures with trivial canonical bundle is still an open question (the Sommese-Blanchard examples always give, on 3-manifolds diffeomorphic to tori, complex structures with effective anticanonical bundle). One might believe that the previous pathologies occur because we did not restrict ourselves to the class of Kahler manifolds. However, even the Kahler condition is not sufficient to ensure that deformation and diffeomorphism type coincide. In fact, and it is striking that we have (non rigid) examples already in dimension 2, we illustrate in the last sections (cf. [20,21] for full proofs), that there are Kahler surfaces which are K(ir, l)'s, and for which there are different deformation types, for a fixed differentiable structure. These final examples tie up perfectly with the beginning of our journey, that is, the classification problem of real varieties. The original point of view illustrated in the lecture was to zoom the focus, considering wider and wider classes of objects: • • • • • •
Smooth real algebraic Varieties/Deformation Smooth complex algebraic Varieties/Deformation Complex Kahler Manifolds/Deformation Symplectic Manifolds Differentiable Manifolds Topological Manifolds
In the article we could not illustrate all the possible aspects of the problem (we omitted for instance the topic of symplectic manifolds, see [5] for a nice survey), we tried however to cover most of them.
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Types of Real and Complex Manifolds
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The paper is organized as follows: in Sec. 1 we recall the definition and the main properties and examples of real varieties, and we report on recent results [24] on the Enriques classification of real surfaces. Section 2 is devoted to some open questions, and illustrates the simplest classification problem for real varieties, namely the case of real curves of genus one. In the next section we illustrate the role of hyperelliptic surfaces in the Enriques classification, and explain the main techniques used in the classification of the real hyperelliptic surfaces (done in [24]). Section 4 is mainly devoted, as already mentioned, to complex tori and to products C x T, and there we barely mention the constructions of Sommese, Blanchard and Calabi. We end up with the classification and deformation theory of real tori (this is also a new result). Section 5 is devoted to a detailed description of what we call the Blanchard-Calabi 3-folds (they are obtained starting from curves in the Grassmann variety G(l,3), via a generalization of the previous constructions). This approach allows to construct smooth submanifolds of the Kuranishi family of deformations of a Blanchard-Calabi 3-fold, corresponding to the deformations of X which preserve the fibration with fibres 2-dimensional tori. We are moreover able to show that these submanifolds coincide with the Kuranishi family for the Sommese—Blanchard 3-folds, and more generally for the Blanchard-Calabi 3-folds whose associated ruled surface is non developable. In this way we obtain infinitely many families whose dimension, for a fixed differentiable structure, tends to infinity. Finally, we sketch the Calabi construction of almost complex structures, without proving in detail that the Calabi examples are a special case of the Sommese—Blanchard 3-folds. In Sec. 6 we recall the topological classification of simply connected algebraic surfaces and briefly mention recent counterexamples to the FriedmanMorgan speculation that for surfaces of general type the deformation type should be determined by the differentiable type. Section 7 is devoted to some results and examples of triangle curves, whereas the last section explains some details of the construction of our counterexamples, obtained by suitable quotients of products of curves. For these, the moduli space for a fixed differentiable type is the same as the moduli space for a fixed topological type (and even the moduli space for
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a given fundamental group and Euler number, actually!), and it has two connected components, exchanged by complex conjugation.
1. W h a t is a Real Variety? Let's then start with the first class, explaining what is a real variety, and what are the problems one is interested in (cf. [34] for a broader survey). In general, e.g. in real life, one wants to solve polynomial equations with real coefficients, and find real solutions. Some theory is needed for this. First of all, given a system of polynomial equations, in order to have some continuity of the dependence of the solutions upon the choice of the coefficients, one has to reduce it to a system of homogeneous equations h(zo,z\,...
,zn) = 0,
< . fr{Z(),Zl,-
• • ,Zn)
= 0.
The set X of non trivial complex solutions of this system is called an algebraic set of the projective space P£, and the set X(M.) of real solutions will be the intersection X H Pg. One notices that the set X of complex conjugate points is also an algebraic set, corresponding to the system where we take the polynomials fj(z0,...
,z-n) = 0
(i.e., where we conjugate the coefficients of the f'js). X is said to be real if we may assume the / j s to have real coefficients, and in this case X = X. Assume that X is real: for reasons stemming from Lefschstz' s topological investigations, it is better to look at X(R) as the subset of X of the points which are left fixed by the self mapping o" given by complex conjugation (curiously enough, Andre' Weil showed that one should use a similar idea to study equations over finite fields, in this case one lets a be the map raising each variable z; to its g-th power). The simplest example of a real projective variety (a variety is an algebraic set which is not the union of two proper algebraic subsets) is a
Deformation Types of Real and Complex Manifolds
199
plane conic C C Pf. It is denned by a single quadratic equation, and if it is smooth, after a linear change of variables, its equation can be written (possibly multiplying it by —1) as
zl + z\ + zl = Q or as z
o
—• z i
— %2 — " •
In the first case X(R) = 0, in the second case we have that X(W) is a circle. Indeed, the above cases exhaust the classification of smooth real curves of genus = 0. We encounter genus = 1 when we proceed to an equation of degree 3. For instance, if we consider the family of real cubic curves with affine equation y2 — x2(x + I) — t, we obtain • two ovals for t < 0, • one oval for t > 0. However, every polynomial of odd degree has a real root, whence every real cubic curve has X(M) ^ 0. On the other hand, there are curves of genus 1 and without real points, as we were taught by Felix Klein [54] and his famous Klein's bottle. Simply look at a curve of genus 1 as a complex torus X = C/Z + TZ. For instance, take r = i and let a be the antiholomorphic self map induced by cr(z) = z + 1/2. By looking at the real part, we see that there are no fixpoints, thus X(M) = 0: indeed the well known Klein bottle is exactly the quotient X/a, as an easy picture shows. [Remark: this real curve X is the locus of zeros of two real quadratic polynomials in P^..] The previous example shows once more the usefulness of the notion of an abstract manifold (or variety), which is indeed one of the keypoints in the classification theory. Definition 1.1. A smooth real variety is a pair {X, a) consisting of a smooth complex variety X of complex dimension n and of an antiholomorphic involution a : X —> X (involution means: cr2 = Id).
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Let M be the differentiable manifold underlying X and J be the complex structure of X ( J is the linear map on real tangent vectors provided by multiplication by i): then the complex structure —J determines a complex manifold which is called the conjugate of X and is denoted by X, and a is said to be antiholomorphic if it provides a holomorphic map between the complex manifolds X and X. Main problems (assume X is compact) • Describe the isomorphism classes of such pairs [X, a). • Or, at least describe the possible topological or differentiable types of the pair [X, a). • At least describe the possible topological types for the real part X' := X(R) = Fix(o-). Remark 1.2. One can generalize the last problem and consider real pairs (Z C X, a). Recall indeed that Hilbert's 16th problem is a special case of the last question for the special case where X = P 2 , and Z is a smooth curve. In practice, the problem consists then in finding how many ovals Z(M) can have, and what is their mutual disposition (one inside the other, or not). For real algebraic curves one has the following nice theorem. Theorem 1.3 (Harnack's Inequality). Let (C,cr) be a real curve of genus g. Then the real part C(W) consists of a disjoint union of t ovals = circles (topologically: S1), where 0 < t < g + 1. Remark 1.4. Curves with g + 1 ovals are called M (aximal)-CURVES. An easy example of M-curves is provided by the hyperelliptic curves: take a real polynomial P29+2 (x) of degree 2g + 2 and with all the roots real, and consider the hyperelliptic curves with affine equation Z2 = P2g+2(x)
.
The study of real algebraic curves has been the object of many deep investigations, and although many questions remain still open, one has a good knowledge of them, for instance Seppala and Silhol [73] and later Frediani [41] proved
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Theorem 1.5. Given an integer g, the subset of the moduli space of complex curves of genus g, given by the curves which admit a real structure, is connected. A clue point to understand the meaning of the above theorem is that a complex variety can have several real structures, or none. In fact the group Aut(X) of biholomorphic automorphisms sits as a subgroup of index at most 2 in the group Dian(X) consisting of Aut(X) and of the antiholomorphic automorphisms. The real structures are precisely the elements of order 2 in Dian(X)-Aut(X). Therefore, as we shall see, the map of a moduli space of real varieties to the real part of the moduli space of complex structures can have positive degree over some points, in particular, although the moduli space of real curves of genus g is not connected, yet its image in the moduli space of complex curves is connected! In higher dimensions, there are many fascinating questions, and the next natural step is the investigation of the case of algebraic surfaces, which is deeply linked to the intriguing mystery of smooth 4-manifolds. Indeed, for complex projective surfaces, we have the Enriques' classification of surfaces up to birational equivalence (equivalently, we have the classification of minimal surfaces, i.e., of those surfaces S such that any holomorphic map S —> S' of degree 1 is an isomorphism). The Enriques (-Kodaira) classification of algebraic varieties should consist in subdividing the varieties X according to their so-called Kodaira dimension, which is a number kod(X) G {—oo, 0 , 1 , . . . , dim(X)}, and then giving a detailed description of the varieties of special type, those for which kod(X) < dim(X) (the varieties for which kod(X) = dim(X) are called of general type). As such, it has been achieved for curves: Curve
Fl
Kod
9
—oo
0
Elliptic curve: C/Z + TZ
0
1
Curve of general type
1
>2
and for surfaces, for instance the following is the Enriques' classification of complex projective surfaces (where Cg stands for a curve of genus g):
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Surface
Kod
Pl2
Structure
Ruled surface with q — g Complex torus K3 Surface Enriques surface
— 00
0 1
FixCg C 2 /A 4 homeo to X\ C P£
Hyperelliptic surface Properly elliptic surface Surface of general type
0 0
1
0
1
0 1 2
1 >2 >2
K3/(Z/2) ( d x C[)/G dim (j>i2{S) = 1 , . . . dim^ 1 2 (S) = 2 , ?
Now, the Enriques classification of real algebraic surfaces has not yet been achieved in its strongest form, however it has been achieved for Kodaira dimension 0, thanks to Comessatti (around 1911, cf. [28,29] and [30]), Silhol, Nikulin, Kharlamov and Degtyarev (cf. [32,33,34] and [75]) and was finished in our joint work with Paola Frediani [24] where we proved: T h e o r e m 1.6. Let (S,a) be a real hyperelliptic surface. (1) Then the differentiable type of the pair (S, a) is completely determined by the orbifold fundamental group exact sequence. (2) Fix the topological type of (S, a) corresponding to a real hyperelliptic surface. Then the moduli space of the real surfaces (S',o~') with the given topological type is irreducible [and connected). (3) Real hyperelliptic surfaces fall into 78 topological types. In particular, the real part S(R)) of a real hyperelliptic surface is one of the following: • • • •
a disjoint union of c a disjoint union of b the disjoint union of the disjoint union of
tori, where 0 < c < 4, Klein bottles, where 1 < b < 4, a torus and of a Klein bottle, a torus and of two Klein bottles.
This result confirms a conjecture of Kharlamov that for real surfaces of Kodaira dimension < 1 the deformation type of (S, a) is determined by the topological type of (S, a). Our method consists in (1) Rerunning the classification theorem with special attention to the real involution. (2) Finding out the primary role of the orbifold fundamental group, which is defined as follows:
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For X real smooth, we have a double covering 7r : X —> Y = X/(a) (Y is called the Klein variety of (X, a)), ramified on the real part of X, X{R) = Fix(a). In the case where X(W) = 0, n°rb(Y) is just defined as the fundamental group 7ri(Y). Otherwise, pick a point xo G -X"(R), thus a(xo) = XQ and the action of a on it\(X, XQ) allows to define irfrb(Y) as a semidirect product of ~K\(X,XQ) with Z / 2 . One checks that the definition is independent of the choice of xo. We have thus in all cases an exact sequence 1 -> 7n(X) ->• < r b ( y ) -> Z/2 -> 1. This sequence is very important when X is a K(n, 1), i.e., when the universal cover of X is contractible. 2. Complex Manifolds which are K (ir, l ) ' s and Real Curves of Genus 1 Some interesting questions are, both for complex and real varieties Question 1. To what extent, if X is a K(n,l), does then iti{X) termine the differentiable type of X, not only its homotopy type?
also de-
Question 2. Analogously, if (X, a) is a K(TT, 1) and is real, how much from the differentiable viewpoint is (X, a) determined by the orbifold fundamental group sequence? There are for instance, beyond the case of hyperelliptic surfaces, other similar instances in the Kodaira classification of real surfaces (joint work in progress with Paola Frediani). Question 3. Determine, for the real varieties whose differentiable type is determined by the orbifold fundamental group, those for which the moduli spaces are irreducible and connected. Remark 2.1. However, already for complex manifolds, the question whether (fixed the differentiable structure) there is a unique deformation type, has several negative answers, as we shall see in the later sections. To explain the seemingly superfluous statement irreducible and connected, let us observe that: a hyperbola {(x,y) € R 2 |xy = 1} is irreducible but not connected.
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I want to show now an easy example, leading to the quotient of the above hyperbola by the involution (a;, y) —• (—a;, —y) as a moduli space, and explaining the basic philosophy underlying the two above mentioned theorems concerning the orbifold fundamental group. 2.1. Real curves of genus
one
Classically, the topological type of real curves of genus 1 is classified according to the number v = 0, 1, or 2 of connected components (= ovals = homeomorphic to circles) of their real part. By abuse of language we shall also say: real elliptic curves, instead of curves of genus 1, although for many authors an elliptic curve comes provided with one point defined over the base field (viz.: the origin!). The orbifold fundamental group sequence is in this case 1 -»• Hi(C,Z)
^ Z 2 -> < r f e (C) -»• Z/2 -> 1.
and it splits iff C(R) ^ 0 (since 7r°r6(C) has a representation as a group of affine transformations of the plane). Step 1. If there are no fixed points, the action s of Z/2 on Z 2 (given by conjugation) is diagonalizable. Proof. Let a be represented by the affine transformation (x, y) —¥ s(x, y) + (a, b). Now, s is not diagonalizable if and only if for a suitable basis choice, s(x, y) — (y, x). In this case the square of a is the transformation (x, y) —> (x, y) + (a + b,a + b), thus a + b is an integer, and therefore the points (x, x — a) yield a fixed S1 on the elliptic curve. • Step 2. If there are no fixed points, moreover, the translation vector of the affine transformation inducing a can be chosen to be 1/2 of the 4-1eigenvector e\ of s. Thus we have exactly one normal form. Step 3. There are exactly 3 normal forms, and they are distinguished by the values 0, 1, 2 for v. Moreover, s is diagonalizable if and only if v is even. Proof. If there are fixed points, then a may be assumed to be linear, so there are exactly two normal forms, according to whether a is diagonalizable or not. One sees immediately that v takes then the respective values 2, 1.! Moreover, we have then verified that s is diagonalizable if and only if v is even. •
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2.2. The description
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of the moduli space for the case v = 1
Let a act as follows: (x, y) —¥ (y, x). We look then for the complex structures J which make a antiholomorphic, i.e., we seek for the matrices J with J 2 = — 1 and with Js = — sJ. The latter condition singles out the matrices (_^ _ o ) while the first condition is equivalent to requiring that the characteristic polynomial be equal to A2 + 1, whence, equivalently, b2 — a2 = 1. We get a hyperbola with two branches which are exchanged under the involution J —> —J, but, as we already remarked, J and —J yield isomorphic real elliptic curves, thus the moduli space is irreducible and connected. 3. Real Hyperelliptic Surfaces on the Scene The treatment of real hyperelliptic surfaces is similar to the above sketched one of elliptic curves, and very much related to it, because the orbifold group has an affine representation. Recall: Definition 3.1. A complex surface S is said to be hyperelliptic if S = (E x F)/G, where E and F are elliptic curves and G is a finite group of translations of E with a faithful action on F such that F/G = P 1 . Historical Remark. The points of elliptic curves can be parametrized by meromorphic functions of z £ C. Around 1880, thanks to the work of Appell, Humbert and Picard, there was much interest for the hyperelliptic varieties of higher dimension n, whose points can be parametrized by meromorphic functions of 2 e C , but not by rational functions of z £ C n . The classification of hyperelliptic surfaces was finished by Bagnera and De Pranchis [6,7] who were awarded the Bordin Prize in 1908 for this important achievement. Theorem 3.2 (Bagnera— de Franchis). Every hyperelliptic surface is one of the following, where E, F are elliptic curves and G is a group of translations of E acting on F as specified (p is a primitive third root of unity) : (1) (E x F)/G, G = Z/2 acts on F by x ^ -x. (2) (E x F)/G, G = Z/2 © Z/2 acts on F by x i-> -x, x H* X + e, where e is a half period. (3) (E x Fi)/G, G = Z/4 acts on Ft by x M- ix.
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(4) (5) (6) (7)
(E (£ (£ (E
x Fi)/G, x Fp)/G, x F,)/G, x F p )/G,
G= G= G= G=
Z/4 © Z/2 acts Z/3 aete on Fp Z/3 0 Z/3 acis Z/6 acts on Fp
on Fi by x M- ix, x •-»• x + (1 + i)/2. by x (-»• px. on F„ 6y a; H» pa;, a; i-> a; + (1 - p)/3. by x M- - p x .
In fact, the characterization of hyperelliptic surfaces is one of the two key steps of Enriques' classification of algebraic surfaces, we have namely: Theorem 3.3. The complex surfaces S with K nef, K2 = 0, pg = 0, and such that either S is algebraic with q = 1, or, more generally, such that 6i = 2, are hyperelliptic surfaces if and only ifkod(S) — 0 (equivalently, iff the Albanese fibres are smooth of genus 1). I want now to illustrate the main technical ideas used for the classification of real Hyperelliptic Surfaces. Definition 3.4. The extended symmetry group G is the group generated by G and a lift a of a. Rerunning the classification theorem yields Theorem 3.5. Let (S,a), (S,a) be isomorphic real hyperelliptic surfaces: then the respective extended symmetry groups G are the same and given two Bagnera — De Franchis realizations S = (Ex F)/G, S = (Ex F)/G, there is an isomorphism $: ExF-tExF, of product type, commuting with the action ofG, and inducing the given isomorphism ip : S —» S. Moreover, let a : E x F —> E x F be a lift of a. Then the antiholomorphic map a is of product type. We need to give the list of all the possible groups G. Lemma 3.6. Let us consider the extension 0 -» G -> G -> Z/2 = (o-) -> 0.
(*)
We have the following possibilities for the action of a on G : in what follows the subgroup T of G will be the subgroup acting by translations on both factors. (1) If G = Z/q, q = 2, 4, 3, 6 then a acts as —Id on G and G = Dq. (2) IfG = Z/2 x Z/2, then either (2.1) a acts as the identity on G, (*) splits, and G = Z/2 x Z/2 x Z/2 or
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(2.2) a acts as the identity on G, (*) does not split, and G = Z/4 x Z/2 (and in this latter case the square of a is the generator ofT) or (2.3) a acts as (j j j , (*) splits, G = DA, the dihedral group, and again the square of a generator of Z/4Z is the generator of T. (3) If G = Z/4 x Z/2, then either G = T x DA ^ Z/2 x DA, or G is isomorphic to the group G\ := (a, g, t |er2 = 1, gA = 1, t2 = 1, ta = at, tg = gt, o~g = g~1ta), and its action on the second elliptic curve F is generated by the following transformations: o~(z) = z + 1/2, g(z) = iz, t(z) = z + l / 2 ( l + i ) . [The group Gi is classically denoted byc\ (cf. Atlas of Groups)]. In particular, in both cases (*) splits. (4) If G = Z/3 x Z/3, then we may choose a subgroup G' of G such that a acts as -Id x Id on G' x T = G and we have G = D3 x Z / 3 . Main ideas in the classification of real hyperelliptic surfaces. The first point is to show that the orbifold fundamental group has a unique faithful representation as a group of affine transformations with rational coefficients. And the second is to distinguish the several cases, using byproducts of the orbifold fundamental group, such as the structure of the extended Bagnera de Franchis group G, parities of the (invariant u of the) involutions in G, their actions on the fixed point sets of transformations in G, the topology of the real part, and, in some special cases, some more refined invariants such as the translation parts of all possible lifts of a given element of G to the orbifold fundamental group. How to calculate the topology of their real part? We simply use the Albanese variety and what we said about the real parts of elliptic curves, that they consist of v < 2 circles. Since also the fibres are elliptic curves, we get up to 4 circle bundles over circles, in particular the connected components are either 2-Tori or Klein bottles. For every connected component V of Fix(cr) = 5(R) its inverse image 7r _ 1 (y) in E x F splits as the G-orbit of any of its connected components. Let W be one such: then one can easily see that there is a lift a of a such that a is an involution and W is in the fixed locus of a: moreover a is unique. Thus the connected components of Fix(
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Then we let C be the set of equivalence classes of connected components of UFix(
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Lemma 4.2. On a compact complex manifold X one has an injection H°(dOx)
© H°(dOx)
-»• HhR(X, C ) .
Proof. It is sufficient to show the injection H°(dOx) the map sending
->• Hj}R{X,R)
by
U) —¥ U + U) .
Else, there is a fucntion / with df = u> + Q, whence df = u> and therefore ddf = d(ui) = 0. Thus / is pluriharmonic, hence constant by the maximum principle. Follows that u) = 0. • Lemma 4.3. Assume that {Xt}teA is a 1-parameter family of compact complex manifolds over the 1-dimensional disk, such that there is a sequence tv —» 0 with Xt„ Kahler. Then the weak 1-Kahler decomposition HhR(X0, C) = H°(dOXo)
©
H°(dOXo),
holds also on the central fibre Xo. Proof. We have / : E —> A which is proper and smooth, and /*(Oi, A ) is torsion free, whence (A is smooth of dimension 1) it is locally free of rank q := (1/2)6 1 (X 0 ). In fact, there is (cf. [71], II. 5) a complex of Vector Bundles on A, E° - • E1 -»• E2 -»• • • • En
(**)
such that (1) Rlf*(^lhi^) is the i-th cohomology group of (**), whereas (2) Hl(Xt, £lx ) ^s t n e *" tn cohomology group of (**) ® Cf. Claim. There are holomorphic 1-forms w i ( t ) , . . . ,u)q(t) defined in the inverse image f~l(Uo) of a neighbourhood (Uo) of 0, and linearly independent for each t €UQProof of the Claim. Assume that u>\(t),... ,<Jq(t) generate the direct image sheaf /*(0|;| A ), but a>x(0),... ,w g (0) be linearly dependent. Then, w.l.o.g. we may assume Wi(0) = 0, i.e., there is a maximal m such that u>i(t) := u>i(t)/tm is holomorphic. Then, since &i(t) is a section of /*(fii| A ), there are holomorphic functions a; such that u>i(t) = Si=i,... iqati(t)wi(t), whence it follows that wi(i)(l — tmcti(t)) = Ei=2,...,qOti(t)wi(t).
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This however contradicts the fact that f*(£lhiA) is locally free of rank q and completes the proof of the claim. Let dv be the vertical part of exterior differentiation, i.e., the composition of d with the projection ( f i | -¥ fi||A). By our asumption, 0Ji(tu) is enclosed, whence, by continuity, also Wj(0) G H°(dOx0)- It follows from the previous lemma that, being h = 2q, HlR(X0,C) = H°(dOXo) © H°(dOXo). • Recall that for a compact complex manifold X, the Albanese Variety Alb(X) is the quotient of the complex dual vector space of H°(dOx) by the minimal closed complex Lie subgroup containing the image of Hi(X,Z). The Albanese map ax : X -> Alb(X) is given as usual by fixing a base point XQ, and defining ctx{x) as the integration on any path connecting XQ with x. One says the the Albanese Variety is good if the image of H\ (X, Z) is discrete in H°(dOx), and very good if it is a lattice. Moreover, the Albanese dimension of X is defined as the dimension of the image of the Albanese map. With this terminology, we can state an important consequence of our assumptions. Corollary 4.4. Assume that {Xt}teA is a I-parameter family of compact complex manifolds over the 1-dimensional disk, such that there is a sequence tv —• 0 with Xt„ Kdhler, and moreover such that Xtv has maximal Albanese dimension. Then the central fibre XQ has a very good Albanese Variety, and has also maximal Albanese dimension. Proof. We use the fact (cf. [17]) that, when the Albanese Variety is good, then the Albanese dimension of X is equal to m&x{i\AlH°(dOx) <8> A*(jff°(dOx)) -» H%R(X,C) has non zero image}. If the weak 1-Kahler decomposition holds for X, then the Albanese dimension of X equals (l/2)max{,7'|A-7\H'1(A',C) has non zero image in H^XjC)}. But this number is clearly invariant by homeomorphisms. Finally, the Albanese Variety for Xo is very good since it is very good for Xtv and the weak 1-Kahler decomposition holds for XQ. •
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Remark 4.5. As observed in ([19, 1.9]), if a complex manifold X has a generically finite map to a Kahler manifold, then X is bimeromorphic to a Kahler manifold. This applies in particular to the Albanese map. Recall that (cf. [58]) a small deformation of a Kahler manifold is again Kahler. This is however (cf. [49]) false for deformations in the large, and shows that the following theorem (which pretty much follows the lines of Corollary C of [19]) is not entirely obvious (it answers indeed in the affirmative a problem raised by Kodaira and Spencer [59, p. 464], where the case n = 2 was solved in Theorem 20.2) Theorem 4.6. A deformation in the large of complex tori is a complex torus. This follows from the following more precise statement: let XQ he a compact complex manifold such that its Kuranishi family of deformations TV: S —> B enjoys the property that the set B(torus) := {b\Xb is isomorphic to a complex torus} has 0 as a limit point. Then Xo is a complex torus. We will use the following folklore lemma. Lemma 4.7. Let Y be a connected complex analytic space, and Z be an open set ofY such that Z is closed for holomorphic 1-parameter limits {i.e., given any holomorphic map of the 1-disk f : A —> Y, if there is a sequence U->0 with f(U) G Z, then also /(0) € Z). Then Z = Y. Proof. By choosing an appropriate stratification of Y by smooth manifolds, it suffices to show that the statement holds for Y a connected manifold. Since it suffices to show that Z is closed, let P be a point in the closure of Z, and let us take coordinates such that a neighbourhood of P corresponds to a compact poly cylinder H in C n . Given a point in Z, let H' be a maximal coordinate polycylinder contained in Z. We claim that H' must contain H, else, by the holomorphic 1-parameter limit property, the boundary of H' is contained in Z, and since Z is open, by compactness we find a bigger polycylinder contained in Z, a contradiction which proves the lemma. D
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Proof of Theorem 4.6. It suffices to consider a 1-parameter family (B = A) whence we may assume w.l.o.g. that the weak 1-Lefschetz decomposition holds for each t s A . By integration of the holomorphic 1-forms on the fibres (which are closed for tv and for 0), we get a family of Albanese maps at : Xt -¥ Jt, which fit together in a relative map a : H -> J over A (Jt is the complex torus
(H°(dOXt)y/Hi(Xt,Z)). Apply once more vertical exterior differentiation to the forms <Ji(t) : dv(wi{t) vanishes identically on XQ and on Xtu, whence it vanishes identically in a neighbourhood of Xo, and therefore these forms Wi(t) are closed for each t. Therefore our map a : S —> J is defined everywhere and it is an isomorphism for t = tv. Whence, for each t, at is surjective and has degree 1. To show that at is an isomorphism for each t it suffices therefore to show that a is finite. Assume the contrary: then there is a ramification divisor R of a, which is exceptional (i.e., if B = a(R), then dim B < dim R). By our hypothesis at is an isomorphism for t = tu, thus R is contained in a union of fibres, and since it has the same dimension, it is a finite union of fibres. But if R is not empty, we reach a contradiction, since then there are some t's such that at is not surjective. • Proposition 4.8. Assume that X has the same integral cohomology algebra of a complex torus and that H°(dOx) has dimension equal to n = dim(X). Then X is a complex torus. Proof. Since b\(X) = 2n, it follows from Lemma 4.2 that the weak 1Lefschetz decomposition holds for X and that the Albanese Variety of X is very good. That is, we have the Albanese map ax '• X —> J, where J is the complex torus J = Alb(X). We want to show that the Albanese map is an isomorphism. It is a morphism of degree 1, since ax induces an isomorphism between the respective fundamental classes of H2n(X, Z) = H2n(J, Z). There remains to show that the bimeromorphic morphism ax is finite. To this purpose, let R be the ramification divisor of ax '• X —> J, and B its branch locus, which has codimension at least 2. By means of blowing ups of J with non singular centres we can dominate X by a Kahler manifold g:Z -+X (cf. [19, 1.8, 1.9]).
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Let I ^ b e a fibre of ax of positive dimension such that g_1W is isomorphic to W. Since Z is Kahler, g~lW is not homologically trivial, whence we find a differentiable submanifold Y of complementary dimension which has a positive intersection number with it. But then, by the projection formula , the image g*Y has positive intersection with W, whence W is also not homologically trivial. However, the image of the class of W is 0 on J, contradicting that ax induces an isomorphism of cohomology (whence also homology) groups. • Remark 4.9. The second condition holds true as soon as the complex dimension n is at most 2. For n = 1 this is well known, for n = 2 this is also known, and due to Kodaira (cf. [56]): in fact for n = 2 the holomorphic 1-forms are closed, and moreover h°(dOx) is at least [(l/2)bi(X)\. For n > 3, the real dimension of X is greater than 5, whence, by the s-cobordism theorem [68], the assumption that X be homeomorphic to a complex torus is equivalent to the assumption that X be diffeomorphic to a complex torus. Andre Blanchard [10] constructed in the early 50's an example of a non Kahler complex structure on the product of a rational curve with a two dimensional complex torus. In particular his construction (cf. [77]), was rediscovered by Sommese, with a more clear and more general presentation, who pointed out that in this way one would produce exotic complex structures on complex tori. The Sommese-Blanchard examples (cf. [79], where we learnt about them) are particularly relevant to show that the plurigenera of non Kahler manifolds are not invariant by deformation. In the following section 5 we will adopt the presentation of [19]): the possible psychological reason why we had forgotten about these 3-folds (and we are very thankful to Andrew Sommese for pointing out their relevance) is that these complex structures do not have a trivial canonical bundle. Thus remains open the following Question 4. Let X be a compact complex manifold of dimension n > 3 and with trivial canonical bundle such that X is diffeomorphic to a complex torus: is then X a complex torus? The main problem is to show the existence of holomorphic 1-forms, so it may well happen that also this question has a negative answer. On the other hand, as was already pointed out, the examples of Blanchard, Calabi and Sommese show that the answer to a similar question is
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negative also in the case of a product C xT, where C is a curve of genus g > 2, and T is a complex torus. For this class of varieties we still have the same result as for tori, concerning global deformations. Theorem 4.10. A limit of products C xT of a curve of genus g > 2 with a complex torus is again a product of this type. This clearly follows from the more precise statement: let g > 2 a fixed integer, and assume that Xo be a compact complex manifold such that its Kuranishi family of deformations TT : S —> B enjoys the property that the set B" := {b\Xb is isomorphic to the product of a curve of genus g with a complex torus} has 0 as a limit point. Then XQ is isomorphic to a product C xT of a curve of genus g with a complex torus T. Proof. By Lemma 4.7 we can limit ourselves to consider the situation where B is a 1-dimensional disk, and every fibre Xt is, for t ^ 0, a product of the desired form. Step I. We claim that there is a morphism F : S —> C, where C —• B is a smooth family of curves of genus g. We use, for the purpose of proving our claim, the isotropic subspace theorem of [17], observing that the validity of this theorem does not require the full hypothesis that a variety X be Kahler, but that weaker hypotheses, e.g., the weak 1-Kahler and 2-Kahler decompositions, do indeed suffice. The first important property to this purpose is that the cohomology algebra H*(Xt,C) is generated by ^(X^C). The second important property is that for a product C x T as above, the subspace p\{Hl{C, C)) is the unique maximal subspace V, of dimension 2g, such that the image of A3(V) -> H3(C x T,C) is zero. This is the algebraic counterpart of the geometrical fact that the first projection is the only surjective morphism with connected fibre of C x T onto a curve C of genus > 2. From the differentiable triviality of our family E —> A follows that we have a uniquely determined such subspace V of the cohomology of Xo, that we may freely identify to the one of each Xt-
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Now, for t 7^ 0, we have a decomposition V = Ut © Ut, where Ut —
Pl(H°(nht)). By compactness of the Grassmann variety and by the weak 1-Kahler decomposition in the limit, the above decomposition also holds for Xo, and UQ is a maximal isotropic subspace in H°(dOx0). The Castelnuovo de Pranchis theorem applies, and we get the desired morphism F : H —> C to a family of curves. Step II. We produce now a morphism G : H —> T where T —• B is a family of tori. Let a : H -* J be the family of Albanese maps. By construction, or by the universal property of the Albanese map, we have that F is obtained by taking projections of the Albanese maps, whence we have a factorization a : 3 —>• J —• J", where J" —> B is the family of Jacobians of C. The desired family of tori T —> B is therefore the family of kernels of J -> J". To show the existence of the morphism G, observe that for t ^ 0, an isomorphism of Xt with Ct x Tt is given by a projection Gt : Xt —> Tt which, in turn, is provided (through integration) via a complex subspace Wt of H° (fijr,) whose real span Ht = Wt © Wt is M-generated by a subgroup
olH\XuZ). Although there are several choices for such a subgroup, we have at most a countable choice of those. Since for each t there is such a choice, by Baire's theorem there is a choice which holds for each t ^ 0. Let us make such a choice of H — Ht^t: then the corresponding subspace Wj has a limit in H for t = 0, and since the weak 1-Lefschetz decomposition holds for Xo, and this limit is a direct summand for Ut, we easily see that this limit is unique, and the desired morphism G is therefore obtained. Step III. The fibre product F Xg G yields an isomorphism onto C XgT for each fibre over t ^ 0, but since also the fibre over 0 is a torus fibration over Co, and the cohomology algebras are all the same, it follows that we have an isomorphism also for t — 0. • We end this section by giving an application of Theorem 4.6. Theorem 4.11. Let (X,a) be a real variety which is a deformation of a real torus: then also (X, a) is a real torus.
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The orbifold fundamental groupo sequence completely determines the differentiable type of the pair (X, a). The real deformation type is also completely determined by the orbifold fundamental group of (X, a). In turn, the orbifold fundamental group, if s is the linear integral transformation of Hi (X, Z) obtained by conjugation with a, is uniquely determined by the integer r which is the rank of the matrix (s — Id){mod2) acting on Hi (X, Z / 2 ) and, if r =£ n = dim X, by the property whether the orbifold exact sequence does or does not split (if r = n it always splits). P r o o f . T h e first statement is a direct consequence of T h e o r e m 4.6. Concerning t h e second statement, let a be a lifting of t h e antiholomorphic involution a on the universal cover C " : then it follows as usual t h a t t h e first derivatives are bounded, whence constant, a n d a is a n affine m a p . We may moreover assume a to be linear in the case where a admits a fixed point (in this case the orbifold exact sequence splits). Let A be the lattice Hi (X, Z): so we have first of all t h a t a is represented by the affine m a p of A ® E , x —> s(x) + b, where s is the isomorphism of A to itself, given by conjugation with a. T h u s s is an integral m a t r i x whose square is t h e identity, and it is well known ( cf. [22, L e m m a 3.11]) t h a t we can thus split A = U®V®W+®W~, + + where s acts by s(u, v,w ,w~) = (v,u,w , —w~). Since t h e number of (+1) — eigenvalues is equal t o the number of (—1) — eigenvalues, a being antiholomorphic, we get t h a t W+, W~ have t h e same dimension n — r (whence dim U = dim V = r). Since the square of a is the identity, it follows t h a t a1 is a n integral translation. This condition boils down to sb + b S A. If we write b = (bi, 62, b+,b"), the previous condition means t h a t 61 + 6 2 , 26+ are integral vectors. In order to obtain a simple normal form, we are allowed to take a different lift cr, i.e., t o consider a n affine m a p of the form x —¥ s(x) + b + A, where A 6 A, and then to take another point c G A ® 1 as the origin. T h e n we get an affine transformation of the form y —> sy + b + A + sc — c. T h e translation vector is thus (61,62, b+, 6~) + (Ai, A2, A + , A~) + (ci — C2, C2 — ci,0,2c-). We choose c~ — — (l/2)b~ and A - = 0, C2 — Ci = &i + Ai, —A2 = bi + 62 — Ai, and then we choose for A + the opposite of the integral p a r t of&+.
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We obtain a new affine map y ->• sy + b' where b' = (0,0, b'+,0) and all the coordinates of b'+ are either 0 or 1/2. So, either b' — 0, or we can assume that 26'+ is the first basis element of W+. We obtain thus two cases: (1) There exists a lift a represented by the linear map s for a suitable choice of the origin. (2) There exists a represented, for a suitable choice of the origin, and for the choice of a suitable basis of A, by the affine map y —> sy + (l/2)e~l where ef is the first basis element of W+. Case (1) holds if and only if there is a fixed point of a. Whereas, in case (2) we can never obtain that b'+ be zero, therefore the square of any lift a is always given by a translation with third coordinate 2b'+ ^ 0, so the orbifold exact sequence does not split. We conclude that (1) holds if and only if the orbifold exact sequence splits. By the normal forms obtained above, it follows that the integer r and the splitting or non splitting property of the orbifold fundamental group sequence not only determine completely the orbifold fundamental group sequence, but also its affine representation. To finish the proof of the theorem, assume that the normal form of a is preserved, whence also the affine representation of the orbifold fundamental group. We are now looking for al the translation invariant complex structures which make the transformation a antiholomorphic. As before in the case of the elliptic curves, we look for the 2n x 2n matrices J whose square J 2 = — Id, and such that Js = —sJ. The latter condition implies that J exchanges the eigenspaces of s in A ® R. We can calculate the matrices J in a suitable R-basis of Ag>R where s is diagonal, i.e., s(y1,y2) = (2/1,-2/2)- Then J{yi,y2) = (Ay2,Byx) and Js = —sJ is then satisfied. The further condition J2 = —Id is equivalent to AB = BA = —Id, i.e., to B = — A~x. Therefore, these complex structures are parametrized by GL(n,W), which indeed has two connected components, distinguished by the sign of the determinant. Since however we already observed that a provides an isomorphism between (X, a) and (X,a), and X corresponds to the complex structure —J, if J is the complex structure for X, we immediately obtain that A and —A give isomorphic real varieties.
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We are set for n odd, since det(-A) = ( - 1 ) " det A. In the case where n is even, observe that two matrices A and A' yield isomorphic real tori if and only if there is a matrix D € GL(A) such that D commutes with s, and such that D conjugates J to J ' (then the diffeomorphism is orientation preserving for the orientations respectively induced by the complex structures associated to A, resp. A'). Thus D respects the eigenspaces of s, whence D(yi, 1/2) = {Diyi, .D22/2), and A is transformed to A' = D2A(D{)~1. Whence, we see that the sign of the determinant of A' equals the one of (det A)(detD), whence we can change sign in any case by simply choosing D with det£> = — 1. • 5. The Blanchard-Calabi Threefolds The Sommese-Blanchard examples (cf. [10,11,77]) provide non Kahler complex structures X on manifolds diffeomorphic to a product C x T, where C is a compact complex curve and T is a 2-dimensional complex torus. In fact, in these examples, the projection X —> C is holomorphic and all the fibres are 2-dimensional complex tori. Also Calabi [13] showed that there are complex structures on a product C x T (that all these structures are non Kahlerian follows also by the arguments of the previous theorem, else they would produce a complex product structure C x T"). The result of Calabi is the following Theorem 5.1 (cf. [13]). Let C be a hyperelliptic curve of odd genus g > 3, and let T be a two dimensional complex torus. Then the differentiable manifold C x T admits a complex structure with trivial canonical bundle. We shall try to show that the construction of Calabi, although formulated in a different and very interesting general context, yields indeed a very special case of the construction of Sommese-Blanchard, which may instead be formulated and generalized as follows Theorem 5.2 (Jacobian Blanchardr-Calabi 3-Folds). Let C be curve of genus g > 0, and let W be a rank 2 holomorphic vector bundle admitting four holomorphic sections o~i, o~2, 0-3, 04 which are everywhere ^.-linearly independent (for instance, ifW = L@L, where H°(L) has no base points, then four sections as above do exist).
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Then the quotient X of W by the Z 4 -action acting fibrewise by translations: w —> w + E^i,... ,4 n^Oi is a complex manifold diffeomorphic to the differentiate manifold C xT, where T is a two dimensional complex torus, and will be called a Jacobian Blanchard-Calabi 3-fold. The canonical divisor of X equals Kx = K*(KC — detW), where n : X —> C is the canonical projection. Moreover, X is Kdhler if and only if the bundle W is trivial (i.e., iff X is a holomorphic product C x T). Indeed one has h°(flx) = g unless W is trivial, while, if the vector bundle V is defined through the exact sequence
then h}(Ox) = 9 +
h°{Vv).
Proof. The four holomorphic sections o~\, 02, 03, &4 make W a trivial Rvector bundle, and with this triviahzation we obtain that X is diffeomorphic to the product of C with a real four dimensional torus. Let us now show that any vector bundle W = L © L, where H°(L) has no base points, admits such sections a\, 01, 0%, 04. In fact, we have s\, s
:=* (-s 2 ,Si),<7 4 :=* ( i s 2 , i s i ) .
We have the Koszul complex associated to (si, S2): 0 -»• I T 1 -> {Ocf
-> L -»• 0.
Now, an easy calculation shows that, defining V as the kernel subbundle of the linear map given by a := (ci, <J2, U3,04)
then in the special case above we have V — L~x © L~l. Returning to the general situation, since now on a torus T = W /T there are canonical isomorphisms of H°(Q,^) with the space of linear forms on the vector space W, and of HX(OT) with the quotient vector space Horn
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(T,C)/H°(Q,^),
we obtain immediately
(1) the exact sequence 0 -> 7r*(ftc) -»• ^
-» (W v ) -)• 0,
(2) an isomorphism 7?.17r»(Cx) — ( ^ v ) From the Leray spectral sequence follows immediately that hl(Ox)
=
If X is Kahler, then, since the first Betti number of X equals 4 + 2g, then it must hold that /i°(ftx) = g + 2, in particular h?{Wy) > 2 . But (W v ) is a subbundle of a trivial bundle of rank 4, whence two linearly independent sections of {Wy) yield a composition (Oc)2 —> {Ww) —> (Oc) 4 whose image is a trivial bundle of rank 1 or 2. But in the former case the two sections would not be C-linearly independent, whence the image must be a trivial bundle of rank 2 and taking determinants of the composition, we get that (Wv) is trivial if h°(Wv) > 2. Similarly, assume that hP{Wy) = 1: then (W v ) has a trivial summand I of rank 1. Then we have a direct sum (Wv) = I®Q and dually W is a direct sum Oc®Q~1- We use now the hypothesis that there are four holomorphic sections of W which are R-independent at any point: it follows then that Q~l admits two holomorphic sections which are everywhere R-independent. Whence, it follows that also Q~X,Q are trivial. D Definition 5.3. Given a Jacobian Blanchard-Calabi 3-fold 7r : X —• C, any X-principal homogeneous space n' : Y —>• C will be called a BlanchardCalabi 3-fold. Corollary 5.4. The space of complex structures on the product of a curve C with a four dimensional real torus has unbounded dimension. Proof. Let G be the four dimensional Grassmarm variety G(l, 3): observe that the datum of an exact sequence 0 -> V -»• (Oc) 4
-*W->0
is equivalent to the datum of a holomorphic map / : C -> G, since for any such / we let V, W be the respective pull backs of the universal subbundle U and of the quotient bundle Q (of course, for the Blanchard-Calabi
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construction one needs the further open condition that the four induced sections of W be R-linearly independent at each point). We make throughout the assumption that / is not constant, i.e., W is not trivial. Then we have the following exact sequence:
where Nf, the normal sheaf of the morphism / , governs the deformation theory of the morphism / , in the sense that the tangent space to Def(/) is the space H°(Nf), while the obstructions lie in H1(Nj). By virtue of the fact that @G = Hom(?7, Q), and of the cohomology sequence associated to the aboce exact sequence, we get 0 -> H°(QC) -»• H°{Vy ®W)^ ->• H1^
®W)^
H\Nf)
H°(Nf)
->•
Hl(9c)
-> 0,
and we conclude that the deformations of the map are unobstructed provided H\VV ® W) = 0. This holds, in the special case where W = L © L, if the degree d of L satisfies d > g, since then Hl(Vv ®W) = HX{{L ®L)®{L® L)) = 0. If d > g the dimension of the space of deformations of the map / is given, for g > 2, by 3g — 3 + 4/i°(2L) = Ad +1 — g, and this number clearly tends to infinity together with d = deg(L). On the other hand, we want to show that the deformations of the map / yield effective deformations of X as a Lie group principal fibration. This can be seen as follows. Consider the exact sequence 0 -> n*(W) -+QX^>
7r*(Oc) -»• 0
and the derived direct image sequence
o -> (w) -»• 7r»ex ->• (e c ) -»> (V ® w) -> n^ex
-»• ( e c ) ® vy
-> (det(W) ® W) -> 1Z2TT,ex -> (©c) ® det(W) -> 0, where we used that n2-Kt(Ox) ^ A 2 (V V ) S det(W). Notice that 7r,©x is a vector bundle on the curve C, of rank either 2 or 3. In the latter case, since its image M in (©c) is saturated (i.e., (&c)/M is torsion free), then M = (®c) and all the fibres of it are then biholomorphic. In this case, for any holomorphic tangent vector field on C, which we identify to the 0-section of W, we get a corresponding tangent vector field
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on W by using the fiberwise simply transitive action of R 4 corresponding to the four chosen holomorphic sections. Since all the fibres are isomorphic, the corresponding tangent vector field on X is holomorphic, and the bundle W is then trivial. Since we are assuming W not to be trivial, we get thus an exact sequence 0 -> ( 9 C )
-> (Vv
® W)
->•
1V-K.QX
By the Leray spectral sequence follows that we have that if°(V v ® W)/H°((&c)) injects into fi'°(7?.17r»@x) which is a direct summand of H^Qx) = H°{1llTt«Qx) © Hl{W). Therefore the smooth space Def(/) of deformations of the map / embeds into the Kuranishi space Def(X) of deformations of X. The space H1 (W) is the classifying space for principal X homogeneous spaces (cf. [55,74]) and therefore we see that this subspace of H1(@x) corresponds to actual deformations. However for W of large degree we get HX(W) — 0, so then the Blanchard-Calabi 3-folds coincide with the Jacobian Blanchard-Calabi 3folds. D R e m a r k 5.5. (1) In the Blanchard-Calabi examples one gets a trivial canonical bundle iff det(H^) = Kc- This occurs in the special case where W = L(& L and L is a thetacharacteristic such that H°(L) is base point free. In particular, we have the Calabi examples where C is hyperelliptic of odd genus g and L is the l/2(g — l)th power of the hyperelliptic line bundle ofC. (2) Start with a Sommese-Blanchard 3-fold with trivial canonical bundle and deform the curve C and the line bundle L in such a way that the canonical bundle becomes the pull back of a non torsion element of Pic(C): then this is the famous example that the Kodaira dimension is not deformation equivalent for non Kahler manifolds (cf. [79]). (3) In the previous theorem we have followed the approach of [19], correcting however a wrong formula for TV^TT^OX (which would give the dual vector bundle). (4) The approach of Sommese is also quite similar, and related to quaternion multiplication, which is viewed as the C-linear map ip : H = C 2 —> HomR(tf = M4, H = C 2 ), defined by ip{q){q') = qq'. Whence, quaternion multiplication provides four sections of Opi(l) 2 which are M-linearly independent at each point.
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Sommese explores this particular situation as an example of the natural fibration occurring in the more general context of the so called quaternionic manifolds (later on, the word twistor fibration, for the other projection, has become more fashionable). Sommese moreover observes that given a line bundle L such that H°(L) is base point free, the choice of two independent sections yields a holomorphic map to P 1 and one can pull back Opi (l) 2 and the four sections: obtaining exactly the same situation we described above. We learnt from Sommese that Blanchard knew these examples too (they are also described in [79]). We will indeed prove in the sequel a much more precise statement concerning the deformations of Blanchard-Calabi 3-folds: we need for this purpose the following Definition 5.6. A Blanchard-Calabi 3-fold is said to be developable if the corresponding ruled surface is developable, or, in other words, if and only if the derivative of the corresponding map / : C -¥ G yields at each point p of C an element of (V v ig) W)p of rank < 1. Remark 5.7. Observe in fact that (cf. [4, p. 38]) since / is non constant, and if there is no point of P 3 contained in each line of the corresponding family of lines, / is the associated map to a holomorphic map F : C —>• P 3 , so the union of the family of lines is the tangential developable of the image curve of the mapping F. If such a point exists, it means that there is an effective divisor D on the curve C, and the bundle W is given as an extension 0 -»• (Oc)(-D)
-4 {Ocf
-*• W -+ 0
and moreover a fourth section of W is given (which together with the previous three provides the desired trivialization of the underlying real bundle). Remark 5.8. By a small variation of a theorem of E. Horikawa concerning the deformations of holomorphic maps, namely Theorem 4.9 of [17], we obtain in particular that, under the assumption H°((@c) ® Vw) = 0, we have a smooth morphism Def (TT) —> Def (X) . This assumption is however not satisfied in the case of Sommese-Blanchard 3-folds, at least in the case where the degree of L is large.
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Theorem 5.9. Any small deformation of a non developable BlanchardCalabi 3-fold such that H1^^ ® W) = 0 is again a Blanchard-Calabi 3-fold. In particular, a small deformation of a Sommese-Blanchard 3-fold with L of degree d > g is a Blanchard-Calabi 3-fold. Proof. In this context recall once more (cf. [55]) that the infinitesimal deformations corresponding to H1(W) correspond to the deformations of A" as a principal homogenous space fibration over the principal Lie group fibration X, and these are clearly unobstructed. Indeed, for the SommeseBlanchard examples, it will also hold H1(W) = 0, if the degree d of L is large enough. Assume that the vanishing H1^^ ® W) = 0 holds: then the deformations of the map / are unobstructed, and it is then clear that we obtain a larger family of deformations of X, parametrized by an open set in H°(Nf) © if 1 (W r ), and that this family yields deformations of X as a Blanchard-Calabi 3-fold (one can show that these are the deformations of X which preserve the fibration 7r). All that remains is thus to show that the Kodaira Spencer map of this family is surjective. Since we already remarked that we have an isomorphism W = n*(@x), then Hl{W) ^ HX-K*(QX), and we only have to control H°(n1ir*ex). To this purpose we split the long exact sequence of derived direct images into the following exact sequences 0 -> ( 6 c ) -+Vy ®W -> ft -!• 0, 0 -> 11 -> 1V-K*QX
->• K- -> 0,
0 -> /C ->• ( ( 9 C ) O Vy) -> W ® A 2 ( F V ) , and observe that all we need to prove is H°(JC) = 0. In fact, then H°{Nf) = H°(TZ) = H°(J11K*OX) • By the last exact sequence, it would suffice to show the injectivity of the linear map H°((GC) <8 ^ v ) -»• H°(W ® A2(VV). But indeed we will show that the sheaf K. is the zero sheaf. Consider again in fact the beginning of the exact sequence of derived direct images 0 _• (W) -> Tr.e* -> ( 9 c ) -»• ( ^ v ® W)
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and observe then that the homomorphism ( 6 c ) ->• {Vy <8> W) = TZ1nt,(iT*(W)) is indeed the derivative of the map f : C -+ G. Since the homomorphism of sheaves we are considering is ( 6 c ) <£> Vy = TZ^^iQc)) -)• U2Tvt{TT*(W)) = W h?{Vy) the sheaf map ( 6 C ) ® Vw -> W ® A 2 (V V ) is induced by wedge product of ( 6 C ) -)• (V v ® W) with the identity of V v , therefore we will actually have that the sheaf K. is zero if the subsheaf of (V v ® W) given by the image of ( 6 c ) —> (Vy W) consists of tensors of generical rank 2. But this holds by virtue of the hypothesis that X be non developable. Finally, let X be a Sommese-Blanchard 3-fold with L of degree d> g, then we have the desired vanishing i / 1 ( V v W) = 0. The condition that the associated ruled surface is non developable follows from a direct calculation which shows that, if fit) is given as the subspace generated by the columns of a 4 x 2 matrix B(t), then the 4 x 4 matrix B(t)B'{t) has determinant equal to the square (up to sign) of the Wronskian determinant of the section s. But s is non constant, for a Blanchard-Calabi 3-fold, therefore the Wronskian determinant is not identically zero and we are done. • Remark 5.10. How does the developable case occur and which are its small deformations? The condition that the four sections should in every point of C provide a real basis of the fibre of W simply means that there is no real point in the developable surface associated to / : C —> G. A more detailed analysis of the developable Blanchard-Calabi 3-folds and of their deformations could allow a positive answer to the following Question 5. Do the above Blanchard-Calabi 3-folds provide infinitely many deformation types on the same differentiable manifold C x T? {In other words, do these just give infinitely many irreducible components of the moduli space, or also connected components?) Calabi's construction is also quite beautiful, so that we cannot refrain from indicating its main ideas. The first crucial observation is that, interpreting E 7 as the space C l of imaginary Cay ley numbers, for any oriented hypersurface M c R 7 , the Cayley product produces an almost complex structure as follows: J(v) = (v x n ) 1 ,
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where v is a tangent vector at the point x e M, n is the normal vector at x € M, and wl stands for the imaginary part of a Cayley number w. For instance, this definition provides the well known non integrable almost complex structure on the 6-sphere S6. Moreover, Calabi shows that the almost complex structure is special hermitian, i.e., that its canonical bundle is trivial. Second, Calabi analyses when is the given complex structure integrable, proving in particular the following. If we write E 7 = R 3 x R 4 , according to the decomposition C = H ® He^ (here H is the space of Hamilton's quaternions), and M splits accordingly as a product 5 x R 4 , then the given complex structure is integrable if and only if 5 is a minimal surface inR3. The third ingredient is now a classical method used by Schwarz in order to construct minimal surfaces. Namely, let C be a hyperelliptic curve, so that the canonical map is a double cover of a rational normal curve of degree g — 1. If g = 3 we have then a basis of holomorphic differentials u>i, uj2, k>3 such that the sum of their squares equals zero. Similarly, for every odd genus, via an appropriate projection to P 2 , we obtain three linearly independent holomorphic differentials, without common zeros, and satisfying also the relation w\ + u)\ + wf = 0. The integral of the real parts of the uVs provides a multivalued map of C to R 3 which is a local embedding. Since another local determination differs by translation, the tangent space to a point in C is then naturally a subspace of R 3 , whence Cayley multiplication provides a well denned complex structure on C x R 4 . Since moreover this complex structure is also invariant by translations on R 4 , we can descend a complex structure on CxT. From the contructions of Blanchard and Calabi we deduce a negative answer to the problem mentioned in Remark 2.1 and in the introduction. Corollary 5.11. Products of a curve of genus g > 1 with a 2-dimensional complex torus provide examples of complex manifolds which are a K(n, 1), and for which there are different deformation types. We shall see in the next section that the answer continues to be negative even if we restrict to Kahler, and indeed projective manifolds, even in dimension two.
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6. Moduli Spaces of Surfaces of General Type In this section we begin to describe a recent result (cf. [21]), showing the existence of complex surfaces which are K(n, l)'s and for which there are different deformation types. Before we get into the details of the construction, it seems appropriate to give a more general view of the status of the art concerning deformation, differentiable and topological types of algebraic surfaces of general type. Let 5 be a minimal surface of general type: then to S we attach two positive integers x = x(@s), V — Ks which are invariants of the oriented topological type of S. The moduli space of the surfaces with invariants (x,y) is a quasiprojective variety (cf. [48]) denned over the integers, in particular it is a real variety. Fixed (x, y) we have several possible topological types, but indeed only two if moreover the surface S is simply connected. These two cases are distinguished as follows: (1) S is even, i.e., its intersection form is even: then 5 is a connected sum of copies of a K3 surface (possibly with reversed orientation) and of Pi x pi. (2) S is odd: then 5 is a connected sum of copies of P£ and P c o p p . Remark 6.1. P£ o p p stands for the same manifold as P^, but with reversed orientation. It is rather confusing, especially if one has to do with real structures, that some authors use the symbol P£ for P c o p p . In general, the fundamental group is a powerful topological invariant. Invariants of the differentiable structure have been found by Donaldson and Seiberg-Witten, and one can easily show that on a connected component of the moduli space the differentiable structure remains fixed. Up to recently, the converse question DEF = DIFF? was open, but recently counterexamples have been given, by Manetti [64] for simply connected surfaces, by Kharlamov and Kulikov [53] for rigid surfaces, while we have found rather simple examples [21]: Theorem 6.2. Let S be a surface isogenous to a product, i.e., a quotient S = (Ci xC2)/G of aproduct of curves by the free action of a finite group G. Then any surface with the same fundamental group as S and the same Euler number of S is diffeomorphic to S. The corresponding moduli space Msop
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is either irreducible and connected or it contains two connected components which are exchanged by complex conjugation. There are infinitely many examples of the latter case. Corollary 6.3. (1) DEF ^ DIFF. (2) There are moduli spaces without real points. (3) There are complex surfaces whose fundamental group cannot be the fundamental group of a real surface. For the construction of these examples we imitate the hyperelliptic surfaces, in the sense that we take S = (Ci x C-^jG where G acts freely on C\, whereas the quotient C^jG is P£.. Moreover, we assume that the projection <j> : Ci —> P^ is branched in only three points, namely, we have a so called triangle curve. What happens is that if two surfaces of such sort are antiholomorphic, then there would be an antiholomorphism of the second triangle curve (which is rigid). Now, giving such a branched cover <j> amounts to viewing the group G as a quotient of the free group on two elements. Let a, c be the images of the two generators, and set abc = 1. We find such a G with the properties that the respective orders of a, b, c are distinct, whence an antiholomorphism of the triangle curve would be a lift of the standard complex conjugation if the 3 branch points are chosen to be real, e.g., —1, 0, + 1 . Such a lifting exists if and only if the group G admits an automorphism r such that r(a) = a - 1 , T(C) = c - 1 . An appropriate semidirect product will be the desired group for which such a lifting does not exist. For this reason, in the next section we briefly recall the notion and the simplest examples of the so called triangle curves.
7. Some N o n Real Triangle Curves Consider the set B C P£. consisting of three real points B := {—1,0,1}. We choose 2 as a base point in Pj. — JB, and we take the following generators a, (3, 7 of 7Ti(Pc — B, 2) such that a/?7 = 1
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and a, 7 are free generators of 7i"i(Pc — B, 2) with a, /3 as indicated in the following picture.
££}
£D
2
oc
e
With this choice of basis, we have provided an isomorphism of 7Ti (Pj. — B, 2) with the group T00:=(a,/3,7|a/37 = l ) . For each finite group G generated by two elements a, b, passing from Greek to latin letters we obtain a tautological surjection TT-.Too^G, i.e., we set 7t(a) = a, 7r(/3) = b and we define 7r(7) := c. (then abc = 1). To TV we associate the Galois covering / : C —• PJ., branched on B and with group G. Notice that the Fermat curve C := {(x0, x\, x^) € Pcl^o + : E " + a;2 =®) is in two ways a triangle curve, since we can take the quotient of C by the group G := (Z/n) 2 of diagonal projectivities with entries n-th roots of unity, but also by the full group A = Aut(C) of automorphisms, which is a semidirect product of the normal subgroup G by the symmetric group permuting the three coordinates. For G the three branching multiplicities are all equal to n, whereas for A they are equal to (2,3,2n). Another interesting example is provided by the Accola curve (cf. [1,2]), the curve Yg birational to the affine curve of equation y2 =
x2g+2
_
1
If we take the group G = Z/2 x Z/(2g + 2) which acts multiplying y by —1, respectively 1 by a primitive 2g + 2-root of 1, then we realize Yg as a triangle curve with branching multiplicities (2, 2g + 2,2<7 + 2). G is not however the full automorphism group, in fact if we add the involution sending x to 1/x and y to iy/x9+l, then we get the direct product Z/2 x D2g+2 (which is indeed the full group of automorphisms of Yg as it is well known and as also follows from the next lemma), a group which represents Yg as a triangle curve with branching multiplicities (2,4,1g + 2).
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One can get many more examples [61] by taking unramified coverings of the above curves (associated to characteristic subgroups of the fundamental group). The following natural question arises then: which are the curves which admit more than one realization as triangle curves? It is funny to observe: Lemma 7.1. Let f : C —• P£. = C/G be a triangle covering where the branching multiplicities m, n, p are all distinct (thus we assume m < n < p). Then the group G equals the full group A of automorphisms of C. Sketch of the proof. Step I. By Hurwitz's formula the cardinality of G is in general given by the formula \G\ = 2( 5 - 1)(1 - 1/m - l/n - 1/p)" 1 . Step I I . Assume that A ^ G and let F : Pj. = C/G -> P£ = C/A be the induced map. Then / ' : C —&• P^ = C/A is again a triangle covering, otherwise the number of branch points would be > 4 and we would have a non trivial family of such Galois covers with group A. Step III. We claim now that the three branch points of / cannot have distinct images through F: otherwise the branching multiplicities m' < n' < p' for / ' would be not less than the respective multiplicities for / , and by the analogous of formula I for \A\ we would obtain \A\ < \G\, a contradiction. The rest of the proof is complicated. • We come now to our particular triangle curves. Let r, m be positive integers r > 3, m > 4 and set p := rm — 1, n := (i— l ) m . Notice that the three integers m
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Step I. G = A, where A is the group of holomorphic automorphisms of C, A:=Bihol(C,C). This follows from Lemma 7.1. Step II. If a exists, it must be a lift of complex conjugation. In fact a normalizes Aut(C), whence it must induce a antiholomorphism of P,J. which is the identity on B, and therefore must be complex conjugation. Step III. Complex conjugation does not lift. This is purely an argument about covering spaces: complex conjugation acts on 7Ti(Pc — B,2) = Too, as it is immediate to see with our choice of basis, by the automorphism r sending a, 7 to their respective inverses. Thus, complex conjugation lifts if and only if r preserves the normal subgroup K := kenr. In turn, this means that there is an automorphism p: G ->• G with p(a) = a~1,p(c) = c _ 1 . Recall now the relation aca~x = cr: applying p, we would get a~1c~1a = c~ , or, equivalently, r
-1
r
a ca = c . But then we would get aca~x = cT = o _ 1 ca = am~1cam~1 which holds only if
= cr
,
r = 7- m - 1 (modp). Since p = rm — 1 we obtain, after multiplication by r, that we should have r 2 = l(modp) but this is the desired contradiction, because r 2 — 1 < r m - 1 = p. a 8. The Examples of Surfaces Isogenous to a Product Definition 8.1. A projective surface S is said to be isogenous to a (higher) product if it admits a finite unramified covering by a product of curves of genus > 2. Remark 8.2. In this case, (cf. [20, Propositions 3.11 and 3.13]) there exist Galois realizations S = (Ci x C^/G, and each such Galois realization
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dominates a uniquely determined minimal one. S is said to be of nonmixed type if G acts via a product action of two respective actions on C\, C2. Else, G contains a subgroup G° of index 2 such that (Ci x C2)/G° is of nonmixed type. Proposition 8.3. Let S, S' be surfaces isogenous to a higher product, and let a : S —> S' be a antiholomorphic isomorphism. Let moreover S = (Ci x C2)/G, S' = (C[ x C2)/G' be the respective minimal Galois realizations. Then, up to possibly exchanging C[ with C2, there exist antiholomorphic isomorphisms CTJ, i — 1, 2 such that a := d\ x a2 normalizes the action of G, in particular &i normalizes the action of G° on Ci. Proof. Let us view a as yielding a complex isomorphism a : S —> S'. Consider the exact sequence corresponding to the minimal Galois realization S = (Ci x C2)/G, l->H:=Ugix
Ug2 ->• 7^(5) -> G -> 1.
Applying cr» to it, we infer by Theorem 3.4 of [20] that we obtain an exact sequence associated to a Galois realization of S'. Since a is an isomorphism, we get a minimal one, which is however unique. Whence,we get an isomorphism a : (C\ x C-i) -> {C[ x C2), which is of product type by the rigidity lemma (e.g., [20, Lemma 3.8]). Moreover this isomorphism must normalize the action of G = G', which is exactly what we claim. • In ([20] and correction in [27]) we have Theorem 8.4. Let S be a surface isogenous to a product, i.e., a quotient S — {C\ x C2)/G of a product of curves by the free action of a finite group G. Then any surface S' with the same fundamental group as S and the same Euler number of S is diffeomorphic to S. The corresponding moduli space Ms° p = M| l f f is either irreducible and connected or it contains two connected components which are exchanged by complex conjugation. We are now going to explain the construction of our examples: Let G be the semidirect product group we constructed in Sec. 2 and C2 be the corresponding triangle curve. Let moreover g[ be any number greater or equal to 2, and consider the canonical epimorphism rp of Hg> onto a free group of rank g[, such that in terms of the standard bases 01, &i,... ,afl<, 6ff<, respectively 7 1 , . . . , 7 ^ , we have V>(«i) = VK&O = 7»-
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Compose then ip with any epimorphism of the free group onto G, e.g., it suffices to compose with any fi such that (1(71) = a, n(j2) = b (and n("fj) can be chosen whatever we want for j > 3). For any point C[ in the Teichmiiller space we obtain a canonical covering associated to the kernel of the epimorphism fioip : Hg> —> G, call it C\. Definition 8.5. Let S be the surface S := (C\ x C2)/G because G acts freely on the first factor).
(S is smooth
Theorem 8.6. For any two choices C[(I), C[{II) ofC[ in the Teichmiiller space we get surfaces S(I), S(II) such that S(I) is never isomorphic to S(II). Varying C[ we get a connected component of the moduli space, which has only one other connected component, given by the conjugate of the previous one. The last result that we have obtained as an application of these ideas is the following puzzling: Theorem 8.7. Let S be a surface in one of the families constructed above. Assume moreover that X is another complex surface such that T^I{X) = 7Ti(5). Then X does not admit any real structure. Proof (idea). Observe that since 5 is a classifying space for the fundamental group of 7Ti(5), then by the isotropic subspace theorem [17] of the Albanese mapping of X maps onto a curve C"(/)2 of the same genus as C 2 . Consider now the unramified covering X associated to the kernel of the epimorphism iri(X) = TTI(5) -> G. Again by the isotropic subspace theorem, there exists a holomorphic map X —> C(I)\ x C(I)2, where moreover the action o f G o n X induces actions of G on both factors which either have the same topological types as the actions of G o n C i , resp. C2, or have both the topological types of the actions on the respective conjugate curves. By the rigidity of the triangle curve C2, in the former case C(J)2 — C2, in the latter C(I)2 = C2. Assume now that X has a real structure a: then the same argument as in [24] Sec. 2 shows that a induces a product antiholomorphic map a : C(I)i x C(I)2 —> C(I)i x C(I)2- In particular, we get a non costant antiholomorphic map of C2 to itself, contradicting Proposition 7.2. • In [20,21] we have the following definition.
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Definition 8.8. A Beauville surface is a rigid surface isogenous to a product. Beauville [9] gave examples of these surfaces, as quotients of the product C x C where C is the Fermat quintic curve. This example is real. It would be interesting to (1) Classify all the Beauville surfaces, at least those of non mixed type, and (2) Classify all the non real Beauville surfaces, especially those which are not isomorphic to their conjugate surface. Acknowledgments I am grateful to S. T. Yau for pointing out Calabi's example, and for his invitation to Harvard, where some of the results in this paper were obtained. Thanks also to P. Frediani, with whom I started to investigate the subtleties of the real world [23] and [24], for several interesting conversations. I would like moreover to thank Mihai Paun and Hubert Flenner for pointing out a small error, and especially express my gratitude to Andrew Sommese for reminding me about the relevance of Blanchard's examples. The present research took place in the framework of the Schwerpunkt Globale Methode in der komplexen Geometrie, and of EAGER. References [1] R. D. M. Accola, On the number of automorphisms of a closed Riemann surface, Trans. Amer. Math. Soc. 131 (1968) 398-408. [2] R. D. M. Accola, Topics in the theory of Riemann surfaces, Lecture Notes in math. 1595, Springer Verlag (1994). [3] N. L. Ailing and N. Greenleaf, Foundations of the theory of Klein surfaces", Lecture Notes in Mathematics 219, Springer-Verlag (1980). [4] E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, Geometry of algebraic curves I, Grundlehren der math. Wiss. 267, Springer-Verlag (1985). [5] D. Auroux, Symplectic maps to projective spaces and symplectic invariants, Proc. of the Gokova Conf. 1999, Turk. J. Math. 25(1) (2001) 1-42. [6] G. Bagnera and M. de Pranchis, Sopra le superficie algebriche che hanno le coordinate del punto generico esprimibili con funzioni meromorfe 4ente periodiche di 2 parametri, Rendiconti Ace. dei Lincei 16 (1907). [7] G. Bagnera and M. de Pranchis, Le superficie algebriche le quali ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti, Mem. Ace. dei XL 15 (1908) 251-343.
Deformation Types of Real and Complex Manifolds 235 W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag (1984). A. Beauville, Surfaces algebriques complexes, Asterisque 54, Soc. Math. Fr. (1978). A. Blanchard, Recherche de structures analytiques complexes sur certaines varietes, C. R. Acad. Sci. Paris 238 (1953) 657-659. A. Blanchard, Sur les varietes analytiques complexes, Ann. Sci. Ec. Norm. Super., Ill 73 (1956) 157-202. E. Bombieri, Canonical models of surfaces of general type, Publ. Math. I.H.E.S. 42 (1973) 173-219. E. Calabi, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Amer. Math. Soc. 87 (1958) 407-438. F. Catanese, On the Moduli Spaces of Surfaces of General Type, J. Diff. Geom 19 (1984) 483-515. F. Catanese, Automorphisms of Rational Double Points and Moduli Spaces of Surfaces of General Type, Comp. Math. 6 1 (1987) 81-102. F. Catanese, Connected Components of Moduli Spaces, J. Diff. Geom 24 (1986) 395-399. F. Catanese, Moduli and classification of irregular Kahler manifolds (and algebraic varieties) with Albanese general type fibrations; Appendix by Arnaud Beauville, Inv. Math. 104 (1991) 263-289; Appendix 289. F. Catanese, (Some) Old and new Results on Algebraic Surfaces, Joseph, A. (ed.) et al., First European congress of mathematics (ECM), Paris, France, July 6-10, 1992. Volume I: Invited lectures (Part 1). Basel: Birkhaeuser, Prog. Math. 119 (1994) 445-490. F. Catanese, Compact complex manifolds bimeromorphic to tori, Proc. of the Conf. Abelian Varieties, Egloffstein 1993, De Gruyter (1995), 55-62. F. Catanese, Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer. Jour. Math. 122 (2000) 1-44. F. Catanese, Moduli spaces of surfaces and real structures, Florida State Univ. Publications (2001), math.AG/0103071, to appear in Annals of math. F. Catanese, Generalized Kummer surfaces and differentiable structures on Noether-Horikawa surfaces, I, in Manifolds and Geometry, Pisa 1993, pp. 132-177, Symposia Mathematica XXXVI, Cambridge University Press (1996). F. Catanese and P. Frediani, Configurations of real and complex polynomials, in Proc. of the Orsay Conf. on Algebraic Geometry, pp. 61-93, Soc. Math, de France, Asterisque 218 (1993). F. Catanese and P. Frediani, Real hyperelliptic surfaces and the orbifold fundamental group, Math. Gott. 20 (2000), math.AG/0012003. S. S. Chern, Characteristic classes in Hermitian manifolds, Ann. of Math. 47 (1946) 85-121. S. S. Chern, Complex manifolds, Publ. Mat. Univ. Recife (1958).
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[27] C. Ciliberto and C. Pedrini, Real abelian varieties and real algebraic curves, Lectures in Real Geometry, (Madrid, 1994), Fabrizio Broglia ed., pp. 167-256, de Gruyter Exp. Math. 23, Berlin-New York (1996). [28] A. Comessatti, Fondamenti per la geometria sopra superficie razionali dal punto di vista reale, Math. Ann. 73 (1913) 1-72. [29] A. Comessatti, Sulla connessione delle superficie razionali reali, Annali di Mat. p.a. 23(3) (1914) 215-283. [30] A. Comessatti, Reelle Fragen in der algebraischen Geometrie, Jahresbericht d. Deut. Math. Vereinigung 41 (1932) 107-134. [31] A. Comessatti, Sulle varieta abeliane reali I e II, Ann. Mat. Pura Appl. 2 (1924) 67-106 and Ann. Mat. Pura Appl. 4 (1926) 27-71. [32] A. Degtyarev, I. Itenberg and V. Kharlamov, Real Enriques surfaces, Lecture Notes in Math. 1746, Springer (2000). [33] A. Degtyarev and V. Kharlamov, Topological classification of real Enriques surfaces, Topology 35 (1996) 711-729. [34] A. Degtyarev and V. Kharlamov, Topological properties of real algebraic varieties: de cote de chez Rokhlin, Russ. Math. Surveys. 55(4) (2000) 735-814. [35] S. K. Donaldson, An Application of Gauge Theory to Four-Dimensional Topology, J. Diff. Geom. 18 (1983) 279-315. [36] S. K. Donaldson, Connections, cohomology and the intersection forms of 4-manifolds, J. Diff. Geom. 24 (1986) 275-341. [37] S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29(3) (1990) 257-315. [38] S. K. Donaldson, Gauge theory and four-manifold topology, Joseph, A. ed. et al., First European congress of mathematics (ECM), Paris, France, July 6-10, 1992, Volume I: Invited lectures (Part 1), pp. 121-151, Basel: Birkhauser, Prog. Math. 119 (1994). [39] S. K. Donaldson, The Seiberg-Witten Equations and 4-manifold topology, Bull. Am. Math. Soc. 33(1) (1996) 45-70. [40] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, Oxford: Clarendon Press (1990). [41] P. Frediani, Real algebraic functions, real algebraic curves and their moduli spaces, Tesi di dottorato Universita' di Pisa (1997). [42] M. Freedman, On the Topology of 4-Manifolds, J. Diff. Geom. 17 (1982) 357-454. [43] R. Friedman, B. G. Moisezon and J. W. Morgan, On the C°° invariance of the canonical classes of certain algebraic surfaces, Bull. Amer. Math. Soc. 17 (1987) 283-286. [44] R. Friedman and J. W. Morgan, Algebraic surfaces and four-manifolds: some conjectures and speculations, Bull. Amer. Math. Soc. 18 (1988) 1-19. [45] R. Friedman and J. W. Morgan, Complex versus differentiate classification of algebraic surfaces, Topology Appl. 32(2) (1989) 135-139.
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R. Friedman and J. W. Morgan, Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. F. 27, Berlin: Springer-Verlag (1994). R. Friedman and J. W. Morgan, Algebraic surfaces and Seiberg-Witten invariants, J. Algebr. Geom. 6(3) (1997) 445-479. D. Gieseker, Global moduli for surfaces of general type, Inv. Math. 4 3 (1977) 233-282. H. Hironaka, An example of a non-Kahlerian complex-analytic deformation of Kahlerian complex structures, Ann. Math. 75(2) (1962) 190-208. J. Jost and S. T. Yau, Harmonic mappings and Kahler manifolds, Math. Ann. 262 (1983) 145-166. J. Jost and S. T. Yau, A strong rigidity theorem for a certain class of compact complex analytic surfaces, Math. Ann. 271 (1985) 143-152. V. Kharlamov, The topological type of nonsingular surfaces in RP of degree 4, Functional Anal. Appl. 10 (1976) 295-305. V. Kharlamov and V. Kulikov, On real structures of real surfaces, math.AG/0101098. F. Klein, Uber eine neue Art von Riemannschen Flachen, Math. Ann. 10 (1876) 398-416. K. Kodaira, On compact analytic surfaces II-III, Ann. Math. 77 (1963) 563-626, Ann. Math. 78 (1963) 1-40. K. Kodaira, On the structure of complex analytic surfaces I, Amer. J. Math. 86 (1964) 751-798. K. Kodaira, On the structure of complex analytic surfaces IV, Amer. J. Math. 90 (1968) 1048-1066. K. Kodaira and J. Morrow, Complex manifolds, Holt, Rinehart and Winston, New York-Montreal, Que.-London (1971). K. Kodaira and D. Spencer, On deformations of complex analytic structures I-II, Ann. of Math. 67 (1958) 328-466 . J. Kollar, The topology of real algebraic varieties, in Current developments in Mathematics 2000', pp. 175-208, Int. Press (2000). A. Macbeath, On a theorem of Hurwitz, Proc. Glasg. Math. Assoc. 5 (1961) 90-96. M. Manetti, On some Components of the Moduli Space of Surfaces of General Type, Comp. Math. 92 (1994) 285-297. M. Manetti, Degenerations of Algebraic Surfaces and Applications to Moduli Problems, Test di Perfezionamento Scuola Normale Pisa (1996) 1-142. M. Manetti, On the Moduli Space of diffeomorphic algebraic surfaces, Invent. Math. 143(1) (2001) 29-76. F. Mangolte, Cycles algebriques sur les surfaces K3 reelles, Math. Z. 225(4) (1997) 559-576. F. Mangolte and J. van Hamel, Algebraic cycles and topology of real Enriques surfaces, Compositio Math. 110(2) (1998) 215-237.
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F. Mangolte, Surfaces elliptiques reelles et inegalites de Ragsdale-Viro, Math. Z. 235(2) (2000) 213-226. B. Mazur, Differential topology from the point of view of simple homotopy theory, Publ. Math. I.H.E.S. 21 (1963) 1-93, and correction ibidem 22 (1964) 81-92. J. W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes 44 Princeton Univ. Press. G. Mostow, Strong rigidity of locally symmetric spaces, Annals of Math. Stud. 78 Princeton Univ. Press (1978). D. Mumford, Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, Vol. 5, Oxford University Press (1970). M. Seppala , Real algebraic curves in the moduli space of complex curves, Compositio Math. 74(03) (1990) 259-283. M. Seppala and R. Silhol, Moduli spaces for real algebraic curves and real abelian varieties Math. Z. 201(2) (1989) 151-165. I. R. Shafarevich, Principal homogeneous spaces defined over a function field Steklov Math. Inst. 64 (Transl. AMS Vol. 37) (1961) 316-346 (85-115). R. Silhol, Real Algebraic Surfaces, Lectures Notes in Mathematics 1392, Springer-Verlag (1989). Y. T. Siu, The complex analyticity of harmonic maps and the strong rigidity of compact Kahler manifolds, Annals of Math. 112 (1980) 73-111. A. J. Sommese, Quaternionic manifolds, Math. Ann. 212 (1975) 191-214. T. Suwa, On hyperelliptic surfaces, J. Fac. Sci. Univ. Tokyo 16 (1970) 469-476. K. Ueno, Bimeromorphic Geometry of algebraic and analytic threefolds, in C.I.M.E. Algebraic Threefolds, Varenna 1981, pp. 1-34, Lecture Notes in math. 947 (1982) 1-34. S. T. Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sc. USA 74 (1977) 1798-1799. E. Witten, Monopoles and Four-Manifolds, Math. Res. Lett. 1 (1994) 809-822.
Reprinted from Notice of the American Mathematical Society 43(10) (1996) 1117-1118 © 1996 American Mathematical Society
WEI-LIANG CHOW, 1911-1995
S. S. Chern
After a long illness Wei-Liang died on August 10, 1995. He and I first met in Hamburg, Germany, in October 1934, when I had just come from China as an entering student while he was on his way from Gottingen to Leipzig in order to work with van der Waerden. Wei-Liang came from a Mandarin family, one whose leading members realized the importance of the westernization of China. With their resources and enlightened view, the family produced during the turn of the century several leaders of Chinese society in different areas. Perhaps as a result of the family situation Wei-Liang did not attend a Chinese school. However, through private tutoring he was quite familiar with the Chinese language and Chinese history. His family position allowed him to go through college in the U.S., receiving his B.A. from the University of Chicago in 1931. His Chicago years gradually focused him to mathematics, and in 1932 he went to Gottingen, then one of the greatest mathematical centers in the world. Unfortunately political events in Germany during this period made his stay in Gottingen undesirable, and he decided to go to Leipzig to work with van der Waerden. It was at this juncture that we met in Hamburg. The decline of Gottingen had the result of elevating Hamburg to a leading mathematical center in Germany. Her leading attraction was Emil Artin, the young professor who gave excellent lectures and whose interest extended over all areas of mathematics. Although Wei-Liang was a Leipzig student, the German university system allowed him to live in Hamburg. Besides the contacts with Artin, he had a more important objective, which was to win the love of a young lady, Margot Victor. They were married in 1936, and I was fortunate to be present at the wedding. 239
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After their marriage Wei-Liang returned to China and became a professor of mathematics at the Central University in Nanking, then the Chinese capital. The next years China was at war, with the coastal provinces occupied by the Japanese. We next saw each other in 1946 in Shanghai after the war ended. In a decade of war years Wei-Liang had practically stopped his mathematical activities, and the question was whether it was advisable or even possible for him to come back to mathematics. His return to mathematics was most successful; I would consider it a miracle. He began by spending the years 1947-49 at the Institute for Advanced Study, after which he accepted a position at Johns Hopkins University, from which he retired in 1977. At Johns Hopkins he served as chairman for more than ten years. He was also responsible for the American Journal of Mathematics, a Hopkins publication and the oldest American mathematical journal. Wei-Liang was an original and versatile mathematician, although his major field was algebraic geometry. He made several fundamental contributions to mathematics: 1. A fundamental issue in algebraic geometry is intersection theory. The Chow ring has many advantages and is widely used. 2. The Chow associated forms give a description of the moduli space of the algebraic varieties in projective space. It gives a beautiful solution of an important problem. 3. His theorem that a compact analytic variety in a projective space is algebraic is justly famous. The theorem shows the close analogy between algebraic geometry and algebraic number theory. 4. Generalizing a result of Caratheodory on thermodynamics, he formulated a theorem on accessibility of differential systems. The theorem plays a fundamental role in control theory. 5. A lesser-known paper of his on homogeneous spaces gives a beautiful treatment of the geometry known as the projective geometry of matrices and treated by elaborate calculations. His discussions are valid in a more general context. Chow led a simple and secluded life, with complete devotion to mathematics and some other intellectual activities including philately. He was an authority on Chinese stamps and published a book on them. Margot and Wei-Liang had three daughters and a happy family life.
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Chow, 1911-1995
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Undoubtedly he was an algebraic geometer of the first caliber. I once nominated him, with the support of Zariski, for membership in the National Academy of Sciences. Unfortunately it did not meet with success, and I think it was a loss to the Academy.
Reprinted from Notice of the American Mathematical Society 43(10) (1996) 1119-1123 © 1996 American Mathematical Society
C O M M E N T S O N CHOW'S W O R K
Serge Lang Department of Mathematics, Yale University, New Haven, CT 06520, USA
Van der Waerden's prewar series of articles began an algebraization of Italian algebraic geometry. I was born into algebraic geometry in the immediate postwar period. This period was mostly characterized by the work of Chevalley, Chow, Weil (starting with his Foundations and his books on correspondences and abelian varieties), and Zariski. In the fifties there was a constant exchange of manuscripts among the main contributors of that period. I shall describe briefly some of Chow's contributions. I'll comment here mostly on some of Chow's works in algebraic geometry, which I know best. Chow Coordinates One of Chow's most influential works was also his first, namely, the construction of the Chow form, in a paper written jointly with van der Waerden [1]. To each projective variety, Chow saw how to associate a homogeneous polynomial in such a way that the association extends to a homomorphism from the additive monoid of effective cycles in projective space to the multiplicative monoid of homogeneous polynomials, and the association is compatible with the Zariski topology. In other words, if one cycle is a specialization of another, then the associated Chow form is also a specialization. Thus varieties of given degree in a given projective space decompose into a finite number of algebraic families, called Chow families. The coefficients of the Chow form are called the Chow coordinates of the cycle or of the variety. Two decades later he noted that the Chow coordinates 243
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can be used to generate the smallest field of definition of a divisor [12]. He also applied the Chow form to a study of algebraic families when he gives a criterion for local analytic equivalence [13]. He was to use them all his life in various contexts dealing with algebraic families. In Grothendieck's development of algebraic geometry, Chow coordinates were bypassed by Grothendieck's construction of Hilbert schemes whereby two schemes are in the same family whenever they have the same Hilbert polynomial. The Hilbert schemes can be used more advantageously than the Chow families in some cases. However, as frequently happens in mathematics, neither is a substitute for the other in all cases. In recent times, say during the last decade, Chow forms and coordinates have made a reappearance due to a renewed emphasis on explicit constructions needed to make theorems effective (rather than having noneffective existence proofs, say) and for computational aspects of algebraic geometry whereby one wants not only theoretical effectiveness but good bounds for solutions of algebraic geometric problems as functions of bounds on the data. Projective constructions such as Chow's are very well suited for such purposes. Thus Chow coordinates reappeared both in general algebraic geometry and also in Arakelov theory and in diophantine applications. The Chow coordinates can be used, for example, to define the height of a variety, and to compare it to other heights constructed by more intrinsic, nonprojective methods as in [41, 42, 43]. They were used further in Arakelov theory by Bin Wang [44]. Chow coordinates were also used to prove a conjecture of Lie on a converse to Abel's theorem. See the papers by Wirtinger [47] and Chern [38]. Abelian Varieties and Group Varieties (a) Projective construction of the Jacobian variety. In the fifties Chow contributed in a major way to the general algebraic theory of abelian varieties due to Weil (who algebraicized the transcendental arguments of the Italian school, especially Castelnuovo). For one thing, Chow gave a construction of the Jacobian variety by projective methods, giving the projective embedding directly and also effectively [19]. The construction also shows that when a curve moves in an algebraic family, then the Jacobian also moves along in a corresponding family. (b) The Picard variety. Chow complemented Igusa's transcendental construction of the Picard variety by showing how this variety behaves well in algebraic systems, using his "associated form" [15]. He announced
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an algebraic construction of the Picard variety in a "forthcoming paper". Indeed, such a paper circulated as an unpublished manuscript a few years later [22] but was never published as far as I know. (c) Fixed p a r t of a n algebraic s y s t e m . Chow also developed a theory of algebraic systems of abelian varieties, defining the fixed part of such systems, i.e., that part which does not depend genuinely on the parameters [20, 21]. His notion of fixed part was used by others in an essential way, e.g., by Lang-Neron, who proved that for an abelian variety A defined over a function field K, the group of rational points of A in K modulo the group of points of the fixed part is finitely generated [41]. This is a relative version of the Mordell-Weil theorem. (d) Field of definition. Chow gave conditions under which an abelian variety defined over an extension of a field k can actually be defined over k itself [20, 21]. Chow's idea was extended by Lang [40] to give such a criterion for all varieties, not just abelian varieties, and Weil reformulated the criterion in terms of cohomology (splitting a cocycle) [46]. Homogeneous Spaces (a) Projective embedding of homogeneous spaces. Chow extended the Lefschetz-Weil proof of the projective embedding of abelian varieties to the case of homogeneous spaces over arbitrary group varieties, which may not be complete [25]. Chow's proof has been overlooked in recent years, even though interest in projective constructions has been reawakened, but I expect Chow's proof to make it back to the front burner soon just like his other contributions. (b) Algebraic p r o p e r t i e s . Chow's paper [9] dealt with the geometry of homogeneous spaces. The main aim of this paper is to characterize the group by geometric properties. The latter could refer to the lines in a space, as in projective geometry, or to certain kinds of matrices, such as symmetric matrices. For instance, a typical theorem says: Any bijective adjacencepreserving transformation of the space of a polar system with itself is due to a transformation of the basic group, provided that the order of the space is greater than 1. Birational geometry is considered in this context. TheChow Ring In topology, intersection theory holds for the homology ring. In 1956 Chow defined rational equivalence between cycles on an algebraic variety and
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defined the intersection product for such classes, thus obtaining the Chow ring [23], which proved to be just as fundamental in algebraic geometry as its topological counterpart. Algebraic Geometry over Rings In the late fifties began the extension of algebraic geometry over fields to algebraic geometry over rings of various type, partly to deal with algebraic or analytic families, but partly because of the motivation from number theory, where one deals with local Dedekind rings, p-adic rings, and more generally complete Noetherian local rings. Chow contributed to this extension in several ways. Of course, in the sixties Grothendieck vastly and systematically went much further in this direction, but it is often forgotten that the process had begun earlier. I shall mention here some of Chow's contributions in this direction. (a) Connectedness theorem. In 1951 Zariski had proved a general connectedness theorem for specializations of connected algebraic sets. Zariski based his proof on an algebraic theory of holomorphic functions which he developed for this purpose. In [26] and [32] Chow gave a proof of a generalization over arbitrary complete Noetherian local domains, based on much simpler techniques of algebraic geometry, especially the Chow form. (b) Uniqueness of the integral model of a curve. The paper [27] proved the uniqueness of the model of a curve of genus > 1 and an abelian variety over a discrete valuation ring in the case of nondegenerate reduction. (c) Cohomology. Invoking the theory of deformations of complex analytic structures by Kodaira-Spencer, the connectedness theorem, and Igusa's work on moduli spaces of elliptic curves, Chow and Igusa proved the upper semicontinuity of the cohomology over a broad class of Noetherian local domains [30]. Semicontinuity was proved subsequently in the complex analytic case by Grauert and by Grothendieck in more general algebraic settings. However, Chow and Igusa's contribution did not get the credit they deserved (cf. [39], Chapter 3, §12, and the bibliographical references given there, referring to work in the sixties but not to ChowIgusa). (d) Bertini's theorem. During that same period in the late fifties Chow extended Bertini's theorem to local domains [29].
Commentson Chow's Work 247
(e) Unmixedness theorem. A homogeneous ideal defining a projective variety is said to be unmixed if it has no embedded prime divisors. Chow proved that the Segre product of two unmixed ideals is also unmixed, under fairly general conditions, in a ring setting [34]. Algebraicity of Analytic Objects Chow was concerned over many years with the algebraicity of certain complex analytic objects. We mention two important instances. (a) Meromorphic mappings and formal functions. In 1949 Chow proved the fundamental fact, very frequently used from then on, that a complex analytic subvariety of projective space is actually algebraic [8]. Twenty years later he came back to similar questions and proved in the context of homogeneous varieties that a meromorphic map is algebraic [35]. Remarkably and wonderfully, almost twenty years after that he came back once more to the subject and completed it in an important point [36]. I quote from the introduction to this paper, which shows how Chow was still lively mathematically: "Let X be a homogeneous algebraic variety on which a group G acts, and let Z be a subvariety of positive dimension. Assume that Z generates X [in a sense which Chow makes precise}.... One asks whether a formal rational function on X along Z is the restriction along Z of an algebraic function (or even a rational function) on X. In a paper [35] some years ago, the author gave an affirmative answer to this question, under the assumption that the subvariety Z is complete, but only for the complex-analytic case with the formal function replaced by the usual analytic function defined in a neighborhood of Z . The question remains whether the result holds also for the formal functions in the abstract case over any ground field. We had then some thoughts on this question, but we did not pursue them any further as we did not see a way to reach the desired conclusion at the time. In a recent paper [48], Faltings raised this same question and gave a partial answer to it in a slightly different formulation. This result of Faltings led us to reconsider this question again, and this time we are more fortunate. In fact, we have been able not only to solve the problem, but also to do it by using essentially the same method we used in our original paper." (b) Analytic surfaces. In a paper with Kodaira it was proved that a Khler surface with two algebraically independent meromorphic functions is a nonsingular algebraic surface [16].
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Other Works in Algebraic Geometry Chow's papers in algebraic geometry include a number of others, which, as I already asserted, I am less well acquainted with and won't comment upon, such as his paper on the braid group [7], on the fundamental group of a variety [17], on rational dissections [24], and on real traces of varieties [33]. PDE Chow's very early paper on systems of linear partial differential equations of first order [5] gives a generalization of a theorem of Caratheodory on the foundations of thermodynamics. This paper had effects not well known to the present generation of mathematicians, including me. It was only just now brought to my attention. An anonymous colleague wrote to the editor of the present collection of articles on Chow's work: "This paper essentially asserts the identity of the integral submanifold of a set of vector fields and the integral submanifold of the Lie algebra generated by the set of vector fields. This is widely known as 'Chow's theorem' in nonlinear control theory and is the basis for the study of the controllability problem in nonlinear systems. Controllability refers to the existence of an input signal that drives the state of a system from a given initial state to a desired terminal state. A more detailed exposition of the role of Chow's theorem, with several references, is provided in the survey paper [37]." References Works by Wei-Liang Chow are referenced [1] through [36]. [1] (with van der Waerden) Zur algebraische Geome-try IX, Math. Ann. 113 (1937), 692-704. [2] Die geometrische Theorie der algebraischen Funktionen fur beliebige volIkommene Korper, Math. Ann. 114 (1937), 655-682. [3] Einfacher topologischer Beweis des Fundamental-satzes der Algebra, Math. Ann. 116 (1939), 463. [4] Uber die Multiplizitat der Schnittpunkte von Hyperflachen, Math. Ann. 116 (1939), 598-601. [5] Uber systemen von linearen partiellen Differen-tialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98-108. [6] On electrical networks, J. Chinese Math. Soc. 2 (1940), 3-160. [7] On the algebraical braid group, Ann. Math. 49, no. 3 (1948), 654-658. [8] On compact complex analytic varieties, Amer. J. Math. 71, no. 4 (1949), 893-914.
Commentson Chow's Work 249 [9] On the geometry of algebraic homogeneous spaces, Ann. Math. 50, no. 1 (1949), 32-67. [10] Uber die Losbarkeit gewisser algebraischer Gleichungssysteme, Comment. Math. Helv. 23, no. 1 (1949), 76-79. [11] On the genus of curves of an algebraic system, Trans. Am. Math. Soc. 65 (1949), 137-140. [12] On the defining field of a divisor in an algebraic variety, Proc. Amer. Math. Soc. 1, no. 6 (1950), 797-799. [13] Algebraic systems of positive cycles in an algebraic variety, Amer. J. Math. 72, no. 2 (1950), 247-283. [14] On the quotient variety of an abelian variety, Proc. Nat. Acad. Sci. 38 (1952), 1039-1044. [15] On Picard varieties, Amer. J. Math. 74, no. 4 (1952), 895-909. [16] (with Kodaira) On analytic surfaces with two independent meromorphic functions, Proc. Nat. Acad. Sci. 38, no. 4 (1952), 319-325. [17] On the fundamental group of an algebraic variety, Amer. J. Math. 74 (1952), 726-736. [18] On the quotient variety of an abelian variety, Proc. Nat. Acad. Sci. 38 (1952), 1039-1044. [19] The Jacobian variety of an algebraic curve, Amer. J. Math. 76, no. 2 (1954), 453-476. [20] On abelian varieties over function fields, Proc. Nat. Acad. Sci. 4 1 (1955), 582-586. [21] Abelian varieties over function fields, Trans. Amer. Math. Soc. 78 (1955), 253-275. 22] Abstract theory of the Picard and Albanese varieties, unpublished manuscript. 23] On equivalence classes of cycles in an algebraic variety, Ann. Math. 64, no. 3 (1956), 450-479. [24] Algebraic varieties with rational dissections, Proc. Nat. Acad. Sci. 42 (1956), 116-119. [25] On the projective embedding of homogeneous spaces, Lefschetz conference volume, Algebraic Geometry and Topology, Princeton University Press, 1957. 26] On the principle of degeneration in algebraic geometry, Ann. Math. 66 (1957), 70-79. 27] (with S. Lang) On the birational equivalence of curves under specialization, Amer. J. Math. 79 (1952), 649-652. 28] The criterion for unit multiplicity and a generalization of Hensel's lemma, Amer. J. Math. 80, no. 2 (1958), 539-552. [29] On the theorem of Bertini for local domains, Proc. Nat. Acad. Sci. 44, no. 6 (1958), 580-584. [30] (with Igusa) Cohomology theory of varieties over rings, Proc. Nat. Acad. Sci. 44, no. 12 (1958), 1244-1248.
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Remarks on my paper "The Jacobian variety of an algebraic curve", Amer. J. Math. 80 (1958), 238-240. On the connectedness theorem in algebraic geometry, Amer. J. Math. 8 1 , no. 4 (1959), 1033-1074. On the real traces of analytic varieties, Amer. J. Math. 85, no. 4 (1963), 723-733. On the unmixedness theorem, Amer. J. Math. 86 (1964), 799-822. On meromorphic maps of algebraic varieties, Ann. Math. 89, no. 2 (1969), 391-403. Formal functions on homogeneous spaces, Invent. Math. 86 (1986), 115-130. R. W. Brockett, Nonlinear systems and differential geometry, Proc. IEEE 64 (1976), 61-71. S. S. Chern, Web geometry, AMS Proc. Sympos. Pure Math., vol. 39 (1983), 3-10. R. Hartshorne, Algebraic geometry, Springer-Verlag, 1977. S. Lang, Abelian varieties over finite fields, Proc. Nat. Acad. Sci. USA 4 1 , no. 3 (1955), 174-176. S. Lang and A. Neron, Rational points of abelian varieties over function fields, Amer. J. Math. 8 1 , no. 1 (1959), 95-118. P. Philippon, Sur des hauteurs alternatives I, Math. Ann. 289 (1991), 255-283. , Sur des hauteurs alternatives II, Ann. Institut Fourier 44, no. 4 (1994), 1043-1065. , Sur des hauteurs alternatives III, J. Math. Pures Appl. 74 (1995), 345-365. B. Want, A note on Archimedean height pairing and Chow forms, preprint, Brown University, 1996. A. Weil, The field of definition of a variety, Amer. J. Math. 78, no. 3 (1956), 509-524. [47] W. Wirtinger, Lies Translationsmannigfaltigkeiten une Abelsche Integrate, Monat. Math. u. Physik 46 (1938), 384-443. [48] G. Faltings, Formale Geometrie und homogene Raiime, Invent. Math. 64 (1981), 123-165.
Reprinted from Illinois Journal of Mathematics 34(2) (1990) 175-190 © University of Illinois
T H E LIFE AND WORK OF KUO-TSAI CHEN
Kuo-Tsai Chen was born on July 15,1923 in Chekiang, China. He earned a Bachelor of Science degree in mathematics from Southwest Associated University in Kungming in 1946. He then moved to Shanghai, where he became an Assistant at the Mathematics Institute of the Academia Sinica in 1946-47. On the recommendation of its director, Shing-Shen Chern, he went to work with Samuel Eilenberg at Indiana University. After one year there, he followed Eilenberg to Columbia University in New York, where he received his doctorate in 1950. During his graduate studies, he was a mathematics instructor at the National Bible Institute in New York from 1948 to 1950, and an Assistant at Columbia University in 1949-50. After being awarded his Ph.D. degree, he went first to Princeton University as an instructor in 1950-51, and then to the University of Illinois as a Research Associate in 1951-52. His next position was that of a Lecturer at the University of Hong Kong where he stayed from 1952 to 1958. His parents were then living in Taipeh, Taiwan. In the first course he gave in Hong Kong, Chester Chen, as he had become known, met a charming sophomore, Julia Tse-Yee Fong, who became his bride in 1953. His very strict sense of duty did not allow him to give his preferred student special help, which occasionally made her very mad at him. Their happy marriage brought forth three children: Matthew in 1955, who earned a Doctorate in mathematics at the University of California at Berkeley and who is currently an electrical engineer with AT & T; Lydia in 1956, who graduated from Sarah Lawrence College, and is now a painter and editor; Lucia in 1960, who graduated from MIT and is now a graduate student in material science at the University of Illinois. Chen's next position was at the Instituto Tecnologico de Aeronautica in Sao Jose dos Campos, Brazil, first as an Associate Professor in 1958-59, and then as a Professor in 1959-60. He became a member of the Institute for Advanced Study in Princeton in the winter of 1960-61, and again later in the spring of
'it is in fact the inverse limit of the nilpotent Lie group %{A/I)/<&(A/I) n (1 + J"). These are simply connected as they are diffeomorphic to their Lie algebras Q(A/1)/Q{A/1) rrJ" via the exponential map. O 1990 by the Board of Trustees of the University of Illinois Manufactured in the United States of America
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1962, the fall of 1971, and finally in the spring of 1979. He was appointed Associate Professor at Rutgers University in 1962-63, and became a Full Professor there from 1963 to 1965. From there he went to the State University of New York in Buffalo where he stayed from 1965 to 1967. In 1967 he once again joined the faculty of the University of Illinois at Urbana, this time as a Professor. He remained there until his death in August, 1987 which followed a long illness. He was a devout Christian throughout his life. Chen was an outstanding and original mathematician. His work falls naturally into three periods: his early work on group theory and links in the three sphere; his subsequent work on formal differential equations, which gradually developed into his most powerful and important work; his work on iterated integrals and homotopy theory, which occupied him for the last twenty years of his life. The goal of Chen's iterated integrals program, which is a de Rham theory for path spaces, was to study the interaction of topology and analysis through path integration. Chen's early work contains significant contributions to the theory of links in the three sphere. Two smooth links are said to be isotopic if the associated imbeddings of S1 into S 3 can be deformed, one into the other, through smooth imbeddings. In [3] Chen showed that the quotients IT
/It
of the lower central series ir^S3 - L) =vl-=> IT2I
v3 D
of the link group depend only on the isotopy class of the link. Since all knots are isotopic to the trivial knot, one obtains as a special case the well known result that the lower central series of a knot group stabilizes at TT2: m° =
IT2
for all .$ > 2.
This led to Milnor's work on isotopy invariants of links, notably his definition of jS-invariants, which are numerical invariants of the lower central series of a link group and are generalized linking numbers [M]. Even though Chen, in collaboration with Fox and Lyndon, did give an algorithm for computing the quotients of the lower central series of a finitely presented group [10], the problem of computing its quotients remains extremely difficult. In his thesis [1], Chen showed that for any finitely presented group TT, the quotients of the lower central series of ir/tr" are computable, where TT" denotes the second derived subgroup [IT2, IT2] of IT. When IT is a link group, these groups are now known as the Chen groups of the link and are isotopy invariants. Chen's subsequent work is clearly united by the dual threads of formal Lie theory and the theory of connections on bundles whose structure group is a
The Life and Work of Kuo-Tsai Chen 253
"Lie group" of formal power series. The germ of these ideas first appears in his very first paper on group theory [1], although it becomes clearer through the papers [5], [7]—[9], [11]—[21], while from [22] on, iterated integrals become the main theme of his work. To illustrate Chen's original ideas, we present a somewhat revisionist view of his approach to de Rham homotopy theory through formal Lie theory which he developed at about the same time as Dennis Sullivan developed his theory of minimal models [SI], [S2]. Formal Lie theory takes place in a formal power series ring: Denote the ring of formal power series in the non-commuting indeterminates Xv..., Xn over Rby ^ = R«X1,...,X„» This is a complete topological ring. The neighborhoods of 0 are the powers of the maximal ideal J = (power series with trivial constant term}. It can be viewed as a Lie algebra with bracket [U, V] = UV- VU. Let L( Xv..., Xn) denote the Lie sub-algebra of A generated by A^,..., Xn and its closure in A by Q(A)=L{Xl,...,X„y.
The exponential and logarithm maps exp: J -* 1 4- J,
log: 1 + J -> J,
defined using the usual power series, are continuous and mutually inverse. The prototypical example of a "Lie group" of formal power series, called a Malcev group in the literature (cf. [Q]), is <&(A) = {X<=A: logA-e
a(A)}.
This is an infinite dimensional Lie group with Lie algebra Q(A). All other Malcev groups are obtained by replacing A by A /I, where / is a closed ideal of A generated by elements of 8(^4). One then defines Q(A/I) to be the image of Q(A) in A/I and &(A/I) to be the image of ©(^4). Again, the exponential map Q(A/I) -* ®(A/I) is an isomorphism, so that ®(A/I) behaves like a simply connected nilpotent Lie group. 1
'it is in fact the inverse limit of the nilpotent Lie group ®(A/T)/@(A/I) n (1 + / " ) . These are simply connected as they are diffeomorphic to their Lie algebras Q(A/1)/Q(A/I) n / " via the exponential map.
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Chen's basic tool is the transport map of a trivialized bundle. Let M be a manifold, assumed to be smooth as throughout this article. A trivialized principal G-bundle over M is a trivial bundle M X G -» M together with a distinguished trivialization, unique up to the right action of G. Here G may be a subgroup of GL(n) or a Malcev group. A connection on a trivialized bundle corresponds to a g-valued 1-form u on M via the rule Vs = ds — su,
(l)
where s: M -* G is a section. Denote the space of piecewise smooth paths [0,1] -» M by PM. Given a connection w on a tivialized bundle, we obtain a transport map Tu:
PM-+G
which takes the path y to the result of parallel transporting the identity along y. Equivalently, T(y) = A^l), where X(t) is the solution of the initial value problem X'(t)=X{t)A(t),
X(0)=ld.
(2)
Here A: [0,1] -» Q is the function defined by Y*u = A(t)
dt.
From elementary differential geometry, we know that whenever a and /? are composable paths, T(a)T(p) = T(afi). When G is a Malcev group, equation (2) is what Chen called a formal differential equation; cf. [12]-[20]. These were a key ingredient in his earlier work on normal forms of germs of diffeomorphisms (R", 0) -* (R", 0) and their infinitesimal analogue, germs of vector fields at 0 in R". In [15], [17], [18], [19], [21], [24] Chen studied the behaviour of the integral curves of a vector field near a singular point, and the behaviour of a local diffeomorphism near a fixed point building on a line of thought that had originated with Sternberg. One of the main results is a nonlinear decomposition theorem for germs of diffeomorphisms analogous to the semi-simple times unipotent decomposition of matrices [17]. For vector fields he established the infinitesimal analogue in [18], where he showed that the germ of a vector field with an elementary critical point at 0 has a Jordan canonical form; that is, it can be written as the sum of commuting semi-simple and nilpotent vector fields. He used this to show that two vector fields, each with an elementary critical point at the origin, are equivalent if and only if they are formally equivalent. The corresponding result for diffeomorphisms is proved in [19]. The cornerstone of Chen's work in homotopy theory is a "well known" but elegant formula for T . To express it, we need to introduce Chen's iterated
The Life and Work of Kuo-Tsai
Chen
255
integrals which he introduced in [5] in a special case, in [7] for 1-forms, and in [32] for higher dimensional forms. Suppose that wv..., wr are smooth forms on M taking values in an associative algebra A (e.g., gl(w)> UQ or a power series ring), each of degree at least 1, then fwlw2
•••wr
is an A -valued differential form on PM of degree r
E(degw. - i). 1
Representing the standard /--simplex as A'= {(tlt...,*,)
erOsi!<
•••
£fr£l},
define a smooth function 0: Ar X PM -> Mr by
*((r l ,...,0,y) = ( Y ( 0 . - , Y ( 0 ) The iterated integral is then defined by fw1w2 • • • wr = IT^*{W1 X vv2 X • • • X wr), where m denotes the projection of Ar X PM onto PM and w* denotes integration over the fiber of it with respect to the volume form dtx A • • • A dtr. When each Wj is a 1-form, jw^... wr is a function / WJWJ . . . w,: PM
—»^4
which is a very natural generalization of the usual line integral. It takes the path y to /•••/
fl(t1)...fr{tr)dh...dtr,
where g*Wj = fj(t) dt. The transport formula is obtained by solving (2) by Picard iteration. It is analogous to the Dyson exponential of physics. LEMMA (Chen [41], [12], [7]). Suppose that G is a Lie or Malceu group with Lie algebra g. / / the Q-valued 1-form w defines a connection on the bundle
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M X G -» M via(l), then the transport is given by the convergent power series T
u(y)
=l+f(o+fuu>+(
www + . . . .
(3)
A connection w on a trivialized bundle is flat if and only if its curvature vanishes: du + l/2[co,w] = 0. In this case the value of the transport on a path depends only on its homotopy class. It induces a holonomy (or monodromy) homomorphism ®:ir1(M,x)-+G,
{y}
-»
T(y).
To see what one can do with this, consider a manifold M which possesses closed 1-forms wv..., wr, linearly independent in cohomology, satisfying w i A M>. = 0 on the level of forms for each i and j . Let co =
LWJXJ
e E\M)
®
R((Xlt...,Xr)).
The curvature vanishes so that we get a holonomy homomorphism 0 : vx(M,x)
-»R«Xi,...,X,».
Choosing, as we may, loops y 1 ,..., yr at x such that the matrix
h is non singular yields r elements 1 + Uv...,l + Ur of R((Xlt..., Xr)), where U1,...,Ur^ I are linearly independent modulo I2. It is now easy to see that these generate a free subgroup of R((XV..., Xr)) of rank r. (This is essentially a theorem of Magnus.) It follows that Yi, • • •, Yr generate a free subgroup of iTi(M, x) of rank r. Using this Chen gave the first example of how the Hodge numbers hp-
(Chen [31], [32]).
Suppose that M is a compact Kahler manifold.
If hl-° > h2-0 + 1, then tr^M, x) contains a free subgroup of rank > 2 (and hence one of countable rank).
The Life and Work of Kuo- Tsai Chen
257
In fact, Chen went a lot further and developed the theory of power series connections [41], [42], a procedure for computing, from the de Rham complex of M, a presentation of the /-adic completion of the real group ring of the fundamental group of a manifold as a quotient of the power series ring R((H1(M))) by an ideal generated by Lie elements. To understand what this means, recall that the augmentation ideal J of the real group ring of a group IT is the kernel of the augmentation e: Rir -* R which takes each element of n to 1. Its powers define a topology on RTT whose completion is RTT"= lim
RTT/J".
The group ®(Rw") is a Malcev group and is called the Malcev completion of ir^MyX) and its Lie algebra g(Rw"), the Malcev Lie algebra associated to «X{M, x) (cf. [Q]). To see why one should be able to compute Rir^M, * ) * using formal Lie theory, consider the flat ®(R7r1(M, x)")-bundle whose monodromy representation p : « i ( A f , x ) -» A u t ( @ ( R « i ( M , x ) " ) ) is the right regular representation g - » [X-* Xg). This bundle is trivial essentially because ®(R7r1(M, xY), being the inverse limit of simply connected nilpotent Lie groups, is contractible. Choosing a trivialization then yields a g(Rw 1 (M, x) )-valued l-form w on M whose associated transport induces the monodromy representation p and the canonical homomorphism WiiM.x)
-»@(Rv 1 (Jif,jc) A )
obtained by evaluating p at 1. In [41], Chen gives a direct algorithm for finding an R<(// 1 (Af)>)-valued l-form u and relations Rv..., Rm e g ( R « / / 1 ( M ) » ) such that a is flat modulo the closed ideal generated by the R/s and such that the homomorphism
Q:Rnl(M,Xy->R((H1(M)))/(R1,...,Rm) induced by the transport is an isomorphism. The method is a variant of standard deformation theory. One begins with the trivial connection (w = 0) on the bundle MXR«H1(M)»^M whose monodromy representation is the augmentation Po:R^(M,x)
^R.
(4)
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One proceeds to deform this connection through a sequence of connections («„)„ ., 0 and simultaneously find a sequence of Lie elements (R^n) ^i(n)) such that the connection on (4) is flat modulo the ideal (*i(»)
*i(»))
+J"+1
and such that Tn, the transport of «„, induces an isomorphism RWl(M,x)//"+1^R«i/1(M)»/(/?1(«)
i?1(«))+J"+1.
One can interpret Chen's construction [41] of the connection form limw„ as the construction of the dual of the versal deformation of the trivial connection on the bundle M x R - » M . This method is dual to, and equivalent to, Dennis Sullivan's method of computing g(R7r1(Af, x)") using the 1-minimal model of M [SI], [S2]. However, Chen's method is often considerably simpler to apply, such as in the case when M is a finite cell complex; the complement of a union of hyperplanes in C" being a good example where Chen's methods work well (cf. [K]). One of Chen's revolutionary ideas was to extend the method of power series connections to higher homotopy groups which he did in [32], [35], [38], [42]. Suppose that M is a manifold which, for simplicity, we shall assume to be simply connected. Fix a point x of M. Recall the standard isomorphism
=irk{toxM,i}x),
where QXM denotes the space of piecewise smooth loops in M based at x and i)x the constant loop at x. By a well known theorem of Borel and Serre, the Hurewicz homomorphism induces an injection ^(0,^,1,,) ®R-JTt(Q,M,R). Multiplication of loops induces an associative product on the loop space homology. Regarding H.(QXM, R) as a graded Lie algebra with the standard bracket [U,V] = UV-
{-\)6eiVi
•nk(Q,xM, i\x) ® R inherits the structure of a graded Lie algebra. From the point of view of Chen's work, the relation between m.{^lxM, t)x) ® R and H.(SlxM, R) is analogous to the relation between g(R7r1(Af, x) ) and Rw1(M, x). (The latter is naturally isomorphic to H0(QXM, R).) Chen's bold step was to apply formal Lie theory to compute the Lie algebra ir.(QxM, TJX) ® R in a way analogous to his power series method for computing the Lie algebra g(Rw1(M, *)*) in the non-simply connected case. The analogue of the
The Life and Work of Kuo- Tsai Chen 259
completed group ring being the completion 91 of H.(QXM, R) with respect to its augmentation ideal:
The analogue of © ( R ^ A f , x)") being the generalized "group" { X <= 91: log X G n.(QxM, T,X) ® R} , which has a (generally non-associative) multiplication defined using the Baker-Campbell-Hausdorff formula. Abstractly, one can consider the tensor algebra A = R(H) of a graded vector space H. (In the sequel H will be concentrated in degrees <, — 1.) This is a graded algebra EA", where A"=
^Ar'n~r,
and Ar,s consists of those elements of ®C4 of degree s. (Thus, for example, the element ar ® • • • ® as has degree s + the sum of the degrees of the ay.) The elements of finite degree A*= R((H)) in the completion of A with respect to the augmentation ideal (H) can be viewed as a Lie algebra with bracket [U,V] = UV-
{-l)detUdeiVVU.
Denote by L ( # ) the Lie sub algebra of A" generated by H and its closure in A" by Q(^4"). As in the ungraded case, we can define a complete graded Lie algebra Q(A"/I) £ A"/I whenever / is a closed ideal of A" generated by elements of g(/4 A ). We shall call such a Lie algebra a generalized Malcev Lie algebra. The analogue of a connection on a trivialized bundle M x §(A"/I) -* g ( j 4 * / / ) is a connection form, an element w of degree 1 in
E\M)
®Q{Ayi)=]imE\M)
® [a(A'/l)/(jH
n Q(A'/I))]
.
Such a connection form is defined to be integrable if it satisfies the usual integrability condition du + l / 2 [ w , u ] = 0. Associated to each connection form is its transport map
T: C.(PM)
-*A'/I,
a graded R-linear map from the smooth chains on the path space into A"/I.
It
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is denned by evaluating the A "//-valued interated integral of degree 0 l + /W + / « « + /coWW+--on each chain. When w is integrable, its transport induces a chain map T: C.(QXM)
-+A"/I,
and holonomy maps 0 : H.(QXM)
->A"/I,
L o g 0 : -n.t{QxM)
-•
g(A'/l).
The latter map, being the composition of the logarithm of © with the Hurewicz homomorphism, is the analogue of the composite ^(M,x)
->©(R571(M,x)")->0(R7r1(M>x)")
of the natural map with the logarithm. As in the case of the fundamental group, Chen [32], [34] used these ideas to detect large subalgebras of irk(QxM, ij x ): Suppose that wv..., wr are closed differential forms on a manifold M, each of degree > 2, linearly independent in cohomology, satisfying wt A wy = 0 on the level of forms for each i and j . Consider the graded algebra
A
=R(Xv...,Xn),
where Xj has degree 1 - deg Wj. (This is isomorphic to the tensor algebra on the graded vector space spanned by indeterminates X, of degree — deg Wj. As each Xj has degree <, — 2, A is complete.) The form (o = ! > , . * , e E\M)
® R<*i,...,Xr)
has degree 1 and is integrable. It thus defines a holonomy homomorphism LogQ: v.(QxM)
- g(R
Xr).
If there are elements a 1 ( . . . , ar e Trk(UxM, t]x) such that the matrix
is non-singular, then the elements log ©(ax),..., log ©(a,) generate a free Z-Lie subalgebra of L( Xl Xr) by the graded analogue of Magnus's
The Life and Work of Kuo-Tsai
Theorem. It then follows that av...,ar
Chen
261
generate a free Lie sub algebra of
Chen developed this idea into the method of formal power series connections, a method for computing the Lie algebra ir.(SlxM, -qx) ® R and the Hopf algebra H.(QXM,R) directly from the de Rham complex of M. As in the case of the fundamental group, this approach to de Rham homotopy theory is dual to and equivalent to Sullivan's method of minimal models. Apart from generalizing connections by allowing the connection form to take values in a graded Lie algebra, Chen also allowed the graded Lie algebra to have a differential. A formal power series connection on M is a 1-form w on M taking values in a generalized Malcev Lie algebra g with a continuous differential 8: g -» g of degree — 1. The connection (w, 8) is integrable if Du + l / 2 [ w , w ] = 0, where D is the differential d ® 1 + 1 ® 8 of the differential graded Lie algebra E\M) ® g. The transport 7 = 1 + i
+ •• •
of a formal power series connection is then the chain map T: C.(QXM)
-» UQ
obtained by integrating T over chains. This induces a Hopf algebra homomorphism 0 : H.(SlxM)
->
H.{UQ).
Chen's fundamental theorem is: THEOREM [41], [42]. For each connected manifold M, there exists a continuous differential 8 on Q(R((H.(M)))) of degree -1 and a g ( R « t f . ( A / ) > » valued 1-form w such that ( « , 8) is integrable. Moreover, if M is simply connected, then the holonomy map
0 : H.{SlxM)
-+H.(UQ(R((H.(M)))),8)
is a graded Hopf algebra isomorphism. If M is not simply connected, then the homomorphism R
- H0(UQ(R((H.{M)))),
8)
is the J-adic completion of the group ring of the fundamental group as a Hopf algebra.
262 R. Hain & Ph. Tondeur
Applying standard algebra and topology, one obtains the following consequence. COROLLARY. / / (u, 8) is an integrable connection as in the theorem, then, when M is simply connected, the logarithm of the transport of a induces a Lie algebra isomorphism
Log0: w.(fixM) - » / / . ( g ( R « # . ( M ) » ) , 8 ) . When M is not simply connected, the logarithm of T induces an isomorphism of Malcev Lie algebras Log©:
B(Rir(M,x))
^HQ(Q(R((H.(M)))),S).
One could not develop such ideas without pondering the relationship between iterated integrals and the cohomology of the loop space of M. This Chen did in a series of papers [35], [39], [42] interwoven with those in which he developed formal power series connections. Regarding iterated integrals as a subspace ^(PM) of the de Rham complex E\PM) of the path space of M, Chen showed that
-> E\Q,XM X QXM)
induced by multiplication of paths induces a coproduct A: MQXM) fWlw2...
-/-(OJIf) wr -> £ fwr...
®f{SlxM) wt <8>
fwi+l...wr.
With this coproduct, f(toxM) becomes a d.g. Hopf algebra. Integration induces a graded Hopf algebra homomorphism
l:H\/-(QxM))^H\QxM,R). Naively one may expect that I is always an isomorphism. However, if we filter iterated integrals by length: Rs/!S/
2
£
•••
£/-(a,AO,
The Life and Work of Kuo-Tsai Chen 263
then standard properties of iterated integrals imply that, as a tr-^M, x) module, H\/'(QXM)) is a union of the unipotent submodules im{ff*U)->/f(/-(0,Jf))}. So I cannot be an isomorphism in general. However, in the absence of the fundamental group, Chen proved the following result which is a de Rham analogue of Adams fundamental theorem [A]. THEOREM [35], [38], [42]. / / M is simply connected, then the integration map I is a graded Hopf algebra isomorphism.
In the non-simply connected case, he proved the following result which is related to work of Stallings [S]. An elementary proof of it may be found in [H]. THEOREM [39].
For all s ^ 0, the homomorphism
H\fs)
-» H o m ( Z 1 7 1 ( M , x ) / ^ + 1 , R )
induced by integration, is an isomorphism. Consequently, if generated, then the adjoint of the integration map
IT-^M,
x) is finitely
Rir(J#,jc)"-» H o m ( i / ° ( , / ( f l x M ) ) , R ) is an isomorphism of complete Hopf algebras. In his later papers [42], [43], [47], [54], Chen studied the de Rham theory of spaces Pf obtained by pulling back die fibration PM -» M X M along a function / : N -* M X M: P,-^
PM
i
I
N -y* MX M. Chen considered the subcomplex /(Pf) of E\Pf) generated by pullbacks of iterated integrals along F and pullbacks of forms on N along the projection. This complex can be described as the circular bar construction [45] on the de Rham complexes of N and M X M. Chen proved that the cohomology of /(Pf) is isomorphic to H\Pf) when, for example, M is simply connected. Taking N = M and / to be the diagonal, Chen obtained a complex for computing the cohomology of the free loop space of M, anticipating the recent work on cyclic homology and the cohomology of the free loop space.
264
R. Hain & Ph.
Tondeur REFERENCES
[A] J.F. ADAMS, On the cobar construction, Colloque de topologie algebrique (Louvain, 1956), George Thone, Liege, Masson, Paris, 1957, pp. 81-87. [H] R. HAIN, The geometry of the mixed Hodge structure on the fundamental group, Algebraic Geometry, 1985, Proc. Symp. Pure Math., vol. 46 (1987), pp. 247-282. [K] T. KOHNO, On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces, Nagoya J. Math., vol. 92 (1983), pp. 21-37. [M] J. MILNOR, Isotopy of links, A Symposium in Honor of S. Lefschetz 1954, Princeton University Press, 1957, pp. 280-305. [Q] D. QUILLEN, Rational homotopy theory, Ann. of Math., vol. 90 (1969), pp. 205-295. [S] J. STALIJNGS, Quotients of the powers of the augmentation ideal in a group ring. In: L. Neuwirth (ed.), Knots, Groups and 3-manifolds, Papers dedicated to the memory of R.H. Fox, Annals of Math. Studies 84, Princeton, 1975. [SI] D. SULLIVAN, Topology of manifolds and differential forms, Proceedings of a conference on manifolds, Tokyo, 1973. [S2] . Infinitesimal computations in topology, Publ. Math. IHES., vol. 47 (1977), pp. 269-331.
Richard Hain Philippe Tondeur April 1989
The Life and Work of Kuo-Tsai
Chen
265
Students of Kuo-Tsai Chen Gerald John Ciaccai, 1970 Tryggve Fossum, 1972 Salma Shukrallah Wanna, 1976 Larry James Williams, 1976 John Lawrence Cuadrado, 1977 Richard Hain, 1980
Publications of Kuo-Tsai Chen 1. 2. 3. 4.
Integration in free groups, Ann. of Math., vol. 54 (1951), pp. 147-162. Commutator calculus and link invariants, Proc. Amer. Math. Soc, vol. 3 (1952), pp. 44-55. Isotopy invariants of links, Ann. of Math., vol. 56 (1952), pp. 343-353. A group ring method for infinitely generated groups, Trans. Amer. Math. Soc, vol. 76 (1954), pp. 275-287. 5. Interated integrals and exponential homomorphisms, Proc. London Math. Soc, vol. 4 (1954), pp. 502-512. 6 On the composition functions of nilpotent Lie groups, Proc. Amer. Math. Soc, vol. 8 (1957), pp. 1158-1159. 7. Integration of paths, geometric invariants and generalized Baker-Hausdorff formula, Ann. of Math., vol. 65 (1957), pp. 163-178. 8. Integration of paths, a faithful representation of paths by noncommutative formal power series. Trans, Amer. Math. Soc, vol. 89 (1958), pp. 395-407. 9. Exponential isomorphism for vector spaces and its connection with Lie groups, J. London Math. Soc, vol. 33 (1958), pp. 170-177. 10. Free differential calculus IV (With R.H. Fox and R.C. Lyndon), Ann. of Math., vol. 68 (1958), pp. 81-97. 11. Linear independence of exponentials of Lie elements, An. Acad. Brasil. Cienc, vol. 31 (1959), pp. 507-509. 12. Formal differential equations, Ann. of Math., vol. 73 (1961), pp. 110-133. 13. Decomposition of differential equations, Math. Ann., vol. 146 (1962), pp. 263-278. 14. An expansion formula for differential equations, Bull. Amer. Math. Soc, vol. 68 (1962), pp. 341-344. 15. Decomposition and equivalence of local vector fields, Proc. Nat. Acad. Sci., vol. 49 (1963), pp. 740-741. 16. Expansion of solutions of differential systems, Arch. Rational Mech. Anal., vol. 13 (1963), pp. 348-363. 17. On local diffeomorphisms about an elementary fixed point, Bull. Amer. Math. Soc, vol. 69 (1963), pp. 838-840. 18. Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math., vol. 85 (1963), pp. 693-722. 19. Local diffeomorphisms —C 00 realization of formal properties, Amer. J. Math., vol. 87 (1965), pp. 140-157. 20. On a generalization of Picard's approximation, J. Differential Equations, vol. 2 (1966), pp. 438-448. 21. On nonelementary hyperbolic fixed points of diffeomorphisms, Proceedings of International Symposium on Differential Equations and Dynamic Systems, Academic Press, San Diego, 1967, pp. 525-530. 22. Iterated path integrals and generalized paths. Bull. Amer. Math. Soc, vol. 73 (1967), pp. 935-938.
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Tondeur
23. A Igebraization of iterated integration along paths, Bull. Amer. Math. Soc., vol. 73 (1967), pp. 975-978. 24. Normal forms of local diffeomorphism on the real line, Duke. Math. J., vol. 35 (1968), pp. 549-556. 25. Algebraic paths, J. Algebra, vol. 10 (1968), pp. 8-36. 26. Homotopy of algebras, J. Algebra, vol. 10 (1968), pp. 183-193. 27. Convering-space-like algebras, J. Algebra, vol. 13 (1969), pp. 308-326. 28. An algebraic dualization of fundamental groups. Bull. Amer. Math. Soc., vol. 75 (1969), pp. 1020-1024. 29. An exact dynamical system is tree-like and vice versa. Trans. Amer. Math. Soc., vol. 149 (1970), pp. 561-567. 30. A sufficient condition for nonabelianness of fundamental groups of differential manifolds, Proc. Amer. Math. Soc., vol. 26 (1970), pp. 196-198. 31. Algebras of iterated path integrals and fundamental groups, Trans. Amer. Math. Soc., vol. 156 (1971), pp. 359-379. 32. Differential forms and homotopy groups, J. Differential Geom., vol. 6 (1971), pp. 231-246. 33. On Whitehead products, Proc. Amer. Math. Soc., vol. 34 (1972), pp. 257-259. 34. Free subalgebras of loop space homology and Massey products, Topology, vol. 11 (1972), pp. 237-243. 35. Iterated integrals of differential forms and loop space homology, Ann. of Math., vol. 97 (1973), pp. 217-246. 36. Fundamental groups, nilmanifolds and iterated integrals. Bull. Amer. Math. Soc., vol. 79 (1973), pp. 1033-1035. 37. Solvability on manifolds by quadratures, Bull. Amer. Math. Soc., vol. 80 (1974), pp. 1210-1212. 38. Connection, holonomy and path space homology. Proceedings of Symposia in Pure Mathematics, vol. 27, pp. 39-52, American Mathematical Society, Providence, R.I., 1975. 39. Iterated integrals, fundamental groups and covering spaces. Trans. Amer. Math. Soc, vol. 206 (1975), pp. 83-98. 40. "Reduced bar construction on de Rham complex" in A Collection ofpapers in honor of Samuel Eilenberg, Academic Press, San Diego, 1976, pp. 19-32. 41. Extension of C°° function algebra by integrals and Malcev completion of vx, Adv. in Math., vol. 23 (1977), pp. 181-210. 42. Iterated path integrals. Bull. Amer. Math. Soc., vol. 83 (1977), pp. 831-879. 43. Pullback de Rham cohomology of the free path fibration, Trans. Amer. Math. Soc, vol. 242 (1978), pp. 307-318. 44. Path space differential forms and transports of connections, Bull. Inst. Math. Acad. Sinica, vol. 6 (1978), pp. 457-477. 45. Circular bar constructions, J. Algebra, vol. 57 (1979), pp. 466-483. 46. Poles of maps into P„(C) and Whitehead integrals. South Asian Bull. Math. vol. 3 (1979), pp. 116-124. 47. Pullback path fibration, homotopies and iterated integrals, Bull. Inst. Math. Acad. Sinica, vol. 8 (1980), pp. 263-275. 48. The Euler operator. Arch. Rational mech. Anal., vol. 75 (1981), pp. 175-191. 49. On the Hopf Index theorem and the Hopf invariant, Bull. Amer. Math. Soc, vol. 5 (19.81), pp. 57-69. 50. "Pairs of maps into complex projective space" in Contribution to analysis and geometry, Johns Hopkins Press, Baltimore, Maryland, 1980, pp. 51-62. 51. Degeneracy indices and Chern Classes, Adv. in Math., vol. 45 (1982), pp. 73-91. 52. On the Bezout theorem, Amer. J. Math., vol. 106 (1984), pp. 725-744. 53. "Loop spaces and differential forms" in Homotopie Algebrique et Algebre Locale, Asterisque, vols. 113-114, 1984, pp. 725-744. 54. Smooth maps, pullback path spaces, connections and torsions, Trans. Amer. Math. Soc, vol. 297 (1986), pp. 617-627.
