Iterated Integrals and Cycles on Algebraic Manifolds (Nankai Tracts in Mathematics, Vol. 7)

ITERATED INTEGRALS AND CYCLES ON ALGEBRAIC MANIFOLDS NANKAI TRACTS IN MATHEMATICS Series Editors: Shiing-shen Chern, ...

Author: Bruno Harris | K. t Chen

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ITERATED INTEGRALS AND CYCLES ON ALGEBRAIC MANIFOLDS

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 by Y. L.Yu

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

VOl. 5

Contemporary Trends in Algebraic Geometry and Algebraic Topology edited by Shiing-Shen Chern, Lei Fu & Richard Hain

VOl. 7

Iterated Integrals and Cycles on Algebraic Manifolds by Bruno Harris

VOl. 8

Minimal Submanifolds and Related Topics by Yuanlong Xin

Nankai Tracts in Mathematics - Vol. 7

ITERATED INTEGRALS AND CYCLES ON ALGEBRAIC MANIFOLDS

Bruno Harris Department of Mathematics Brown University, USA

orld Scientific NewJersey London Singapore Hong Kong

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ITERATED INTEGRALS AND CYCLES ON ALGEBRAIC MANIFOLDS Nankai Tracts in Mathematics -Vol. 7 Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in anyform or by any means, electronic or mechanical, includingphotocopying, recording or any informationstorage and retrieval system now known or to be invented, without wrinenpermissionfram the Publisher.

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ISBN 981-238-720-X

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To Professor S.S. Chern and

to the memory of Professor K.T. Chen

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P1:e face

This book begins with some work of K.T.Chen, who used iterated integrals of 1-forms on a manifold to study its fundamental group T I , more precisely the quotients of T I by the sequence of groups in its lower central series: 7r1/(7rll T I ) , ~ 1 / ( ( 7 r 1T, I ) ,T I ) , and so on, (,) denoting the commutator subgroup. The first step is well-known: if we choose a vector space (over R) consisting of closed 1-forms and isomorphic to the first deRham cohomology group of the manifold X then integration of these forms over closed paths gives a homomorphism of r l ( X ) to the R-dual of H 1 ( X ) ,which is just H1 ( X ,R) if we assume H1 finite dimensional, and with this assumption 7r1/(7rl, T I ) (mod torsion) embeds as a discrete cocompact subgroup in the Lie group H l ( X , R ) . Chen showed that if one uses iterated integrals of the closed 1-forms, e.g integrals of the form

in the case of two 1-forms, then one can similarly map 7r1/((7r1 , T I ) , .. . , " 1 ) to a nilpotent Lie group (constructed using H 1 ( X ) and H 2 ( X ) ) and obtain information on this quotient of 7r1; again the image is discrete and cocompact . We will explain Chen's results in the most concrete form for a compact Kahler manifold X , where the whole sequence of T I quotients and their homomorphism to Lie groups is obtained by means of a flat connection 8 canonically associated to X . 0 is a 1-form on X with values in an (infinitedimensional) Lie algebra (or more precisely a formal series of 1-forms on X with Lie algebra coefficients). This is done in Chapter 1, and Chapter 2 studies more closely the special case where X is a compact Riemann surface and only the part of the above construction involving iterated integrals of two 1-forms is examined. It turns out that these iterated integrals give vii

viii

Iterated Integrals and Cycles o n Algebraic Manifolds

information about how the Riemann surface X is embedded in its Jacobian J ( X ) . Following A.Weil [Weil 1962, p.3311 we consider both X and its image X - under the map of J ( X ) to itself which is the group-theoretic inverse , and form the algebraic l-cycle X - X - , which is homologous to 0 in J ( X ) . This l-cycle, following Hodge and Weil, has an image in another torus, an intermediate Jacobian of J ( X ) associated to H 3 ( J ( X ) ) .Weil puts this example in a discussion of whether this algebraic l-cycle homologous t o zero is also algebraically equivalent to 0 (roughly speaking, whether it can be deformed to 0 using an algebraic deformation). We show that just this “quadratic part” of Chen’s construction calculates (gives a formula for) the “Abel-Jacobi” image of X - X - in the intermediate Jacobian. This allows us to prove that “in general” X - X - is not algebraically equivalent to 0 and also to give specific examples, the first specific examples of cycles homologous but not algebraically equivalent t o 0 , consisting of algebraic curves over Zsuch as the Fermat curve x4 y4 = 1. We refer to our papers [Harris 1983a], [Harris 1983133. Chapter 3 (partially) generalizes Chapter 2 to higher dimensional Kahler manifolds X . In the Riemann surface case the construction in Chapter 2 associates to three elements of HI ( X ,Z) represented by mutually disjoint 1-cycles C1, C2,C3 a real number obtained by iterated integrals over one of them, say C3, of harmonic forms ~ 1 a2 , Poincare dual t o C1, C2. If we take twice this real number and reduce it mod Z, and do this for all possible triples of homology classes as above, we can regard this set of real numbers mod Zas the Abel-Jacobi image of X - X - . The generalization in Chapter 3 consists in taking k homology classes [Cl],. , . , [ C k ] (of possibly different dimensions) representable by cycles C1, . . . , ck such that any k - 1 of the Ci are disjoint (so k 2 3) and

+

and associating to this k-tuple of homology classes a real number, which depends only on the complex structure of X (if X has a Kahler metric). This real number is constructed by using the heat kernel exp(-tA) of the Laplace operator A on differential forms. We show that this same number can be expresses by iterated integrals involving the harmonic forms Poincare dual to the [Ci],where the domains of integration are now intersections of the cycles Ci. We have attempted throughout the book to give definitions and details of proofs that will make it accessible to, say, second year graduate students, or perhaps even students with less background. We do however

Preface

ix

assume some acquaintance with topology and with Lie groups in Chapter 1. In Chapter 3 we explain a connection between the heat kernel and cycle pairings on a Riemannian manifold and rely considerably on the second chapter of “Heat Kernels and Dirac Operators” by Berline, Getzler, and Vergne. For K.T.Chen’s work we can refer to his collected works and the article by Richard Hain (see Bibliography). I would like to express my gratitude to Professor Chern Shiing-Shen for inviting me to present these results at the Conference on the Work of K.T.Chen and W.L.Chow at the Nankai Mathematical Institute in October 2000, to come back in Fall 2001 to give a course on this subject, and to write the present book. Professor Chern’s warm kindness and that of other faculty and students at Nankai during this visit have made it an unforgettable experience: I would like to mention especially Professors Zhou Xing-Wei, Bai Chengming, Feng Huitao, Long Yiming, Fang Fuquan, Ge Molin, and students Miss Long Jing and Miss Zhu Tong. For a similar course a t Brown in Fall 2002, I would like to thank Amir Jafari, Wang Qingxue, Justin Corvino, and Alan Landman. For some of the basic mathematical knowledge in this book, I am grateful to my friends H.C.Wang, Gerard Washnitzer, and Ezra Getzler.


Contents

Preface

vii

1. Iterated Integrals. Chen’s Flat Connection and

~1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Differential equations . . . . . . . . . . . . . . . . . . . . . . 1.3 Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Chen’s Lie algebra and connection . . . . . . . . . . . . . . 1.6 Some work of Quillen . . . . . . . . . . . . . . . . . . . . . . 1.7 Group homology . . . . . . . . . . . . . . . . . . . . . . . . 1.8 The basic isomorphisms . . . . . . . . . . . . . . . . . . . . 1.9 Lattices in nilpotent Lie groups . . . . . . . . . . . . . . . . 1.10 Some Hodge theory . . . . . . . . . . . . . . . . . . . . . . 2 . Iterated Integrals on Compact Riemann Surfaces

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalities on Riemann surfaces and iterated integrals . . Harmonic volumes and iterated integrals . . . . . . . . . . . Use of the Jacobian . . . . . . . . . . . . . . . . . . . . . . . Variational formula for harmonic volume . . . . . . . . . . . 2.6 Algebraic equivalence and homological equivalence of algebraic cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Calculations for the degree 4 Fermat Curve . . . . . . . . . 2.8 Currents and Hodge theory . . . . . . . . . . . . . . . . . . 2.1 2.2 2.3 2.4 2.5

3. The Generalized Linking Pairing and the Heat Kernel xi

1

1 2 7

8 14 19 21 25 26 28 35 35 35 42 45 47 52 55 67 73


xii

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The main theorem . . . . . . . . . . . . . . . . . . . . . . .

Appendix: Orientations, Fiber Integration

73 84

95

Bibliography

101

List of Notations

103

Index

107

Chapter 1

Iterated Integrals, Chen’s Flat Connection and m1

1.1

Introduction

Iterated integrals in calculus have the form rx

rt

More generally

Interesting examples: take

dt fl(t)dt = 1-t’

f2dt =

...

n=l

Iterated integrals are related to differential equations, which in turn are related to 7r1, the fundamental group of a manifold: this latter relation involves Lie groups and Lie algebras. So we will begin by proving a theorem of K.T. Chen which describes nilpotent quotients of X I : TI/(TI, TI), ~ 1 / ( ( 7 r 1 ,TI), TI), . . . using certain nilpotent Lie groups. The rest of the book will examine more closely the iterated integrals which give this relationship: “Chen’s flat connection” for 7r1 of compact Kahler manifolds , or more specifically complex projective algebraic manifolds, and find 1

2


that we obtain some information on algebraic cycles and on homology of these manifolds. We will introduce a little Hodge theory and the heat operator e - t A , A = Laplacian on forms. The more experienced reader should skip much of the introductory material on Lie theory.

1.2

Differential equations

On the real line R, consider the n x n first order linear system dy' = g(t)A(t) (initial condition y'(0) = E)

dt

(1.3)

y' = ( y l l . . . , y n ) , A ( t ) = given n x n matrix of functions. If A ( t ) is a constant matrix A then the solution is y'(t)= ZeAt. If we take c'= standard basis vector Zi,and y'i(t) = F''exp(At) then we can put these n solution vectors into the n rows of an invertible matrix Y ( t )and rewrite the problem as: find Y ( t )invertible satisfying

d Y ( t ) = Y(t)A(t)dt

(1.4)

Y ( 0 )= I . The solution is given by an infinite series of iterated integrals

+

...+/.../

A(tl)dtlA(t2)dt2* . .A(tn)dtn+ O 1. If g is a general Lie algebra we cannot multiply in this way as g is only closed under the [ , ] product. However we define, for a = ai @ Ci, p = pj @ C; (Ci, E g, ai, pj 1-forms E A ' ( X ) ) , the bracket [a,p] as

c;

If g = M ( N ,R) we calculate that a A a = f [a,a ] . (So, [a,a] need not ] [ a ( w ) , P ( v ) ] .We can also be 0). We define [ a , p ] ( vw, ) = [ a ( v ) , P ( w )define da = Cidai @ Ci as a g-valued 2-forms. As a function of pairs v, w of vector fields.

w)= .(.(W))

- w(a(zI))- a ( [ vw , ]).

Here a ( v ) = C a ( v i ) c i is a g-valued function on X , and w ( a ( v ) )is the g-valued function obtained by acting with w (on the scalar function part of a(v)). The main example is the left-invariant Maurer-Cartan 1-form p on a Lie group G with Lie algebra g. Recall first that we can define g = Te(G)and also identify a tangent vector v, E Te(G)with the vector field vg = Lg,(ve) (Lg,=action on tangent vectors of left translation by g, with L g ) . vg is then a left-invariant vector field: Lgl,(v9) = and we have an isomorphism(of Lie algebras) Te(G)4 Lie algebra of all left invariant vector fields, by ve H Lg(ve) = vg. We now define the Te(G)= g-valued 1-form p on G: for g E G, vg E T,(G), @(us)= L9-i+(vg).So if vg = Lg,(v,) then p(vg) = v e . wglg,

Proposition 1.1

The Maurer-Cartan 1-form p satisfies 1 dp f - [ p , p] = 0 2

(us 2-form on G).

Proof: We first evaluate these 2-forms on pairs of left-invariant vector fields v,w. Note that p ( w ) is a constnat (in g) g-valued functions on G, and so v ( a ( w ) )= 0. Thus

4


Recall that we defined

(the definition Now

is consistent with and by the definition of u for a left-invariant v, so

and

Thus

for left-invariant vector fields, but any vector field E = C fivi, fi functions and vi left-invariant, and 2-forms are linear over (scalar-valued) functions.

0 Remark 1.1 p is frequently written as the 1-form g-ldg on G. This can be thought about as follows. g denotes the map G -+ G which takes g to g , i.e., the identity map dg is then the differential of the identity map, so for vg E T,, dg(v,) = v,. g-ldg then stands f o r the composite L,-I+ o dg : vg H L,-I*(w,). Using this notation, g-ldg A g-ldg is a matrix product (if G = G L ( N ,R)) and

d(g-ldg) = d(g-') A dg = -g-ldg A g-ldg,

+

so d(g-'dg) g-ldg A g-ldg = 0. W e see now that f o r any map Y : X + G , the pull-back Y ' p = a is a g-valued 1-form on X and satisfies da ; [ ~ , C = Y ]0.

+

Theorem 1.1 If X is a manifold, a a g-valued 1-form o n X satisfying d a $ [a,a] = 0, 20 E X a given point, and G is a Lie group with Lie algebra g, then 1. There exists an open neighborhood U of xo in X and a map Y : U 4 G such that

+

i ) Y * ( p )= a

and

ii) Y ( x o )= e.

2. If U is a connected neighborhood of xo in X and Y : U + G satisfies i ) and ii) of I., then Y is unique and is given by the following formula; if

Iterated Integrals, Chen's Flat Connection and

5

7ri

p : [0,1]4 U is a path from xo to x in U , then if G = G L ( N ,W) c M ( N ,W) or if G is a Lie group in an associative algebra A with identity element e , then

Y ( z )= e

+ l l p * a ( t i )+

11

OStl st251

p * a ( t l ) A p*cx(tz)

+ '.

'

(each term is in A, sum is in G). If p,p' are homotopic paths from xo to x in U ('xed end points during homotopy) then the iterated integral formulas o v e r p and overp' are equal. 3. If X is connected and simply connected, xo E X , then there is a unique Y : X --+ G satisfying i ) and ii) of 1. and given by 2. If X is not assumed simply-connected and 7r : X 3 X is its universal covering with ZOE X , ~ ( 5 0 = ) xo given, then we let ZU on X be n*a. Then there exists p : ( X , f o ) -+ ( G , e ) as above and p gives a homomorphism m : n l ( X , x o ) 4 G such that ? is equivariant. More generally, gives a homomorphism of the fundamental groupoid of X into G . An element of the fundamental groupoid is a path p : [0,1] -+ X with p ( 0 ) = x , p ( l ) = x', up to homotopy. The homomorphism sends p into an element g E G as follows: there is a unique map Yp : [O, I] + G such that Y,*(p) = p * ( a ) on [0,1]and Y p ( 0 ) = e; we let g = Y p ( l ) .I f p : [0,1] -+ Y and p' : [0,1]--+ Y are composable paths, i.e., p ( 1 ) = p'(0) and p i s sent to 9 , p' to g' then the composite path pp' is sent to gg'. Proof: We concentrate on proving the following statement: there is a covering space 7r : X 3 X of X and a map p : --+ G such that p * p = 7r*a and Y ( & ) = e for some point 20 6 X above xo E X . The other statements are easy consequences of covering space theory, while the iterated integral formula is just the 1-dimensional case and has already been discussed. The idea of the proof is to replace maps (defined locally) X + G by their graphs in X x G, and to replace the requirements on the maps by specifications on the tangent spaces to these graphs. Existence of such graphs will follow from the Frobenius integrability theorem. pr3 ( p ) Thus on X x G we form the g-valued 1-form A = prT(-a) (where pri, i = 1 , 2 , are the projections to X , G ) . We write for short, A = -a + p. The tangent space to X x G a t any point (x,g) is denoted T,X @I T g G ,with elements ( v x ,w g ) . Let D,,g c T,X@T,G be the subspace of all (v,, w g )satisfying a(v,) = p ( w g ) , i.e., A ( v Zw , g )= 0. Since p : T g ( G )4 g is an isomorphism, the dimension of D,,g is the same as the dimension of T x X , therefore constant.

+

6


In fact Dx,g is the graph of a linear map T x X 4 TgG which pulls back pg to ax. To satisfy the hypotheses of the F’robenius integrability theorem, we have to show that if VI, VZare vector fields on X x G which lie in D x , ga t every (x,g),then [V1,V2]also lies in Dx,g at every ( x , g ) . We can either do this by a direct calculation, or else prove an equivalent condition: the “matrix coefficients” ai of the g-valued 1-form A = -a p , A = C ai @ Ci where the ci are any basis of g, satisfy: each dai E ideal (in the algebra of forms on X x G ) generated by a1 , a2, . . . ; in other words dai = C a j A bji for some 1-forms b j i . To see this we write

+

d A = -da

+ d p = -21( [ a ,a]- [ p ,p ] ) =

1 -2[ ‘ Y - p , a + p ] ,

+

(since [a,p ] = [ p ,a] for any 1-forms a , p ) so d A = -+[A,B ] ,B = a p which implies the condition on the d&i. The Frobenius theorem now says that every (x,g) E X x G has a neighborhood U and a closed submanifold 2 of U , containing (x,g),whose tangent space a t every (x’,g’) E 2 is Now re-topologize X x G so that it becomes a manifold of dimension of dimX with these 2 as open sets. Noting that the projection p r l of Dx,gto T x X is a n isomorphism we see that prl makes the retopologized X x G a covering space of X . (It helps to use the left action of G on X x G ) . Let X be the connected component of ( 5 0 , e) = ZO E X x G. Then X 4X is a covering space and ~ 7 - 2 1 2: X -+ G is a map Y : X 4 G satisfying the required conditions. 0 Exercise 1 Show that in general there is no global map Y : X -+ G with Y * p = a by finding a counterexample - a (very small) manifold X , a 1-form a on it, and a (very small) G , g, such that there is no Y : X -+ G as above.

Question The above proof seems not to have used fully the assumption that G is a Lie group. Can we then state a more general theorem? Prove directly that if Vl,Vz are vector fields on X x G with & ( x , g ) E Dx,gfor every ( x , g ) then [Vl,I41 satisfies the same condition.

Exercise 2

7

Iterated Integrals, Chen's Flat Connection and XI

1.3

Program

We will (following K.T.Chen) construct a special Lie algebra g and a gvalued 1-form 0 ("Chen's connection") for a manifold X - to simplify and make the construction canonical we assume X is compact Kahler. As motivation, recall that for any simplicia1 complex X , the fundamental group can be defined as having generators corresponding to closed edge paths and relations corresponding to 2-dimensional simplices: thus only dimensions 1 and 2 are involved. Here, the Lie algebras 0 will be defined as the free Lie algebra generated by the vector space H1(X; IR) = H I , modulo relations c [HI,H I ]obtained from the reduced diagonal map

A* : H z ( X ; R ) - + H z ( X x X ; I R ) = ( H z @ I R ) @ ( ~ @ . z ) ~ ( H 1 @ -+ Hi @ H I . The image A * ( H z ) c H1@H1 is in fact contained in the skew-symmetric elements C a @ b - b @ a which we identify with [HI,H I ] in the free Lie algebra. There will be a number of technical points involved; g will in general be an infinite dimensional Lie algebra and so 0 will be some kind of infinite series. Similarly its "Lie group" G will be an inverse limit of finite dimensional LIe group. The result will involve also some other associative algebras and another Lie algebra associated to 7r1 = 7rl(X,zO). We refer for the following to [Quillen] and [Lazard]. First, we form the associative algebra R7r1 = group algebra of TI with coefficients IR, then a descendng sequence of 2-sided ideals in R7r1:

I = augmentation ideal > I' > I 3

3

and consider the associated graded algebra Gr(Rr1) = (R7r1/1) @ (1/1') @ . . . @ (In/1"+l)

@.

..

which is furthermore a Hopf algebra. Next we consider the group 7r1 itself and a sequence of normal subgroups (the lower central series ); to indicate groups generated by commutators use round brackets; ( T I , T I ) , ( ( T I , T I ) , 7rl),etc. 7r1 > ( 7 r l , 7 r l ) > ( ( 7 r l , 7 r 1 ) , . i r l ) > . . . > 7 r r n ) > 7 r i n + ' )

>...

where T!') = 7r1 and 7rY") = (7rin),7r1). The quotients 7rin)/7r!"+') are abelian groups and the commutator operation in TI induces a map (a Lie

Iterated Integrals and Cycles on Algebraic Manifolds

8

bracket now)

which makes the associated graded abelian group grr1 = @,"=, 7ry)/7rin+ into a Lie algebra(we will tensor with R). Associated with the Lie algebras g and ( g r r l )@ R are their enveloping associative algebras U ( g ) and U(grr1@R) (also Hopf algebras). The group G associated to g will be contained in a completion, U(g)" of U ( g ) . After all these algebras have been defined we will have: 1) The Chen connection 8, which gives a homomorphism r1 -+ G of groups, will also induce a homomorphism

11 : Gr(Rn1)+ U ( g ) of Hopf algebras. 2) (As shown by Quillen ) for any group (of Hopf algebras)

Q : U(Grn1 @EX)

-+

T I , there

is a homomorphism

Gr(Rr1)

3) Both Q and 11 are isomorphisms

U(Gr(r1)@ R) 3 Gr(Rr1)3 U ( g ) (Quillen showed Q is an isomorphism in general; the proof here will consider only compact Kahler X ) . 1.4

Lie algebras

We begin this program by reviewing the universal enveloping algebra U ( g ) of a Lie algebra g. Analytic definition of U ( g ) where G is a (finite dimensional) Lie group: we recall that the elements of g are first order differential operators on G

in local coordinates which are invariant under left translation. g is closed under [,] but not under usual (associative) product Jv (= second order differential operator). So we consider U ( g ) = the associative algebra of differential operators generated by g= all left invariant differential operators of all orders. However we will give a more algebraic construction of U ( g )


~1

9

(valid over any field); U ( g ) will then be an associative algebra (with unit element 1) containing g (as a sub-Lie algebra) and characterized by the following universal property: given any associative algebra A and any Lie algebra homomorphism h : g -+ A , there is a unique extension of h to an associative algebra homomorphism of U ( g ) to A , agreeing with h on g and taking 1 to 1. To construct U ( g ) we first construct the tensor algebra

on the vector space g, then factor out the 2-sided ideal R c T ( g )generated by all elements of the following form: for z, y E g, r = z 8 y - y 8 z - [z, y] ( E g 8 g @ g) is required to be in R. Thus U ( g ) = T ( g ) / Rand it is easy to see that g 4 T ( g ) U ( g ) is 1-1. The universal property of U ( g ) follows from a universal property of T(g). The Poincark-Birkhoff-Witt theorem describes U (g) as follows: if wl,w2,.. . is any vector space basis of g, then the monomials: 1,zli, wivj(i 5 j ) , . . ' , zli, zliz . . . wik (il 5 . . 5 ik) are a vector space basis for U ( g ) . In this construction, g need not be finite dimensional. In particular we may take g to be the free Lie algebra generated by a vector space V : -+

g=

v @ [V,V ]@ [ [V,V ], V ]@ .

* *

and find that U ( g ) = T ( V )= free associative algebra on V . An important property of U ( g ) is that there is an associative algebra homomorphism

OIn/In+l =H/I

@ 1/12 @

...

is also a Hopf algebra if I is a Hopf ideal.

Definition 1.1

1. An element z E H is called primitive if

A(z) = ~ @ l + l @ z . The primitive elements are a Lie algebra under [,I, denoted P ( H ) . 2. An element g E H will be called "group-like'' if it is invertible and satisfies A(g) = g @ g. The group-like elements are a group under multiplication. Note that g = 1

+ i j , ij E H , if g is group-like.

Example 1.1 a ) The 2-sided ideal I in H generated by a collection of primitive elements z is a Hopf ideal. b) The primitive elements in U ( g ) are exactly g (use the PoincarkBirkhoff-Witt theorem to prove this!) c) The group-like elements in RIT are just IT (for T a group). (Prove this!)

12


We look now at the process of completing certain Lie algebras and their enveloping algebras, to obtain something resembling formal power series, in which we can construct exponential and log functions.(See [Lazard]). First, let L be the free Lie algebra on a vector space V (in our applications V will be finite dimensional). Then U ( L )is the tensor algebra (or free associative algebra) T ( V ) and L is the Lie subalgebra of T ( V )generated by v;

where the subspaces L1 = V , Li+l = [Li,V ] satisfy [Li,L j ] C Li+j. L is called a graded Lie algebra. For each n, $i>nLi denoted L>, - is a Lie ideal. Thus we have a sequence of onto homomorphism of Lie algebras ---t

L / L l n 4 L/LLn-l

--+

.

"

--f

L/L>2 = v

+0

and the inverse limit is a Lie algebra L A whose elements may be written as infinite series

Next consider a graded ideal I L in L ;

I L = ( I L n L1) EB ( I Ln L 2 )EB . . . . Then L / I L is again a graded Lie algebra g and we can form the completion GIA;

g = g 1 $ g 2 CB ... , gi = L i / I L n Li and gA = l i m ( g / g > n )

+ +

with elements Z" = z1 z2 . . . (again formal infinite series). Since g = L / I L , U ( g ) = U ( L ) / R ,R= %sided ideal in U ( L ) generated by I L . So R is again a graded, or homogeneous ideal and U ( g ) = U ( L ) / Ris a graded associative algebra generated as associative algebra by its degree 1 elements g 1 = V / ( I L n V ) . For simplicity we will assume I L n V = (0) so g1 = V = L1 (as will be the case in our examples). Now consider in U ( g ) the ideal of all elements of degree 2 n, which is just (Vl)" = (g)" = U ( g ) l n (note this is larger than the ideal generated by


x1

13

g2n). Then the inverse limit of the U(g)/U(g)>,,will be denoted U(5)": it is an associative algebra with diagonal map homomorphism L

Since (Vl)" n g = g>,,, - g" embeds in U(g)" as Lie subalgebra: both of these have elements which are all the infinite series, with terms in gi or v,Z respectively. 1. The primitive elements of U(g)" with respect to A" are just the elements of g". 2. Let x E U(g)" have constant term 0, and define exp x = 1+x+x2/2!+ . . . , an element o f U ( g ) " . Similarly define log(l+x) = x - x 2 / 2 + x 3 / 3 - . . . . Then: a ) if p E U(g)" is primitive, then exp(p) = g E U(g)" is group-like:

Theorem 1.2

A*(exp P) = exp P c 3 exp P E ( U ( 8 )8 U ( g ) ) " (and expp has constant term 1 and is invertible). b) If g = 1 x E U(g)" is group-like then logg is primitive (and so is in 0"). 3. We have bijections which are inverse to each other

+

exp

Primitives

__f

= g"

1%

Proof: 1. If p = pl +pa

+

*.

G = group-like elements of U ( g ) " .

. , (pi E U ( g ) i then

AYP)

= PeD 1

+ 1eDp

if and only if

for each i, so pi E gi (primitives of U ( 5 )= g). 2. a ) A" (p) = p 8 1 1 @ p implies

+

+

A"(expp) = exp(A"(p)) = exp(p 1 1 8 p ) = e x p ( p 8 1)exp(1eD p) as p 8 1,18p commute = expp 8 expp,

so expp = g is group-like.


14

b) If A''(g) = g @ g (g = 1

+ x) then

so logg is primitive. 0

In the discussion above we can replace g by g / g l n which is an ( n - 1) step nilpotent and graded Lie algebra. Then in the completion U(g/g2n)" we have = g / g 2 n (each series actually has only a finite number < n of terms) and the corresponding group exp(g/gln) has multiplication defined purely by the bracket in exp 2 exp y

= exp(a:

1 + y + z[x, y ] + . . . ).

Suppose further that g is finite dimensional over R: then - is also finite dimensional and the corresponding group is diffeomorphic to g / g > n via exp, log and is a nilpotent Lie group.

1.5

Chen's Lie algebra and connection

Now let us consider a connected manifold X with finite dimensional Nl(X;IW) = H1 and define g as (free Lie algebra on H I ) / ideal (&H2). Let us consider any g'-valued 1-form 6 on X , by which we mean an infinite series

e=e1+e2+ with

ei

=1-form on X with values in g. We will assume that 1 de -t 2

-[e,el = 0,

and that 81 has the following special form: choose any basis h1,i of H1 over let at be closed 1-forms on X whose cohomology classes are dual to the h l $ i ,i.e.,

R,and

Then we assume

6'1

=

xiat

@ h1,i.

Iterated Integrals, Chen's Flat Connection and xi

15

If we fix n > 1 and consider g / g l n = g ( n ) , we get a "homomorphic image" O(n) = 6 mod g2n which again is a flat connection and gives a homomorphism which we denote

II

: 7r1(X,20) -+ G(n) = Lie

group of g / g r n

explicitly given by the iterated integral series whose degree 1 term, for an element y E nl(X,ao) is just Ci(&ai)hl,i = image of y in H,(X;R) = m / [ T , m €9R ] The group homomorphism given by O(n): h(n) : ~ 1 (2x0,)

+

G(n) c U ( B / B ~ ~ ) "

(also denoted I I ) extends to an associative algebra homomorphism, again denoted generically as I I I b l

+

U(B(n))"

which is compatible with the diagonal maps A , A' since group elements in 7r1 go into group-like elements in G(n): it is a "Hopf homomorphism" h ( n ) , which is compatible also with homomorphisms given by decreasing n. Thus, h(n)commutes with homomorphisms to R defining the augmentation ideals, and so induces also a homomorphism of the completion (R7r1)" + U(g(n))" and a homomorphism of the associated graded algebras:

Grh(n) : Gr(Rr1)

-+

Gr(U(B(n))A)= U ( g ( n ) )

(since U ( g ( n ) )is a graded algebra). This last homomorphism induces just the natural isomorphism on the degree 1 elements

(17rl)/(17rl)2 = Tl/(Tl,Tl) €9R -+ H l ( X , R ) = g 1 which are generators, and so Grh(,) is surjective. The Grh(,) for increasing n are compatible and define a Hopf homomorphism(surjective)

I I = Grh : GrR7r1 4 U ( g ) . We also have surjective homomorphisms (R7r1)' -+ U(g)" and U ( ~ ) / ( Bfor) ~all n. We will prove all of these are isomorphisms.

R 7 r 1 / ( 1 ~ 14 )~


16

Before proving this, we want to construct connections 9 satisfying the two conditions above. We will consider only compact oriented (connected) manifolds X satisfying a special condition on 1-forms and 2-forms, which will be satisfied by all compact complex Kahler manifolds (and in particular by all non-singular complex algebraic varieties in projective space). The assumption is that we are given R-linear subspaces:

XI, C’of A1(X),

X2,E20f A2(X) (Ai(X) = all real valued differentiable i-forms) satisfying

1) !Xi C kerd and Xa

--+

kerd/dAi-’ = HhR(X) is an isomorphism.

2) d : A’ 4 A2 induces an isomorphism C’ 4 E2 3a) %’ A X ’ C X2 + E 2 (3a just says that if a ,/3 E X’ so that a A /3 represents a cohomology class in H 2 , then a A p = y q, where y E X2represents this cohomology class and E E2 c dA’ so q is exact).

+

3b) [ ( X ’ + C ’ ) A C ’ ] n k e r d c E2. We remark that 1, 2, 3a hold on any Riemannian (compact) manifold if we write W= harmonic pforms

AP = XP @ dAP-’

@

d*AP+]

and take C1 = d*A2 ( = “coexact” 1-forms) ,

E 2 = dA’ = dC’ = exact 2-forms.

However 3b requires that the metric be Kahler as well (we define this later). A basic theorem of K.T.Chen is: Theorem 1.3

(K. T . Chen) Given the subspaces

XI, C1 c A1(X); X2,E2 c A2(X) satisfying 1, 2, 3 above there is a unique

B

= B1

+Bz

+ .. .

Iterated Integrals, Chen’s Flat Connection and xi

17

(formal infinite series with 8i E A1(X)@J.J~),where J.J is the free Lie algebra on HI, modulo relations A,(H2), satisfying the following 3 conditions I. 81 E 3C18 HI and represents the identity map HI +. H1 if we identify X1 with H1 = vector space dual of H I . ( W e are assuming these spaces are finite-dimensional): 81 = C Qi 8 xi.

III. de

+ +[elel = 0.

Proof: Existence is shown by constructing 81,132,.. . inductively, starting with 81; the proof will also give uniqueness. So we start with any basis x1,x2,.-. of H1 = 81 and “dual” basis a1,a2,... of X1, i.e., 23 ayi = & , j , and take 81 = ai 8 x i . Thus, dB1 = 0, while I11 requires dB2 = -4[81,191]. Here

xi

By 3a), ai A

a? = %(ai A a j ) + aij

where %(ai A a j ) E X 2 and aij E E 2 ;these last two elements are uniquely determined by ai A aj and aij = -aji since a i A aj = -aj A ai. By 2), d : C1 -+ E 2 being an isomorphism, there exists a unique aij E C’ with

so 03% . . - -a.. 23’ Now the requirement 111 says that 1 de2 = --[el,el1 2

We will show that

K(ai A a j ) 8 [xi,x j ] = 0 in A2 8 8 2 so that the above


18

equation reduces to

Since d : C1 ---f E2 is an isomorphism, this equation for solution in C1@ gz, namely

02

has a unique

Now we show that for the basis ai of X1 dual to basis xi of Hl we have

C X ( a i A aj)8 [xi,

zjl=

o E 3~’ CZJ8 2

( X ( a iA a j ) = harmonic part of cri A a j ) : the left hand side can be regarded as a linear transformation HZ --+ gz (identifying X 2 and H 2 ) which takes a homology class TZ E H2 to ( 2 2 , ai A a j )\xi, x j ]. (( z2, ai A a j ) means Jz2 ai A aj = J,, X ( a i A aj).) It is more convenient now to write

ci,j

and show

. if and we have to show that x i , j ( a i 8 a j ) 8 z i z jvanishes on all A , ( z ~ ) But we consider zizj as being the image of z i @ z j ,which is in the free associative algebra on H1 , then C(ai 8 a j ) 8 (xi 8 zj) is the linear transformations of H1 8 H I to itself taking h 8 h’ to C ai(h)zi8 aj(h’)xj = h 8 h’, ie., the identity linear transformation. Thus C(ai8 a j ) 8 (xi 8 zj) takes A,(+) to itself and C ai 8 aj 8 X i X j takes 6 , ( z 2 ) to its image in 8 2 , which was expressly designed to be 0. Now for On, n 1 3, we already have 01 E 2-C1 8 01 and we assume we have also defined Oi E C 18 gi for 1 < i < n satisfying: for 1 5 j 5 n - 1,

Iterated Integrals, Chen's Flat Connection and a1

19

Consider n-1

18i

A Bn-i

E

( X 1+ C ' ) A C1 8 gn.

i=l We want to show that d(CY1: 8i A On-i) = 0. Then assumption 3b will give us that the 2-form coefficients will be in ((XI C1) A C') n ker d = E 2 , and E2 is isomorphic to C1 via d , so there will exist a unique 8, E C' 8 gn such that den = 8i A On-i which satisfies condition I11 for On. But,

+

zyI:

= a sum of terms f8, A 8 b A gc.

Consider triples ( p , q , r ) with p , q , r 2 1 and p + q + r = n. Then 8, A Oq A 8, occurs in this sum: once with coefficient -1, in (do,+,) A 8, once with coefficient +1, in -8, A doq+, and nowhere else. So the sum is 0 concluding the proof of the theorem.0 We come back to our program of constructing homomorphisms of Lie algebras, associative algebras, and Hopf algebras which we will show are isomorphisms. First of all, from the connection 8 we have homomorphisms, denoted II(=iterated integral) of groups: 7r1 --+

G = group-like elements in U(8)''

of Hopf algebras:

of graded algebras:

1.6 Some work of Quillen Next, following Quillen and Lazard, for any group 7r we consider its descending central series and associated graded abelian group Grn, which is a Lie algebra (using group commutator in n).

20


s,

The function q : 7r 4 q(y) = y - 1 induces a homomorphism Q of a Lie algebra to an associative algebra

Gr7r t GrR7r and so a homomorphism

Q : U(Gr7r8 R) -+Gr(R7r) of Hopf algebras. We will now give the details of Q. We will use the following construction studied in [Quillen]: for any group 7 r , form the descending central series ((, ) denoting group commutator) 7.p) = 7r

= (r, ).

3

J

. . . 2 n(n+l) = (&I,

).

J -

and consider the graded abelian group Gr(7r) = algebra using group commutator in T (see [Lazard]). We want to construct a Lie algebra homomorphism

as a Lie

(Gr7r)8 R -+ Gr(&) = $ n >-l ( & ) n / ( & ) n f l , where [, 1 in Gr(%) is given by commutator in this associative algebra. We start with the function q(y)=y-1

q:7r+&,

inducing the abelian group homomorphism 7r(1)/7r(2) (Y7r(2))

4

rwn/(&)2

4

q(y)

+ (Ey.

This induces an isomorphism over R: 7r(1)/7r(2)

8R

4

&/(&)2.

These identities will be used:

+ ( 7 2 - 1)+ (71- I)(%- I),

a ) 4(YlYZ) = 7 1 7 2 - 1 = (71 - 1) b) -1

dYlY2Y1

-1 72

-

) - ((71 =

= elements =

- 1) - (72 - l)(n-

1))r;'r;l

[(n - I), ( 7 2 - 1)](1+(rl1r;l - 1) of

[(n- I), ( 7 2 - l)]

mod 13.


~1

Next, assume inductively that for a given n 2 2, q ( x p ) ) E dn-'), 7 2 E 7r then

21

and if

y1 E

Q((Y1,72)) =

[ q ( n ) , 4(^12)1 mod

( W n + l ) .

Using u ) and then b) we prove that ----71

q : TI"' --i Rx

-+I

/Rx

is a group homomorphism, induces q : 7p/x;"+l)

-+I Rx /R7r I I

--i

TI"),

also a homomorphism, and if y1 E 72 E x then q ( ( n , y 2 ) )belongs -+2 to Thus q(7ry")) c and = [yl- l , y 2 - 11 mod Rx ----n ----n+l q induces 7rin)/xin+') 4R7r /R7r and takes group commutator (,) E -n+l ----n+2 ( 7 r p ) , x 1 )into algebra commutator [,I E Rx . Finally, one can /Rn show that q induces a graded Lie algebra homomorphism Gr(7r)4 Gr(Rn) - -2 which is an isomorphism on x / d 2 ) @I R + R7r/Rx . Thus one gets a homomorphism of associative algebras and even of Hopf algebras

sf'

.

Q : U((Gr7r)@I R) + Gr(R7r). Also (see [Quillen]), Gr(Rn) is the universal enveloping algebra U(P(GrRr)), where P denotes the Lie algebra of all primitive elements (the Hopf structure in Gr(R7r) arises from that in Rx). The Hopf algebra map Q thus induces a Lie algebra map Q : (Grx)@I R 4P(GrR7r). Similarly I I induces Lie algebra map

I1 : P ( G r ( R x ) )--i g = Gr(8'')

= P(U(g)).

We aim t o show that all these Q , I1 are isomorphisms.

1.7 Group homology We will use some homology theory of discrete groups x = 7r1 (X). Let B7r be the classifying space of x; it is a K ( x ,1) space, i.e., its fundamental group is x and its higher homotopy groups xn for n 2 2 vanish. The homology

22


groups with coefficients in any Rx module M , Hi(B7r,M) are denoted Hi(n,M ) (and depend only on x). For i = 0, Ho(x,M ) = M / T G M . For the (connected) manifold X and x = 7r1(X1xo),we construct Y = Bxl as Y = XU cells of dimension 2 3. Thus the inclusion X -+ Y induces an isomorphism 7rl(X) 4 x1(Y). Up to homotopy we can replace the inclusion X c Y by a fibration p : X -+ Y , with X -+ X a homotopy equivalence, with fiber p-l(y0) = F (where yo= base point of Y ) . Notationally we will write X instead of X (we know H i ( X ) = Hi(*) and 7rz(X) = .Z(X).) For any fibration F 4 X -+ Y as above (a “Serre fibration”), we have some exact sequences of homology groups, called exact sequences of low order terms in a spectral sequence: see H. Cartan and S. Eilenberg’s book “Homological Algebra” (p. 328 Th.5.12a with n = 1). In these H I (F) = H I ( F ;R) is a 7r1 = x1 ( X ,5 0 ) module and H I (F),l means

Hl(F)/&Hl(F). The exact sequences are

Hz(X)4 H z ( Y )

+

Hi(F),,

+

Hi(X) 4 H i ( Y )

0.

(1.7)

Hz(X) H z ( Y ) 4 0

(1.8)

-+

If further H1(F) = 0 then

Hs(X)-+ H 3 ( Y )

-+

Hz(F),,

-+

-+

is exact. Further, there is the long exact sequence of homotopy groups (for all i 2 0);

. . . 4 Ti+l(Y)-+ T i p ) 4 T i ( X )

-+

xz(Y)4 . . . .

7ri

(1.9)

We have assumed xi+l(Y) = 0 for i L 1 and xl(X)4 xl(Y)is an isomorphism, so xz(F) 1x z ( X ) (- means isomorphism). and xl(F)= (0). The Hurewicz theorem now gives an isomorphism 7rz(F) H2(F) and a commutative diagram

m(F)

-=-+Hz(F)

1

-1

Thus image of H z ( F ) -+ H z ( X ) ( = image of Hz(F)/&Hz(F) = image

-+

Hz(X))

x z ( X )4 HZ(X) spherical

= (definition) : H ,

(X)= Hz”Ph(X),


23

~1

Now (1.7) and (1.8) together become H 3 ( X ) -+ H 3 ( Y ) --* H2(F),,

-+

H 2 ( X ) -+.H2(Y)

+

0

---*

HIP)

f f l ( Y )+ 0.

So we get a short exact sequence 0 -+ H2"Ph(X)+ & ( X )

due to Hopf. Now we apply A, : H2

-+

4

H 2 ( Y ) -+ 0

H I €4 H I , to the various spaces, and note that

6 * H 2 ( S 2 )= 0 as H l ( S 2 )= 0.

So, A , H l p h ( X ) = 0 , giving a diagram 0

-

Hz"Ph(X)

1 0

-

H2(X)

1

lT*

-

HdY)

I

-

0

x*

H l ( X ) 63 H l ( X ) A H l ( Y ) 63 H l ( Y ) .

So image 6, c H 1 ( X ) @ H 1 ( X can ) be identified with image A, C H l ( Y ) @ Hl(Y), y = qm(x)). Next we look at spaces Y = B(group): for any normal subgroup N of a group (discrete), we have a fibration

We write Hi(B(group);Z) = Hi(group; Z). Then the exact sequence (1.7) becomes, with coefficients Z,

Here

Iterated Integrals and Cycles on Algebraic Manafolds

24

Using this in (l.lO), we get a sequence with exact rows and diagonals 0

0

I

T N / N n (I-,r)

T

~ ~ p - H7 +

~

~

N

+

N/F, N) )

I W,r) ( r / ~ ) / (r r// ~~ )0 -+

-+

--f

I1

r/N(r,r) NOW take r = n l ( x ) = T , N = (n, n),r / N . = nab, N/(r, N ) = nf?)/ny).The exact sequence ( l . l O ) , with coefficients in R (but Q would also work) gives the exact upper row of the following diagram, whose lower row is exact by definition:

I (Hl(T1)= H l ( X ) = Hl(Y) = H1).

Lemma 1.1

”;

~ 2 ( n : b ; ~ )~ l ( n r CQR ~ ; H1(TTb;R)skew ~ )

is an isomorphism.

Proof: We may assume n1 is finitely generated, then pass to direct limit. So nyb is also finitely generated. Now nyb = ZT @ Torsion so H2(nrb;IR) 2 H2(ZT;R). Since B ( Z ) = S 1 , B ( Z T )= S1 x ... x S1(r times) =Tr: r-dimensional torus. Thus, H i ( T T ) = A i ( H 1 ( T r ) ) . Passing to the dual vector space H i ( T T )we see that &*=cup product: A 2 ( H 1 ( T r ) )-+ H2(T‘) is an isomorphism. 0

So in the diagram (1.11) preceding this lemma the vertical A, is an isomorphism, and we conclude that the last vertical arrow is also an isomorphism(induced by A, on H2(nyb)) (n1( 2 )/n1(3) ) @

+

A2(H1)/A*(H2).

Iterated Integrals, Chen’s Flat Connection and

TI

25

(Hi are Hi(X;R), x1 = x l ( X ) We now conclude: (xl(2) /xl )@R = Gr(x1)(2)has the same R-dimension

h

as A 2 ( H i ( ~ ) ) / & H z (=~8)2 . 1.8

The basic isomorphisms

As consequence of this calculation, we have: Assuming x1 ( X ) a b= H I ( X ) is finitely generated, the previously defined surjective linear map

II

0

Q :Gr(~i)(z)

-+

gz

is an isomorphism. We previously knew that

II

0

Q : Gr(ri)(i)= xi xi, xi)

+

81 = H 1

is an isomorphism (natural isomorphism), and induces I I o Q in degree 2 (as deg 1 elements generate deg 2 by commutator). Finally, we recall that g is the free Lie algebra L(H1) on H I modulo the relations &,(Hz) contained in [HI,H i ] . So the “identity map” Hi -+ (Grxl)(l) extends to a Lie algebra homomorphism cp : L(H1) -+ (Grx1)@ R = Gr .

We also have the natural quotient (onto) homomorphism $J : L(H1) -+ g, all of these fitting into a triangle of Lie algebra homomorphisms, which commutes since it commutes on generators=degree 1 elements:

L( HI) -5 Gr(x1) @ R = G r

JII~Q

$\ g

In degree 2, 11 o Q is an isomorphism and so ker$Jz = A*Hz = kervz. Since q(&H2) = 0, cp induces a Lie algebra homomorphism g 4 Gr(x1) @ R, which is inverse to I I o Q. Thus

Theorem 1.4 I I o Q , 11, and Q are all Lie algebra isomorphisms (of the associated graded algebras):

Q : (Grxl) @ R -+ P(Gr(&)) I I : P(Gr(&))

+g

= L(H1)/(A*H2)

26


and so also isomorphisms of the universal enveloping algebras

Q : U((Gr7r1)@ R) + Gr(R7rl)= U(P(GrR7r)) 11 : Gr(R7r1)

---f

U ( g ) = Gr(U(g)”).

Finally, since 11 arises from a (Hopf) algebra homomorphism R7r1 -+ U(g)”, and is an isomorphism on associated graded algebras, it is also an isomorphism on completions:

11 : (R7rl)A

---f

U(g)A

is an isomorphism. 1.9

Lattices in nilpotent Lie groups

The term “lattice” here means a subgroup D of a (connected, simplyconnected) nilpotent Lie group N such that D is discrete in N and N / D is compact.

Theorem 1.5 Assume 7 r l ( X , x o ) i s finitely generated. Let G(,) = exp(g/g2n) = exp(g(,)) as before, and 11 : 7 r l ( X ; z o ) + G G(nl the monodromy map given by iterated integration of Chen’s connection 8. Then, the image of 7rl(X,zo) in G(,) is discrete and co-compact; i.e., G(,)/(image) is a compact manifold, for each n L 2. Further, this image is just 7rl(X,20) modulo and torsion elements mod 7rr”’). ---f

(.in’

Proof: We use induction on n and the following lemma:

Let G be any finite dimensional connected Lie group and N a connected closed normal subgroup of G (which we may assume is also a Lie group), GIN = H the quotient Lie group. Suppose D C G is a subgroup such that D n N is a discrete co-compact subgroup of N , and DID n N c GIN is also discrete co-compact. Then D c G is discrete co-compact.

Proof of Lemma We consider the principal fiber bundle N f G 1:GIN. Since D l D n N is discrete in G I N , there is a neighborhood U of the identity eN in GIN such that U n ( D N I N ) = eN. If U is small enough, there


AI

27

is a section s : U --t p-'(U), s ( e N ) = e. Then p - ' ( U ) = s(CI)N is diffeomorphic to U x N (by multiplication) and D n s ( U ) N = D n N . Since D n N is discrete in N , there is a neighborhood V of e in N so that D n V = (e). Now s ( U ) V is a neighborhood of e in G and D n s ( U ) V = ( D n N ) n V = D n V = ( e ) , so D is discrete in N . To see G / D is compact, note that GIN is the union of the D I D n N translates of finitely many compact sets, and N is the union of the D n N translates of finitely many compact sets. 0 Next we prove the theorem. Recall our notation: gqn) = g / g ~ G(nl ~ , = e x p q n ) Furthermore g n = [

gin] = g 2 n , and g(") = g/g["l, G(n)= G/G(") (G(n)denoting the iterated commutator subgroup). We now have two exact sequences of groups related by the group homomorphisms I 1 : n + G(n). [ [ g l l g l ] , . . .,611 (n factors) denoted gyl. And so

(1)

(1)

-

-

G(")/G("+l) TI1

7r(4/7r(n+1)

__f

G(,+I)

T

Ir

7r/n(n+l)

- TI1 __f

G(n)

(1)

7r/n(n)

(1).

We need to prove that the leftmost vertical arrow is an isomorphism when we tensor ~ ( ~ ) / 7 r ( ~ +with ' ) R. To see this we show that the following diagram commutes (all maps are group homomorphisms as the groups and Lie algebras are abelian):

Here GrII is the isomorphism GrW7r -+ U ( g ) , Prim denotes primitive elements (which map isomorphically to 0). I I ( y ) E G(") since I I is a group homomorphism, and For y E dn), Gcn) = expg["l, so

with


28

On the other hand,

Q ( y ~ ( ~ +=l (y ) ) - 1)

mod

(G)"+l,

and

G T I I ( Q ( ~ ~= ~ I+I~( )-~)1) = I I ( ~-) 1 = x, again.

mod g["+'l

We conclude by recalling that G r I I is an isomorphism for each n. 0

1.10

Some Hodge theory

We have still to construct the differential forms 3c1, C1,3c2,E 2 used in the connection 0 and for this purpose we will give an introduction to Hodge theory on Riemannian and Kahlerian manifolds, then define these 1- and 2-forms. This will complete the proofs of the discussion of 7r1 and nilpotent Lie groups.

A. Compact Riemannian and Kahlerian Manifolds 1. Riemannian manifolds X . Start with X any C" manifold of dimension n , A P ( X ) = all C" pforms. We assume that for each x E X with T,X the tangent space, we are given a positive definite symmetric bilinear form g z = g : T,X x T z X -+ W. g gives an isomorphism G : T, -+ T: = cotangent space, by G ( v ) ( w )= g ( v ,w) and so we can define g as a bilinear form T * x T* + W, g ( G ( v ) ,G ( w ) ) = g ( v,w). We assume g is differentiable with respect to x. For each integer p , 0 5 p 5 n = dimX we can define g 1 APT* @APT* -+ R,again positive definite symmetric, by

g ( l l A . . . A ,Z 1'1 A . . . A Zb) = det[g(li,1;)]. Next we assume X is oriented, i.e., we have a (continuous) choice of one of the two connected components of (AnT:) - (0), called the "orientation class".

Definition 1.2 The volume element dvol E AnT: is the unique element in the orientation class satisfying g(dvo1,dvol) = 1. If e1,. . . en is an orthonormal basis of T* with e1 A tation class, then this n-covector is dvol.

. - .en in the orien-


29

~1

Next we can define the Hodge +-operator APT: An-PT:, a linear isomorphism. First we recall the interior product operator -+

il : APT*-+ AP-'T*, for 1 E T* given by il(l1 A

. . . A l p ) = g(1,11)12 A -9(1,

/2)11

* * '

A

1,

A 13 A

. . . A 1,

' *

.

+ ( - l ) p - l g ( l , l p ) ~ ~A . * * A l p - l . Now we define * as follows: for *(I1 A

A main property of

3

* *

11

A

. . . A 1,

A l p ) = ilpilp-l *

*

E

A" T * ,let

. ill (dvol).

* is: for a,P both E APT*,

a A *P

= g ( a , @duo1 = p A

*a.

Also, * : AP -+ An-P, * : An-p -+ AP and -M( = ( - l ) P ( " - P ) on AP. We may let the point 2 vary in the above constructions which take place for each T:, and obtain the operator * taking pforms Ap(X) to n - p forms An-P(X) (all forms are real valued). Thus, for a,P E AP(X),g ( a , P ) is a Co3function and g(a,P)dvol = CY A *P E A n ( X ) . Also, g ( a , a ) 2 0 and g(a,a ) = 0 implies a = 0. Suppose further X is compact. Then each Ap(X) is a pre-Hilbert space with inner product

*D Next we bring in the exterior differential d , and define an operator

d*=f*d*:Ap+'-+Ap so that for a E AP(X),y E Ap+l(X),

(da,Y)X = ( a ,d * y ) x .

The exact sign in d* = Ifr * d* is determined by using - -1 P ( n - P ) a P *n-p(*paP ) ( )

and Stokes's theorem. For n even d* = - *d*.

Iterated Integrals and Cycles o n Algebraic Manafolds

30

The Laplace operator A : A P ( X ) + A p ( X ) is defined as A = dd* id*d.

Definition 1.3

3CP = kerA = { h E A

p : dd*h

+ d * d h = 0).

Equivalently, 3P = { h E A P : d h = 0 and d * h = 0). It is easy to calculate that the three subspaces XP, d*Ap+l =image d*, dAp-' = image d are mutually orthogonal. We will assume known (or accepted) the following basic theorem from elliptic PDE.

Theorem 1.6 Let X be compact oriented Riemannian and let ,8 E A P be orthogonal t o W ; that is, for all h E 3P, /3 A *h = 0. T h e n there is a unique a E AP such that

sx

A a = 0.

Further, 3l

= ker A

is finite dimensional.

Assuming this theorem, we prove that AP is the orthogonal direct sum of the three subspaces 3P, dAp-', d*Ap+l:

AP = XP

@

dAp-l

@

d*Ap+l

(Hodge decomposition).

To see this, start with any y E AP and let

/3 = y - (orthogonal projection of y on 3-c) i

where hi are an orthonormal basis of 3P. By the theorem, there is a unique a so that

so

y = P(y)

+ d(d*a) + d*(da).

Next we show that d : d*Ap+l -+ d A p is an isomorphism, and d* : d A P -+ d * A p f l is an isomorphism. To show d is onto, we just note that d(3-cP) = 0, d(dAp-') = 0 so dAP = d(d*AP+'). To show d is 1-1on d*Ap+l, let y E d*Ap+' and d y = 0. Then d*y = 0 as d*d* = 0 (d*d* = ( d d ) * ) . Now dy = 0, d*y = 0 implies y E X,but 3c and d * A p f l are orthogonal, so y = 0. Similarly d* is an isomorphism.

Iterated Integrals, Chen’s Flat Connection and x i

31

We conclude that ker d = 3cP @ dAp-’ and so 3cP -.+ kerd -+ kerd/Imd = H g R ( X ) is an isomorphism. In other words, the natural onto map kerd 4 kerd/Imd = HP is “split” by the choice of a complement to Imd in ker d, namely the orthogonal complement XP. The Hodge * operator sends 3cP isomorphically t o Xn-P, dAp-’ to d*An-Pfl and d*AP+l to dAn-P+1. Note that * does not behave simply with respect t o wedge products or d , and in general the wedge product of two elements of X need not lie in X:if Q,P E X then Q A ,B = y dq, y E X,d v E dA.

+

B. Complex manifolds, Kahler manifolds First consider C” with complex valued coordinate fuctions z j = x ~+j i y j . By tangent space at a point p we mean the tangent space of the underlying real manifold with real coordinates 21, y1, . . . , x,, y,. Thus, Tp has R-basis

a

a d ... -

-

a -

ax, ’ By,

aX1’ayl’

’

We define the (R-linear) operator J : Tp -+Tp by

a i = l , . . . n. dyi axi axi dyi (so Tp is n-dimensional over the field R + RJ isomorphic to C). a

J(-)

a

= -,

a

J(-)=--,

On the dual vector space Tp*with dual basis dxl, d y l , . . . , dx,, dy,, the transpose of J, denoted J again, acts by J(dyi) = dxi,

J ( d ~ i= ) -dyi.

Let U , V be open subsets of C” and F : U -i V a CO” map. We say that F is holomorphic if the differential d F : T p ( U )4 T F ~ ~ , satisfies (V) dFoJ= JodF

(allpEU).

These are the Cauchy-Riemann equations. We can now define a complex manifold X to be a real 2n dimensional differentiable manifold with linear automorphisms J of T p ( X )for each p, with J2 = -Id, such that the coordinate charts F taking open sets U in Cn to open sets V in X satisfy d F o J = J o d F (and so the “change of coordinates” FC1 o Fz, mapping an open set in Cn to another open set in Cn, are holomorphic.)

32


Next we proceed to the definition of Kahler structure on a complex manifold X with J as above. First of all we assume given a Riemannian metric g on X such that if we restrict g to any T p thern J preserve g (or J is orthogonal with respect to g), meaning g(Jv,Jw)= g(v, w) for all v,w E T p .

The standard example is Cnwith the previous J and

&,. .

*

, ayn a being

orthonormal with respect to g. Next we want to construct a 2-form w on X using g and J : we define w(v, w)= -g(v, Jw) for v,w E Tp.

Equivalently, w(v, Jw)= g(v,w ) . Then, w(w,v) = -w(v, w) because W ( U , v) =

-g(v, Jv) = -g(Jv, J 2 v )

= g(Jv,v) = g(v,J v ) = 0.

Note that if v # 0, g(v,v) > 0 is equivalent to w(v, Jv) > 0: this says that w restricted to the 2-dimensional space with ordered basis v, J v is an orientation. In C" with Euclidean g as before, w(&, &)= 1 (and w = 0 on other pairs) says that w = d x l A d y l . . * d x , A dy,.

+ +

Definition 1.4 A complex manifold ( X , J ) with Riemannian metric g such that J preserves g, is called a Kahler manifold (or has a "Kahler structure") if the 2-form w(v,w ) = -g(v, J w ) is a closed 2-form: dw = 0.

The simplest examples are Cn or open subsets U of Cn with Euclidean g , and quotients of Cn by a group of holomorphic isometries acting discon-

tinuously with no fixed points. Every Kahler manifold is locally "like Cn", where locally is to be interpreted in a suitable infinitesimal sense. For an explanation of this one may consult the book by Griffith and Harris, and we will just say that to prove certain identities involving d, J, d * , on a Kahler manifold it is only necessary to check them in the standard case of C". The identities we will need involve the following: Define dC= (J-' o d o J)/47r : AP -+ AP+l (SO d" = i(8- 8)/47r and dd" = ia8/2~).

Iterated Integrals, Chen's Flat Connection and x i

33

Note that dC as well as d takes real valued forms to real valued forms. Define (d")* = adjoint of d"(re1ative to ( a ,p ) =~ a A *P) so

,s

(dC)*= J-'d*J/4n

(as J* = J-').

The identities referred to above are: dd" = -d"d, dd"* = -dc*d, d"d* = -d*d"

+

+-

(however dd* d*d = A # 0, d"d"* d"*d" = A/167? # 0). J is defined to act on forms so that J(a A 0) = Ja A J p . Also

As an easy consequence of these identities, i.e., of the fact that d and d* each commutes (up to a - sign) with d" and dc*, we have a further decomposition of each Ap(X) into orthogonal subspaces: Ap(X) = 3Cp @ ddcAp-' @ ddc*AP@ d*dCAP@ d*d"*Ap+' (here dc* = J-ld* J/4n). Further, ddc maps d*dc*Ap+2isomorphically onto ddCAPc AP+' and is 0 on the other subspaces, and similarly each of the operators ddc*, d " P , d*dc*is an isomorphism on one of the above subspaces and is 0 on the others. This implies the important

Lemma 1.2 (d, d" lemma) Suppose Q E AP is d exact and d" closed, or else Q is dC exact and d closed. Then Q is dd" exact. Proof: We note that

Imd

= dd"Ap-2 @

ddc*AP

ker dC= 3Cp @ ddcAp-' @ d*dCAp, so Imd n ker dC= dd"Ap-'. Similarly Imd" n ker d = ddcAp-'. Another simple but important fact is: ker d n ker dC= 3Cp @ dd"Ap-' so that (kerd n kerdC)/ImddCis naturally isomorphic both to 3 C P and to H & R ( X ) .The isomorphism of this quotient vector space with H g R involves only the comples structure J of X and not the choice of a Kahler metric g (but assumes a Kahler metric exists).

34


We can now complete the proof of Chen's theorem in the Kahler case by exhibiting the subspaces W , CP,Ep satisfying the condition l., 2 . , 3a., 3b needed. We begin by considering the subalgebra kerdC of AP for p = 1 , 2 , so kerdC = XP @ dcdAp-2 @ dCd*Ap. The subspace XP of Chen's theorem is taken to be just the harmonic pforms XP, the subspace C1 is defined to be dCd*AP,for p = 1, and the subspace E 2 is defined to be dcdAp-2, for p = 2, i.e., dcdAo. Thus d : C1 -+ E2 is an isomorphism. As noted previously, the conditions I.,2 . , 3a are valid in a Riemannian manifold, if we take X, C ,E to mean: harmonic, coexact , exact. Condition 3b was [(X1+C1)AC1]nkerdc E2.

To see this, we write

(X@ dcd*A) A dcd*A c dcA (since dc is a super-derivation and d C X= 0), and dcA = dcdA @ dcd*A, sodCAnkerd=dcdA=E.

Chapter 2

Iterated Integrals on Compact Riernann Surfaces

2.1

Introduction

Our next main purpose is to study Chen's connection 8 more closely in the lowest-dimensional case, namely when X is a compact Riemann surface with complex structure J and (Kahler) metric g (note that the Kahler condition dw = 0 is automatic here since X has real dimension 2). To do this we will make many simplifying assumptions on the terms of 8 on which we will concentrate, in particular we construct a simplified version 8 of 8; more precisely the series defining the homomorphism of 7r1 ( X ,xo) given by 8 will contain some of the terms of the series given by 8. A main reference will be [Harris, 1983~1.

2.2

Generalities on Riemann surfaces and iterated integrals

Note that the * operator on 1-forms of X is conformally invariant, so depends only on the complex structure and not on the choice of metric in its conformal equivalence class (this is always true in the middle dimension), in fact * = J-' on 1-forms here. Thus,

X1 = k e r d n k e r d * = kerdnkerd', C1= d"d*A' = d'dA' = $A2 = d"Ao, E 2 = dC' = dd"Ao, so all these subspaces are definable by d,d" alone, when X is a Riemann surface. Now let a1,a2 E X1 satisfy

35

36


equivalently,

is exact, so

a1 A a2

01

A

a2

+ dQ12 = 0

for a unique a12 E C1 = dcAo. Let now A be the associative algebra with 2 generators x,y and relations x2 = y2 = xyx = yxy = 0.

A has R-basis l , x , y,xy,yx and contains the Lie algebra L = basis x,y, [x,y] (L=free Lie algebra on generators x,y). Let

with

e be the L

valued 1-form

Then 1 - -[eye] =o 2

de+ and so

e defines a homomorphism

t : 7rl(X,xo)-+ G = exp(L) = elements of the form exp(ax

+ by + c[z, y]) in A .

t is in fact a homomorphism of the fundamental groupoid (not necessarily closed paths, modulo end-point preserving homotopy) into G. This implies (1) (2)

J7172ai = J,, ai + J

s,

-

-

(3)

J,-l

72

ai,

+ Q12

(Q19 Q Z )

J,, (a1,a2) + a 1 2 + J

72

(a17 a2)

+ Q12 + J,, a1

JT2

a21

+ a12 = - J,(a1, a?,)+ a 1 2 + J, a1 J, 0 2 .

(a1,m)

Using (2), (3) we derive the following formula for changing base point in 7rl: let 20,xi be base points, 1 a path x1 to 20 (up to homotopy) y E 7r1(X,xo), 1yZ-l E 7rl(X,X I ) , then

Our aim is to simplify the algebraic situation by making further asai € Z for all sumptions on a1 , QZ, y. First, we assume a1,a2 satisfy: y E 7r1(X,xo). We denote this by ai E 3C:. We retain the assumption

&

37

Iterated Integrals on Compact Riemann Surfaces

Jx a1 A a2 = 0 and introduce a subgroup the kernel of

Thus

(Xi8 Xi)’of Xi €3 Xi,namely

and as before,

satisfies

Such a form a12 is defined for any element of that if we write

I ( % , a2;Y) =

I

(01, a2)

+ a12,

(Xi8 Xi)’.

mod

Z,

in

We now note

R/Z,

then by (2), I defines a function of 2 variables

I : (Xi€3 Xi)’x

7r1 (

X ,zo)

+

R/Z

which is “bi-multiplicative”, and since R/Z is an abelian group, factors through a homomorphism still denoted I , or I z o ,

IZ0 : (Xi€3 Xi)’€3 7r1 ( X ,z o y b

+

R/Z.

This homomorphism depends on the base point 5 0 , as shown by formula (4). In order to remove the dependence on 20, we restrict IZo to a smaller subgroup: namely the kernel of the homomorphism

(4) then shows that the definition of I on the subgroup is unchanged if we replace each yi by 17il-l. We can then replace 7r1(X,20)by H i ( X ;Z) and use the Poincar6 duality isomorphism H1 ( X ;Z)+ Xi (which we define so that [r]H a y = a3 implies

for all a E XI). Finally we denote the domain of I (the kernel just defined) as (Xi8 Xi €3 Xi)‘.


38

This group is the kernel of

Xi €3 Xi c3 Xi -+ Xi @ Xi @ Xi

We note that if a ~az, , a3 E satisfy then a1 @ 0 2 €3 a3 E (!Xi 8 Xi 8 Xi)’and

I(a1 @ a2 €3 0 3 ) =

I

5’

( a ~a2) ,

+

a( A

a12,

aj

= 0 for i , j = 1,2,3

mod Z

(where y E 7r1 (X, ZO) corresponds to ag). Next we would like to replace the domain of I by a subgroup (A3%;)’ of the third exterior power A3(Xi),defined as the kernel of a homomorphism to Thus consider the commutative diagram

x;.

0

- - 1@3 (XZ )

I

(Xp3

X;@x;@X;

-

0

Here

and j3(a @ p @ y) = cy

+ /3 + y. j z is the usual homomorphism and induces

31.

We will show that 21 can be defined as a homomorphism (A3%;)’

+

R/Z such that for triples al,az,ayg E Xi satisfying the special condition that for i = 1,2,3, ai is Poincar6 dual to a simple closed curve Ci,and C1, Cz, C3 are mutually disjoint, then

Lemma 2.1 Let Ci, i = 1 , 2 , 3 be disjoint oriented simple closed curves o n X . Then their harmonic Poincart! dual forms ai satisfy ai €3 aj @ (Yk E (Xi@’)’ and I ( a l @a2 @ a3) is invariant under cyclic permutations and is 0 (in R/Z) i f any two of the a’s are equal. Proof: Consider the surface Xi with two boundary components Ci,C: obtained by cutting X along Ci. More precisely, the boundary d X i is the

39

Iterated Integrals on Compact Riemann Surfaces

disjoint union of C,! and -C!, and to a point p on C, correspond two points p’ on Ci, p” on Cr.

Now integration of a( on X i from a base point 50 to a variable point 5 defines a single-valued (harmonic) function hi such that u ) hi@”)- hi($) = 1, and b) dhi = ai, (the result of integration is independent of path 20 to 5 on

Xi). Let 7 be any 1-form on X (continuous). By u ) we have c)

Let 77 = - a j , k , which is coexact and so orthogonal to the harmonic form *a,; then a 0 so that for each x E X,there is a Ci,2 5 i 5 Ic, such that dist(x,Ci) 2 6. Then Kci(x,t ) tends to 0 like e--e2/4t as t -+ 0, and the other factors are a t most tm. So the integrand is uniformly bounded on X and rapidly convergent to 0. 0 We need to extend this lemma to the case where X is replaced by a cycle C1 . . . Ci-1, assuming this intersection is defined and has the correct dimension - the main case being where C1,. . . , Ci-1 are submanifolds intersecting transversely.

n n

nCi-1 satisfying the condition above and C1n.. . nCi-1 nCi+l n. . . nCk being empty (i.e. the support is empty)

Lemma 3.3

With C1n.. .

we have:

Proof: Again there exists E > 0 such that each x E (support of) CI 0. . . Ci-lhas distance 2 E from a t least one of ci+l, .. . , c k l so the previous proof applies. 0

n

We need now information on the behavior of r ( x ,y, t ) as t 0 for (x,y) in a small neighborhood of the diagonal. Following [BGV], page 82 -+

and Theorem 2.30, write

r(x1d2 = 11t1I2

(3.16)

where E E T y ( X )and x = exp,(c) is the image of [ under the exponential map from Tvto a neighborhood of y. In Rn we could just write 5 = x - y. Also write N

k y ( x , y) = (47rt)-n/2 exp(-r2/4t)

tiGi(x,y )

(3.17)

i=O

for x near y and k r = 0 for ( x , y ) outside the neighborhood of the diagonal, (we are omitting factors ld~1'/~, Idyl1/' used in [BGV]), where @i(x, y) are C" sections (everywhere) of the bundle H o r n ( A * ( T i )A , *(Ti)) with G o ( z , x )= identity: we will write, using Iw" x Iw" notation,

The Generalized Linking Pairing and the Heat Kernel

79

Q0(z,y) = C(*dzr)A dyr modulo higher order terms in ~ ( zy), (where

I

denote multi-indices i l < . . . < ip). Theorem 2.30 gives the following pointwise, in (x,y), estimate: let N = (n/2) 1, then

+

d*K(z,y, t ) - d*k,N(s,y) = o p 2 ) uniformly for

( 2 ,y) E

(3.18)

X x XI as t --+ 0. Recall that:

r(z,Y1t ) =

r

- d * q x , y17)dT =

l+r

(3.19)

The second integral defines a C" function of (x,y) on X x X ([BGV] Proposition 2.37, page 93). We thus have to study s,' d * K ( z ,y, t ) , which estimate can be written as S,'[d*kT(x,y)]d7 0(1) by the above 0(t1l2) (as t -+ 0), (where we write kt for :k with N = (n/2) 1). Thus we have to calculate:

+

+

A. The Euclidean case, which is the "initial term" of d*kt, i.e.

(3.20) where Q(z,y, t ) is the Euclidean heat kernel Qt(z,y) =

Q(z,~ , t=) ( 4 ~ t ) - " / exp(-lz ~ - yI2/4t) x d(y1 - 21) A .. . A d ( ~ n ' -5").

(3.21)

B. The influence on A , from adding to Q extra terms Q.(xi -yi).'pi(z, y) or Q.tj.cpj(z,y) where j 2 1 and cp(z,y) are differentiable functions. We start with A and note that if we define

f

:IWn

XR"

f(z,y) = Y - z = E, then Q(z,y, t ) = f * I c ( [ , t ) where k([, t ) = ( 4 ~ t ) -e~p(-1[1~/4t)d[l ~ / ~ . . . den.

f * does not commute with d*

=-

* d*

because f * does not induce an isometry of T,'R with the normal to the diagonal at (x,5):

80


f * ( d [ i )= dyi - dxi has length 2, SO (3.22) Thus we have to calculate

To calculate d * k ( & , . . . ,tn,t ) we use the following notation: (3.24) where the ( n - 1)-form de denotes the unnormalized angular form around the origin of R" (rotation invariant form on any sphere S"-' with center 0) with

(I'(n/2) here denotes the gamma function, not the kernel). We let dB = r ( n / 2 ) ( 2 ~ " / ~ ) - ' & 9 be the normalized angular form on R" L' on R".

\ (0) (Jsn-,d0

= 1). This form is

We can now calculate (helped by [Hein]) d*lc(t,t ) = - *. d * [ ( 4 ~ t ) - ~ exp(/' C.$/4t)d 0, require a calculation of

which is almost the same as what was just done with Ic = 0 but with r - n / 2 replaced by T - ~ / ~ +The ~ .same change of variable v = r 2 / 4 r now gives the integral

82

Iterated Integrals and Cycles o n Algebraic Manafolds

so we can estimate such terms as bounded by the initial Euclidean integral times r 2 k . In conclusion, the asymptotic expansion of K ( z ,y, t ) with respect to powers o f t gives for r ( x , y , t ) a "leading term" which as t -+ 0 is bounded above by the Euclidean angular form f*dO (which is locally L1 and of order 1/rn-' in the normal direction to the diagonal, r being

the distance to the diagonal) and approaches this angular form as T -+ 0, with the rest of the terms after the leading one being of orders l / r n - k - l with k > 0. In particular, r ( z , y , t ) as t -+ 0 approaches an L1 form on X x X\ (diagonal) defining a current r(z,y) that satisfies

on X x X , A = diagonal, b~ = Dirac current, W A = harmonic Poincare dual form to the diagonal, r(z,y) = "angular current" . We want to see now that if the angular current I?($, y) for A c X x X is restricted to X x C for an oriented submanifold C of X and then integrated over y E C, one obtains an angular form for C c X I denoted rc. Using the asymptotic behavior argument, we are reduced to looking at X = R",n even, C = R"2 where n = nl n 2 and Wn = Rnl @ Rn2 is written as X = CL @ C (everything being in a neighborhood of the origin in Rn). C'- has coordinates (1c1 , . . . , zn, ,0 , . . . ,0) and normal bundle orientation given by WCI = dx,,+l A . . . A d x , (see appendix on orientations). C has the usual tangential orientation TC = w C l and normal orientation wc = ( - l ) n ' d x l A . . . A dz,, = ( - 1 ) " ' ~ c ~( S O TC A CJC = d ~ Al . . . A ds,). The angular form around C, d o c has, as current, differential d(d6'c) = b c = limit of Gaussian forms (Gaussian function of r1,t) x w c , where r: = + . . . x i , = square of distance to C. Thus d o c when restricted to CL = Rnl is (-1)"1d&, d& denoting the (normalized) angular form in Wnl around the origin.

+

XI


83

Consider now the commutative diagram

cl x c - x

ixl

x

c-x

f

(3.25)

where i = inclusion of C l to X, f(z,y) = y - z, p = orthogonal projection of to C'. Note that f o (i x I ) : Cl x C -+ X is an isomorphism but does not preserve orientation; however it does preserve the orientation in the second factor C alone, i.e. in the fibers of the vertical maps. Similarly, - p o i : C' -+ C' is multiplication by (-1) and so multiplies the orientation of C'- by (-1)"l. Now we recall that the angular form on X x X was f *dB and its restriction to X x C and image under prl, is p q * f * ( d B ) . Pulling back to C' by i* , we get i*(prl,f*dB) = prl,(i x I ) * f * d B (since the maps i x I, f map fibers C homeomorphically, preserving orientation of C), which is also = i*(-p)*p,(dB) = (image) p*(dB) on C' = Rnll under the map x -+ -x of R". It remains to show that p : R" -+ Rnl (orthogonal projection) sends d6' to p*(dB) = dB1 on CL = R"1. We can then conclude that i*(-p)*(dBl) = i*prl,(f*dO) = (-1)"ldBl on Cl,which is the restriction t o CL of the angular form d o c around C.

x

This last step for dB on R" = R"' @ R"* comes from factoring dB as follows, using T' = zf . . . xi = rf -I-r;:

+ +

s,'(l-

where B(nl/2,n2/2)= X)n1/2-1Xn2/2-1dX and dBi are the normalized angular forms of R"'. Then p*(dB) = integral of dB with respect to the variables xnl+lr...,xnl+nz = d B 1 J P J d & =do].

84


This concludes the discussion of I ' c ( X ) .

3.2 The main theorem We can now state and prove our main result of this chapter, expressing a generalized linking number by iterated integrals. This can be done on a (compact, even dimensional) Riemannian manifold but if the manifold is complex and Kahler then the real number obtained is independent of the choice of Kahler metric and depends only on the complex structure. Let Y be a smooth compact oriented Riemannian manifold of even dimension n, and let CI,. . . , ck,k 2 3, be smooth submanifolds of Y ,or more generally let each Ci be a cycle which is an integer linear combination of smooth submanifolds. Let Ci have codimension pi and assume Pi

+ . . . + P k = n + 1.

n

We will denote the support of Ci by Ci again and denote by the intersection of these supports. We make two assumptions: 1. The intersection of any Ic - 1 of the Ci is empty: c1 .rick = 0 for i = I,.. . ,IC. 2. The intersection of any set of Ci is transverse (that is, if Ci = C nijCij with Cij smooth manifold and if i l , . . . i, are distinct indices then Ciljl, . . . , CiTj,.intersect transversely). For simplicity we will write as if each Ci is an oriented manifold rather than a linear combination. Clearly 1. is the main assumption and we will see that 2. is not a real restriction. Let X = Y k= Y x . . . x Y with product Riemannian metric. i.e. { (y, . . . ,y)} be denoted Y , and let C1 x Let the diagonal in Y k , . . . x ck be the Cartesian product cycle in Yk.Then the generalized linking number

n...nci-,nci+,n..

(Y,c1 x ... x

ck)

(3.26)

is defined. It will be expressed using the following iterated integrals: let

wi be the harmonic (on Y)Poincare dual to the homology class of Ci. Then w1 A . . . A wi-1 is a closed form Poincare dual to the (homology class of) the intersection cycle of C1, . . . , Ci-1 , denoted C1 . . . 0 Ci-1. By the assumption l., this last cycle has disjoint support from the intersection cycle Ci+l . . . 0 ck.To include the case i = Ic in the notation, we define

T h e Generalized Linking Pairing and the Heat Kernel

85

= Y and write the two intersection cycles (with disjoint supports) as C 1 e . . .eCi-l and Ci+l . . . e C k + l for i = 1,. , , k (both being 0 for i = 1 ) . The closed form w 1 A . . A w i - 1 when restricted to the support of

Ck+l

.

.

ci+l

...

fying d q i - 1

ck+l

= w1 A

is exact: there exist forms q i - 1 on this support satis. . . A w i - 1 there. We then write:

n..

on this set Ci+l . n C k + l . Since w1 A . . . A w k = 0 we choose q k = 0 ( q k = 0 is all we need), Write q = Cr<s p T p , ( 1 5 r < s 5 k ) . We will need q only modulo 2: if j denotes the number of odd p i , then q f j(j - 1)/2 mod 2 ( j is odd since cpi = n 1 is odd). The result is then ([Harris, 20021):

sy

+

Theorem 3.2 With the above notation and for any choice of the forms q i - 1 (with q k = 0 ) , we have equality between the generalized linking number on X = Y k o n the left and iterated integrals on the right (recall k 2 3): k

(Y,ClX . . .x

c k )

=

(-1)q+'c/ i=2

.

[(WlA.. Wi-1,wi)-(w1A..

.AWi,

Ci+l*,.,*Ck+l

(3.27) (q = c r < s p T p s and the last term has ( w 1 A . . . W k , 1) = 0. This expression (either side of the above equality) is unchanged if any , Ci is replaced by a homologous cycle Ci satisfying the same intersection conditions, i.e either side of the equation is a function of the homology classes of the Ci; af any two Ci,say C,, C, for r # s are interchanged, the expression is multiplied by ( - 1 ) P r P s .

If Y is complex and the metric is Kahler then the expression is unchanged zf another Kahler metric on Y is used. Proof: We denote points in X x X as ( x , x ' ) . X = Y k and we write x = ( y 1 , . . . , yk), 2' = ( y i , . . . , y;). Since has product metric, its Laplacian A satisfies A = A 1 + . . . + A,, Ai = Laplacian on the ith factor Y , and the Ai commute as operators on X . Thus exp(-tA) is the product of the exp(-tAi) and the kernel K x for X is also a product.

x

K x ( 2 ,2',t>= P ~ K AY . . . A p r i K y = Ky(?4i, Y ; , t ) A . . . A X y ( Y k , Y k 1t )

I)]

86


(we can also see this from the eigenform expansion). We abbreviate this as

K

= K 1 . . . Kk.

Similarly the harmonic part of K on X x X is H X = H = H1 . . . Hk. Recall that rx,t = r(z,x’,t ) satisfied two conditions: (a) d r x , t = K x - H x (b) rx,t is coexact (in image d*), If we change rx,t by adding an exact form

r>,t= rx,t + dE, then for cycles A, B on X with disjoint supports: SAxB ‘ i , t = JAxB

rx,t

-+

( A ,B , as

O’

Thus to define the linking number (A,B ) we can replace r by I” provided I” satisfies (a) and is orthogonal to harmonic forms on X x X . For X = Y k we can choose

.

.

= r1,tKz.. . Kk iHir2,tKs.. .Kk i. . iH I . . Hk-lrk,t

where ri,t= r ( y i , y:,t). The formula (a) for d r’ is easily checked since the Ki and Hi are even degree forms and d-closed. Orthogonality to harmonic forms on X x X is also obvious, since we can assume these to be products of harmonic forms on the factors Y x Y (corresponding to the eigenvalues 0 of A) and for each i the ith term of I?’ has a factor ri,t which is orthogonal to harmonic forms on the ith factor Y x Y . We now have to integrate F‘ = r ’ ( y 1 , . . . ,yn, y;, . . . ,y;, t ) over (yi, . . . , yk) E C1 x . . . Ck and ( y l , . . . ,yn) E diagonal of Y k . However the terms on the right hand side of the expression for I?’ list the y’s and y”s in the order y1, y;, .. . ,y n ry; and so we have to first move the yi to the right, leaving the y j on the left. In the j t h factor ( j = 1,.. . , k) of the i t h term we only need the differential forms of degree n - p j = dimension of Cj in the coordinates (yi, , . . . , yin) of yi , and so of degree p j in y j coordinates (for the factor Hj or K j ) or p j - 1 in y j (for the factor rj).Thus in the ith term moving all dy‘ to the right introduces the sign (-1)Q+ni

where q = C r < s p r p s and IT^ = El-’ p j (since n - p j = p j mod 2).


87

We can now integrate the ith term over CI x . . . x c k . According to the Appendix on Orientations] this means that we integrate the j t h factor over yi. E Cjand multiply the resulting forms in the variables y j . By the lemma in this appendix] this integration] denoted p r l , I gives prl,(Hj) = wc, = w j (harmonic Poincare dual form to Cj in Y) prl*Kj = KC,

where drc, = KC, - WC, on Y (rc,and KC, also depend on t > 0). Thus denoting prl : X x X + X the first projection, we have: k

c ( - l ) q + T ' u w l ( y i ) A . .. ~ W ~ - i ( ~ Z - i ) A \ r c , , t A K c , + l , , A.AKcL,t. ..

pTi*r\,t

a=1

(3.28) Next this has to be restricted to the diagonal Y : y l = . . . = Yk = y of Y k l giving the wedge product on Y, and finally it has to be integrated over y E Y, giving (Yl c 1 x . . . c k ) when the limit t + 0 is taken. This last integral is then k

sy n

where L J ~= W C , . In this sum the first term is (-1)q J7Cl,tKCz,t... KCk,t which by lemma 3.2 approaches 0 as t 4 0 (since C2 . . . c k is empty). For i 2 2 we write C1 . Ci-1 = Cl..,i-l and wl...i-l for the Poincare dual harmonic form. Thus

n.. n

wl..,i-l

= w1 A

. . . A wi-1

+ dai-1

(ai-1 =

smooth form) .

Also we have, as currents, W l ...2 - 1

= &!I ...2-1 - Dl ...2-1

n n

where I'l. ..i-1 is smooth on the complement of C1 . . . Ci-1 and in particular is smooth on Ci+l . . . c k c k + l which we define as X for i = Ic (we take c k + l = X and for i = k, r l . . . k -1 is smooth on X ) . Thus

n n n

Iterated Integrals and Cycles o n Algebraic Manajolds

88

and the ith term in the integral is

(-

l)P+Ti

s y bC1 ...i-1

(-1)QfTi

Jy

A

rci,t KC,+lJ l l

d ( r 1...i - 1 +

*

a

A

*

KCk,t

-

a i - i ) r c i , t K c i + l , t ..Kck,t. .

The first of these two terms is, by definition of the Dirac current b,

* ,-,...n ci-lrci,tKci+l,t. . .KCkrt SC,

n n

n

n n

and since C1 . . . Ci-l Ci+l . . . c k is empty this term is 0 for i = k and for 1 < i < k it is exactly the same as for i = 1 but with X replaced by Cl 0,.. nCiP1;thus as t -+ 0 this term approaches 0. Thus we are ai-1) is a current left with the second of the above terms: here d ( r i...i-1 and the meaning of the integral (including the sign in front), since degree of l?l...i-l + ai-l is 7ri - 1, is:

+

(-l)qslJ y ( r 1.,.i - 1

Write

rl,,,i-1 -tai-1

(-1)'Jy

Vi-lKci,t

A

+ ai-l)(drc,,t)

Kc,+l,t A KCk,t.

= -qi-l, then this integral is a sum of two terms:

. . . A Kck,t - (-1)'

Jy V i - l W i A

Kci+l,t A . . . A KCL,t.

n n

In the first integral, vi-l is an L1 form, the angular form for C1 . . . Ci-1, and is smooth outside a neighborhood of this set (which does not meet Ci+l . . . Ck). The integral over this neighborhood approaches 0 as t --+ 0 since Kci,tA KC,,^,^ A . . . A KCk,tapproaches 0 rapidly here. ~ 1 . . . i - 1 being smooth over the complement of this neighborhood, which includes Ci . . . C k on which Kc,,t . . . KC,,^ approaches ~c,...c,,this integral vi--1. then approaches ( - l ) q In exactly the same way, but using disjointness of C1 . . . Ci-1 and Ci+l . . . c k , the second term approaches

n n

n n

sci,,,ck

n n

n n

-(-I)'

Jci+l,,,ck Vi-lwi.

In each of these integrals, qi-1 is a smooth form on the domain of integration which is a manifold (or linear combination of manifolds) and wi, Poincare dual to in is also Poincare dual to . . . c k in ci+l . . . c k (by our transversality hypothesis). By Poincare duality in Ci+l 0 . . . 0 ck,if cp is a closed form on this manifold (or linear combination) then

ci x,

ci

sci...ck 'p - sci+l...ck cp A W i

= O.

is not a closed form: instead dqi-1 = w1 A . . . A wi-1 on Ci+l . . . c and the above argument with closed forms cp says that

qi-l

k ,


sci . . . C k% - I

89

- sCi+l.,.Ck Vi-1 A W i

depends only on d v i - 1 , i.e. only on wl,. . .LJi-1 and ci+l,. . . , c k . We can then write the above difference of two integrals as

Jc.*... C,(WI A . . .

AWi-lrl)-sci+l,,,Clr(W1

A...Awi-l,Wi).

We recall that w 1 A . . . A wk = 0 (its degree is higher than n = dimY) and we choose q l , , , k = 0. Now adding up the k. integrals, the first being 0, we get the expression in the theorem (the last term in the sum being (-1)"'

S,(Ul

A . . .A Wk-l,Wk))

.

If k = 3 we find: for C1, CZ, C3 mutually disjoint, (YlCI x c z XC3)=

(-l)"f"~c,(W1,W2)-(W1AW2,1)+sy(~1AWz,W3)l.

We note now that the iterated integral expression does not involve the cycle C 1 , but only involves its harmonic Poincare dual w 1 . The iterated integrals are over Ci+l . . . ck+l for i 1 2. We conclude that the linking number as well depends only on the homology class of C1 (and on the metric). Next we will prove a (super) symmetry of the linking number under interchanges of the Ci, which will also prove its dependence only on the homology classes of all the Ci. We will only give details for interchange of C 1 , CZ,that is,

(Y,c2 x

c1 x c3 x . . . x c k ) = ( - l ) p * p(Y, zc 1 x cz x . . . x ck)

(3.30)

(recall pi = codim(Ci)). To do this we just note that instead of carrying out the integration with the form r>.tabove, we can use

Then

and the same calculation as before gives this as

Finally we assume X is complex with Kahler metric and examine the effect of changing to another Kahler metric with the same complex structure.

90


Denote with ' quantities depending on the second metric. Then wi is harmonic in the original metric and w: is the corresponding harmonic form in the second metric. Then dwi = 0 = dCwi and dw: = 0 = d"w: (d,d" depend only on the complex structure) and wl - wi is d-exact. Since wi - wi is also d" closed, the dd" lemma says that there is a form X i on Y satisfying wi - wz = dd"Xi with i = 1,.. . , Ic (recall dd" = (i/27r)L@). Thus W;

A

. . .W :

= W1 A..

. A W i + dd" Ci=1~1 . . .wj-1Xjw;+i . . .W : .

n n

Having chosen any qi on Ci+2 . . . Ck+l satisfying q k = 0 or at least J y q k = 0) we can choose: 7; = qi

. On c k + 1

dqi

...

+ d " C j = , w l . . . ~ j _ 1 X j ~ ~ + l w:.

= X we get from q k = 0 that

s,

and so q; = 0 since Y is a complex manifold. For i 5 k, start with

and multiply by w: = wi

+

dd"Xi,

getting

Subtracting this from the equation for q:, we have

Since dcdqi-1

= d"w1

. . . wi-1

= 0, this is

and so the integral over a cycle, e.g.

Ci+l

. . . c k + 1 is 0:

= w1

. . .w, (and


91

This concludes the proof of the theorem. 0 The following question now arises: is it possible to replace the disjointness hypothesis by a hypothesis on products of cohomology classes? Even the answer to the following basic question seems unknown: if X is a compact oriented manifold, simply connected, and Kahler, and a , p are cohomology classes with real coefficients whose cup product is 0, can their Poincare dual homology classes be represented by cycles with disjoint supports? We will now state a “complex version” of this last theorem. X is again a compact complex Kahler manifold and K ( z ,y,t), H ( z ,y) are the heat kernel and the kernel for orthogonal projection to harmonic forms. y(z, y, t ) is the coexact form on X x X (for each t > 0) satisfying:

y is in image of (dd“)”, and

(thus if X has complex dimension 1 then y is of type (1 - 1 , l - 1)). For some formulas and details on y,see [Harris, 19931 and [Harris, 20021 . Let (A,B ) be complex-analytic cycles on X which are disjoint and whose complex dimensions satisfy

dimCA + dimCB = dim@X- 1 . We define the Archimedean Height Pairing ( A ,B ) as

(3.31) If we integrate y over A first we obtain a form y ~ ( yt,) and as t -+ 0 one can show that y ~ ( yt ,) approaches a current YA which is a smooth form outside of A and so can be integrated over B. Such currents are sometimes called “Green’s forms” for A . However we need not elaborate on this and just use

r(x,Y,t ) . Suppose now X = Y k = Y x . . . x Y where Y is complex Kahler of complex dimension m and c1,.. . , ck (k 2 3) are compact complex submanifolds of Y (or cycles which are linear combinations of submanifolds) and satisfy the following conditions as cycles in Y :


92

+

k

codim(Ci) = ( d i m Y ) 1 (all dimensions and codimensions are (a) complex). (b) Any k - 1 of the Ci have empty intersection. (c) All intersections are in “general position”: if each Ci as cycle is Ci = C a i j C i j (Cij being submanifolds of Y ) , then for any distinct i l , . . . , i,, ciljl, . . . ,Cirjr intersect in general position (and so the intersections are submanifolds). Let a l l . .. , a k be the harmonic Poincare dual forms to the homology classes of the Ci. Condition (b) implies that for i = 2 , . . . ,k there exist forms pi-1 on Ci+l 0.. . c k such that a1 A . . . A = ddCpi-l on this intersection. We set p k = 0. (For instance, we could use the dd“ lemma, or else take pi-1 = -rz,where 2 = C1 0... 0 Ci-1). For any choice of such pi-1, we write

n

We then have

Theorem 3.3 Let X = Y k with product metric ( X , Y are compact Kahler). Let c1,. . . ,c k be complex cycles on Y satisfying conditions a , b , c above, and let ( A , B ) denote the Archimedean pairing on X . Let C1 x . , . x c k be the product cycle on Y k and Y be the diagonal cycle. Then k

(Y,c1x . . . X C k ) =

.

(a1A . . A a i - l

i=2

c,+1 . . ..Ck+

, ai)- (a1A. . .A a i , 1)

1

(3.31) (here c k + 1 = Y ) . This real number (either side of the above equation) depends only on the homology classes of the Ci, and is unchanged under permutation of C1, . . . , c k .

Proof: The proof is essentially the same as the proof of the previous theorem involving real cycles (but with fewer or - signs as all dimensions are even). However the last statement of the previous theorem (independence of the metric) is not known to us to be true here since the proof does not carry over. We remark that in the case where all the Ci are divisors we can choose as Green’s currents yci = log (Ioill’,ui being meromorphic sections of

+


93

metrized analytic line bundles on Y (with the metric 11 11 normalized so that dd'yc, = 6ci - aci, aci = ai harmonic as before). Comparing the calculation of the height pairing above with [Deligne 1987, Sec. 81 we see that he is studying the same height pairing for divisors Ci. Deligne does not assume the intersection condition (b) above; consequently our statement on homology classes does not hold in his situation.


Appendix: Orientations, Fiber Integration

It is necessary to orient a compact n-manifold in order to integrate an n-form over it. We make explicit here how we make various choices of orientations . All manifolds will be C”. Let X be a compact oriented n-manifold and S a compact k-dimensional submanifold. An orientation of S is equivalent to an orientation of its normal bundle N = Ns by the following convention: a t x E S we choose an ordered basis of the tangent space Tx(S) in the orientation class of S and follow it by an ordered basis of N,. The resulting ordered basis of Tx(X is required to be in the orientation class of Tx(X). Our Poincare duality convention is then as follows: let ws be a closed form on X such that for every closed form LY on X we have Js~ = Jxa !A ~ s.

Such ws exist and will be called “Poincare dual forms to 5’”. For instance, w could be a Thom form, Gaussian shaped in the normal direction to

S

and defining the normal orientation to S. Further, w s could depend on a real parameter t > 0 and as t -+ 0, w s could approach the Dirac current 6s of integration over 5’: this means that for all forms (Y (not necessarily closed) on X we have:

Jx

a!

A ws

4

Js a! (written also as Jx a! A 6s)

as t 4 0. Next we consider two oriented submanifolds S1,S2 of X which intersect transversally so that their intersection T is a submanifold of dimension dimT = dim& dim& - n. We orient T so that an oriented ordered basis for the normal bundle to T in X consists of an oriented ordered basis for the normal bundle to T in Sz (which then also orients the normal bundle

+

95


96

to S1 in X ) followed by a basis orienting the normal bundle to S2 in X . Poincare dual forms w s l , ws2 t o S1,Sz in X then determine a Poincare dual form w T : WT

= US1 A WS2 .

Also, the restriction of wsl to S2 is a Poincare dual form to T in S2. We also have to orient Cartesian products of manifolds and of submanifolds. If X 1 , X2 are oriented manifolds of dimensions n 1 , n ~we write p r i : X 1 x X 2 -+ X i for the projection, i = 1 , 2 , and for a1,a2 forms on X I ,X z , we write pr;crl or a(x1) and prZa2 or C Y ~ ( X Z for ) the corresponding forms on X1 x X2. If X I ,X2 are oriented tangentially by top degree forms w1, w2, we orient X 1 x X2 by p r i w l Apr,*wz. So

sxlxx2

(Pr;771 A P a r l a ) = sx,771

sx,

772

.

Let A l , A2 be oriented submanifolds of X I ,X2 of dimensions a l , a2 and with Poincare dual closed forms W A ~W, A ~ . We would like to define the orientation and Poincare dual form W A x ~ ~ of A1 x A2 so that for any closed forms ai on X i , d e g a i = ai, SA1 x Az prrcrl A

prlaz =

sA1 sA2 01

a2

=

sxl

xx2

p r ; a A pr,*az A W

A X~A ? ,

For this we have to take WA1x A 2 = (-1)(n1-a1)a2pr;WA1 A pr,*wA2

.

In particular, for X 1 = A1 we get w X 1 x A z = pr$wA2

For X Z = A2 we get: W A l x X z = (-1)(n1-a1)n2pr;wAl

.

Multiplying, W X i X A2 WAl X X Z = WA1 x A ~ .

For n even W A x~ ~ =z ( - l ) a 1 a 2 p r ; W A 1 A praWA2. As a n example we consider in X x X the diagonal A (A should not be confused here with the Laplacian). Using local coordinates 51,. . . , z, in X , X is oriented by the n-form dxl A . . . A dxn. We take a copy Y of X with coordinates y 1 , . . . , yn (p*yi = xi if p is the identification map p : X + Y ) and write the coordinates in X x X as ( X I , . . . , x,, y1,. . . ,y,). We use the

2

Orientations, Fiber Integration

97

diagonal map i : X -+ X x X , i(z1,.. . ,z,) = ( 5 1 , . . . , x n , x l , .. . ,x,) and orient X and A so that i is orientation preserving as map from X to A. We orient X as above and orient the tangent bundle of X by the ordered basis a , the tangent bundle of A by the basis . . . , aaz, da~ , a . . .,K and the normal bundle of A in X x X by . . ,- K a Then a a a X x X is oriented by . . , =, . . . , The projections p r j , = 1 , 2 , of X x X to X then induce orientation preserving diffeomorphisms of A with X . Let now A l , A2 be oriented manifolds of dimensions kl , k2 and let f i : Ai 4 X be differentiable maps. We assume that f l x f 2 : A1 x A2 + X x X is transverse to A c X x X . For example, this transversality holds if A1 = X , f l = Identity. Then the fiber product of fl x f 2 with the inclusion of A, in other words the inverse image of A in A1 x A2, is a submanifold A1 x x A2 of A1 x A2 with normal bundle in A1 x A2 oriented by ( f l x fi)*(wa)where WA orients the normal bundle of A in X x X . Assume now A1 = X , f 1 = Identity map I , write F = I x f : X x A2 + X x X . Then F-l(A) = graph o f f = ( ( f ( a z ) , a z ): a2 E Az}, denote this by g r ( f ) . We claim: A2 + gr(f), induced by F , is an orientation preserving diffeomorphism (taking a to ( f ( a ) , a ) . To see this, we take any top degree form p on A2 that orients A2 and show first that under pr2 : gr(f) -+ A2, we have that pr;p is a top degree form on gr(f) that gives an orientation of g r ( f ) . Namely, if the tangent space to A2 at a point a2 has oriented basis , . . .&, k = dirn(A2),then a t ( f ( a z ) , a 2 ) ,g r ( f ) has tangent space basis ( f * ( € ~ @ ) & , . . . , f*(&) @ &). Assuming p(C1 A . . . A &) = 1, we find that

&& + &-, + -&+&. +&. =, x.

=,

&:.

Pr,'(P)[f*( 0 a smooth f o r m on X x X satisfying d r = kt,A - W A then rt,AZ= p r l , F * ( r ) is a C" f o r m on X satisfying n t , A Z = kt,A2 -wA,. Proof: 1.To prove the statement about W A , let a be any closed form on X , then Jx a A p r l * F * ( u A )= & x ~ z P r ; aA F * ( u A ) = . f g r ( f ) p 6 a (since F * ( ~ Ais) a Poincare dual form to gr( f)). Now using the orientation preserving diffeomorphism f x I d : A2 + gr( f) and the map pr1 o ( f x I d ) = f : A2 3 X we get: S g r ( f ) p r T a= s A , ( f Id)*pr;a = J A , f * a = f * ( d A z ) ( a ) .

2. For any form a on

X,

Jx a A P r b ( F * k t , A )= J X x A z Pr;a A (F*kt,A). Write F * ( k t , A ) = I c t , g r ( f ) : since F is transverse to A, kt,gr(fl is a Gaussian shaped form on X x A2 peaking on g r ( f ) . Thus the last integral above approaches J g r ( f ) p r ; a as t -+ 0, and as in l., this equals J A f*(a) ~ = f*(dAz)Q*

3. Starting with dI'(x, y, t ) = kt,A - W A on X x X we apply F * and then prl*, both of which commute with d, obtaining the statement for rt, kt,Az, wA2 on x . 0

Orientations, Fiber Integration

99

Integration over the fiber Denote by X I , X z compact oriented manifolds of dimensions n1,nz and by p r i : X1 x X 2 -+ X i the projection. Denote by A* the differential forms on these manifolds. We define: :A*(X1 x

X z ) -+ A * ( X 1 )

in such a way that for ai E A * ( X i ) ,

pri* (pr;ai A P r l 0 2 ) = a1

(sx,

az)

[similarlyprz+(pr;al A prlaz) = (J,, a1)az]. Then for cp E A * ( X 1 ) ,$ E A*(X1 x X Z )

Prl*[(Pr;cp)A $1 = cp A P T l * ( $ )

J,,

x x z[(Pr;cp)A $1

E

A*(Xl)

= sx,['p A P T l * ( $ ) I .

If X I ,X2 have no boundary then d o prl, = prl* o d

d oprz* = (-l)n1pr2* o d

(nl

= dimX1).

We extend prt to a map from currents Ti on X i to currents p r f ( T i ) on X1 x X2 in such a way that if Ti is given by a form ~i on X i (thus Ti(ai)= oiA T ~ then ) prf is given by the form p r t ( q ) . Thus we define, for ai E A * ( X i ) ,

I,,

Pr;(Tz)(Pr;W

APT;~Z= ) (JX,

[in particular, p r l ( 6 ~=~ S)

~I)Tz(~z) = Tz(P~z*(W;WP G W ) )

X ~ ~ and A ~ ]

pr;(Tl)[pr;al A p r , * a ~=] ( - l ) ( d e g T l ) ( d e g a 2 ) T ~ ( a l )

(assuming deg

[In particular pr;6A1 = xxz .I We thus have pr;(Ti) = Tio p r i , : A*(X1 x X z ) -+ Iw for all n1 and even nz. For a general map f : X -+ Y we define, for a current T on X , form a on

y,

f*(T)(a) = T(f*a).


Bibliography

NOTES We would like to point out in particular the following references: 1. K.T. Chen’s “Collected Papers” and the summary in it of his life and work by R. Hain and P. Tondeur. 2. The survey article by R. Hain, “Iterated Integrals and Algebraic Cyles: Examples and Prospects” in the volume ‘‘Contemporary Trends in Algebraic Geometry and Algebraic Topology” edited by S.S. Chern, L. Fu and R. Hain (Nankai Tracts in Mathematics voi. 5, World Scientific, 2002), and many other valuable papers by R. Hain. 3. The book “Heat Kernels and Dirac Operators” by N.Berline, E.Getzler, and M. Vergne. 4.The article “Groupes Fondamentaux Motiviques de Tate Mixte” by P. Deligne and A.B. Goncharov, giving some different viewpoints and directions from those in this book. 5. The survey article “The Ubiquitous Heat Kernel” by Jorgenson and Lang. 6. Many articles by R. Harvey and B. Lawson on currents, differential characters and singularities.

THE REFERENCES Berline, N., Getzler, E., Vergne, M. (1992). Heat Kernels and Dirac Operators. Springer, Berlin. Bloch, S. (1984). Algebraic Cycles and Values of L-functions. J.Reine.Angew.Math. 350, pp. 899-912. Chen, K.T. (2000). Collected Papers of K.T.Chen. P.Tondeur, Editor. Birkhauser, Boston. Deligne, P. (1987). Le Determinant de la Cohomologie. Contemporary Mathematics 67, pp. 93-177. Deligne, P. et Goncharov, A.B. (2003). Groupes Fondamentaux Motiviques de Tate Mixte. ArXiv:Math,NT/0302267. Griffiths, P. and Harris, J. (1978). Principles of Algebraic Geometry. Wiley. Hain, R. (2002). Iterated Integrals and Algebraic Cycles: Examples and Prospects. In: Contemporary Trends in Algebraic Geometry and Algebraic Topology, Editors S.S. Chern, L. Fu,R. Hain. World Scientific. 101

102


Harris, B. (1983). Harmonic Volumes. Acta Mathematica 150, pp. 91-123. Harris, B. (1983). Homological vs Algebraic Equivalence in a Jacobian. Proc.Nat.Acad. of Sciences USA 80, pp. 1157-1158. Harris, B. (1989). Differential Characters and the Abel-Jacobi map, pp. 69-86. Algebraic K-Theory: Connections with Geometry and Topology, Ed. by J.Jardine and V.Snaith. Kluwer. Harris, B. (1990). Iterated Integrals and Epstein Zeta Function with Harmonic Rational Fucntion Coefficients. Illinois J.Math 34. pp. 325-336. Harris, B. (1993). Cycle Pairings and the Heat Equation. Topology 32. pp. 225238. Harris, B. (2002). Chen’s Iterated Integrals and Algebraic Cycles. pp. 119-134. In: Contemporary Trends in Algebraic Geometry and Algebraic Topology, Ed. by S.S. Chern, L. Fu, and R.Hain. World Scientific. Hein, G. (2001). Computing Green Currents via the Heat Kernel. J.Reine und Angew.Math. 540, pp. 87-104. Jorgenson, J. and Lang, S. (2001). The Ubiquitous Heat Kernel, pp. 655-682. In: Mathematics Unlimited: 2001 and Beyond, Ed. by B. Engquist, W. Schmid. Springer. Kodaira, K. and Morrow, J. (1971). Complex Manifolds. Holt, Rinehart and Winston. Lazard, M. (1954). Sur les Groupes Nilpotents et les Anneaux de Lie. Ann.Ecole Norm. Super. 71, pp. 101-190. Quillen, D.G. (1968). On the Associated Graded Ring of a Group Ring. Journal of Algebra 10, pp. 411-418. Schiffer, M. and Spencer, D.C. (1954). Functionals of Finite Riemann Surfaces. Princeton University Press. Weil, A. (1962). Foundations of Algebraic Geometry (2nd edition), pp. 331. American Mathematical Society. Weil, A. (1979). Collected Papers. Paper 1952e and Comments. Springer.

List of Notations

I

integral, iterated integral 1 fundamental group 1 commutator subgroup terms of lower central series 1 ((n, n),. . . ), n) A Laplace operator on forms 2 ,-tA Heat operator 2 G Lie group 2 0 Lie algebra (of G) 2 TdG) tangent space at e 2 Ai(X) differential i-forms on X 3 CY 1-form, with values in g 3 1% PI bracket of Lie algebra valued 1-forms 3 U Maurer-Cartan g-valued 1-form on G 3 7 Chen's connection: a g-valued 1-froms on a manifold X e A, reduced diagonal map: H 2 ( X ) -+ H l ( X ) @ H 1 ( X ) 7 Rn1 group algebra of n1 7 7 I,E Augmentation ideal in Rn (kernel of Rn -3 R) Gr(Rni) associated graded algebra: @F'-o((IW.rr)n/(G)n+l 7 7 n1 3 (7r1,Tl) 3 . * . lower central series of a group n1 U(e) universal enveloping (associative) algebra of a Lie algebra 0 8,9 iterated integral homomorphisms given by 11 Chen's connection 8 8, 15, 19, 21 Q Quillen homomorphism U(Gr7r18 R) -+Gr(Rn) 8 , 20 function n -+ E, q ( y ) = y 1 20 4 tensor algebra T ( V ) R1@ V @( V @V )@ . . . (free associative algebra) generated by a vector space V 9 A diagonal homomorphism H -+ H @ H of a Hopf algebra H 9,lO n1

1

103

104


L free Lie algebra 10 H Hopf algebra 10, 11 P(H) primitive elements of a Hopf algebra H 12 completions of the graded Lie algebras L , g L"10" (formal infinite series) 12 all elements of degree L n, g" = l@(g/g,n) L>n10>n 12, 13 A extension of A : U ( g ) 4 U ( g ) I8 U ( g ) to a homomorphism U(dA ( U ( 8 ) U(€l))A 13 B(n) B / B > n (Lie algebra) 15 % Lie group of ( g / g > n ) 15 h(n) group homomorphism T I -+ G(n) 15 IT same as i E 16 Xi, C 1 ,E2 special subspaces of the differential forms A i ( X ) 16, 34 XP harmonic pforms 17 Q Lie algebra homomorphism induced by q 20 ,(a, ,(n) ( T , T ), (dn-l) , T ) terms of lower central series 20 B ( T ) ,K(T11) classifying space of a discrete group T 22 Ti i-th homotopy group 22 image of 7r2 ( X ) -+ Hz ( X ) ,spherical homology classes 23 HiPh( X )

"

+

Tp

Tl/(Tl,

m)

Ai

i-th exterior power 25 41 11, Lie algebra homomorphisms 25, 26 g[nl " e 1 , L l 1 I 1 . . . ,s11(nfactors) 27 ($4 exp g In] 9 Riemannian metric 29 d vol volume element 29 * Hodge star operator A P ( X ) -+ An-p (4 29 d* adjoint of d 30 A Laplace operator:dd* d*d (sometimes denotes diagonal). P(Y) orthogonal projection of y in A p ( X ) to XP 30 almost complex structure operator, T , ( X ) ---f T , ( X ) J 4% w) Kahler 2-form 32 dC operator &(J-' o d o J ) on forms, also = i(8 - a)/47r (dC)* adjoint of d", also = J-'d* J/47r 33

+

e

simplified version of 0

35, 36

Xi harmonic 1-forms with periods in Z 36 (Xic3 Xi)' kernel of intersection number pairing

37

30 31 33

List of Notations

105

, az;7)

iterated integral over y 37 homomorphism defined by I(a1, 0 2 ; y) 37 (Xi8 Xi 8 Xi)' a subgroup of (Xi)@' 37 (A3Xi)' a subgroup of A3Xi 38 Ail Bi, C i simple closed curves 38, 40 P same as (A3%;)' (later) Lefschetz-primitive cohomology classes 40, 46 v Y = 2 1 : P --f R/Z 40 A Alel- Jacobi map 42 D a 3-chain in R3/Z3 42 Jacobian of a Riemannn surface X 45 J a c ( X ) ,4x1 Ximage of A ( X ) ( i e . , X C J a c ( X ) ) under map g + g-' of J a c ( X ) 45 W Kahler 2-form on J a c ( X ) 45 S Schiffer-Spencer coordinates (s1, . , ~ 3 ~ 4 ) 48 Q map P @z JR + H o ( X ;K 2 ) 49, 50 j3 complex structure 41 T current 67 6, Dirac current 67

I

(4B )

Real valued linking number 73, 76 exp( -tA) Heat operator 73 K ( z ,Y,t ) l K kernel of the Heat operator 74 75 form satisfying dI'= K - H r(z1Y,t ) ,r eigenform of A 74 76 square of Riemann distance dist(z,Y)2, d z , Qt (5, Y) Q(z,Y,t ) Euclidean heat kernel 76 ci chains, usually cycles 77, 78, 84 K c , rc forms attached to C 77 asymptotic expansion of K ( z ,y, t ) to order N k,"(XI Y) dB (unnormalized) angular form around origin in R" dl3 normalized angular form 80 Pr* integration over the fiber of pr 87, 98 c 1u . . . u c i intersection of supports (sets) 84 c1. ... . c a intersection cycle 84 (wlA*..AWi_l,~i) vi-1 Awi where dqi-1 = w1 A ' . . A w i - l (w1 A . * .A ~ i - 1 , l ) vi-1 85 86 K11..' 1Kk K i = p r f K ( z 12/, t ) = K ( z i ,y i , t )

4

78 80

85

106

Iterated Integrals and Cycles o n Algebraic Manijololds

H i 1. . . Hk Hi = p r f H ( x ,9) = H ( x ~yi) , Y(X1Y't)' Y dd"7 = K - H 91 ?A Green's form for complex cycle A

86 91

Index

L’ current (and L1 form), 72 d,d‘ lemma, 34

harmonic forms, 30 harmonic volume, 42 heat equation, 75 heat kernel, 74 heat operator, 73, 74 Hodge *-operator, 29 Hodge theory, 28 Hopf algebra, 10, 11 hyperelliptic Riemann surfaces, 50

Abel-Jacobi map, 42 algebraic equivalence of cycles, 52 angular current, 82 angular form, 76, 80-83, 88 Archimedean height pairing, 91 asymptotic expansion, 78, 82

integration over the fiber, 99 intermediate Jacobian, 46 iterated integral, 1

Chen’s connection, 7, 16 Chen’s Lie algebra, 14 coexact forms, 16, 34 completion (inverse limit), 12, 14 current, 67 cycle, 78, 84

Jacobian manifold, 45 Jacobian variety, 45 Kahler manifold, 32 Kahler metric, 34

descending central series, 19 diagonal homomorphism, 9, 11 Dirac current, 67

Laplace operator, 30 lattice (discrete cocompact) subgroup, 26 linking number, 73, 76, 84-86, 89 lower central series, 7

Euclidean heat kernel, 76 Fermat quartic curve, 54 first Chern form, 69 free Lie algebra, 9

Maurer-Cartan I-form, 3 modular curve, 54

Green’s currents, 93 Green’s forms, 91 group homology, 22 group-like element, 11

orientations, 95 period matrix, 60 107

108

Iterated I n t e p l s and Cycles on Algebraic Manifolds

Poincare-Lelong formula, 69 primitive element (of Kahler manifold cohomology), 46 primitive elements (in a Hopf algebra), 11 quadratic differentials, 48 Quillen homomorphism Q, 8, 20, 21 Regularity theorem, 70 Schiffer variation, 47 spherical homology classes, 23 Torelli space, 47 universal enveloping algebra, 8-10

Iterated Integrals and Cycles on Algebraic Manifolds (Nankai Tracts in Mathematics, Vol. 7)

Iterated integrals and cycles on algebraic manifolds

Iterated Integrals and Cycles on Algebra

Lectures on algebraic cycles

Minimal Submanifolds and Related Topics (Nankai Tracts in Mathematics 8)

Lectures on algebraic cycles

Algebraic Graph Theory (Cambridge Tracts in Mathematics, Vol 67)

Metric Diophantine Approximation on Manifolds (Cambridge Tracts in Mathematics, 137)

Algebraic Graph Theory (Cambridge Tracts in Mathematics, Vol 67)

Algebraic cycles and motives

Polynomials and Vanishing Cycles (Cambridge Tracts in Mathematics)

Fourier Integrals in Classical Analysis (Cambridge Tracts in Mathematics)

Fourier Integrals in Classical Analysis (Cambridge Tracts in Mathematics)

Function Spaces and Wavelets on Domains (EMS tracts in mathematics, vol.7)

Lectures on Algebraic Cycles (Duke University mathematics series ; 4)

Group Cohomology and Algebraic Cycles

Algebraic cycles and Hodge theory

Synthetic Geometry of Manifolds (Cambridge Tracts in Mathematics 180)

Geometrisation of 3-Manifolds (EMS Tracts in Mathematics)

Inspired by S S Chern: A Memorial Volume in Honor of a Great Mathematician (Nankai Tracts in Mathematics (Paperback)) (Nankai Tracts in Mathematics)

Geometrisation of 3-Manifolds (EMS Tracts in Mathematics)

Geometrisation of 3-Manifolds (EMS Tracts in Mathematics)

Structures on Manifolds (Series in Pure Mathematics)

Lectures on Buildings (Perspectives in Mathematics, Vol 7)

New Research on Three-Manifolds And Mathematics

New Research on Three-Manifolds And Mathematics

Volume and Surface Integrals Used in Physics (Cambridge Tracts in Mathematics and Mathematical Physics, No. 1)

Divisors (Cambridge Tracts in Mathematics)

Convexity (Cambridge Tracts in Mathematics)

Divisors (Cambridge Tracts in Mathematics)

Scissors Congruences, Group Homology and Characteristic Classes (Nankai Tracts in Mathematics, V. 1.)

Iterated Integrals and Cycles on Algebraic Manifolds (Nankai Tracts in Mathematics, Vol. 7)