Lectures on vector bundles over Riemann surfaces

LECTURES ON VECTOR BUNDLES OVER RIEMANN SURFACES BY R. C. GUNNING PRINCETON UNIVERSITY PRESS AND THE UNIVERSITY OF T...

Author: Robert Clifford Gunning

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LECTURES ON VECTOR BUNDLES OVER RIEMANN SURFACES BY

R. C. GUNNING

PRINCETON UNIVERSITY PRESS AND THE

UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY 1967

Copyright (

1967, by Princeton University Press All Rights Reserved

Published in Japan exclusively by the University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

Preface.

These are notes based on a course of lectures given at Princeton University during the academic year 1966-67.

The topic

is the analytic theory of complex vector bundles over compact Riemann surfaces.

During the preceding academic year, I gave an

introductory course on compact Riemann surfaces.

The notes for

that course have appeared in the same Mathematical Notes series under the title "Lectures on Riemann Surfaces"; they are sufficient, but not necessary, background for reading this set of notes.

The present course is not really intended as a natural

sequel to the preceding course, though.

It is not a systematic

presentation of the theory of complex vector bundles, taking up the thread of the discussion of compact Riemann surfaces from the previous year; rather it is a set of lectures on some topics which I found interesting and suggestive of further developments.

The

aim is to introduce students to an area in which possible research topics lurk, and to provide them with some hunting gear.

In a bit more detail, the topics covered in these lectures are as follows.

Sections 1 through 1+ contain a general discussion

of complex analytic vector bundles over compact Riemann surfaces, from the point of view of sheaf theory.

In the preceding course,

only sheaves of groins were considered, since that is all that is really needed in one complex variable; but I decided to take this opportunity to introduce the students to some broader classes of sheaves, sheaves of modules over sheaves of rings, and in particular, analytic sheaves on complex manifolds.

The relevant defi-

nitions, and the connections with complex vector bundles and complex line bundles, are given in section 1; the notion of a coherent analytic sheaf is introduced. and discussed in some detail as well.

Section 2 contains a discussion of the general

structure of coherent analytic sheaves over subdomains of the

complex line C , and over the complex projective line

IP

.

In

section 3 these results are extended to coherent analytic sheaves

over arbitrary compact Riemann surfaces, by considering a Riemann

surface as a branched covering of

IP, and examining the behavior

of sheaves under such covering mappings.

The principal results

are the representations of arbitrary coherent analytic sheaves in terms of locally free sheaves, and the existence of meromorphic sections of such sheaves.

These results are applied in section 4

to prove the Riemann-Roch theorem for complex analytic vector bundles, and to show the analytic reducibility of vector bundles. Section 5 is devoted to a rather unsatisfactory descrip-

tive classification of complex analytic vector bundles of rank 2 on a compact Riemann surface.

Any such vector bundle can be

viewed as an extension of one complex analytic line bundle by another, and the possible extensions are quite easily classified;

the difficulty lies in determining which line bundles can be subbundles of a given vector bundle.

Mumford's notion of stability

comes into the discussion quite naturally here; for unstable bundles the classification can be carried through quite easily, while for stable bundles this approach seems not very satisfactory. No attempt was made to treat stability thoroughly or in detail,

since I did not intend to go into the discussion of analytic families of complex vector bundles; that would merit a full year's lectures by itself.

The classification was only carried far

enough to obtain some results needed for the last part of the course.

Sections 6 through 9 contain a discussion of flat vector bundles over compact Riemann surfaces.

There was not time enough

to get very far, so this is more an introduction to the subject than a complete discussion; actually, the theory has not yet been developed to the point that a complete discussion is possible.

The definition of flat vector bundles and a general description of their relation to complex analytic vector bundles are covered in section 6; the main result is of course Weil's theorem, (Theorem 16).

Cohomology with coefficients in a flat sheaf is

treated in section 7; and the exact sequence relating this to the cohomology with coefficients in the associated analytic sheaf is

introduced in section 8.

The concluding section 9 is a preliminary

treatment of families of flat vector bundles, including some further details on the analytic equivalence relation among such bundles.

The two appendices cover some questions which came up during

the lectures, and which led to brief digressions. The formalism of cohomology with coefficients in a locally free analytic sheaf seemed to be rather confusing at times; the first appendix is an attempt to clarify matters.

The analytically trivial flat line

bundles on a compact Riemann surface can be described quite directly in terms of the period matrices of the abelian differentials on the surface, while the situation is rather more complicated in the case of vector bundles and the general picture is still incomplete; the second appendix gives an indication of why the vector bundle case is necessarily more complicated.

It must be emphasized that these really are preliminary and informal lecture notes, as claimed on the cover; they are not intended as a complete and polished treatment of the material covered, but rather merely as a set of notes on the lectures, for the convenience of students who attended the course or are interested in this area.

Were I to give the same course again, not

only would I hope to get much further, but also should I make several changes in the presentation; for instance I would perhaps discuss analytic structures on families of flat vector bundles directly in local terms rather than referring everything back to the characteristic representations of the bundles, following somewhat the lines sketched in other lectures, (see Rice University Pamphlets, vol.54, Fall 1968).

It seemed to me, though, that it

would be better to make these notes available now for whatever use they might have, rather than to wait for some years to polish and complete the discussion.

I should like to express my thanks here to the students who attended the lectures, for their interest and assistance, and to Elizabeth Epstein, for a beautiful job of typing. Princeton, New Jersey July, 1967

R. C. Gunning

Contents Page

§l.

Analytic sheaves

. .

. .

.

.

.

.

. .

.

.

.

.

.

.

.

1

.

.

26

free and locally a. sheaves of modules; b. free sheaves; c. analytic sheaves; d. coherent analytic sheaves. §2.

Local structure of coherent analytic sheaves a. c.

§3.

.

local structure; b. semi-local structure; global structure over the projective line.

Induced mappings of analytic sheaves

.

.

.

.

.

4+5

. .

.

.

58

.

71

.

.

inverse image sheaf; direct image b. sheaf; some applications. c. a.

Riemann-Rock theorem .

.

.

.

.

. . .

.

.

.

reducible vector bundles; b. Riemaam-Roch theorem; c. Serre duality for vector bundles. a.

§5.

A classification of vector bundles of rank two . classification.of extensions; b. divisor classification of unstable bundles; c. d. remarks on stable bundles; e. surfaces of low genus. a.

order;

§6.

Flat vector bundles . . . . . . . . . . . a. criterion for flatness; b. Weil's

.

.

.

.

96

.

.

.

.

123

c. connections and flat theorem; representatives.

§7.

Flat sheaves: geometric aspects

.

.

.

.

.

cohomology with flat definitions; b. sheaf coefficients; c. defham isomorphism and duality; d. role of the universal covering space; e. duality explicitly. a.

§8.

Flat sheaves: analytic aspects

.

.

.

.

.

.

.

.

Prym differentials and their periods; some special properties; c. meromorphic Prym differentials. a.

b.

.

.

157

Page §9.

Families of flat vector bundles

.

.

.

.

.

.

.

.

.

179

.

space of irreducible representations of the fundamental group; b. space of irreducible flat vector bundles; c. space of equivalence classes of connections; d. bundles of rank two in detail; e. analytic equivalence classes. a.

Appendix 1.

. .

. .

.

. .

.

.

. .

.

.

.

.

. .

.

.

.

231

. .

.

.

235

.

The formalism of cohomology with coefficients in a locally free analytic sheaf.

Appendix 2 .

.

.

.

.

.

.

.

.

.

.

. .

. .

.

. .

.

.

Some complications in describing classes of flat vector bundles.

§1.

Analytic sheaves.

(a)

Sheaves provide a very convenient and useful bit of machinery

in complex analysis, and will be used unhesitantly throughout these lectures.

Those readers not already familiar with sheaves and their

most elementary properties are referred to §2 of last year's Lectures on Riemann Surfaces, which contains all that will be pre-, Only sheaves of abelian groups were treated

supposed in this section.

there; but more general classes of sheaves are also of importance, so we shall begin by considering some of these.

The definition of a sheaf of rings over a topological space parallels that of a sheaf of abelian groups, except of course that each stalk has the structure of a ring, and that both algebraic operations (addition and multiplication) are continuous.

All the

rings involved here will be assumed to be commutative, and to possess an identity element.

There are thus two canonical sections over any

open set, the zero section and the identity section.

Considering

only the additive structure, a sheaf of rings can be viewed also as

a sheaf of abelian groups. Let

T. be a sheaf of rings and J be a sheaf of abelian

groups over a topological space M , with respective projections

p:

--> M and 7r: , ---, M . Viewing

abelian groups, the Cartesian product

merely as a sheaf of

FL XJ has the structure of

a sheaf of abelian groups over M X M , with the projection p X 77-:

X X,1 --? M X M

.

The restriction of this sheaf to the

diagonal M C M X M is then a sheaf of abelian groups over M , which will be denoted by X e,d

.

Definition.

The sheaf

.de

of abelian groups over M is

called a sheaf of modules over the sheaf

of rings (or more briefly,

a sheaf of V -modules) if there is given a sheaf homomorphism

'k eJ -- J such that for each point on stalks

p e M the induced mapping

X 1p --A J p defines on J p the structure of

IL p

an 'fop-module. For any open set U C M a section f e r(u, ne) )

is

readily seen to be the restriction to the diagonal U C U X U of a section

7. XJ ) ; that is, there are sections

(r,s) e.r(u x u,

s e r(u, 2) such that

r e r(U,

and

all p e U .

The sections

r(U, Ft )

f(p) = (r(p),s(p))

for

form a ring, and the sections

form an abelian group; and the homomorphism ReJ T I

r(U,,J )

exhibiting . as a sheaf of it -modules leads to a mapping

r(u, 7Q a on

J ) = r(u, IL ) x r(u, j ) ---. r(u, 1) which clearly defines

r(u, )) _ ,1 U the structure of a

r(U, 9C) = 19 U - module.

Thus (J U)

is in the obvious sense a presheaf of modules over

the presheaf

(7(U}

of rings.

a presheaf of rings and such t h a t J U

(.d U)

Conversely, whenever

( R U)

is

is a presheaf of abelian groups,

is an X U module and restriction mappings are

module homomorphisms, the associated sheaf I is a sheaf of

X -modules. The notions of sheaf homomorphisms, and related concepts, introduced last year for sheaves of abelian groups, extend readily to sheaves of modules.

If j is a sheaf of R -modules, a subset

J C J is called a subsheaf of W -modules if 3 is a abelian groups and, for each point p e M ,

9p

of

is an K p-submodule

In the obvious fashion,

of .d p .

a subsheaf J C X of in

t is itself a sheaf of It-modules;

-modules is also called a sheaf of ideals

, since for each point p e M the stalk ,8 p

an ideal in Xp .

If 3 C j is a subsheaf of '' -modules, the is also a sheaf Of R -modules.

quotient sheaf

mapping

,1 --A

cp:

is necessarily

A sheaf

between two sheaves of R -modules is called

an R -homomorphism if for each point p e M the induced mapping

cpp: j p ---D

)( C) of an cp( 2 ) C 3

sheaves; and

is a homomorphism of R

p

'k -homomorphism

The kernel

Pmodules.

cp: J --? 3 , and the image

, are subsheaves of

Ig -modules of their respective

induces an 'R -isomorphism J/?C = cp( A ) .

cp

An

exact sequence of sheaves of ',-modules is a sequence

...

of sheaves

i-1

of

i

that for each

,A

(Pi

> Ali

> 1 i+l

; ...

-modules and of It -homomorphisms

the image of

i

(Pi

cpi-1

cpi

such

is precisely the kernel of

cpi

A short exact sequence is an exact sequence of the form

0-4 where

11'Pl

,822 0 ,d3

0 denotes the zero sheaf of

way of writing the

.

-isomorphism

00,

-modules; it is an equivalent

t .e 3 = J 2/cp1( J 1)

One additional construction of some importance should be mentioned as well.

If .8 and 3 are sheaves of

tensor product I QX7

{ Ju ®

-modules, their

is the sheaf defined by the presheaf

U} ; this is also a sheaf of ' -modules, (recalling

U that all rings considered here are commutative).

It is a simple

matter to verify that for each point p e M the stalk

Jig®' P) p .

(J ®,,,) )P =

For notational convenience, the p

tensor product will be denoted by ,p ®) when there is no danger Tensor products of sheaves of modules satisfy many

of confusion.

of the familiar properties of tensor products of modules, as is readily verified by considering the separate stalks.

for any sheaf 0 of '' -modules, R e

In particular,

(recalling that all

rings considered here have units); and if

o -> ,/ l is an exact sequence of sheaves of

9

0 .d 3 0 0

2

.

-modules, then tensoring with

yields an exact sequence

'1l eR 3

'f

J2

3

0.

eW j

(Note especially that it is not claimed that the latter sequence is a full short exact sequence; for if 0 -- 1O 1 --- 1 2 is

exact, it is not necessarily true that 0 --> 3 e 11 ___;;

2

is exact. The sheaf 3 is called flat if

®j

0

1

' 9 ej 2

is exact whenever

is exact.)

If J and 0 are sheaves of W -modules, their direct sum

(b)

is the sheaf of

Z. -modules defined by the presheaf

and will be denoted by I ® 3

{

®3 U}

J U ; this sheaf can be identified in

the obvious manner with the sheaf J &.7

considered earlier. A

particularly simple example is the sheaf of 19 -modules m =

®

...

®k ,

the direct sum of m copies of the sheaf

W, ; any sheaf of X -modules isomorphic to Xm will be called a free sheaf of 'k -modules of rank m . Note that the stalk 'k

at a point p

p

is just the set

and elements R e R. P will be

of m-tuples of elements of considered as column vectors

R =

where

E

rj e

ILP

The sheaf `R,m has m canonical sections

.

a r(M, 'R m) , where

the identity section

1 e r(M, '& )

0 e r(M, I) .

section

is the column vector with 3-th entry

EJ

and all other entries the zero

These sections are free generators of the

in the sense that any element R e 'k

sheaf

uniquely in the form R = r1E1(p) + ... + rmEE(p)

rj e X module

p

for some elements

; in a similar sense, they are free generators for the

r(M, X m)

An

can be written

p

of sections of the sheaf Z

X -homomorphiam

very simply as follows.

M

n can be described

(pe

The image

q)Ej

of the section

is a section of X n , hence can be written n uniquely as qE = Z fi4Ei for some sections fi4 a r(M, ) . Ej a r(M, gym)

i=1 These sections

(fib)

the ring L M = r(M,

can be viewed as forming a matrix

cp

.

The matrix fully determines

For any element R e IR

Z r4E4(p)

j.l

can be represented

p

m in the form R =

over

this matrix will be called the matrix

representing the homomorphism the homomorphism.

0

for some r a A

-5-

;

p

and since

cp

is an IL -hom morphism,

iEi rJfij()Ei(P) .

cp(R) =

E In other words, using matrix terminology,

versely any n X m matrix

-homomorphism

0

ip(R) = 0(p)-R

.

Con-

over the ring ZM determines an

cp: m --> In . If cp: Zm -? jLn

and

ILn --> a are two IL -homcmorphisms, represented by. matrices

yr:

and ir,cp

T respectively, then it is evident that the composition

is represented by the matrix product TO .

homomorphism matrix

0

Consequently the

cp: Wm --> Xm is an isomorphism if and only if the

representing that homomorphism is an invertible,matrix

over the ring k M :

(The matrix

0

is of course invertible pre-

cisely when its determinant is a unit of the ring km , that is,

is invertible in the ring TM . matrices over the ring

The set of all m x m invertible

M form a group which will be denoted by

GL(m, It M) , and called the general linear group over the ring R M.)

A sheaf J of R -modules over the topological space M is called a locally free, sheaf of R -modules of rank m if for some open covering 74 = (Ua)

of M the restrictions I 1Ua

free sheaves of rank m over the various sets

Ua .

are

The above

description of homomorphisms of free sheaves can be used to derive a convenient description of locally free sheaves; for this purpose the following bit of machinery is required. If let

S :t (m,

is a sheaf of rings over a topological space M , be the sheaf of groups defined by the presheaf

(GL(m, R U)} ; this is a sheaf of not-necessarily abelian groups of course, but there is no difficulty in the way of the definition,

and since nothing is needed but the notation, no further fuss will

-6-

be made.

_ (Ua)

Let

be an open covering of the space

a one-cocycle of 1f[ with coefficients in IT (m, collection of elements

0a3 a GL(m, RUa n u.)

triple intersection Ua n U. n p

M.

By

is meant a

such that for any

(p0a3)(pOf,) = (p&,,) , where

y ,

denotes the natural restriction homomorphism into the group

GL(ma W

The set of all these one-cocycles will be

Ua n u n U) y 0

.

denoted by Zl( )Il , .!1 Z (m,' )) in

elements

where

U

®a a GL(m,

Two one-cocyles

(Oar), (TCO)

will be called equivalent if there are

.A x (m, 7Z ))

Zl(Z'

.

)

such that

'Ya, = (p8a)0a3(p8)-1 ,

p now denotes the natural restriction homomorphism into

the group

It is a straightforward matter to

GL(m, 'U

a n U)

show that this is an equivalence relation; the set of equivalence

classes will be denoted by H1(?X ,

S Z (m, Z )) , and will be

called the first cohomology set of A with coefficients in x (m, W. )

.

Suppose that V = (V(X)

is another open covering

of M , which is a refinement of Ut with refining mapping µ that is, suppose that

µ: T -> VL is a mapping such that

Va C µVa for each Va e if IL:

Z1( Vt ,

.

Then µ induces a mapping

ly (m, /L )) > Z1(1C ,

the mapping which takes a cocycle

A/z (m, a. ))

a

0UCFPeZl(vt aJx(maT- ))

into the cocyle

(o)VaVP = PVa n V. 0µVa,µV,

denotes the restriction mapping to 0 Va n V. C µVa n µV, . It is clear that the image is again a cocyle

where' pV n

a

and that

v

µ takes equivalent cocycles into equivalent cocycles;

so that

induces a mapping

µ

µ*: n (V`. , A 4 (m, 'k ) )

> $l(4 ,

If IP is a refinement of

Lemma 1.

* are two refining mappings, then

Proof. for each set

X (ma K ))

g

defined element of

GL(m,

'

eVa

V a

g

and

V

* = V

Considering any cocycle Va a 1(' define

, and if

0U

e

II ap

PV 0µV

a

a

, since

Zl(u , P x (m, x) ),

a

this is a well-

Then

va C µva n Vva

(go) V V = PV n V ogv , µV = pV n V (oPV , VV ovv , VV ovv µV ) a a a a aO s =

and

ao that

(aeva)(v )vavv(aeeO)-1

pva n VO

are equivalent; hence

µo

g* = V* , as asserted.

Now for any two coverings u( , Ir of M , write It < Vi 1r if µ is a refinement of 7 ; the set of all coverings is partially

ordered under this relation, and by Lemma 1 there is a well-defined

mapping Hl(rt , v It (m, X )) --> Hl(In , .a x (m, I )) )f
J Z(m, G ) , and these in turn to inclusion mappings

H1(M, Ax (m, s. ))

T H1(M, I x (m, C" )) T H1(M, b;t (m, 4 )) .

In the case m = 1 , these inclusions were used last year in classifying complex line bundles, and hence locally free analytic sheaves of rank 1; (recall in particular §7(a) and §8(a)).

For the case

m > 1 the situation is considerably more complicated, with no

satisfactory classification theorem yet in sight; indeed, the idea of a classification theorem for the set of all complex analytic vector bundles of the same form as that for complex line bundles, may not be a reasonable one.

(d)

The class of locally free sheaves is not closed under

completion of short exact sequences; and many of the analytic sheaves naturally arising, more especially in several complex variables, are no`t.locally free.

For these reasons it is convenient

to introduce a somewhat broader class of analytic sheaves.

For

the purpose of the present lectures, it will suffice to consider only analytic sheaves (sheaves of

over a Riemane

surface M An analytic sheaf

is called a coherent analytic sheaf

over the Riemann surface M if to each point p e M 9there is an

open neighborhood U such that the restriction of J to U

is

the cokernel of an Q1-homomorphism of free analytic sheaves, that is to say, such that there is an exact sequence of analytic

sheaves of the form

(1)

SL

IU

The canonical sections

19mlu

%

Ei a r(u, B-m1U)

generate the sheaf d mJU

as a sheaf of & -modules, hence their imagea

generate the sheaf ) jU

V. 0 .

W1. J JU

AEj e r(u, j lU )

as a sheaf of B--modules; so in this

sense, a coherent analytic sheaf is finitely generated.

Letting

Si = AEI a r(u, j JU)

sheaf P , the homomorphism X

be the generators of the

is described by.

Z

A(Z

kernel of A m-tuples

(f

for all m-tuples

is thus the subsheaf of (t m,U a (9 F for points

(fj) a

; the

consisting of those

p e U such that E f1.Si(p) = 0

and is called the sheaf of relations among the generators

Si

The exact sequence (1) shows that the sheaf of relations is also finitely generated.

Thus a coherent analytic sheaf can be described

as a finitely generated analytic sheaf such that the sheaf of relations among the generators is also finitely generated.

It is

convenient to derive a few conditions guaranteeing that an analytic sheaf be coherent; for the sake of simplicity, only the case of one complex variable will be considered. Lemma 3.

homomorphism

The kernel and image of any

(Oka's Lemma)

(m

X:

- Q n of free sheaves are coherent ana-

lytic sheaves. Proof.

Selecting any point p

in the Riemann surface,

it suffices to find an open neighborhood U of p homomorphism

NI:

(2)

(Lm11U .

(L mIU such that the sequence

(qm,U I . OnjU

!9-mljU

is exact.

and a sheaf

For then the image of A

is the cokernel of

is coherent by definition; and the kernel of

A

11

,

hence

is the image of

, so is coherent (locally at least, which suffices) by the first observation.

As for the proof of the existence of the homomorphism

consider first the case n = 1 ; then k m X 1 matrix A = (11,...11 m)

is represented by an

of holomorphic functions.

-17-

1.1

Choose

a coordinate mapping that

z(p) = 0 .

Bi(z) = z4&uj(z) r

z

in an open neighborhood U of p

The functions

assumed that

uj(z) # 0

uj(z)

such that

sufficiently small, it can be

for all

rl = min(rl,...,rm) .

that

can then be written

functions

for an

0 ; and by taking U

u1(0)

Bi(z)

such

By relabeling, suppose

z e U .

Then if

(f

is an element of

e

p

the kernel of X , necessarily

zr-rluu(z)fi(z)

0 = Z 1j(z)fj(z) = zr1 E

so that fl(z)

177 JZ

r

zJ r

uu(z)fj(z)

Consequently, it is evident that X: (.hlJU --.;- Q1'IU , defined as the mapping sending the (m-l)-tuple

(f

a

(gj) a LI

-' to the m-tuple

P where

fl(z)

Tz)

fi(z) = gj(z)

m

J=2 for

zrJ ruj(z)gj(z)

,

j = 2,...,m ,

is the desired mapping.

The remainder of the proof will be by induction on the inex n ; the case

n = 1 having just been treated, suppose that the re-

sult holds also for n - 1 , and consider the mapping aLm .:

In = e-1 ®cj n

.

This homomorphism can be split

into the direct sum X = X' ® X" X,

:

OL m

61

n-1 and J': LI m

of homomorphisms

_ (9 1 ; and the kernel of X

is the intersection of the kernels of

-18-

X'

and of

V"

.

By the

induction hypothesis, for some open neighborhood U of the point p

there will be an exact sequence of analytic sheaves 61

'IU X1

> alMIU

0 n-lIU

for some homomorphism %I ; and by the result proved for the case

n = 1 , by reducing the size of U

if necessary, there will be an

exact sequence of analytic sheaves

one

X°

6 m1IU

DmIU

for some homomorphism

> a1IU

Then the exact sequence

.

aIU xl

B

t

nIU

@mIU

n

which results serves to conclude the proof.

If .8 and ) are coherent analytic subsheaves

Lemma 4.

of 0- n , then J n 0 is also a coherent analytic subsheaf of Proof.

Restricting attention to a sufficiently small open

neighborhood U of a point p

a:-

sheaf homomorplhisms that I = a( U'-s)

phism a ®T: B

and

C m n .

on the Riemann surface, there are 6L n

3 = t((1 t)

Ot

and .

m n

(F,G) a (Lfl ®S

T(G) a Jq n

by a (F,G) = a(F) ,8 n

-

s

s+t = j9 a ®

note that an element d(F)

(

T:

e

__

n such

Consider then the homomor, and let X be its kernel;

lies in ,x if and only if

q q , hence the map

a': X -- (R n defined

has as its image precisely the subsheaf

Since )

is coherent by Lemma 3, upon restricting

U still further if necessary there will be a homomorphism.

.

Otm

6-8+t with image precisely X

q):

at.p:

(¢m

.

But then

-=, a n has image precisely J no , so that 4 n 0 is

coherent by Lemma 3, thus concluding the proof.

The following properties of coherent analytic sheaves are straightforward consequence of the definition and of Okats lemma. Theorem 2.

(a)

If ) is a coherent analytic sheaf,

then an analytic subsheaf

C 9 is coherent if and only if it is

locally finitely generated. (b)

If, in an exact sequence of analytic sheaves of the

form

any two of the sheaves are coherent, then the third is also coherent. (c)

If J and 3

are coherent analytic sheaves, then

the kernel, image, and cokernel of any sheaf homomorphism cp: ) ---> ' are also coherent analytic sheaves. (d)

If

V, and

coherent analytic sheaf

are coherent analytic subsheaves of a , then

Z+ ,d and X fl J are also co.

herent. (e)

® J and

If

R. and A are coherent analytic sheaves, so are

(Z ®a .d .

Proof.

(a)

It is only necessary to show that an analytic

subsheaf ,o c J which is locally finitely generated is coherent. Restricting attention to a sufficiently small open neighborhood U of a point

p , the hypothesis is that J is the image of a sheaf

homomorphism p: &m - J ; so clearly it suffices to show that the kernel

of T

-')

lCLs aj

'>J

y

., *>0

Letting *l = *a , it is clear that

)

?0

is the image of *l ; an

easy diagram chase shows that the kernel of

is (kernel *l) = 4rl

= (image p1) + (kernel a) , hence (kernel *l) is finitely generated

since both (image p) and (kernel a) are coherent.

Thus-0 is

coherent.

Next suppose that j and ) are coherent; for a suitably small open neighborhood U there is a sheaf homomorphism a:

s

--? A which is onto.

r IPl

s

P,

"

t

V 10 0

'AO 0

Letting *1 = *a , the image of *1

is coherent by part (a); in-

deed, the proof of that part shows that the kernel of *1

is

finitely generated, so upon restricting U further if necessary, there will be a sheaf homomorphism

q)l: (Lr

making the

diagonal sequence in the above diagram exact. aq)l

lies in the kernel of

sheaf homomorphism p:

ir

Clearly then

(or the image of q)), there is a

LV r

Vso that the diagram remains

commutative; needless to say, further.

p

Since the image of

U may have to be reduced still

is onto, so that

V,

is a finitely gen-

erated subsheaf of J , hence It is coherent by part (a).

Finally suppose that k and .3

are coherent; for a suit-

ably small. open neighborhood U there are sheaf homomorphisms

p:

r

;;-

P, and t:

alt --k

which are both onto. t------> Q t------> 0

0------> B r ----- ^>

J

0 ---->

L;

0

0

0

0

let

q)l = (Pp

; and let

*1: Ct

> J be a sheaf homomorphism

such that the above diagram is commutative.

the mapping pl ®*l; m r+t

It is then clear that

=r®

is onto, so then

full diagram above is exact and commutative.

This shows that .d

is finitely generated; and repeating the argument on the kernels

in the exact sequence of the top row shows that A

is actually

coherent.

Since o is locally finitely generated, so is the

(c)

image T(J )'C 3 ; but then from part (a) it follows that is coherent.

q)(,J )

Considering next the exact sequences

0 --? (kernel p) ---? J --? p(,d ) --? 0 and

0 --? p(,j ) --? J -0 (cokernel p) --? 0 , it follows from (b) that both the kernel of p and the cokernel

of T are coherent. .(d)

Note that

hence is coherent by (a).

.

+,g

is clearly locally finitely generated,

Note further that

(b); so considering the natural homomorphism the image of

/ 'Q

is coherent by

p:

will be a locally finitely generated subsheaf of

3 / 'k , hence is coherent by (a).

Since p( ,o ) _

fl

it follows from (b) that I fl X is coherent also. (e)

Since

0 - '. T 'k®,, T J T 0 , it follows

immediately from (b) that R

coherent.

Over some open

neighborhood U of any point there will be an exact sequence of analytic sheaves of the form

-24-

r

l? lp

Then tensoring with Al

rl ®b Nov

noting that

that

0

Qr

Jrl and j r

r-p? R--?0

yields the exact sequence

-- (¢r ) j --?

® . ---> 0

10

Qr ® ,d = ,9 r , and

djJl r

and

are both coherent, it follows from (c) that

is coherent, as the cokernel of a homomorphism of coherent

analytic sheaves.

With this observation, the proof of the theorem

is concluded.

-25-

Local structure of coherent analytic sheaves

§2.

Over a Riemann surface, any coherent analytic sheaf can be

(a)

described quite simply in terms of complex vector bundles; the present section will be devoted to a local and semi-local version of this relationship on a general Riemann surface, and the global version over the complex projective line. Theorem 3.

On a Riemann surface, every coherent analytic

subsheaf of a locally free sheaf is locally free.

The assertion being local, it suffices to prove

Proof.

that a coherent analytic subsheaf d C any point p

p

is locally free.

For

on the surface, there is an open neighborhood U

of

over which the sheaf is finitely generated; thus there are sec-

tions

Si E r(U, B-m) , i = 1'...'r , such that the sheaf homomor-

phism

T:

B'r,U --1 (

JU

, defined by

fjSj(q) , has as image precisely

(fl'...,fz) E (4-q --

It is evident that, restricting U

JIU

to be sufficiently small, there

is no loss of generality in supposing that no germ

Si(p) E J-M

can be written as a linear combination of the remaining elements,

with coefficients in the ring & p . morphism if

(p

and if

be the element having a zero at

and hence

p

then has as its stalk at

(fl'." ,fr) E N

fjIfi E /}p

The kernel

for all Si(p) = -

j

E

.

N( of the homo-

the zero stalk.

(For

(fl'...' r / (0,...,0) , let

p

fi

of least order; thus

But then (f./fi)S.(p)

0 = (p(fl'...,fr) = Z '

a contradiction.)

The

j#i

kernel x is also coherent, so will be generated over U

ing U

further if necessary) by some sections

Fi E r(U,

(restrict'9'r)

;

but since

p = 0 , necessarily all the germs

and hence

throughout U .

0

Thus

Fi(p) = (0,...,0)

U - , IU

Cp:

,

is an

isomorphism, and the proof is thereby concluded.

If j is a coherent analytic sheaf over a

Corollary.

Riemann surface M , then each point

p e M has an open neighbor-

hood U over which there is an exact sequence of analytic sheaves of the form

0

(1)

Moreover, if U

>

(

4--,j jU

lU 1 > L

0

is a coordinate neighborhood, then whenever there

is an exact sequence of the form (1), Hq(U, ,J) = 0

for all

q > 0 ;

and corresponding to the sheaf sequence (1) is the exact sequence of sections

o -> r(u,

(2)

Proof.

,jLm1)

> r(u, gym) --P-> r(u, j ) --> o

Since J is coherent, each point

open neighborhood U cp:

T1

p e M has an

for which there is a sheaf homomorphism

c9-mjU --> J IU which is onto; and the kernel of

(P

is coherent.

The kernel is then locally free by Theorem 4, so that there is an

exact sheaf sequence of the form (1) if U

is sufficiently small.

Recall from last year's lectures (Corollary to Theorem 4, page 44)

that for a coordinate neighborhood U ,

Hq(U, q) = 0

for all

q > 0 ; but then the exact cohomology sequence corresponding to (1) begins with the sequence (2), and for the higher terms, all vanish except perhaps proof.

Hq(U,

This suffices to conclude the

With a little more effort, and a few more preliminaries, it is possible to establish a semi-local form of the exact sequence (1).

(b)

Let M be an arbitrary Riemann surface, and consider an

open subset U C M . linear subspace

The bounded holomorphic functions form a

r0(U, -') C r(u, (9-)

phic functions on U .

of the space of all holomor-

For any element

put

f e r0(u, L-)

II fll = psU'pU I f(p) I ;

becomes a Banach space.

r0(U, 6-)

it is only necessary to show that

under which

r0(U,

this is clearly a norm on the vector space

(To verify the latter assertion r0(U, &-)

is complete in norm;

but this is obvious, since convergence in norm is equivalent to

uniform convergence on U .)

The spaces

r0(U, fin) = r0(U, 0-)n

can be given a corresponding Bsnach space structure, defining for

any element

F = (fl,...,fn) e r0(U, .n)

Of particular interest will be the space

the norm IIFII = max IIf1 II i m)

r0(U, 6L" m)

of bounded

holomorphic m X m-matrix valued functions, which-can be identified 2

with the space

(The vector space of all complex

r0(U, j5 m

m X m matrices will be denoted by G X m , and can be identified

with

2 C1°

; and r0(U, 8mX m) is the set of bounded, complex

analytic mappings from U

into C X m .)

If

U, Ul

are two open

subsets of M and U C Ul , it is clear that the restriction to U of an element

f e r0(Ul, B-)

indeed, the restriction mapping

is an element pU(f) e r0(U, 0-) ;

Pu: P0(Ul, . ) T r0(U, (l )

a continuous linear mapping between these Banach spaces.

is

pU will also be used for the restriction

The notation

mapping between other spaces of analytic functions. stance,

pU: GL(m, OU ) --> GL(m, (9 U)

Thus, for in-

is a group homomorphism,

1

where as before

GL(m,

Theorem 4.

U) = r(u, rlx (m, 0- ) ) Ul, U2

Let

.

be open subsets of a Riemann

surface M , with intersection U = Ul fl U2 ; and assume that the

linear mapping e: r0(UI, 4 ) ® ro(U2,

) ---> ro(U, &)

defined by ®(fl,f2) = pU(fl) + pU(f2) , is onto.

Let

0: GL(m, (9 U ) X GL(m, 6U2 ) --> GL(m, 1

neighborhood A of the identity element 0

U

Then there is an open

be defined by O(F1,F2) =

the image of

(IL

I e GL(m,C)

such that

contains at least the set

(F a GL(m, B U)IF(p) e 6 for all p e U)

Recall that the matrix exponential mapping

Proof. exp: Cm X m

.

> GL(m,C)

defines a complex analytic homeomorphism

between an open neighborhood

DG

of the origin in

open neighborhood 4 of the identity exp-1: AO --> DG be its inverse.

Cm X m and an

I e GL(m,C) ; let

(For the definitions and ele-

mentary properties of the matrix exponential function, see for instance C. Chevalley, Theory of Lie Groups, I, (Princeton Univ. Press, 1946), especially Chapter I.) neighborhoods of the origin in

Cm

X m

exp Xl . exp X2 e AO whenever Xi a Di

Let

Dl, D2 C DC be open

such that .

it

0'm X m)

S2j C r0(Uj,

be the open subset of that Banach

space defined by

S2j = (G e rG(Uj, (¢ m X m) I G(p) a Dj

for all p e Uj )

.

It is then possible to define a mapping LILm xm)

121 ® 122 -> rG(U,

`Y:

by putting

T(G1,G2)=exp-'(expG1- expG2); and it is evident that

T

is a continuous mapping from the open

subset 12l ® 122 C r0(U1, 0) ®rG(U2, C9 ) into the Banach space r0(U, (QmXm)

.

To prove the desired theorem, it is sufficient

to show that the image of the mapping borhood of the origin in

contains an open neigh-

`Y

r0(U, 0 m x m)

.

For the image of

would then contain a basic open neighborhood U C r0(U,(fCm

`Y

Xm)

of

the form

for all p e U) ,

12 - (G e r0(U, (amXm)IG(p) e D where

D C D0

is an open subneighborhood of the origin.

= exp (D) C GL(m,C) , whenever

F E GL(m, Q1 U)

F(p) e 0 for all p e u , there is an element

F = exp G ; but then

is such that G e 12

such that

G = !( G1, G2)pa for some elements

a 12i ,

and putting F. = exp G. e r(UJ, /a x (m, follows that

Letting

)) = GL(m, J2 UJ ) ,

F = exp `Y (G1,G2) = O(F1,F2) ,

as desired.

it

The

proof will be completed by establishing some properties of the

mapping

`Y

, and using some general results about mappings between

Banach spaces.

-30-

*: Dl X D2 -> DO C d Xm be the complex analytic

Let

mapping defined by *(Z1,Z2) = exp-1(exp Zl

exp Z2)

.

In the

Taylor series expansion of this flznction at the origin

(0,0) s D1 X D2 , the constant term is the matrix *(0,0) = 0 The first-order terms can be viewed as a linear mapping

A: dnxm ® nXm --> nXm i and recalling the series expansion for the matrix exponential function, note that

exp *(Z1,Z2) = I+ A(Z1,Z2) + (higher order terms) and that exp Z2

exp 4,(z1,Z2) = exp Z1

= I +(Z1 + Z2) + (higher order terms)

Writing

x(Z1,Z2) = Zl + Z2 .

hence that

(Z1,Z2)e C M

X m. ® do X m'

,

note then that

I*(z1,Z2) - *(0,0) - A(z1,Z2) lim

(3)

(Z1,Z2)

for

I(Z1,Z2)I = max(IZ11,IZ21)

any matrix Z = (zip) a &° X m , and for any matrix pair

IZI = maxi JlziJI

- (0,0)

"1'"2

=0

1

Defining then the linear mapping

A: ro(Ul, by A(G1,G2)

61mxm)

x ro(u2, B mxm)

= A(PUG1,PUG2) = PUG,

->

ro(u, Imxm)

+ PUG2 , it follows directly

from (3) that

lira

1Ih(G1,G2)

- T(o,o) G1, G2) 11

(G1, G2) a (0'0)

-31-

A(G1,G2)II

=0

derivative is

The same argument, with its obvious

d'Y(0,0) = A .

modifications, shows that

is indeed strictly differentiable.

'Y

d'Y(0,0) = A , on the other hand, can be identified

The derivative

with the direct sum of m2

copies of the mapping ® ; and by

hypothesis it is necessarily onto. image

This suffices to show that the

contains an open neighborhood of the origin,

'Y(S11 ®a2)

and hence to conclude the proof, in view of the following general result.

Let

Lemma 5.

E

and

F be Banach spaces,

open neighborhood of the origin, and mapping.

If

'Y:

12 C E

be an

Sl -- F be a continuous

is strictly differentiable at the origin, and if

'Y

its derivative

d'Y(0): E ---> F

is onto, then the image

contains an open neighborhood of the image

'Y(0)

'Y(a) C F

.

Before turning to the proof of the lemma, it

Remarks.

might be convenient to recall the relevant definitions. A continuous mapping

'Y:

Sl -- F

is said to be differentiable at the

origin if there is a continuous linear mapping

A: E - F

such

that

lim

x-40 the mapping

IIT(X lx¶)---A(X) = 0

A , which is evidently unique, is called the derivaThe

tive of

'Y

at the origin, and is denoted by A = d'Y(0)

mapping

'Y

is said to be strictly differentiable at the origin if

it is differentiable with derivative

lim Xl, X 2 -40

A

and if moreover

IIT(xl) - ''(x2) - A(x1 X2)" NX

X2

=0.

.

To simplify the notation, suppose that

Proof.

Since the continuous linear mapping A = d!(0)

'Y(0) = 0

.

is onto, it follows

from the open mapping theorem (see Dunford-Schwartz, Linear Operators I, (Interscience, New York, 1964), pages 55 ff.), that the image

under

A of the open ball

contains an open ball

origin in

of radius 1 centered at the origin

B1

of some radius

Bc

c > 0

centered at the

F ; hence for any element y e F there exists an element

x c E such that A(x) = y and

Letting 'y0 = T -A

11x11 < c Ilyll .

it follows immediately from the strict differentiability of that for a sufficiently small constant

r > 0 ,

IITO(xl) - 'Yo(x2)II = IIT(xl) - I(x2) - A(xl - x2)II < ' whenever

'Y

11x1

-

x211

xl, x2 e Br C a .

The proof will be concluded by showing that

'Y(Br) D

Bcr/4

For any point y E Bcr/4 , there exists a sequence of points xn a Br (i)

(ii)

(iii)

such that the following conditions hold: x0 = 0 ;

A(xn+l) = y - TO(xn) '

llxn+l - xnll < n1l 2

(To see this, note that

llyll c

but as above, there exists such a point with obtained the points so selected that xn+2 - xn+l = yn+2

_ - 'Y0(xn+l) +

A(xl) = y

xl must be so chosen that 11x111
ro(U, f) defined'by

©(f1,f2) = pI(f1) + pU(f2) , is onto.

Proof.

Since U1-U

sets of C , there is a such that

r(z) = 0

and

U2-U

are disjoint closed sub-

Cm real-valued function

r(z)

in an open neighborhood of U1-U

in an open neighborhood of U2-U .

If

f E r0(U, d-)

in C and

r(z) - 1

is any bounded

holomorphic function, set

r(z)f(z) if Bl(z)

0

z E Ui U'

if

It is then clear that and that

{(l_r(z(z) if

z E U

gj(z)

is a

pU(gl) + pU(g2) = f .

cp(z) c r(u 1

U U2,

(' 0'l)

0

92(z)

C°°

if

z E U z E U2-U.

bounded function in U

The differential form

defined by ag1(z)

if

z E U

-ag2(z)

if

z E U2

1

(P(7,) =

is then a well-defined e differential form. last year's lectures that there is a

Ul U U2

such that

Coo

function h(z)

this proof). and

cp

h

can be chosen to

is bounded, (but see the remarks at the end of

The functions

f2(z) = g2(z) + h(z)

that

in

ih = ep , (Theorem 4+, page 4+2); and by examining

the proof of that theorem, it is evident that be bounded if

It was proved in

fi e r0(Ui'

and

fl(z) = gl(z) - h(z)

for

z E Ul

z c U2 , are clearly so defined

for

©(fl'f2) = f , thus concluding the

proof.

Remarks.

The proof of Theorem 4 of last year's lectures

was complicated by the necessity of allowing for unbounded differential forms; for the case of a bounded differential form

desired function h

h(z)

can be taken simply as

= 211-1i ffU

1

UU 2

*(t) - d5

cp , the

Lemma 7.

the complex line

Let C ,

V be a simply-connected open subset of

and let K C V be a compact subset of V .

Then any non-singular holomorphic matrix-valued function F E GL(m, B-V)

H E GL(m,

can be uniformly approximated on K by an element

19 C)

Proof.

Again let

D C Cm X m be an open neighborhood of

the origin such that the matrix exponential function establishes

an analytic homeomorphism

exp: D - A , where A is an open

neighborhood of the identity F E GL(m, 6-V)

I E GL(m,C )

G E 6

m X m,

First, suppose that

F(p) E A for all points p

has the property that

in an open neighborhood of K ; thus

.

F = exp G

for some function

for some open neighborhood W of K.

G can be uniformly approxi-

ordinary Runge theorem, the function

mated on K by a polynomial P ; and then H = exp P singular holomorphic matrix-valued function in

F uniformly on K .

mates

By the

is a non-

C which approxi-

Next, for an arbitrary function

F E GL(m, a V) , there will exist finitely many functions Fi E GL(m, m V)

such that

F1(p) E A for all points

open neighborhood of K , and such that the set

GL(m, 0 V)

F = Fl

...

p

in an

Fn .

(For

is a topological group with the compact-open

topology; it is readily noted that the group is connected, hence it is generated by an open neighborhood of the identity. open neighborhood consists of all elements that

Fo(p) E A for all points

p

F E GL(m, U V)

product of elements from this open subset.) Fi

Fo E GL(m, O-V)

such

in a compact set containing K

in its interior; and thus any element

argument to each function

One such

is a finite

Applying the preceding

leads to the desired result, thus

concluding the proof. Theorem 5.

Let

D C C be an open subset of the complex

plane such that its closure

D

is compact and simply connected;

and let J be a coherent analytic sheaf over an open neighborhood

of D .

Then there is an exact sequence of analytic sheaves of

the form

0 --b ?'ID 'P1 ell) --T->J ID -> 0

(1k)

Proof.

By the Riemann mapping theorem, there is no loss

of generality in supposing that

domain of C .

.

D

is a bounded rectangular sub-

The corollary to Theorem 3 shows that there is an

exact sequence of the form (4+) over an open neighborhood of each point

p e D .

Thus

D

can be decomposed by a sufficiently fine

rectangular grating (that is, by a finite number of lines parallel to the real or imaginary axis) into smaller rectangular segments, on an open neighborhood of each of which there is an exact sequence of the desired form.

To complete the proof, it is merely necessary

to describe how to patch together exact sequences over two neighboring rectangles into an exact sequence over their union; for this process can be used to patch together the sequences in each horizontal row first, and then to patch together the various rows into an exact sequence over Thus suppose that

D .

Ul, U2

are open rectangular neighbor-

hoods of two adjacent rectangles, as in the following diagram, and

let u = Ul f1 U2 .

Ul

r------------ r----t -----------i I

I

U2

I

i

i I I

I

I

I

I

--------------

I

I

------------a----------U

Suppose further that over an open neighborhood VV

of Ui

there

is an exact sequence of analytic sheaves of the form

- CILrj 0

VJ ---> 6sj I Vi -i

0

_1 I Vi

There are thus two such sequences over the intersection V = v1 fl V2 and it is clear that there are sheaf homomorphisms the following diagram is commutative, where

id

a,P , such that

denotes the iden-

tity homomorphism.

o

---> o- 1 IV

0 ->

r

pl >

0sl v

a

v -> o

id) 4s I VIV > 0 2-> 2 )___ p

(To see this, let Ei E r(v, ts1. 1) be tithe canonical generating sections, and put Si = a1(Ei) E r(V, J ) . As in the corollary to s

Theorem l+, there are sections

a2(Fi) = Si

Fi c r(V, a 2)

such that

Letting a be the homomorphism defined by

a(Ei) = Fi , it is evident that 0

is constructed similarly.)

B:

s1 ® ( 2

a a = al , as desired. 2

The mapping

Now define sheaf homomorphisms

(s2 ® ms1 ,

e. :

s2

®

6sl

( 1 ® Q s2

by

= (G+C& - 0$G, F - RG)

B(F,G)

0'(K, L) = (L+OK - c'L, K It is readily verified that B

is an isomorphism with

- CLL)

0'0 = 00' = identity, and hence that. as its inverse;

0'

and the following

diagram of exact sequences is also commutative.

r

0 - -1 ®

&'2IV

a1

(pl(Pl., ) s1

& ®t9 s2 I v -- ,8 I V -- 0 jid

I°

0 --

4r2 ® (9 s1v

The mapping

0

(p ,id)

s

T

a

s

2 ® (Q 11 v 2

o

l V --> 0

is defined by a non-singular holomorphic matrix-

valued function over V , that is, by an element © e GL(s1 + s2, &V) .

By Lemma 7, the matrix

© can be approxi-

mated uniformly over the compact subset

U C V by a non-singular

holomorphic matrix-valued function over

V1 U V2 ; thus there is

an element

©0 E GL(s1 + s2, S

p c U the matrix

such that for all points ) 2 1 ©(p)©0(p)-1 c A , where A is an open neighbor-

hood of the identity

V U V

I c GL(s1 + s2, c)

In particular, select A

.

sufficiently small that Theorem 4 applies.

Then, recalling by

Lemma 6 that all the hypotheses of Theorem 4 are fulfilled, it follows that there are elements such that

©2 E GL(s1 +s2 ,

81 e GL(s1 +s2

,

l9 U) and

©(p)'©0(p)-1 = ©2(p)'©1(p)

for

2

all p e U ©(p) =

Writing

©1

for all

=

©1©0 , this condition can be written p e U .

isomorphisms defined by the matrices

Letting

©

,

01

be the sheaf

consider then the exact

sequences

r +

l

0 ---?

Jul

l 2 1Ul 1 T

r2+s

o ---?

l

0.

a B"1

s +s

---?

s +s 2 l iU2

J Ul --? 0

,9

TJ

a B 2

IU2 --- 6-

,0 1U2

Since on the subset U the construction yields the fact that al = a20 = a202Bl , hence that

a10i1

= x202 , these two sequences

coincide over U , and therefore determine an exact sequence of the desired form, over Ul U U2 .

Considering now the special case of the complex projective

(c)

line

This then concludes the proof.

1P, the structure of general coherent analytic sheaves can

be'described quite easily in terms of locally free analytic sheaves. To see this, recall that

1P has a standard complex analytic coor-

dinate covering V _ ((Ul,z1),(U2,z2)) ; here

zj: Uj -- C are homeomorphisms from Uj

subsets,

complex line, the points zl(p) # 0

and

are defined by

z1(p) = 1/z2(p)

projective line over

1P

p e Ul fl U2

If

1P

are open

onto the entire

are precisely those for which

0 , and the coordinate transition functions

z2(p)

Theorem 6.

31

Uj C 1P

when p e Ul fl U2

.

,eQ is a coherent analytic sheaf over the

, then there are locally free analytic sheaves

for which the following is an exact sequence of

analytic sheaves.

0 Proof.

>31l

>3

(P

'J

,

;- 0.

In terms of the standard coordinate covering I& of

the projective line, let

Dj C Uj

be open subsets homeomorphic to

./

the disc

(zi a C lzjl < 2)

under the coordinate mappings

the intersection D = D1 fl D2 coordinate system.

zj ;

is thus a finite annulus in either

By Theorem 5 there are exact sequences of

analytic sheaves of the form

0>

(5)

(¢rjlDi

j> SsjlDj -;-. lD1

00 .

This then provides two such exact sequences over the intersection D = Dl fl D2

.

Just as in the proof of Theorem 5, the sequences (5)

D there is an isomorphism

can be so modified that over

GB2ID

B: O s'ID

for which the following is a commutative

diagram.

0---> CLr1lD1? 6s1lD al>

0

--

Note that necessarily tion of

0

morphism

pp2

to the subsheaf

'r2ID

and

-L2-.;J ID a

p1( & r1ID) C 6Ls11D

Now the elements

© e GL(s, 01 D)

s2 l D

0- 0

and rl = r2 ; and that the restric-

s1 = s2

01: &11D

commutative.

matrices

0 r2 ID

lD0 0

,

0

defines an iso-

for which the diagram is still

and 1 can be represented by

©1 a GL(r, CAD) ; and these matrices

define complex fibre bundles for the covering

(U1,U2)

or equiva-

lently, locally free analytic sheaves 6 , a1 respectively, over ]P

.

By construction, these are just the sheaves desired to com-

plete the proof of the theorem, however.

Having demonstrated this result, the basic existence theorem follows readily for the special case of the projective line, by a

similar argument. .

Any vector bundle over the projective line

Theorem 7.

7P

admits non-trivial meromorphic sections. Let 2A = ((U1,z1),(U2,z2)}

Proof.

plex analytic coordinate covering of

the coordinate mappings

zj

be

under

As in the proof of Theorem 6, a

..

can be defined by

(D1,D2) , hence by an ele-

a coordinate bundle for the covering

012 a GL(m, aD ) where D = D1 fl D2 .

coordinate, for example,

D C Ui

(z. E cffzjf < 2)

0 e H-(1P, /S X (m, B"))

complex vector bundle

ment

, and let

7P

open subsets homeomorphic to the disc

be the standard com-

In terms of the zi

is a holomorphic, non-singular

012(z1)

matrix-valued function in the annulus

< 1,11 < 2 ; and the

D:

2 Laurent expansions of the various entries yield a matrix a of rational functions on

which approximate

7P

any compact subset of the annulus

012

uniformly on

(Recall that a rational

D .

function is a quotient of polynomial functions; of course, the only singularities of

©

are at the two points

8 in

Thus there is a rational matrix most at the points

z1 = 0

singular in the annulus

D:

ti

for all points

identity

and

z1 = 0

z2 = 0 .)

1P , with singularities at

z2 = 0 , such that

< I zi

and

r(M,

d (e)) -a ...

,

so that there are natural mappings from the space of holomox'phic sections of a subbundle

0 C `!

to the space of holomorphic sections

of the ambient bundle tions of a bundle quotient bundle

`Y , and from the space of holomorphic sec-

to the space of holomorphic sections of a

`Y

`Y/O

A vector bundle

.

it has a proper subbundle; otherwise A vector bundle

`Y

is called reducible if

`Y

is called irreducible.

`Y

is said to be decomposable into the direct sum

and 8, written

0

of two vector bundles

`Y = i ®©, if the

is given as the direct sum

corresponding locally free sheaf C9(!)

6-(`Y) = e (0) ® 0 (©) ; if no such decomposition is possible, the

bundle

`Y

is decomposable as so that

`Y

If the vector bundle

is said to be indecomposable. `Y = 0 ® 8 , then both

0

`Y

and © are subbundles,

is certainly reducible; or what is the same thing, an

irreducible bundle is indecomposable.

One of the complicating fac-

tors in the study of complex vector bundles is that the converse assertion is generally false; that is, an indecomposable bundle need not be irreducible.

To examine these latter definitions a bit further, let

(0C43) E Z1(Z, b'k(m, &- )) and

(T

)) be

) e Z1(14 , h T, (n, (

defining cocycles.for complex vector bundles

0

Riemann surface M ; and suppose that

and

0 C `Y

and

`Y

over a Over

`Y/O = © .

each coordinate neighborhood Ua there are then isomorphisms (O)JUa = LtmJUa 6L

mlUU C Q

rank r

njU

LV-(`Y)I

Ua = 9 nIUa ; and the imbedding

is such that the quotient is a free bundle of

over U. .

ality in putting

and

It is evident that there is no loss of genern,Ua = O'mIUa ® L rI Ua .

canonical generating sections of ment if necessary, let

Fi

(9rjUa

,

(For let

Ei

be the

and, passing to a refine-

be sections of Q1njUa which are the

-6o-

images of Ej

under the mapping to the quotient sheaf.

There is

an injection !9'r I Ua --- an JUCe defined by taking F j --> E j and it is easily seen that Q nIUU and the image of C'rJUa .) to an n-tuple

is the direct sum of

Thus an element

must preserve the subspaces m P C

The matrices

0a$ , and their effect

on the quotient spaces must be given by the matrices

(2)

`Yak

`Y

P ; indeed, their restrictions

to the subspaces Q P must be the matrices

fore the matrices

(Q mJUa

F e L9 P corresponds

under this imbedding.

(F) e a n

;

There-

©a$ .

must necessarily have the form

(11

a4 Aa

Ko

©a$

TC43 _

The converse being apparent, the condition that

0 C `Y

and

`Y/O =

is therefore just that the bundles can be represented by cocycles The condition that

such that (2) holds.

the direct sum

be decomposable into

`Y

`Y = 0 ® 8 is correspondingly that the bundles can

be represented by cocycles of the form (2) with Aa3 = 0 . Lemma 11.

m > 1

Let

`Y

be a complex vector bundle of rank

over a Riemann surface M , and suppose that

non-trivial meromorphic section complex line bundle

* ; and r

F .

Then

`Y

admits a

has as subbundle a

`Y

has a meromorphic section

f

which determines the section F under the natural injection

0 -? IN- (*) -> )q (`Y) (That is,

f = F when

f

-

is considered as a section of the

larger bundle .)

-61-

Proof.

Let `Yao E Zl( U1 , /J x (m, dL )) be a defining

cocycle for the complex vector bundle

dinate covering

_ [tJ)

`Y

in terms of some coor-

,

of M ; and let

trivial meromorphic section of this bundle.

is not identically zero, and

F = [Fa) Thus

be a non-

Fa E r(ua, km) whenever

Fa(p) _ `!

p E ua n uP ; here

Fa

morphic functions.

Refining the covering ]/Z if necessary, suppose

that

Fa

is envisaged as a column vector of mero-

is holomorphic and non-singular (that is, not all com-

ponents vanish) at all points in ua except perhaps one point;

za be a coordinate mapping in ua such that the excep-

and let

tional point is the origin zaaFa(za)

Then for some integer ra

za = 0 .

is a holomorphic, non-singular vector function on all

Again refining IQ if necessary, there is a holomorphic

of ua .

non-singular matrix valued function

Ta E GL(m, (Q u) such that

a

T. - za a = E1 where as before

E1

is the constant vector

1

El =

... 0 0

The vector bundle

`!

"I\ ) E Cm

'/

is also defined by the equivalent cocycle

and the meromorphic section

TV

Fa

terms of this cocycle by the functions Since

matrix

za aEl TI

= `!

zPrE1 in ua

must be of the form

-62-

flTJ

,

F

is expressed in

-raE - TaFa = za 1

it is apparent that the

* ...

*Q13

*

0

*

...

I

here

(*

defines a line bundle

)

QI3

section

* C `Y

with a meromorphic

-r

inducing the given section

(fa) = (za a)

F = (F)a

, and

the proof is thereby concluded. Theorem 10.

Every complex vector bundle of rank m > 1

over a compact Riemann surface contains a line bundle as a subbundle; so only line bundles are irreducible. Proof.

The Corollary to Theorem 9 shows that every com-

plex vector bundle over a compact Riemann surface admits a nontrivial meromorphic section; so the theorem follows as an immediate consequence of Lemma 11. Corollary.

A complex vector bundle

`Y E Hl(M, b

(m, 6-))

over a compact Riemann surface M can always be defined by a cocycle

(fir

)

E Zl( 1 4 , ,

'( (m, m ))

of the form

* lob

* (3)

Proof.

*1

0

*2Q

0

0

`0

0

* *3a$

The vector bundle

0

...

1mm

T must contain a line bundle

as a subbundle, by Theorem 10; so the defining cocycle can be

given by matrices having as first column the vector

\o (brim)

where

/ )

defines the line bundle

*1 .

The (m-1) x (m-1)

matrix block along the diagonal (omitting the first row and column) defines the quotient bundle, and then must have the indicated form by the obvious induction.

The Riemann-Rock theorem, for complex line bundles over a

(b)

compact Riemann surface, was proved in last year's lectures,

(Theorem 13, page 111); recall that the theorem can be stated in the following form.

If M

-1 a H(M, (g-)

g , and

is a compact Riemann surface of genus

is a complex line bundle of Chern class

c(g) e Z , then

dim HO(M, 6-W) - dim H1(M, O-W) = cW + 1 - g This theorem extends easily to complex vector bundles, in the following form. Theorem il.

(Riemann-Roch theorem for vector bundles).

If M is a compact Riemann surface of genus 0 e IJ(M,

X (m,

))

g

and

is a complex analytic vector bundle of rank

m over M s then dim HO(M, &(0)) - dim H'(M, 6 (0)) = c(det 0) + m(1-g) ,

the cohomology groups being finite-dimensional complex vector spaces. Proof.

The case m = 1

The proof will be by induction on the rank m . is just the Riemann-Roch Theorem for complex line

bundles, and is known.

Assuming that the theorem has also been

proved for complex vector bundles of rank m-1 , consider a vector bundle

0 e Hl(M,

bundle

0

bundle

8 = O/cp

of rank m .

x (m, 64 ))

contains a line bundle

cp

By Theorem 10, the

as subbundle; so the quotient

is a well-defined vector bundle of rank m-l

To the exact sequence of sheaves

0 -.? B-(Cp) -- -

(0) -.? 9- (©)

0

there corresponds an exact sequence of cohomology groups, beginning as follows.

0 --- H0(M, B ((p)) --- H0(M, S (0)) -- - HO(M, 0 (e)) --(4)

-- R1(M, 6 (CO) -- x1(M, CQ (0)) -- -> H (M, m

((p))

--.;.

R1(M, 6L (e))

...

It follows immediately from this exact sequence and the induction hypothesis that all the cohomology groups are finite-dimensional complex vector spaces; and, recalling that any line bundle

cp

1,2 (M'

d'((p)) = 0

for

, the alternating sum of the dimensions of the

complex vector spaces in the exact sequence (4), up to the term ico2i2(M, a (cp))

, must be zero.

Writing that sum out, and regrouping

terms, it follows from the induction hypothesis that

-65-

dim HO(M, 6 (0)) - dim Hl(M, 6-(0)) [dim HO(M, 61 ((p)) - dim Hl(M,

((p))) +

+ [dim H0(M, 0 (e)) - dim Hl(M, d (e)))

[c((p) + 1- g) + [c(det ©) + (m-1)(1-g)) ; but since

det 0

(det

and hence

c(det 0) =

= c(cp) + c(det ©) , the induction is completed and the proof therewith concluded. For any complex vector bundle

Corollary. 1

0 E Hl(M,

(m, 62 ))

over a compact Riemann surface M ,

ON, ® (0)) = 0

The proof is again by induction on the rank m

Proof.

of

0 , the case m = 1

lectures.

for all q > 1 .

having been demonstrated in last year's

With the notation as in the proof of Theorem 11, the

exact sequence (4) contains segments of the form

--> ON, c (CO) ---> ON, 6 (0)) --> ON, 0 (Q)) --> ...

for all q > 1 ; but since Hq(M,

Hq(M, 6L (cp)) = 0

and

Ll (e)) = 0 by the induction hypothesis, the desired result

follows immediately. Remarks.

For any coherent analytic sheaf

o on a com-

pact Riemann surface M , it follows from Theorem 8 that there are complex vector bundles

(5)

0, 01 , forming an exact sequence

0 ---> 0-(01) ---> 61 (o) -.-> j ---;- o

.

It is an immediate consequence of the above results and the exact

cohomology sequence associated to (5) that the groups

Hq(M, ) )

are finite-dimensional complex vector spaces, and that whenever

q > 1 .

Hq(M, , ) = 0

Writing

x( a ) = dim H0(M,,d) - dim

H'(M, j ) ,

the exact cohomology sequence associated to (5) also shows that

x( J) =X(O) -x(Ol); this provides a form of the Riemann-Roch theorem for arbitrary coherent analytic sheaves.

(In this connection, see also the

article by A. Borel and J.-P. Serre, Ie theoreme de Riemann-Roch, Bull. Soc. Math., France 86(1958), 97-136.)

The Serre duality theorem also extends readily to complex

(c)

vector bundles.

(The line bundle case was treated in last year's

lectures, §4 and §5.)

Recall that over any Riemann surface M

there is the Dolbeault exact sequence of analytic sheaves, of the form

0

(6)

where type

a

(r,s) .

sheaves a

r,s

-C r,s

-, at

a 0, 0

-

is the sheaf of germs of

> 00,1 CO*

0,

differential forms of

(See page 72 of last year's lectures; that these can be viewed as sheaves of

L IL-modules., and that

is a homomorphism of sheaves of N -modules, are quite evident.)

For any complex analytic vector bundle (9 (0)

0

over M , the sheaf

is locally free, hence flat, so tensoring with (6) yields

the exact sequence of analytic sheaves

(7)

(0)

o

here F r,s(,)

er,s 0 -(0)

_

differential forms of type bundle that

0 .

e°'1(0) -;. 0

a°'°(0)

(r,s)

is the sheaf of germs of

Coo

which are sections of the vector

(See page 73 of last year's lectures.)

gr's(,)

;

It is clear

is a fine sheaf; and so, recalling Theorem 3 of

last year's lectures,

r(M, P °'l(o))

Hl(M, m (0)) =

(8)

ar(M, a °'0(0))

Letting % be the sheaf of germs of distributions on the Riemann

surface M , (recalling §5 of last year's lectures), there is an

C" differential

exact sequence parallel to (7), replacing the

forms by distribution differential forms; and there follows an analogue of formula (8) for distributions. The spaces

r(M, e 1'1(o))

can be given the structures of Let

topological vector spaces, as follows.

(Ua,za)

be a coor-

dinate covering of the compact surface M by a finite number of coordinate neighborhoods which are small enough that &(0) free over each Ua .

A section F e r(M, 61"(0))

written F = (Fa) , where

Fa e r(Ua,( a

rank of the vector bundle

0 ; thus

fia e

r(Ua, 1° 0°0)

.

1'1)m)

can then be

and m

Fa = (fip a

is

is the

daa) , where

For any integer n > 0 , put

Pn(F) = Z Z

E

sup JDVfia(za)J ,

aiV1+v2 (Q(cp2) --.> o

are called equivalent if there is such that the following

diagram is commutative:

o

> ftpl)

> 6L (o)

Ie

o --> (9-((vl)

> S(w2) --> o Jid

(cp2) -> o ,

where id denotes the identity isomorphism; it is evidently only the equivalence classes of extensions that are of importance here.

Theorem 13.

If

(pl,

are complex line bundles over an

cp2

arbitrary Riemann surface M , the set of equivalence classes of extensions of the line bundle

cpl

by the line.bundle

is in a

cp2

natural one-to-one correspondence with the set of elements of the

eohomology group H'(M, o' (cplcp2l))

; the trivial extension

cpl ®cp2

corresponds to the zero element of the group. Proof.

bundle

T2

(Ua)

If

0

is an extension of the bundle

cpl

by the

, select a sufficiently fine coordinate covering

of the Riemann surface M that the bundles involved can

be represented by cocycles

(cgoCV) e Zl( i(, c*) The condition that

0

and

(0C43 ) a Z1(vl , AY (2, & ))

is an extension of

cpl

by

cp2

, that is,

that there is an exact sequence of the form (1), is just that the cocycle

(0C3)

has the form

_

la 0

for some elements elements

(%ov)

Xap a

Lv-

u

a

n u

CP2Q

The only restriction upon the

is that they are so chosen that the matrices

satisfy the cocycle condition, namely, ever

Xa

p e Ua n up n U7 .

0C3(p)O07(p) = 0ay (p)

0C43

when-

Writing this out explicitly, the elements

V must satisfy

X

(3)

xCI(p) = c)lcp(p)x0y(p) +X (p)c)2,y(p) whenever p c uanu, nu, .

It should be noted that the set of functions

(%ao)

for a covering

induces in a natural manner a corresponding set of functions

describing the same extension for any refinement A

of

..

Thus,

all possible extensions are described by classes of families of functions

(x

Suppose next that

satisfying (3).

)

0

and

0' are

two extensions of the same line bundles, defined by sets of functions (x

)

and

and LI (0')

morphism

in terms of the same covering of M .

(%'a3)

are both free sheaves over the open set

Since 6L(O)

Ua , any iso-

is represented over each set Ua by

e: 6-(0)

a holomorphic matrix a e GL(2, 6-U

and these matrices clearly

satisfy

(4)

D

whenever

(P).8 (P) = ®a(P)-0C43 (P)

The condition that

0

and

p e Ua (1 U0

be equivalent extensions is that

0'

e: -(0) -> 4 (01)

there exists an isomorphism

such that (2) is

e must be the identity on the subsheaf

commutative; thus

((pl) C 6L(O) , and must induce the identity on the quotient sheaf (cp2) _ 6(0)/ B (cpl)

the isomorphism

.

In terms of the matrix representation of

e , this condition is just that the matrices a

be of the form

ha

CO for some functions

ha e 0

U

.

1

Condition (4) then takes the form

a

(5) q)1CO(P)hO(P) + X (P) = X (P) + ha(P)cP2 Thus extensions described by functions NP) and

(P)

whenever p e UUf u . 0

(x'

)

are equiva-

lent if and only if, possibly after passing to a refinement of the covering, there are holomorphic functions such that (5) holds.

ha

in the various sets Ua

Considering in place of the flanctions

XOP

the functions

vaP="lapa% p condition (3) takes the form

(6)

a ,,7(p) = a 07(p) +a,,(P)(P1ly(P)cg2, ,(P)

To the function germ

at any.point vat

ciated an element of the sheaf

hood UO if p e

containing p

p

whenever

p e ua nu. nu, .

there is canonically assoin a coordinate neighbor-

the element is defined by

aa3(z0) , while

as well, the element is given by vao(zy)= 'P170(GOaa3(zB) U7

in the coordinate neighborhood U7 .

With this interpretation, (6)

takes the form

ca7(z7) = cO,(zy) +vaB(zO)'P1O7T2BY = a 7(zy) +vco(zy) thus these functions form a cocycle In terms of the functions

.

(aa3) a Z1(Vt, S (cP1cP21))

ca, , the equivalence condition (5) for

two extensions takes the form

+ vaO(P) = c (7) h(P) 0 Letting ha(zes)

to the functions

cC43(z0) - c

(P) + ha(P)cP1

(P)cP2cO(P)

whenever p e Ua nuo .

be the elements of the sheaf 0.(c,1cg21)

associated

ha(p) , condition (7) becomes

(zo) = ho(zo) -ha(za)cc 03 (P)gg2CO(P) = h,(z.) -ha(zo) ;

thus the extensions are equivalent precisely when their defining cocycles are cohomologous, which suffices to conclude the proof.

Note

that the zero element of the cohomology group obviously corresponds to the trivial extension

0 = cpl ®cP2

-74-

as a direct sum.

One immediate consequence of this description of extensions is the following result. Corollary. g

and

0

Let M be a compact Riemann surface of genus

be a complex analytic vector bundle of rank two over

If there is a line bundle

M..

cpl C 0 with quotient bundle cp2 = 0/cpl

and if c(cp1) - c(cp2) > 2g -2 , then

0

is decomposable into the

direct sum

" = cpl®cp2 Proof.

The set of all extensions of the line. bundle

by the line bundle

cp1

correspond to the elements of the group

cp2

; and by the Serre duality theorem for complex

H1(M, C' (cplgp2')) line bundles,

H1(M, 4 ((Pp21)) = HO(M, 8- (K(P11(P2) )

If

c((P1) - c((P2) > 2g -2 , then

c(KCP11g2)

= c(K) - c((P1) + c((P2) _

= 2g-2 - c((p1) + c(cp2) < 0 , so that necessarily H0(M, e'(Kcpilcp2)) = 0 ; this gives the desired result.

As an amusing sidelight, the cohomology class associated to an extension can be described in the following manner.

and hence that

that k (cp1cp21) = QZ (cplcp2l) ® Hl(M' N (Cp1q'1)) = H1(M, 1Yt) = 0 2

a = (aCO) e z1(Ul, , I (cp1cp21)) extension of

cp1

by

cp2

Recall

Thus if

is a cocycle describing a particular

, there will always exist a meromorphic

cochain h - (ha) a CO( 2(, ,)Y( (cplcp2]')) such that 5h = a .

All

extensions of meromorphic complex line bundles are thus trivial; and since all meromorphic line bundles are trivial, all meromorphic

-75-

vector bundles are trivial - thus providing an extension to arbitrary Riemann surfaces of the remarks following Theorem 7, conIn terms of these meromorphic func-

cerning the projective line. tions

ha , the meromorphic matrices

®a(za) = 0 .1

1 ha(zes)

/

provide an explicit meromorphic change of coordinates, reducing the extension to the trivial extension; so

0C43

= ®«1°P

By the Serre duality theorem, H1(M, 6- (cplcp2l)) is canoni-

cally dual to the space

r(M, B (Kcpllcp2)) = r(M,

thus the cohomology class

the vector space

is canonically a linear functional on

a

of holomorphic differential

r(M, LI-1'0(cpllT2))

forms which are sections of the bundle

cp lcp2

a

on the section w

§5(b) of last year's lectures. 0,0(cp1cp21))

g = (ga) a C0( the elements

.

1

w = (wa) a r(M, 4 1,O(cpllcp2)) , the value

associated to

al' O(cpllc,)) ;

a(w)

If

of the functional

is given as follows, recalling

Letting be any cochain such that

Sg = v

3ga define a global section

(aga) a r(M, e

0°1(cplcp2l))

,

and therefore '5ga .. wa a

P(M,

now a(w) = ffM ga .. wa .

Suppose that the covering VL = {Ua) larities of the meromorphic functions joint open sets

Ua

is so chosen that the singu-

ha

are contained in dis-

, and are not contained in any intersection

i

U. n U0 .

For each

i

select a

Coo

function ri

in U0 i

-c

which

1

is identically

in each intersection Ua n U. , and which 1

vanishes identically in a neighborhood of the singularities of ha i

in

H Vl

Uai ; and put .

ra = 1

for the remaining sets of the covering

Then in the construction of the Serre duality mapping it is

possible to take

a(w) = j"M

ga = raga , so that

Ua

However, except for the sets

Wa

, the function ha

is holomor-

i

phic and ra = 1 , so that

c(w) = E IIU

i

i

a(rcpa) = 0

wa = E IIU

i i

i i

a(ra ha )

i

a.

d (raihaiwai

=

by Stokes' theorem; and since ra = 1

i

j

i

Note however that h.a e C0(Vl ,

i

a.

iZ f

Uai

a(w) =

Therefore, since 7wa = 0,

.

i i i

a(ra ha wa.)

r ai hsiwsi

tai

UUa

on

i

,

it follows that

a hawai l i

l'0)

and

S(walla)= c

e Zl(z ,a 1,0),

so that the singularities of haws are globally defined; hence the total residue R (hawa)

of these singularities is well defined.

Therefore,

(8)

o(w) = 2T1 R (hawa) ,

which is the result desired.

(b)

Having classified the extensions of one line bundle by

another, the next problem is that of determining which line bundles are possible as subbundles of a given vector bundle.

This can be

approached through an extension of the notion of the divisor of a

meromorphic function, in the following manner. is an m-tuple of meromorphic functions in a neigh-

If` F(z)

at the point p

borhood of a point p e C , the order of F(z) such that

V

that integer

is

is an m-tuple of holomor-

(z-p)VF(z)

phic functions with no common zeros in an open neighborhood of p;

is the least of the orders of the m

equivalently of course,

V

component functions of

F(z)

.

The order will be denoted by Vp(F).

Note that if 8 is a holomorphic invertible m Xm matrix in an

open neighborhood of p , then p(F) = P(8F) .

is a vector bundle over M , and if

0 e H1(M, 6 % (m, 6L ))

is a meromorphic section of

F e r(M,%1 (0))

that the order

Vp(F)

of

F

(F) -

E

p This divi-

in the usual sense.

sor is associated to a unique line bundle cp

is then defined by

V

and is a divisor on the surface M

sense that

0 , it is apparent

at a point p e M is a well-defined

The divisor of the section F

concept.

Now if

cp a H-(M,

admits a meromorphic section

*)

,

in the

f e r(M, 7(pp))

with

4 (f) = J (F) Lemma 13.

If

0

is a complex analytic vector bundle (of

arbitrary rank) over a Riemann surface M , then a complex analytic line bundle

cp

is a subbundle of

divisor j associated to

Fe r(M,

1

(O))

Proof. cp

Ay

cp

0

if and only if, for every

, there is a meromorphic section

such that A-$(F) . If cp C 0 , and if

is any divisor associated to

, there is a meromorphic section f e r(M, 7r( (pp)) such that =

$(f) ; but by Lemma 11, f determines a section F er(M,W[(o)),

and since ,J (F) = J (f) = n9- , the implication in the one direction is demonstrated.

Conversely, if

M associated to a divisor A9

,

is any line bundle over

cp

and if F e r(M,

(cp))

is a mero-

morphic section such that n¢ _ ,9(F) , then there is by Lemma 11 a subbundle

C 0

(f)

hence

with a section

f

such that

but then necessarily

(F)

determines

f

F,

and the

= cp ,

result is thereby proved.

Now consider an arbitrary complex analytic vector bundle

over a compact Riemann surface; and suppose

Y' E H1(M, )b't (m, C)) that

`Y

cocycle

is fully reduced, in the sense that it is defined by a (`Y

) e Z1(

,

-h X (m, W ))

Corollary to Theorem 10.

of the form given in the

If F e r(M, d1 (Y'))

is a non-trivial

holomorphic section, write (fla

f2a

F = (Fa) =

and suppose that

(fm)

is the last component of F that is not

identically zero, so that It is apparent then that

fra A 0 but

fia = 0 whenever

(fra) a r(M, & (*r)) , so that at least

one of the diagonal line bundles in the reduction of a holomorphic section.

that

cp C ì ,

Now if

it follows that

trivial line bundle.

r < i <m.

cp

`Y

must have

is any line bundle over M such

1 C cp 1 ® `Y

, where 1 denotes the

By Lemma 13, the vector bundle

cp l ® `Y

ad-

mits therefore a holomorphic, non-trivial section F ; and therefore, as noted above, at least one of the diagonal bundles must admit a non-trivial holomorphic section, so that

cp l ®*r

r

1

® *r) = c(*r) - c((P) > 0 . Hence, for any line bundle cp C Y', it follows ithat c(cP) < maxjc(*j) , where (*i) are the diagonal

c(CP

line bundles in any fixed complete reduction of the bundle in other words, the Chern classes

c((P)

of line bundles

are subbundles of a fixed complex vector bundle from above.

Y'

; or

which

cp

Y', are bounded

Having made this observation, the divisor order of a

complex analytic vector bundle

pact Riemann surface M

over a com-

Y' e Hl(M, b -k (m, Q1 ))

is the integer, denoted by div '' , defined

by

div Y' = max c((P) (PC''

(9)

where

f(p)

are the line bundles which are subbundles of

'i'

.

The elementary properties of the divisor order of a vector bundle are easily established.

First, recall from Theorem 17. of

last year's lectures that for any line bundle

cp e H1(M, (9-*)

over a compact Riemann surface M and any non-trivial section

f e r(M, 1((p)) it follows that

e(cp) =

P em Vp(f)

.

It then

follows immediately from Lemma 13 and the definitions, that for

any vector bundle

Y' e Hl(M, 2

(m, (Q ))

over a compact Riemann

surface,

(10)

div Y' =

bundles

I c H1(M,

max

E

V (F)

(Fer(M, 'nt (Y')), FA0) peM p cp,

g

and a vector bundle

) t (m, Ql )) , note that

cp C T

if and only if

9 ®cp C g ® I ; and therefore div(9 ® Y') = c(g) +div Y' Finally, suppose that

I e H '(M, h x(m, Q ))

is fully reduced,

in the sense that it is defined by a cocycle of the form given in

the Corollary to Theorem 10; and let *1,*2,...,*m be the diagonal line bundles in that reduction.

(Thus

sive extensions by the line bundles

is the result of succes-

`Y

*11*2,..., m , in that order.)

As noted in the paragraph immediately above,

for any line bundle

(p C `Y

c(cp) < maxi c(*j)

; and therefore,

divi`<maxi c(* )

.

In particular, it is clear that

(13)

div(*1 ®..® Vim) = maxj

since each bundle

*i

c(*

is in this case a subbundle.

The following useful observation is also quite easy. e H1(M, N X(m, 67 )) cp e H1(M, m

)

Let

be a complex vector bundle and

be a complex line bundle on the compact Riemann

surface M ; then (14)

For if

implies that

r(M, 0 (cp"1 (9 0)) # 0

cp l ® 0

div 0 > c(T)

has a non-trivial holomorphic section, it follows

from Lemma 13 that there is a complex line bundle

g C cQ

1

(90

which has a non-trivial holomorphic section, hence such that

c(g) > 0 ; but then gcp C 0 , and thus div 0 > c(g) + c(cp) > c(cp) . Lemma 14.

g ; and let

Let M be a compact Riemann surface of genus

0 e H-(M, lY (m, m ))

bundle of rank m over M .

be a complex analytic vector

The divisor order of

fies the inequality

div 0 >

c(det 0) - 9

m

;

0

then satis-

0

and if

is decomposable, it further follows that div 0 >

c(det 0)

m For any complex line bundle

Proof.

cp a H1(M, (I *)

over

the surface M , it follows from the Riemann-Roch theorem for complex vector bundles that

dim HO(Ms (9.(p 1 ® 0)) = dim H1(M, C ((P -l ® 0)) +

+ c(det(cp 1 ® 0)) + m(l-g) > c(det((p'l ® 0)) + m(1-g) > c(det 0) + m(1-g-c(cp))

Whenever

c(cp)
0 , and therefore by (14) that div 0 > c(cp) since

c(cp)

takes on arbitrary integral values, it follows that

div 0 > n where n is any integer satisfying n < and the first inequality follows easily from this.

posable as a direct sum

c(det 0) +1- g, m 0 If

0 = cpl ED. ..® cpm , then since

= c((pl) +...+ c((pm) , necessarily

c(cpj) > m c(det 0)

dex j; but from (13) it follows that

is decom-

c(det 0) = for some in-

div 0 = maxj c(cpj) > m c(det 0),

concluding the proof.

(c)

Let us now return to the consideration of the special case

of complex analytic vector bundles of rank 2 over a compact Riemsnn

surface M .

(15)

For any complex line bundle

, set

11'(M,g) _ (0 a H1(M, .b ';C (2, L ))Idet 0

Note that for any other line bundle

I

it'follows that

Y (M,0 ®n = )f(M,t, 2) ; thus to describe all vector bundles, it

suffices merely to describe the sets if (mv) for fixed line bundles

9v

of Chern classes

cQv) = Y

taking just the values

In particular, it suffices to describe the sets INM,l)

v = -1,0

and I

where p is the point bundle associated to a p e M .

fixed base point

If the description behaves reasonably

upon tensoring with line bundles, the classification problem will have been settled.

In carrying out this classification, the set

of vector bundles naturally decomposes into two components.

A complex vector bundle

0 e N1(M, b x(2, QI ))

is called

stable if

c(cp) < 2 c(det 0) for every line bundle

cp C 0 ; otherwise,

0

is called unstable.

(This terminology and classification was introduced by D. Mumford; see for instance Proc. Int. Congress of Math., Stockholm (1962), 526-530.)

only if

Note for any line bundle * that

4r-lcp C 0 .

2

it

c(det 0) ; hence c((p) = c(*-lcp) + c(*) < 2 c(det(*

c(det 0) + c(*) = 2

stable.

if and

is stable and cp C * ®0 ,

Now if 0

follows that c(* lcp) <
'fO(M, ) defined by

cp1 E

xl(M,

*) -->( PP] ® g11g) E If ON, g) .

It is evident that two distinct line bundles image under this mapping precisely when

have the same

cpl,cp2

cp2 = cpilg

; thus i 0(M,

is naturally identified with the quotient sp ace of

under the involution

cp

-->

p' lg

The group

Ii-(M, t9

(M, m

structure of a complex analytic manifold of dimension

has the

) g

)

(with

infinitely many components), as noted in last year's lectures; and the involution

0 If

(Mg)

cp --> cp lg

is an analytic mapping.

Therefore

also has the structure of a complex analytic variety of

dimension

g .

(This variety has singularities at the fixed points

of the involution.)

For an indecomposable unstable vector bundle in V that is to say, for a bundle

0 E 4 "(M,I) - Y O(M,t) , it follows

directly from Lemma 15 that there is a unique line bundle

that

cp C 0

and

cp

such

c((p) = div 0 . Therefore, letting

(17) 1r"(M,9;(p) _ (0e')("(M,9)- 110(M, 9) IcPCO and c(cq) = div 0), the sets (17) are disjoint, for different bundles

g,

cp ; so the

set of unstable vector bundles over M can be written as the disjoint union

(18)

IrO(M,g) U

(M,g,(P)

( C (T) Q 2C(g)

for every compact Riemann surface M ; the union is over all the line bundles

cp

satisfying

c(cp) > 2 (g)

.

There remains the pro-

blem of describing in more detail the sets 1("(Mg,(p) Theorem 14.

On a compact Riemann surface M of genus

g

there is a natural one-to-one correspondence between the elements of V "(M,9;(p) , for any complex line bundles

g, (p

such that

,n

c(cp) > 2 (g) , and the points of the complex projective space

of dimension n = dim H1(M, 9 (cp2g-l)) -1 . Given the line bundles? g,cp , subject to the con-

Proof.

dition that c(cp) > 2 (g) extension of

cp

by

, let

0 E H1(M, JJ

cp2 = gcp l .

Since

it follows readily from (12) that and therefore

0

is unstable.

(2, c- )) be any

c(cp2) - c(g) -c((P) < 2 (g),

div 0 = c(cp) > 2 (g) = 2 (det 0)

That is, then,

(M, g; (P) = (0 E H1(M, . (2, Q1) ) I (P C 0 and det 0 = g)

fl

Cindecomposable vector bundles) The set of equivalence classes of extensions of

-87-

cp

by

cp2 = gcp l

was determined in Theorem 13; but this classification cannot be applied directly, since for the present purposes the weaker equivalence of vector bundles is of interest, rather than the equivalence of extensions.

(That is to say, there is no restriction the way cp 0 .)

is realized as a subbundle of However, suppose that

of

cp

is any indecompQsable extension

0

by cp2 = g9_1 ; and choose a cocycle 0aO a Z1(l/i

0 , in terms of some covering

representing the vector bundle

11t = (Ua)

x(2, CA) )

of M . This cocycle can be taken in the form I-aO

C'V

(19)

0

where

a Z1( V.

o

92a$

represents

CL *)

and

(p

(cp2ao)

, CO)

Z1(VL

represents cp2 = 90 -1 ; and, as in the proof of Theorem 13, the elements

XUO

are arbitrary analytic functions in Ua n U. , sub-

ject only to the condition that the elements

a,(z,) = cp

(p)N

(p),

043

for

z

= zP(p) , form a cocycle

If

= Z1( VL , (n-(cp29-1)) .

(o.

)

Zl(?A , CL (cpT21)) =

a

is another extension of

0'

cp

after passing to a refinement of the covering if necessary, will be defined by a cocycle some other functions

X''

the cocycle associated to

(0'a,)

; and let (x

cp2

0'

of the form (19), but with (a''

)

a Z1(71f.,

The bundles

)

by

0

-(p2g-1)) be and

0'

are

the same (that is, are analytically equivalent) if and only if, after passing to another refinement of the covering perhaps, there such that

are functions a e GL(2, C'Ua )

(20)

O0 = a0a0-l

-88-

in

Ua n U

Writing the matrices

©a explicitly as

E

f 11a12a f

la

2la

22a

equation (20) takes the form

X4 1a$

(21)

0

I1/f11R

2a0

\f21P f22P

\f21a

Considering at first the functions that

(f2la)

f22a/I

,

c(g)p

0

'P2a

it follows from (21)

can be non-trivial only when

(

But if

) = c(q2g-1) = 0 .

c(g)T; ) = 0

non-trivial holomorphic section, necessarily

and

tion

xao

)) _

Note however that c((p2g-1) = 2c(cp) -c(g) < 0 ;

so the section

cp

Pap

(f2la) e r(M, (.(qxp

p2a,3f21 = f2lacpap ' hence that

= r(M,

so

I\( na f12cr

=

f12P

cp2

f2

qxp

c((p) =

and

qxp

7c

(9)

has a

= 1 ; and letting

both be defined by the same cocycle ((ps) o the secbecomes a globally defined holomorphic function on the

compact Riemann surface M , hence a non-zero constant

then further follows from (21) that

c

It

+ f22cr 2

cp2aPf22P = cX

or equivalently that -

_

f22P - f22a

-

This last equation shows that C (qxp )), and since

0

in

c # 0 , necessarily

H1(M, (cp2

X

CO V

)=0

as well; but then, recalling the correspondence of Theorem 13, it follows that the bundle

is decomposable, a contradiction.

Therefore it is necessary that

f2la

-

0

Using this fact, equa-

tion (21) clearly reduces to the following equations:

=fup; (22)

f22a = f22P ;

f22P = fiia ap + fl2ar2c

'pVf12P + X

The first equation in (22) shows that the various functions define a global holomorphic function

(flla)

over M , which must be

a.1

constant since M is compact; and similarly, the functions reduce to a constant

a2 .

Necessarily

f22a

a.1a2 # 0 , of course.

Thus (21) reduces to the condition that

alga - a2 X = cpaP P - facp20

(23) where

all a2

phic functions in

(23) through by cps

a1Q

,

it finally reduces to the equation

in Ua n UP .

(z ) - a2a' (z.) = f,(z.) - fa(ze)

The vector bundles

and

associated to the functions

defined by fa(za(p))= fa(p) ; and multiplying

fa(p) a r(Ua,

(21k)

0

(a0) in z1(7

and

Of

defined by the cocycles

, Q. (0 21)) = z1(?!i ,

and a zero-cochain (fa) a CO(I&

LV_ ((p29-1))

0'

H'(M, Ql (cp29-1)) .

al, a2 , such

Since the bundles

are indecomposable, the cohomology classes

non-zero, as in Theorem 13; and thus

a

and

a1, a2

such thgt (21+) holds,

that is, if and only if there are non-zero constants

that a.1a - a2a' = 0 in

(aoP)

S (eg-1)) are thus

equivalent if and only if there are non-zero constants

and

are holomor-

Again, as in Theorem 13, consider the sec-

Ua .

(fa(za)) e r(Ua,

tions

fa = f22a

are non-zero constants, and

a'

as representing elements in the projective space

a

and

0

a' are

can be considered

1Pn = H1(M, dL( ciated to

a , a'

The final result is that bundles assoare the same if and only if

sent the same point in the projective space

a

and

a'

repre-

1Pn , where

n = dim H1(M, ( (cp2g-1)) - 1 ; that concludes the proof of the theorem. In one sense, formula (18) and Theorem 14 provide a descriptive classification of the sets

bundles over M .

of unstable vector

1f "(M,E)

All the components in (14) are complex analytic

space of dimension < g .

(By the Serre duality theorem,

H°(M, & (KIP 2g)) ; and since

H1(M,

= c(K) - 2c(cp) + c(g) < 2g-2 ,

c(Kpp-29) _

it follows from the Riemann-Roch

theorem (recall formula (14) on page 113 of last year's lectures) it

that

dim H1(M, ( g-1) ) < g .

Therefore

dim n

The question arises, whether the entire set

(M,9;(p) < g - 1)

(Mg)

can be given

the structure of a complex analytic variety, with all these components imbedded as complex analytic subvarieties; and any complete classification theory must provide an answer to this question.

We

shall return to this question later.

(d)

This approach to the descriptive classification of complex

vector bundles does not seem to be of use for stable bundles.

For

in the stable case, there is no uniqueness result analogous to Lemma 15; indeed, in the extreme case, exactly the opposite of Lemma 15 is true. if

0 e If (Mg)

Recall from the definitions and Lemma 11+ that

is a stable bundle, then 2 (g)-g < div 0 < 2

the lowest possible value of

div 0

is thus

l] - g , where

the square brackets denote the largest integer function. Leimna 16.

Let M be a compact Riemann surface of genus g ,

0 e 1f(M,g)

and let

be a stable vector bundle such that

div 0 _ [c(g2) + l - g Then for any line bundle cp e I(M, it follows that

bundle cp

1

cp

a

cp C 0 .

Proof.

such that

such that c(cp) = div 0

*)

To show that

cp C 0 , where

cp

is a line bundle

d.(cp) = div 0 , it suffices to show that the vector 1

® 0

has a non-trivial holomorphic section.

®0 has such a section

i C cp 1 0 0

For if

F , there is by lemma 11 a line bundle

with a non-trivial holomorphic section

f er(M, 61 (T)))

c(n) > 0 .

inducing the section F ; and of course,

Now since

cpr) C 0 , it follows that c(cp) = div 0 > c(ncp) = c(n) + c((P) , so that actually

0 = 1 , and

phic section, necessarily

eses of the Lemma, let

c(p1) = div 0 ; and let cp

cp1

be a line bundle such that

cp2 = OAP, = gcp-l

cpl C 0

norphic section. V(,= {uaI

cp-1 0 0

and

be the quotient bundle.

be any line bundle with c(cp) = div 0 = e(cp1) and

thus it is required to show that

cocycle

cp C 0 .

0 e '(MA) satisfying the hypoth-

Given the vector bundle

Let

has a non-trivial holomor-

c(n) = 0 ; but since

cp # cpl;

has a non trivial holo-

In terms of a suitable coordinate covering

of the surface M , the bundle a Z1(1JL ,

,P1 ( (2, (S ))

0

(D

can be defined by a

of the form

T2ap

where

(cpJao) a Z1 ( 7/L, 9 *)

line bundle

defines the line bundle

can be defined by a cocycle

cp

cp

J

; and the

((p.) a Zl( ?/'L, &-*) .

A section F e r(M, & ((p -l ®0)) then consists of pairs of functions fla,f2a a r(Ua, LA )

(25)

f1a(P) = cPa1(P)cp

f

f2 e r(M, CA 1

(P)flP(P) +

f2a(P) = cP(P)P2(P)f2P(P)

whenever p e Ua n U .

'Pap

satisfying

(l))) .

(p)kaO(p)f2P(p)

The functions

(f2a)

form a section

For each such section f2 , the functions

are readily seen to form a cocycle in

Zl( x, t (cp lcpl)) ; and the condition that there exist holomorphic

functions

(fla)

satisfying the first line in (25), is Just that

this cocycle be cohomologous to zero. the proof of Theorem 13.) section

(26)

f2

(Recall in this connection

The mapping which associates to each

this cocycle thus yields a linear mapping

r(M, 6. ((P 1pP2))

> H (M, Q (cP-1cp1)) ;

and a given section f2 a r(M, at (0-1 cP2)) can be extended to a section F = (f1) e r(M, (3.(cp 1 ®0)) precisely when

f2

is in

2

the kernel of the linear mapping (26).

In particular, to show that

there is a non-trivial holomorphic section

F e r(M, (9 (cp 1 (& 0)),

it clearly suffices to show that there is a non-trivial element in the kernel of (26); and than1 will obviously be the case whenever

dim r(M,

(cp-1T2)) > dim H'(M, I (cp lcp1)) .

The proof of the

Lemma is thereby reduced to demonstrating this last inequality. By the Riemann-Roch theorem for complex line bundles (page 111 and following, in last year's lectures),

dim r(M, N (W 1p ))

c(p

lcp2)

-(g-1) = c(g)-2 div 0+1-g.

From the Serre duality theorem for line bundles,

=

H1(M, QZ

1P1))

r(M, O.(Kqxp11)) ; and since c(Kgxpll) = c(K) +

+ c(cp) - c(cp1) = 2g - 2 but

Kgxpil

K

(since by assumption cp # cpl),

it follows again from the Riemann-Roch theorem that dim H1(M, 6 (cP lw1) ). = dim r(M, 61 (Kgxpil)) - g-1

.

Therefore

dim r(M, 4 ((r-p2)) - dim H1(M, Ql (p lcpl)) > c(9) - 2 div 0+2-2g > 0, which concludes the proof.

We shall return to the classification of stable bundles later, using rather a different approach.

(e)

It is perhaps of some interest to examine more closely the

preceding approach to the classification of vector bundles, over Riemann surfaces of low genus.

surface M = IP

Consider firstly a compact Riemann

of genus zero.

')f'(M,g) C 1J (M, g)

The stable bundles

are characterized by 2 (g) > div 0 > 2

according to Lemma 11+ and the definitions; so '),("1(M,g) = 0 .

so, for any line bundle

cp1

the bundles

"(M,9;(p)

are in on-

to-one correspondence with the points of the protect 9e space

where

Al-

IPn

n = dim HI'(M, (Q (cp2g-l)) - 1 ; but by the Serre duality

theorem H1(M, (A (g,2g-l)) = r(M, 0 (KT- 2g)) , and since c(KCp 2g) _ = -2 - 2c(cp) + c(g) < -2 ,

that

r(M, Ql (K(P2g)) = 0

it follows from the Riemann-Roch theorem .

Thus

n = -1 , and 1f"(M,9;cp)

Altogether then,

(27)

u(]P,g) _ j0(jp,g)

that is, all vector bundles in 1f(]P,9)

are decomposable.

It is clear that this extends readily to vector bundles of all ranks over

1P.

(This observation was first made by A. Grothendieck in

Amer. Jour. Math. 79(1957), 121-138-) Next consider a compact Riemann surface of genus

g = 1

The stable bundles 'C '(M, 9) C ').r (M, 9) are then. characterized by 2 (g) > div 0 > 2

1 ; thus necessarily div 0 _

[c(2) + 1]_ 1

and all stable bundles belong to the extreme class discussed in Lemma 16.

(That lemma can then be applied to describe this partic-

ular set of bundles in more detail; this was carried out by M. F.

We shall not

Atiyah in Proc. London Math. Soc. 7(1957), 1+11+ -1+52.

pursue this approach further here, preferring a different technique for the treatment of stable bundles.) cp

Again, for any line bundle

, there is a natural one-to-one correspondence 1r'(M,g;p) 4-? it

where n = dim H1(M, LY (pF-1))

- 1 ; here

H1(M, 6t (pg-1) )

r(M, (9 (K(p2g)) = r(M, 6L (0_2g)) , since K = 1 , and from c(cp 2g)= and the Riemann-Roch theorem, it follows that

= c(g)- 2c(cp)

r(M, Q< (9-2 9)) = 0 except when

cp2

=g .

When

cp2

= 9 , of course

r(M, (Q ((P-2g)) = r(M, C) = C . Therefore '[("(M, 9; (p) _ 0 when-

ever

g

ever

= g

, while '1("(M,g;cp) consists of a single element when.

Thus, over a compact Riemann surface M of genus

g = 1 , (28)

(MI9) = 1,('' (M, 9) u u'0(M, 9) U

2U )." (M, g; (P) (P =g

where

X0

{0 E Ir (M,9)1div 0

[c(g2) +1]_1}

and

1( " (M,

g;cp)

has a single element whenever

p_9

.

§6.

Flat vector bundles.

(a)

Over an arbitrary Riemann surface M , the constant sheaf

GL(m,C) X M can be viewed as the subsheaf of the sheaf

b C(m, (¢)

of germs of complex analytic mappings from M into the complex Lie group

GL(m,C) , consisting of germs of locally constant mappings.

The inclusion mapping of sheaves

is GL(m,C) -

(m, Q )

induces a mapping.of the cohomology sets

i*: H1(M,GL(m,C)) -? H1(M, 6x (m, (R ))

(1)

The elements of the set

H1(M,GL(m,C))

.

will be called flat complex

vector bundles of rank m over the Riemann surface M . analytic vector bundle which lies in the image of to possess a flat representative;

0

i* will be said

for such a complex analytic vec-

tor bundle can be defined by a cocycle in which all of the functions

A complex

(0

)

alp

a

Z1(YL , 'h x(m, (a))

are constants.

The set of all

flat complex vector bundles mapping onto a given complex analytic vector bundle

will be called the set of

0 e H1(M, b x (m, 61 ))

flat representatives of that bundle

0 .

The aim of the present

chapter is an investigation of flat complex vector bundles, and of their relationships with complex analytic vector bundle The .case of flat complex line bundles (of flat vector

bundles of rank 1, in other words) was discussed in last year's lectures, in connection with the classification of complex analytic line bundles; recall in particular §8.

In that discussion, one

began with the exact sequence of sheaves of abelian groups

0--C i>

(2)

where

i

l'0-> 0,

is again the inclusion mapping and

defined by M(f) =

2RY

d2

is the mapping

d log f , for any germ f e & * .

The

exact cohomology sequence associated to (2) contains the segment

(3)

H1(M,C) -i- xl(M, d.*)

r(M, 91'°)

and by the Serre duality theorem,

= Hl(M,.l'O) c

H1(M, o1'0)

d2*(g)

of

As noted in last year's lectures (Lemma 19), the image a complex line bundle class of

g

and only if

is essentially the Chern

9 e H (M, 8*)

; hence a line bundle

has a flat representative if

g

c(g) = 0 , the set of all

Further, if

c(g) = 0 .

S*r(M,

flat representatives of

is the coset

g

g +

the space of cosets is the Picard variety of M .

.lo) , and The dis-

cussion of flat complex vector bundles of arbitrary rank is roughly parallel to the discussion of flat complex line bundles, as reviewed above; the non-abelian character of general vector bundles is rather a complicating factor, however. see for instance A. Well,

(For other treatments of this topic,

"Generalization des fonctions abeliennes,"

J. Math. Pores Appl. 17(1938), 1+7-87; and M. F. Atiyah,

analytic connections in fibre bundles,"

"Complex

Trans. Amer. Math. Soc. 85

(1957), 181-207.)

One preliminary construction should first be discussed. Suppose that

µ: GL(m,C) T GL(n,C)

sentation; that is,

µ

is a complex analytic mapping between these

two complex manifolds, and µ structures.

is a complex analytic repre-

is a homomorphism of the group

It is obvious that

-97-

µ

induces a sheaf homomorphism

:

.t

0- ,

-,

--,.

,h n.tn, 61 ) ; and this in turn leads to a map-

ping between cohomology sets, of the form µ: Hl(M,

,

(m, C )) -- H1(M, h,k (n,

Thus to any complex analytic vector bundle

67 ))

0 e H1(M, )J :( (m, -))

there corresponds a complex analytic vector bundle µO a H1(M, 1) X-(n, al ))

This construction will arise in partic-

.

ular for the adjoint representation Ad: GL(m,C) --? GL(m ,C) which is defined as follows.

For any matrix A e GL(m,C) ,

Ad(A)

is the linear transformation on the complex vector space

2

CMXi = Cm

of m X m complex matrices, which associates to a

matrix X e d "m the matrix

e e Xm defined by AXA-1

Thus there arises the cohomology mapping Ad: H1(M, I t°(m, B )) ---. H1(M, fix (M2, Lit ))

Now, to obtain the analogue of the exact sequence (2) for treating vector bundles of arbitrary rank, consider the differ-

ential operator D defined as follows.

F E GL(m, 6 U) over an open subset

For any matrix

U C M , let DF = F 1dF ;

thus DF is an m X m matrix of complex analytic differential forms of type (1,0) over the set U , which will be written

DF a (L9 U11 0) )m Xm

=

(110) ®(9-U Xm

leads to a sheaf mapping =

LI (1, 0) ®

61 m X m

sub sheaf GL(m,(

D:

.

The

mapping D then

(m, CD) T (Qt (1, 0))M Xm

; and the kernel of the mapping D is the

)Ci

. (m,

CD

) .

Over any Riemarn surface M , there is a twisted

Lemma 17.

exact sequence of sheaves of groups of the form

(4+)

GL(m,C) i ?

O-

the inclusion mapping

\o;

(1,0) ®Smxm

D

(m,

is a homomorphism of sheaves of groups,

i

while the sheaf mapping D satisfies

D(FG) = Ad(G

(5)

Remark:

DG .

To say that (1+) is a twisted exact sequence of

sheaves of (non-abelian) groups just means that sheaf mappings, with

i

a homomorphism and D

i

and D

are

satisfying a rela-

tion of the form (5), and that at each stage the kernel from the right is the image from the left, as usual. Proof.

It is clear that

i

precisely the subsheaf of .h..(m, (-) F e j) a (m, (9

A

e

such that DF = 0 .

)

is an isomorphism, with image

consisting of germs

Moreover, if

LO (l,0) ® d_rXm = ( dL(l,0))mXm , there is a germ

p F e d m Xm

p

p

satisfying the differential equation dF = FA and

the initial condition

F(p) = I ; but then

DF = A , so the mapping D F,G C 1) .(m, 6-)

,

p

is onto.

it follows that

F e

,a 'k (m,

and

Finally, whenever

D(FG) = (FG)-1d(FG) _

= G 1F-1(dFG + FdG) = G DFG + DG = Ad(G) DF + DG as desired, to conclude the proof. Theorem 15.

Let M be an arbitrary Riemann surface.

every complex analytic vector bundle is associated a cohomology class

0 e H1(M, b t (m, 9 ))

To

there

A a HZ(M, 6 (K ®Ad')) , where

0

is the canonical bundle; and

K E H'`(M, &.*)

sentative if and only if D*) = 0 Suppose that

Proof.

.

(0ap) a Zl(Z/L, h ( (m, 6L ))

cocycle representing the vector bundle covering

_ CU.)

Dap a GL(m,

(1U

has a flat repre-

0 ,

is a

in terms of an open

of the Riemann surface M ; thus the matrices

) a n uP

satisfy

0C413(p)-0P7 (p) =

0M(p) whenever

Applying the differential operator D , it

p e Ua n UP n Uy .

follows from (5) that

DO07 (p) = Ad(VP7(p)-1).Doap(p) +DOP,(p)

but this is just the condition that the elements cocycle

=

(1'0)

Doa,3 e Zl( WL o d- (K (9 AdO) ) ® L9- (Add)

cocycle formalism.)

,

p e Ua n U. n U?, ;

whenever

DOC43

form a one-

since ® (K ® AdO) =

(See the discussion in Appendix 1, for the

The cocycle

is equivalent to a flat

(Oa4)

cocycle precisely when there is a zero-cochain

(Fa) a CO( Zf 1, , hence such that

.b0 (m, 0 ) ) such that D'

- 0

O

= FaCC4,3F-1

is flat,

Applying (5) again, this condition is

.

just that 0 =

D(Faaepl)

= Ad(Fpoc

_1

D(F-l)

DOa - DFP 3J

=

Putting X a = DFa , this condition can be rewritten

a

Ad(c ) .a ;

but this is equivalent to the assertion that the cocycle DOS is the coboundary of the one-cochain

(Xa) E CO( Za , 61 (K

) AdO)) ,

where

Xa = DFa for some functions

matrices

X

a

can be written

X

a

Fa a GL(m, 0-Ua)

Since any

= DFa , perhaps after passing to a

refinement of the covering v'(,, it follows that

representative precisely when

.

DOa13 - 0

in

0

has a flat

Hl(M, 0- (K ® Ad 0))

Finally, it should be demonstrated that the construction is canonical, in the sense that the cohomology class associated to the bundle 0

is independent of the choice of representative cocycle

(Oa$)

of the bundle; this is a straightforward but uninteresting calculation, which will be left to the reader.

That concludes the proof

of the theorem.

Recalling the notation introduced earlier (see p.69), let

(b)

0* denote the dual vector bundle to a vector bundle

is defined by the cocycle (t0) , whenever defining

0

.

(0C413)

0 ; thus

0*

is a cocycle

It is then evident that Ad 0 is the dual vector

bundle to the vector bundle Ad 0

Now from the Serre duality

.

theorem for complex analytic vector bundles (Theorem ]2), it follows that the cohomology groups

HH(M, 01 (K (&Ad 0))

and

HO(M, 6 (Ad 0*))

are canonically dual to one another; thus every

cohomology class

a

determines a linear functional (also denoted

by a ) on the vector space

r(M, 6 (Ad o*)) , and

only if this associated linear functional is zero.

a = 0

if and

It is of in-

terest to have a more explicit form for this linear functional associated to a cohomology class

a

.

Recall that the Serre dual-

ity can be described explicitly as follows.

be an

Let lit = [Ua)

open covering of the Riemann surface M so that the vector bundle 0

is described by acocycle

(0ao) a Z1( UL, 6 X.(m, (V ))

;

and.

let

(ca43 ) a Z1( 11L, o' (K G) Ad 0)) = Zl(VL

cycle representing the cohomology class

61"0(Ad 0))

o

be a ..-

The elements

c

can

a a-16

be viewed as column vectors of holomorphic differential forms of type

that cay =

over Ua fl U fl Uy .

c07

(%a) a

E

1'0(Ad 0)) = CO(V/L,

cochain with coboundary

Gap

ential forms of type

7%Ce

(l,l)

X

a be vectors of Ua such

in the various sets

(1,0)

= T.0 - Ad(O-1) 2à in u,, fl Up acap = 0 ; thus

morphic,

Let

(K ®Ad 0)) be a zero-

(cap) ; that is, let

e differential forms of type

that

Ua n U. , such

associated to the various intersections

(1,0)

Since

cap

are column vectors of

such that

are holodiffer-

)app in

aXa = Ad(O coo

ing T. as column vectors of holomorphic functions in the various sets

Ua such that Ta = Ad(O**p)Tp

is

Ua fl Up ,

are scalar differential forms of type various sets

Ua , and that

c

(l,l)

in the

Then

in Ua fl Up .

tTa-aXa =

the linear functional associated to

it follows that

is given explicitly by

c(T) = 21-1 IM tTa. Xa

Actually, it is more convenient to view the various elements

and Ta as 2

length m

la

m X m matrices rather than as column vectors of , since the adjoint representation is then easier to

describe; and with this convention in mind, the linear functional associated to ment

(6)

c e H1(M, a (K ® Ad 0))

T e r(M, 19 (Ad 0*))

takes the value on the ele-

given explicitly by

c(T) = 2.-11 IM tr(tTa'axa)

where

tr

denotes the trace of a matrix.

The cohomology class

a = D** a H1(M, 4 (K ® Ad 0))

is of

course of particular interest, and it can be described explicitly as a linear functional on

matrices Fa a GL(m, I' u

a

r(M, 01 (Ad 0*))

)

as follows.

such that oao = FQF

P1

Select

over Ua fl U

this is always possible, since all meromorphic vector bundles are trivial (as pointed out on p.1+3).

covering v(.= (Ua) matrices

Fa

Furthermore, assume that the

is so chosen that the singularities of the

(the poles of any entry or the points where the ma-

trix has zero determinant) lie in disjoint open sets not lie in any intersection

u, fl Ua .

Ui , and do

Now from (5) it follows

that

= DOao = D(FaP-p1)

Cr

=

Ad(FDFa + D(F1)

= Ad(O_l)Ad(Fa)DFa

- Ad(F)DF

thus

Na = Ad(Fa)DFa are m X m matrices of meromorphic differential forms of type (1,0)

Ua , such that

in the sets

For each set

Ua select a

Coo

a

_

- Ad(O

)Xa

function ra such that

each intersection Ua fl U. , that

rj

in

Uaf u

ra = 1

on

vanishes identically in an

open neighborhood of each singularity of the meromorphic differ-

ti ential form

Xi

ti in

Uj , and that

ra = 1

if

Na

is non-singular

in Ua ; this is always possible, in view of the special form of

the covering Ul. in each set

Then the differential forms

%a = rd a are

Coo

Ua , and also satisfy the relation aa =% - Ad(O 1)%a

in

Ua fl U. .

a = D 0

Thus by (6) the linear functional associated to

is given by

A (T) for any element

= ,1-,1 fM

T = (Ta) a r(M, 19 (Ad 0*))

holomorphic except for those sets

U.

Now since

.

Xa

is

containing the singularities

ti

of

Xi , and the elements

Ta are all holomorphic, this expression

can be rewritten 27MD *(T) = E 1U atr(tTj Xj) = E fu dtr(tT3 S

J

S

J

J)

= E 1au tr(tTj"j) = E f3u tr(tTj J

3

S

S

= E 27fi *tr(

T,ti%j)

,

S

where X denotes the total residue of the meromorphic differential form at all its singularities; therefore the linear functional associated to the cohomology class

D*0

has the value

tr(tTa Ad(Fa)-DFa)

D *(T) _ -

(7) - Z tr(tTa dFa a1) for any section

T = (Ta) a r(M, 9- (Ad 0*))

,

.

With this observa-

tion, Theorem 15 can be given the following restatement. Corollary 1.

It 0 e H1(M, h - (m, C. )'3-be a complex

analytic vector bundle over the compact Riemann surface M ; let

(0a0) a Z1( lit , h 'f (m, ( ) ) be a representative cocycle for

and let (Fa) a CC( YL

,

that

Ua fl U

Fa = Oa.,F,

if and only if

in

9J

.

0

(m, bi )) be meromorphic matrices such Then

0

has flat representatives

rltr(tTa dFa Fa) = 0

for all sections

T = (Ta) a r(M, 0. (Ad 0*)) , where t. denotes

the total residue of the differential form tr(tTa dFa Fat) e

The condition for a bundle to have flat representatives, as restated in Corollary 1, is actually quite useful, after a few preliminary observations about the space any complex analytic vector bundle

0 , an endomorphism of

defined to be a sheaf homomorphism

T: & (0) T d (0)

A = {Ua)

0

(0)

to each set

sheaf,

Let

.

Ta:

such

Then the restriction of the sheaf

.

Ua has an isomorphic representation as a free

(.(O)IUa = e.mIUU ; so that an endomorphism T

homomorphisms

is

has a representative cocycle

(0ap) a Zl( Vt , fj a°` (m, S )) .

0

be an open covering of the Riemann surface M ,

that the vector bundle

For

r(M, O- (Ad 0*))'.

6LmIUU of free sheaves and these

(} MIUa

are described by matrices

determines

Ta a r(Ua, (¢

m Xm)

.

It is readily seen

that these matrices satisfy the conditions

(8)

Ta(p)oao(p) _ 0a13 (p)TP(p)

and conversely, any set of matrices

whenever p e Ua fl UP ;

(Ta) e

C0( A ,

6 m Xm)

fying (8) determines an endomorphism of the vector bundle sented by the Gocycle

morphisms of

(0a3 )

.

0

repre-

It is clear that the set of endo-

0 , which will be denoted by End 0 , has the struc-

ture of an algebra over the complex numbers; if T = (Ta)

satis-

are endomorphisms, and

aS + bT e End 0

a,b

S = (Sa)

and

are complex numbers, let

be the endomorphism represented by the matrices

\aDa + ova) , and let by the matrices

ST E End 0

(SaTa)

R = (Ra) a r(M,

Now consider any section-

.

))

(Q- (Ad

be the endomorphism represented

Ra can be viewed as

; the elements

Ua , and

holomorphic m X m matrices defined over the sets matrices satisfy

Ra = Ad(O*)-RP = 0* RP (0* )-l

P

013

in UaflU . so that

Note then that Ta

T = (Ta) a End 0 ; it is thus clear that there is a natural

one-to-one correspondence between the sets End 0

.

a$

TaVT 0-1 P ok3 ,

satisfies

t'RCC

these

= t0-l R t0

r(M, a (Ad 0*))

and

With this observation, Corollary 1 to Theorem 15 can be

restated as follows. Corollary 2.

Let

be a complex

0 e H1(M, bJ X (m, (9 ))

analytic vector bundle over the compact Riemann surface M ; let (0C413 ) a z-( ZQ

and let that

,

b C (m, a

(Fa) a CC(1TL

Fa = 0a13 FP

,

be a representative cocycle for

))

ht (m, )k ))

in Ua fl UP .

0

be meromorphic matrices such 0

Then

has a flat representative

if and only if T, tr(Ta dFa Fa1) = 0

for all endomorphisms

T = (Ta) a End 0

.

Some further consequences of Theorem 15 now follow rather readily, upon looking more closely at the algeebra

morphisms of the bundle Corollary 3.

End 0

of endo-

0

0i a R1(M, N x (mi, Ol ))

let

,

i = 1,...,r,

be complex analytic vector bundles over a compact Riemann surface M , and let

0 _ 0l ®. ..® Or

.

if and only if all the bundles

Then

0

has flat representatives

0 i have flat representatives.

In terms of a suitable coordinate covering VZ.= (Ua)

Proof. 0

the bundle

where

can be represented by a cocycle

are cocycles representing

01013

Fia a CC(111.,

1

k (mi,

that the matrices if

T = (Ta)

is any endomorphism of the bundle blocks

r2

corresponding to the decomposition of be an endomorphism of trary endomorphisms of

will be an endomorphism of

tr(Ta dFa

Fat)

0 , the matrices

0 ; and Tii = (Tiia) Tii

T = T11 ®...ED Trr

Oi 0

.

Now

FP .

Ta = Tail, 1 0

The sheaves

FP(X)

,

are

F p , hence are fine sheaves; so letting

be the subsheaf which is the kernel of

d ,

(the

subsheaf of closed differential forms), it follows that

(10)

H1(M, 3 (x)) = r(M, a l(x))/dr(M, f °(X))

,

2(x))/ar(M, f1(x))

,

H2(M, 3(x)) = r(M,

{

Hq(M, a (x)) = 0

L

c

whenever q> 3.

(See Theorem 3 of last year's lectures for the proof of these assertions; the isomorphisms (10) will be called the deRham isomorphisms, as a convenient abbreviation.) in

r(M, a

"(X))

Given any differential form

, the cohomology class associated to that differ-

ential form by means of the isomorphism (10) is called the period class of the differential form.

1' his aeham isomorphism is particularly useful in describing

a duality for cohomology with coefficients in the sheaves 3 (X) If

1(M,GL(n,C)) is a flat vector bundle over the Riemann sur-

X e 1

face M , let

if X 3t*

be the complex conjugate of its dual bundle; so

3t

is defined by a one-cocycle

(XU$) a Z1(Vt ,GL(n,C)) , then

is the flat vector bundle defined by the one-cocycle

() e Z1(Ja,GL(n,C)) .

By the deRham isomorphism (10), any

cohomology class A e Hp(M, differential form

(X))

can be represented by a closed

cp a r(M, b p(X)) ; this representing differential

form is not unique, but the most general such is given by for arbitrary differential forms

that actually

cp'

a r(M, F p-1(X))

.

cp + dept

Recall

is given by vectors of differential forms

cp

in

cpa

the various open sets Ua of a suitable open covering of the surface, such that vectors

cpa = Xa$ -

cps

in each intersection Ua n u, ; the

are viewed as column vectors, as usual.

cpa

In a similar

H2-p

manner,, any cohomolo gy class

B e

(M, 51

( 3t* ))

can be repre-

sented by a closed differential form * e r(M. F 2-p(3Z*)) , or more generally by the closed differential forms fir'

e r(M. E

1-p(3c*))

.

* + d'r'

The exterior product

,.cp

for any

is then a

global scalar differential form of degree 2 on the Riemann surface

M ; for in Ua fl U *P

cp,

.

it is evident that

ira

cpa =

i

X1 'X

q)

UP P

_

Upon choosing different representative differential

forms for the same cohomology classes, the exterior product form is

modified to become

t(T +

(cp + dept) _

the class of the form r(M, e 2)/dr(M, F 1)

e p + d( j ,. ep' + F ' _q) + *' dept) ; thus

cp in the quotient space depends only on the original cohomology classes.

_136-

Recall from the standard deRham theorems that

r(M, F 2)/dr(M, F 1) = H22 (M, C) ; when the surface m is compact, C and the deRham correspondence reduces to integrating

R2(M,C)

the differential forms in

over the surface M .

r(M, E 2)

In

summary then, there is a bilinear Hermitian mapping Hp (M'

g (X) ) ® Hrp(M, 9

C

which associates to the cohomology classes A and

B

the complex

constant

= fm W..cp, over any compact Riemann surface

The duality theorem is the

M.

assertion that this is a dual pairing.

Theorem 20. It X e H1(M,GL(n,C )) bundle over a compact Riemann surface M . Hp(M, a (X))

and

H2-p(M, 3 (e))

The cohomology spaces

p = 0,1,2 .

The proof is just a straightforward adaption of the

proof of the Serre duality theorem. r(M, t p(X))

.

are canonically dual to one

another, under the pairing (11), for Proof.

be a flat vector

introduce the norms

On the vector spaces pn , as defined on page 68.

These norms determine the structure of a topological vector space, actually a Frechet space, on

r(M, EP(X) ; and the dual vector

*))

space is

r(M, x 2-p(

, where

x is the sheaf of germs of

distributions on the compact Riemann surface M , 12).

(12)

(recalling lemma

Now consider the sequence of vector spaces

r(M, e p-1(X)) -a-- r(M, f p(X)) a-> r(M, F P+l(x))

d

2he linear mappings

are obviously continuous in the topologies

introduced on these spaces, and the image dr(M,

P-1(X)) C r(M, f P(X))

is a closed linear subspace; (the

latter follows directly from the fact that the quotient space

ker dim d = HP(M, 3 (X)) last year's lectures).

is finite dimensional, as on page 95 of

Thus in the dual of the exact sequence

(22),

namely

(13)

r(M, x 3-P(X*)) 4_ r(M, x 2-P(R*)) 4 d* r(M, '( 1-P(e)) ,

it follows readily that

ker dim d = HP(M, 9 (X))

ker d*/im d* is the dual vector space to .

There is an exact sequence of sheaves

of germs of distributions over M of the form

0 -- ' (X*) -- x0(7*) -- 3(1(32*) -T

X2('X*)

--> 0 ,

(see the following Lemma 20); the corresponding exact sequence of sections contains the segment (13), since it follows readily from the definition of the derivative of a distribution that the operator

d*

of (13) is just the exterior derivative on distributions.

The

sheaves of germs of distributions are fine sheaves, hence

* -P(Ms

which shows that

I

3 (X ) )'

ti =

r(M, X c -P(X*))

dr(M, x -P(X ) )

-P(M, 3. (R*))

ker d in

is the dual space of HP(M,3 (X)).

It is an easy matter, which will be left to the readers, to verify that this duality is that given by (11); the proof is then completed, except for the following result.

-138-

Lemma 20.

Over any Riemann surface M there is an exact

sequence of sheaves

x 0 d> }( 1 d

0 -> C where

2

0

is the exterior derivative.

d

Proof. It U be a product neighborhood of the origin in the complex plane, so that U = I X I

for some open interval

I .

It is rather apparent that it suffices to prove the following three assertions; the notation and terminology are as in §6 of last year's lectures. (i)

then T

If

T E X

U

is a distribution such that

aTaT_ 'E

ZF

0

is a constant.

First select an auxiliary function h E o C fh(t)dt = 1 .

For any function

such that

f E o C U set

f(x,y) = fl(x,y) + h(x) f f(s,y)ds

(14)

The function f1

is also

C"

in U , has compact support in U ,

and moreover satisfies f fl(x,y)dx = 0 ; thus there is a e func-

tion g c o C U such that

fl - 6g/6x , and therefore

T(fl) = T(6g/6x) _ -(aT/6x)(g) = 0 .

Applying the same idea to the

integrand in (14), write f(x,y) = fl(x,y) + h(x) f f2(s,y)ds + h(x)h(y) ff f(s,t)dsdt

where

T(f2) = 0 .

Then T(f) = T(h(x)h(y)ff f(a,t)dsdt)

= c f f f(s,t)dsdt

where

c = T(h(x)h(y)) ; but this shows that T

distribution

c ,

is the constant

as desired.

For any distribution T e XU there is a distribution

(ii)

such that

S e a(U

T = as/ax

Given any function

U , consider again the decom-

f e o

position (14+) as in part (a) above.

S(f) yields a distribution S e XU . for a function (aS/ax)(g)

aSjax = T

g e o C U , then

Setting

T(fo fl(s,y)ds) Note that whenever

f = 6g/ax

f = fl ; so that

S(age) = T(fo ay(s,y)/as ds) = T(g) ,

and.

as desired.

it should be remarked in passing that (14) yields immediately

a description of the most general such distribution S

.

For in-

is any distribution

stance if

T = 0 , or in other words if

S

such that

?S/ax = 0 , then applying S

to (14) it follows that

S(f) = S(fl) + S(h(x)f f(s,y)ds) = R1(f f(s,y)ds) , where R1 is

a distribution in % (iii)

If

.

Tx y e XU are distributions such that aTay =

= aTy/ax , there is a distribution S e X U

such that x = aS/ax

and y = By part (b) above there will be a distribution S1 e x U such that

aSl/ax = T

; indeed, as remarked above, the most general

such will be of the form S(f) = S1(f) + R1(f f(s,y)ds)

for any distribution R1 e x 2 . that

aS/ayy = y .

The problem is to choose

R1 so

Note that

aT

AS

AS

AS

X)

=0

so that aS

(y - - 1)(f) = 1 (f f(s,Y)ds) for some distribution R2 e % 2 ; thus the condition to be imposed on the distribution Rl

0 = (Ty -

as

is that for every function

as

)(f) = (y - - 1)(f)

f e o C U

IR, - -cry( f f(s,Y)ds)

IR,

= R2(f f(s,Y)ds) - -NY( f f(s,Y)ds) , or just that

jayy = R2

in X I .

$r part (b) above there always

exists such a distribution R1 a x2 , and the proof is thereby concluded.

Remarks. A flat vector bundle, and its corresponding sheaf of flat sections, are clearly of a more purely topological than analytical-topological nature.

One would expect that there would

exist a proof of Theorem 20 of a purely topological sort, as indeed there does.

(See Glen E. Rredon, Sheaf Theory (McGraw-Hill, 1967);

the discussion there is restricted to flat line bundles, but the extension should be straightforward.)

However, since the analytical

machinery is at hand, and has been used similarly before, it seemed more reasonable to prove the theorem by that means than to digress further on general sheaf theory.

As a first application, the duality theorem together with Theorem 18 permit the easy calculation of the cohomology group H2(M, 3 (X))

.

If M is a compact Riemann surface and

X is a

flat vector bundle over M , it follows Prom Theorem 20 that H2(M, 5 (X)) = HO(M, 9. (f)) ; and therefore, applying Theorem 18,

(15)

dim

C

H2(M, % (X)) = r

where

is the largest

r

integer such that the trivial bundle of rank contained as a subbundle of

a 3Z

r

is

, for any compact

Riemann surface M .

A more interesting application of the duality theorem is to p =11 ; the assertion then is that for any flat vector

the case bundle

X e H1(M,GL(n,C))

cohomology groups

over a compact Riemann surface M , the

and H '(M, 9 (e))

H'(M, .4 (X))

dual to one another.

are canonically

Using the isomorphism of Theorem 19, this

duality takes the form of a dual pairing

(16)

Hl(,Cl(M), X)

1

1(M),

and it is of interest to see the explicit form that this pairing takes.

Note in particular that when the representation

tary, so that

is uni-

X = X* , this dual pairing becomes a nonsingular

Hermitian-bilinear form on the complex vector space the bundle

X

H1(al(M),X) ;

X is called a unitary flat vector bundle in this case,

since it can be defined by a cocycle consisting entirely of unitary matrices.

(Real]. from last year's lectures that for the

special case that

X is the trivial line bundle, this is just the

intersection matrix of the surface.) tween the cohomology group r(M,

C 1(X)/dr(M, f 0(X))

The direct relationship be-

Hl(3c1(M),X)

and the deRham group

can be handled most easily by intro-

ducing the universal covering surface of M , and transferring the The explicit

bundles and differential forms to that covering space.

cohomology structure of the surface must eventually be used, of course.

(d)

It M be the universal covering space of the Riemann sur-

face M , and

f: M_7--.> M be the covering mapping.

vector bundle

X e H1(M,GL(n,C))

of germs of flat sections image sheaf

For any flat

over M , with associated sheaf

3'(X)

,

it is clear that the inverse

f-'( ,/ (X)) , as defined in §3, is a flat sheaf of

rank n over the covering surface M ; hence f(X)) _ .3 (X) for some flat vector bundle

'

a Hd l'(i,GL(n,C))

bundle X is the trivial bundle, since

sheaf 9 (X)

.

3c1(M) = 0 ; so that the

3

is the trivial sheaf, that is,

Cn X 3

Recall that the fundamental group gl(M) as a group of transformations of the space

mapping

Actually the

31

can be viewed

commuting with the

f: i( - M ; to avoid confusion, this representation Select a base point

should be made quite explicit.

consider the fundamental group as the group

classes of closed paths in M based at the composition

7172 e3t1(M,po)

traversing first the path 71

p

e M , and

of homotopy

3c1(M,po)

po ; if

0

71i72 a Icl(MPo) ,

is the closed path obtained by

and then the path 72

.

(The fact

that this group is isomorphic to the fundamental group defined by

means of open coverings of M , as in last years lectures, is

left as an exercise to the reader.) 3(

The universal covering space

can be defined as the space of homotopy classes (with fixed end

points) of paths in M based at the point p

0

; the mapping

f: Nf --> M

is that which assigns to any path its end point.

t e ! and

y e al(M,po) , then the path

Tic

If

, obtained by tra-

versing first the closed path 7 then the path a , is also an element of

Nf ; and the mapping a - ? yc

a group of transformations on the space

exhibits

Nf

3cl(M,po)

The points of

.

as

Nf

will generally be denoted by p or z ; and the mapping takes the point z e RI

7: r --> r

to the point denoted by 7 - 'Z'

It is clear that f(7 -'Z) = f(z) for all z e

It PO

.

path at p

0

y e mcl(M,po)

be the point of A corresponding to the trivial

, that is to say, to the homotopy class of null-

homotopic closed paths at of the covering space

Nf

p .

0

; this will be called the base point

Any closed path

7 e 9 l(M,po)

if pl e

that the end point of the path y is just is another point such that

f(pl) = po , there is an element

a e ICl(M,po) such that Pi = a unique path beginning at covering mapping

N

a y . po = a7a

Pi

a

f , and the end point of

is the

y C Nf

7 under the

a 7 is

-1 N

Pi

covering mapping

to sheaf

po ; the path

and covering the path

Since the transformations

,

is

beginning at po ; and it is clear

covered by a unique path y C r

= f-l( -(X))

and all

commute with the

y:

f: Nf --> M for all

7 e 3cl(M,po) , and

it/` is apparent that these transformations

y:

('k) .

connected components of the set fl(U) C Nf

(39) _

7

extend

For if (1i) are the for a contractible open

subset U C M , then by definition 3 (v)IV _ 51 (X)IUi ; the transformation of

associated to any

3(

7 e 3cl(M,po)

merely permutes

the sets ?i j, so can be extended to be the same permutation of the

restrictions

IVi

.

In terms of the isomorphism

9-(k)

' Cu X

the automorphism of sheaves associated to the group element

7 e 'cl(M'po) is a mapping y: Cu X 3( ---> Cu X 3( which must be of

the form

for some matrix

and z e 1 phism

v,y Z)

y ' (V,Z)

(17)

.

X(y) a GL(n,C) , where V e Cu

The mapping

y --? X(y)

is a column vector

is clearly a group homomor-

X

X: 3cl(M,po) --? GL(n,C) ; note that the representation

is only determined up to an inner automorphism of the isomorphism

a'(X) = Cu

Lemma 21.

is of course not unique.

X 3(

The homomorphism X is the characteristic repre-

sentation of the flat vector bundle Proof.

GL(n,C) , since

X .

This is a straightforward matter of examining more

explicitly the above construction.

ing of M such that the sets

Let M= (Ua)

be an open cover-

U. and all their intersections are

contractible; thus 1Jt. is a Leray covering for flat sheaves, and Icl(M,po) _ gl(V.,U0)

for fixed base points

set U. a lJj, the inverse image

f 1(Ua)

po a U0

.

For any

in the covering space

is the union of countably many open components tai , and these

components form an open covering K= (Uai)

of M ;

let

Uoi

be

0

that component of f(U0) containing the base point p0

of the

covering.

The flat vector bundle

cycle

) a Z1(VL,GL(n,C)) ; this corresponds to a choice of

(x

X can be represented by a

co-

an isomorphism 9-(X)IUa = On X Ua for each open set Ua . ing for the induced sheaf

the corresponding isomorphism

,7 (X)

X is represented

(3t)IVai = On X Uai , it is clear that the bundle

N

(Iai'PP j ) e Z1(1/

by the cocycle

whenever Uai fl U,j

9$

, GL(n, C)) , where

such that

are constant matrices Cai e GL(n,C)

exhibit explicitly the reduction of the sheaf

Col

9' _ (UaO'u

= I ; all the matrices

'.. .,U

Cai

are then uniquely

there is a unique chain

) e 91( V.,Uo)

in

covering 7

under the mapping f

closed, but

f(V

Vi. based at Ua0i0 = U

0i0

U0 ; and the transformation of M associ-

i

cycle

On , when the bundle

y

X

7

is viewed as defined by the coto reduce the

Cad

to the trivial sheaf, the automorphism associated to

has on the factor

However, since

Ca

Cn the form

X(y) = Ca

j j

Ca

qiq 0i0

C i

j+1 j+1

j j

j+1 j+1

=

Ca

qiq

Ua i Xa.a j j

it follows that (18)

Uagiq

to

a0i0

is the identity mapping on

(Xai,0j) ; so applying the isomorphisms

sheaf '(x)

N

N

is the covering translation taking U

The sheaf automorphism associated to the factor

and

The chain y need not be

-

q q

7

to the trivial

9"(X)

Now for any closed chain

y = (a0i0 ,Ua1i11...,Vagiq )

ated to

ai Pi Pi = I

There is no loss of generality in suppos-

0

determined.

Cai94

the isomorphisms Cia: On X Uai --> Cn xUai

whenever Uai fl UP j # 9$ ;

ing that

= Xa$

Xais 13j

Since the bundle X is trivial,.there

.

product sheaf On X M .

Select-

y) = Cagiq = XaOalXala2 ... q-1 q

j j+l

it

but this is just the characteristic representation of the bundle X , recalling the construction in last year's lectures, and the proof is thereby concluded.

Now consider a cohomology class A e RI(M, ?'(X))

Under

.

the deRham isomorphism (10) this cohomology class can be represented

by a closed differential form p e r(M, e c 1(X)) ; and p lifts to a closed differential form f p e r(M, mapping associated to the covering

E1(X))

f: M

M .

under the induced

The form f p

is clearly invariant under the automorphism

lation

y e n1(M,po)

isomorphism

associated to a covering trans-

1(X))

y: r(i, C 1(X))

Since the bundle X is trivial, under the

.

1)n the differential form f p

C 1(X) _ (

viewed as an n-tuple

9)

of differential forms on the manifold M e T(M,(

in the ordinary sense, that is, as an element

the form

9)

can be

e 1)n,

is clearly closed, and satisfies

(7 .,Z) = X(7) . 45R)

(19)

for every covering translation 7 e nl(M,po)

.

The cohomology class

A can also be represented by a cohomology class A e RI(al(M),X) under the isomorphism of Theorem 19; the cohomology class A and the differential form p are related quite simply, as follows.

Letting PO e M be the base point of the

Lemma 22.

covering space M , and for each loop

be the path in M covering 7

)

letting y

and based at the point po , the

cohomology class A e RI(n1(M),X) (Ay) a Zl(n1(M),

7 e nl(M,po)

is represented by the cocycle

, where

A'Y = _X(Y)_ltif ' y

-1l+7-

Select an open covering U= {Ua)

Proof.

of the surface

M as in'the proof of Lemma 21, and continue with the constructions and notation as in that proof.

(ACO) e Z1(D , 9 (X))

Let

be a

cocycle representing the cohomology class A , and select e

Ua such that

vector-valued functions Fa in the various sets

Aap = Fo - Xa'a in Ua fl UP ; the differential form

cp

repre-

senting A under the deRham isomorphism (10) can be taken to be

in Ua .

a = dFa

X to the trivial bundle,

Reducing the bundle

as in the proof of Lemma 21, the differential form

95

ai .

given by Tai = Caicpa = %aidFa in each set

is clearly

For any closed

path 7 e nl(M,po) , the lifted path y can be covered by a chain

i i all 0o

(Zfa

qq

of the covering ilL , where

o0

0

f(y) = (Ua 'U s...IUa ) e nl(?j(, o) . Then the path 7 can o al q for j= 0,1,...,q, be decomposed into non-overlapping segments 7 and

N such that 71

Ua

lies entirely within the set

; the end points 3

of the segment 71

will be denoted by p1

and

pJ+l , so that

PO and pq+l are the end points of the full. path IV

lu

Pi a Uaj-1 i

for

fl Ua i

j-1

i

1=o

7

Za

i

J 1

E fN Ca i

J=o

o Cajij[Fai(pi+l)

_

, and

Nov

.

i

f,,, P= E fw 7

j = 1,2,...,q

7

71

-

dF

i i

J

Fa,(Pi)J

for

Pj = f(ps) ,

j

= agiq F aq(Pq+l)

-

oio F

+ E (2'a Fa (Pj) J=l j-1 J-1 j-1

o(po) +

Fa (Pi)] j

Here of course

Fa (p q+1) = Fa (p0) , and from the definition of q

0

these functions it follows that Faj-1 (P j) = Xa

j-1

a (Fa (Pj) - Aaj j

j

-1

a

.

)

j

Furthermore, as in the proof of Lemma 21, the constants Ca i

are

Sj so chosen that Ca

i

0 0

= I

and j-1ij-1

Xa

=

j-1ij-1,a ji

j

2'a

Xa

i

j-1 j-1

and Ca i

a = Ca i

= X(y)

.

thus

;

1

j j

j-1 j

= Xa

x

.. Xa

oai ai`

j j

a

j-1 j

It then follows that

q q

f3 = X(y) Fa (Po) - Fa (po) - Z Ca i Aa 7

j j J-1aj

j=1

0

0

so that -X(y)-l

fy q _ -Fo(Po) + X(y)-1 Fo(Po) +

q Z (Xajaj+l ...

+

_

Fa (Po) + X(Y) -1 - Fa (Po) + Ay

o

where A

Aaj-1aj

Xaq-1 4

j

0

is as defined in the proof of Theorem 19, and represents 7 -X(y)-1

the cohomology class A ; thus

fN q)

is cohomologous to Ay

.. y hence represents the cohomology class A as well, which concludes the proof. Remarks.

begins at p0 e 11

Recall that the path y was so chosen that it and ends at

7 p0 e 31.

Since the differential

form p is closed and the space M is simply-connected, the integral

f

is unchanged if y is replaced by any other path Prom

y

PO

to y po

; thus Lemma 22 can be restated as the assertion that

the cohomology class A


(A7 )

where

N (20)

Ay =

X(y)-1 fy .

PO

9

po F: M

More generally, introduce the e function

(21)

F(Z) =

- q )

given by

Ca

,

PO

noting as above that this is well-defined.

Ay = -X(y) 1

Thus

F(y' po) ; and further, for every z e

recalling

(19), N

N

N

y'P 7-Z F(y'Z) = f,,, ro + f N ro = fN ° q) + X(y) PO y'po PO

N z -

..

..

f,,, ro = X(y)(F(Z) -Ay] PO

so that

Ay = F(Z) - X(y)-1 It is apparent from this that the choice of the base point PO e 11

For if Pi

is completely immaterial.

is any other point of

and

G(z) = fz q) , then

G(z) = F(z) + C where

C = go

; and

p1

p1

By = G(z) - X(y)-1 G(y Z) = A7 + (c - X(y)-1 so the cocycles

11

N

N

(Ay)

and

sent the same element in

(By)

c) ,

are cohomologous and thus repre-

H1(nl(M),X)

.

As for terminology, the cohomology class A = (Ay) called the period class of the differential form 9 ;

will be

this is the

cohomology class defined by the cocycle A7 given by (21) and (22), where

9)

is any closed differential form on

11

satisfying (19).

When X is the trivial bundle of rank 1, the period class is just

the homomorphism A: 3tl(M) - C determined by the periods of the differential form 9), whence the terminology.

To determine the explicit form of the dual pairing (16),

(e)

it is necessary to use rather explicitly the topological structure of the surface.

and p

0

If M is a compact Riemann surface of genus

g

e M is a fixed but arbitrary base point, let j = 1,...,g , be a standard set of return cuts

aJ,Ti e X l(M,po) ,

on the surface.

Thza

aj,Ti

disjoint except for the point

are closed loops on M , which are

po and which dissect M into a

simply-connected surface, in the sense that 0 = M - Ui(ai U Tj) is simply connected. for instance.)

(See Seifert-Threlfall, Lehrbuch der Topologie,

The loops will be assumed to be labeled in the

order shown on the following diagram.

Upon tracing along these loops in order, as indicated by the numbers

on the preceding diagram, it is apparent that 0 can be viewed as a polygon of

1+g

sides, each loop

u.

r

determining two separate

sides; and the surface M can be recovered from the (closed) polygon 0 by identifying appropriate pairs of opposite sides. ting PO a 31

be the base point of the covering, where

f:

Let-

M

is the universal covering space, the polygon 0 C M can be lifted to a unique polygon

C 31, where

is bounded by a loop begin-

ning at

PO

and following along the covering of the loops

in the order indicated.

Thus

C AW

a,,T

is an open 2-cell, and its

boundary consists of 4g paths covering the loops

a,,T, ;

this

is clearer upon considering the following diagram.

Here

are paths in

J

31

which cover the loop

with the orientations as indicated on the diagram.

that tracing along the boundary of

to M , determines a loop in

T

a a(M,po),

It is clear

and projecting that path in-

,1(M,po)

which is homotopic to zero;

and thus one secures the relation

T111T1 011 ... Tga9Tgagl in the group lation

1

,l(M,po) , which is of course equivalent to the re-

(5). Let T

a,

be paths in

NA

-152-

beginning at po

and covering

the loops TVcj respectively; as noted earlier, the endpoint of the path a,

T j

is the point

is the point

e,

IV

T po , and the endpoint of the path viewing

PO

of transformations on l .

al = T1 al a

and ai ends at

begins at Tlcl

po ;

T1cl po , necessarily 1 =

the point

as acting as a group

Note then that ii = T1 , so ends at

Tlpo ; and since al

the point

,cl(M,po)

Tlpo , necessarily

and since

T1a1Til

ends at

1 and T1 be-

Continuing in this way around

gins at the point

T a T

the boundary of

, all the arcs and vertices can readily be iden-

I I I

po .

tified as suitable translations of the basic paths iia, point

PO

.

and the

In general, let

(23)

A. i _ [T1ol] ... [T,a,] a nl(M,po) ,

where

[T,o] = TOT-lo-l

as before.

It is then readily verified

that

?t,-lI a T,I T ; (24)

X3-iT, o, ; ; = lead for

j - 1,...,g .

With this notation, the dual pairing (16) takes

the following form. Theorem 21.

and

Let M be a compact Riemann surface of guns g , be a flat vector bundle over M ; and con-

X e lll(M,CL(n,C))

sider cohomology classes A e lil(M, In terms of representative cocycles

'

(X))

and B e H1(M, 3(X*)).

(A7) a Zl(,cl(M),X)

and

x* (B7) C Z1 (ifI(M), X) ,

the dual pairing associates to these coho-

mology classes the value

E

< A,B > =

I BTJ X(T

J=1

1 [BT

+

J=1

-lA J

-

B X(a )-1 A

) X(aJT,)-l(AX - X(ai)AX J J-1

J

it

Proof.

J

aJ

aJ

TJ

- Q X(TJaJ)-1(AX

(p e r(M, E '(X))

J

be a differential form repre-

and let p be the induced differential form on

A

for all

11

;

thus

y E al(M,po) , and the cocycle

representing A

E Z1(it l(M),X)

J

A under the deRhem isomorphism (10),

senting the cohomology class

X(7)- p(z)

J-1

is given explicitly by

7

= F(z) - X(y)-1 F(7

A

dF = p and

where

)

7

larly, let

* E r(M, E '(X*))

F(po) = 0

Simi-

.

represent the cohomology class

B

and y be its induced differential form on I1 ; thus (^Z)

X*(y)

and

and

C(po) - 0 .

and

B

By = G() - X (y)-1

G(y z)

where

dG

The dual pairing of the cohomology classes A

is given by

< A, B > = IM f .. lp = J

f tdG .. dF

W ..

Applying Stokes' theorem and (24), g

< A,B > _ -1 C)

tdC . F = - E J=l

T,aT

tdG

f

- F

J-0

J

J_

J

g

E

Iti

J=1 zeTJ

-E J=1

f

zea

[tdd(X

Ti-^Z')

F(XJ_1TJ Z) -

tdZ!()J

Z) F(XJ Z)) ;

J

and then, recalling the above functional equations for the functions

F

and

G ,

< A,B :>

J-1

E J,

-A1

jr]. zCOJ Since

T a T-1)] J-1 J J J

E Jti J=1 ZeTJ

T

J-1 J

fT td-5

) - td-('') ' (F(z) -AX J

BT X(T)-1

C(p0)

the above further

,

reduces to

E

< A,B >

X(TJ)-1 [AX

BT

Jr.]. g

- E o X(a J=1

J-1T JaJ T-1] J

-A X

- X(cJ)-1A), - Ac ] J

X(aJ)-1 IX(TJ)-1 AX J-1

[BT X(T)-1 Aa - a J

J

BT X(cJT

J=1

]

J

J-1

J

J=1 F

T

X(T J)-1 [AX J

- F to J=1

+

[AX

AA

J-1 J

BT

J-1

g

J)-1

J

-E

-

J-1

J

J)-1

+A

_AX ]

T J

J

X(aJ)-1 AT ]

J

J

[AX - X(aj )AX

J

J

]

J-1

J

g

B Ba X(T Ja J )-1 [A J=1.

- X(T J )AX

J-1

J

J

which is the desired result. Remarks.

The formula for the dual pairing as given in

Theorem 21 can be simplified somewhat by using the formal properties of cocycler.

Recall that Ay

Ay

By 1y2

-

Ay-1 = -

By1

X(y2) .i B72 and

for all

tBy-1 =

and

1

1 2_

yl,y2 a nl(M) 1)

.

;

thus

Recall further that

hence that AXE = X([a Ti]) A"1-1+A[aj,TJ]

Xi =

.

Thus

8

(B a-1 ATJ- B T-1

E

J-1

t-

-1 -1 BTUX(ai Ti

1

.

)

_-

"

A

cj )

1

1 a_1 )

Al

J

"J-1 X(T

-a

-1

J

E1 rB + aj X(Tj){ X(TA[aj,Tj] L JJ

a (I-X(Ti))]

+

8

__

-

X(T

jZlttBajTj

tB[a -

1 -1 aj ) A[a

Tj ]

,T ]

+

AX J -1

AX ,

so 8 (25)

< A,B > = Z B j=1

a

B i

T

Ac - B (aT)_ i

[c ,T ]+ B[c ,T ]A i

j-1

at sheaves: analytic aspects

§8.

Consider a flat vector bundle

(a)

X e H1(M,GL(n,C))

compact Riemarn surface M of genus g .

over a

The complex analytic

version of the deRham sequence considered in §7(c) above is the following exact sequence of sheaves:

o -- & (x)

(1)

d (x)

d-1' °(x) --> 0

The associated exact cohomology sequence over M , which in a sense is the basis for the complex analytic properties of flat sheaves,

has the form

-

o --> r(M, 9.(x)) --- r(M,t4-(x)) d-.. r(M, ml'0(x)) --

(2)

--4

H1(M, 9'(X))

- Hl(M'a(x)

Hl(M' d1'°(x))

--- I2(M, 3-(x)) - 0 , I2(M, B (X)) = 0 .

since

Lemma 23.

Over a compact Riemann surface M , the kernel

of the homomorphism d: H1(M, 9 (X)) -->

(M, 0 1'°(X))

in the

exact sequence (2) is canonically isomorphic to the space

Cr M a 1'0 d M, & t X-)) X where

j

'

denotes the dual complex vector space. Proof.

Considering the exact sequence (2) and the corre-

sponding exact sequence associated to the dual flat vector bundle

X*

a

H1(M,GL(n,C)) , the Serre duality theorem exhibits the dual

pairings indicated by 4-- in the following diagram of exact sequences.

0 --? K

d> Hl(M, &1'°(x))

H1(M, &(x))

--k ...

0 4- L 4.._- r(M, 41' °(x*)) 4--d r(M, d (x*)) ._ In this diagram,

K and

...

.

L are defined as the kernel and cokernel

To show that K and L are dual

of the respective mappings d.

vector spaces, which is the desired result, it is clearly sufficient to show that this diagram is commutative, in the sense that (F,dG) _ (dF,G)

for all

F e I'(M, LV (X))

and

here,

G E r(M,

notes the inner product expressing Serre duality.

de-

The result is

immediate, upon recalling the explicit form of the Serre duality. Letting V.= CTJa)

be a Leray covering of M for coherent analytic

sheaves, an element

has a representative cocycle

F e u'(M, fi(x))

(Fad) a Z1(Vt ,C-(X)) ; and there are a cochain

(Fa) a Co(IX

,

a (X))

C° functions

such that

Fa forming

(F,,,) = S(Fa) , that

Then for any sec-

is, such that

Fa, = F - Xa11Fa

in Ua fl u, .

tion

a r(M, dt 1,°(X*))

the dual pairing is defined by

(F,*) = JM t*a .. IFa . Similarly, an element p e Hl(M, SL'-'O(X))

cocycle ((Pa)

has a representative

(pa4) a Z1('lit, 6t'°(X)) ; and there is a cochain

. Co(4

e 1'°(x))

such that

roa, = Pp -

,lcpa

in Ua n U.

Then for any section

G = (Ga) e r(M,(9-(X*)) , the dual pairing is

defined by

(9), G) = fMtGa .. 4a Now if

F e u1(M,(9(X))

cochain

and

(Fa) e C°(VI, 5(X))

(Fap) a Z102

,

(dF) a Co(,t

6-(X))

Z1(X

tGG'JFP

with coboundary as the cocycle

representing F ; clearly the zero-cochain

el'O(X)) ,

(dFrO) e

G e r(M,(V(X*)) , select a zero-

, S 110(X))

has as its coboundary the cocycle representing

dF .

Note that

t

GjFa

in Ua fl U. , so this is a global differential form

tGa Fa a r(M, e 0,1).. Thus by Stokes' theorem, (F,dG) = fM tdGa .. IJFa = fM(d(tGaZFa)

fM tGaJ(dF(X) _ (dF,G) concluding the proof. Theorem 22.

If M is a compact Riemann surface and is a flat complex vector bundle over M , there

X e $1 (M,GL(n,C))

is an exact sequence of complex vector spaces of the form

(3) 0-dr(M r Proof.

1,0( m(X)) )

1,0

* *

dr(M B(XX)))

-0 .

The exact sequence of vector spaces (2) can be re-

written as an exact sequence

0 --- r(M, 91'°(X))/dr(M, ®(X)) -k u1(M, 9-(X)) -> K -- - 0 where K is the kernel of the mapping

d: E1(M,(9-(X)) ---> E1(M, S 1'0(X));

and since that kernel K is described as in Lemma 23, the result is demonstrated. The cohomology group

Remarks.

H1(M, 3(X))

is in a

sense a purely geometrical entity; the complex structure of the

Riemann surface M enters only in the form of the exact sequence (3).

Clearly the main problem is that of describing explicitly

the exact sequence (3), or just that of describing the image of

the mapping S-.1 When

X is the identity bundle of rank 1, then

H1(M, 3 (X))= H1(M,C)

is the ordinary cohomology of the surface M ;

the image of

S

consists of those cohomology classes which are

represented by the abelian differentials on the surface M , and is described by the period matrix of the surface.

flat vector bundle

X , the differential forms

For an arbitrary

e e r(M,C 1'0(X))

will be called (generalized) Prym differentials on the surface M. (Classically, the Prym differentials are such differential forms

when X is a unitary flat line bundle on the surface M ; see for instance H. Weyl, Die Idee der Riemannschen Flche.)

The mapping

S

in (3) is just that which associates to any Prym differential

e

its period class, as defined in §7(c); for the Prym differentials

are closed differential forms of degree 1 on the surface.

morphism

p

The homo-

in (3) associates to each cohomology class

A e R'(M, a (X))

a linear functional

p(A): r(M, 6'1'°(X ))

C;

and the cohomology class A

is the period class of a Prym differ-

ential if and only if

is the zero mapping.

p(A)

The period

classes of the Prym differentials will also be called the holomorphic cohomology classes in

H1(M, 3 (X))

In order firmly to fix the conventions, the mappings and

p will be described explicitly as follows.

S

Select an open

covering Vt = (t J) , which is a Leray covering for coherent ana-

lytic sheaves and for flat sheaves. A Prym differential e e r(M, B 1'0(X))

is described by n-tuples

ential f o r m s in the various open sets

ea of abelian differ-

U a , such that

ea = x e

in Ua n UP . such that

In each Ua select a holomorphic function F.

ea = dFa ; the constants

Aa5 = FP - X Fa in Ua fl UP

(4+)

form a one-cocycle

(AC43 ) e Z1( Vi, 6- (X)) , which represents the

period class A = S(e)

of the Prym differential.

cohomology class A e E11(M, 9 (X)) (Aa5) e Z1(ilL, 3'(X))

functions F. ; and then

(5)

p(A)

represented by a cocycle

can be written in the form (f+) for some e aFa a r(M, e 0'1(X))

ferential p e r(M, B1'0(X )) the mapping

Now an arbitrary

For any Prym dif-

.

of. the dual flat vector bundle

X

,

associates to p the value

p(A)-9) = fMtq)a .. aFa

If A is a holomorphic cohomology class, the function Fa can be taken to be holomorphic, so

aFa = 0

and

p ;

conversely, by the exactness of the sequence (3), if d 1,0(X*)) , then the cocycle p e r(M,

sented in the form (f+) where

(AC43 )

F. is holomorphic.

can be repre-

The explicit mappings in the exact sequence (3) can also be described in the following slightly different form.

In additon

to (3), consider the corresponding exact sequence for the dual vector bundle

X

;

and write the two sequences out in the follow-

ing form.

0-

rvii

r(h

(6)

r,,. ml, Orv*r%I

la 10 [I...f,&l1o(x)l*4..LHl(M,

0 4-

F 3 (X *))

There are the dual pairings a and

y

rdM

1, 0 X

0

and

C X follows

In terms of a suitable covering 1R-= (Ua) of M ,

from Leans. 15.

the analytic bundles

and

(XCP )

and

g

XCP

, where

is as in (l0); the dual bundle

then be defined by a cocycle 1

unstable as well; and bundle such that

n

(L )

X can be defined by cocycles

(x

as in (11).

)

= 9 det

X

X*

can

Note that X is

is the unique complex line

c(i 1) = div X > 0

n -l C X*

and

.

Now con-

sider an analytic section

a

=

Pa Since

and

ga a r(M,(.(n))

ga = 0 unless

e r(M, &(X))

ga c(1) =-c(g) < 0 , it follows that

in H1(M, 0*) ; but if

n = 1

and

n = 1

ga 10

it follows as in the proof of part (ii) of Leffina 15 that the bundle

X is analytically decomposable, which is impossible by assumption. Thus necessarily then

ga = 0 , and so

r(M, -a-(x)) = r(M,

Q)) ; and

r(M, m (X)) = r(M,

as well.

plays the same role for the dual bundle

1

= ® det X* * X , the desired concluSince

rj

sion in this case follows immediately from condition (9). (iv)

If

X

is a unitary flat vector bundle, then the

complex analytic sections of X are necessarily all constants; (see the following Lemma 25).

That is to say,

r(M,(L(x)) =

= r(M, e'(x)) ; therefore the condition (9) that there be an even splitting in the exact sequence (3) reduces to the condition that

dim r(M, & (X)) = dim r(M, 9(X*)) . follows that

r = dim r(M, 9- (X))

Now from Theorem 18 it

is the largest integer such

that the identity representation of degree r be contained in the unitary representation

of the group

a1(M) ; it is familiar,

however, that reducible unitary representations are fully decomas well.

posable, so that r = dim r(M, .4-'(X*))

The proof of the

theorem is then completed, except for the proof of the following result.

If

lemma 25.

X

is a unitary flat vector bundle over a

compact Riemann surface M , the complex analytic sections of are necessarily all constants; that is to say,

X

r(M,is (X))=

= r(M, 9-(X)) Proof.

(Fa) a r(M, ®(X))

Consider a section

in terms of an open covering

of the surface M ; the

= (Ua)

2'

expressed

elements F. will be viewed as column vectors of holomorphic

functions on Ua of length n, where n

tion in Ua ; since the matrices

XCP

of a cocycle defining the

X are unitary, it follows that

hence that

X

a real-valued func-

Introduce the norm

bundle

is the rank of

IIFjI = IIF0II

in Ua n u.

is a well-defined, real-valued function on the

entire surface M .

Since, M is compact, this function will at-

tain its maximum at some point

p0 e Ua C M ; upon multiplying 0

all the vectors

F. by the same unitary constant matrix, it can

be assumed that

Fa (P0) _

0

0

Now the function fla (p) 0

is holomorphic in the open neighborhood

Ua

of po , and

0

Ifla (P)1 5 IIFa (P)II 5 IIFa (Po)ll = Ifla (Po)l for all p e Ua ; 0

0

0

0

so by the maximum modulus theorem,

fla (p)

0

Furthermore, since

fill

(P) = fla (Po)

0

for all p e Ua

Ua

and

as well; thus

is constant in Ua 0 IIFa (P)II < IIFa (Po)ll

0

0

, necessarily f2a (P) _ 0 0

0

= fna (p) - 0

in

0

is constant in Ua , hence is constant on 0

Fa 0

0

0

the entire (connected) Riemann surface M , as desired. The observation that stable and unitary bundles behave similarly will appear in its true light later, when the relations between these two classes of bundles are discussed in more detail. Upon considering ease (iii) of Theorem 23 more closely, it is easy to construct examples of such bundles for which the exact sequence (3) is not an even splitting.

Let

9,n

be any two complex ana-

lytic line bundles over the surface M such that

0 < c(g) = c(rj 1),

that dim r(M, 01 (a)) # dim r(M, a (1 1)) , and that there is an indecomposable complex analytic vector bundle

tension of

9

by

bundles

, i

(That there exist such line bundles on

i .

surfaces of genus

g > 4

such that

a point bundle

is quite trivial.

1 < c(g) = c(i 1)

Select any two line and that

1)) ; for instance, select for

dim

dim

X which is an ex-

, and select for

9 =

T I-1

a bundle of Chern

'P

class

c(i 1) = 1 which is not a point bundle.

extensions of cohomology group

by

n

The set of all

is in one-to-one correspondence with the 1)) , by Theorem 13; so to guarantee

the existence of a non-trivial extension, it suffices merely to ensure that that cohomology group be non-trivial.

By Serre's dual-

ity theorem, H1(M, 4(g1 1) ) = r(M, G (K9-1n)) ; and since c(Kg-1rj) = 2g-2-2c(g) , it follows from the Riemann-Rock theorem for line

bundles (see page 112 in last year's lectures) that c(Kg-lij)

= 2g-2-2c(g) 7 g , hence

r(M, &(Kg-11')) # 0 whenever 2c(g) < g-2 .)

whenever that

c(g) = c(rj 1) > 0 , it is clear

Since

r(M, 6 (g)) = r(M, 31 (n- l)) = 0 ; so in view of the criterion

of part (iii) of Theorem 23, the exact sequence (3) is evidently not an even splitting for this vector bundle

X .

On the other

hand, though, there are analytically indecomposable unstable flat vector bundles splitting.

X such that the exact sequence (3) is an even

Fbr instance, let

div X - g-1 , where g

is the genus of the surface M .

be that line bundle for which

and let g

g > 1

X be any such bundle for which

by

r1 =

n .

g-1

det X , so that

c(g) = div X = g-1

X

Let

C X ;

and

is a non-trivial extension of

By Theorem 13, there exists such an extension if and

only if H1'(M, B (grt 1)) = r(M, 4 (Kg-1'q)) # 0 , hence if and only g = Kr1 , since r(M,

c(Kg-1r1) = 2g-2-2c(g) = 0 .

Now clearly

(g)) = r(M, 9-(r1)) = 0 ; and from the Riemann-Roch theorem,

dim r(M,B (g)) = dim r(M,B (K9-1)) + c(g) + 1-g = dim r(M,B (n-1)) Therefore, by the criterion of part (iii) of Theorem 23, the exact sequence (3) is an even splitting for this vector bundle

X .

This

latter particular case is of some interest in uniformization questions, as noted in last year's lectures.

Again, when

X

is the identity bundle, the exact sequence

_

(3) is an even splitting, and moreover, the cohomology group H1(M, 3-(X))

is the direct sum of the image of

plex conjugate.

r(M,01,0(1))

B

and of its com-

In general, the Prym differentials in

have periods in

; and complex conjuga-

H1(M,

tion establishes a conjugate-linear isomorphism H1(M, 3- (v)) H1(M, 9 (X))

Combining these two mappings leads to the isomor-

.

phism (into)

r(M' B 1' 0()) dr(M, 0 (Z) )

,> H1(M, -9 (X))

and one can ask whether there is an isomorphism

where

$) ,

H1(M,a (X)) ° (im S) ( D

(12) im S

denotes the image of the homomorphism

Theorem 24.

S

If M is a compact Riemann surface and

1 X e H1(M,GL(n,C))

is a unitary flat vector bundle over M , then

there is an isomorphism of the form (12). Proof.

It follows immediately from Theorem 23(iv) that

dim (im S) = dim(im necessary to show that

dim H1(M, (X)) ; so it is merely (im S) fl (im $) = 0

.

Suppose contrari-

wise that there is a non-trivial cohomology class

A e (irn S) fl (im S) C H1(M, 3' (X)) .

In terms of a suitable open

covering Vt= (Ua) of the surface M that cohomology class is represented by a cocycle

(AC43 ) a Z1( 7 , 3 (X))

AC43 = F0 - Xap1Fa = G0 for some cochains

such that

in Ua fl u,

(Fa) a C°(1R ,®(X)), (Ga) a C°(1Q

thus A = 8(dFa) = 6(dGa) .

Now the functions H. = Fa - Ga are

harmonic in Ua , and ga = X

O

in Ua n u, ; harmonic functions

also satisfy the maximum modulus principle though, so as in the

proof of Ienma 25 it follows that Ha is constant.

Fa = Ga + Ha and both F. and lows that both F. and

Since

G. are holomorphic, it then fol-

Ga are also constants; but then the coho-

mology class A is trivial, which is a contradiction.

That serves

to complete the proof. Remark.

If X

is a real unitary (orthogonal) bundle, then

X = X , and it follows that

ticular when X

(im $) = (im $) ; this is true in par-

is the identity bundle.

Finally, there is the question of the construction of generalizations of the Picard variety of a Riemann surface, involving the cohomology groups

H1(M,9.(X))

interest, where the representation

.

In many cases of arithmetic

X is essentially rational, it

is possible to define a lattice subgroup of H-(M, 3(X))

and

parallel the construction of the Picard variety, as given in last year's lectures.

[See the paper by Goro Shimura, "Sur lea inte-

grales attachees aux formes automorphes," Jour. Math. Soc. Japan 11(1959), 291-311.7

This method fails in the general case; we

shall return to this question later, from another point of view.

(c)

The study of'the analytic properties of flat sheaves can

be approached in a slightly different way, by considering instead of the exact sequence (1) the following exact sequence of sheaves:

(13)

0 -- 9(X) --.j(X) -1'0(X)

where '"L

denotes the sheaf of germs of meromorphic functions on

the Riemann surface M . entire sheaf ')J 1'°(X)

The mapping d

of germs of meromorphic differential forms

which are sections of the vector bundle image

in (13) is not onto the

X ; as is familiar, the

d rn(X) C 1'°(X) consists of those germs which have zero

residues.

The exact cohomology sequence associated to (13) has

the form

o --,

a r(M,d'

T r(M,'hj (x))

(X.)) -?

H '(M, -5 (X)) --> 0 since

0

as noted earlier.

Therefore the flat co-

homology has the analytic representation

H1(M,a (x)) n,

r(M,d

X) i

indeed, this holds for an arbitrary Riemann surface M , although we shall consider here only compact Riemann surfaces. X , the sections

For any flat vector bundle

e e r(M,d'

(X))

will be called meromorphic Prym differentials on the surface M . It should be emphasized that the meromorphic Prym differentials are those meromorphic differential forms

e e r(M,')

'0(X)) which have

zero residues at each point of the surface; in the classical terminology, these would be known as differential forms of the second kind.

With this in mind, these meromorphic Prym differentials

have well-defined period classes

e

Be a H1(M, 9'(X)) , as in §7(c);

the period mapping S: r(M,d)(X)) --? H1(M, 9(X)) is precisely the mapping arising in the exact cohomology sequence (14).

Thus

the analytic representation (l4) can be interpreted as the assertion that each cohomology class A e Hl(M, $ (X))

is the period

class of some meromorphic Prym differential on the surface M ; and that two meromorphic Prym differentials have the same periods precisely when their difference is the exterior derivative of a meromorphic section of H-(M, 9 (X))

f e r(M,?) (X))

The distinguished subspace

.

of greatest interest is the space of those coho-

mology classes which are the periods of holomorphic Prym differentials.

There is a slight variation of this analytic representation of the flat cohomology based on the observation, made earlier during these lectures, that all vector bundles are meromorphically trivial.

Let u = (U.)

be an open covering of the Riemann sur-

face M which is a Lersy covering both for flat sheaves and for analytic sheaves; and select a representative cocycle (x

) a Z1( V, GL (n, C))

for the flat vector bundle

X .

There

are elements Pa a GL(n, U) = r(Uu, h (n,'(.) ) such that

a

Pa(z) = XM*P0(z) whenever matrices the isomorphism

z e Ua fl U

; and in terms of these

P: N (X) ---? "ln

P(Fa) = Pa1Fa

for

is given by

)!'a e m (X) .

The same mapping of course yields the isomorphism P:

ryy,1,0(X)

(,1,0)n

Applying these isomorphisms to the

exact sequence (13), there results the commutative diagram of exact sequences

-174-

o --- . 3 (X) - te a % (X) d 1k:"o(X) PI.v

id. f n,

(16)

'41

o ---> 3 (x) note that

ip

P1:1

dP

%l 1,0

-

is Just the mapping P

itself, while

dp

is the

mapping given by

1(Fa) = Pald(PaFa)

dp(Fa) = (17)

= dFa +

where

F. e')Yt and d

is the ordinary exterior derivative.

The

exact cohomology sequence associated to the second line in (16) has

the form

dp ? 0 ---> r(M,a (X)) ip P. r(m7)

r(M,") . s>

? H1(M, 3 (X)) --- . 0 , since again H1(M,7

) = 0 ; therefore the flat cohomology has the

analytic representation

(18)

R1(M,

(X))

r(M,dp,'l )

dP-T

.

The advantage of this representation is that the sections are global meromorphic differential forms on the Riemamn surface; the vector bundle appears only in the differential --operator

dp .

--11

The dual pairing

H1(M,-3'(X)) ® H1(M, 9(X*)) ---> C con-

sidered on page 162, which assigns to cohomology classes

and B e

Ae

Ii-(M,

by

[A,Bi, has an interesting form in terms of this analytic repre-

(X))

H'(M, 91(k*))

sentation of the flat cohomology.

are represented by

If the cohomologr classes A,B

and * e r(M,dk(X*)) respec-

(X))

cp a r(M,d

the complex number denoted

tively under the isomorphism (15), recall that in a suitable coordinate neighborhood of any point,

cp

and * are represented by

n-tuples of ordinary meromorphic differential forms, viewed as column vectors; moreover these vector-valued meromorphic differential forms can be expressed locally as the exterior derivatives of some vector-valued meromorphic functions, which functions will be

denoted by

fcp

and

Theorem 25.

pairing

.

On a compact Riemann surface M the dual

H1(M, .4 (X))

[A,B] = 27ri

(19) where

fi

cp a r(M,d't1(X))

-27ri W [t*- (POI, and * e r(M,d?k(X*))

represent the coho-

and B e '1'M, 3 (X* ))

mology classes A e H'(M, 9 (X)) under the isomorphism (15), and

respectively

denotes the total residue

of the differential form in brackets. Proof.

Note that the residues in (19) are independent of

the choices of the integrals of the differential forms; for any two integrals differ by a constant, and the differential forms

* both have zero resides at each point.

classes A e Hl(M, 9 (X)), B e cocycles

H'(M,

9

cp,

Now consider cohomology

(X*))

,

and repreaentetive

(A.) a Zl(14, 3(X)), (B,,,,) a Z1(j/L, -9(X )) respec-

tively, in terms of a suitable open covering

= CUa)

of the

surface M . The differential forms cp a r(Mdlk (X)), r e

representing A,B respectively under the isomor-

phism (15) are given by

*a = dGa in Ua , when Fa,Ga

gpa = dFa,

are meromorphic vector-valued functions in Ua such that (XaO)-1Ga

Aap = Fp

X-

in %, fl u,

Bad = Gp -

The cover-

gx'

ing UI can be so chosen that each singularity of Fa or

G. has

an open neighborhood meeting no other set of the covering

T!(, but

Ua ; and modifying these functions

F,,Ga in these neighborhoods

of their singularities leads in the obvious manner to

Fa,Ga in Ua which also satisfy the relations BC43 = GP - (x

)'1Ga

C'O

AC43 - F -

in Ua n u, , since F. = F

etc., ea

functions X1_:1

'a

in

The differential forms cp'a = dra a r(M, a 1(x)) ,

Ua fl UP .

then represent the cohomology classes

*'a = dGa E r(M, a l(X*))

A,B respectively, under the defham isomorphism of §7(c).

Recalling

pages 137 and 16e, the dual pairing of the Theorem is given by

> = jM *a

[A, B] = < A ,

since t*a, ^

t*a

gaa

-

(pa = o

(P., = E fu t*a ,,

a

(Pa

a

in Ua fl U0 for any other set U

Assuming the sets Ua have smooth boundaries,

of the covering.

it follows from Stokes' Theorem that

jat*1..gpa=jUa(dGa)..gaal =j but since G. = Ga and gaa - gaa on aU. , it further follows that tGa (Pa jaUa

where r{U

a

=

fluxtGa -(pa = 2Wri RU [tGa gaa]

a

a

denotes the total residue in Ua .

Altogether then,

[A,B] =

E 27t1

a

[tG'(Pa] = 2iri 7e[tGa-cpa]

since there are no residues in the intersections that

[A,B] = -[B,A] , this can be rewritten

[A,B] = -27ri'R [t* .Fa] and this suffices to conclude the proof.

Ua n U0 .

Noting

§9.

Families of flat vector bundles

(a)

On any surface M , the mapping which associates to a flat I'X

vector bundle

X

its characteristic representation

establishes

a one-to-one correspondence

H1(M, GL(n, C)) - Hom(7r1(M), GL(n, C))/GL(n, C) ;

(1)

and this can be used to describe in a reasonably convenient manner the family of all flat vector bundles over the surface.

Moreover,

this description provides a natural complex analytic structure associated to the family of flat vector bundles over a compact surface.

Suppose that M is a compact Riemann surface of genus as noted earlier, the fundamental group

a group with 2g

generators

1r1(M)

g ;

can be described as and one

a1,...ag, r1,...,T9

relation

(2)

[ag,Tg] ... [Q2,T2][a1,T1] = 1

where the commutator is written respondingly, an element

,

[a,T] = QTU 1T-1 , as usual.

p e Hom(7r1(M),GL(n,C))

Cor-

is completely

described by the 2g matrices p(a1) = Si e GL(n,C)

p(T 1 )

= T a GL(n,C)

i

and these can be arbitrary matrices, subject only to the restriction that

[SgTg] ... [S2,T2][S1,T1] = I

Thus the set

Hom(7r1(M),GL(n,C))

subset of the product space GL(n,C)

can be identified with a certain

GL(n,C)2g

.

Recall that the group

has the natural structure of a complex analytic manifold

of complex dimension n ; (see for instance C. Chevalley, Theory of Lie Groups I, (Princeton University Press, 1946)).

the mapping

Z T

origin in the space

Explicitly,

Z takes an open neighborhood D of the Cn

xn

of all complex matrices

Z homeomor-

phicall.y onto an open neighborhood of the matrix A e GL(n,C)

the components of the matrix Z

neighborhood of A .

are local coordinates in that

GL(n,C)28 thus has the

The product space

structure of a complex analytic manifold of dimension 2gn2 .

Introduce the matrix-valued function F on the manifold

GL(n,C) 2g

defined by F(51,...,S9 T11...IT9) = [Sg,Tg) ... [S1,T1) ;

(3)

it is obvious that this is actually a complex analytic mapping

F: GL(n, C)2g -> SL(n,C) , where

SL(n,C) C GL(n,C)

determinant one, since

(4)

is the subgroup consisting of matrices of det[s ,TiI = 1

.

The subset

R = ((Sj,Ti) a GL(n,C)2gjF(Sj,Ti) = I) C GL(n,C) 2g

is then a complex analytic subvariety of the complex manifold GL(n,C)29 ; and the mapping

p e Hom(1r1(M),GL(n,C)) --> (P(a ),P('ri)) e RC GL(n,C)2g identifies

Hom(7r1(M),GL(n,C)) with this subvariety, and thus

establishes a complex analytic structure on the set

Hom('rr1(M),(M(n,C)) Remarks.

At several points, the study of Riemann surfaces

inevitably leads to constructions or problems involving several complex variables.

This was noted in §8 of last year's lectures,

in the preliminary discussion of divisors on a Riemann surface;

and it has now arisen in the discussion of flat vector bundles.

So

far as these lectures are concerned, really not much is involved except the definitions of a complex analytic manifold and of an analytic subvariety.

(A complex analytic manifold is just the obvious

generalization of a Riemann surface; an analytic subvariety is a closed subset defined locally as the set of common zeros of a finite number of complex analytic functions.)

The reader should be able

to follow the discussion with no further prerequisites than needed for the preceding parts of these lectures; but doubtless the discussion will be clearer to those readers having some familiarity with several complex variables.

For the case of line bundles (vector bundles of rank n = 1),

the mapping F

is clearly the trivial mapping F(SVTi) = 1 ; thus

in this case R = GL(l,C)2g = (C ) 2g , and R itself has the structure of a complex analytic manifold of complex dimension 2g For the case of vector bundles of rank n > 1 , the mapping F

non-trivial, and R GL(n,C)2g

.

is a proper subset of the complex manifold

Although R

is a complex analytic subvariety, it is

not a complex analytic submanifold; the subvariety R singularities.

is

contains

To investigate this situation, it is natural to

Yntroauce the differential of the mapping F .

Recall that

F

is

a complex analytic mapping between two complex analytic manifolds;

plex manifold.

point

The differential

GL(n,C) 2g

at the point

SL(n,C)

differential

of this mapping F at a

dFp

is the induced linear mapping from the tan-

p e GL(n,C)2g

gent space of of

is a Lie subgroup, hence is itself a com-

SL(n,C) C GL(n,C)

for

p , to the tangent space

at the point

F(p) ; in more primitive terms, the

is just the homogeneous linear part of the

dFp

Taylor expansion of the mapping F centered at the points

p

(Xi,Yi

local coordinates

and

in terms of local coordinates

F(p)

a (Cp

To be explicit, introduce

.

xn)2g

centered at the point

p = (Sj,Tj ) a GL(n,C) 2g by the mapping

(xiIYi) ->

XJ,TT'exP YJ) ;

Z e Cn

and introduce local coordinates

Xn-l

(viewing

as com-

Z

plex matrices of trace zero) by the mapping

Z --> recalling that dFp(Xj,Yj)

Z

det(exp Z) = exp(tr Z)

The differential

.

is just the homogeneous linear part of the Taylor ex-

pansion at the origin of the function Z(Xj,YY)

Y

F(p)-exp Z(XiYY) =

)

given by

.

j

Lemma 26.

With the notation as above, the differential

Up of the mapping

F

at the point

p = (Sj,Tj) a GL(n,C)2g

g (5)

dFF(Xj,YY) _

lAd([S1,T1]-1

...

ESJ-I,Tj-1]-ITJgJ).

-(I-AdT31)'Xj3

.

is

Proof.

As a preliminary, recalling the Taylor series

expansion of the exponential function, note that there is a Taylor expansion of the following form: l Y] = S(exp X)T(exp Y) (exp - X)S-1(exp - Y)T

X,

= [S, TI + higher powers of the variables. This leads directly to the Taylor expansion

XgT9-exp

Y ) _

F(Si*exp

X1]

i

S E ([Sg,Tg]...[Sj+I,TJ+I]S3Tj[(I - Ad

= F(SiTJ)+ J

(I -Ad Ti1).Xi]Si T 1[SJ-7TT-1]...[S1,T1])

+ higher powers.

On the other hand, the function

Z(X_jJ.Yj)

has the Taylor expansion

Z(Xj, J) = 0 + dFp(X_J,Yj ) + higher powers;

and hence F(p) + Since

Z(XjYj) = F(Sj-exp Xj,

higher powers. Yj) , the desired

result follows immediately upon comparing terms, and the proof is thereby concluded.

The mapping point

F: GL(n,C)2g --> SL(n,C)

is regular at a

p e GL(n,C)2g precisely when its differential dFp

at

p

is a linear mapping of maximal rank, that is, when rank(dFp) = n -l; we are of course only considering surfaces of genus

,Q.,

g > 1.

An

open neighborhood U

of a regular point

can be viewed as the

p

p

Cartesian2product Bad an open set in C set in

(2 g -1)n2+1

C" -1 ; and the mapping F in Up

onto the second factor.

it follows that

Ro

is just the projection

Letting

Ro = (p a RI F

(6)

and as an open

is regular at

is a complex analytic submanifold of

GL(n,C) 2g of complex dimension

(2g-1)n +1

,

if Ro

Moreover, the tangent space to the manifold Ro

is non-empty.

at a point

p e Ro

can be viewed as that vector subspace of the tangent space to GL(n,C)2g

at

p

which is the kernel of the linear mapping dFp

With these remarks in mind, the following is an almost immediate consequence of the preceding lemma. Theorem 26.

The manifold Ro

is the subset of the ana-

lytic variety R formed by the irreducible representations of the group p E Ro

71-1(M)

.

The tangent space to the manifold Ro

can be identified with the space

cycles of the group Cn X n

rrl(M)

at a point

Z1(7rl(M),Ad p) of co-

with coefficients in the 7rl(M)-module

of all n X n matrices under the group representation Ad p,

(here n > 1). Proof.

a point

p e R

The condition that the mapping F be regular at is that

rank(dFF) = n2-1 , or equivalently, that

dimC(kernel dFp) _ (2g-1)n +1 , viewing the differential as a linear

mapping

dFp :

The kernel of

dFp

C2

2

-gn

?e

-l

is the space of all matrices (Xi 'y ) a (Cn)28 i

such that

dFP(Xi,Yy) = 0

.

However, comparing the explicit formula

(5) in Lemma 26 with formulas (6) and (7) in §7, it is evident that

X ,Y

there is a one-to-one correspondence between points the kernel of

the group

dFP

and cocycles

of

(Aa ,AT ) e Zl(7r1(M),Ad p )

i

7r1(M)

in

with coefficients in the 7r1(M)-module

Cu

Xn

of

all n Xn matrices under the representation Ad p; the correspondence is of course given by AQ = Xj,

= Yi

-

Thus the kernel

ATE

of UP can be identified with the vector space and the condition that

F be regular at

Z1(7r1(M),Ad p)

is just that

p

Now recall that in the proof of

dim Zl(7r1(M),Ad p) = (2g-1)n +1 .

the Corollary to Theorem 19, on pages 133 ff., it was demonstrated

that for any representation X of

7r1(M)

dim Z1(7r1(M),X) = (2g-l)n+q where

q

of rank n

is the largest integer such

that the identity representation of rank q is contained in the

representation X = t^l ; since Ad p has rank

n2

,

dim Z1(7r1(M),Ad p) = (2g-l)n2+q , where the identity representation of rank

q

is contained in

be regular at q = 1

p

(Ad p )* , and the condition that

is just that

q = 1 .

F

However the condition that

is in turn equivalent to the condition that the dimension

of the vector space consisting of all complex matrices A e for all

such that A = (Ad P(7))*.A =

Cn X n

7 E 7r1(M)

is precisely 1 ; and this can be rewritten dimC(A a Cn X nlP(V),tA = tA.P(7)

,

all

7 e 71(M)) = 1 .

The latter vector space always includes the space of scalar matrices, hence has dimension at least 1.

If the representation

p

is

irreducible, it follows from Schur's lemma that this vector space consists,preeisely of the scalar matrices, hence that

while if

p

q - 1 ;

is reducible, it is evident that this is a larger

vector space, and that

be regular at

p e R

q > 1 .

Therefore the condition that F

is precisely that the representation

p

be

irreducible; and this, together with the earlier remarks in the course of the proof, suffices to conclude the proof. Remarks.

The Jacobian matrix of the mapping F , which

describes the differential function on the manifold p

in

GL(n,C) 2g

dF ,

is evidently a complex analytic

Gi,(n,C)2g ; so the set of those points

at which the differential

dFp

does not attain

its maximal rack is a proper analytic subvariety of Thus

R

0

GL(n,C)2g

is the complement in R of a complex analytic sub-

variety of R ; indeed, clearly R - Ro

is the analytic subvariety

consisting precisely of the singular points of the variety R .

The more detailed study of the analytic space R prospect, but would lead too far afield here.

is an interesting

It should also be

remarked that explicit form of the fundamental group is not really needed for the general results established here, but does simplify the treatment somewhat; the more general situation has been discussed elsewhere, (see the Rice University Studies (Summer 1968),

Proceedings of the Complex Analysis Conference, Rice, 1967).

(b)

Having shown that the set

Hom(7r1(M),GL(n,C)) = R has a

natural structure as a complex analytic variety, and that the subset of irreducible representations form a complex analytic manifold,

there arises the problem of investigating the quotient spaces of these varieties modulo inner automorphisms of

GL(n,C)

convenient to begin with a more general situation.

Cartesian product manifold r > 1 , a point

GL(n,C)r

there is a proper linear subspace of linear transformations

Considering the

for any integers

(Sl,...,Sr) E GL(n,C)r

It is

.

n > 1 ,

is called reducible if

Cu preserved by all the

Si ; all other points are called irreducible.

The irreducible points form a subset

GL(n,C)o C GL(n,C)r ; note in

passing that Ro = R fl GL(n,C)og For any integers

Lemma 27.

points in

GL(n,C)r

hence the set

Proof.

the reducible

form a proper complex analytic subvariety;

GL(n,C)o

set of the manifold

r > 1, n > 1

of irreducible points is a dense open sub-

GL(n,C)r

Any matrix

S e GL(n,c)

can be viewed as a complex

analytic homeomorphism of the (n-l)-dimensional complex projective

space F

n-1

, and the action of

be denoted by

S

on a point

the fixed points in

dimensional linear subspaces of CA

GL(n,C)r X lp n-l

V1 =

will

correspond to one-

left fixed setwise by the

linear transformation represented by S . manifold

7P n-l

n-1

z e 1P

In the complex analytic

consider the subset z

for

j = 1,...,r) i

since the group action of the transformations is complex analytic,

clearly V1

is a complex analytic subvariety.

The obvious pro-

jection mapping

Tr: GL(n,C)r X lPn-1 -

GL(n,C)r

is a complex analytic mapping which is proper, in the sense that the inverse image under

7r

of any compact set is again compact;

this is an immediate consequence of the observation that the mani-

fold P n-l

is compact.

Now it is known that under these circum-

7r(v1) C GL(n,C)r is a complex analytic sub-

stances the image set

variety; this is the Remmert proper mapping theorem, (see for instance R. Gunning and H. Rossi, Analytic Functions of Several Complex Variables (Prentice-Hall, l965), p.160; or R. Narasimhan, Introduction to the Theory of Analytic Spaces (Springer, Lecture Notes in Mathematics No.25, 1966), p.329). consists precisely of those points

that

z

the subset of

for some point

The variety 7r(V1)

(51,...,Sr) a GL(n,C)r

z e P

n-1

and

such

j = 1,...,r ; thus

GL(n,C)r , consisting of r-tuples of linear trans-

formations with one-dimensional common fixed sets, is a complex The set of all k-dimensional linear subspaces

analytic subvariety.

of

Cn

fold for

also has the structure of a compact complex analytic mani-

k > 1 ; these are the Grassmann manifolds Mn,k , (see

for instance F. Hirzebruch, New Topological Methods in Algebraic Geometry, Springer, 1966).

Grassmann manifold n

,k

Repeating the above argument with the in place of the complex projective space

P n-1 = n,l , it follows that the subset of

GL(n,C)r , consisting

of r-tuples of linear transformations with k-dimensional common fixed sets, is a complex analytic subvariety.

GL(n,C)r - GL(n,C)r

Since

is the union of these varieties for

k = 1,...,n-1 , the proof of the lemma is completed.

Remark.

Actually the only part of the preceding lemma

which will be used is the assertion that set of the manifold

GL(n,C)r

is an open sub-

GL(n,C)r ; readers wishing to avoid the

machinery used in the proof of the lemma can construct a direct proof of this assertion.

The fact that the reducible points form

a proper subvariety is of course obvious, since it is well known that there do exist irreducible r-tuples of such matrices.

The Lie group

GL(n,C)

acts as a group of complex analytic

homeomorphisms of the complex manifold

GL(n,

C)r

through the ad-

joint representation; that is, the complex analytic mapping

GL(n,C) X GL(n,C)r -> GL(n,C)r defined by (7)

(T;S1,...,Sr) -> (TSlT 11 ...ITSrT 1) =

exhibits

GL(n,C)

as a complex analytic Lie transformation group

acting on the complex manifold

GL(n,C)r

(For the definition

.

and general properties of Lie transformation groups, see for example S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, 1962), Pages 110 ff..)

the subgroup

The center of

GL(n,C)

Z(n,C) C GL(n,C) , consists of scalar diagonal

matrices; and clearly each matrix T e Z(n,C) identity mapping on the manifold GL(n,C)r .

determines the Thus it is more to

the point to consider the action of the quotient group PGL(n,C) = GL(n,C)/Z(n,C) , the projective general linear group, as a transformation group on the manifold

this quotient group can also be written

GL(n,C)r

.

Note that

PGL(n, C) = SL(n, C)/SL(n, c) fl Z(n, C) , GL(n,C) =

since

SL(n,C) fl Z(n,C)

of matrices of the form

and

a

where

discrete subgroup of SL(n,C) , so group locally isomorphic to The group

PGL(n,C)

is the identity matrix

I

n

is a complex number such that

consists

= 1 ; this is a finite is a complex Lie

PGL(n,C)

SL(n,C) , hence of dimension n -1

acts on the manifold

GL(n,C)r

in a rather

complicated -manner, and no attempt will be made to give a complete

discussion of this matter; as usual, though, the situation can be considerably simplified by restricting attention to suitable invariant subsets of the manifold

GL(n,C)r

.

(Compare with the

discussion in David Mumford, Geometric Invariant Theory (Springer, 1965).)

The subset

GL(n,C)o C GL(n,C)r

consisting of irreducible

points is mapped onto itself under this group action; and in the sequel, only the group action

PGL(n,C) X GL(n,C)r -? GL(n,C)r

(8)

defined as in (7) will be considered. T e PGL(n,C)

except for the identity acts without fixed points,

that is to say, that the group fold

Note that each element

PGL(n,C)

GL(n,C)o ; for if T E GL(n,C) TSiT 1 = Si

(Si) a GL(n,C)r

is a matrix such that

for an irreducible set of matrices

it follows from Schur's lemma that Lemma 28.

acts freely on the mani-

For any integers

T e Z(n,C)

.

r > 1, n > 1 , each point

has an open neighborhood U such that the set

(T a PGL(n,C) jAd(T) U fl u # 01

has compact closure in PGL(n,C) Proof.

.

Recall from Lemma 27 that

subset of the manifold

GL(n,C)r

GL(n,C)r

Each point

.

is an open

(S) d E GL(n,C)o

has an open neighborhood U such that the point set closure U compact and U C GL(n,C)r

Note that the unitary matrices form a

.

compact subgroup U(n) C GL(n,C) ; so replacing the set U by Ad(U(n))-U

if necessary, there is no loss of generality in assum-

ing further that the neighborhood U

is mapped onto itself by the

of unitary matrices.

subgroup U(n) C GL(n,C)

U then has the desired properties.

This neighborhood

For suppose that, in contra-

diction of the desired result, there exists a sequence of elements such that

TV a PGL(n,C)

fl u # 0

TV

have no limit point in

the elements

TV

can be viewed as matrices

in the statements.

Each matrix

where AV, BV

TV =

TV

V , but that

PGL(n,C) ; of course,

the elements

than as elements of the quotient group

for each

TV a SL(n,C)

rather

PGL(n,C) , with no change

can be written in the form

are unitary matrices and DV

is a

diagonal matrix of determinant one, say DV = diag(dV,...,dn) Now on the one hand, the matrices in

SL(n,C)

Dv will have no limit points

either; so after passing to a subsequence if neces-

sary, and relabeling the matrices, it can be assumed that as

idii

bounded as

V -?

V --?

for

for

i = m+l,...,n

it necessarily follows that fl U # 0 for each

i = 1,...,m , but that

1 < m < n .

Idii

Since

remain 1

On the other hand,

V , since the set U

is mapped onto

itself by any unitary matrix; therefore there are points

e U

(SV) j

E U for each index

so that'

V

Since U is compact,

.

after passing to a further subsequence if necessary, it can be assumed that the matrices as

(Sj) E U C GL(n,C)r

explicitly as

(SV) _

(SV)

V -?

(svj ),

converge to a point Writing these matrices out

(S j

(sue) = lira (sue) , note V

that

> -co

V dv sk2

Ad(D,V) - (SV ) = d.

Since the points

belong to the compact set U , all the com-

(SV)

ponents of these matrices are bounded; that is, V

dk Vj

dvaV

< M < m for all

V,j,k,.E

.

d2

Letting and

V --? co , note that

Idk/d'I ---> - whenever k = 1,...,m

L = m+l,...,n ; and therefore sj =

lim

sVJ = 0 whenever

V -'m k = 1,...,m and

L = m+l,...,n .

This means that the point

is reducible, which is impossible since

(Sj)

(Sj) E U C GL(n,C )o ; and

this contradiction serves to conclude the proof. Theorem 27.

For any integers

r > 1, n > 1 , the quotient

space

H(n,C)r = GL(n,C)r/PGL(n,C)

under the group action (7) has the structure of a complex analytic manifold such that the natural projection

Tr: GL(n,C)r -> H(n,C)r

bundle.

is a complex analytic principal PGL(n,C) Remarks.

The conclusion of the theorem is just that the

H(n,C)r , with the natural quotient space topology,

quotient space

can be given the structure of a complex analytic manifold, in such

a manner that each point of H(n,C)r for which the inverse image

7r

has an open neighborhood V

1'(V) C GL(n,C)0

is analytically

homeomorphic to the product manifold PGL(n,C) X V ; and the homeomorphism 7r 1(V) = PGL(n,C) X V commutes with the obvious actions PGL(n,C)

of the group

For a general discussion

on the two sets.

of such group actions, see Richard S. Palais, "On the existence of slices for actions of non-compact Lie groups," Annals of Math. 73 (1961), 295-323Proof.

Select a fixed point

(Sd) a GL(n,C)o , and con-

sider the complex analytic mapping

G: PGL(n,C) -> GL(n,C)r defined by group

G(T) =

PGL(n,C)

for any T E PGL(n,C)

.

Since the

acts freely, this is a regular mapping at each

point of PGL(n,C) , in the sense that the differential dGT has maximal rank at each point

T e PGL(n,C)

.

This is true quite

generally; but for the sake of completeness, and since the explicit formulas will be needed later anyway, a special proof will be given for this instance of the result.

For the result in general, see

for instance L. P. Eisenhart, Continuous Groups of Transformations (Dover, 1961; Princeton University Press, 1933), especially Chapter I.

Introduce local coordinates

(Xi ) e (,pX

n)r

centered at

G(T) = (TSiT ') , by the mapping (Y --a. (TSiT 1.exp xx);

the point

Z e CU

and introduce local coordinates

Xn-1

(viewing

Z

as complex

matrices of trace zero) centered at the point T e PGL(n,C) , by the

mapping Z -?

Z .

identified with the group dinates, the mapping

The group SL(n,C)

is again locally

PGL(n,C) .

In terms of these local coor-

G is described by the coordinate functions

(Gi (Z)) a (Lj, xn)r , where TSiT-1.exp GG(Z) = (T exp Z)Si(T exp Z)-1

;

expanding both sides of this equality in a Taylor series in the variable

Z , note that

TST-1(I + G (Z) + higher powers of G (Z)) _

= T(I+ Z +...)s (I - Z +...

)T-1

i

The differential dGT(Z)

is just the homogeneous linear part of

the Taylor expansion of the function

(Gi(Z)) , so that

dG, (Z) = T(S11ZS -Z)T 1 (9)

=

To show that the linear mapping

dGT,

is of maximal rank, it suf-

fices to show that it has trivial kernel.

Z

If

is a matrix such__ ucLGSJ.

dGT(Z) = 0 , it follows from (9) that

that Since

(Sd)

is an irreducible point, Schur's lemma shows that

is a scalar matrix; and since so

Z = Ad(Si1)-Z = Si Z

tr Z = 0 , necessarily Z = 0 , and

dGT has trivial kernel as desired. Now since the mapping

neighborhood t

G

of the identity

is regular, there is an open I e PGL(n,C)

such that

G is a

complex analytic homeomorphism between submanifold

GSA)

and a complex analytic

t

of an open neighborhood U of

Select an analytic submanifold V C U such that that the tangent spaces of the submanifolds point

GL(n,C)

(Si) e V , and

and V at the

G(0)

are linearly independent subspaces which span the fall

(SJ)

tangent space of the manifold

GL(n,C)o -

The complex analytic Ad(T)-(Xi)

mapping t X V --? GL(n,C)r , defined by for

in

(S

T e t C PGL(n,C)

and

regular mapping at the point

e V C GL(n,C)r , is then also a

(X

(I,(Si)) ; hence, after restricting

the neighborhoods suitably, this mapping is a complex analytic

homeomorphism t X V = U .

To complete the proof of the theorem,

it is only necessary to show that this mapping extends to a complex

analytic homeomorphism from PGL(n,C) X V into

GL(n,C)r ; and it

is clear that for this purpose it suffices to show that, after restricting the neighborhoods further if necessary, no two points

of the submanifold V are equivalent under the action of the group PGL(n,C)

points

Suppose, contrariwise that there are sequences of

.

(X3V), (X'',) e V , and of transformations

lim (X3v) = lim (X''V) _ (Si) V -4 00 V -4 00

such that for each

V

.

= lim

and

(1''V)

From Lemma 28 it is clear that upon choosing a suit-

able subsequence, the transformations

to some element

TV a PGL(n,C)

TV

can be assumed to converge

T e PGL(n,C) ; and since Ad(T)-(S i) =

necessarily T

is the identity

V -4 Go

transformation.

However this means that

ciently large, and it is impossible that = Ad(I).(

'.V)

TV e A for

V

suffi(X'i'V) _

since the mapping is a homeomorphism from t X V

,nr,

.

onto

This contradiction serves to complete the proof of the

U .

theorem.

To apply this theorem to the particular case at hand, coneider again the complex analytic submanifold

HO = R fl GL(n,C)og C GL(n,C)og defined by (4).

For any element

P e PGL(n,C)

and any point

(S,)T,) a GL(n,C)2g , note that

,

it is thus evident that the submanifold Ro C GL(n,C)og served by the adjoint action of the group

is pre-

PGL(n,C) , and that the

quotient space So = R0/PGL(n,C) C GL(n,C)og/PGL(n,C) = 11(n,C)og

(10)

is a complex analytic submanifold of the complex manifold Indeed,

has the inherited structure of a complex analytic

110

principal

H(n,C)og.

bundle over the manifold

PGL(n,C)

So

.

Note in pass-

ing that

dim H(n,C)og = dim GL(n,C)og - dim PGL(n,C) = (2g-l)n2+1 ,

and that

dim So = dim R0 - dim PGL(n,C) = 2(g-1)n +2

.

In summary,

the following holds: Theorem 28. so = Ro/PGT,(n,C)

For

n > 1 , the quotient space

has the structure of a complex analytic manifold

such that the natural projection principal

PGL(n,C)

at a point

p e So

bundle.

module

is a complex analytic

The tangent space to the manifold

S

0

can be identified with the eohomology group

w1(7r1(M),Ad p) of the group Cn X n

R0 -? So

of all

n X n

7r1(M)

with coefficients in the irl(M)-

matrices under the group representa-

tion Ad p .

-196-

The first assertion, as noted above, is an almost

Proof.

immediate corollary of Theorem 27.

For the second assertion,

recall from Theorem 26 that the tangent apace to the manifold at a point

p e R

Zl(T1(M),Ad p)

;

can be identified with the cocycle group

0

indeed, using local coordinates to identify the

tangent space to the manifold the vector space

GL(n,C)og

no C GL(n,C)og

consisting of those 2g-tuples of matrices - X,,

at the point

with

p

(Cn xn)2g of 2g-tuples of complex matrices, the

tangent space to the submanifold

AQ

R0

= Yi

,

is the subspace

(Xj,Yi)

such that

is a cocycle A e Zl(1r1(M),Ad p)

.

The tan-

'rj gent space to the quotient manifold corresponding to a representation space of

So = R0/PGL(n,C) p e Ho

at a point

is just the quotient

Zl(1r1(M),Ad p) modulo the vector subspace which is the

at the point

tangent space to the orbit

p

.

The

tangent spaces to the orbits were essentially calculated in the course of the proof of Theorem 27, however, recalling formula (9); the tangent space to the orbit at

matrices

is just the vector space of

p

(X,, Y,) e (f xn)2g of the form Yi = (Ad(T,l)-

X, = (Ad(S-1) -I) - Z ,

and Z

where

p = (SV,Ti)

zero.

Recalling formula (8) of §7, these matrices correspond pre-

cisely to the group

are arbitrary complex matrices of trace

B1(,r1(M),Ad p)

therefore the tangent space to

S

0

of one-coboundaries; and at

p

is precisely

Zl('R1(M),Ad p )/Bl(1r1(M),Ad P) - Hl(7r1(M),Ad p) , which serves to complete the proof.

This theorem then establishes a natural structure of a complex analytic manifold of dimension 2(g-1)n2+2 Rl(M,GL(n,C))0 C I1(M,GL(n,C))

bundles of rank n > 1

on the set

of irreducible flat complex vector

over the compact Riemann surface M ; the

same assertion holds for the case

n = 1 , since all flat line

bundles over M are irreducible, and the set of all such has the natural structure of the complex manifold (C )2g

(c)

of dimension 2g.

On the quotient manifold So = RJPGL(n,C) there is a

further equivalence relation to be investigated, the complex analytic equivalence of flat vector bundles.

Recall that in the case

of bundles of rank n = 1 , the analytically trivial flat line bundles form the Lie subgroup

Sr(M, &1'G) C 111(M, C*) ; and the set

of analytic equivalence classes of flat line bundles is the quotient

11

Lie group Hl(M, C*)/Sr(M, 8 1'G) , which is the Picard variety of the Riemann surface M , (recalling §8 of last year's lectures).

The set of flat vector bundles of rank n > 1

is not a group, so

the investigation of the corresponding space of equivalence classes We shall begin by examining individual

is rather more complicated.

equivalence classes, as represented by complex analytic connections.

Let VL _ (Ua)

be an open covering of the compact Riemann

surface M , such that VI is a Zeray covering for both flat sheaves and analytic sheaves. X C Zi1(M,GL(n,C))

For a given flat vector bundle

select a representative cocycle

(Xup ) a Z1( 12,GL(n,C))

.

Recall from Theorem 17 in §6(c) that

there is a natural one-to-one correspondence between the set of flat

-198-

vector bundles analytically equivalent to

X and the set

A*(1./,X)

of equivalence classes of complex analytic connections for the bundle

Explicitly, as in §6(c) again, a connection X e AM X)

X .

a zero-cochain since

X

(%a) a c°(Jfj,61'0(Ad x))

is a flat bundle DX = 0

for a flat vector bundle

X

such that

and hence

is

&% = DX ;

Therefore

8% = 0 .

it follows that

A(?A,X) = Z°(M, B1'°(Ad x)) = r(M,c1'°(Ad x)) ; or equivalently, the complex analytic connections for a flat vector bundle

X can

be identified with the space of Prym differentials for the bundle

Ad X tial

Now to each complex analytic connection or Prym differenthere is canonically associated a flat

X e r(M, $11 O(Ad X))

HX(X) a H1(M,GL(n,C))

vector bundle

analytically equivalent to X,

Select holomorphic functions

as follows.

Fa a GL(n,9-U)

in the

a various sets Ua such that DFa = Xa , and set (X'a,) a Z1(O,GL(n,C))

in Ua fl u, ; the cocycle

flat vector bundle

Hx(X)

X',,

.

=

represents the

There is thus a well-defined mapping

Hx: r(M, cP 1' °(Ad X)) --. H1(M, GL(n, C)) ,

Cu)

such that the image of HX consists precisely of the flat vector bundles analytically equivalent to the given bundle in particular,

Hx(0) = X , where

0

X ; note that

denotes the trivial Prym

differential. If

for some connection

X' = Hx(X') a H1(M,GL(n,C))

a r(M, (91'0(Ad X)) , there is a corresponding mapping

HX':

r(M,m1'°(Ad X'))

-199-

-

H1(M,GL(n,C))

and the mappings Hx and Hx,

have precisely the same images;

the relation between these two mappings can be described as follows Select holomorphic functions

Ga E GL(n,CU )

in the various sets

a Ua such that DGa = Xa ; thus the cocycle

-G-1)

'

represents the flat vector bundle

E

Z1(t.t,GL(n,c))

X'

The mapping

.

G: r(M, m l' G(Ad X)) -? r(M, d1' G(Ad X')) defined by (12)

(GX)a =

Xa),

for any

X = (%a) E r(M, 6Ll,O(Ad x))

is clearly a non-singular affine mapping from the space r(M, 9 1'O(Ad X))

onto the space

r(M, o 1'0(Ad x')) ; and moreover,

EX =Hx, G.

(13)

To see this, note that for any connection bundle

Hx(X)

X = (Xa)

the image 1)


Fa E GL(n, 4U) are holomorphic functions such that

, where

DFa = Xa

a The same cocycle can also be written since

D(FaGa1) =

Hx(X) = Hx,(GX)

DGa) as desired.

and =

(GX)a ,

it follows that

Since the principal interest here

lies in the image of the mapping Hx , rather than in the mapping itself, the relation (13) is useful for obtaining a number of different representations for that image set.

In particular, it

is clear that for studying the local properties of the mapping,

it suffices to restrict attention to neighborhoods of the zero

element of

r(M, 8 1'0(Ad X))

.

Returning to the mapping (11) again, note that r(M,(L1'0(Ad x))

is a finite-dimensional complex vector space,

hence has the natural structure of __1a complex analytic manifold.

Recalling Theorem 28, the subset o(M,GL(n,C)) C H1(M,GL(n,C)) consisting of irreducible flat vector bundles also has the structure of a complex analytic manifold, indeed, can be identified with the manifold

described above.

So

r0 (M, B 11 0(Ad X)) C r(M, ® 1'0(Ad X))

Letting

be the subset consisting of

those complex analytic connections which determine irreducible flat vector bundles, the restriction of the mapping (11) is the mapping

HX: r0(M, (S!l'0(Ad X)) - Ho(M,GL(n,C)) = So from a subset of the complex manifold complex manifold

So

Lemma 29.

r(M, m 1'0(Ad X))

into the

.

The subset

r0(M, Qv1'0(Ad X)) C r(M, dt1'°(Ad x))

is the complement of a complex analytic subvariety of r(M,01'0(Ad X)), hence is itself a connected complex analytic manifold; and the mapping

HX

is a complex analytic mapping.

Proof.

Select any point Xo a r(M, (91' 0(Ad X) ) , and let

Xi e r(M, d11'0(Ad x))

,

1 < i < r , be a basis for this space of

Prym differentials; then local complex analytic coordinates t = (ti,...Itr)

centered at

X°

can be introduced by the mapping r

(t1,...,tr)

0 (X° + Et i=l

.

In each coordinate neighborhood a C M with local coordinate

za,

select-a point pa ; and let 0 C Cr be an open neighborhood of the origin in the space

Cr with coordinates

the open subset Ua X 0 C C1+r

(t1,...,tr)

.

in

consider the system of partial

differential equations a

(14)

Fa(za,t) =

E ti?a(za)]

aza where

i=l

Fa: Ua X A - Cu

X n

are matrix-valued functions subject

to the initial conditions

Fa(pa,t) = I

(15)

for all t e t

.

It follows from the Cauchy-Kawalewsky Theorem (see for instance Courant-Hilbert, Methods of Mathematical Physics (Interscience,1962), vol. II), that there are unique holomorphic solutions

Fa(za,t)

of the differential equation (14+) satisfying the initial conditions

(15), provided that the neighborhoods Ua and t sufficiently small; since the covering 2

M being compact, the same set

Ua

t

both are chosen

can be assumed finite,

can be used with all the sets

The function F. can of course be assumed to be non-singular

t h r o u g h o u t Ua X 0 .

For any fixed point

t = (t1,...,tr) e t ,

the cocycle

-F(zt)) a Z1(VL,GL(n,C)) r

represents the flat vector bundle HX(X° + E ti?i) ; and the components of the matrices

in the set 0 .

(XV (t))

are holomorphic functions of t

Passing to the characteristic representation of

the bundle, it is obvious that there results a complex analytic

mapping

(16)

A -k R = Hom(7f1(M), GL(n, C)) C GL(n, C)2g .

The subset R C R of irreducible representations is the comple0

ment of a complex analytic subvariety of R , as seen earlier; so, since the mapping (16) is complex analytic, the inverse image

of the set R

0

o

is either empty, or all of A , or the complement

of a proper complex analytic subvariety of A . proves the first assertion of the lemma.

Note that this

Upon restricting the

mapping (16) to the subset o C t , and following it with the complex analytic projection R

0

--? R0/PGL(n,C) = So , it follows

that the mapping HX is a complex analytic mapping in the set and the proof is thereby concluded. Remarks. subset

If the given bundle

r0(M, m 1'0(Ad X)) C r(M,

X

is not irreducible, the

1'0(Ad x))

may be empty, insofar

as the preceding lemma goes; this point will be taken up again later.

Of course, in view of the remarks immediately preceding

the lemma, it would have been sufficient to take mapping (12) is clearly complex analytic.

Xo = 0 ; for the

This was not done,

merely because the saving in effort would have been negligible.

The mapping HX of (ll) is not always a one-to-one mapping onto its image; as proved in Theorem 17, two complex analytic

connections have the same image under HX if and only if they are equivalent in the sense of the definition on page 115.

to study the image of HX ,

In order

it is convenient to pass to the set of

equivalence classes of connections, and to consider the mapping

induced by H. on this set; and that requires a more detailed examination of the equivalence relation.

For any flat vector bundle End(X)

X e Ir11(M,GL(n,C)) , the set

X

of complex analytic endomorphisms of

is a finite dimen-

sional complex vector space, which can be identified with the space r(M,61(Ad X)) ; in terms of an open covering 2

surface M , an element matrix-valued functions

T e End(X) Ta E

IgU X n

= {Ua)

of the

is described by holomorphic

such that TaX = Xa$T , or

a equivalently Ta =

in each intersection Ua f U

Actually of course,

End(X)

is a finite dimensional algebra over

the complex numbers, as discussed on page 105.

ible endomorphisms is the group Aut(X) C End(X) lytic automorphisms of group of dimension Aut(X)

s , where

of complex ana-

is a complex Lie

s = dim r(M, B (Ad X))

.

I e GL(n,C)

is the identity matrix and

is an arbitrary non-zero constant; this is a normal, even

a central, subgroup of Aut(X) , and is isomorphic to quotient group

Aut(X)/C*

= P Aut(X)

P Aut(X)

is a complex Lie group of dimension

page 115, to each T E Aut(X)

C

.

The

will be called the projective

group of complex analytic automorphisms of the bundle that

The group

contains as a Lie subgroup the set of automorphisms of the

form T. = cI , where c e C

X ; clearly Aut(X)

The set of invert-

X ; note s - 1

As on

there is associated the mapping

Ad*(T): r(M,011'0(Ad X)) -k r(M,(Q1'0(Ad X)) defined by (17)

DTa) = Ad(Ta)Xa- dTaTa1

for any X = (Xa) a r(M, 01'0(Ad X)) exhibits

Aut(X)

.

It is easily seen that this

as a complex Lie group of nonsingular complex

affine transformations of the vector space

r(M,a1'0(Ad X))

;

actually, since it is obvious that the automorphisms of the form Ta = cI

act trivially, this also defines a similar action of the

quotient group P Aut(X)

.

The quotient space

r(M, m 1' 0(Ad X) )/ P Aut(X) under this group action is precisely the space

A (vj,X)

of equiv-

alence classes of complex analytic connections for the vector bundle

X As a slight digression, consider an equivalent bundle

X' = Hx(X') , for some connection

X' a r(M, d 1'0(Ad X)) ;

and as

before introduce the mapping

G: r(M, m 1' 0(Ad x)) -> r(M, (91' G(Ad x') ) defined by (]2), for some functions DGa = Xa .

Ga a GL(n,0 U) such that

There is a corresponding mapping

G: r(M, B (Ad x)) ---> r(M, m (Ad x') ) defined by

for any T=(Ta) a r(M,0(Ad x))

(19)

and this of course induces a mapping G: Aut(X) -> Aut(X') Now it is easy to see that (20)

GAd*(T) = Ad*(GT)G

,

or equivalently, that the following diagram is commutative for any

T c Aut (X)

:

Ad (T)

r(M,(q" 0(Ad X))

r(M, m1'0(Ad x,)). For given any connection

> r(M, 0 1'0(Ad X))

Ad* cam')

r(M,(91'0(Ad x-))

X = (%a) c r(M, 0 1'0(Ad x)) ,

it follows

that Ad(GCFa%

DGa)-Ad(GWa_ )DGa Ad(Ga)DTa + Ad(Ga)DG.]

=

= Ad(Ga)(Ad(Ta)(%a - DTa) - Xal =

Thus the mappings

group Aut(X)

X'

GAd*(T)X

.

G transform the action of the transformation

on the space

into the action of

r(M, 9 1'0(Ad X))

the transformation group Aut(X') whenever

aa1)l

D(Gc

is equivalent to

on the space

r(M,(Q"0(Ad X')),

X ; once again, this is a useful

observation, enabling local questions to be considered near the

neighborhood of the trivial connection

X = 0

alone.

Now the question arises how this action of the transfor-

mation group Aut(X)

affects the special subset

ro(M, (P 1'0(Ad X)) C r(M, o-1'0(Ad X)) , as above; and the answer is

provided by the following simple but interesting observation. Lemma 30.

A connection

fixed by a transformation

X c r(M,6L1'0(Ad X))

T c P Aut(X) ,

T # I ,

is left

if and only if

X corresponds to a reducible flat vector bundle. P0(M, LvL,O(Ad X)) C P(M, 071'0(Ad x))

The subset

of connections corresponding

to irreducible flat vector bundles is thus preserved under the

transformation group P Aut(X) ; and

acts freely as a

P Aut(X)

complex Lie group of nonsingular affine transformations on the subset

P0(M,(R1'O(Ad X)) Proof.

of the complex vector space P(M, CD1'0(Ad X)).

Suppose that a connection

left fixed by an element

T c P Aut(X),

Xa = Ad(Ta)-(%a - DTa) , where

X c P(M, m 1'0(Ad X))

T

is

I ; thus

is an automorphism

(Ta) c Aut(X)

which is not given by a scalar matrix, that is, which is not of

the form cI .

Write

X. = DFa for some functions

Fa a GL(n,6LU

a so that t h e cocycle

(X'

(Fa X

-P-1)

represents the flat

vector bundle corresponding to the connection Sa =

in each set U. .

X ; and set

Note that

DSa = Ad(FTAd(FaT; so that the matrices

Ad(Ta)-(Xa - DTa)] = o

S. are nonsingular constant matrices; and

note further that

Sa X' (18)

The matrices

=

FFaFal

_

Faxpl

1 -1 = FaXWD FpTep = X'' Up s P

Sa are not scalar matrices, since

Ta

are not scalar

matrices; and it follows immediately from (18) then that the flat vector bundle represented by the cocycle

(X'

00

)

is reducible.

Conversely, if the flat vector bundle represented by the cocycle

(XI ),associated to the connection X nonsingular constant matrices

is reducible, there exist

Sa which are not scalar matrices

but which satisfy

Reversing the preceding

argument, the matrices

represent a non-scalar

Ta =

automorphism of the bundle

(x

)

such that Ad*(T)'% = X .

This

proves the first statement of the lemma; and since the second statement is an immediate consequence of the first, the proof is thereby concluded.

Since the complex Lie group P Aut(X)

acts freely as a

group of complex analytic automorphisms of the complex manifold r0(M,(S 1'°(Ad X))

,

the natural supposition is that the analogue

of Theorem 27 holds, that is, that the quotient space

(21)

Ao(M,x) = ro(M, 0 1'C(Ad X))/P Aut(X)

has the structure of a complex analytic manifold such that the natural projection

(22)

ro(M, (9 1'0(Ad x)) --- A*(M,ic)

is a complex analytic principal P Aut(X)

bundle.

orbits locally are submanifolds, and the manifold

The individual r0(M, &

1,0

(Ad X))

locally has such a product structure, on general principles.

In

order to prove the supposition, referring back to the proof of Theorem 27, it is only necessary to establish an analogue of

Lemma 28 for the action of the group P Aut(X)

this result for bundles of rank n = 2

.

We shall establish

in the course of a more

explicit, analysis in the subsequent paragraphs.

Before turning to

this, however, it is interesting to see what can be said about the

complex manifold A*(M,X)

in general, assuming the truth of this

supposition, and with as little work as possible. First, it is easy to see that the complex manifold A0(M' X)

has complex dimension n2(g-l) + 1 where

g

is the genus of the compact Riemann surface M .

is also assuming that the manifold course.)

for any bundle X e H1(M,GL(n,C)),

Ao(M,X)

(This

is not empty, of

For from the fibration (22) and earlier observations it

follows that dim A:(M,X) = dim r0(M, 0 1'0(Ad X)) - dim P Aut(X)

= dim r(M, 61' 0(Ad X)) - dim r(M, & (Ad X)) + 1 while from the Riemann-Roch theorem in the form given by formula (9) of §1+, applied to the bundle Ad X , it follows that dim r(M,& (Ad x)) - dim r(M,a 1'°(Ad X*)) = n2(1-g)

* Since the bundles Ad X and Ad X

.

are canonically isomorphic,

the desired result follows immediately.

By the way, if one is

interested only in bundles of determinant one, hence in connections

of trace zero, the corresponding space has dimension n2(g-1)

;

in

many ways, this is the more natural space to consider.

Next, the tangent space to the manifold Ao(M,X) point corresponding to the trivial connection

at the

X = 0 e r(M, 4 1'0(Ad X))

can be identified in a natural manner with the vector space

(23)

r(M, d 11 0(Ad X))/dr(M, B (Ad x))

.

On the one hand, since complex vector space

r(M, C9 1'0(Ad X))

,

is an open subset of the the tangent space to

at any point can be identified with the vector

r0(M, 011 O(Ad X))

space

ro(M, 8 1'O(Ad X))

r(M, B 1'0(Ad X))

itself.

On the other hand, since Aut(X)

is the group of invertible elements in the algebra End(X) , the tangent space to the space

Aut(X)

at the identity can be identified with

End(X) = r(M, 61(Ad x)) ; explicitly, if

T(t) = (Ta(t))

is a one-parameter subgroup of Aut(X) , then

a r(M,0(Ad X))

Aa = dt Ta(t)I

t=0

Now the orbit of the trivial connection

X = 0 e r(M, & 1'0(Ad X))

under the action of a one-parameter subgroup

given by

T(t) C Aut(X) and since

-1

dt(Ad(T(t))'X)aIt

=0

the tangents to the orbits of the group X = 0

-1

= L_a(_) Ta + dTaTa dta Ta

= -dAa a dr(M,6 (Ad x))

connection

is

Aut(X)

t=0

,

at the trivial

form the vector subspace

dr(M,(P(Ad X)) C r(M, 61'0(Ad x)) , which suffices to conclude the proof of the result.

In connection with this observation, it should

be noted that the assertion is vacuous unless the bundle itself irreducible.

of the bundle

Ad(X)

X is

If X is irreducible, the constant sections form a one-dimensional family; thus

dim dr(M, 6(Ad x)) = dim r(M,0 (Ad X))- 1 , and the dimension of the tangent space (23) agrees with the dimension of the manifold

A:(M,X)

as calculated in the preceding paragraph.

Considering in more detail the special case of bundles of

(d)

rank n = 2 , recall from §5 that the complex analytic vector bundle corresponding to a flat complex vector bundle

X c H1(M,GL(2,C))

can be represented by a cocycle of the form

T

C'O )

(O ?VA) _

(24+)

E Z1(Vt, b (2, & ))

;

cp

((pi)

the components

bundles

cpi

assumed that

are cocycles representing complex line

such that cpl C X and T2 = X/cpl , and it can be c((pl) = div X

(Recall that div X

.

is the maximum

value of the Chern classes of complex line bundles which can appear

as subbundles of X .)

X

Since

c((pl) + c((p2) = 0 , and that

X

is a flat bundle, it follows that

is analytically indecomposable

unless c((pl) = c((p2) = 0 ; and furthermore, -g < c((pl) < g-l is the genus of the Riemann surface M .

where

g

or if

c((pl) = 0

and the bundle

then the line bundle

cp1

X

If

c((pl) > 0

is analytically indecomposable,

is uniquely determined; indeed,

is 1P1

the unique line bundle such that

cp1 C X and

c((pl) > 0 .

Paralleling the discussion on page 200, there is a complex analytic mapping

A(?A,X) --> A(V,(0C )) commuting with the actions (17) of the complex analytic transfor-

mation group Aut(X) ; so the discussion of the structure of the

set, of equivalence classes of complex analytic connections for the

X can be translated into terms of complex analytic connec-

bundle

This is the key to the following dis-

tions for the cocycle (24). cussion. Remarks.

In the discussion of complex analytic connections

in §6(c), connections were actually only defined for a specific cocycle representing the complex analytic vector bundle in terms of the given open covering 1R. ; thus the set of connections pro-

rather than by

bably should'have been denoted by A(V( ()C43 )) A(VL ,0)

Whenever cocycles

.

and

(0aO)

(0'

)

represent the

same complex analytic vector bundle, formula (12) can be used to establish an isomorphism between the sets

A(VL,(0CO))

and

A(VL,(0)) ; the temptation is to call the connections related by this isomorphism equivalent, and to define the set of connections

A(TL ,0)

0

for the vector bundle

as the set of equiva-

lence classes of connections for all the cocycles representing the bundle

0

.

The mapping (12) is not always uniquely defined, how-

ever; so that it is first necessary to pass to the set of equivalence classes

A(Vt

(0

))/Aut(o) = A*(1JL,()C43 ))

is a flat cocycle, the set

the set

.

When

(0aO)

CO

r(M,(TLO(Ad c)))

A(V/l,(0CO))

can be identified with

of Prym differentials for the flat vector

bundle, and the complications are less.

If X = (%a) 6 A(Vt ,(0

))

tion for the cocycle (24), the terms

is a complex analytic

Xa and

2 X 2

connec

matrices of

holomorphic differential forms in the various open sets

Ua of the

, such that in each intersection Ua fl U

covering

DO

= 6(%a) = Xp - Ad(O

).%

This equation can be rewritten more explicitly as

(25)

Clap TOO (P2aO

"11p X12 1)

2

'21P

x11a `12a l 221a

CP2a0 T

fdq 0

'122a

d'r

dP2a0

A more detailed analysis of the component functions of such a connection is not of great interest in general; but the following simple observations will be of use later. Lemma 31.

If there exists a complex analytic connection

X = (%a) a A(7K,()C43 ))

for the cocycle (24+) such that

"2la = 0

then necessarily c(cpl) = c((p2) = 0 ; and if c((pl) = c((p2) = 0 there exist complex analytic connections

' = (%a) e A(2 ,()at3))

for

the cocycle (24+) such that '121a = 0 and Alla # X22a , on surfaces of genus

g>1. Proof.

which

If

? = (%a)

is a complex analytic connection for

'121a = 0 , then writing out equation (25) in detail, it follows

that the remaining components of the connection

"a are arbitrary

analytic differential forms subject only to the conditions

"lip - %,,a = d log (Pla0 (26)

k22P

X22a = d log (P2aO

X12 " T10092a0>Ì2a = (%I]

"X22 )'PlaoTao +

'P1

dTao

The first line in (26) shows that the cohomology class in

represented by the cocycle (d log (plCO)

Hl(M, (S1'0)

is trivial,

hence as in Lemma 19 of last year's lectures, necessarily c((pl) = 0 ; applying the same argument to the second line in (26),

or recalling that

c ((p2) = 0 . and

cp2

c((pl) + c((p2) = 0

o

it also follows that

Now assume that c (cpl) = c (cp2) = 0 ; the bundles

have flat representatives, so that the cocycles

be taken to be constants, hence d log cpl

7122a =

P

lLl =

can

cpi

= d log (P2CvP = 0 -

first two lines in (26) then merely assert that

and

cpl

a

The

=

are global holomorphic differential forms

on the Riemann surface; and in order that there should exist holomorphic differential forms

a

the difference

must be an abelian differential such

a =

- 722

satisfying the third line in (26),

that the cohomology class in H1( M,O1'0((p11cp2)) represented by the cocycle (27)

cp14dTCO

a

is trivial. By the Serre duality theorem, r(M, m ((P1(P21)) .

that

a

if

cpl

cp2

, then

can be completely arbitrary.

111(M, B 1, O(cpllcp2)) ti

r(M, d ((P1w21)) = o

If

cpl = cp2

, so

, then

r(M,8 ((p1(p21)) = r(M, 0) = C ; the mapping which takes an abelian differential

a

into the cohomology class represented by the co-

cycle (27) is a linear mapping

surface M has genus differentials

a

r(M, d

110)

--> C , and since the

g > 1 , there will be nontrivial abelian

in the kernel of this mapping.

conclude the proof of the lemma.

That suffices to

Now consider an element

T e P Aut(X)

If the bundle

.

X

is represented by the cocycle (24+), then a representative (Ta) E Aut(X) = Aut(&CO )

matrices

T. E GL(2,C1U)

of T will consist of a collection of such that

in Ua fl U

0CO-TP

a This can be written more explicitly as

(tlla

(28)

t12a

t21a t22

play

Tai

G

2a$

Using this, the group P Aut(X)

_

play Tai

tl1

G

t2l

Let

If

t22

X , as follows.

X E H1(M,GL(2,C))

be a flat vector bundle

of rank 2 on a compact Riemann surface M of genus (i)

12

can be described quite simply in

terms of the analytic invariants of the bundle Theorem 29.

t

g > 1

.

X represents a stable complex analytic vector bundle,

then P Aut(X) = 1 , the trivial group. (ii)

If

X represents an indecomposable unstable complex ana-

lytic vector bundle, then P Aut(X) = r(M, o ((pl(p2l) )

(iii)

.

If X = cpl ®cp2 is a decomposable flat vector bundle,

then

P Aut(X)

Prcof.

(i)

if tP0L(2,) if

cpl

T2

cpl = p2

For any non-trivial element

T e P Aut(X) , it

follows from Lemma 19 that there are representatives (0C43 )

of

E

Z1(Vt, h X (2, Ot )) of X and

T such that the matrices

0CO

(Ta) E C°(fl, hl (2, B ))

and T. are all in upper tri-

angular form; the matrices

0Op will still be written out with

the notation used in (24+), although of course it is not necessarily true that

t21a

and remembering that

(29)

Writing out equation (28) in detail,

c((pl) = div(X)

I

= t21p =

0 , it follows that

t22a = t22P ' -1 t12P - p1ap'P2apt12a = (tlla -

tiia = tllp '

The first line in (29) shows that t22 = t22a = t22P

-1

PaP

t11 = t1a = t11

are holomorphic functions on the entire Riemann

surface, hence constants.

If

t11 1 t22

,

the second line in (29)

shows that the cohomology class in H-(MLQ ((pll(p2)) by the cocycle

((pilT

)

represented

is trivial; it then follows from

Theorem 13 that the bundle is unstable.

and

X is analytically decomposable, hence

t11 = t22 , the second line in (29) shows that

If

tea = cpl cp2't12

Since

.

is non-trivial,

T E P Aut(X)

t12a # 0 ; hence 2c((pl) = c(cpl(p2l) > 0 , and the bundle again unstable.

Altogether then, if P Aut(X)

group, the complex analytic vector bundle

X

is

X

is a non-trivial

is necessarily

unstable; and this proves part (i) of the theorem. (ii)

If

X

is an unstable complex analytic vector bundle,

select a representative cocycle (0

) a

standard form (24+); and for any element

representative

Z1(Vt, b a^( (2, m ))

T E P Aut(X) , select a

explicitly as in (28).

(Ta)

in the

Considering the com-

ponents in the second row and first column of (28), note that

t21a

= p-a5q)2aO.t21P

Since the bundle .

X is unstable,

= -2c((p1) < 0 ; therefore t2la = 0 , except in the case

c(cpil(p2) that

cp1 = cp2

constant.

, when

t21

= t2la = t21P

is an arbitrary complex

In this latter case though, considering the components

in the first row and first column of (28), it follows that

t

tllp-tlla = and as before, if posable.

t21 1 0

X

the bundle

;

is analytically decom-

In amy case then, necessarily t21a = 0 ; so the condi-

tions (28) again reduce to the conditions (29).

above, the first line in (29) shows that

As in part (i)

and

tl1 = tlla = t1

X

are complex constants; and since the bundle

t22 = t22a ' t220

is analytically indecomposable, the second line in (29) shows that

t11 = t22 and that (t12a) a P(M,61((p1(p21)) matrices

Multiplying the

Ta by a complex constant, the element

a representative

Ta = 0

C1

T E P Aut(X)

has

in the form

(Ta)

t12a) (30)

-

, for any tl2a E P(M, 6. ((P1IP21) )

1

and it is obvious that this establishes a one-to-one correspondence, and

indeed a Lie group isomorphism, between the groups P Aut(X)

P(M, a ((P1(P21)) (iii)

If

X

is an analytically decomposable vector bundle,

select a representative cocycle standard form (24+), with

and consider an element

as in (28). If

cpi

(0ap )

E

in the

Z1(l/l, b -4 (2, 61))

complex constants, and

T

= 0

T E P Aut(X) , with a representative

cp1 # cp2 ,

it follows as in part (ii) that

(Ta) t21a=0,

and that

tll = tlla = tllp and

are constants;

t22 = t22a = t22P

and since (t12a) c r(M, m ((pl(ppl')) = 0 , it further follows that t12a = 0

T. by a complex constant,

Multiplying the matrices

.

T c P Aut(X)

the element

has a representative

1

0

0

t22

Ta

(31)

for any

equation (28) reduces to a GL(2,C)

T = Ta = T PGL(2,C)

.

T. = T

of the form

t22 e C

and this establishes the isomorphism P Aut(X) = cpl = tp2

(Ta)

C

.

Finally, if

; so the matrices

are arbitrary, hence clearly P Aut(X)

The proof of the theorem is thereby concluded.

Finally, consider the action of the complex Lie group P Aut(X)

as a group of complex analytic automorphisms of the com-

plex manifold A(V(,X) = A(v1,(0a0)) sented by a transformation X = (Xa) e A(Vt,(Oa0))

If

.

(Ta) c Aut(Oa9)

T e P Aut(X) ,

is repre-

and if

is a complex analytic connection, the group

action as defined by (17) is

(Ad*(T)'X)a = TaX

a1

-

dTaTa

Using the detailed description of the group P Aut(X)

as provided

by Theorem 29, the supposition of page 208 can be verified in this case as follows. Theorem 30.

If X C H1'(M, GL(2,C))

is a flat vector bundle

of rank 2 on a compact Riemann surface M of genus the quotient space

Ao(M,X) = o(M, (D1'0(Ad X))/P Aut(X)

g > 1 , then

has the structure of a complex analytic manifold of complex dimension

l-g - 3

such that the natural projection

n:

r0 (M, 61' 0(Ad X)) ---? Ao (M, X)

is a complex analytic principal P Aut(X) Proof.

If

bundle.

X represents a stable complex analytic vector

bundle, then from part (i) of Theorem 29 the group P Aut(X) trivial; the desired result is an immediate consequence.

Note that

X are neces-

all flat vector bundles analytically equivalent to sarily irreducible, so that

is

r0(M, (Q1'0(Ad x)) = r(M, O 1'0(Ad X))

is non-empty; the formula for the dimension of the space

A0(M,X)

was derived on page 209. If

X represents an unstable indecomposable complex ana-

lytic vector bundle, then from part (ii) of Theorem 29 there is an

isomorphism P Aut(X) = r(M,o ((pl(p2l)) ; the isomorphism is given explicitly by (30). for the element

Writing out the group action (17) in detail,

T e P Aut(X)

corresponding to a section

(ta) a r(M, & ((pl(ppl)) and for any connection X = (%a) e A(?/C, (0aO) 1 (Ad*(T)-X)a

to V "71a

0

72a

1

-ta

0

1

0

dt

-ta

(32)

("Ila+ taX2la '21a First, consider the case that

X12a+ ta(-22a - Xlla) - t 7 1a - dta X22a - ta-11-21a c((pl) = div(X) > 0 ; all flat vector

bundles analytically equivalent to

X are irreducible, so that

r0(M, 0 1'0(Ad x)) = r(M, (9 1'0(Ad x))

is non-empty.

(To see the

same thing in a different manner, by Lemma 30 the connections cor-

responding to reducible flat vector bundles are fixed points for

some transformations of the group P Aut(X)

if X = (%a)

.

is

fixed for a non-trivial element

T E P Aut(X) ,

(32) that

c((p1) > 0 , lemma 31 shows that

'21a = 0 ; but since

this is impossible.)

it is clear from

acts freely as a

The Lie group P Aut(X)

group of complex analytic automorphisms of the complex manifold. A(1/L,(0C43 ))

Upon examining the proof of Theorem 27, it is clear

.

that it suffices merely to prove an analogue of lemma 28 in this case.

since

The proof of this analogue is quite trivial, though; for '121a

# 0 by lemma 31, it follows from (32) that the group

P Aut(X) = r(M, d ((p1(p21))

acts as a non-trivial group of trans-

lations on the components in the first row and first column of the connections, and the desired result is an immediate consequence.

Next, considering the case that

c((p1) = c((p2) = div(X) = 0

if

0 ti=

P Aut(X)

(p1p21))

r(M, d

cp1 = cp2

.

X = (Xa)

cp2

, there only remains

is given by (32), where

an arbitrary complex constant.

t # 0

cpl

The action of the group P Aut(X) = C on

the manifold A(Vt,(Oa$))

transformation

T2

C if cpl=T2

the desired result being trivial if the subcase

cpl

t = to E C is

Note that the fixed points of a

are those complex analytic connections

such that '121a = 0 and

22a =

77.1a

; Iemm 31 shows

that there are connections which are not left fixed by any nontrivial transformation, so that there are always irreducible flat vector bundles equivalent to the given bundle

X .

Restricting

attention to these connections, it again follows immediately from

(32) that the group P Aut(X)

always acts as a non-trivial group

of translations on some component of the connections, and the proof is completed as above.

X represents a decomposable complex analytic

Finally, if vector bundle

X = cp1

for the case that

, then upon writing (25) out explicitly

cp2

TC43 = 0 , the set

A(UL,(0CO ))

of complex

analytic connections has the form

X = (%a) = (Xi ja) , for arbitrary Xija a P(M, (L 1, 0((pi(p;l) ) Note that dim r(M, o 1,0((pi(p31)) = g or g-l according as cpi = cp3

or

cpi # cp j

First, suppose that

; hence dim r(M, Q 1' 0((pi(pl)) > 0 always. cp1 # cp2 ; so by part (iii) of Theorem 30, note

that P Aut(X) = C* , the isomorphism being given explicitly by (31) Writing out the group action (17) in detail, for the element T e P Aut(X) X = (%a) c A(

c E C and for any connection

corresponding to ,(0CO))

,

C

1

0 Y "71a `12a

0

c

II 7"ry

X]1a

c 2a

°'21a

`22a

1 A0

01

1/c)

The set of connections which are not fixed points of any nontrivial

T e X Aut(X)

is described by X21a

12a # 0 , and is

obviously a non-empty set; the action of the group

C* obviously

satisfies the analogue of Lemma 28, which completes the proof of this case.

lastly, suppose that

cpl = T2

,

so that from part (iii)

P Aut(X) = PGL(2,C)

of Theorem 30 again,

The connections are

arbitrary matrices of abelian differentials, and the group PGL(2,C) acts on these matrices by inner automorphism.

The set of connec-

tions which are not fixed points of any non-trivial P c P Aut(X) _ = PGL(2,C)

is just the set of irreducible matrices of abelian dif-

ferentials, and is clearly non-empty since

g > 1 .

This case of

the theorem reduces almost immediately to Theorem 27 itself; and with that, the proof of the entire theorem is concluded. Remarks.

In the course of the above proof it was demon-

strated that the sets

A0(M,X)

are always non-empty.

This can be

restated as the assertion that every flat vector bundle is analytically equivalent to an irreducible

X e 1it(M,GL(2,C))

It follows from this that in examining the

flat vector bundle.

set of complex analytic equivalence classes of flat vector bundles X C Hf(M,GL(2,C))

,

there is really no loss of generality at all

in restricting attention to the set of irreducible flat vector bundles; thus the fact that the discussion in §9(b) was restricted to the subset

Ro C R , is of no great concern after all.

The manifolds (lg-3

Ao(M,X)

all have the same complex dimension

for bundles of rank n = 2), as noted on page 209; but they[

are not all analytically homeomorphic.

If X is a stable complex

analytic vector bundle, or more generally if

that P Aut(X) = 1

X

is a bundle such

(a class of bundles called simple complex

vector bundles, for the obvious reason), then Cog-3

A*(,X) 1, P(M,(O 1'0(Ad X) )

.

If X represents a decom-

posable complex analytic vector bundle. X = cpl ®cp2 cp1 # cp2

,

where

then as noted in the proof of Theorem 30,

Ao(M, X) 2_' (Cg XC XEg-1 XEg-1)/C

(33) Cg-l

where

E

=

g-1

- (0,...,0)

is the set of non-zero elements in

C-1 and the action of the group e is given by 1

c (z1)z 2,w1,w2) = (z1oz2)cw1,cw2) for

c c

C

,

zi a C, wi a Eg-1 .

It is evident that the space Cog-3

(33) is not even topologically homeomorphic to the space

It is quite possible to read off from the proof of Theorem 30 a description of the complex manifolds

Ao(M,X) ; the group actions

are also interesting, remembering for instance the quadratic term in (32).

But there is not time enough here to pursue this matter

further. The discussion on pages 197-201 of last year's lectures should be looped at in the light of the above discussion; see the appendix in R. C. Gunning, "Special coordinate coverings of Riemann surfaces," (Math. Annalen 170(1967), 67-86), for more details.

(e)

Summarizing briefly the conclusions of the preceding parts

(c) and (d), it is apparent that the mapping (11) induces a one-to-

one complex analytic mapping between the following two complex manifolds

r (M, &1'0(Ad X)) (311.)

HX: Ao (M, X) =

--? So = Ho (M, GL(n, C) )

P ut

o

at least in the case of flat complex vector bundles

n = 2 ; and the image of HX

X of rank

is the set of irreducible flat vector

bundles analytically equivalent to

X .

Now it is a quite straight-

forward matter to describe the differential of this mapping; and the following observation then results. Theorem 31.

If

X E H (M,GL(2,C))

is a flat complex

vector bundle of rank 2 on a compact Riemann surface M of genus g > 1 , then the mapping (34) is a regular mapping (has a non-

singular differential); thus the image of HX is locally a complex analytic submanifold of the complex manifold So , (in the sense that the image of a relatively compact subset of Ao(M,X)

Identifying the tangent space of So

submanifold).

the space

image (4)

H1(M, V'(Ad X))

,

X with

the tangent space to the submanifold

is the subspace of H1(M, 3 (Ad X))

the period classes of the Prym differentials Proof.

at

is a

consisting of

r(M, (51'0(Ad X))

It suffices merely to consider an open neighbor-

hood of the trivial connection pages 200 and 205 .

X = 0 , in view of the remarks on

The tangent space to

at

r0(M, 6 1'0(Ad X))

X = 0 is identified with the vector space r(M, 01'0(Ad X))

,

as

For any connection

before.

X = (%a) e r(M, 6L 1'0(Ad x))

HX(t?)

the image

t F C ;

of this family under the mapping (11) is a difSo , and the tangent vector to

ferentiable curve in the manifold this curve at the point dHX(X)

for

tX = (tXa)

the one-parameter family of connections

consider

,

HX(0) = X

is just the image vector

This latter tangent vector is of course just the deriva-

.

t = 0

at the point

tive of the vector-valued function HX(t?)

X

when that function is expressed in local coordinates near

To carry out this calculation explicitly, select a suitable open coordinate covering with local coordinate

of the Riemann surface

VL = (Ua)

M,

za in Ua ; and select a base point As in the proof of Lemma 29, for

pa a Ua for each neighborhood.

a suitable open neighborhood A of the origin in the complex t-plane, choose holomorphic functions

Fa(za,t) 6 GL(2,0-U

a such that

S- Fa(za,t) dza = Fa(za,t) t X a(za)

,

and

a (35)

L

Fa(pa,t) = I

t e A

for all

The cocycle

(36)

E

(Xto(t)) =

for each

then represents the image

HX(t?)

Fa(za,0) = I , hence that

Xao(0) = Xto X .

representing the flat vector bundle morphic functions

(37)

Ga(za) a (2

X2 a

dGa(za) = Xa(za)

Z1(

t e A ; note that

is the given cocycle Furthermore, choose holo-

such that and

,GL(2,C))

Ga(pa) = 0

As on page 161, the constants

(38)

Aa =

form a one-cocycle

(Aap) a Zl('UL, 9'(Ad X)) , which represents the

period class A = 8(x)

of the Prym differential

X e r(M,l -'0(Ad X))

The cocycles (36) and (38) are related in

.

the following manner. from (35) that to

t

As noted already, it follows immediately

Fa(za,0) = I

.

Differentiating (35) with respect

t = 0 , note that

and setting

as / 6Fa

az 1\ a- (za,0) dza = Xa(za) ,

6Fa (pa,0) = 0

6Fa consequently it is clear that

(zaO) = Ga(za)

t = 0

.

Then

= Xa$LGa(za)X00 - Xa$Gp(zp)]

1

(39)

_-AC43 . To express the mapping HX(tX)

in terms of the complex coordinates

introduced on the manifold So = Ho(M,GL(2,C))

in parts (a) and

(b) above, it is necessary to go from the cocycle (36) to the corresponding characteristic representation of the bundle.

Let

,...,Ua ) be a closed chain in 1f1(Vt, Uo) represental q ing one of the standard generators of the fundamental group of the

a = (Ua ,U o

surface.

The matrix associated to

a

describing the characteristic

representation corresponding to the cocycle (36) is Q(t) = X oal(t) ...'Xaq-laq(t)

and the matrix associated to

describing the cocycle in

a

corresponding to the cocycle (38) under the isomorphism

Zl(tr1(M),R)

of Theorem 19 is q

E Ad(Xa

AQ =

j=1

Introducing coordinates X e Cn Xa(0)

... Xa

a

a )-1-Aa.

q-1 q

J J+1

Xn

in a neighborhood of the matrix

by the mapping

as on page 182, the curve

by the curve

H.X(t%)

is described in these coordinates

such that

X(t)

Xa(0)'exp X;t) = a(t)

.

Differentiating this last equation with respect to t = 0 ,

t

and setting

it follows that q

dt X(t)

a

1-1 J

It=0

=

dat

Xa a ...Xa

a(0)-1

J=l o 1 q

E Ad(Xa a J=1

J

a

...Xa -

j-2 J-1 1

... Xaq-laq

1 J+1"'

Xaj1 -1a

J+l

dXaJ-1aJ (0) dt

=-Aa, by (39) and (l+0).

where

It is thus clear that

dH.X(%)_-8(X) ,

(4+1) 8(X)

denotes the period class of the Prym differential

and the tangent space to the manifold

S

0

at the point

X

is

a

q 1q

X

identified with the cohomology group

H1(7r1(M),Ad(X))

as in

Theorem 28.

Now to apply formula (41), recall from the remark on page

209 that the tangent space to the manifold Ao(M,X) identified with the vector space

is naturally

r(M, (31-'°(Ad X))/dr(M,(P (Ad X))

The differential of the complex analytic mapping

Hox: A:(M, X) -> So then coincides with the negative of the period mapping

r(M, A 1' 0(Ad X))

> H1(M, 3- (Ad X) )

dr(M, m (Ad x)) of the Prym differentials.

It follows from Theorem 22 that this

mapping is always an isomorphism into, and the proof of the theorem is therewith concluded.

This theorem shows that the complex analytic manifold (of complex dimension 2(g-1 )n +2

where n = 2)

So

is the disjoint

union of the complex analytic submanifolds (each of complex dimension

(g-1)n +1 where

n = 2) consisting of the analytic equiva-

lence classes of flat vector bundles; it must be recalled that these submanifolds have not been shown to be closed subsets, so they must provisionally be viewed as submanifolds in an extended sense.

This splitting can also be described as the decomposition

of the manifold

So

into integral submanifolds for the differen-

tial system 8r(M, 671'0(Ad X)) C H1(M, a (Ad X))

,

where

is identified with the tangent space to the

H3-(M,3 (Ad X))

manifold

S

0

at the point

X E So .

Upon identifying each of these

submanifolds to a point, the resulting quotient space can be identified with the subset

H'-(M, h &0 (2, 69 ) ) C H1(M, A a` (2, G)) consisting of those complex analytic vector bundles over M which admit flat representatives, a subset which was described quite explicitly in Weil's theorem, (Theorem 16); recall that it has been shown that all the vector bundles in this subset have irreducible flat representatives.

The quotient mapping

µ: S0

;0 Hl_ (M,

(2,

can be viewed as a form of complex analytic fibration of the complex manifold

So , but it must be a singular fibration in soms

sense since not all the fibres are even topologically the same;

and this fibration induces some sort of complex analytic structure on the quotient space

H*(M,

(2,(9))

It is just at this stage, when things at last begin to look rather interesting and there are a considerable number of questions begging to be looked into, that time has unfortunately run out, and these lectures must be called to a halt.

I hope to

have an opportunity to continue the discussion of this subject in the near future.

I cannot close without mentioning another

approach to the imposition of a complex analytic structure, on the

subspace of

Hl_(M, o 3

(2,O ))

consisting of those complex analytic

vector bundles admitting unitary flat representatives, which can

be found in the papers by M. S. Narasimhan and. C. S. Seshadri

("Holomorphic vector bundles on a compact Riemann surface," Math.

Annalen 155(1964), 69-80) and by C. S. Seshadri ("Space of unitary vector bundles on a compact Riemann surface," Annals of Math. 85 (1967), 303-336).

An excellent survey of the literature and of

the present general state of knowledge of complex vector bundles over arbitrary Riemann surfaces can be found in the paper by

H. R8hrl ("Holomorphic fibre bundles over Riemann surfaces," Bul. Amer. Math. Soc. 68(1962), 125-160); the readers can find references there for the manly topics not treated in these lectures.

The formalism of cohomology with coefficients in a

Appendix 1.

locally free analytic sheaf. Several times in the present discussion an explicit description of the cohomology groups

locally free analytic sheaf .8

with coefficients in a

over a Riemann surface M , has been

required; the description involves the local isomorphism J IU = Q m1U , and is sometimes a bit confusing notationa.11y, so an

attempt will be made here to straighten things out.

Each locally free sheaf J of rank m is of course given o

by

for some complex analytic vector bundle

(9(Z)

an open covering

= (Ua)

Select

0 .

of the Riemann surface M , such that

the sheaf .1 is free over each set Ua ; and further, select an

isomorphism J IUa N amIUa

.

Having made these choices, compari-

son of the isomorphisms over the intersections Ua n u, yields a cocycle

) a Zl(li( , ,u x (m, (L ))

(?

describing the vector bundle

043

0 e H1(M, $ X (m, C(! ))

such that J _ (Q (0) , as in §2. The point

now is that there is a useful description of the cohomology groups

Hq(7n , ) ) = Hq(7JC ,

a (cz)) , as follows.

Recall that a cocycle f e Zq(71L ,

j ) = Zq(7/1 , &

(.D))

is

given by a collection of sections

er(Ua n...nua,J) fa...a o q o q

(1)

such that

q+l (2)

Z (-1)Jf J=O

o...(Xj-laj+l...aq+l

(p) = 0

whenever p eU

fl.. .n U

o

Since Ua n...n Ua C Ua , under the selected isomorphism o q q

aq+l

J

U"' q

a Uaq

o

element of the module

denoted by fa o from the set

fa

the section

..a q

can be identified with an

r(ua n...n Ua , 0 o q

)

; this section will be

a (za ) , and is Just a complex analytic mapping q q

zq(ua n...n ua )CC into Cm, where o

za q

q

coordinate mapping in Ua q

are many other such

Of course, there

.

fa

representations possible for the section

a

it should be

;

q

o

emphasized that a choice has been made here.

If ua n...n Ua c uP, 9

0

then the same section

fa

.

under the identification

fa

has a representation

.a

q

o

o

r(ua n...n

o

is the

...a q

(zJ3)

m)

, Ua ,I ) = r(Ua n.. .n ua o q q

provided by the coordinate neighborhood UP J Ua n...n Ua ; here o q fa (zn) ...a o q

is a complex analytic mapping from the set

Cm

n...n Ua ) C c into

These representations are of

.

q

0

course related by

fa ...a (zi(p)) _ ooa (p) fa ...a (za(p)),

(3)

0

9

9

p c Ua n...n ua n UP.

o

q

o

q

The cocycle condition (2) can be rewritten as merely a condition on complex analytic functions, by using the identification m)

r(Ua n...n Uo+l , j ) = r(uo n...n u

Qt

provided by the

aq+l'

o

; it has the form J Ua n...n Ua coordinate neighborhood U. q+l q+l o

q+l ) = 0 , E ('l)jfa .. a ..a a q+l (zaq+l 0 3 -1 3+1' 3=0

z

6 z

aq+1

(U

aq+l

o

n...n Ua ) q+l

.

Note that all except the last term in the sum (4) have a natural form, as an analytic function in terms of the coordinate corresponding to the last index.

f

(5)

The last term can be rewritten

)=11a

o... q (zq+l )

fa

o

zaq

fa ...a E r(ua n...n ua J) with o q o q

Now, identifying the section the analytic function

)

( q.fao.' 9

-

... a

q

E r(ua n.. n Ua

= fa ...a (zq)

q

o

q

o

the cocycle condition can be written

q (6)

+

(p) +(-1)q a a. ..a o" J-1 j+l q+l

J=o

n.

a fa ...a (P) = 0 q+l q o q

z (-1) a

a

Similarly, the coboundary condition q+l (7)

(sf)a ...a o

(P) =

q+l

E

a .. a o

J=o

a.

-1

+1'

..a

(P)

q+l

in terms of the analytic functions representing the sections takes

the form

(8)

(P) (sf)ao...a q+l

q

E (-l)jfa ..a a .. a q+l o J=o J-1 3+1

(p) +

(-1)q+l.D

a fa ... a

q+1 q

'

In particular, the group

H1( Vt

in terms of a representative cocycle as follows.

('D a3 )

The cocycles z1(, CQ(?))

lytic mappings

(9)

, 61 (0))

o

(P)

q

can be described,

E z1( Vt , h x, (m, CA

))

are given by complex ana-

fa$:.Ua n U. --> C such that

fPy(P) - fay(P) + oyp(P)-fas(P) = 0

for

p c ua n u. n uy ,

or equivalently,

(9')

fo(p) =

The cocycle h = (ha)

7(p)

f = (fa$)

when the

for

p e Ua f UP fl U7

is the coboundary of a zero-cochain

ha are complex analytic mappings

ha: Ua ->

such that

(10)

fV(p) = hp(p) - 0,0(p)- a(p) for p e Ua f UP .

Appendix 2.

Some complications in describing classes of flat vector

bundles.

Consider in particular the problem of describing all ana-

lytically trivial flat complex vector bundles of rank n over a compact Riemann surface M .

According to Theorem 17, there is a

one-to-one correpondence between this set of flat bundles and the of equivalence classes of complex analytic connec-

A (jfl.,I)

set

tions for the identity bundle

of the space M .

I

for a suitable open covering A

For a general complex vector bundle

0

defined

by a cocycle (0ap) a z1( Vt , h;( (n, a )) , a connection is given Aa of holomorphic differential forms of degree 1 in

by matrices

Ua , such that

the various sets

AP - 0a1p%a0ap = m ap =

(1)

0

is the identity bundle, defined by the cocycle 0a = I ,

this condition reduces to

a = %P in Ua fl u, ,

(1' )

so that

A(11l ,c )

T = (Ta) a Aut (0) matrices

Ta

Ta0ap = -DaPTP

when

Ua , such that

in Ua fl UP

;

0a43 = I , this condition reduces to

Ta = Te

(2' )

M

An automorphism

is a collection of holomorphic.non-singular

in the various sets

(2)

so

r (M' ( (91,0)n Xn)

T = Ta

in Ua fl u, ,

is a global holomorphic matrix function on M .

is compact,

T

is necessarily constant; so that actually

Since

Aut (I) _ (GL(n,C)) . Two connections (Xa) and

(x)

in

A(

are equivalent precisely when there is an automorphism T - (Ta) a Aut (0)

such that

(3)

)à = TcXaTa - dTaTa1

in U.

and when 0a$ = I , this condition reduces to (3')

X'a = T7XaT-1

for T e GL(n,C)

Thus the analytically trivial flat vector bundles of rank n over M are in one-to-one correspondence with the equivalence classes (under conjugation by matrices

T e GL(n,C))

of abelian differential forms on M . 1,0)n X n)

% e r(M,(

,

Given any matrix

select non-singular holomorphic functions

F. in the various sets Ua such that flat vector bundle associated to cycle

of n X n matrices

X

XJUa = DFa = Fa1dFa ; the

is that described by the co-

(Xap) e Z1(Vt,GL(n,C)) , where Xap = FaFp1

(4)

in Ua I Up .

It is quite easy to describe the characteristic representation

X of the bundle

X in a parallel manner. It

f: A --> M be the universal covering space of the Riemann surface M , and view

as in V. It ential form on

form

as the group of covering transformations,

a1(M,po)

= f*X

e r(1, ( .1, 0)n Xn) be the matrix differ-

induced from X by the covering mapping; the

then satisfies (7. z) = 3(z)

for all

7 e nl(M,po)

Further, let F be a holomorphic non-singular matrix-valued differential form on

31

such that

D)

(5)

this function then satisfies F(y' z) = X(Y)- F(z)

(6)

where

for all

7 e al(MDpo)

is a homomorphism representing the

X: n1(M,po) --- > GL(n,C)

characteristic representation of the bundle

(The function F

X .

can be viewed as arising from the functions on

Pf

induced from the

functions

Fa under the covering mapping

bundle

on A is reduced to the trivial bundle; compare with

the discussion on page 145.)

f , after the-induced

Note that the function F

is uniquely

determined only up to a constant factor C C. F for any C e GL(n,C) ; this corresponds to the fact that the homomorphism X only up to an inner automorphism in

GL(n,C)

.

is determined

Recalling the dis-

cussion in §7 again, especially (22), the period class A e H1(al(M),I) = Hom(al(M),Cn X e r(M,(

1,0)n X n)

X

n)

of the differential form

can be described in a similar manner as well.

Selecting a holomorphic, matrix-valued function H on A such that dH

(7)

this function then satisfies (8)

where

H(Z) - H(Y

(Ay) E Z1(a1(M),I)

the function H

z) = A7 for all

Y e al(M,Po)

is the period class of X .

Note that

is uniquely determined up to an additive constant,

and hence the homomorphism A

is uniquely determined by X

For flat bundles of rank n = 1 , inner automorphisms are trivial, so both the characteristic representation

X and the

period class

a

ferential form

are homomorphisms uniquely determined by the difX e r(M, & l'0)

.

In this case, the relation be-

tween these two homomorphisms is particularly simple, which can be Comparing the differential equations (5) and (7),

seen as follows.

= dh = f-1 df = d log f , so that the functions

related by f = ceh

for some constant

f

and h are

c # 0 ; but then, applying

equations (6) and (8), it follows immediately that

x(7) = e

(9)

-a 7

for all

7 e al(M'Po)

Thus the analytically trivial flat line bundles are determined very directly in terms of the period classes of the abelian differentials; recalling Lemma 22, this can be rewritten in the more familiar form

fX X(7) = e 7

(10)

where

X e r(M, (9 l'0)

.

for all

7 e ir l(M,po)

(Recall also the discussion in §8 of last

year's lectures.)

For flat bundles of rank n > 1 , the characteristic representation GL(n,C)

,

X

is determined only up to inner automorphisms in

so one could not expect such a simple relationship as (9)

to hold between the characteristic representation and the period class of a matrix

X e r(M,( 8.1,0)n

X n)

of abelian differentials.

One might hope at least that the character of the representation

X

is determined directly by the period class A , as a weaker form

of (9) ; but unfortunately that is a vain hope as well.

The char-

acteristic representation X (which is actually an equivalence class of representations of the group

rcl(M)

) is of course uniquely

determined by the period class A , since both are uniquely determined by the matrix

X of abelian differentials.

But the class

cannot be described directly as a function of the class A

X and A must involve the matrix

the relation between

X

alone;

X , and

hence the global structure of the compact Riemann surface. To see that this is so, it suffices to examine the following simpler situation.

Suppose

X(z)

is an n X n matrix of

holomorphic differential forms of degree 1 in an open neighborhood

U of the real axis in the complex plane, such that and let

F(z)

X(z+l) = X(z)

be a non-singular holomorphic matrix function in U

such that

dF = F% , and H

such that

dH - X .

= X.F(z)

for some matrix

be a holomorphic matrix function in U

These two functions then satisfy F(z+l) _

for some matrix A e Cn

X e GL(n,C) , and H(z+l) = A+ H(z)

X n

constant factor

;

F

is uniquely determined up to a

C e GL(n,C)

and H

is uniquely deter-

mined up to a constant term H+B , so that the matrix determined up to an inner automorphism in

GL(n,C)

uniquely determined by the differential form X .

X

is

while A

is

(In the compact

Riemann surface case as above, the universal covering space A can be taken to be the upper half-plane, and the transformation 7

can be taken in the form 7

= cz

for some real constant

c

the exponential mapping reduces this to the special case envisaged here.)

The problem is to find the extent to which the conjugacy

class of the matrix

X can be determined as a function of the

matrix A alone. Select a constant matrix

S

such that

e 5 = X , and

introduce holomorphic functions

F(z) =

(u)

and K(z)

G(z)

H(z) =

and

it is easy to see that these functions

such that

K(z) ;

G(z) and K(z)

then satisfy

and K(z+1) = K(z)

G(z+1) = G(z)

(12)

in U

Such functions admit a Fourier expansion in a neighborhood of the

real line, of the form +00

K(z) =

(13)

K

E n=-00

for suitable matrices Kn a Cp

e2ninz

n

In

Recalling that

.

X(z) =

dG(z) + e'S-z S-dzG(z)] =

= F(z)-l dF(z) = G(z)-l a

and

= G(z)-l dG(z) + G(z)-1 SG(z)dz

X(z) = dH(z) = Adz + dK(z)

it further follows that

A + dz K(z) = G(z)-l d- G(z) + G(z)-1' SG(z)

(14)

Again both sides of (14) are invariant under the translation

z T z+l , so admit Fourier expansions of the form (13).

For the

left-hand side in particular, the expansion is just +00

A+

E

21t in

Kne2n inz

n=-00

so that A

is the constant term; and bus

const[G(z)-l

(15)

where

A =

G(z) + G(z)-1' SG(z)]

denotes the constant term of the Fourier expansion

of the expression in brackets.

of the matrix

dz

This formula expresses A

S , hence of course in terms of

in terms

X ; but the expression

involves in addition the function

G(z) , which can be an arbitrary

nowhere-singular analytic matrix function invariant under the translation

z T z+l .

The discussion is now reduced to seeing the

extent to which the choice of the flxnction

tionship between A and

S

G(z)

affects the rela-

in (15).

Note first as a consequence of (15), that, letting tr

de-

note the trace of a matrix

tr A = tr S + const [tr G(z)-l dz G(z)]

(16)

However

tr G(z)-1

dz

G(z) =

d log det G(z) ; and since log det G(z)

is invariant under the translation

z --> z+l , its derivative has

a Fourier expansion in which no constant term appears.

Therefore

(16) reduces to

trA=trS;

(17)

det X =

or in other words,

A and

etr S = etr A

is a relation between

X which does not depend on the choice of function

G(z)

.

To see that this is in general the only such relation, consider in the

2 X 2

(18)

matrix case the function

G(z) =

Me-2niz +Z+Me2aiz

where M =

a -a2 a:

1

for some constant the translation expansion.

a e C ; the function

G(z)

is invariant under

z -9 z+l , and indeed, (18) is just its Fourier

This can be rewritten

11- 2a cos 21tz G(z)

2 cos 21tz from which it is apparent that

-2a2 cos 21tz 1

1+2a cos 2itz

G(z) = 1 .

J

,

For any matrix

it is obvious that

det G = 1

with

G = (y

G 1 = ( y a) ; and

therefore M'e-2niz

G(z)-1 =

(19)

+ I +

M,e2niz

2

M' = (a

where

-1

a -a

Now it is clear that

const [G(z)-l dz G(z)]

[(M'e-27tiz

= const

= M'

2,ti M + M'

+ I + M'e21tiz)(-2ni Me-2niz + 2ni Me

2niz)]

(-2iti M) = 0

and that

const [G(z)-l SG(z)] [(M'e-2niz +

= const

I +

M'e2aiz)S(Me-2niz + I +

2niz Me

=S+2M'SM . Therefore (15) reduces to the equality

A = S + 2M'SM

(20)

which explicitly involves the matrix M depending on an arbitrary parameter

In particular, taking

a .

S = (O1

s )

for example,

2

(20) becomes

sl -2a2(sls2

A

(21)

=(2 a(si Thus,

2

s2 +2a

(81-82)

tr A = sl+s2 = tr S , and

det A = det S + 2a

(22)

so if

s2)

-2a3(sl-s2)

sl # s2 ,

of the parameter

det A a ,

(sl_s2)2

;

can be made arbitrary by suitable choice

showing that the eigenvalues of A

can be

arbitrary subject only to the restriction that

tr A = tr S

.

This observation shows that one cannot expect the description of analytically trivial flat vector bundles to be as straightforward as the description of the analytically trivial flat line bundles, as given in §8 of last year's lectures for instance; and this may explain some of the complications in the present discussion.

Lectures on vector bundles over Riemann surfaces

Lectures on Vector Bundles over Riemann Surfaces

Lectures on vector bundles on Riemann surfaces