Lectures on Vector Bundles over Riemann Surfaces. (MN-6), Volume 6 9780691218212

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LECTURES ON VECTOR BUNDLES OVER RIEMANN SURFACES BY R. C. GUNNING

PRINCETON UNIVEPSITY PRESS AND THE UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY 1967

Copyright © 1967, by Princeton University Press All Rights Reserved Published by Princeton University Press, Princeton, New Jersey; in the United Kingdom, by Princeton University Press, Chichester, West Sussex 5

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Printed in the United States of America

ISBN 0-691-07998-6 (pbk)

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 Motes 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 h 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 groups 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 h

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 descriptive 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. Ho 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 l6).

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.5^, 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. „ . . . . x Princeton, New Jersey July, 1967

R. C. Gunning

-iii-

Contents §1.

Analytic sheaves

1

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

Local structure of coherent analytic sheaves . . . a. c.

§3-

2.6

local structure; "b. semi-local structure; global structure over the protective line.

Induced mappings of analytic sheaves

45

a. inverse image sheaf; b. direct image sheaf; c. some applications. §4.

Riemann-Roch theorem

58

a. reducible vector bundles; b. Riemann-Roch theorem; c. Serre duality for vector bundles. §5-

A classification of vector bundles of rank two . .

71

a. classification of extensions; b. divisor order; c. classification of unstable bundles; d. remarks on stable bundles; e. surfaces of low genus. §6.

Flat vector bundles

9^

a. criterion for flatness; b. Weil's theorem; c. connections and flat representatives. §7-

Flat sheaves: geometric aspects

123

a. definitions; b. cohomology with flat sheaf coefficients; c. deRham isomorphism and duality; d. role of the universal covering space; e. duality explicitly. §8.

Flat sheaves: analytic aspects a. Prym differentials and their periods; b. some special properties; c. meromorphic Pr-ym differentials.

-iv-

157

§9.

Families of flat vector bundles

179

a. 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. Appendix 1

231

The formalism of cohomology with coefficients in a locally free analytic sheaf. Appendix 2 Some complications in describing classes of flat vector bundles.

235

§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 presupposed in this section.

Only sheaves of abelian groups were treated

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

){. tie a sheaf of rings and ja be a sheaf of abelian

groups over a topological space p: y\ — > M

and

M , with respective projections

V. JL — > M . Viewing

1K_ merely as a sheaf of

abelian groups, the Cartesian product ?t xj. has the structure of a sheaf of abelian groups over p xir: K X J diagonal

M X M , with the projection

— > M X M . The restriction of this sheaf to the

M C M X M

is then a sheaf of abelian groups over

which will be denoted by H o A

-1-

M ,

Definition.

The sheaf d. of abelian groups over

M. is

For any open set

J

p 6 M

defines on J

P

uC M

a section

P

the induced mapping the structure of

f € r(u, 7loj! ) is

readily seen to be the restriction to the diagonal section

(r,s) e r(U X U, ??. X J

r e r(u, ft ) and

of a

) ; that is, there are sections

s e r(u, J ) such that

all p e U . The sections

UC U X U

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

r(u, 2f^ ) form a ring, and the sections

f(u, a ) form an abelian group; and the homomorphism f(o Jl — > _J exhibiting A r(u, H o J on Thus

as a sheaf of \ -modules leads to a mapping

) = r(u, TSL ) X r(u, J ) — > r(u, J ) which clearly defines

r(u, J ) = J ( 2 TJ}

the presheaf

the structure of a

-module.

is in 'the obvious sense a presheaf of modules over { ?ll } of rings.

a presheaf of rings and such that J

r(U, %. ) = 'K

Conversely, whenever

{ Ji }

{^

3 is

is a presheaf of abelian groups,

is an ft ..-module and restriction mappings are

module homomorphisms, the associated sheaf

I

is a sheaf of

•J^ -modules. The notions of sheaf homomorphisms, and related concepts, introduced last year for sheaves of abelian groups, extend readily to sheaves of modules. 3 C J

If ^i is a sheaf of 3? -modules, a subset

is called a subsheaf of $> -modules if D

abelian groups and, for each point

-2-

peM,

"J

is a subsheaf of isantfl

-submodule

of Jl

. I n the obvious fashion, "R. is itself a sheaf of ^-modules;

a subsheaf j C 1R. of 1R, -modules is also called a sheaf of ideals in ov , since for each point an ideal in

(L . If 3 C J

quotient sheaf J /3 mapping an

p e M

the stalk Ji

is a subsheaf of ^ -modules, the

is also a sheaf of K -modules.

cp: ^ — > J

—>

"J

J( C y& of an cp( ^

)C 0

sheaves j and

p s M

is a homomorphism of

^-homomorphism >

are

A sheaf

between two sheaves of K -modules is called

(K -homomorphism if for each point

cp : J

is necessarily

the induced mapping

(ft -modules.

The kernel

cp: Jl — S * D , and the image

subsheaves of

H{ -modules of their respective

• For notational convenience, the

tensor product will be denoted by s$ ® •) when there is no danger of confusion.

Tensor products of sheaves of modules satisfy many

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

J of ^ -modules, K ®~J

= J

In particular,

, (recalling that all

rings considered here have units); and if n

r>

o

is an exact sequence of sheaves of K. -modules, then tensoring with "J yields an exact sequence

(Note especially that it is not claimed that the latter sequence , —?* J p exact, it is not necessarily true that is exact. The sheaf 0 —>

J ® J

—>

0 — > J ^ — > J2

(b)

If J

J

J

—>

0®*°

is called flat if

J ®J

„

is exact whenever

is exact.)

and 0

is the sheaf of

0 —S» 0®

is

are sheaves of A. -modules, their direct sum

K -modules defined by the presheaf

and will be denoted by J © d

{ J!

© J „} ,

; this sheaf can be identified in

the obvious manner with the sheaf s$ o D

considered earlier. A

particularly simple example is the sheaf of (R -modules .. © 1L , the direct sum of

-h-

m

copies of the sheaf

$_ ; any sheaf of

0y -modules isomorphic to

a free sheaf of 1jt_ -modules of rank Note that the stalk \ of m-tuples of elements of

^

m

^_m

will be called

p

is just the set

m .

at a point

; and elements

T^ m

Re

will be

considered as column vectors

R =

where

r. e

*]?

E. e r(M, Vy. J

• The sheaf ??. ) , where

the identity section section

canonical sections

is the column vector with j-th entry

1 e r(M, \ ) and all other entries the zero

, in the sense that any element

uniquely in the form

module

m

0 e r(M, % ) . These sections are free generators of the

sheaf %

r. e \

E. 0

has

B E )[

can be written

R = r-E (p) + ... + r E (p) for some elements

; in a similar sense, they are free generators for the r(M, Hi™)

of sections of the sheaf

An K. -homomorphism very simply as follows.

cp: 8[_ — >

The image

o(_

Xm

.

can be described

cpE. of the section j

E. e F(M, %. ) is a section of If n , hence can be written 3 n uniquely as CpE. = 2 f..E. for some sections f. , e r(M, £ ) . J

These sections the ring Hi

i = l "*•" ^

(f..)

"

can be viewed as forming a matrix

over

= r(M, 1R. ) ; this matrix will be called the matrix

representing the homomorphism

9 . The matrix fully determines

the homomorphism. in the form

$

For any element R e l ! can be represented m H = 2 r.E.(p) for some r. e TR. ; and since cp i"l J P

-5-

is an l!L -homomorphism,

elements

Oa

where

now denotes the natural restriction homomorphism into

p

the group

6

GL(m, 'K.

„

) .

It is a straightforward matter to

show that this is an equivalence relation; the set of equivalence classes will be denoted by

M (Vl , Jo K. (in, K. )) , and will be

called the first cohomology set of Vi with coefficients in }} »C (m> \. ) • Suppose that If = {V } of

is another open covering

M , which is a refinement of V\ with refining mapping

that is, suppose that

[i: 11 — > uL

V C M-V

e If . 'inen \± induces a mapping

for each

V

n ;

is a mapping such that

n: zHft , hi (m, t )) — > ZX(V , J>% (m, Ji. )) , the mapping which takes a cocycle

e Z ( JTL , A) X (m, K. ))

U

into the cocyle

where

p

.. denotes the restriction mapping to v p n Vg C HV a 0 uVg • It is clear that the image is again a cocyle n

a

Va

and that

\i takes equivalent coeycles into equivalent cocycles;

-7"

so that

n

induces a mapping

Lemma 1.

If IT is a refinement of Vt , and if

are two refining mappings, then Proof.

for each set

\i — v

Considering'any cocycle e

V

]f define

defined element of

B

GL(m, \~

= p

VO

*

u

) , since

are equivalent; hence

n

u

e Z1( VI , J>% (m,

a p

* and

\x and

; V C HV

this is a wellfl VV

. Then

* = V

, as asserted.

Wow for any tvro coverings VI , if of M , write U < 1TL if Jl is a refinement of IS} ; the set of all coverings is partially ordered under this relation, and by Lemma 1 there is a well-defined

mapping H

1

^ , jb t (m, )?.))-> H1( If , i X (m, \ )) whenever

)f < uX • It is clear that these mappings are transitive, so that it is possible to introduce the direct limit set

ffV

E1(V[ , A t (m, Z ) ) ,

JbX On, X ) ) = d i r . l i m . 1A

which is called the first cohomology set of in

il^(m, ^ ) .

M

with coefficients

(Recall that to define this direct limit intro-

duce the union

U

H ( V\ , h'Z. (m, X )) ; two elements

f e iiHVl ,h%

(m, X )) and

g e H1(lf , i

#

|u

M

such that ^j |u

there is an W. -iso-

U

; and restricting to an intersec-

fl U B ^ $ , it follows that there is an isomorphism

*Pft= ""Wft • !R. I" ^ U R

>

^

~ a ^ "a p ^ "„ 7 . Letting

|u

fl Ufl , and these isomorphisms

Pf)o(Pfiv = (Pfyy when restricted to


&