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 <
.
whenever
It is clear that these mappings are transitive, so that
it is possible to introduce the direct limit set
)) = dir.lim.
$1(M, 1x (m,
Hl( X11
24
, ,8 x (m, Z) ),
which is called the first cohort logy set of M with coefficients
in b'(m, X) .
(Recall that to define this direct limit intro-
Hl(bI
duce the union U
,
AX (m, x )) ; two elements
in
f e H'('UZ , ,b X (m,?, )) and g e Hl(f , .a
(m,
)) will be
called equivalent if there is a common refinement )( < IL , 7U' < 14
in which f
and
g have the same image, and the set of
equivalence classes is the direct limit.)
With this machinery at hand, returning to the original
question, consider a locally free sheaf J of m ; and let'
k,-modules, of rank
be an open covering of M such that JIUa
(Ua}
is free for each Ua .
Thus for each set
morphism 9a: JIUa
Ua there is an
V% -iso-
V MIUa ; and restricting to an intersec-
tion Ua n U. # 0 , it follows that there is an isomorphism
(per = cpc pol:
.mIUa n U0
- > eju n U , and these isomorphisms
obviously satisfy the identity Ua n U. n u7,
Letting
.
(p
= (p
(p
when restricted to
OC43 a GL(m, Va n u.)
be the matrices
representing these isomorphisms of free sheaves, it is thus evident that these matrices form a one-cocycle The isomorphisms
(&
) a Zl(24 , ,o x (m,
are of course not unique, the most general
(pa
where
such being of the form
ea(pa
arbitrary isomorphism.
Letting
ea: VIUa --? XmjUa
is an
9a a GL(m, A U) be the matrix
a representing the isomorphism
ea , the most general one-cocycle
associated to the sheaf .1 is then of the form
for arbitrary such 9
.
(p9(,)O,(p90)-1
All these cocycles are equivalent how-
ever, so the construction associates to each locally free sheaf
and suitably fine covering 7n a unique element
0,4 ( ,o) a a1(?/! , A
(m,
)) ; this element is called the
coordinate bundle representing the locally free sheaf
Note
that whenever C is a refinement of. with a refining mapping
then
P:
Lemma 2.
rank
mpp
0
Ir
(J) = µ** A (.1 )
Let J and .J'
be locally free sheaves of
over a topological space M , and
(,6)aH1(74,6x(m,X)) and
0 0
r( 1 ') a H-( Ile, A;( (m, P )) VE
.
be coordinate bundles
)).
representing these sheaves in terms of coverings 74 and
and J '
Then
pectivelys
'
such that
2/1'
Let if be a common refinement of the coverings
Proof.
Vt and
are isomorphic if and only if there
of u and
is a common refinement it
res-
72'
, and let
,f I Va --> 'mIVa and
cps:
cpa: ,a IVa T ?ml Va be isomorphisms.
If there is an isomorphism
e to
e: J --? ) ' , then for each set Va the restriction of
V. leads to an isomorphism
1
ea = roar e.roa Letting a e GL(m, V, V
: k mI va --> WmVa .
be the matrix representing
follows readily that %, _ (p6s)Osp(p®p)-1 ; thus
.
(
= 4+ ,u. (2) in If'( )!' a b Z (m, Z))
.
( Oaf)
V
are equivalent, and- hence
(,8
0
if
0
,a
(0`1
it
,
)
and
') _ (0.) _ ( Oaf) _
Conversely, if
(J) there are matrices ea satisfying the
above equality, and defining isomorphisms follows readily that
from j to J '
e(,
.
e = ((pa)-loearroa
ea ; and then it
is the desired isomorphism
This concludes the proof.
For any locally free sheaf .9
over M , the various coor-
dinate bundles representing I in all suitably fine coverings all lead to -the same element
0
V
0( J) e &(M, b
(m, P,,)) , since
(J ) = u*0 n (.d ) for any refining mapping µ: )f - .
VL.
This element is called the fibre bundle representing the locally
free sheaf , .
Theorem 1.
If 'R is a sheaf of rings over a topological
space M , the mapping which associates to a locally free sheaf of
'k -modules of rank m its representative fibre bundle establishes a one-to-one correspondence between the set of such sheaves and the
b (m, Z. )) .
cohomology set Hl(M, Proof.
It is an immediate consequence of Lemma 2 and the
definition of the direct limit that the mapping 2 -? 0(2 )
is
a one-to-one mapping from the set of locally free sheaves of rank
m into the cohomology set H'(M, h 't (m, `k ) ) . To complete the For
proof it is only necessary to show that the mapping is onto.
any given cohomology class 0 e Hl(M, b x (m, `n..)) covering i and a representative cocycle for
(0
select La
) a Z1(u , h;r(m, z is a
There is no loss of generality in assuming that u(.
0 .
basis for the topology of M , since there is always ao basis refining any open covering.
Introduce a
presheaf.
( .a Ua) I
Ua a V(, defined as follows: for each Ua put 'dU
a and for each containment U C Ua let
a
PU U :
= mIUa , U
-J U
a
be the homomorphism of free sheaves associated to the matrix
000 a GL(m, RUa n 0
u
)
.
The cocycle conditions on the matrices
show that this is indeed a well-defined presheaf.
Letting
,J be the sheaf associated to this presheaf, it is immediately evident that (0 hence that
)
represents the coordinate bundle of the sheaf
0 _ 0(J )
.
This therefore serves to conclude
the proof.
As a notational convention, for any locally free sheaf
of X -modules of rank m over the topological space M , let
-11-
ss
O(J ) a Hl(M,h ;C (m, ?)) denote the fibre bundle representing that sheafS and for any sufficiently fine open covering let
0
u
(J ) e Hl(Iq
,
4
(m, P, ))
itt of
M.,
denote the coordinate bundle
representing that sheaf, or equivalently, representing the fibre
bundle (I) .
Conversely, for any fibre bundle )) , let
0 E Hl(M, b Z (m,
(0)
denote the locally free sheaf
of R -modules represented by the fibre bundle
case m = 1
its a bit further attention.
0 .
The special
Note that ,FJZ(1,
is the subsheaf of IZ formed of the invertible elements in I hence is a sheaf of abelian groups; the notation .7 x (1, 'K ) = `,* C 7( will sometimes be used as well.
case, since
V,*
is commutative, the cohomology set
= H1(M, h bt (l, R ))
In this
Hl(M, x*) =
is a group as well; indeed, this is Just the
first cohomology group of M with coefficients in the sheaf of abelian groups, as defined in §3 of last year's Lectures on Riemann Surfaces.
The elements of H1(M,
will usually be
denoted by lower case Greek letters.
If ,p and j are locally free sheaves of A -modules of ranks m and n respectively, it is clear that the direct sum ®7 is a locally free sheaf of . -modules of rank m+n and that the tensor product J ®,7
Z -modules of rank mn .
,
is a locally free sheaf of
Both operations are associative, and
the sheaf J is the identity for the tensor product operation. Similar operations can be introduced in the cohomology set
(M,
(*, )) _
0 e Hl(M, h
(m, 'k )) l and
representative cocycles
(m, Z)); if `Y a Hl(M, (n,)) , selecting
(M, ,8
U
(0C413 )
and
(`YC44B)
for a common open
)
covering 7 of M , let
by the cocycle
0 ® `Y
(0C413 ®`YC413 )
defined by'the cocycle
be the cohomology class defined
and 0 0 `Y be the cohomology class
(0C413 0
)
.
The direct sum and tensor
product (or Kronecker product) of matrices are defined as usual, and it is easy to see that the definitions are independent of the choices of, cocycles.
It is clear from the definitions that, con-
sidering the fibre bundles representing locally free sheaves,
O(J®,)=O(J)®O(O) and 0(.J®J)=0(J)®O(,7). It is left to the reader also to verify that the tensor product is a group operation in the dubset of locally free sheaves of rank 1; this corresponds to the group structure in Jjl(M,
p ((l, (For
H1(M, R*) _
this reason, locally free sheaves of
rank 1 are also sometimes called invertible sheaves.)
that j ® ( 0 1 ®,, 2) = (J ®
1) ®l ®' 2
Note further
, as one verifies
by considering separate stalks. It should be remarked in passing that a locally free sheaf
is flat; that is to say, if .O is a locally free sheaf of
Ik -modules and 0 T 1 --> sheaves of
2
is any exact sequence of
-modules, then 0 --> j® 0 1 T A® D 2
an exact sequence.
is also
(The result is of a local nature, so it suf-
fices to prove it for free sheaves; but since
RIn00 = ( 7X®... ®X) ®0 _ ( It®D) ®... ®(It0 = ) 0 ... ® 3
, the assertion is trivially true.)
The principal interest of the preceding constructions for
(c)
the present lectures lies in their usefulness in studying complex analytic manifolds, and in particular, one-dimensional complex analytic manifolds, (Riemann surfaces).
On a Riemann surface M ,
note that the sheaf 0- of germs of holomorphic functions is actually a sheaf of rings, (commutative rings with identity). sheaf of modules over the sheaf- of rings
A
0' is called an analytic
sheaf over the Riemann surface. According to Theorem 1, there is a one-to-one corre-
spondence between the locally free analytic sheaves of rank m over the Riemann surface M and the acohomology set
In this case, ,1JZ(m, 6-) is the sheaf
H1(M,
defined by the pre-sheaf
{GL(m, QZU} , where
3-U
is the ring of
holomorphic functions over the open set U C M ; but since GL(m,
0'U
)
can be viewed as the group of complex analytic mappinge
from U into the complex analytic manifold GL(m,C) , the sheaf .,b x (m, 9) can be considered as the sheaf of germs of complex
analytic mappings from M into H1(M, 1)
(m, 0-) )
GL(m,C)
.
The elements of
will be called complex analytic vector bundles
of rank m over the Riemann surface M ; the special case m = 1 was considered last year, and the elements of the set (actually
the group) Hl(M, h 4 (1, 1')) = Hl'(M, line bundles.
*) were called complex
Recall from last year's lectures (in particular
§l, page 54), that to any complex line bundle there was canonically associated the sheaf
g e H1(M,
C9- )
of germs of
holomorphic cross-sections of that bundle; and the construction given there was precisely the construction used in the proof of
, 1.
Theorem 1 in the paragraph above.
Thus the sheaves
are
&-(g)
precisely the locally free analytic sheaves of rank 1 over the
Riemann surface M ; and the fibre bundle representing the sheaf
0"(g)
is precisely the complex line bundle
.
The locally
free analytic sheaves of higher ranks thus form a straightforward generalization of the sheaves considered last year, with a parallel
notation; and for this reason, a locally free analytic sheaf of
rank m represented by a complex analytic vector bundle 0 e H1(M,
x(m, (0 ))
will sometimes be called the sheaf of
germs of holomorphic cross-sections of the vector bundle
The sheaf
0 .
of continuous complex-valued functions on
the Riemann surface M is also a sheaf of rings, as is the sheaf 00
of infinitely differentiable functions; recall the natural
inclusions CLC the sheaf of rings set
H1(M,
C
Locally free sheaves of rank m over are in one-to-one correspondence with the
Z (m, 4 )) , called the set of continuous complex
vector bundles over M ; and the locally free sheaves of
C set
-modules of rank m are in one-to-one correspondence with the
H1(M, 1 x(m, 0")) , called the set of Ow
bundles.
The inclusion mappings
6L--;;b
complex vector
T G lead to
inclusion mappings ,b-C(m, (a) T A x(m, {p C) > 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
-')<
is finitely generated.
Since j is coherent,
further restricting U if necessary, there are homomorphisms
a,a
such that the vertical column in the accompanying diagram is an exact sequence.
p1-sl
al
Ll
I'
r
>-
61S Jo
m I
such that the above diagram is comma-
There is a homomorphism pl
tative, reducing U further if necessary; (this is left as an
easy
exercise to the reader - recall that mappings of free sheaves are determined by what they do to the canonical sections
Ei ).
By
F e B-q is in the kernel
?S of
q)
commutativity, an element
and only if image al .
if
apl(F) = 0 , that is, if and only if pl(F) a kernel a = Since
image
q)l fl
image
al
Lemmas 3 and 1+, there is a sheaf homomorphism
is coherent by ,g_r 1V:
-;;- a9
such that image 1V = image q)l fl image al ; and since image * C image q)l , there is a sheaf homomorphism
that the above diagram commutes. cpl(F) _ *(G)
for some element
0:
B-r
? Bm such
Now F e ( if and only if G e 0 r , hence if and only if
(pl(F - 9(G)) = 0 ; thus x = (image e) + (kernel pl) , and )' is
finitely generated since image 0 and kernel q)l are both coherent
by Lemma 3.
First suppose that X and
(b)
8 are coherent; then
over a sufficiently small open neighborhood U there are sheaf homomorphisms are onto.
morphism
p:
6jr
-
R and
a: 68 - J , both, of which
Restricting U if necessary, there is clearly a homoq)l
such that the following diagram is commutative and
exact.
r G.
PI
0 -->
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 <
Ilyll
c xn+2
xl,...,xn+l , the next point
Having
must be
A(xn+2) = y - 'YO(xn) ; the difference
must then be such that
'YO(xn) .
Note that
A(yn+2)
= A(xn+2) - A(xn+l)=
IIA(yn+2)11 =
= ll'YO(xn+1) - 'YO(xn)II < c llxn+l - xn11 < n+1 2
Ilyll
; but there exists
such a point and
xn+2
for which
yn+2
IIyn+211 <
c-
2n+1
11A1
=
2nc Ilyll
- xn+1 + yn+2 is the desired next element.) It should
be observed by the way that 11x n+111 < 11xn+l - xnll + .+ IIx2 - xlll + + 11x1 '. x011 <
c
so that all these points do lie in Br C a
,
Ilyll
It follows from (iii) that the sequence of points
a limit point x e E ; and note that
11x11 <
xn
11yl1 < r ,
converge to
so that
But it follows from (ii) that then A(x) = y - 'YO(x)
x E Br .
so that
'Y(x) = A(x) + 'YO(x) = y ; thus
y E 'Y(Br) , and the proof
is thereby concluded. Remarks.
The demonstration of Theorem 4 given here is
essentially due to Malgrange and Grauert; compare also the notes of the Seminaire d'Analyse (Semiinaire Frenkel), Strasbourg, 19641965.
The first statement of a theorem of this sort, with quite a
different proof, was due to H. Cartan, (J. Math. pures et appl. 19 The statement and proof given here carry over al-
(1910), 1-26).
most verbatim for several complex variables. Lemma 6.
Let
U1, U2
be bounded open subsets of the
C , with intersection U = U1 n U2 ; and suppose that
complex line
Then the mapping
U17U n U2-U
o: ro(U1,Q)ED ro(U2,(P)->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
<
e
is non-
and 12(p)©(p)-1 c t,
3 2
p e D , where D is an open neighborhood of the
I e GL(m,C)
Let D C D be the open subset homeomorphic to the disc IV
IV
` . (zj E CIIz,l < 2) , so that D = D1 n D2 f e r0(D, (.9.)
f e PO(D
can be written
f = f1 + f2
Note that any function for some functions
(9) ; this is an immediate consequence of the Laurent
expansion of the function
f(zi)
in terms of either coordinate z
Then, applying Theorem 4, there are holomorphic, non-singular matrixvalued functions _ ®1(P)®2(p)
®j e GL(m, 6-D )
Im(p)®(p)-1
such that
=
in D , hence such that ti
012(P) _ 81(P)82(P)e(P)
for
p e D
Now, to construct a meromorphic section of the complex analytic vector bundle
0 , begin with any non-trivial (that is, not iden-
tically vanishing) meromorphic vector-valued function F E r(7P,m).
E r(D1, l m) and F2 = 8-le2 1F E r(D2, 971 m) s1
The functions Fl =
are meromorphic in their respective neighborhoods; for identically singular, and is rational on
morphic on P. By construction then,
7P, so
8-1
®
is not
is mero-
F1(p) _ ®1(p)F(p) = '32(p)F2(p)
ti
whenever
p e D , so that this is the desired meromorphic section,
concluding the proof. Remarks.
It is apparent that the proof of Theorem 7 fl
actually yields somewhat more, namely, that
HI(7P, l z (m,
)) = 0 ,
where 7Yt is again the sheaf of germs of meromorphic functions. (In other words, all complex meromorphic vector bundles over
7P
are trivial, or all locally free sheaves of m -modules over
lP
Thus for any complex analytic vector bundle
are globally free.) 0 E H-(P, ,
quently, that
(m,
.))
r(3P,
it follows that 1(c) = Y7 m , and conse-
ow
r(P ,
the latter clearly being
nonzero.
With these observations in mind, Theorem 6 can be extended as follows.
jective line
Let J be any coherent analytic sheaf over the proI P, represented by an exact sequence of analytic
sheaves of the form
where a and
-1 are locally free.
Tensoring this exact
sequence with the sheaf ?I , considered as a sheaf of
-modules,
it follows readily from the condition that g and
are locally
free that the following is also an exact sequence of analytic
sheaves. (6)
0
'Yq ®
0
If 3 is represented by the complex analytic vector bundle
it is evident that
®
LR (,D) _
(,D)
m
(
so
that (6) reduces to the exact sequence of sheaves of '_modules
(7)
0-->
The homomorphism
B
e 7Y!'m-
0
in (7) is represented by an ml Xm non-
singular matrix ® of rank m1 over the field P(p , m ) ; and after choosing coordinates such that this matrix has the form
® = (®1,0) , where
®1 c GL(m1,
®d
"'(p
)
,
it follows from (7) that
m1
J
m
Thus, upon tensoring with ryyi, every coherent analytic sheaf over IP
reduces to a free sheaf of
-modules. -(
Induced mappings of analytic sheaves
§3.
Suppose that M and N
(a)
are two arbitrary Riemann surfaces,
with M 0- and N&- as their respective sheaves of germs of holo-
f: M T N is a non-trivial complex
morphic functions, and that
(Here non-trivial means that M
analytic mapping.
to a single point.
is not mapped
Recalling the discussion in §10(a) of last
year's lectures, such a mapping is continuous and open, and exhibits
M
as a local branched covering of N .)
the surfaces, the mapping
Given a sheaf over one of
induces a sheaf over the other sur-
f
face in a natural manner; these induced sheaves are the subject matter of the present section.
Consider firstly a given sheaf J of abelian groups over the Riemann surface
N ; there is then an induced sheaf F= f-14)
of abelian groups over the Riemann surface M , defined as follows. Let
Ua a
be a basis for the topology of M , and to each set
_ (Ua)
associate the abelian group V a = r(f(Ua),
; whenever
per: N --? k be the homomorphism defined by
Ua C Up , let
restricting an element of Y f(Ua) C f(UP)
)
.
= r(f(UP), ,8)
It is evident that
{Ua'
a,pa43) pR
of abelian groups over M ; and X = f1( o ) the associated sheaf.
to the subset
P
is a presheaf
is defined to be
The same construction also applies to
sheaves of rings, of course; so in particular,
f1(N (Q)
well-defined sheaf of rings over the Riemann surface M .
is a It
should be noted that f1(N d-)
can be identified in a natural
manner as a subsheaf of M Lf
(For letting X = f-1(N (R )
defined by the pre sheaf
.
CUa, Vt
-45-
C43 )
be
as above, to any element
ha a I-a = r(f(Ua),N (Q ) associate the element ha this mapping clearly yields an isomorphism desired.)
f e r(Ua,Md);
1(N 0 ) T M
,
This identification will always be made, without further
Since f1(N (p.) C M & is a subsheaf of rings,
comment.
as
(9
M(SL
can
be viewed as a sheaf of modules over the sheaf of rings r'-(N) This construction is not quite satisfactory for sheaves of modules over the sheaf of rings sheaves over the Riemann surface
be defined by the presheaf
Ra =
r(f(Ua), 2 )
(Ua'
N
6
, that is, for analytic
For letting
N .
as above, each
pats}
is a module over the ring r(f(Ua),N 6-) ,
where the latter rings form the presheaf leading to the sheaf fl(N Cu
)
; thus 1 has only the natural structure of a sheaf of
modules over the sheaf of rings fl(N (4
)
.
Thus it is more
natural in this context to consider the inverse image sheaf f°(J) over the Riemann surface M , defined by
f°(.j )
(1)
= M m ®-1(
This is then a sheaf of modules over the sheaf of rings M LV that is, an analytic sheaf over the Riemann surface M . particular that
f°(N 19
M m ®_l( .)fl(N f
,
Note in
M
N
The properties of this operation on sheaves are summarized in the following assertions. Lemma 8.
Let
f: M T N be a non-trivial complex
analytic mapping between two Riemann surfaces, and let
)' ill J21 ...
be analytic sheaves over the Riemann surface N.
(i) Then fo(j 1 ®j 2) = fo(A 1) ®fo( ,p2) fo( )l ®ns .d 2)
_
fo( J 1) ®
, and
fo(.d 2) 1416
If j1 T 2 -?
(ii)
3 --? 0 is an exact sequence of
analytic sheaves over N , then
fo(,p 2) --? fo(j 3
1)
-;;b 0 is an exact sequence of
analytic sheaves over M If J is a locally free analytic sheaf of rank m over
(iii)
N , then fo() )
is a locally free analytic sheaf of rank m
over M . If 2 is a coherent analytic sheaf over
(iv)
N , then
fo(.J)
is a coherent analytic sheaf over M . Proof.
(i)
It is evident that
f_](
=f.01)®f1(J2) and fl(j 1(9 N
p
= f-1( )
1
f -l(
a
2)
J1
2)
g p 2)
; this can be seen by considering
the stalks at any point of the surface, since
(f-'( 'I ))p=if(p)
Then
fo( Jl ®J2) = Md
(D f
l( ,J1 ®J 2) = fo(4 1) ®fo( .1 2) ;
and, recalling the standard properties of the tensor products of modules,
f°(
1 ® Mm
ti
.02) = Mm ®-1(.N
Jl) ®
f 1(
ti
-1( N
1( 'J1)0-1( f Nm
ti (f-1( A 1)
®
-1(Nm
f l( )1) 0f-1 ( f-1( 0
®-1(
)M
N® )f -1(
J2)
2
N
(M LI 0
at
f l( '12)
M0 ) ® -1( N NO
)M0 M61
)
(MLV
®-1( N 15 )f
1(,d 2))
fo( p2)
n
It is an immediate consequence of the hypotheses that
(ii) p
i( ,A 1) -? f-'( 02) -? f -l( ,d 3) -? 0 is an exact sequence of sheaves over M ; tensoring with M 0
, and recalling the
properties of the tensor product (especially page 4), it follows
that M® ® f-'( J 1) --> M
2
--? M®
f -l( 13) - 0
is exact as desired. As noted earlier,
(iii)
f°(N ' ) = M 0
this and part (a) above that
f°(N 9 m) = M
that
is locally free of rank m over
p E M .
Since
of
f
; and it follows from
m m
.
Now suppose
N , and consider a point
is continuous, there is an open neighborhood U
sufficiently small that G If (U) = N LV mIf(U) ; but then
p
f°(J )IU
If(U)) = ?(.NCO mlf(U)) = M 0 m , so that
is locally free of rank m over M . (iv)
over
of
Finally, suppose that A is a coherent analytic sheaf
N , and consider a point f1
p E M .
Again, by the continuity
there is an open neighborhood U of
p with image as small
as desired, in particular, so small that there is an exact sequence of analytic sheaves
N a mlIf(U) -- ? N ® mlf(U)
T
) f(U)
. Then,
using parts (b) and (c), there is an exact sequence of analytic
sheaves over U of the form m1IU mIU
M
0
T f°( J ) IU T 0 , so that f°() )
--? M (g
is a
coherent analytic sheaf over M ,completing the proof of the lemma.
Consider next a given sheaf
(b)
of abelian groups over the
,p
Riemann surface M , and assume that the mapping
f: M T N
is
onto all of N ; there is then an induced sheaf 3 = fo(J) of N , called the direct
abelian groups over the Riemann surface
Let
defined as follows.
image of the sheaf
CU.)
basis for the topology of N , and to each set Ua e
element of
3a
be the homomorphism defined by restricting an
r(f-l(UP), ) )
is evident-that
associate
a = r(f l(Ua),2 ) ; whenever Ua C UP , let
the abelian group
p0$' ' P T
be a
fl(Ua) C fl(UP)
to the subset
(Ua' 3 apc)
.
it
is a presheaf of abelian groups
over N ; and 0 = fo( J) is defined to be the associated sheaf. Note that this presheaf is a complete presheaf. Uo
are sets of the covering 2 such that U
(Up)
and
Firstly, if h
0
, then ho
and hence necessarily ho = 0 .
fl(U
02
P
ho
7
1
2
fl u 01
is the zero section over fl(U
Secondly, if hP e 3
, then the sections
fl U0
= U UP
= r(fl(US), 1 )
py0 (h0 ) = py0 (h0) whenever 2 2 1 1
are elements such that
1
0
is an element such that
e J o = r(f1 (U0), J )
p(h0) = 0 for all
U7 C U0
(For suppose that
h0
agree over
2
) = f'(U ) n fl(T 02) ; there is therefore a section
l
ho e 1
= r(f l(U0), J )
o
such that
pPo(h0) = hP
for all
As a consequence, there are natural identifications = 3 a = r(f-l(Ua), . )
P
r(Ua,fo( ,d )) _
In particular, there is a natural iso-
morphism
r(N,f0( J ))
(2)
'
r(M, .1 )
The same construction also applies to sheaves of rings,
o(M
of course; so in particular,
rings over the Riemann surface
N .
)
is a well-defined sheaf of
It should be noted that
fo(M
can be identified in a natural manner as a subsheaf of (For letting (UQ, ) a,pa4}
= fo(M (l )
N
be defined by the presheaf
as above, to any element ha e r(Ua,N
associate
)
ha o f e r(fl(Ua),M dV ) _ J a ; this mapping clearly
the element
extends to an isomorphism
N
(9
T 0(M
)
,
This
as desired.)
identification will always be made, without further comment.
Since NO C fo(M Q1 )
is a subsheaf of rings,
f
o
(
M
viewed as a sheaf of modules over the sheaf of rings If ,p
is an analytic sheaf over M
then the direct image sheaf fo(I )
(Q
N
)
can be
0
.
(a sheaf of M (L -modules),
is an analytic sheaf over
N
(a sheaf of N C1 -modules); for it is evident from the definition that
f0( 8 )
o(M S )
,
is a sheaf of modules over the sheaf of rings
and as noted above,
o(M S ) D N CO
The operation .
of passing to the direct image of an analytic sheaf is basically more complicated than that of passing to the inverse image; so to establish the properties of the direct image mapping
f0
which will
be required, further restrictions will be imposed on the mapping f .
Let
Lemma 9-
f: M -> N be a complex analytic mapping
between two Riemann surfaces, which exhibits 2M
as an r-sheeted
p
branched covering of N ; and let J, .0, A ,...
be analytic
sheaves over the Riemann surface M .
(i)
fo ( -Jl ®,d2) = fo( .O l) ®fo('J2) .
Then 0
J
if
(ii)
j
an exact sequence
are coherent analytic sheaves over M forming
0 --> ,9l ---;. J2 --? ,J3 --> 0 , then the
following is an exact sequence of analytic sheaves over N .
0 -> fo( J l)
fO( J2) -> fo(J3) -> 0
The direct image sheaf
(iii)
analytic sheaf of rank r
over
fo(M41 )
f0() ) Proof.
N .
is a coherent analytic sheaf over (i)
(ii)
of p
the set
N
This assertion is. obvious, and actually holds
for arbitrary complex analytic mappings
U
is a locally free
If ) is a coherent analytic sheaf over M , the direct,
(iv)
image
;-
For any point
f .
p E N there is an open neighborhood
sufficiently small that each connected component f -'(U) C M
and that the sheaves
Vi
of
isp contained in a coordinate neighborhood,
J jIvi
are cokernels of injections of free
sheaves over pVi i thus, by the Corollary to Theorem 3, it follows that
H-(Vi, ,p j) = 0 .
For each set
Vi , the exact cohomology
sequence associated to the exact sequence of sheaves
0 -- J l
vi --> ,d 2 Vi --> d 3 Vi --> 0 therefore begins as
follows:
0->r(vi,1l)---;- r(vi,.d2)-->r(vi,13)->0.
Taking the direct sum of these sequences over all the neighborhoods vi
the following is also an exsct sequence.
o --> r(f-1(u), J 1) --> r(f 1(u),
,a 2) --> r(f-1(U), J 3) --> 0
.
Taking the direct limit of these exact sequences in turn, leads immediately to the desired result.
p e N is not the image of a branch
If the point
(iii)
point under the mapping
f , then for any sufficiently small con-
nected open neighborhood U of p
the inverse image
decompose into the disjoint union of r
connected sets
of which is analytically equivalent to U ;
r(f-1(U),M ( ) = E r(vj,M ( )
and thus
j=l
In a neighborhood of such a point, it follows
r(U,N 0 )-modules. (V
fo(M a) ^' N
If
r .
is the image of a branch point,
p
there is still a decomposition
r(f-1(U),M a ) = z r(vj,M (Q )
but now the number of components of them are branched coverings st sider a single component that case.
vj , each
E r(U,N (D ) , considered as
j=l
that
will
f_1(U)
is less than
vj
p .
;
r , and some
It suffices then to con-
vj , and prove the desired result in
Suppose therefore that
f: M --> N exhibits M
as
an r-sheeted branched covering of N , with a branch point of order
r-l
hood of p
over
p ; restricting to a sufficiently small neighbor-
and choosing coordinates properly, it can be assumed
that M = (z a CJJzJ < 1) w = f(z) = zr
.
p: z = 0 , and if
Now if U
,
N= (w a cJJwJ < 1) , and is any open disc around the point
h e r(f 1(U),M m ) , then
00
avzv
00
h(z)
(avrz
+
X,+lz
zr-l h-1 (w)
= h1(w) +
Thus
+ ... + zr-layr+r-lz
)
V=o
V=o
.
P(f 1(U),M 0 ) = P(U,N CO )r as P(U,N )-modules, the isomor-
phism being that which associates to (h1(w),...,h,(w))
.
the r-tuple
h(z)
This isomorphism is compatible with the
isomorphisms introduced above at all the regular points
and therefore fo(M 0 (iv)
N C!1 r
,
z
0 ,
as desired.
Finally, suppose that j is a coherent analytic
sheaf over M .
For each point
open neighborhood U of
p e N and each sufficiently small
p , there is then an exact sequence of
analytic sheaves of the following form.
If-1 M --> 0 .
0 -> MC. elf l(U) --> Ma mJf l(U)
Applying (ii), there results the following exact sequence of analytic sheaves over
U :
fo(M fl MI)
0
fo(M a m)
fo(M Q m) = fo(M O)m
By (i),
by (iii).
Thus
f0( ) )
> fo(,a)
0
; and this sheaf is then coherent,
is coherent, and the proof is thereby
concluded. Remark.
The analogous result in several complex variables
is considerably more difficult, but of great importance; see for instance H. Grauert and R. Remmert, Bilder and Urbilder Ana].ytischer
Garben, (Annals of Math. 68(1958), 393-443)
The operations of passing to inverse images and to direct
(c)
images of sheaves are of course not inverse to one another; there are several interesting relations between the two operations, but for the present purposes, only the following observations need be made. Lemma 10.
Let
f: M --> N be a complex analytic mapping
between two Riemann surfaces, which pexhibits M as an r-sheeted
branched covering of N ; and let
be an analytic sheaf over M.
,p
There is a homomorphism of analytic sheaves
(i)
(p: fo(fo(J )) -4 A , which is onto. (ii)
If
fo(fo( 6-
N
is a locally free sheaf of rank 1 over N , then
A ) = 3 N a fo( 0) . N
Proof.
To begin, there is a homomorphism
(i)
cp': f-1(fo(J )) --> J of sheaves of f-'( as follows.
h e (fo(,0 ))f(p)
resentative
of
ha e r(fl(Va), J )
p e M , then
.
The germ h will have a rep-
for some open neighborhood Va
f(p) , by the definition of the direct image sheaf.
P E f-1(Va) , the section ha
determines an element
and this element is defined to be the image evident that
cp'
f-1(N 0- )-modules.
if
N )-modules, defined
for some point
If h e f l(fpo(J )) p
by definition,
N
.
ha(p) Ej p It is
is a well-defined homomorphism of sheaves of Note also that the mapping
s e,4 p , and sa a r(Ua, 2 )
through
cp'(h)
Since
s , then sa
cp'
is onto.
For
is a section of A passing
can be extended to a section
sa a I'(fr1(f(Ua)), I ) , by setting
sa s 0
on the components of
f-1(f(Ua)) ment
other than Ua .
The section
sa
h e (fo(,O ))f(p) = f l(fo( ) ))p ,
Now recalling that f°(fo(6 )) define the homomorphism
and clearly
(PI (h) = s
0
f-1 (f
M
f-1 (N
cp: f°(fo( I )) -
)),
0
by putting
ge
cp(g ® h) =
determines an ele-
J
e
p
a well-defined homomorphism of ana-
lytic sheaves, and (ii)
is onto since
cp
is onto.
cp'
After restricting attention to a sufficiently small
open neighborhood on N , it can be assumed that ;! = N(k -
Then,
on the one hand, 5o p f0(, ) N ® if fo(j) f0( J N
= f°(N(
and on the other hand, f°9
f°( 5 ) also.
M.
0 = MQ ® A .0 = J and
M
so that
f0(f°( 3) 0a J) = fo(D ) C
toMconclude
This suffices
the proof.
By using the various properties of induced sheaves which have been established above, it is quite easy to extend to arbitrary compact Riemann surfaces the basic existence and representation theorems established in §2 for the projective line. Recall from §10 of last year's lectures that any compact Riemann surface
M
can be represented as an r-sheeted branched covering
f: M -- IP
of the projective line. Theorem 8.
If J is a coherent analytic sheaf over a
compact Riemann surface M , then there are locally free analytic sheaves _' , _+ 1
over M for which there is an exact sequence of
the form
0 --? 3' 1
? 9 -. J
0.
Proof.
Let
f: M - P be a complex analytic mapping
exhibiting M as an r-sheeted branched covering of the projective line
3P.
The direct image sheaf fo(J )
is a coherent analytic
f°(J )
sheaf over P , by Lemma 9(iv); so by Theorem 6,
the image of a hommorphism
will be
fo( , ) , for some locally free
analytic sheaf R over the projective line.
That is, there is an
exact sequence of analytic sheaves over P of the form
--;- fo(J ) T 0 .
The sheaf
= f°(T, )
is a locally free
sheaf by Lemma 8(iii); and by Lemma 8(11), the following is also an exact sequence of analytic sheaves:
Combining the homomorphism
cp
W-;? f°(fo(,p )) -- 0
with the homomorphism of Lemma 10(i)
leads finally to the exact sequence
9 -*-;-k ,J
T 0 .
The kernel
of 4 is coherent by Theorem 2; and as a coherent subsheaf of a locally free sheaf, it must also be locally free, by Theorem 3. This therefore provides the desired exact sequence
0 --
0 .
l Theorem 9.
If ,d is a coherent analytic sheaf over a
compact Riemann surface M , there is a complex line bundle
over M such that Proof.
Let
I
(R (g) 0 J admits a non-trivial global section.
f: M -- P be a complex analytic mapping
exhibiting M as an r-sheeted branched yycovering of the projective
line P .
The direct image sheaf
is coherent by
Lemma 9(iv), so upon recalling the remarks after Theorem 7, it
follows that 3, ® d fo( ,d ) = II,9( m , (where II, h is the ]P sheaf of germs of meromorphic functions over fore a non-trivial section
g
3P); there is there-
of this tensor product sheaf.
For
asuitable finite open covering representation
of F, this section has a
(Ua)
g = {ma ® sa) , where ma is a meromorphic func-
tion in Ua and sae r(Ua,fo(J ))
.
The functions
ma will
have altogether finitely many poles over P ; so there is a locally free sheaf such that
14" of rank 1 over P with a section h = (ha) a r(P, -)
hdna is holomorphic in
Ua .
Then hg= {hna ®sa) 3s
a non-trivial holomorphic section of 3 ® fo(J ) , so that
r(3p,3®f0
By Lemma 8(iii)thesheaf f°(3') will be locally free of rank 1 over M , so that f°(3 ) =QC(s) for some complex line bundle
I
over M . Then from formula (2) and
Lemma io(ii) it follows that r(M, (¢ Q) ®J) = r(M, f°(9) ® J) ' .o )) r(a', -91 ®g fo(iJ )) suffices to conclude the proof. Corollary.
0 , which
Any complex analytic vector bundle over a
compact Riemann surface M admits non-trivial meromorphic sections. Proof.
If
0 e Hl(M,
.
(m, 6L ))
is a vector bundle of
rank m over M , it follows from Theorem 9 that there is a com-
plex line bundle
a H1(M, (Q *)
such that LV (g) ®(!{ (0) = U (g 00)
admits a non-trivial holomorphic section h ; letting
g
trivial meromorphic section of
gh
9-1 , it follows that
non-trivial meromorphic section of
0 , as desired.
be a nonis a
Riemsnn-Rock theorem
§4.
Before considering the Riemann-Roch theorem itself, it is
(a)
necessary to examine some further general properties of complex analytic vector bundles.
Let
0 e H1(M, A x (m, CI-))
be a complex
vector bundle of `rank m over a Riemann surface M ; and let
(0
)a
Z1(ti'L
, b t (m, a-)) be a representative cocycle, in terms
of some open covering V0
a GL(m, 4S'U
C43
{Ua)
of M .
Since the elements
are matrices over a commutative ring, their
)
1
a
determinants
a GL(l, U
= det 0
qi
a
fl
Ua n U ,3 are
U)
well-defined; and it is evident that these determinants form a cocycle
(q
) e Z
1 (1/(
,
(
)
.
The cohomology class
cp
of the
cocycle is clearly determined uniquely by the cohomology class
0
and is thus a line bundle canonically associated to the vector
bundle
The line bundle
0 .
cp
line bundle of the vector bundle cp = det 0 .
will be called the determinant 0 , and will be denoted by
The elementary properties of the determinant line from the definition.
bundle follow
Thus for two vector
bundles 01 a H'(M, ht (ml, (Q-)) and 02 a H1(M, 2t x.
6))
,
note that
det (01 ®ID2) _ (det 01) (det 02) , and (1)
det (01 ®02) _ (det
01)M2
(det 02) M, ,
where the tensor product of line bundles (the group operation in Hl(M, (Q-*))
is written as multiplication.
a complex line bundle and
0
In particular, if I is
is a complex vector bundle of rank m,
then det(g 0 0) = fm then
det 0 . Flarther, if 0 C T
det !' _ (det o) - (det e) .
and
!'/@ = 8 ,
The Chern class, or characteristic
class, of a complex line bundle
considered as an element of
the group 8 of the integers, will be denoted by c(g) .
(For the
general properties and the definition of the Chern class of a line bundle, recall §7(a) of last year's lectures.) vector bundle
0
c(det 0) , the Chern
is associated the integer
class of its determinant line bundle.
Thus to a complex
(This integer is sometimes
called the degree of the line bundle, but that terminology will not be used in the present lectures.)
A complex vector bundle
0 e H1(M, h t (m, 0. ))
a subbundle of a complex vector bundle
0 C
and written
is called
`F a H1(M, b' (n, (A )) ,
`F , if the corresponding locally free sheaf t(O)
is given as a subsheaf of
sheaf &-(!)/ 0(0)
6L(T) , in such a manner that the quotient
is also locally free.
then associated to a complex vector bundle
The quotient sheaf is e e Hl(M, b 'k (r, 0 )),
which will be called the quotient bundle and denoted by e = `F/@ The bundle
`F
the bundle
e .
is also called an extension of the bundle
0 by
In such a situation, there is an exact sequence
of locally free analytic sheaves of the form
0 -^-;- 6' W --? 6- (y) _.- 6L (e) --? 0 i the associated exact cohomology sequence then begins
o --o r(M,
.---a r(M, 6 (y)) --> 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
with the notation as in §5 of last year's lectures. norms on
r(M, e1'1(0)) , and thus determine
The
pn are
the structure of a
topological vector space. = r(M,
r(M, a
Of course, since
e 1,1(0 ® Kl-r ® K1.-s)
, all the spaces
r,s(o)) _ can
r(M, e r,s(o))
likewise be topologized.
Nov, for the application of these considerations, let
0 E R1'(M, J T- (m, (P )) be any complex vector bundle over the compact Riemann surface M , and let
(0a$) E z1"(?A ;
)`L J
x (m, ( ))
be
a defining cocycle in terms of a coordinate covering '/C = (U,za)
of M .
The matrices
n u) , where
0aP = t0-l a GL(m, L9U
a
t
P
denotes the transposed matrix, clearly also define a cocycle for ; the bundle defined by this cocycle
that covering
be called the dual bundle to the complex vector bundle will be denoted by
(0Q;3)
0 ,
will and
0
On a compact Riemann surface M , the space
Lemma 12.
is the space of continuous linear functionals on the
r(M, K (o ))
r(M, F
topological vector space
1,1(x))
.
In terms of a suitable coordinate covering
Proof.
VL = (Uceza) , let T = (Ta) a r(M, 7( (o*)) and F = (Fa) a r(M,
£1,1(0))
; and let
(ra)
unity subordinate to the covering R .
Thus
Ta will be an
Tia over the open set Ua , and Fa
m-tuple of distributions
will be an m-tuple of
Co partition of
be a
C"
Bia of type (1,1)
differential forms
over the open set Ua ; and writing the matrices 0aP = t0
l
0a$ =
in each intersection Ua n UP
= (cp ija
that and
Tia
J
Bia =
iJa;B jP
J
-69-
it follows
Each Tia will be a continuous linear functional on the subspace of
r(M, uP1,1)
consisting of those'forms with support in
as noted last year; and hence
Tia(raeia)
UCe ,
is well-defined.
Note
further that, restricting attention to the subset Ua n UT ,
Z Tia(rc peia) = E ijk "ij
i
E
ijk
*
TJB(raxBp
cp
eke
TjB(rcxBeko) = Ej TjO(raroej6)
It follows readily that defining
T(F) = exhibits
r(M, )((fi*))
tionals on
,
as desired.
(Serre duality for vector bundles.)
be a compact Riemann surface, and
complex vector bundle over M .
and H°(M,
'U(fi*))
Proof.
Ta(r 'a)
as the space of continuous linear func-
r(M, e 1'1(0))
Theorem ]2.
a
Let M be a
0 e H'(M, .bZ (m, &' ))
Then the vector spaces
H1(M, a (-w
are canonically dual to one another.
The proof is the same as that of Theorem 9 of last
year's lectures, (replacing Lemma 13 of last year's lectures by the preceding Lemma 12). Remarks.
Thus no further details will be given here.
It should be observed that, applying the Serre
duality theorem in the form it is given in Theorem 12, the Riemann-
Roch theorem for complex vector bundles can be rewritten as
(9)
dim HO(M, 19(0)) - dim HO(M, Qtl'U(-D*)) = c(det 0) + m(l-g),
for any vector bundle
0 e H1(M, . X (m, Ql ))
Riemazua surface M of genus g .
over a compact
§5.
A classification of vector bundles of rank two
(a)
In discussing the classification of complex analytic vector
bundles over a compact Riemann surface, we shall initially restrict attention to vector bundles of rank two; this case is both sufficiently complicated and sufficiently useful to be of interest.
(As
to the usefulness, recall the discussion in §9 of last year's lectures.)
Perhaps the first topic that comes to mind in considering the classification problem is that of determining the possible extension of one complex line bundle by another.
Let
TV
complex analytic line bundles over a Riemann surface M . from §4(a) that a complex analytic vector bundle called an extension of
cpl
by
0
cp2
be two
Recall
of rank two is
if there is given an exact se-
cp2
quence of analytic sheaves of the form
(1)
o
> 6-(Cpl)
> 4f 0)
Two such extensions
0
a sheaf isomorphism
e: &(0) --> 9(01)
and
0'
> (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 `i ,
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) <
c(det 0) + 1 - g
.
it thus follows that
dim HO(M, 61 (0 -1 ® 0)) > 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) < <
cp C * ® 0
® 0)) , so that
®0
is also
The notion of stability is therefore well-behaved under
tensoring with line bundles.
Let I (M, 9)
denote the set of denote the set
stable vector bundles in "f (M,9) , and u "(M,9)
of unstable vector bundles in if (Mg) ; so 1f (Mg)
can be decom-
posed into the disjoint union
?I(M,0 = 1I(M,0 U 4 "(M,9) Then T'(M,g) ® i = V'(M,gn2) and V"(M,9) ® Tl = for any line bundle
I
.
Note that
0 e u (M, g)
2r"(M,g12)
is stable pre-
cisely when div 0 < 2 c(g) ; for by definition,div 0 = max c(O) (PC0 Then
V'(M,g) = {m E V(M,g)dbvo0 <
c(g))
2
(16)
E }.f(M,gdiv 0 >
I1(M,g)
c(g)) 2
Let M be a compact Riemann surface, and con-
Lemma 15.
sider a vector bundle (i)
If
div 0 >
If
Proof.
(i)
cpl
0
is indecomposable, there is
such that cpl C 0 and c(cpl) = div 0
Suppose that
div 0 >
c(g) ; let
2
a subbundle for which c(cpl) = div 0 , and let quotient bundle.
Thus the vector bundle
0
cp2 = O/cpl
If
cp C 0
c(cp) = div 0 , the vector bundle
cpl C 0
be
be the
has a full reduction,
in the sense of the Corollary to Theorem 10, where diagonal line bundles.
cp1
c(cpl) = div 0
div 0 = 2 c(g) , and
a unique line bundle
.
c(g) , there is a unique line bundle
and2
such that cpl C 0 (ii)
0 E V (M, g)
cpl,cp2
are the
is another subbundle for which cp l ® 0
admits a non-trivial
holomorphic section, by Lemma 13; but then, as in the discussion
on page 79, at least one of the diagonal bundles must also admit a non-trivial holomorphic section.
cp lcp1' cplcp2
Since
c(cp lcp2) = c(cpllcp2) = -c(p1) +c(pp2) _ -2c((pl) +c(det 0) = = -2c((pl)+ c(g) _ -2 div 0 + c(g) < 0 , that bundle has no non-
trivial holomorphic sections; so the bundle
cp lcpl
must have such
sections. Since c(cp 1cp1) = 0 , necessarily (p- lcpl = 1 , and so cp = p1 , as desired.
(ii)
If
1
div 0 =
c(g) , the argument begins as in part (i);
2
the only change is that now Q- 1Q) = 0 .
To continue from that
F f be a holomorphic non-trivial section of
point, let
2 it is merely necessary to show that
ment as in part (i).
0 = 0 , and conclude the argu-
Suppose, contrariwise, that
tically zero; since f2 is a section of necessarily
tp 1tp2 = 1
tp 1 ®O;
f2
is not iden-
T-1T2 , and
c(q lcp2) =o,
In terms of a suitable coordinate covering
.
lJt.= (Ua) , the bundle `Y = 9 -1 ®0 can be defined by a cocycle (`Ya3)
of the form `Y
ap
1),
=
?b.
ap
C0
and it has then a holomorphic section of the form
F = (FCC) =( fla c
where
is a non-zero constant.
c
The functions
(fla)
are thus
holomorphic in Ua , and satisfy
fla(p) = w(p)f10(p)+cxa,(p) But then the cocycle
`1
=
T-1 ®O
self decomposable. Now let
,
div 0 >
2
itself,
(*)) ; as in Theorem 13, 0
is it-
This contradiction serves to conclude the proof.
If 0(M,g)
c(g)
H-(M,
(x0)
is then decomposable, so that
denote the subset of ?l(M,g)
ing of decomposable bundles. that
p e Ua (1 U. .
(cXX) , hence the cocycle
must be cohomologous to zero in the bundle
for
whenever
consist-
It follows immediately from Lemma. 14 0 e if 0(M,i) , hence that every
decomposable bundle is unstable; that is to say
V (M,0 C if"(M,0 The set rO(M,g)
.
can be described quite easily.
Note firstly
that cp1 ®cp2 = cpi E) (P2' = 0 e 1°(M, g) if and only if either (For suppose that
or cpl = cp2 .
1Pl = p1
Chern class of these four bundles. that
cpi C 4D , it follows
cpll ®0 admits
1 C cpll ®0 ; hence by lemma 13, the bundle
a non-trivial holomorphic section.
cpilcpi
or
cpilep2
j =2 .)
Then one of the two bundles
admits a holomorphic non-trivial section; but
since c (cpilcp!) < 0 , or
Since
has the largest
cp1
it is clear that
cpl = cps
for either j =1
Now consider the mapping
H(M, C*) '-->'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) 2`a 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
Conversely, if
.
0i
it follows
= 0iv Fip ,
Fa = Fla ®.. .® Fra satisfy Fa = '
Ta can be decomposed into
®...® o
0i ; and selecting matrices
satisfying Fia
l))
Dap = 01V
= (Tiia)
will
are arbi-
(Tlla ®.
Trra)
Note that
= tr(Ta (dFa Fla ®...® dFra F ra )) r
tr(Tiia i=1
via
IF-J-)
But then the desired result follows immediately from Corollary 2, and the proof is thereby concluded.
Now consider any endomorphism T = (Ta) a End 0 ; here
Ta
are holomorphic m X m matrices in the various sets
Ua , and
Ta(p) = 0ap(p)TP(p)Oap(p)-1 whenever p e Ua fl UP .
The charac-
teristic polynomial of the matrix Ta thus coincides with the characteristic polynomial of the matrix
TP
in Ua fl UP ; so that
the characteristic polynomial of the endomorphism
T
is a well-
defined polynomial of degree m , with coefficients which are holomorphic functions on the Riemann surface M .
If M
is compact,
the coefficients must be constant; but then the roots of the poly-
,
nomial, which are the eigenvalues of the matrices constants.
Ia
The identity endomorphism
Ta , are also
I c End 0 , where the matrices
are all identity matrices; all eigenvalues of the identity are 1.
An endomorphism T c End 0 integer
r , where
0
is called nilpotent if
Tr = 0
for some
is the zero endomorphism; it is obvious that
the nilpotent endomorphisms are precisely those for which all the eigenvalues are zero. Lemma 18.
Let
be an indecomposable
0 c R1(M, .h '.(m, ( ))
complex analytic vector bundle over a compact Riemann surface M
Then any endomorphism T = (Ta) c End 0 a sum T = cI + N , where
and N
c e ( ,
is the identity endomorphism,
is a nilpotent endomorphism.
Note that an endomorphism of the form T = cI + N
Proof.
has all eigenvalues equal to
of T
I
can be written uniquely as
are equal to
values equal to
c ; and conversely, if all eigenvalues
c , the endomorphism N = T -ci has all eigen-
0 , hence N
is nilpotent.
Therefore the desired
lemma is equivalent to the assertion that, with the same hypotheses,
an endomorphism T c End 0
has all eigenvalues coinciding.
To prove that this is so, suppose cQnntrariwise that an
endomorphism T has at least two distinct eigenvalues. terms of a suitable coordinate covering ZL= (Ua) cocycle
(0a$) a Z1( V,, h ;((m, (9))
the endomorphism T = (Ta) a End 0 of the form T. = Tla ® T2a , where eigenvalues.
and a suitable
representing the bundle
0
will be represented by matrices
T1a and T2a have different
Now decomposing the matrices
sponding blocks, the condition that
Then, in
0a3
into four corre-
T be an endomorphism of
0
is just that
Tla 0
022x)
021QQ
in each intersection Ua fl u, .
= 012aVT20
0llap 012a
011OP 0120P
T2a
0
-D21ap
022ap
0
0
T2P
But this implies that
and T2a02lCP = 021OPT1p .
Since
T1
together different eigenvalues, it follows that 021 = 0 , hence that the bundle
T10
0
Tla@12
and
T2
@Ip = 0
=
have aland
is decomposable; this is a con-
tradiction, so the proof is thereby concluded. Lemma 19.
0 e H1(M, )5;((m, 0 ))
Let
be a complex analytic
vector bundle over an arbitrary Riemann surface M , and let T e End 0
be an endomorphism of the bundle
0
.
Then there are
representatives 0 = (@a3 ) a Zl('l 1. , 1 x(m, Ol ) ) and T = (Ta) such that the matrices
and Ta are all in upper triangular
0
form.
Proof.
p C 0
Let
tained as a subbundle
be a complex analytic line bundle con-
p C 0
and with
c((P) = div 0
.
The set
is then a non-trivial finite-dimensional complex
r(M, CL ((P1 ® c))
vector space, and all the non-trivial sections are non-singular holomorphic vector-valued functions, (that is, have trivial divisors).
T
It is clear that so
T
also determines an endomorphism of
q)-1 ®@ ;
acts as a linear transformation on the vector space
r(M, & (q)-1 ® c))
.
Let
F c r(M, Ol(q)-1 ® 'b))
section which is an eigenvector of the operator
be a non-trivial T ; and choose
QQ
representatives T = (Ta) a C0(
0 _ (0a3 ) E Z1( , /J x (m, LI ))
z ,
a m Xm)
and
for a suitable open covering 11(_, such
F
that the section
is represented by Fa = E1
(where as before,
1
El = (.9.)
El = Fa = "Cap =
). Now since
0
it follows
tj
that the matrices
have the form (dap 0
*
*
...
*
*
1
Dap
and since TTl = TaPa Ta for the eigenvector
= I''F'a = 1,E,
, where
X
is the eigenvalue of
F , it also follows that the matrices Ta
have the form
Ta 0
The desired result then follows from the obvious induction. With these two observations made, the most important consequence of Theorem 15 follows quite readily. Theorem 16.
surface, and
(Weil's theorem)
0 = 01 ®...® 0r
Let M be a compact Riemann
be a complex analytic vector bundle
over that Riemann surface, where the bundles Then
0
for all
@i
has flat representatives if and only if i
are indecomposable.
c(det @i) = 0
.
Proof.
It follows immediately from Corollary 3 of Theorem 15
that it suffices to prove that an indecomposable vector bundle has flat representatives if and only if
c(det 0) = 0
.
0
So suppose
that 0 c HI(M,
is indecomposable, and let
be
D*O
D*O
the cohomology class given in Theorem 15, viewing functional on the complex vector space 0
.
End 0
as a linear
of endomorphisms of
In view of Lemma 18, the functional A is fully determined
once its values on the identity endomorphism and on the nilpotent endomorphisms are determined.
On the one hand, note that
D*0(I) =
= - 'R.tr(dFaP ) = - lt d log det Fa , recalling formula (7); for it
is familiar - or an easy calculation if not familiar - that
tr(dFF 1) = d log det F for any non-singular matrix-valued function F .
The functions
fa = det Fa , using the notation in the discus-
sion preceding formula (7), form a meromorphic section of the line bundle
det 0 ; and R d log fa = p
e
M
Theorem 11 of last year's lectures,
Vp(fa)
Therefore, by
d log fa = c(det 0)
D*@(I) = - c(det 0)
(9)
.
;
so that
.
On the other hand, for a nilpotent endomorphism T , apply Lemma 19 to choose representatives T = (Ta)
0 = (0ap) a Z1(l/7L, b x (m, S ))
and
which are upper triangular matrices; and choose meromor-
phic non-singular matrices
Fa also in upper triangular form and
Ta, dFa, F1
Then since all the matrices
satisfying Fa = (D '
are in upper triangular form, the diagonal entries in Ta d'a Fa will be the products of the corresponding diagonal terms in the
factors; but the diagonal entries of Ta are its eigenvalues, and are all zero since (10)
Ta
D@(T) = 0
is nilpotent. if
T c End 0
Thus is nilpotent.
* Combining assertions (9) a n d (10), it follows that D O = 0
if
and only if that
c(det 0) = 0 ; then by Theorem 15, it further follows c(det 0) = 0
has flat representatives if and only if
0
which suffices to conclude the proof.
is a complex analytic vector bundle admitting flat
0
If
(C)
representatives, there naturally arises the problem of describing all of the flat representatives of if
0
0
.
Note that from Theorem 16,
has flat representatives, necessarily
line bundle
det 0
tensoring by
c(det_10) = 0 ; so the
has a flat representative
p E ;i (M,C*)
.
Then
establishes a one-to-one correspondence between all
cp
the flat representatives of
q)-1 ® 0
and all those of
0 ; so for
the further discussion, it suffices to restrict consideration to the
set of vector bundles (17 )
Um =
1/m(1)
_ (0 E
a )) Jdet 0 = 1)
H1(M,
The problem is to describe the set of flat representatives of any
bundle
0 E Ir
m
.
For any bundle
0 E
select a representative cocycle
(@a3 ) E Z1( ?A, h .(m, (')) , and consider the one-cocycle DO = (DOap) E Z1( V., l7' (K ® Ad @)-)- introduced in Theorem 15. 0
complex analytic connection for the bundle covering such that
?11 is a zero-cochain
in terms of the
X = (Xa) E CO( u , Cl (K ® Ad @))
5). = DO ; such connections exist, in view of Theorem 15,
precisely when the bundle
0
has flat representatives.
all such connections will be denoted by A(Z are two such connections, the differences
cochain
A
Tc
C0( 7/L
, 0 (K ® Ad 0) )
,0)
Ta =
such that
Xa
If
.
-
The set of %, X1 E A(1R,0)
Xa form a one-
ST = 8W' -5%=DO-DO = 0
that is to say, these elements determine a section
E r(M, (Y (K ® Ad @))
.
The set A(VL m) C CO( Ut , lV (K ® Ad @))
of all such connections is thus a coset of the subgroup r(M, (P (K ® Ad @)) C CC(VL
of the vector bundle
,
I (K ® Ad m))
of sections (zero-cocyles)
K ® Ad @ ; so that, on a compact Riemann sur-
face M , the set of connections dimensional linear space.
A(1/[ ,@)
is naturally a finite-
(As to the terminology, the problem of
describing the flat representatives of a complex analytic vector bundle
@
can be viewed as a problem of reducing the pseudogroup
structure on the bundle space, viewed as a complex manifold.
A
general class of similar pseudogroup problems can be treated by basically the same formalism that is involved here; this includes problems leading to the usual affine and projective connections on manifolds, and connections in vector bundles, and the terminology has been adapted from the geometers.
For a discussion of connec-
tions from this point of view, see R. C. Gunning, "Connections for a class of pseudogroup structure," Proc. Conf. on Complex Analysis (Minnesota, 1964), Springer-Verlag, 1965.) If
T
is any automorphism of the vector bundle
dition that
T = (Ta)
Ta C GL(m, 3-Ua)
T
of the vector
induces in a natural manner an automorphism Ad T
bundle Ad @ , or of the vector bundle KO Ad @
@ ,
.
(For the con-
be an automorphism is just that the matrices
satisfy TT(p)maP(p) _ 'D a$(p)TP(p)
whenever
p E Ua fl U. ; and then it is evident that Ad Ta(p)-Ad Oap(p)
Ad
TP(p) , or that the same relation holds upon tensor-
ing with any line bundle.)
The mapping Ad T
in turn induces
isomorphisms Ad T: Cq( 1Jt , tY (K ® Ad m))
Cq( Uz , v2 (K ® Ad m))
which commute with the coboundary mappings. see an explicit form for these mappings.
It is of interest to
In terms of the natural
identifications, a zero-cochain X e C0( 21L, (J- (K ®Ad 0) ) can be represented by holomorphic differential forms
Xa of type (1,0) in
Ua , and Ad T-X will be represented by the dif-
the various sets
ferential forms Ad Ta-Xa ; and a one-cochain Cr
C
C1(lj(,, -(K (&Ad 0))
ential forms
can be represented by holomorphic differ-
QaP of type (1,0) in the various intersections will be represented by the differential
Ua fl U. , and Ad
forms Ad
(See also the discussion in Appendix 1.)
It
is of particular interest to consider the effect of the mapping
Ad T
on the cocycle
morphism,
DO =
From the definition of an auto-
0aP = TaOapTPl
the differential operator
in each intersection Ua flTT. ; applying D ,
in view of (5) it follows that
=
DO0
V
,6
DT
=
and viewing these elements as forming a
Setting
Ta = Ad(Ta)-DT
cochain
T = (Ta) a C0(VZ, 3 (K ® Ad m)) , the above condition can
be rewritten
Ad(TP)'DOaP
- DOa = T -(O 1 )'Ta ;
or, in other words, DO = ST
(12)
Thus the cocycles
DO
and Ad(T)-DO
are cohomologous; indeed,
formula (12) gives an explicit coboundary relating these two cocycles,
in terms of the automorphism T . Now if )'. = (xa) c A( j/L, c) is a connection, so that DO = 5X
upon applying the operator Ad T
it follows that Ad(T)'DO =
S
therefore by (12),
DO = Ad(T) -DO - BT
= 5[Ad(T)'%- T) so that
T
is also a connection for the bundle
to each automorphism T c Aut 0
0
Thus
.
there is associated a mapping
Ad*(T): A(VL ,0) -k A( 12,O)
Ad*(T) - x = Ad(T)'X - T ; it is an easy matter to verify that the mapping (13) exhibits as a group of transformations on left to the reader.
A('UL ,0) , but the details will be
Two connections
equivalent, denoted by X N XI
Aut 0
if
X,X
e A(1JL,m)
will be called
XI = Ad *(T.) X for some auto-
morphism T c Aut 0 ; it is clear that this is an equivalence relation, indeed, is just the equivalence relation associated to the
group action of Aut 0
on the set A(VLA) .
The set of equiva-
lence classes of connections will be denoted by A (1!L ,0)
.
Theorem 17. It M be an arbitrary Riemann surface, and 0 e
Nm(1)
be a complex analytic vector bundle over M represented
by a cocycle (0V) c Zl(1j(, ,5 ' L (m, 6 )) covering jfj,= (Ua)
that all the sets
in terms of an open
of M ; and suppose the covering VL is such Ua are contractible.
Then there is a one-to-
one correspondence between the set of flat representatives of in the covering Ik , and the set of connections for the bundle
0
A*(2/{,,0) .
0
of equivalence classes
If X = (Xa) e A( V,O) is any complex analytic
Proof.
connection, there are functions Fa c GL(m, C U )
such that DFa =
a Xa in each set Ua ; that there exist local solutions of the equation DFa = Ila , or equivalently of the equation dFa = FaXa ,
is
familiar, and these yield global solutions in any simply-connected
set Ua . form
The most general solution is readily seen to be of the
Ca c GL(m,C) ; for if DFa = DGa , it
CaFa , for some matrix
follows from equation (5) that D(FaGG) = Ad(Ga)'DFa + DGa =
FaG;
= Ad(Ga)'[DFa - DGa] = 0 , hence
= Ca is constant.
in the proof of Theorem 15, it follows that
Then as
Dap = FaOa P1
repre-
sents the same vector bundle, and since it readily follows from (5)
that
DO''
= 0 , this yields a flat representative.
Of course,
by C.F. yields the same flat representative, as an
replacing
element of H1(M,GL(m,C))
.
Thus each connection X c A( UL m)
yields a well-defined flat representative of the bundle
0 ; and
conversely, as in Theorem 15, all the flat representatives of
0
in terms of the covering V. so arise. Now two connections
x = (xa) _ (DFa)
and X _ (X)=(DFa)
yield the same flat representative of the bundle there are matrices
Ca a GL(m,C)
0
if and only if
such that
in Ua n UP
Fa0a ')-1 =
recall that flat vector bundles are elements of the cohomology set H1(M,GL(m,C))
.
This can evidently be rephrased as the condition
that there exist constant matrices
represents an automorphism of
Ca such that Ta = (Fa)-1CaFa
0 ; or equivalently, that there
exists an automorphism T = (Ta) c Aut 0
such that FaTaFF are
Further, the condition that
constant matrices.
F'«TaF
be con-
stant is just that
0 = D(FaTaF&1 ) = Ad(FaT; )'DF'a + Ad(Fc)'DTa + DFa Ad(Fa).[Ad(T-
=
or
Ad(Ta)-DTa = Ad*(Ta)'ka ;
Xa =
and by definition, this is equivalent to the condition that X be equivalent to
X'
Thus, connections yield the same flat repre-
.
sentative precisely when they are equivalent, and the theorem is thereby demonstrated. Remarks.
If If is a refinement of the covering M , any
refining mapping
µ: if - X induces a mapping of connections
µ*: A*(1k ,0)
A*(3f ,0) ; and the correspondence between con-
nections and flat representatives established in. Theorem 17 is preserved upon passing to a refinement. connections
A*(0)
Thus it is possible to define
by passing to the direct limit over all cover-
ings, and to extend Theorem 17 to a correspondence between A (0) and the set of all flat representatives of
theorem
0 .
However, by Izray's
(see §3(f) of last year's lectures), all flat bundles can
be described by H1(111, GL(m,C))
for any covering 2A = [Ua) such
that all sets Ua and Ua n UP are contractible; and thus it is possible to take a fixed covering for all the discussion, to finesse this point altogether.
One slight extension of Theorem 17 should be noted here.
-117-
Beginning with a complex analytic vector bundle
0 c u1 m(i) =
= H1(M,aJ'j (m, 6 )) , one can consider the set of special flat
representatives in the cohomology set H1(M,SL(m,C)) ; here denotes the special linear group, the subgroup
SL(m,*)
SL(m,*) C GL(m,*)
consisting of matrices of determinant 1, and
the natural inclusion
is SL(m,C)
(m,
)
induces the mapping of cohomology sets
i*: H1(M,SL(m,C)) -> Hl(M, d.((m, 0 )) Note that whenever the one cocycle
(0a3 ) a Z1(1(, J x (m, L ))
represents the complex analytic vector bundle
lows that
0 = d log det 0
X _ (%a) e A(X 0) bundle
= tr(O
dO
0 c 'Wm(l) , it fol-
) = tr(DOap)
.
If
is any complex analytic connection for the
0 , then S(tr %) = tr(S)) = tr(DO) = 0 , so that
tr x e CO(?11, (0 (K)) tr X e r(M, (Q- (K))
.
is actually a section, an element
A connection
). e A(2L,0) will be called a
special complex analytic connection if
tr ). = 0 ; and the set of
such special connections will be denoted by A0(V ,@) T = (Ta) e Aut 0 , it follows that so on a compact Riemann surface, so
.
det Ta = det TP
If
X,X' c A(?A ,0)
tr T = tr X .
in Ua fl U. ;
tr Ta are connections, and if
A ti V , then by definition V = = tr
Now if
det Ta will be a constant, and
0 = d log (iet Ta = tr(dTa T-1) =
Ta =
.
so that tr X' =
The set A0(M ,O)
is thus preserved
under the equivalence relation introduced on connections; and the equivalence classes of elements of A0(?JI.,c)
AO( A 0)
will be denoted by
Corollary. It M be a compact Riemann surface, and 0 e lm(l) be a complex analytic vector bundle over M represented by a cocycle
(@a3 ) a Z1(Vt
,
4 x(m, Q ))
in terms of an open
covering V L= (TJa) of M ; and suppose that the covering T/Z is such that the sets
Ua are contractible.
Then there is a one-to-
one correspondence between the set of special flat representatives of
0
in the covering
, and the set A0( ,0)
classes of special connections for the bundle Proof.
0
Following the proof of Theorem 17, if
e A0(1/L,0)
X
of equivalence
and if Fa C GL(m, 61 Ua )
is such that DFa=Xy
then d det log Fa = tr DFa = tr Xa = 0 , so that det Fa is a constant; since
CoFa
is also a solution of the differential equation
DFa = Xa for any constant matrix Ca a GL(m,C) , there are solutions
it follows that
Fa a SL(m, 6 U) , and that the most general
a such solution is of the form CoFa for
.
Then
is a special flat bundle, uniquely determined by
O'4 =
the special connection X . and in
Ca a SL(m,C)
H1(M,GL(m,C))
Since equivalence in
H1(M,SL(m,C))
coincides, connections determine the same
bundle precisely when they are equivalent, and that serves to complete the proof of the Corollary.
It was noted in Corollary 3 to Theorem 15 that a decomposable complex analytic vector bundle
0 = 01 ®..® 0r
resentatives if and only if all of the bundles sentatives.
of
0
@i
has flat rep-
have flat repre-
It should be pointed out that the flat representatives
are not necessarily just the direct sums of the flat repre-
sentatives of the various bundles
@i , though.
To illustrate this,
consider the particular case of a decomposable vector bundle of rank 2, say
0 _
T1
e T2 , where
are line bundles with
q)1,q)2
Suppose that
c(cp1) = c(p2) = 0 , and
q)1 # q)2
(Cad) a zl(VZ,,GL(2,C))
is a flat representative of the bundle 0,
.
in terms of a coordinate covering Zk _ CUa)
surface M ; and that of a connection
corresponds to the equivalence class
(Cad)
under the correspondence of
X = (Ac,) e A(?A ,0)
Note that the cocycle
Theorem 17.
of the compact Riemann
(c
)
is decomposable as a
CLP
if and only if the connection
direct sum Oaf = c1CP ® c2a43
X _ ()) cocycle
is decomposable correspondingly.
Recall that under the correspond-
is decomposable.
(Cad)
(For suppose that the
ence of Theorem 17, this cocycle has the form Cad = Fa"aPF-l
where
.h'( (2, (9
(Fa) a CO(
satisfy DFa = %a . Explicitly
))
then,
(f1.i
r°la3 0 l
0
c2
f12P P22
f2
_
(flla 21' f12a (Pic$ 0 f
1.
22 a
0
2
so that '
°1a3 f12a = 92a$ f12P '
=
11a - q)1ap -lip '
_ °2a3 f2la
If the line bundles
cp1aP
(c1a)
°2aP
f22a = 92
P21 '
and
(p1ap)
are equivalent, as can
clearly be supposed, it follows immediately that f21a - 0 ; for
P12a and
f220
f12a - 0
and
P21, are holomorphic sections of non-
trivial complex line bundles of Chern class zero.
-120-
Then of course
DFa = Aa is decomposable as well.
The other direction is trivial.)
Thus to exhibit the existence of flat representatives of
0
which
do not decompose into the direct sum of flat representatives of and
,
P1
it suffices to show the existence of complex analytic
(P2
connections
X e A(1JL.,O)
which are not decomposable.
nections A = (Ace) are ccchains A e CO(A , that
These con-
(A (K (&Ad @))
such
SA = DO , or more explicitly, such that
A-
D@ap ;
writing this out in terms of the matrix components
Aiia , viewed
as analytic differential forms, the condition is that
1
710
01a"3
x`.12 0
'210 '220
0
0
921
71a x.12 a q)lap 0
'1a '-22a
'P2aP
0
flap 0 0
D'P2
hence that
'2la ='P1o'210 ' Thus
bundle
Aii
'22a ='220 - D'P2ap
is any complex analytic connection for the complex line
Ti ; while
'12a andla are sections,
(%
.121) e P(M'
(2'211) e r(M,
Since
P1#q)2
but
1'0(q)1021))
1'0(q)2911)
c(cp1) = c(cp2) = 0 ,
-121-
,
it follows from the Riemann-
Roch theorem that dim r(M, m 1'°(cp102l)) = dim r(M, (91'°((p2 1)) = g - 1 ; and therefore, on a Riemann surface of genus
g > 1 , there
are connections k e A( V1,,(P1 6) %) which are not decomposable. (Since the fundamental group of a surface of genus 1 is abelian, all flat bundles are necessarily decomposable in that case; recall the discussion in §9(b) of last year's lectures.
The restriction
in the preceding observation is-thus to be expected.)
-122-
g > 1
P.
at sheaves: geometric aspects.
(a)
The last chapter was devoted to a discussion of the complex
analytic vector bundles which admit flat representatives; the present chapter will take up the discussion of flat vector bundles themselves.
Paralleling the terminology introduced earlier for complex analytic vector bundles (see §4(a)), a flat vector bundle X1 e__URZ(M,GL(n1,C))
X C H-(M,GL(n,C))
is called a subbundle of a flat vector bundle if there are representative cocycles
(X.) c Zl(jk,GL(n1,C))
X1aP X120
X*0 =
(1)
(XV) a Z1(V. ,GL(n,C)) of the form
and
o
Xza4
for some covering 74 of the space M ; the cocycle
(%) e Z1(VL , GL(n2, C)) represents the quotient bundle 2
= X/)1 a Hl'(M,GL(n2,C))
.
A flat vector bundle, X
is called
reducible if it contains a proper flat subbundle; otherwise called irreducible.
X is
X is said to be decom-
A flat vector bundle
posable into the direct sum of two flat vector bundles
X1
and X ,
written X = X1 ®X2 , if there are representative cocycles of the form (1) in which
XL2CtO a 0 ; if no such decomposition is possible,
the bundle is said to be indecomposable.
Again, a decomposable
bundle is reducible, but the converse does not generally hold. Recall from last year's lectures (pages 184-189) that to each flat vector bundle
X e H1(M,GL(n,C))
homomorphism % nl(M) --- GL(n, C)
called the characteristic repre-
sentation of that vector bundle, and that
up to an inner automorphism in
there is associated a
X is uniquely determined
GL(n,C) ; here
n1(M)
denotes the
fundamental group of the surface, with a fixed but arbitrary base
point. .The mapping X -> X establishes a one-to-one correspondence H1(M,GL(n,C)) --> Hom(rtl(M),GL(n,C))/GL(n,C)
(2)
,
where the quotient on the right denotes equivalence classes of
homomorphisms under inner automorphisms of
In particular,
GL(n,C)
every flat vector bundle over a simply-connected surface M
is
trivial; or more generally, the restriction of a flat vector bundle
to any simply-connected open subset of M is a trivial vector It is also easy to see that a flat vector
bundle over that subset.
bundle
X c H (M,GL(n,C))
is irreducible (or indecomposable) pre-
cisely when its characteristic representation X C Hom(rtl(M),GL(n,C))
is irreducible (or indecomposable, respec-
tively); details will be left to the readers. A flat vector bundle vector bundle, so the sheaves
X and (.(X) as usual.
X represents a complex analytic 10(X)
of germs of
CO*
sections of
of germs of holomorphic sections of X are defined
Moreover, since the bundle
X has a representative co-
cycle consisting entirely of constant matrices, the subset of C-(X) consisting of germs of locally constant functions is a non-trivial subsheaf of
6-(X) ; this will be called the sheaf of germs of flat
sections of the bundle
that the she basis cocycle
3"(X) {Ua}
X , and will be denoted by '(X) . can thus be defined as follows.
(Note
Select a
of the open sets of M , and a representative
(Xa*) c zl(VZ ,GL(n,C))
for the flat vector bundle
X
To each open set Ua c(Jl associate the complex vector space
aa
= Cn , viewed as the space of column vectors of length n ;
and to each inclusion relation Ua C UP
associate the linear trans-
It is
formation Xap: .P -> 9'a defined by the matrix XCP . readily seen that the collection
{Ua, a a, XX}
is a complete pre-
sheaf of complex vector spaces over M ; the associated sheaf is
defined to be the sheaf 3.(X) ) In general, a sheaf
of complex
vector spaces over M will be called a flat sheaf of rank n
if
each point p c M has an open neighborhood U such that the
restriction of 9 to U
is the trivial sheaf of vector spaces of
rank n over U , that is, such that 3 IU = (t X U .
The sheaf
of germs of flat sections of a flat vector bundle is obviously a flat sheaf; and conversely, it is clear that any flat sheaf S'
is
the sheaf of germs of flat sections of a flat vector bundle which
will be denoted by X( 9- )
.
Of course, this is merely another special case of the general structure of sheaves of modules over sheaves of rings, as discussed in §1.
Taking for R the trivial sheaf of rings*
I = C X M over M , the flat sheaves of rank n
are just the
locally free sheaves of R -modules of rank n ; and as in Theorem 1, such sheaves are described by their representative fibre bundles in
H1(M, h 7-(n,V, )) = H1(M,GL(n,C))
.
The notation is
a bit different from that introduced in the earlier general discussion, as a matter of convenience only.
Note that
Flat sheaves can also
C X M viewed as a sheaf has the product topology of
the discrete topology on C and the usual topology on M .
be viewed as sheaves of local coefficients in the sense of Steenrod; (see N. E. Steenrod, The Topology of Fibre Bundles, (Princeton University Press, 1951), especially §30).
For a simply-connected manifold M any flat vector bundle
(b)
X
is trivial, hence
Hq(M,
3
(X)) =
for all dimensions
q ; thus for a contractible manifold M it follows that Hq(M, 3 (X)) = 0
for all dimensions
A compact manifold
q > 0 .
such that the sets Ua
M admits a finite open covering 1/C = (Ua)
and all their intersections are contractible; and by the above
remark, such a covering is a Leray covering of M for the sheaf 9 (X) , q
.
so that Hq(j , 3 (X)) = Hq(M, 9 (X))
Now a q-cochain in
Cq(17
,
3'(X))
for all dimensions
is given by an n-dimen-
sional complex vector associated to each intersection
Ua n...n Ua # 0 ; so it is evident that
Cq(1R
(X))
,
is a
q
0
Thus, for a compact mani-
finite-dimensional complex vector space.
fold M , all the cohomology groups
Hq(M,
are finite-
(X))
dimensional complex vector spaces, whenever
is a sheaf of
9.(X)
germs of flat sections of a flat vector bundle
X .
The lowest
cohomology groups can be described explicitly quite easily. Theorem 18.
Let
be a flat vector
X e H-(M,GL(n,C))
bundle over any (connected) manifold M .
If
then H°(M, 9 (X)) = 0 ; and more generally,
X
is irreducible,
H°(M, 3 (X))
Cr
where r is the largest integer such that the trivial bundle of
rank r
is contained as a subbundle of Proof.
the bundle
X
Let 7n = (Ua)
X
be an open covering of M such that
is defined by a cocycle
(x
) a Z1( Vl ,GL(n,C))
If Ho(M, a (x)) = r(M, 3 (x)) _ d , select Fi e r(M, 9- (X)) ; each section Fi
sections vector
r
linearly independent is given by a column
Fia associated to each set U. such that
whenever Ua fl U0 j 0 .
For each Ua the
r
Fia
= X
.Fip
Fia
vectors
'
i = 1,...,r , are linearly independent; so there is a nonsingular
matrix 8a a GL(n,C)
such that ®a Ei = Fia , where
Ei
is the
column vector having the entry 1 in the i-th place and entries elsewhere.
The cocycle
X'
Ba
=
®0
X00
is equivalent to the co-
cycle
Xa$ , hence defines the same flat bundle
XI
= Ei
X'
for
0
X .
Since
however, it follows that the matrices
i
have the form
0
0*
1
* T i
0
0 ... l * ...
0
0... 0*... *
0
0 ... 0 * ... * /
n-i
thus the identity bundle of rank r , the bundle defined by the cocycle formed of identity matrices of rank r , appears as a sub-
bundle of X . r = 0 .
In particular, if
X
is irreducible, necessarily
That concludes the proof.
The corresponding one-dimensional cohomology group can be described purely algebraically in terms of the characteristic representation; the machinery involved is the cohomology theory of abstract groups, but a very simple-minded approach is quite adequate.
Selecting a particular mapping
X: ic1(M) --? GL(n,C)
for the
X
characteristic representation,
can be viewed as exhibiting
as a group of operators on the vector space same thing,
ic1(M)
CP , or what is the
is a 3c1(M)-module under the action described by X .
Cu
A 1-cocycle of the group
3c1(M)
with coefficients in this icl(M)-
module is a mapping A: 3c1(M) --? Cu , associating to each element
a column vector A7 a Cu , such that
7 e Ic1(M)
for all
Xy1 Ay +Ay
Ay
(3)
1y2
1
2
2
71,72 a 3c1(M) ;
the set of these cocycles form a complex vector space denoted by Zl('c1(M),X)
.
A 1-cocycle A e
if there is a vector
B e Cu
Ay = B - Xy
is called a 1-coboundary
such that B
for all
7 e IC1M ;
it is easy to see that (4+) defines a 1-cocycle A for any vector B e Cu , and that thus the set of 1-coboundaries forms a vector subspace
B1(3c1(M),X)
.
The quotient space
H1(3t1(M),X) =
is called the first cohomology group of the group
Ic1(M)
with co-
efficients in the 3c1(M)-module associated to the representation X It is easy to see further that this cohomology group is unchanged
when the mapping X is replaced by a conjugate representation, in the sense that the two cohomology groups are canonically isomorphic;
thus the cdhomology group can be considered as associated to the characteristic representation itself. Theorem 19.
Let
X e H1'(M,GL(n,C))
be a flat vector bundle
over any (connected) manifold M , with characteristic representation
X .
Then there is a natural isomorphism
H'(M, 3.(X)) = Hl(3tl(M),X)
Proof. It V1 = (Ua)
be a Leray covering of the manifold
M for the sheaf 3 (X) ; it is apparent that the covering can also
be so selected that
, where U0
3c1(M) _ 3t1( Vt. ,Uo)
X has a
, and that the bundle
for the chains of the covering ?Jl,
representative cocycle
is a base
a
(XU$) a Z1( j/ ,GL(n,C))
(For the termi-
.
nology and notation, see pages 184-189 of last year's lectures.)
Consider an arbitrary cocycle A = (Aa$) a Z1(ut , 3-(X)) ; thus AaP a Cu
are vectors (viewed as column vectors) associated to
Ua n U
intersections
such that
ACAS + Any when-
Acty = X
07
ever Ua n 0 n Uy #
To each closed chain
7 = (Ua ,Ua,,..-,U P) of the covering lit based at
associate
Uo
0
the vector P)-1 A
Ay =
Dal + (X a3 ..
+
Note that Ay = Ay whenever
1
yl
the chain
y
.
and
p-2 are homotopic chains.
y
results from a simple jerk on
n Ua
Suppose then that Ua i-1
is the closed chain
fact that Ay
= Ay
n 0 j A , and that i
,UO,Ua ,..-,Ua )
y1 = (Ua 1...,Ua
0
a2 +
P-1+Ap-lp
aP-1 PA
(It suffices to show this when y1
y1
P-1P)-1 A
P-1
.
i
-1
The
.
p
in this case follows immediately from the co-
cycle condition; details will be left to the readers.) can be viewed as a mapping A: jrl(7/t,U0) --;- Cu
.
If
Thus A yl,y2
two closed chains, it follows by an easy calculation that
are
thus actually A e Z1(3cl(Uo),X) .
The mapping A --? A
21(Vt, 5 (X))
well-defined linear mapping from the vector space to the vector space
; we shall see that it
Z1(3[1(lit ,uo),x^)
establishes the desired isomorphism.
A
is a
First suppose that the cocyle
is cohomologous to zero, that is, that there is a zero-cochain
B = (Ba) a Co( 11(. , 3 (X)) such that Aa = B - X1 Ua fl U
.
It follows readily that then for any closed chain
y^Ba0 - X7 Bao
the mapping A has the value A =
y = (Ua ,...;Ua )
o^
so that A
Ba whenever
p
The mapping A --? A thus in-
is also a coboundary.
duces a well defined linear mapping from Hl( VI , $ (X))
^
1
H (g l( 211 , o), X)
A e Zl( 71.. ,
.
(X))
into
Next suppose that for some cocycle
is
the image cocycle A e Zl(g 1( 1/L ,U0),X)
cohomologous to zero, that is, that there is a vector
that Ay = B - yl B
7 of the covering
for every closed chain
For any element U. e VI select a chain
and ending at U. ; and if
such
B e GIn
3c(a) based at Uo
with Ua = Ua
IC(a) = (Ua
p
0
p
set B
a = (XaDal
..
X
p-1 p
)-l B + (x
... +
)-1 A
X
1a2
..
p-1 p
1
+A
aP-1 p A p-2p-1 It is a straightforward matter to verify that independent of the choice of the chain
3c(a)
p-1p
B. so defined is ; and that the cochain
(Ba)ne
B =
so that A
Co( U(, $ (X))
has the cocycle A
is cohomologous to zero.
+ .. .
aoal
as its coboundary,
Thus the mapping from
(X))
to H (,cl( R ,UO), X) is an isomorphism into.
Finally, suppose given an arbitrary one-cocycle
For any element U. a
A = (AC,) e Z1(g1( U(, ,U0), X)
chain ,c(a) based at
Uo
X
Xc(a) = X a a
01
select a
and ending at Ua ; and set xa
where
a p-1p
if Ua n u, j 0 for two subsets
3c(a) _ (Ua ,Ua ,...,Ua )
0
1
p
Ua,U, a VL , then the chain
is a closed chain based at
Uo ; set
1
a(p)
Aa$
3t(a). (p)"l
^
Since A
is a cocycle, it follows readily that whenever
Ua n 0 n Uy
the element
XA
1
A m - ic(y)
Aa+A
A
Ic(y) so the vectors
ACAS
form a one-cocycle A = (A a$) a Z1(
, 3 (X))
It is easily verified that this cocycle maps onto the given coho-
H'(Vi, 9. (X))
nioology class A under the isomorphism from
to
,u0), X) ; the latter isomorphism is therefore onto, and
H1(a1('
the proof is thereby concluded.
An application of Theorem 19 leads to a simple explicit calculation of the cohomology group
Riemann surface M .
H1(M, 9'(X))
Recall that the fundamental group
a compact orientable surface M of genus
a group with 2g relation
over a compact
generators
g
Ic1(M)
of
can be described as
a1,...,ag'Til ...ITg
and the one
;
agTgaglTgl
(5)
...
a1T1a11T11 = 1 ;
(see for instance H. Seifert and W. Threlfall, lehrbuch der Topologie
Selecting any homomorphism
(Chelsea, 1911.7)).
X: 3t1(M) --? GL(n,C)
as the characteristic representation of a flat vector bundle X C Hl(M,GL(n,C)) A c Z1(,c1(M),X)
over the surface M , a one-cocycle
is described by any 2g
Ac AT
vectors
i
a Ca i
subject to one relation corresponding to the group relation (5).
To make this explicit, note from (3) that A 7 e g 1(M) ; and setting
[a,T] = aTa-1T-1
y1
=
-Xy Ay for
any
for any a,T c ic1(M)
it further follows from (3) that
+ TX
A[a, T ]
TaT-l . Aa
(6)
a I-)C1
=
where
I
Ta'
QAT -
denotes the identity matrix.
vectors Aa ,AT: e Cn
AT
XTa Aa - XT
AT
(I-XTl).Aa]
Thus the condition that the
form a one-cocycle is just that
0 = A[ag,Tg] ... [a1,T1] (7)
A A[ag,Tg] +...+ X[all T1]
X[
+A[a1,T1] A[a2,T2]
Tg-1]...[allT1
g-11 where the terms A
are given by equation (6).
Such a one-
[ai,Ti]
cocycle is a one-coboundary if and only if there is a vector such that A
(8)
for
i = 1,...,g .
=B-X-1
B
B e Cn
Corollary. it x e H-I(M,GL(n,C)) over a__cclompact Riemann surface of genus
dim
C
p
such that the trivial bundle of rank p X , and
bundle of rank
X
bundle
q
Then
g .
H1(M, g (X)) = 2(g-l)n+p+q , where
bundle of
be a flat vector bundle
is the largest integer
is contained as a sub-
is the largest integer such that the trivial
q
is contained as a subbundle of the dual vector
.
The condition that the vectors Aa , AT
Proof.
i
determine i
a one-cocycle A e Zl(3t1(M),X) , as expressed in equation (7), is a system of
n linear constraints on the dim
vectors; and so
C
2gn
components of those
Z1(Itl(M),X) = 2gn - r , where
of the system of linear equations (7).
r
is the rank
Viewing that system of
equations as being of the form
g
(e
E
i=1
ai
A +© A ai
©a , ©
for suitable matrices
i
of the space of row vectors for
i = 1,...,g .
R
X-
i
1
F.
[ai,Ti]
R 8ai
[a1'
= R
= R
i
= 0 8 Ti
Ri.
ai
.
, the corank n-r
is the dimension
such that R ©
R e d'
1
ai
thus
F.
Ti
)=0 T
i
For any row vector
X[ai-l,T1-1
Ri = R
Ti
X-
X-1 Ti
1
ai
= R
R e
G-U
= R 8Ti
set
R1 = R , and
Nov the conditions
i+l
are readily seen to be of the form
X
a.
(I- _l)
(I-XT1)
= R3 ' XT a
0,
or equivalently
Ri ' T .
..
= Ri ' XTT.
..
..
ai = Ri TT.
.. X
.. 1 .
TT.
;
= 0
but this clearly implies that Ri = Ri T = Ri
Xc
i
also that Ri isfies
Therefore whenever the row vector R sat-
= Ri+l '
R 8
= R 8 ci
, and hence i
= 0 , the column vector
tR e Cu
is such
Ti
that
tX-1 tR
tR
=
for all
7 Consequently the corank
is the largest integer
q = n-r
that the identity representation of rank or equivalently, the largest integer bundle of rank bundle
X
q
7 e gl(M)
q
q
.
such
t^-1
X
is contained in
such that the trivial
q
is contained as a subbundle of the dual vector
.
Now the coboundaries
of the form Ay = B- yl B
B1(nl(M),X) for any vector
dim C B1(,cl(M),X) = n-p , where
space of all those vectors
are just the cocycles
p
B e Cu
7 e nl(M) ; equivalently of course,
B e Cu .
Thus
is the dimension of the sub-
such that y B = B p
for all
is the largest integer such
that the identity representation of rank p
is contained in X
or the largest integer such that the trivial bundle of rank p contained as a subbundle of the vector bundle follows that
X
is
It therefore
dim H1(M, 51 (X)) = dim Hl(nl(M),X) = dim Z1(nl(M),X)-
- dim B1(nl(M),X) = 2(g-l)n+p+q , which is the desired result. Remarks. q
Note that saying that the trivial bundle of rank
is contained as a subbundle of the dual vector bundle
X*
,equivalent to saying that the trivial vector bundle of rank a quotient bundle of the vector bundle
if
X itself.
is q
Note also that
X is the trivial bundle of rank n , then 9-(X) = Cu X M
Theorem 18 and the Corollary to Theorem 19 assert that
H°(M, 3 (X)) = Cu
is
and Hl(M, }(X)) _ egn , as they should.
There is a form of the deRham isomorphism which can be used
(c)
to describe the cohomology groups of a manifold with coefficients in the sheaf of germs of flat sections of a flat vector bundle.
(For
the usual deRham isomorphism, and the required background material about differential forms, see §5(a) of last year's lectures.) C p(x)
denotes the sheaf of germs of C"
If
differential forms
which are sections of the flat vector bundle
X e H1(M,GL(n,C))
over the Riemann surface M , the operation of exterior differentiation is a well-defined sheaf homomorphism
d: e p(X) -->
G p+1(X)
.
There then arises the exact sequence of
sheaves (the deRham sequence)
0 -> .4 (X) --> a°(X) d a el(X)
(9)
recalling that M is two-dimensional. locally isomorphic to
'(X) C C 1(X)
F 2(X) -> 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
is represented by the cocycle
(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
in (6), which are the obvious
In addition, it is clear that the operation of complex conju-
ones.
gation is an isomorphism (or more precisely, a conjugate-linear isomorphism)
combining this with the
H1(M, 3' (X*)) = Hl(M,
Hermitian dual pairing o a~
H1(M, -3 (X)) ® 111(M, 3 (A*))
of Theorem 20 establishes a dual pairing and
cit, if A e H1(14, 9 (X)) ing
13
in (6).
To be expli-
B e H1(M, 3 (X*)) , the dual pair-
is defined by
[A, BI = < A,B> ,
(7)
where to
p
B e H1(M, 5 (e))
is the complex conjugate cohomology class
is defined as in equation (11) of W.
B , and Lemma 24.
Over a compact Riemann surface, (6) is a commu-
tative diagram, in the sense that
p(B)'q) = [B,bp]
and
-162-
p(A)'* = [A,5*]
whenever
(p a I'(M,B1'0(X)),
* e r(M,d'l,O(X*)), A s HI(M, 0 (X)),
and Be1l(M, 9'(x)) Since the diagram (6) is symmetric, it suffices to
Proof.
prove conmutativity only on one side.
Consider then a cohomology
class A e H1(M, 3-(X)) , represented by a cocycle (Aa4) a Zl( Vt, 9' (X)) , and a Prym differential * _ (*a) a r(M, B 1'0(X*)) ; select A s s = Fp - X-11Fa
cf'
in U a n u,3, and let
functions
Fa satisfying
B = b* a 111(M, 9' (X*))
Then, from the discussion on page 136 and from (5), it follows that
[A, B] = < A, B > = JM t*a A dFa
= JM t*a .. cF = p(A)'* , as desired.
Note now that the condition that a cohomology class A e H1(M, 9-(X))
be analytic, that is, That it be the period class
of a Prym differential in = [A,B$]
r(M, o41'0(x)) , is just that
for all the Prym differentials
dual pairing
[A,b*]
0 = p(A)*
* e r(M, 6 l'0(X*)) .
The
is given explicitly, in terms of the flat co-
cycles, by Theorem 21.
(b)
The exact sequence (3), for the special case that
X
is
the identity bundle of rank 1, plays an important role in the treatment of the Picard-Jacobi variety attached to a compact Riemann surface.
It is perhaps of some interest to digress briefly, to see
the extent to which some of the standard observations about the special case of that sequence carry over to the general case.
-163-
When
X is the identity bundle,
H1(M, 9(X)) = HI'(M,C)
is
an even-dimensional complex vector space; and the exact sequence (3) is an even splitting of the vector space
Hn(M, 3 (X)) , in the
sense that
dim
(8)
r(M, 61'0(x)
@l,C = dim
(S1
In general, the space
Hl(M, ? (x))
r (m,
d M, G
x
)
need not be even-dimensional;
the condition that this be so is of course just that
p+q be even,
with the notation as in the Corollary to Theorem 19 (page 133).
Even when H1(M, 9-(X))
is even-dimensional, the exact sequence
(3) need not be an even splitting.
For some special cases, however,
the even splitting does occur; these results are su+mmgrized in the next theorem.
Note firstly, however, that the condition (8) can
be simplified somewhat.
On the one hand, recalling the exact se-
quence (2), it follows that dim dr(M,LV(x)) = dim r(M,G (x)) - dim r(M,
(X)) ; and on the other hand, from the Riemann-Roch
theorem it follows that = n(1- g) , where
n
dim r(M, 9 (X)) - dim r(M, 0 11 C(X )) _
is the rank of the bundle
genus of the surface M .
X and
g
is the
Thus the condition (8) for an even
splitting can be rewritten as the condition 2 dim r(M, m (x)) - dim r(M, 9 (x)) =
(9)
= 2 dim r(M, 6L(x*)) - dim r(M, 9' (X*))
Theorem 23. X a
l
(M,GL(n,C))
surface.
Let M be a compact Riemann surface, and be a flat vector bundle of rank n over that
If n = 1 , the exact sequence (3) is always an even
(i)
splitting.
If n = 2
(ii)
and X is a stable vector bundle, the exact
sequence (3) is always an even splitting.
If n = 2
(iii)
X
and
stable vector bundle, let such that
a
c(l) = div X and
is an analytically indecomposable unbe the unique complex line bundle a C X .
Then the exact sequence (3)
is an even splitting if and only if 2 dim r(M,8 (g)) - dim r(m, 9 (e)) _
2 dim r(M, d Q ® det X*)) - dim r(M, (iv)
® det X*))
If X is a unitary flat vector bundle of any rank, the
exact sequence (3) is always an even splitting.
Proof.
(i)
When n = 1 , it follows from the Riemaxm-Roch
Theorem for complex line bundles that
dim r(m, 0 (x)) _
t
X = 1
1
if
0
otherwise
in
H1(M, m*)
,
and it follows from Theorem 18 that
1
if
0
otherwise.
X = 1 in H'(MC*)
dim r(m, $ (x)) -
Since in either cohomology group
X = 1
if and only if
* X = 1 ,
condition (9) is verified and the desired result proved in this case. (ii)
thus
X
For a flat vector bundle
is stable precisely when
X of course
c(l) < 0
c(det X) = 0;
for any complex
C X .
analytic line bundle such that
r(M, G (X)) = 0 whenever
instance, it follows that
r(M, 3'(x)) = 0
and necessarily
Recalling (10) in §5 for
as well.
(9), it is then sufficient to observe that
when
X is stable.
In view of condition
X
is stable precisely
To see this, consider any line bundle
the analytic line bundle
and the analytic vector bundle
9
can then be represented by cocycles
XQP _
(Lad)
and
(x
)
9 C X;
X
in terms
of M , such that
of a suitable covering v(,= (Ua)
(10)
X is stable;
,
0 ICO
for some line bundle
n = (r
)
.
The dual bundle
X* is then
Xa4 , where
represented by the cocycle
-1
0
XC43 _
(XCI:
upon interchanging rows and columns, the same bundle is represented
by the cocycle
(ll)
Xa '
P
It is thus apparent that
g C X precisely when
11-1 = g(det X)-1 C X* ; and since inmiediately that
c(rj 1) = cW , it follows
X is stable precisely when X
is stable, which
proves the desired result in this case. The fact that there is a unique complex analytic
(iii)
line bundle
9
such that
c(g) = div 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)
is represented by the cocycle
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>`I2a = (%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)
)`a = 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.
