Chapter VII.

Compact Riemann Surfaces

In the theory of compact Riemann surfaces it is possible to make particularly

elegant applications of the finiteness theorem. For such considerations we will
always let X denote a connected, compact Riemann surface with structure sheaf(!).
With script letters like !/ we will denote, as before, coherent analytic sheaves over
X. If the support of such a sheaf is finite then !Y will usually be written. For such a
sheaf it is easy to see that H 1 (X, !Y) = (0). The symbols !F, f'§ are reserved for
locally free {!}-sheaves. The letter !l' is usual exclusively for locally free sheaves of
rank 1. All tensor products are formed over(!).
Since X is 1-dimensiona~ every stalk .A" of the sheaf .A of germs of meromor-
phic functions on X is a discrete valuation field with respect to the order function
o". Recall that, for he .A" and t em" a local coordinate, ox(h) = n, where h = f'e
and e e (!)"is a unit. The order of the identically zero germ is defined to be infinity.
The valuation ring associated to o" is (!)" with m x as maximal ideal.
The goal of this chapter is the derivation of the Riemann-Roch theorem along
with the Serre duality theorem. Further a criterion for the splitting of locally free
sheaves is proved. An immediate corollary of this is the classification of the locally
free sheaves over the Riemann sphere.

§ 1. Divisors and Locally Free Sheaves

Every locally free sheaf !F is a subsheaf of the {!}-sheaf,

of germs of meromorphic sections of !F. We identify every stalk !F': with

.A"!F" = U
f'!F", where t em" is a local coordinate at x. The sheaf !F 00 is not
neZ
coherent. However it contains important locally free subsheaves which are related
to !F. These will be introduced in this section. We write !F00 (X)* (resp. !F(X)*) for
/F 00 (X) minus the identically zero section (resp. !F(X) minus the identically zero
section).

H. Grauert et al., Theory of Stein Spaces

© Springer Science+Business Media New York 1979
§ 1. Divisors and Locally Free Sheaves 205

0. Divisors. In Chapter V.2.2, we considered the exact sequence

0 --+ (!)* --+ .,{(* --+ !'} --+ 0

for arbitrary complex spaces. The sheaf !lJ •= .H* j(!)* is called the sheaf of germs of
divisors. In the case of a Riemann surface, every stalk !lJ" is isomorphic to l., and
every non-trivial s e !lJ(U) over an open set U, has discrete support Is I in U. This
is the so-called skyscraper property of !lJ.
For a compact Riemann surface X, the divisor group,

Div X·=~(X1

is canonically isomorphic to the free abelian group generated by the points x e X.

Consequently every divisor D is of the form

D= L nxx,
xeX

where nx E l. and n" = 0 for almost all x. Throughout we write Qx(D) instead ofn".
The integer
deg D •= L Qx(D)
xeX

is called the degree of D. The mapping Div X --+ l, which associates deg D to D, is
a group epimorphism.
If Qx(D) ~ 0 for every x eX then the divisor Dis called positive. For divisors
D1o D2 e Div X, we write D 1 ~ D2 when D2 - D1 is positive. The group of divi-
sors is directed with respect to this relation. In other words, given two divisors D 1o
D 2 e Div X, there exists D 3 e Div X so that D 1 ~ D3 and D 2 ~ D 3 •
The set ID I •= {x e X: Qx(D) ::/= 0}, called the support of D, is always finite.

1. Divisors of Meromorphic Sections. Let F be a locally free sheaf. Given a

local coordinate t e m", every germ s" e F': can be uniquely written in the form

where s" E Fx \mxFx. The exponent m is uniquely determined by sx. We define

to be the order of s" with respect to F. If g(sx) > 0 (resp. g(sx) < 0) then we call x a
zero (resp. a pole) of s". The situation g(sx) = oo only occurs for the identically
zero germ. It follows that F" = {sx E F':: g(sx) ~ 0} and mxFx = {sx E F'::
g(sx) > 0}.
206 Chapter VII. Compact Riemann Surfaces

Each sections e F 00 (X), s ¢ 0, is called a global meromorphic section of F. If

s ¢ 0 then it has only finitely many zeros and poles. Thus the following definition
makes sense.

Definition 1. (The divisor and the degree of meromorphic sections). Given

S E F 00 (X),

(s)•= L o(sx) ·X e Div X

xeX

is called the divisor (with respect to F) of s. The integer deg(s) is called the degree
(with respect to F) of s.
It follows that (s) is positive if and only if s has no poles, or equivalently, if
s E F(X).

Warning: The order functions o and the divisors (s) of sections s e F 00 (X)
depend heavily on the sheaf with which one starts out. For example, with respect
to(!), the zero divisor is associated to the section 1 e (!) 00 (X). On the other hand, if
(!)is replaced by (!)(D) (see Paragraph 3) for some De Div X then (1) =D. In the
following it will always be completely clear with respect to which sheaf we are
forming divisors.
From the above remarks it follows that every meromorphic function
he (!) 00 (X)* has an associated divisor (h). Such divisors are called principal divi-
sors. The map .H(X)*--+ Div X, which sends h to its divisor (h), is a group homo-
morphism. For all he (!) 00 (X)* and s e F 00 (X)*, it follows easily that
(hs) = (h) + (s ). The image group,

P(X) •= Im(.H(X)*--+ Div X) c Div X,

is called the group of principal divisors and the quotient group

Div X/P(X)

is called the group of divisor classes on X. Following the classical language of

algebraic geometry, one says that two divisors D and D' are linearly equivalent if
each is a representative of the same class of divisors. In other words D and D' are
linearly equivalent if D - D' e P(X).

l. The Sheaves F(D). Given a locally free sheaf F and a divisor D, we define an
analytic subsheaf F(D) of F 00 by

F(D)x•={sx E F;': o(sx) ~ -ox(D)} and F(D)•= U F(D)x c F 00 •

xeX

Except for points in ID I, F and F(D) agree. For x e ID I, F x is made smaller

(resp. larger) when ox(D) < 0 (resp. ox(D) > 0). More precisely, it follows that, if
§ 1. Divisors and Locally Free Sheaves 207

t e m x is a local coordinate at x, then

One obtains an l!/-isomorphism of §"(D) onto§" at x by multiplication by t".l.. 0 >. In

particular §"(D) is a locally free sheaf with the same rank as §"and it is always
true that §"CX>(D) =§"<X>.
In the following lemma we summarize the laws which follow immediately from
the general sheaf calculus:

Lemma. Let§",§" 1> and§" 2 be locally free sheaves and D, D~> D 2 divisors on a
compact Riemann surface X. Then
1) Every exact l!/-sequence 0 -+ §" 1 -+ §" -+ §" 2 -+ 0 determines in a natural
way an exact l!/-sequence 0-+ §" 1 (D)-+ §"(D)-+§" 2 (D)-+ 0.
2) If§"= §" 1 + §"2 then §"(D)= §" 1 (D) + §" 2 (D).
3) There is a naturall!/-isomorphism §"(D 1 )(D 2 ) ~ §"(D 1 + D 2 ).
4) If D 1 :::;; D 2 then F(D 1 ) is an analytic subsheaf of F(D 2 ).
The reader can easily carry out the proof. We note here that property 4) will
play an important role in the next section.

3. The Sheaves l!?(D). The above considerations are in particular valid for
§" = l!l. All sheaves l!/(D), De Div X, are locally free of rank 1. Two sheaves, l!I(Dt)
and l!I(D 2 ), are analytically isomorphic if and only if D 1 and D2 are linearly
equivalent.

Proof: The sheaves l!/(Dt) and l!I(D 2) are analytically isomorphic if and only if
l!I(D 1 - D 2) ~ l!l. Let D •= D 1 - D 2. Assuming that D 1 and D 2 are linearly equiv-
alent, D =(h) with he Jt'(X)*. In this case we obtain an l!/-isomorphism
l!i-+ l!?(D) by fx 1--+fx hx. Conversely, given an l!/-isomorphism (!;-+ l!?(D), the image
of 1 e ~(X) in l!I(D)(X) is a meromorphic function he Jt'(X)* with D =(h). 0
By tensoring F with l!/(D), one gets all sheaves §"(D). This is seen from the
natural @-isomorphism F ®~(D)-+ F(D), defined by sx ® hx~--+hxsx.

Remark: If Y is a coherent sheaf and De Div X then Y(D) •= Y ® l!I(D) is

coherent. The reader should note that the statements 1)-3) in the lemma of Para-
graph 2 are in fact valid for coherent sheaves as well as locally free sheaves. The
tensor product of two sheaves of the type l'D(D) is again a sheaf of that type.
Given two divisors D 1 and D 2 , there is a natural isomorphism,

defined stalkwise by s~x®s 2 xt-+Stx · S2x·

One usually identifies the group H 1 (X, l!?*) with the group of analytic isomor-
phism classes of locally free sheaves of rank 1 over X. In this way, the group
operation in H 1 (X, l!l*) corresponds to the tensor product of sheaves. The homo-
208 Chapter VII. Compact Riemann Surfaces

morphism ~ in the long exact cohomology sequence,

0---+ C*---+ vlt*(X)---+ Div X~ H 1(X, (!)*),

related to the short exact sequence of sheaves, 0 --+ (9* --+ vlt* --+ f0 --+ 0, is in fact
defined by D ~----+(!)(D). The kernel of~ is just the group P(X) of principal divisors
and therefore one has a natural injection,

Div X/P(X) c. H 1 (X, (!)*),

of the group of divisor classes on X into H 1 (X, (!)*).

§ 2. The Existence of Global Meromorphic Sections

We will show here that every locally free sheaf (not identically zero) on a
compact Riemann surface has "many" global meromorphic sections. This follows
from a "characteristic theorem" which will give rise in the next section to a
preliminary version of the Riemann-Roch theorem. In particular it is proved that
for every p E X there is a non-constant holomorphic function on X\P which has
a pole at p. This shows that X\P is Stein, from which it follows that Hq(X, 9') = 0,
q ~ 2, for any coherent sheaf 9' on X.

1. The Sequence 0--+ ff(D)--+ ff(D')--+ §"--+ 0. Let D, D' be divisors with
D ::::;; D' and let ff be a locally free sheaf of rank r. Then there is a natural exact
sequence
0--+ ff(D)--+ ff(D')--+ §"--+ 0,
where§":= ff(D')/ff(D). Since this sequence plays such an important role in our
considerations, we will now write down its basic properties: From the definitions
it follows that for every x E X

(1)

The support of §" is therefore the support of D' - D and

(2) dime ff(X) = r deg(D'- D).

Since ff has finite support,

Consequently the first part of the cohomology sequence associated to (•) is the
exact sequence

(3) 0--+ ff(D)(X)--+ ff(D')(X)--+ ff(X)--+ H 1 (X, ff(D))--+ H 1 (X, ff(D'))--+ 0.

§ 2. The Existence of Global Meromorphic Sections 209

Thus

(4) if D::; D' then dime H 1 (X, ff(D)) ~dime H 1 (X, ff(D')).

2. The Characteristic Theorem and an Existence Theorem. Let Y be a coherent

sheaf on X. The characteristic of Y, Xo(Y), is defined by

It will be shown (Paragraph 4) that Xo(Y) is the Euler-Poincare characteristic.

In particular it is proved that for every p E X there is a non-constant holomor-
phic function on X\p which has a pole at p. This shows that X\p is Stein, from
which it follows that Hq(X, Y) = 0, q ~ 2, for any coherent sheaf Yon X.

Lemma 1 (The characteristic theorem). Let ff be a locally free sheaf of rank r

and D a divisor on a compact Riemann surface X. Then

Xo(ff(D)) = r deg D + Xo(ff).

Proof: We will show that, for arbitrary divisors D and D',

Xo(ff(D))- r deg D = Xo(ff(D'))- r deg D'

The claim of the lemma will then follow with D' == 0. First we suppose that
D::; D'. The alternating sum of the dimensions of the vector spaces in the exact
cohomology sequence (3) is therefore zero. In other words

0 = Xo(ff(D))- Xo(ff(D')) +dime ff(X).

If one substitutes r deg(D'- D) for dime ff(X) (see (2) above) then (o) follows
immediately.
If D' is arbitrary, then one chooses D" E Div X with D ::; D" and D' ::; D". Then
it follows from the above that

Xo(ff(D))- r deg D = Xo(ff(D"))- r deg D" = Xo(ff(D'))- r deg D'. 0

Theorem 2 (Existence theorem). Let ff be a locally free sheaf of rank r and D a

divisor on a compact Riemann surface X. Then

dime ff(D)(X) ~ r deg D + Xo(ff).

In particular if JF =I= 0 and deg D > 0, then

lim dime ff(nD)(X) = oo.
n-+oo
210 Chapter VII. Compact Riemann Surfaces

We have therefore established that every locally free sheaf fF 0 has non- +
identically zero meromorphic sections. This is clear from Theorem 2, since fF(D)(X)
is always contained in fF""(X).

3. The Vanishing Theorem. By Theorem 2,

dime l!J(np)(X);?: n + Xo(l!J)

for any p eX, and all n e 7L.. Hence there exists n0 e N so that l!J(np)(X) contains a
non-constant function h for all n > n0 • Since (h) + np ;?: 0, every such function is
non-constant, and holomorphic on X\p, and has a pole of order at most n at p.
This shows the following:
For every p e X there is a non-constant meromorphic function on X which is
holomorphic on X\p. 1 Moreover, X\p is Stein. In particular every compact Riemann
surface can be covered by two Stein domains.

Proof: Let h be as above. Then h: X\p-+ CIs a finite holomorphic map. Thus
X\p is Stein. If p 17 p2 e X are different points, then {X\p 17 X\p2 } is a Stein cover of
X. D
The following is now a consequence of the general theory.

Theorem 3 (Vanishing theorem). Let f/' be a coherent analytic sheaf on X. Then

Hq(X, f/') = 0,

For every compact complex space X and every coherent sheaf f/' on X, almost
all of the groups Hq(X, f/') vanish. Thus the Finiteness Theorem allows us to
define the Euler-Poincare Characteristic,

x(f/') = L"" (-1 )i dime IP(X, f/') E 7L..

i=O

Hence Theorem 3 shows that for compact Riemann surfaces

x(f/') = Xo(f/').

4. The Degree Equation. An amusing consequence of the characteristic

formula, xo(l!J(D)) = deg D + xo(l!J), is the degree equation: For linearly equivalent
divisors D and D',deg D = deg D'. In particulardeg D = Oforall principal divisors.

1 With a bit more effort one can show at this point that for every p e X there exists n0 e N so that
for every n > n0 there ish e .K{X) which is holomorphic on X\P, and which has a pole of order nat p
(a forerunner of the Weierstrass gap Theorem~
§ 3. The Riemann-Roch Theorem (Preliminary Version) 211

Proof: If D and D' are linearly equivalent then, from Paragraph 1.3, @(D)~
@(D') and consequently Xo(@(D)) = x0 (@(D')). The characteristic formula yields

deg D + Xo((9) = deg D' + Xo(@). 0

The reader should note that if X =I= !P' 1, then not every divisor D with deg D = 0
is a principal divisor.

Remark: The degree equation can also be interpreted mapping theoretically, and proved in this way
as well. For this, note that every hE A'(X) defines a branched covering h: X--+ P 1 (the case of h
identically constant is trivial). Certainly, if every point of h- 1 (0) and h- 1 (oo) is counted with its
branching multiplicity, then (h)= h- 1(0)- h- 1 (oo ). For every p E P 1, the sum of the multiplicities of
the points of h- 1 (p) is the sheet number s of the covering h: X --+ P 1. It follows that deg h = s - s = 0.

§ 3. The Riemann-Roch Theorem (Preliminary Version)

The classical problem ofRiemann-Roch consists of determining the dimension
of the i!:>vector space, H 0 (X, @(D)), of all global sections of the sheaf @(D). The
characteristic theorem gives a preliminary solution of this problem.

1. The Genus Theorem of Riemann-Roch. The following notation is standard:

For linearly equivalent divisors D and D', l(D) = l(D') and i(D) = i(D'). We further
note that the dimension l(D) > 0 if and only if there is a positive divisor D' which is
linearly equivalent to D. In particular l(D) = 0 for every D with deg D < 0.

Proof: The first statement is clear, since the divisors which are linearly equiv-
alent to Dare of the form D +(h) for hE .A*(X). The second statement follows
from the first, since linearly equivalent divisors have the same degree. 0
For the zero divisor @(X)~ C and thus 1(0) = 1. The natural number

is called the genus of X. From this definition it only follows that g is a complex
analytic invariant of X. However in Paragraph 7.1 we will show that g is in fact the
topological genus of X (i.e. H 1 (X, C)~ 1[; 2 9).
For every divisor D it follows that

Xo(@(D)) = l(D)- i(D),

and in particular

Xo(@) = 1 - g.
212 Chapter VII. Compact Riemann Surfaces

Thus the characteristic formula Xo(l!7(D)) = deg D + xo(l!7) can be restated as

follows:

Theorem 1 (Riemann-Roch, preliminary version). If Dis a divisor on a compact

Riemann surface X of genus g, then

l(D)- i(D) = deg D + 1- g.

Remark: The Riemann-Roch problem (i.e. the determination of the number

l(D)) is not satisfactorily solved by Theorem 1, because the term i(D) appears as a
dimension of a first cohomology group. However, by means of Serre duality it can
be interpreted as the dimension of a Oth cohomology group (see Paragraph 6). The
final solution of the Riemann-Roch problem is given by Theorem 7.2.

2. Applications. The following Riemann Inequality is a special case of

Theorem 1:

l(D) ~ deg D + 1 - g

This inequality yields the first classical existence theorems. For example, since
l!7(D)(X) contains a non-constant meromorphic function whenever l(D) ~ 2, we
have the following:
For every divisor D with deg D ~ g + 1 there exists a non-constant meromorphic
function h with (h) + D ~ 0.
In particular, given p EX, there always exist non-constant functions which
have poles of order at most g + 1 at p, and which are holomorphic on X\p. One
can state this as a theorem about coverings of 1? 1 : Every compact Riemann surface
X of genus g is realizable as a branched cover with at most g + 1 sheets of the
Riemann sphere 1? 1 . In particular, if g = 0 then X= 1? 1 .
Since l(D) = 0 when deg D < 0, it follows from Theorem 1 that, for every
D E Div X with deg D < 0,

i(D) = g- 1- deg D.

One sees that, with the exception of the case g = 0 and deg D = - 1,
H 1 (X, l!7(D)) ::/= (0) for all divisors of negative degree. Furthermore,

lim dime H 1 (X, l!7(D)) = oo.

deg D-+ -ao

Remark: The existence of meromorphic functions which have poles (perhaps of

high order) only at a prescribed point is guaranteed by the results in Paragraph 2.
The improvement here is that the minimum order which can be prescribed can be
estimated independent of the point p by the genus.
§ 4. The Structure of Locally Free Sheaves 213

§ 4. The Structure of Locally Free Sheaves

We will show that every locally free sheaf, which is not the zero sheaf, contains
locally free subsheaves of the form ((}(D). This theorem is the most important aid in
the study of general locally free sheaves (see, for example, the supplement of this
section as well as Section 8 ).

1. Locally Free Subsheaves. The considerations of this section are formal in

nature. The following language is useful:

Definition 1 (Locally free subsheaves ). An analytic subsheaf ff' of a locally free

sheaf ff is called a locally free subsheaf of ff if
0) ff' is itself locally free.
1) The quotient sheaf ff/ff' is locally free.
The rank equation for a locally free subsheaf ff' of a locally free sheaf ff,

rk ff = rk ff' + rk ff/ff',
is an immediate consequence of the definition. Thus every locally free sheaf !l' of
rank 1 contains only 0 and !l' as locally free subsheaves.
The requirement 1) in the above definition is quite restrictive. For example a
germ tx E ff x always generates a free submodule tx ((} x in ff x• but the quotient
module ff x ftx ((} x is in general not free. For an explicit example, take ff = ((}and tx
a non-unit. On the other hand if tx is a unit then there is no problem: If tx E ff x
and o(tx) = 0 then ff x ftx ((} x is a free ((} x-module.

Proof: Let ffx =((}~and tx = (t 1 , ... , t,), t; E ((}x· Sinceo(tx) = O,some t;,say t 1 ,
is a unit. Let e == t1 1 and define a: ((}~--+ ((}~-l by

Thus a is an epimorphism with kernel ((} x tx. D

It is easy to find locally free subsheaves in ffoo:

Theorem 2. Let ff be locally free and D the divisor of a meromorphic section

s E ff 00 (X). Assume s =I= 0. Then the sheaf !l', defined by !l' == ((}(D)s ~ ((}(D) is a
locally free subsheaf of ff. The sections is always in !l' 00 (X) and, when s E ff(X),
s E !l'(X).
Proof: Since o(hxsx) = o(hx) + ox(D) ~ 0 for every hx E ((}(D)x, ((}(D)xsx c ffx-
Thus !l' is an analytic subsheaf of ff. Due to the fact that s =I= 0, !l' is isomorphic
to ((}(D). Furthermore s E !l'( -D)(X) c !l' 00 (X) and, when D ~ 0, s E !l'(X).
Finally ((}(D)x = ((}xgx, where o(gx) = -ox(D). Defining tx == OxSx, it follows that
!l' x = ((}xtx with o(tx) = 0 for all x EX. Therefore, by the above remark, ffj!l' has
everywhere free stalks. D
214 Chapter VII. Compact Riemann Surfaces

Remark: The sheaf !l' = m(D )s is the only locally free subsheaf of ff of rank 1
with s E !l'"' (X). To see this, let !l' be such a sheaf. Then !l' x = (9 x vx for some
Vx E ff X' where Sx = mx Vx and mx E Ax· Let hx := m; 1 . Then Vx = hx Sx and, since
o(vx) ~ 0, hx E m(D)x· Consequently Vx E m(D)xsx or !}X c !l'x. The mx-module
ff xI!l' x contains therefore a submodule which is isomorphic to !l' xI!l' x· Since
ff xI!l' x is free, and since !l'I!l' x is in any case finite, !l' xI!l' x = 0. Hence !l' x = !l' x
for all x and !l' = !l'. D

2. The Existence of Locally Free Subsheaves. The foundation for the study of
the structure of locally free sheaves is the following theorem.

Theorem 3 (Subsheaf theorem). Every locally free sheaf ff =I= 0 contains a locally
free subsheaf which is isomorphic to m(D) for some DE Div X. One can choose
D = (s), where s E ff"'(X)*.

Proof: From Paragraph 2.2 it follows that ff has a non-trivial global merom or-
phic section. Thus the theorem is an immediate consequence of Theorem 2. D

The following is an immediate corollary of Theorem 3.

Theorem 4 (Structure theorem for locally free sheaves of rank 1). Every locally
free sheaf !l' of rank 1 is isomorphic to a sheafm(D) with DE Div X. Furthermore
one can choose D = (s) with s E !l'"'(X)*.
This theorem says that in the cohomology sequence which is associated to the
short exact sequence of sheaves 0 --+ (9* --+A* --+ ~ --+ 0,

the homomorphism {> is surjective (see Paragraph 1.3 ). Thus one has a natural
group isomorphism,
Div XIP(X) ~ H 1 (X, (9*),

of the group of divisor classes on X onto H 1 (X, (9*).

Remark: Since is surjective, the map H 1 (X, A*)--+ H 1 (X, ~) is injective.

{>
Clearly~ is a soft sheaf and thus H 1 (X, ~) = (0). Thus Theorem 4 is equivalent
to the equation

3. The Canonical Divisors. The sheaf of germs of holomorphic 1-forms over X

is a locally free sheaf of rank 1. Thus one can apply Theorem 4:

Theorem 5. There is a unique divisor class on X so that for every divisor K in this
class, Q 1 ~ m(K).
Supplement to Section 4: The Riemann-Roch Theorem for Locally Free Sheaves 215

One calls K a canonical divisor and its class the canonical divisor class on X.
The significance of canonical divisors appears in Section 6.
If X= lfl> 1 then every divisor -2x 0 , x 0 EX, is canonical. This follows from the
fact that, if z is a coordinate on X\x 0 , then dz is a differential form on X which is
holomorphic and nowhere vanishing on X\x 0 and has a pole of order 2 at x 0 . Of
course one must use the fact that on lfl> 1 the degree of a divisor determines its class.
In the case of ellipti<,: curves the zero divisor is canonical.

Supplement to Section 4: The Riemann-Roch Theorem

for Locally Free Sheaves
The generalized Riemann-Roch problem consists of determining the dimension
of H 0 (X, F(D)) for every locally free sheaf F and every divisor D. In order to do
this one needs to carry the idea of degree over to the case of locally free sheaves.
We use the fact that Xo coincides with the Euler-Poincare characteristic X·
Moreover the additivity of x (i.e. for every exact sequence of coherent sheaves
0--+ !/"'--+ !/"--+ !/""--+ 0, one has x(!/") = x(!/"') + x(!/"")) plays an important role.

1. The Chern Function. We denote with LF(X) the set of analytic isomorphism
classes of locally free sheaves over X. A function c: LF(X)--+ 7L is called a Chern
function if
1) ForDE Div X, c(ll?(D)) = deg D.
2) For every exact sequence 0--+ F'--+ F--+ F"--+ 0 oflocally free sheaves, one
has c(F) = c(F') + c(F").
The following is straight forward:

Theorem 1. The function c: LF(X)--+ Z, defined by

3) c(F) == x(F) - rank F · x(lD)

is a Chern function.

Proof: From Lemma 2.1 we have

c(lD(D)) = x(lD(D))- x(lD) = deg D.

The additivity follows from the additivity of both x and rank. D

Remark: The results in Paragraph 4.2 imply that this c is the only Chern
function. To see this note first that if y is any such function then, since the locally
free sheaves of rank 1 are just the divisor sheaves, y(2) = c(2) for every locally
free sheaf .9' of rank 1. Suppose now that y = c for all locally free sheaves of rank
less than r and let F of rank r ~ 2 be given. Then there is an exact sequence of
216 Chapter VII. Compact Riemann Surfaces

locally free sheaves

where !l' and t'§ have rank 1 and r - 1 respectively. The induction hypothesis and
the additivity imply that y(ff) = c(ff).

2. Properties of the Chern Function. The Chern function behaves nicely with
respect to tensor products:

4) c(ff ® !l') =rank ff · c(!l') + c(ff), when rank !l' = 1.

Proof: Let !l' = l!7(D). Then ff ® !l' = ff(D) and c(!l') = deg D. Thus

c(ff ® !l') = x(ff(D))- rank ff(D) · x(l!7)

= rank ff · deg D + x(ff)- rank ff · x(l!7)
= rank ff · c(!l') + c(ff). D

When !l' 1 and !l' 2 are locally free sheaves of rank 1, it follows from 4) that
c(!l' 1 ® !l' 2) = c(!l' t} + c(!l' 2 ). Thus the map H 1 (X, l!7*)--+ ?L, defined by
!l' --+ c(!l'), is a group homomorphism from the analytic isomorphism classes of
locally free sheaves of rank 1 to ?L. The reader can check that the map y in the
exact cohomology sequence

is the Chern function, provided H 2 (X, 7L) is identified with 7L in the natural way.
D
We remark without proof that, if ff is locally free of rank r greater than 1,
r
c(ff) = c(det ff), where det ff := 1\ ff is the locally free determinant sheaf of
1
rank 1. Thus, letting F be the vector bundle associated to ff, c(ff) is just the first
Chern class ofF, c 1(F) E H 2 (X, ?L) = 7L.

3. The Riemann-Roch Theorem. Equation 3) in Proposition 1 above can be

rewritten as a Riemann-Roch theorem:

Theorem 2 (The Riemann-Roch theorem for locally free sheaves). Let ff be a

locally free sheaf of rank r and D a divisor on a compact Riemann surface X of genus
g. Then

dime H 0 (X, ff(D))- dime H 1 (X, ff(D)) = r(deg D + 1 -g)+ c(ff).

Proof: On the left hand side we have x(ff(D)) which, by the characteristic
theorem, is the same as r deg D + x(ff). If one writes r · x(l!7) + c(ff) for x(ff),
then, since x(l!7) = 1 - g, the claim follows immediately. D
§ 5. The Equation H 1 (X, .A')= 0 217

§ 5. The Equation H 1 (X, .A)= 0

The inequality from Paragraph 3.2, l(np) ~ n + 1 - g, is already strong enough
to show that for every divisor D on X and every point p E X the cohomology
groups H 1 (X, l!i(D + np)) vanish for all n ~ 0 (we already know that the dimen-
sion of these vector spaces is constant for n ~ 0 and is unbounded for n < 0 ). This
"Theorem B for compact Riemann surfaces," which will be made even more
precise in Section 7, has an immediate consequence the fact that H 1 (X, .R) = 0.

1. The C-homomorphism l!i(np)(X)-+ Hom(H 1 (X, l!i(D)), H 1 (X, l!i(D + np))).

Let D be a divisor in Div X, p a point in X, and n ~ 0 a fixed integer. Every
function fe l!i(np)(X) determines the l!i-homomorphism ({) 1 : l!i(D)-+l!i(D + np~
hx t-+ fx hx. Iff =F 0, then qJ1 is injective and the sheaf Jm qJ1 in the exact sequence

has .finite support ((J,~qJ1 )x = 0 for every x ¢ ID I u {p}, wherefx is a unit in l!i x)·
Thus H 1 (X, Jm qJ1 ) = 0 and
every function f =I= 0 in l!i(np)(X) induces a C-vector space epimorphism

Iff= 0, then we define qJ1 to be the zero mapping.

Lemma 1. The map

is C-linear.

Proof: The claim becomes clear when one looks at how qJ1 is defined via the
Cech complex. Let U = {Uj}, i E /,be a cover of X with cochain groups

C 1 (U, l!i(D)) = 0 l!i(D)(Uioi 1 )

io,it

and

C 1 (U, l!i(D + np)) = 0 l!i(D + np)(UioiJ

io,it

Associated to every l!i-homomorphism qJ1 ,fe l!i(np), are the following homomor-
phisms at the section level:
218 Chapter VII. Compact Riemann Surfaces

The collection '¥1 •={rp1 (i 0 , i 1 )} is a homomorphism

which induces the homomorphism 1/11 : H 1 (X, l!J(D))-+ H 1 (X, l!J(D + np)). From
the definition of tp J{i 0 , i 1 ) in (*) it is clear that

Consequently 'Paf+bg = a'¥1 + b'¥, and thus 1/laf+bg = al/11 + bl/J,. D

2. The Equation H 1 (X, l!J(D + np)) = 0. The following is an easy corollary to

the Riemann inequality and Lemma 1:

Theorem 2. Let D e Div X and p e X. Then

H 1 (X, l!J(D + np)) = 0 for all n ~ n0 •= (dime H 1 (X, l!J(D)))l +g

Proof: Let d •=dime H 1 (X, l!J(D)). Then, by 2.2(4), it follows that
dime H 1 (X, l!J(D + np)):::;; d for all n ~ 0. Hence the map 1/J (in Paragraph 1
above) maps l!J(np)(X) into a C-vector space having dimension at most d2 • But d
does not depend on nand, by 3.2, dime l!l(np)(X) ~ n + 1- g. Therefore, when-
ever n ~ n0 = d2 + g, the C-linear map 1/1 is not injective. Thus for such an n there
exists f:f 0 in l!J(np)(X) with rp1 = 0. Since rp1 must be an isomorphism,
H 1 (X, f9(D + np)) = 0. 0

3. The Equation H 1 (X, .,H)= 0. We can now prove the following fundamental
theorem:

Theorem 3. H 1 (X, vH) = 0.

Proof: Let~ e H 1 (X, .,H) be an arbitrary cohomology class. We choose a finite

cover U = {U1} of X, a shrinking of U, m= {Vi}, with v;~ Ui> and a cocycle
~ e Z 1 (U, .,H) which is a representative of ~- We let ~ = (~ 101 J, where
~1 01 1 E vH(U101 J. Then

is likewise a representative of~. Since Vj ~ U 1for all i, each function ~ 1011 has finitely
many zeros and poles on Vj011 • Thus one can find a divisor De Div X so that
'' e Z 1 (m, l!I(D)). Let p eX\ ID I· Then, since &(D) c &(D + np) for all n ~ 0, it
follows that ~' c Z 1 (m, l!I(D + np)) for all such n. Since &(D + np) c vH and
H 1 (X, &(D + np)) = 0 for n ~ n0 , it follows that~, is cohomologous to zero as an
element of Z 1 (m, vH). Thus ~ = 0, and consequently H 1 (X, vH) = 0. D
§ 6. The Duality Theorem of Serre 219

The most important applications of the equation H 1 (X, .H)= 0 are found in the next section. In
closing this section we note in passing a rather simple consequence. Let 0"' denote the sheaf of germs
of merom orphic 1-forms. The differential d: M -+ 0"', which is defined in local coordinates by
dhx•=dhx/dz dz, is C-linear. The C-sheaf d.H c 0"' (it is not an (I)-sheaf!) consists of all residue free
germs of meromorphic 1-forms (i.e. germs of abelian differentials of the second kind; for the idea of a
residue see Paragraph 6.5). Since H 1 (X, .H)= 0, the short exact C-sequence 0-+ C-+ .H-+ d.H-+ 0
induces the following exact cohomology sequence:

This means that

In words,
the cohomology group H 1 (X, C) is isomorphic to the quotient space of global abelian differentials of the
second kind modulo differentials of global meromorphic functions.

§ 6. The Duality Theorem of Serre

In this section we will establish a natural isomorphism between the space of
differential forms H 0 (X, n(D)) and the dual space of the first cohomology group
H 1 (X, (!)(-D)), where Dis a given divisor on X:

Given wE n(D)(X), the associated linearform E>(w): H 1 (X, (!)(-D))..-. C will

be obtained as a residue map by applying the residue theorem. We reproduce here
the algebraic proof of Serre ([GACC], Chapter II), making use of the equation
H 1 (X, Jt) = (0).

1. The Principal Part Distributions with Respect to a Divisor. Defining

Jt = A j(!) as the sheaf of germs of principal parts, we considered in Chapter V.2.1
the exact sequence 0 ..-. (!) ..-. A ..-. Jt ..-. 0. In the following we generalize this
slightly. Let DE Div X and define Jt(D)==A/@(D) as "the sheaf of germs of
principal parts with respect to D." Then we have the exact sequence

(1) 0--.(!)(D)..-.A --.Jt(D)--.0.

It follows immediately that Jt(D), like ::0, is soft. In fact, every section over an
open set U has discrete support in U. Since every stalk Jt(D)x is the quotient
module A x/@(D)x, this implies that the C-vector space Jt(D)(X) of global distribu-
tions of principal parts is canonically isomorphic to the direct sum ffi Ax j(!)(D )x:
xeX

(2) Jt(D)(X) ~ ffi --"x/@(D)x ~ ffi --"x/ ffi (!)(D)x·

xeX xeX xeX
220 Chapter VII. Compact Riemann Surfaces

2. The Equation H 1 (X, (I)(D)) = J(D). Let R be the set of all maps F = (fx)
which assign to every x E X a germfx E .Ax so that almost allfx are holomorphic.
Clearly R is a C-vector space. Further, given D E Div X,

R(D)•={F E R:fx E (I)(D)x}

is a subspace of R. Every element of the direct sum Et> .Ax (resp. Et> (I)(D)x) is a
xeX xeX
family Ux} x EX withfx E Ax (resp.fx E (I)(D)x), where almost allfx vanish. Thus
we have the natural C-linear injection Et> .Ax-+ R which maps Et> (I)(D)x into
xeX xeX
R(D), and which induces a C-isomorphism

(3) .*'(D)(X) ~ Ef> A xl Ef) (I)(D)x ~ R/R(D).

xeX xeX

Every meromorphic function h determines an element (hx) E R. Thus, identify-

ing .A(X) with its image in R, .A(X) n R(D) = (I)(D)(X). We now set

J(D) •= R/(R(D) + .A(X)).

The standard isomorphism theorems of linear algebra yield

(4) J(D) ~ (R/R(D))/(.A(X)/(I)(D)(X)).

The following is an easy consequence of the definitions.

Theorem 1. For every D E Div X there is a natural C-isomorphism

Proof: Associated to the short exact sequence (1) we have the exact cohomo-
logy sequence

0--+ (I)(D )(X)--+ .A (X)~ JF(D )(X)

--+ H 1 (X, (I)(D)) --+H 1 (X, .A)--+···

By Theorem 5.3, H 1 (X, A)= (0). Thus

H 1 (X, (I)(D)) ~ JF(D)(X)/Im e,

where Im e ~ .A(X)/(I)(D)(X). The claim now follows from (3) and (4) above.
0
The reader should note that the spaces R, R(D) and .A(X) have very large
infinite dimensions, but that the finiteness theorem implies that J(D) is finite
dimensional.
§ 6. The Duality Theorem of Serre 221

3. Linear Forms. For maps F = Ux) and G = (g,J E R, we define the product
FG == Ux gx). Equipped with this product, R is of course a ~::-algebra, but it is more
importantly also an algebra over the field vH(X) cR. Let tX: R-+ C be a C-linear
form and h E vH(X). Then one defines the C-linear form htX: R-+ C by htX(F) =
tX(hF). Thus Homc(R, C) becomes an vH(X)-vector space. We explicitly state two
simple, but important, properties:
a) If vH(X) c ker tX then vH(X) c ker htX.
b) If R(D) c ker tX then R(D +(h)) c ker htX.

Proof: The statement a) is trivial. IfF= Ux) E R(D + (h)) thenfx E (I)(D + (h ))x
and therefore u(hxfx) ~ -ux(D). Hence hF E R(D). This proves b). 0

We denote by J(D) the dual space of J(D). It follows from a) and b) that every
meromorphic function hE vH(X)* determines a natural C-linear mapping,

J(D)-+J(D +h),

defined by A.~---+ hA..

Proof: Every A. E J(D) is a C-linear form A.: R/(R(D) + vH(X))-+ C and is there-
fore liftable to a C-linear form tX: R-+ C such that tX vanishes on R(D) + vH(X).
From a) and b) it follows that htX: R-+ C vanishes on R(D +(h))+ vH(X). Clearly
htX induces a C-linear form hA. E J(D + (h)), which is uniquely determined by A. and
h. It is obvious that the map A.-+ hA. is C-linear. 0

Since hA. is no longer in J(D), we go over to the space

J==U J(D),
D

the union of all J(D)'s for DE Div X. For two divisors D 1 ::; D 2 , we have
R(D 1 )::; R(D 2 ). Thus J(D 1 ) ~ J(D 2 ) when D 1 ::; D2 . This immediately implies
that every finite subset of J is contained in some J(D).
The set J is thus a C-vector space which is filtered by the subspaces J(D),
DE Div X. The above defined map, J(D)-+ J(D +(h)), gives us a mapping
vH(X) x J-+ J. Since Homc(R, C) is an vH(X)-vector space, the following remark
is obvious.

Theorem 2. The set J is, with respect to the operation vH(X) x J-+ J, a vector
space over vH(X).

4. The Inequality DimAt(Xl J ::; 1. The critical point of the proof of the duality
theorem is the following surprising dimension estimate. It is obtained by taking a
limit of the preliminary form of the Riemann-Roch formula.

Theorem 3. The vH(X)-vector space J is at most !-dimensional.

222 Chapter VII. Compact Riemann Surfaces

Proof: Let A., Jl e J. We choose De Div X with A., Jl e J(D). Let p eX be fixed.
For every f e lP(np)(X) it follows that, since D- np ~ D +(!),fl. e J(D + (!)) c
J(D- np). Similarly gp. e J(D- np) for all g e lP(np)(X). The C-linear map
Jt(X) $ Jt(X)--+ J, (J, g)~-+fl.+ gp.,

therefore induces by restriction a C-linear mapping

(~) lV(np)(X) $ lP(np)(X)--+ J(D- np)

for all n e 7L. If A. and J1. were linearly independent then both maps (o) and(~) would
be injective. This would mean that

( +) 2 dime lP(np)(X) ~dime J(D- np)

for all n e 7L. From the Riemann inequality (Paragraph 3.2) it follows that

( + +) dime lP(np)(X) = l(np) ~ deg(np) +1- g = n + 1 -g.

Furthermore (Paragraph 3.2), as soon as deg(D- np) = deg D- n is negative,

dime J(D- np) =dime I(D- np) = i(D- np) = g- 1 - deg(D- np).

Thus, for large n,

dime J(D- np) = g- 1 - deg D + n.

From ( + +) and ( +\) one infers that, for n large enough,

2 dime lV(np)(X) >dime J(D- np).
This is contrary to ( +) and therefore A. and Jl must be linearly dependent over
.A(X). D

Remark: There are divisors D such that H 1 (X, lV(D)) =I= (0). In other words,
J(D)' =1= (0). Thus J is a 1-dimensional Jt(X)-vector space.

5. The Residue Calculus. We write Q for the sheaf 0 1 of germs of holomorphic

1-forms on X. Note that, since dime X= 1, Qi = 0 fori> 1. The sheafQ is locally
free of rank 1. Thus, given a local coordinate t e mx at x, every germ Wx e 0;' is
uniquely written as Wx = hx dt with hx E mx. The residue, Resx Wx, of Wx at x is
invariantly defined as the coefficient of t- 1 in the Laurent development of hx with
respect to t. In other words

Resx Wx = 2~i J h dt,

i!H

where H is a small disk about x, and h e lD(R) is a representative of hx. If

(J)x E nx then it is clear that Resx (J)x = 0.
§ 6. The Duality Theorem of Serre 223

Now let wE Q 00 (X) be a global meromorphic differential form and

F = Ux) E R. Then, for almost all X E X, fx Wx E nx. Thus the sum

<w, F) •= L Resx(fxwx) E C
XEX

is finite. We summarize some properties of this pairing in the following:

Theorem 4. The map

<, ) : goo (X) X R ___. C,

defined by (w, F)H <w, F) is a C-bilinear form. Furthermore

0) for all h E .A(X), <hw, F)= <w, hF)

and

1) if wE!l(D)(X) and FER(-D) then <w,F)=O.

Proof: The C-bilinearity of < , ), as well as 0), is clear by definition. Let

w E n(D )(X) and F = Ux) E R(- D). Then

Q(fxwx) = Q(fx) + Q(wx) ~ Qx(D)- Qx(D) = 0.

That is, for all X E X,fxw E nx, and ResxUxw) = 0. This proves 1). D
The following theorem is essential for our further considerations.

Theorem 5 (Residue theorem). If wE Q 00 (X) and hE A(X) then <w, h)= 0.

Proof: Since hw E noo(X), it is enough to give a proof for h 1. It must be =

shown, therefore, that L Resx Wx = 0. Let x 1 , ..• , Xn EX be the poles of wand
XEX
let H 1, ... , Hn be pairwise disjoint "closed disks" about the x;s. Applying Stokes'
theorem, we have

n 1 1 1
L Resxwx= L ~ J w= -~ J w= - - J dw=O,
21ti X\uH,
XEX v=l 1tl oH, 1tl o(X\uH,)

since w is holomorphic outside of UHv, and thus dw vanishes identically on this

~ D
6. The Duality Theorem. Every differential form wE !l00 (X) determines a C-
linear form,
w*: R-.C,
224 Chapter VII. Compact Riemann Surfaces

defined by F~-+(ro, F). The C-vector spaces 0 00 (X) and Homc(R, C) are ..R(X)-
vector spaces as well. In fact, the mapping 0 00 (X)-+ Homc(R, C), defined by
ro 1-+ ro* is ..R (X)-linear.

Proof: This mapping is obviously additive. Let he ..R(X), FeR and

roE fl 00 (X) be given. Then by 0) in Theorem 4 and by the definition of hro*,

(hro)*(F) = (hro, F)= (ro, hF) = w*(hF) = hw*(F). D

From the residue theorem we see that ..R(X) c ker ro*. Moreover, if
roe Q(D)(X) then R( -D) c ker ro* (Theorem 4, 1)). Thus, if roe Q(D)(X~ ro*
induces a C-linear form,

9D(ro): R/(R( -D)+ ..R(X))-+ C.

In other words, if roe Q(D)(X) then 9D(ro) e J( -D)*= J( -D).

Thus, for every divisor D e Div X, we obtain a C-linear map

9D: Q(D)(X)-+ J(- D).

This mapping extends in a unique way to a C-linear map

The reader should note that, if roe Q(D) n Q(D')(X), the elements E>D(ro) and
E)D.(ro) agree in J.
The following lemma is the preparatory step in the duality theorem:

Lemma 6. The mapping e is ..R(X)-linear. If 9(ro) E J( -D), then

roE Q(D)(X).
Proof: The ..R(X)-linearity of 9 follows from the definition of the ..R(X)-vector
space structure on J (see Theorem 2) and the ..R(X)-linearity of rot-+ ro* (see the
above remarks). It remains to prove the last claim.
Let p EX and n •=oP(w) + 1. Define F 0 •= U~) E R by fx •=0 for x ::/= p and
fP = r", where t e mP is a local coordinate at p. Clearly o(!PwP) = -1 and thus
ro*(F 0 )=ResP(!ProP)::fO. Now, since 9(ro)eJ(-D~ ro* vanishes on R(-D).
Thus F 0 ¢R(-D). Equivalently fP¢{!)(-D)P or, in other words, o(fP)+
oP(- D) < 0. This is the same as saying - n - oP(D) < 0. In other words oP(ro) +
pD (D)~ 0 for all p. Thus (1) E n(D)(X). D
The following is now immediate.

Theorem 7. The maps 9: Q 00 (X)-+ J and 9D: Q(D)(X)-+ J( -D) are bijective.

Proof: If 9(ro) = 0 then, for every De Div X, 9(ro) e J( -D). Thus, by

Lemma 6, (1) E Q(D)(X) for all such D. Since n
Q(D)(X) = (0), (1) = 0 and e is
therefore injective. D
§ 7. The Riemann-Roch Theorem (Final Version) 225

By Theorem 3, dim.«<X> J ~ 1. Since !l""(X) =I= (0), the Jt(X)-monomorphism

8 is automatically surjective.
Let DE Div X be given. Then, as the restriction of8 to Q(D)(X), 8v is injec-
tive. Every A. E J( -D) has a pre-image under 8, wE !l""(X). Again by Lemma 6,
wE !l(D)(X). Thus 8v: O(D)(X)-+ J(- D) is bijective. D
Finally the duality theorem is an easy consequence of Theorems 1 and 7.

Theorem 8 (The duality theorem). Let X be a compact Riemann surface and

DE Div X. Then there exists a natural C-isomorphism

Proof: By Theorem 1 there is a natural C-isomorphism between H 1{X, CD(- D))

and I(- D). This induces a C-isomorphism, J(- D)-+ H 1 (X, CD(- D))*, ofthe dual
spaces. Composition with 8v gives us our isomorphism:

In this form the duality theorem is a classical theorem in the subject of alge-
braic curves. The more general form (for complex manifolds of higher dimension
which are not necessarily compact) was first formulated and proved by J.-P. Serre
in 1954 (Un theoreme de dualite, Comm. Math. Helv. 29, 9-26 (1955)).

§ 7. The Riemann-Roch Theorem (Final Version)

The results of this section are consequences of Theorem 3.1 and the duality
theorem. The strength of the Riemann-Roch theorem will be demonstrated by
some selected (classical) applications.
We will always use K to denote the canonical divisor. The notation l(D) and
i(D), which was introduced in Section 3, will be consistently applied.

1. The Equation i(D) = l(K- D). Since every finite dimensional vector space
has the same dimension as its dual space, the duality theorem implies the
following:

Thus i(D) is the number of linearly independent meromorphic differential forms w

such that (w) ~D. In particular i(O) = g: On a Riemann surface X with genus g
there are exactly g linearly independent global holomorphic 1-forms:

For every positive D, H 0 (X, Q( -D)) c H 0 (X, n). Thus using Theorem 3.1,
we can estimate i(D) and l(D) from above: If the divisor class of D contains a
positive divisor then i(D) ~ g and l(D) ~ deg D + 1.
226 Chapter VII. Compact Riemann Surfaces

Since Q ~ ll'!(K) (see Theorem 4.5), it follows that 0( -D)~ ll'!(K- D) for all
De Div X. Thus the above dimension equation can be written in the form

(1) i(D) = l(K- D).

In the case of D = 0 this leads to l(K) = g and similarly, in the case of D = K,

we have i(K) = 1(0) = 1. In other words H 1 (X, 0) ~C. Hence it turns out that

(2) x(n) = l(K)- i(K) = g - 1 = - x(&).

It follows easily now that g is only a topological invariant:

Theorem 1. The genus g of a compact Riemann surface X is a topological invar-

iant. In fact

Proof: We consider the exact sequence of sheaves

0---+ C---+ ll'! d=o 0---+ 0.

Due to (2) above,

x(x, c)= x(&)- x(n) = 2- 2g.

Since H 0 (X, C) and H 2 (X, C) are both isomorphic to C,

2- 2g = x(X, C)= 1 -dime H 1 (X, C)+ 1. 0

We will make further remarks about the structure of H 1 (X, C) in Section 7.

2. The Formula of Riemann-Roch. It follows from Theorem 3.1 that, for all
D e Div X, l(D)- i(D) = deg D + 1 -g. If one writes l(K - D) instead of i(D)
then one obtains the formula of Riemann-Roch.

Theorem 2 (Riemann-Roch, final version). For every divisor D on a compact

Riemann surface X of genus g,

l(D) -l(K- D)= deg D + 1- g.

Since l(K) = g and 1(0) = 1, we find by setting D = K that g- 1 = deg K +

1 - g. Thus we have the degree equation for differential forms: For every differential
form we O""(X)*,

degK=degw=2g-2.
§ 7. The Riemann-Roch Theorem (Final Version) 227

This equation contains for example the fact that H 1 (1P' 1 , C)= (0). One sees this by
noting that the differential dz, where z is an inhomogeneous coordinate, has degree
- 2. Thus, since 2g - 2 = - 2, IP' 1 has genus 0.
From the above it follows that every non-trivial w e O"'(X) has degree - x(X), where x(X) is the
topological Euler-Poincare characteristic of X. Consequently, if IX: X-+ X' is an s-sheeted ramified
covering map between compact Riemann surfaces X and X', and We Div X is the ramification divisor
of IX, then
x(X) + deg w = s · x(X')
Proof: Let w' e O"'(X'), and define w = IX*(w'). A direct calculation shows that deg(w) =
s · deg(w') + deg W. Since deg(w) = -x(X) and deg(w') = -x(X), the claim follows immediately.
0
In particular this shows that deg W is always even, and, when X' = IP' 1 with x(X') = 2, it follows
that
deg W = 2(s + g - 1)

3. Theorem B for Sheaves lD(D). The following important application of the

Riemann-Roch formula uses the simple fact that, when deg D < 0, l(D) vanishes.

Theorem 3 (Theorem B). 1£t D e Div X with deg D ~ 2g - 1. Then

a) H 1 (X, lD(D)) = (O)
b) l(D) = deg D + 1- g.

Proof: a) Since deg K = 2g- 2, deg(K- D)< 0 and therefore i(D) =

l(K- D)= 0. b) This follows immediately from a) and the Riemann-Roch
formula. D
Theorem 3 is the optimal form of Theorem B in the sense that if the degree of a
divisor D is less than 2g- 1, then the cohomology group H 1 (X to lD(D)) may not
vanish. For example, by Serre duality, H 1 (X, lD(K)) ~ H 0 (X, lD) ~ C and
degK=2g-2.

4. Theorem A for Sheaves lD(D ). Let 2 be a locally free sheaf of rank 1 over X,
and let x eX. Then the following are equivalent:
i) The module of sections 2(X) generates the stalk 2" as an (!)"-module.
ii) There is a section s e 2(X) with o(sx) = 0.
iii) There is a sections e 9'(X) with 9'" = lD"s".
iv) dime H 1 (X, 9'( -x)) ~dime H 1 (X, 9').

Proof: i)::;. ii): Lett" e 9'" be such that 9'" = (!)" · t". By i) there exist sections
m
s,. e 9'(X) and germs f"" e (!)"' 1 ~ Jl ~ m, so that t" = "f.f,.xs,.". Thus some
1
s,."(x) e C must differ from 0.
Hence for that section lD(s"") = lD.
ii)::;. iii): Every germ t" e 9'" with o(tx) = 0 generates 9'" as an (!)"-module.
228 Chapter VII. Compact Riemann Surfaces

iii)=>i): Trivial.
ii)~iv): The sheaf ff which is defined by 0-+ !l'( -x)-+ !l'-+ ff-+ 0 has sup-
port only at x where its stalk is C. Thus we have the following
cohomology sequence:

0---+ !l'( -x)(X)---+ !l'(X)~C--

H1(X, !l'(- x))---+ H 1(X, !l')---+ 0

Therefore there is a section s E !l'(X) with Q(sx) = 0 if and only if

!l'( -x)(x) !i !l'(X), This is equivalent to H 1(x, !l'( -x))-+ H 1(x, !l') being
injective. 0

Theorem 4 (Theorem A). Let D E Div X with deg D ;e:: 2g. Then for every x < X
there exists a sections E &(D)(X) with &(D)x = (Dxsx.

Proof: Let !l' :=&(D). If H 1(X, !l'( -x)) = 0, then the above equivalences guar-
antee the existence of such a section. Now !l'(- x) = &(D - x ). But by assump-
tion, deg(D- x) ;e:: 2g- 1. Hence Theorem B implies that H 1(X, &(D- x)) = 0.
0
Theorem 4 is the optimal form a Theorem A in the sense that for deg D < 2g
the sheaf &(D) may not be generated by its global sections. For example, let
D = K + {p). Then &(D)= !l(p). A section in &(D)(X) is just a differentialform w
which is holomorphic on X\P, and, if it would generate &(D)P, would have a pole
of order 1 at p. This would contradict the residue theorem.
There are divisors with deg D < 2g, and for which Theorem A, however, is
valid, namely K: If g =I= 0 then Theorem A holds for the sheafQ ~ &(K). In other
words, given x E X there exists a holomorphic differentia/form w E !l(X) which does
not vanish at x.

Proof: By the above equivalences, we only need to show that

dimG:: H 1(X, !l( -x)):::; dimG:: H 1(X, !l). Since dimG:: H 1(X, Q) = 1, it is enough
then to prove that i(K- x):::; 1. Suppose /(x) = i(K- x) ;e:: 2. Then there exists
h E .A(X) n &(X\x) with a pole of order 1 at x. The map h: X-+ IP> 1 would be
biholomorphic, and, since · g =I= 0, we have reached a contradiction. Thus
i(K-x):::;l. 0

5. The Existence of Meromorphic Differential Forms. The following existence

theorems are immediate consequences of Theorem A for differential forms: Let
DE Div X with deg D ;e:: 2. Then every stalk of Q(D) is generated by a global
meromorphic differentia/form wE Q(D)(X).

Proof: Since !l(D) ~ &(K +D) and deg(K +D) ;e:: 2g, the result follows from
Theorem 4. 0
In particular if D := mp, m ;e:: 2, then we have the following:
Let p E X and m ;e:: 2 be given. Then there exists a meromorphic differential form
§ 7. The Riemann-Roch Theorem (Final Version) 229

on X which is holomorphic on X\P and has a pole of order m at p. Furthermore,for

every p 1 and p2 E X with p 1 =I= p2 , there exists a meromorphic form on X which is
holomorphic on X\{p 1 , p2 } and has poles of order 1 at p1 and p2 .

Proof: Define D := p 1 + p2 • Then there exists w E Q(D )(X) which generates the
stalk Q(D)p 1 • This form must be holomorphic on X\{Pt. p 2 }, have a pole of order 1
at p 1 and have a pole of at most order 1 at p2 • But the residue theorem requires
that it has a pole of exactly order 1 at p 2 . D

6. The Gap Theorem. A natural number w ~ 1 is called a gap value at p E X if

there is no holomorphic function on X\P which has a pole of order w at p. If
X = 1P 1 then there are no gap values. But if X =I= IP 1 then g =I= 0 and w = 1 is
always a gap value.
We write 1. := l(vp) for each v ~ 0 and note that
I.:-::;; lv+ 1 :-::;;I.+ 1 and w is a gap value if and only if lw = lw_ 1 •
Proof: In the exact sequence

0-+ H 0 (X, @(vp))-+ H 0 (X, @((v + l)p))-+ ff(X)-+ · · ·,

the space ff(X) = §"P = @((v + l)p)p/@(vp)P is !-dimensional. This implies that
1. :-::;; 1.+ 1 :-::;; 1. + 1. By definition w is a gap value if and only if every hE @(wp)(X)
is already in @((w- l)p)(X). That is, @(wp)(X) = @((w- l)p)(X) or equivalently
lw=lw-1• D
If v ~ 2g - 1 it follows from Theorem 3 that 1. = v + 1 - g. Thus for v ~ 2g,
1. = 1. _ 1 + 1 and there are no gap values greater than 2g - 1. One can improve
this remark:

Theorem 5 (Weierstrass gap theorem). Let X be a compact Riemann surface with

genus g > 0. Then, for every p E X, there exist exactly g gap values, w1, ... , wg,
which we order so that

1 = w1 < w2 < .. · < wg :-::;; 2g- 1.

Proof: By Theorem 3, l2 g- 1 = g. Hence 1 :-::;; 10 :-::;; 11 :-::;; • • · < l2 g- 1 = g. Since

I;+ 1 - I;~ 1, there must be exactly g- 1 indices i such that li+ 1 - l; = 1. Thus,
since 10 = 1 and l2 g_ 1 = g, there must be g indices i such that l;+ 1 = l;. That is,
there are exactly g gap values. 0

7. Theorems A and B for Locally Free Sheaves. We call a locally free sheaf
"Stein" if Theorems A and B are valid:
A) H 0 (X, ff) generates every stalk ff x as an (!) x-module.
B) H 1 (X, ff) = (0).
230 Chapter VII. Compact Riemann Surfaces

Using this terminology we have the following:

Lemma 6. Let 0 -4 !l' -4 !F -4 <§ -4 0 be a sequence of locally free sheaves.

Assume further that !l' and <§ are Stein. Then !F is also Stein.

Proof: Since H 1 (X, !l') = H 1 (X, <'§) = (0), it follows immediately from the
exact cohomology sequence that H 1 (X, !F)= (0). Further we consider the com-
mutative diagram

where the rows are exact and the maps in the columns associate to a section its
germ at x. By assumption, the images of Ax and 'l'x generate (as lllx-modules) !l'x
and <§x respectively. It follows therefore that the image of H 0 (X, !F) under cpx
generates !F X as an llJ X-module. D

Theorem 7. Given a locally free sheaf!F over a Riemann surface X, there exists a
natural number n+ so that,for every DE Div X with deg D ~ n+, the sheaf!F(D) is
Stein.

Proof: (by induction on the rank r of !F). Since every locally free sheaf of
rank 1 is isomorphic to a sheaf of the type lP(D), the case of r = 1 handled by
Theorem 3 and 4. Now we assume that r > 1 and that the statement holds for
sheaves of rank at most r - 1. By the "subsheaf theorem" (Theorem 4.3 ~ there
exist locally free sheaves !l' and <§ of rank 1 and r - 1 so that the following
sequence is exact for every D E Div X:

0-4 !l'(D) -4 !F(D) -4 <§(D) -4 0.

By the induction assumption and Lemma 6 above, there exists n+ E 7L. so that,
for deg D ~ n+, !F(D) is Stein. D
The following is analogous to the fact that if deg D < 0, then
H0 (X, C9(D)) = (0).

Theorem 8. Let !F be a locally free sheaf over a compact Riemann surface X.

Then there exists n- E 7L. so that for every D E Div X with deg D < n-,
H0 (X, :F(D)) = (0).

Proof: (by induction on the rank r of !F). If the rank of !F is 1 then !F = lli(D')
for some D' E Div X and n- •= - deg D' has the desired property. If rank !F > 1
then, as in the proof of Theorem 7, we choose locally free sheaves !l' and <§ of rank
§ 7. The Riemann-Roch Theorem (Final Version) 231

1 and r- 1 respectively so that, for every De Div X, we have the exact sequence

0-+ .P(D)-+ ~(D)-+ ~(D)-+ 0.

The claim follows immediately from the induction hypothesis and the exact
cohomology sequence. 0

8. The Hodge Decomposition of H 1 (X, C). We want to develop a better under-

standing of the vector space H 1(X, C)~ C 29• As earlier, we begin with the resolu-
tion of the constant sheaf C:

o-c-l!J d={) n-o.

Since H 0 (X, C)= H 0 (X, l!J) = C and H 2 (X, l!J) = (0), the associated cohomology
sequence yields

But H 2 (X, C)= H 1 (X, 0) =C. Hence p is bijective and we therefore have the
exact sequence

The map oc can be explicitely described in the following way: Let ro e O(X) and
U = {U;} a covering of X by contractable neighborhoods. Then there exists
J; e l!J(U;) so that dJ; = ro luc
The function ocii == jj- J; is therefore constant on Uii (we take these intersec-
tions to be connected). The family {oc;i(w)} forms a 1-cocycle in Z 1 (U, C). This
cocycle represents the cohomology class oc(ro) e H 1 (X, C).
Since l!;i c C, H 1 (X, i~;i) is an l!;i-vector subspace of H 1 (X, C). We now show

Lemma 9. Im(oc) n H 1 (X, i~;i) = (0).

Proof: Suppose that oc(ro) e H 1 (X, i~;i). From the above description of oc it fol-
lows that there is a covering {U;} of X and functionsJ; e l!J(U;) such that dJ; = ro lui
and every function jj - J; is constant and real on Uii· Let g; == exp(2n.J=Tt;).
Then lg;j 2 = l9il 2 on U;i and therefore {lg;j 2} determines a real valued contin-
uous function, g, on X. By the maximum principle g is identically constant and
thus ro = 0. 0
There are of course many C-vector spaces Vin H 1 (X, C) so that V ffi Im oc =
H 1 (X, C). However there is one particular V that is quite natural. For this we
consider the "conjugate" resolution of C,
- d=B -
o - c ---+l!J- n-o.
232 Chapter VII. Compact Riemann Surfaces

Here fi •= 0 1 (see 11.2.3, in particular the diagram). For reasons similar to those
above, the associated cohomology sequence is

Using this we prove

Theorem 10 (The Hodge decomposition). The C-vector space H 1 (X, C) is the

direct sum of the spaces Im ex and Im &, where&: fi(X)-+ H 1 (X, C)= ex(O(X))€9
&(fi(X)).

Proof: Since H 1 (X, C)~ C 29 and Im ex~ Im &~ C9 , it is enough to show that
lm ex n Im &= (0). The conjugation C -+ C, defined by c1--+ c, determines an ~­
linear involution a: H 1 (X, C)-+ H 1 (X, C) which has H 1 (X, ~) as a fixed space.
Obviously a leaves every element of Im ex n lm & fixed. Thus Im ex n Im &c:
Im ex n H 1 (X, ~). Hence, by Lemma 9,

Im ex n Im &= (0). 0

§ 8. The Splitting of Locally Free Sheaves

By means of a formal splitting criterion we give a sufficient condition for a
locally free sheaf fF over a compact Riemann surface to contain a locally free
subsheaf of rank 1 which is a direct summand of fF (Theorem 4 ). Every such locally
free sheaf contains maximal subsheaves of rank 1. On the Riemann sphere I?~> the
maximal subsheaves are direct summands (Splitting Lemma). As a corollary, it
follows that every locally free sheaf fF of rank r over IJl> 1 is isomorphic to a sheaf
r
EB (!')(nip), where n~> .. . , n, E 7L. are uniquely determined up to order by fF (a
1
theorem of Grothendieck). The presentation here follows along the lines o£[21].

1. The Number JL(§"). Let !l' be a locally free sheaf of rank 1 over a compact
Riemann surface X. Then, for every s E !l''"'(X)*, deg s only depends on !l' (see
Paragraph 4.2 ). For locally free sheaves of rank r > 1 this is not in general true.
For example, consider§"= (!')(n 1 p) E9 · · · E9 (!')(n,p) for some p EX. Then§" has
sections of degree n~> n2, ... , n,.
In order to define an integer which can be used in place of the "degree", we
make the following observations:
Every homomorphism n: §"-+ f§ between locally free sheaves induces a homomor-
phism n: §"'"'(X)-+ ~'"'(X). If
§ 8. The Splitting of Locally Free Sheaves 233

is an exact sequence of non-zero locally free sheaves, then, for every section
s E .?" 00 (X)*, either
a) n(s) = 0, in which cases= i(s') with s' E .?' (X)* and deg s' = deg s,
00

or
b) n(s) =I= 0, in which case deg(s):::;; deg(n(s)).

Proof: If .?~ is complementary to i(.?~) in .?"" then for every germ tx E .?':
there exist uniquely determined germs t~ E .?~00 , t~ E .?~ 00 so that

o(tx) = min{o(t~), o(t~)}, xEX.

If n(s) = 0, then s E i(.?'(X)) (i.e. s = i(s') with s' E .?' 00 (X)*). But o(sx) = o(s~) for
all x EX. Hence deg s = deg s'.
Now suppose that n(s) =I= 0, and let sx = i(t~) + t~. Since nx maps.?~ isomor-
phically onto '§x, it follows that o(n(s)x) = o(t~). Hence o(sx):::;; o(n(s)x) for all
x EX, and consequently deg(s):::;; deg(n(s)). D
For every locally free sheaf.? over X, we define J.l(.?) as follows:

J.l(.?") := sup{deg(s) Is E .?" 00 (X)*}.

The following direct consequence of the above remarks is useful in showing

that J.l(.?") < oo:
Let 0-+ .?' -+ .? -+ '§ -+ 0 be an exact sequence of non-zero locally free sheaves.
Then

J.l(.?"') :::;; J.l(.?"):::;; max{J.l(.?"'), J.l('§)}.

Now it is easy to prove that Jl(ff) is bounded.

Theorem 1. Let .? be a locally free sheafon a compact Riemann surface X. Then

the degree function, J.l(.?), is finite.

Proof: (by induction on the rank r of.?). For r = 1 the statement is trivially
true. There exists an exact sequence of locally free sheaves 0-+ 2 -+ .? -+ '§ -+ 0,
where 2 and'§ have rank 1 and r - 1 respectively. The proof follows immediately
by the induction hypothesis and the above estimates. D

2. Maximal Subsheaves. The number J.l(.?") can be characterized by the Chern

numbers of the locally free sheaves of rank 1:

(2) J.l(.?") = max{c(2): 2 is a locally free subsheaf of.? of rank 1}.

This follows from the facts that every sections E .?" 00 (X)* determines a locally free
subsheaf 2 of rank 1 with c(2) = deg s, and conversely that a locally free sub-
sheaf determines such a section (see Section 4 ). We call a locally free subsheaf 2 of
234 Chapter VII. Compact Riemann Surfaces

rank 1 in :F maximal if c(.!l') = Jl(ff'). Such sheaves are generated by maximal

sections (i.e. sections s E ff«>(X)* with deg s = Jl(ff')). We have seen that
every non-zero locally free sheaf :F possesses maximal subsheaves.

Theorem 2. Let :F be a locally free sheaf on a compact Riemann surface X. Let

.!l' be a maximal subsheaf in :F and D E Div X. Then .!l'(D) is a maximal subsheaf in
:F(D). In general,

Jl(ff) + deg D ~ Jl(:ff'(D)).

Proof: Since .!l'(D) is a locally free subsheaf of ff(D), c(.!l'(D)) ~ Jl(:ff'(D)). Of

course c(.!l'(D)) = c(.!l') + deg D. Thus, since c(.!l') = Jl(ff),

Jl(:ff') + deg D ~ Jl(:ff'(D)).

We now apply(*) with :F replaced by ff(D) and D by -D. Thus

Jl(:ff'(D)) + deg( -D)~ Jl(ff).

Hence there is equality in (•) and, in particular, c(.!l'(D)) = Jl(ff(D)). 0

3. The Inequality Jl(~) ~ Jl(ff) + 2g. We begin with a consequence of the

Riemann inequality:

Lemma. Let ff be locally free of rank 2 and .!l' a maximal subsheaf which is
isomorphic to@. Then, defining~:= ff/.!1', c(~) ~ 2g.

Proof: Since H 1 (X, r2) ~ U, the cohomology sequence associated to 0 .__. (2 .__.
0 starts out like
ff .__. ~ .__.

0---+ C ---+ ff(X)---+ ~(X)_.!__._. Cg---+ · · · .

Now the sheaf~ is locally free of rank 1. Thus if c(~) > 2g, then, by the Riemann
inequality,

dime ~(X) ~ c(~) + 1 - g ~ g + 2.

Hence the kernel of q> would be at least 2-dimensional, and ff(X) is at least
3-dimensional. However, since the rank of ff is 2, every C-vector space ff x fmx ff x
is 2-dimensional. Thus the restriction map ff(X) .__. ff x fmx ff x has a non-trivial
kernel. In other words, there would be a section s =/= 0 in ff(X) which vanishes at
x. This implies that deg s ~ 1. But that is contrary to the assumption that

Jl(ff) = c(.!l') = c(r2) = 0. D

§ 8. The Splitting of Locally Free Sheaves 235

We now show

Theorem 3. Let fF be a locally free sheaf of rank r ~ 2 and let !£ be a maximal

subsheaf of fF. Then, defining t'§ •= fF /.P,

Proof: As usual we start with the exact sequence

0 -----+ !£-----+ fF ~ t'§ -----+ 0.

At first we consider the case r = 2. There exists a divisor D e Div X with

!£;;;;;;(!)(D) and deg D = c(.P) = J.t(fF). Applying the above Lemma to the seq-
uence 0--. (!)--. fF( -D)--. t'§( -D)--. 0, we find that c(t'§(- D))~ 2g. Since
c(t'§( -D))= c(t'§)- deg D, the claim follows by noting that J.t(t'§) = c(t'§) and
deg D = J.t(fF).
Now consider the case ofr > 2. Let!£' be a maximal subsheaf oft'§. We have an
induced exact sequence 0--. !£--. fF'--. !£'--. 0, where fF' •= n- 1 (.:£'). For all
xeX,

Thus fF' is a locally free sheaf of rank 2. Clearly!£ is a maximal subsheaf of fF',
and therefore

J.t(fF) = c(.P) = J.t(fF').

Consequently

J.t(t'§) ~ J.t(fF') + 2g = J.t(fF) + 2g. 0

4. The Splitting Criterion. One says that the short exact (!)-sequence,
i
0 ---+ f/ 1 -----+ f/ -----+ f/ 2 -----+ 0

e:
splits if there exists an (!)-homomorphism f/ 2 - f/ with 1t e
= id. Thus, in this
0

case, f/ = i(f/ t) ffi e(f/ 2);;;;;; f/ 1 ffi f/ 2· There is a simple formal splitting criterion:
An exact (!)-sequence of locally free sheaves 0 --. f/ 1 --. f/ --. t'§ --. 0 splits when

Proof: Since t'§ is locally free, the above short exact sequence induces the exact
(!)-sequence of sheaves,

0--. .Ytom(t'§, f/ 1 } .._. .Ytom(t'§, f/) .._. .Ytom(t'§, t'§) .._. 0,

236 Chapter VII. Compact Riemann Surfaces

which in tum induces an exact cohomology sequence

· ·· -H 0 (X, Jf Mn(~, .9')) ~ H 0 (X, Jf om(~, ~))-

H 1 (X, Jf om(~, .9' 1 ) ) - • · ·.

If 1t* is surjective, then there exists a global section e E Hom(~, 9') with 1t*(e) =
e
1t o = id. o
Remark: The splitting criterion is valid for any complex space. In the case of Stein spaces the
relevant cohomology group is always zero. Thus we have the following observation.
Every locally free subsheqf of a locally free sheaf over a Stein space is a direct summand.
For !-dimensional Stein manifolds (i.e. non-compact Riemann surfaces~ it follows that every
locally free sheaf of rank r is isomorphic to the sheaf (!J•. (Recall that H 1 (X, (!J) = H 2 (X, Z) = 0. Thus
H 1 (X, tr*) = 0, and thus every locally free sheaf of rank 1 is free.)

The application ofthe splitting criterion which is relevant to us is the following:

Theorem 4. Let iF be a locally free sheaf of rank r over a compact Riemann

surface of genus g. Suppose that 2 is a locally free subsheaf ofrank 1 in iF having the
following properties:
1) The quotient sheaf~ •= ili'/2 is a direct sum 2 2 E9 · · · E9 2r of locally free
sheaves of rank 1.
2) For i = 2, ... , r, c(2) - c(2i) ~ 2g - 1.
Then 2 is a direct summand of iF.
r
Proof: We first note that Jf om(~, 2) = E9 Jf om( !£i, 2). Let 2 ~ lP(D) and
i=2
2i ~ lP(Di), 2 ::::;; i ::::;; r. Then JfMn(2h 2) ~ lP(D - DJ Hence, since

for i = 2, ... , r, it follows (by Theorem B) that, for all such i,

Thus
r
H 1 (X, Jf Mn(~, 2)) ~ E9 H 1 (X, Jf om(2i, 2)) = (0)
i=2

and, by the splitting criterion, iF is a direct summand. 0

In the above proof we used the following fact:
For all divisors D, D' e Div X there is a natural £'9-isomorphism
§ 8. The Splitting of Locally Free Sheaves 237

Proof: Let t E mx be a local coordinate at x. Then @(D)x = t- Ox{D) · (!)x and

(!)(D')x = t- o,(D') (!)X' Thus every germ hx E @(D'- D)x = t-o,{D'-D). (!)X deter-
0

mines, by multiplication, an {9x-homomorphism (homothety),

defined by gx.-hxgx. (Observe that Q(hxgx) = Q(hx) + g(gx) = -Qx(D'- D)-

Qx(D) = -gx(D')). Since @(D)x and @(D')x are free {9x-modules of rank 1, every
homomorphism @(D)x-+ @(D')x is such a "homothety". Thus the map
@(D'- D)-+ ff Qm(I)(@(D), @(D')), defined by associating to each germ the hom-
othety defined above, is surjective. The injectivity of this map is trivial and thus we
have established an isomorphism. 0

5. Grothendieck's Theorem. We fix a point p "at infinity" in IP 1 and set

@(n) == @(np)

for every n E 7!... Then n is the Chern number of @(n) and @(n) ;;; @(m) if and only if
n = m. Furthermore, if ff' is a locally free sheaf of rank 1 over 1P 1 then ff' ;;; @(n ),
where n = c(ff').

Proof: Certainly ff' ;;; @(D) for some D E Div 1P 1 . On IP 1, however, two divisors
are linearly equivalent if and only if they have the same degree. Hence @(D) is
isomorphic to @(deg D) and, since deg D = c(ff'), we have the desired result.
0
Every sheaf @(nt) EB @(n 2 ) EB · · · EB @(n,), nt> ... , n, E 7!.., is locally free of rank r.
The splitting theorem of Grothendieck says that one obtains all locally free
sheaves over IP 1 in this way:

Theorem 5. (Grothendieck ). Let !F be a locally free sheafof rank r ;;::: 1 over IP 1 .

Then there exist integers nt> ... , n, (uniquely determined up to a permutation) such
that !F;;; @(nt) EB · · · EB @(n,).

Remark: The essense of this theorem can be formulated with matrices:

Lett> 1, U 1 '={ZE 1P' 1 IIzl < t}, and U 2 '={ZE IP'dlzl > 1}.
Let A E GL(r, (I)(U 12 )) be a holomorphic, invertible r x r- matrix. Then there exist holomorphic,
invertible matrices P E GL(r, (I)(U 1)), Q E GL(r, (I)(U 2)) so that the matrix D '= PAQ is a diagonal matrix
with z"1 , ••• , z"•, n; E Z, as diagonal terms.

6. Existence of the Splitting. Using the following lemma, both the existence and
uniqueness are proved by successively "splitting off" maximal subsheaves.

Splitting Lemma 6. Let !F be a non-zero locally free sheafover 1P t> and let ff' be a
locally free subsheaf of rank 1 with ~.-\ff');;::: Jl(ff/ff')- 1. Then ff' is a direct sum-
mand of !F.
In particular every maximal subsheaf of !F is a direct summand.
238 Chapter VII. Compact Riemann Surfaces

Proof: We proceed by induction on the rank r ofF, where the induction

hypothesis is the existence part of Theorem 5. If r = 1, then the lemma is trivially
true, because !l' =F. If r > 1, then the sheaf Ff!l' is locally free of rank r- 1,
e
and by assumption is therefore a direct sum !l' 2 ffi ... !l'r of locally free sheaves
I, of rank 1. Now,

max{c(!l' 2 ), ••• , c(!l'r)} ;:5; p,(F/!l') ;:5; c(!l') + 1.

In particular, c(!l')- c(!l'1) ~ -1, 2 ;:5; i ;:5; r. By Theorem 4 it follows that !l' is a
direct sum of F.
If !l' is a maximal subsheaf ofF, then from Theorem 3 it follows that c(!l') =
p,(F) ~ p,(F/!l'). (The genus is zero!). D

7. Uniqueness of the Splitting. For every locally free sheaf F on lfJ 1 we let
m •= p,(F). Then
F( -m)(lfJt) = {s e F 00 (1F» 1 )Ideg(s) = m} u {0} +0,
so

The sections F in F( -m)(!F» 1 ) generate a non-zero l!l-subsheaf ff ofF( -m).

Thus
F•=ff(m)=fO

is an invariantly (by F alone) determined l!J-subsheaf of F. With this language we

now prove the uniqueness part of Theorem 5:

Uniqueness Lemma 7. Let F = !l' 1 ffi · · · ffi !l' r = !l'J. E9 · · · E9 !l'~ be two split-
tings ofF with !l'1 ~ l!l(n1), !l'J. ~ l!l(n1), 1 ;:5; i ;:5; r. Assume that n 1 ~ n2 ~ • • • ~ nr,
and n't ~ n2 ~ ··· ~ n~. Then, with d•=dimc F(-m)(lfJ 1 ) ~ 1,
1) !l' 1 $ · · · ffi !l'd = !l'J. $ · · · ffi !l'4 = .#"; !l'; ~ l!l(m) ~ !l'i, for i = 1, ... , d,
and
2) n1 = ··· = nd = n1 = ··· = n:, = m; n1 = nifor i = d + 1, ... , r.
Proof: Since F=!l' 1 $···E9!l'r, it follows that F(-m)(IP 1 )=
r
EB !l'1( - m)(IP 1 ~ where !l'1( - m)~ l!l(l1) with 11 •= n1 - m ;:5; 0 (by the definition
i=1
ofm). Now
l!l(l)(!Pt) = 0 for l < 0, and l!l(IP 1 ) =C.

Thus there must be d equations n1 = m. Since the n;'s are monotonically decreas-
ing, n 1 = · · · = nd = m > nd+ 1 and !l' 1( - m) ~ 0, 1 ;:5; i ;:5; d.
For every i ;:5; d, !l'1( -m)(IP 1 ) generates the sheaf !l'1( -m). Thus
§ 8. The Splitting of Locally Free Sheaves 239

d d
E9
i= 1
..2"1( -m)(IPt} generates the sheaf E9 ..2"1( -m). Since §"( -m)(IPt) =
i= 1
d d d
E9 ..2"1( -m)(IPt), it follows that :T = E9 ..2"1( -m). Hence # = :T(m) = E9 ..2"1•
i=1 i=1 i=1
The above can obviously be repeated for the splitting §" = ..2'~ EB · · · EB ..2"~.
Thus 1) has been proved as well as n1 = · · · = n;, = m. It now follows that

Since §"j# has rank r- d < r, and since n1, n; are monotonically decreasing, it
follows by induction that n; = ni for i = d + 1, ... , r. D

Corollary. Let§"= C§ EB Yl', where C§, Yl' are non-zero locally free sheaves over
IP 1 . Then

Proof: Since§"~ @(nt) EB · · · EB @(n,), iffollows that J.L(§") = max{n 1, ... , n,}.
D
We see now that every locally free subsheaf ..2' of rank 1 in §"with c(..2') ~
J.L(§"/..2') is necessarily maximal, because the Splitting Lemma implies that
§" = C§ EB 9', and the above corollary shows that J.L(§") = max(J.L(C§),
J.L(..2')) = c(..2').
The following remark is also quite easy to see.
Let § = C§ EB Yl', where C§, Yl' are non-zero locally free sheaves with J.L(§") >
J.L(C§). Let ..2' be a locally free subsheafof rank 1 in§ so that c(9') > J.L(C§). Then ..2' is
a locally free subsheaf of Yl'.

Proof: Let ..2' = (!)s with s E §" 00 (1Pd*, and deg(s) = c(..2'). Let n: ff--+ C§
denote the natural sheaf projection. Then by the rematlk in Paragraph 1, either
n(s) = 0 or deg(s)::;; deg(n(s)). The latter is not possible, because deg(n(s))::;; J.L(C§)
and c(..2') > J.L(C§). Thus s E (~ e-t n)(IP t) = Yl'(IP1 ). Hence ..2' = (!)s c Yl'. D
In particular we have the following:
A sheaf§";;:; m(nt) EB · · · EB m(n,) with n 1 > max{n 2 , ••. , n,} has only one locally
free subsheaf ..2' which is isomorphic to m(n 1 ), and there are no such with ..2' ~ m(n),
n 1 > n > max{n 2 , ... , n,}.
However, for every n::;; max{n 2 , •.. , n,}, there are in the above setting locally
free subsheaves ..2' ;;:; @(n) in§". For this, see Prop. 2.4 in "On holomorphic fields
of complex line elements with isolated singularities", Ann. Inst. Fourier 14,99-130
(1964), by A van de Van.