INTRODUCTION TO RIEMANN SURFACES.
Contents
5. Divisors of meromorphic functions
5.1. Divisors.
5.2. The Riemann-Roch theorem.
5.3. The Weierstrass problem.
5.4. Period lattices and Jacobi varieties.
5.5. The Jacobi theorem.
5.6. Vector bundles and divisors.
5.7. Divisors and vector bundles.
5.8. Chern classes.
5.9. Riemann-Roch theorem for vector bundles.
5.10. Embeddings into Pn .
5.11. Algebraic curves revisited.
5.12. Exercises.
1
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2
4
5
5
6
7
8
10
11
13
14
5. Divisors of meromorphic functions
5.1. Divisors.
P Divisors on a Riemann surface are traditionally viewed as formal locally finite
sums D = nj [pj ], where pj ∈ M , and nj ∈ Z. We may also think of them as currents of degree
2 with integer coefficients. Divisors D1 and D2 are called equivalent if there exists a meromorphic
function f ∈ M∗ (M ) such that D1 − D2 = (f ), the divisor of f .
Example 5.1. Let φ1 , φ2 ∈ M1∗ (M ). Then (φ1 ) − (φ2 ) = (φ1 /φ2 ), φ1 /φ2 = f ∈ M∗ (M ). Thus,
divisors of meromorphic 1-forms belong to an equivalence class, which is called the canonical class
of the surface M , and denoted by K = KM .
A divisor D is called positive, (D ≥ 0), if all the coefficients nj ≥ 0. Any
P divisor can be
expressed as the difference of two positive divisors, D = D+ − D− , further, if
nj [pj ] is reduced
(i.e., all pj are different), then we may assume that D+ and D− (i.e., the points where they are
not zero) are disjoint. Such a decomposition will be called reduced, and we assume that both the
divisor, and the decomposition are reduced.
P
On a set of divisors there is a well defined linear function deg D =
nj , which is constant on
classes for compact Riemann surfaces, because deg(f ) = 0 for any f ∈ M∗ (M ) (see Section 4.3).
The degree of of the canonical divisor will be computed in later in this chapter.
With every divisor on M there is an associated space
OD = {0} ∪ {f ∈ M∗ (M ) : (f ) + D ≥ 0}.
Date: March 31, 2009.
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If D = D+ − D− is a reduced decomposition into positive divisors, then for functions in OD we
have the following: (i) at points in the support of D+ there may exist poles of order at most nj ,
and (ii) at points in the support of D− there must be zeros of order at least |nj |.
The dimension (over C) of the space OD is traditionally denoted by l(D).
Example 5.2. Let M be a compact Riemann surface. If deg D < 0, then for any f ∈ M∗ (M ),
deg((f ) + D) = 0 + deg D < 0.
It follows that l(D) = 0. If D = 0, then OD = C, and l(D) = 1.
Lemma 5.1. If D1 ∼ D2 , then l(D1 ) = l(D2 ). Further, l(D) > 0 if and only if D is an effective
divisor, i.e., if D is equivalent to a positive divisor.
Proof. The first assertion is obvious: if D2 = (f ) + D1 , then OD2 = {f1 f : f1 ∈ OD1 }.
As for the second assertion, if D ≥ 0, then l(D) > 0, since OD contains constants. If D+(f ) ≥ 0,
for some f ∈ M∗ (M ), then OD contains the one dimensional space of functions of the form cf ,
c ∈ C.
On the other hand, if l(D) > 0, then there exists a non-zero f ∈ OD . If f =const, then in
the given decomposition D = D+ − D− , the divisor D− must be zero (since f does not have any
zeros). If f 6=const, then (f ) + D ≥ 0 by the definition of OD , and again, D is equivalent to a
positive divisor.
5.2. The Riemann-Roch theorem. Let M be a compact Riemann surface, and let {p1 , . . . , pd }
be a finite subset of M , zj be the local coordinates centred at pj , and fj = cj /zj , j = 1, . . . , d. The
goal is to construct a meromorphic function on M with these principal parts. By Proposition 4.7,
the sufficient and necessary condition for solvability of the problem is the condition
d
X
cj respj
1
Ω1 (M ).
αk
= 0,
zj
(5.1)
where α1 , . . . , αg form a basis in Ω1 =
If g = 0, then the condition is vacuous, and the
problem always has a solution, therefore, we assume here that g > 0. Let α(pj ) = (α/dzj )(pj ).
Then (5.1) can be written as a system of g homogenous linear equations on the coefficients cj :
d
X
cj αk (pj ) = 0.
1
The dimension of the space of the solutions of the system equals d − r, where r = rk(αk (pj )), the
rank of the matrix. As before, we use the fact that d − r ≥ d − g. Let us now take a closer look
P
at this. Assume first that D = d1 [pj ], i.e., the Mittag-Leffler data has only simple poles.
Denote by C a column of cj . Then C 7→ (αk (pj )) · C is a map into an r-dimensional subspace,
Cd 7→ Lr ⊂ Cg . Therefore, there exists an invertible g × g matrix A such that ALr = Cr , the
coordinate subspace {cj = 0, for any j > r}. This means that instead of {αk } we may consider
a new basis
 
 
β1
α1
 .. 
 .. 
 .  = A . .
βg
αg
For the new basis, βk (pj ) = 0, for k > r, i.e., all the rows of the matrix (βk (pj )) with the index
k > r vanish.
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Let i(D) = g − r. This is the maximal number of linearly independent α ∈ Ω1 such that
α(pj ) = 0, j = 1, . . . , d. Such α form a linear subspace
Ω1−D = {0} ∪ {α ∈ Ω1∗ (M ) : (α) − D ≥ 0}.
Thus, i(D) = dimC Ω1−D . Non-zero solutions of our system of equations are in one-to-one correspondence with non-constant meromorphic functions of class OD . Therefore, dim OD = d − r + 1,
the dimension of the space of solutions of the system including constants (for now D ≥ 0). Since
d − r = d − g + i(D), and d = deg D, we get the first version of the Riemann-Roch formula:
dim OD − dim Ω1−D = deg D − g + 1.
P
So far this is proved for simple divisors D = d1 [pj ], but the condition on the poles is not essential.
In the Mittag-Leffler problem for functions with poles of arbitrary order nj , instead of the matrix
(αk (pj )), one needs to consider the matrix that for each j has nj columns
αk (pj ),
dnj −1 αk
dαk
(pj ), . . . ,
n −1 (pj ),
dzj
dz j
j
and repeat the previous argument.
Let us analyze Ω1−D . By fixing a non-zero holomorphic form α0 with positive divisor K0 , we
observe that Ω1 consists of forms αα0 α0 . Therefore, α ∈ Ω1−D if and only if α/α0 ∈ OK0 −D . Hence,
dim Ω1−D = l(K0 −D) = l(K−D), and we obtain the second version of the Riemann-Roch formula,
l(D) − l(K − D) = deg D − g + 1.
Thanks to Lemma 5.1, this is proved for an arbitrary effective (equivalently positive) divisor,
and therefore, for any divisor with l(D) > 0. Hence, l(K0 − D) − l(D) = deg(K0 − D) − g + 1, if
l(K0 − D) > 0. Apply this to the divisor D = 0 when both formulas hold, since K0 ≥ 0. Adding
them we get a formula for the degree of a canonical divisor
deg K = 2g − 2.
Taking into account the Riemann-Roch formula, we may write symmetrically,
1
1
l(D) − deg D = l(K − D) − deg(K − D),
2
2
from which it follows that it is proved for all D such that l(D) > 0 or l(K − D) > 0. It remains
to check the formula when l(D) = l(K − D) = 0, i.e., that in this case deg D = g − 1.
Suppose d := deg D ≥ g. Let D = D+ − D− . Then d = d+ − d− . From the above,
l(D+ ) ≥ d+ − g + 1 = d + d− − g + 1 ≥ 1 + d− .
Let f1 , . . . , fs be a basis in OD+ . The conditions
(k)
c1 f1 (q) + · · · + cs fs(k)(q) = 0,
for 0 ≤ k ≤ multq D− − 1, at a point q ∈ supp D− , for all such q together give d− homogenous
equations on c1 , . . . , cs . Since s ≥ g + d− , for g > 0 there exists a non-zero solution (cj ).
Therefore, the corresponding function f = c1 f1 + · · · + cs fs vanishes on the support of D−
counting multiplicities. It follows that (f ) + D+ − D− ≥ 0 (as fj ∈ OD+ ), and so l(D) > 0, which
is a contradiction.
If deg D < g − 1, then deg(K − D) ≥ g, which is also a contradiction.
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Thus, we proved the Riemann-Roch formula for surface with g > 0. For the sphere it is
equivalent to the statement that the dimension of the space of polynomials of degree at most d
equals d + 1, the number of coefficients. The proof of that is left as an exercise for the reader.
Corollary 5.1 (Riemann’s inequality). l(D) ≥ deg D−g +1, where the equality holds for deg D >
2g − 2.
Indeed, if deg D > 2g − 2, then deg(K − D) < 0, and therefore, l(K − D) = 0.
Corollary 5.2. If g = 0, then the Riemann surface M is biholomorphic to P1 . If g ≤ 2, then
there exists a two-sheeted holomorphic map M 7→ P1 , i.e., M is a hyperelliptic Riemann surface.
Proof. For g = 0 take D = [p]. Then l(D) ≥ 2, and therefore, there exists a meromorphic function
f with a single simple pole at p. It follows that f : M → P1 is biholomorphic.
If g = 1, then let D = [p1 ] + [p2 ]. Then l(D) ≥ 2, and a nonconstant function f ∈ OD realizes
a ramified two-sheeted cover f : M → P1 .
If g = 2, then dim Ω1 = 2. Let α1 , α2 be a basis. Then deg(αj ) = 2g − 2 = 2, i.e., αj has
two zeros (with multiplicities). Hence, the meromorphic function α1 /α2 is a two-sheeted cover
M → P1 .
5.3. The Weierstrass problem. Given a divisor D, the problem asks for a function f ∈ M∗ (D)
such that (f ) = D.
Theorem 5.1. On a non-compact Riemann surface the Weierstrass problem is always solvable,
i.e., any divisor is a divisor of some meromorphic function.
P
Proof. Let D =
nj [pj ] ∈ Λ′ 2 (M ). We seek f ∈ M∗ (M ) which is a solution of the PoincareLelong equation πi ∂∂ ln |f | = D (see Theorem 4.3). Choose a smooth path γj from pj to infinity
so that d[γj ] = −[pj ]. The decomposition into homogeneous bidegree becomes [γj ] = γj1,0 + γj0,1 =
P
i
2Re γj0,1 . From Section 4.7 there exists a function h ∈ Λ′ 0 such that 2π
∂h = − nj γj0,1 . Thus,
S
h ∈ O(M \ γj ) and the jump of the boundary values of h on γj is h±
j = 2πinj . Hence, in a
coordinate neighbourhood (Uj , zj ) centred at pj ,
h = nj ln zj + hj ,
where hj ∈ O(Uj ). It follows that the function f = eh does not have any jumps on γj , and so
f ∈ M∗ (M ) and (f ) = D.
On a compact Riemann surface there is a necessary condition deg D = 0 for the divisors of
meromorphic functions. When g = 0, it is also a sufficient condition. In the general case there
are further obstructions.
Theorem 5.2 (Abel). The divisor D of degree 0 on a compact Riemann surface M is the divisor
if and only if there exists a 1-chain σ on M such that D = d[σ] and
Rof a meromorphic function
1 (M ).
α
=
0
for
any
α
∈
Ω
σ
P
Equivalently, this can be formulated as follows: D =
([pj ] − [qj ]) is the divisor
PofR a meromorphic function if and only if there exist smooth paths γj from qj to pj such that j γj α = 0
for any α ∈ Ω1 (M ).
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Proof. The proof is the same as that of the above theorem, except that the equation
X 0,1
i
∂h = −
γj
2π
has a solution if and only if the current on the right hand side vanishes on all holomorphic forms
(see Section 4.7).
5.4. Period lattices and Jacobi varieties. Let M be a compact
S Riemann surface of genus
g > 0. Let Γ1 , . . . , Γ2g be a basis of 1-cycles with connected M \ Γj , and let α1 , . . . , αg be a
basis in Ω1 (M ). Then the vectors
Z
Z
αg ∈ Cg , k = 1, 2, . . . , 2g,
α1 , . . . ,
Γk
Γk
are R-linearly independent. Therefore, linear combinations of these vectors with integer coefficients form a discrete lattice Π ⊂ Cg , in particular, there exists a neighbourhood U ∋ 0 in Cg
such that Π ∩ U = {0}. The factor, Jac M := Cg /Π is a compact complex manifold (a torus of
dimC = g), which is called the Jacobi variety of the Riemann surface M .
Proposition 5.1. For an arbitrary fixed point q ∈ M the map
Z p
Z p αg mod Π ∈ Jac M
α1 , . . . ,
M ∋ p 7→
q
q
is a holomorphic embedding.
Proof. The identity
Z
p
αk =
q
Z
p′
αk
q
!
mod Π, k = 1, . . . , g
R
holds if and only if there exists a 1-chain σ such that d[σ] = [p] − [p′ ] and σ αk = 0, for any k. In
this case, by Abel’s theorem, there exists a function f ∈ M(M ) with a single simple pole at the
point p. Then, f : M → P1 is a biholomorphism and g = 0. But this contradicts g > 0.
Let z be a local coordinate centred at p. The Jacobian of the corresponding map from M into
Cg (not mod Π) at this point can be easily computed to be (α1 /dz, . . . , αg /dz)(p) 6= 0. From this
the proposition follows.
Corollary 5.3. Any Riemann surface of genus g = 1 is biholomorphically equivalent to some
torus C/Π.
5.5. The Jacobi theorem.
Theorem 5.3 (Jacobi). Let M be a compact Riemann surface of genus g > 0, and let q1 , . . . , qg ∈
M be some fixed points, not necessarily different, and α1 , . . . , αg a basis in Ω1 (M ). Then for any
g
vector
P R (c1 , . . . , cg ) ∈ C there exist points p1 , . . . , pg ∈ M and paths γj from qj to pj such that
j γj αk = ck , k = 1, 2, . . . , g.
Proof. The existence of such points (pj ) for any (ck ) is equivalent to surjectivity of the map


XZ
M g ∋ (pj ) 7→ 
αk  mod Π ∈ Jac M
j
γj
with arbitrary γj starting from qj .
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In the coordinate charts (Uj , zj ) around arbitrarily chosen pj ∈ M , αk = (αk (pj ) + o(1))dzj ,
and therefore for a path δj in Uj from pj (corresponding to zj = 0) to a point ∆zj ,
Z
1
α → αk (pj ), as ∆zj → 0.
∆zj δj
Hence, the given map M g → Jac M is holomorphic, and the Jacobian matrix of the corresponding
map M g → Cg at the point (p1 , . . . , pg ) equals (αk (pj )). Since det(αk (pj )) 6≡ 0 on M g (exercise),
the C-dimension of the image under the map M g → Jac M equals g. Since the manifolds M g and
Jac M are compact, this map is proper. By the Riemann Proper mapping theorem from several
complex variables, the image of M g must be all of Jac M .
About the uniqueness: If (pj ), (p′j ) ∈ M g give the same vector (c1 , . . . , cg ) ∈ Cg , then by Abel’s
P
theorem, there exists a function f ∈ M∗ (M ) with the divisor (f ) = ([p′j ] − [pj ]). If (f ) 6= 0 (i.e.,
P
(p′j ) is not a permutation of (pj )), then for D = [pj ], OD contains constants and a function
f 6≡const. Hence, strict Riemann’s inequality (Corollary 5.1) holds for D. Such divisors are called
special. From Section 5.2 they are characterized by the condition dim Ω1−D > 0, i.e., there exists
α ∈ Ω1 (M ) \ {0} such that α(pj ) = 0 for all j. The setPof such (pj ) ∈ M g is nowhere dense, and
therefore, for a generic choice of (cj ) ∈ Cg the divisor [pj ] is uniquely defined.
At first sight, the Jacobi theorem is a negative result in the sense that any (ck ) ∈ Cg is an
Pd
obstruction in Abel’s theorem. Its ”positive” content is the number g: if D+ =
1 [pj ] and
d > g, then for the construction of a meromorphic function with the divisor D = D+ − D− , where
P
D− = d1 [qj ] is fixed, d − g points pj , j = g + 1, . . . , d, can be chosen arbitrarily. Jacobi’s theorem
guarantees that there exist some other g points pj , j = 1, . . . , g to compensate the random choice
of d−g points, that is, it guarantees that D is the divisor of some meromorphic function. A rather
transcendental problem of localizing these g undetermined points, or even finding them explicitly,
is essentially at the heart of the Jacobi inversion problem. Nevertheless, in applications, if d > g,
this indeterminacy can be either controlled, or is not crucial.
5.6. Vector bundles and divisors. A standard construction
of a vector bundle over a manifold
S
M consists of exhibiting a trivializing open cover M = Uj , and the transition functions gkj on
non-empty intersections Uk ∩ Uj . The transition functions are the r × r matrices with coefficients
that are holomorphic (smooth, continuous, etc.) on Uk ∩Uj , which further satisfy the compatibility
conditions
F gkj gjk = I, and gkj gjl glk = I, whenever the compositions are defined. On a disjoint
union (Uj × Crwj ) (non-connected complex manifold with a natural projection onto M ) there is a
well-defined equivalence relation: (p, wk ) ∼ (p, wj ) whenever wk = gkj (p)wj (here wj is a column
(wj1 , . . . , wjr )). The quotient space
G
L = (Uj × Crwj )/ ∼
is a holomorphic vector bundle over M of rank r.
Freedom of this construction (especially in the choice of the open cover) is compensated by
π
π′
the equivalence relation of bundles. Holomorphic vector bundles L −
→ M and L′ −→ M are
′
called equivalent, if there exists a biholomorphic map F : L → L that commutes with the
projection, π ′ ◦ F = π, and such that all maps F |Lp : Lp → L′p , p ∈ M are linear. As a rule,
all equivalent bundles are considered equal, and their individual constructions are though of as
different representations of the same bundle. For example, without changing the bundle, the
elements of the trivializing cover can be shrunk without changing the transition functions. The
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Sect. 5
S
cover can be refined, for example, by setting Uj = Ujν , with the identity transition maps on
Ujν ∩ Ujν and leaving the original gkj on Uj ∩ Uk unchanged.
F
The process of local trivialization opposite to the defined above factorization of (Uj × Crwj ) is
far from being unique: if Fj is an arbitrary invertible r × r matrix with coefficients holomorphic
in the corresponding Uj and wj′ = Fj wj , then π −1 (Uj ) is also biholomorphic to Uj × Crw′ , with
j
the transition functions wk′ = (Fk gkj Fj−1 )wj′ . Therefore, the cocycles {gkj } and {Fk gkj Fj−1 } are
considered to be equivalent. It is readily verified now that vector bundles L and L′ with a common
cover (Uj ) are equivalent if and only if the cocycles corresponding to any their trivializations are
equivalent.
Further note, that if (Uj ) is a trivializing cover for L, and (Uν′ ) for L′ , then (Uj ∩ Uν′ ) is their
common trivialization, so establishing the equivalence using cocycles is quite constructive.
Vector bundles of rank r = 1 are called line bundles. In the remaining part of the book we will
deal only with
P line bundles.
Let D =
nj [pj ] be an arbitrary divisor on a Riemann surface M and let (Uj ) be a cover by
non-compact open sets. By the Weierstrass problem (Section 5.3) in each Uj there is a meromorphic function fj with divisor (fj ) = D|Uj . Therefore, fk /fj =: gkj ∈ O(Uj ∩ Uk ), and clearly,
{gkj } is a cocycle. The line bundle defined by this cocycle will be denoted by LD and will be
called the line bundle of the divisor D. By construction, LD admits global meromorphic sections
{fj } in the coordinate representation.
If f˜j are other solutions of the Weierstrass problem, then the functions Fj = f˜j /fj are holomorphic and do not vanish on Uj , and so the cocycle
o
n
g̃kj := f˜k /f˜j = Fk gkj Fj−1
is equivalent to {gkj }. It is clear that change of the cover yields an equivalent bundle, and so LD
is well-defined and depends only on D. Moreover, divisors D and D̃ are equivalent if and only if
the corresponding line bundles LD and LD̃ are equivalent.
Indeed, let (Uj ) be a common trivializing cover of LD and LD̃ , and let gkj = fk /fj , g̃kj = f˜k /f˜j
be the cocycles corresponding to this trivialization.
If D̃ = D + (f ), with f ∈ M∗ (M ), then (f˜j ) = (fj f ), and therefore, Fj := f˜j /(fj f ) ∈ O(Uj )
together with 1/Fj , and
g̃kj = (Fk fk f )/(Fj fj f ) = Fk gkj Fj−1 .
From this it follows that the cocycles are equivalent.
Conversely, if the cocycles are equivalent, then f˜k /f˜j = Fk (fk /fj )Fj−1 , and thus, (f˜j /(fj Fj ) on Uj )
form a meromorphic function f on M , with (f ) = D̃ − D.
5.7. Divisors and vector bundles. Which line bundles can be represented as the bundles of
some divisors? It turns out that all of them.
Proposition 5.2. (i) Any holomorphic line bundle L on a compact Riemann surface has a global
non-zero meromorphic section.
(ii) If D is the divisor of this section, then L = LD .
Proof. (i) If OL 6= 0, then these are global meromorphic (even holomorphic) sections. Let now
OL =S0, (Uj )∞
1 be a trivializing cover, {gkj } be the cocycle of the local trivialization, and p ∈
∞
U1 \( 2 Uj ) (we may assume that this set is non-empty). Then clearly, [p] is a current of bidegree
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Riemann Surfaces
Sect. 5
(1, 1) with the values in L∗ (continuous linear functional on smooth sections of L over M ). By
the Serre duality, HL1,1
∗ = 0, and therefore, there exists a current S of bidegree (1, 0) with values
∗
in L such that [p] = ∂S. Thus, S is a holomorphic L∗ -valued form on M \ {p}, which has a
simple pole at p. From the results of Section 4.14, there exists a meromorphic form φ on M with
a unique pole of order 2 (and residue 0) at p. By the definition of forms with values in bundles
∗ = 1/g , and so
(Section 4.8), the quotient S/φ is a global meromorphic section of L∗ . Thus, gkj
kj
s := φ/S is a global meromorphic section of L.
(ii) Let s = {sj } be a coordinate representation and D = (s) be the divisor of s. By the
construction of LD , the transition functions are sk /sj = (φ/Sk )/(φ/Sj ) = gkj . Hence, LD =
L.
Corollary 5.4. If L = LD , then L∗ = L−D .
Corollary 5.5. If s and s̃ are nonzero global meromorphic sections of L, then the divisors (s)
and (s̃) are equivalent.
Thus, the map [D] 7→ [LD ] defines a bijective correspondence between the equivalence classes
of divisors and holomorphic line bundles on a Riemann surface M (on a non-compact Riemann
surfaces both of these sets are trivial, see Exercise 5.21).
Divisors form an additive group, moreover, this operation extends to equivalence classes by
setting [D] + [D̃] = [D + D̃], i.e., the set of equivalence classes of divisors form an abelian group,
which will be denoted by Div M .
If {sj } (resp. {s̃j }) is a section of LD (resp. LD̃ ), then {sj s̃j } is a section of LD+D̃ (the
divisor of the product equals the sum of the divisors). Therefore, the transition functions of
LD+D̃ are {gkj g̃kj }. The bundle with these transition functions is called the tensor product of
L and L̃ (denoted by L ⊗ L̃). With respect to this tensor product, the equivalence classes of
holomorphic line bundles form an abelian group Pic M , which is called the Picard group of M .
Thus, the correspondence [D] 7→ [LD ] between the equivalence classes of divisors and holomorphic
line bundles is an isomorphism of groups, Div M ∼
= Pic M .
In view of this close relationship, we deduce the following
Proposition 5.3. (i) The space of holomorphic sections OL of the bundle L = LD over M is
isomorphic to the space OD of meromorphic functions f on M such that (f ) + D ≥ 0.
(ii) The space Ω1L of holomorphic forms on M with values in L is isomorphic to the space Ω1D
of meromorphic forms φ on M such that (φ) + D ≥ 0.
Proof. (i) Let s = {sj } be a holomorphic section of L, and let {fj } be a meromorphic section
of the bundle LD from its construction (over all M , see above). Then sk = (fk /fj )sj , i.e.,
f = (sj /fj ) in Uj is a global meromorphic function on M , and furthermore, (f )+D = (f )+(fj ) =
(sj ) ≥ 0, i.e., f ∈ OD .
Conversely, if f ∈ OD , then {f fj =: sj } is a global holomorphic section of L.
(ii) If α = {αj } ∈ Ω1L , then φ = {αj /fj } is a meromorphic form on M with the divisor
(φ) = (αj ) − D in Uj . Therefore, (φ) + D ≥ 0. Conversely, if φ ∈ Ω1D , then {φfj =: αj } is a global
meromorphic form with values in L.
5.8. Chern classes. Since any divisor D admits a global meromorphic section LD , we may
try to find analytic relationship between D and sections of LD in the spirit of Poincaré-Lelong
formula. Note that there cannot be a one-to-one correspondence because the vector bundle is not
8
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Sect. 5
determined by the divisor, but rather by its equivalence class in Div M , i.e., the divisor of the
section may be equivalent to D, but not equal to D. The question can be formulated differently:
is it possible the express analytically a divisor of a given meromorphic section in terms of the
section itself, i.e., is there an analog of Poincaré-Lelong formula?
To answer this question we need to have a notion of absolute value of a section. Again, it
cannot be defined canonically, because local trivializations (vertical coordinates) are not uniquely
defined, as already discussed. Therefore, we will use the following commonly used definition. A
metric in a line bundle L with a trivializing cover (Uj ) and cocycle of transition functions {gkj }
is a collection h = {hk } of positive smooth functions hj ∈ C ∞ (Uj ) such that hk = |gjk |2 hj in
Uk ∩ Uj . If (λj ) is a partition of unity subordinate to (Uj ), then we may take hj = Πν |gνj |2λν .
Thus, a metric exists on any line bundle. (Clearly, h is a section of an R-line bundle corresponding
to cocycle |gkj |2 , which is trivial in the class of smooth vector bundles.)
The absolute value (pointwise norm) of a section s = {sj } of a bundle L with metric h is, by
definition, a non-negative function |s|h , the square of which equals |sj |2 hj in Uj .
If the section s global and meromorphic, then according to the Poincaré-Lelong formula in every
Uj the divisor of this section as a current of degree 2 (bidegree (1, 1)) on M can be represented as
i
∂∂ ln |s|h + c1 (h).
π
Here c1 (h) is a smooth 2-form on M such that in Uj ,
(s) =
c1 (h) =
(5.2)
1
∂∂ ln hj
2πi
(this is correctly defined, because ∂∂ ln |gjk | = 0 on Uk ∩ Uj ). It is called the Chern form of the
metric h, and it compensates in (5.2) the free choice of the metric.
On the first sight, randomness in the choice of h should imply randomness of c1 (h). However,
this is not the case. If we compute the degree of (s) (which is equal to [s](1)) in (5.2), we get
Z
Z
i
∂∂ ln |s|h +
deg(s) = [s](1) =
c1 (h).
M π
M
The first integral vanishes by Stokes’ theorem, and the second is [c1 (h)](1). Therefore, deg(s) =
[c1 (h)](1). In particular, if h and h̃ are different metrics on L, then [c1 (h) − c1 (h̃)](1) = 0, and
thus, the form c1 (h) − c1 (h̃) is exact. Thus, the equivalence class of c1 (h) in H 2 (M, R) does not
depend on the choice of h. It is called the first Chern class of the line bundle L and is denoted by
c1 (L). (Since [c1 (h)](1) = deg(s) is an integer, c1 (L) is an integer-valued class, i.e., it is contained
in the image of H 2 (M, Z) (Čech cohomology, which we did not discuss here), under a natural
inclusion H 2 (M, Z) → H 2 (M, R).) The obtained above identity can be rewritten in the form
deg(s) = [c1 (L)](1), and this general value is called the degree of a line bundle, which is denoted
by deg L, or the Chern number, denoted by c1 (L, M ).
Example 5.3.
(1) T 1,0 (M ) is a holomorphic cotangent bundle with the transition function
gkj = dzj /dzk , corresponding to the cover of M by coordinate charts (Uj , zj ). Any holomorphic form α ∈ Ω1 (M ) in its coordinate representation α = αj dzj defines a global holomorphic section {αj } of the bundle, and the other way round. Thus, OT 1,0 M = Ω1 (M ).
Since (α) = K, is the canonical divisor (if α 6≡ 0), then T 1,0 M = LK is called the canonical
line bundle of the Riemann surface M .
9
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Sect. 5
The dual bundle T1,0 (M ) (with the cocycle of transition maps dzk /dzj ) is called a
holomorphic tangent bundle (in the class of
line bundles it is equivalent to the
smooth
dzj
∂
∂
usual tangent bundle of M ). Since ∂zk = dzk ∂zj , the sections of this line bundle are
holomorphic vector fields of type (1, 0), τ = τj ∂/∂zj , in particular, holomorphic vector
fields.
If h = {hj } is a metric in L, then { h1j } is a metric in L∗ . Therefore, c1 (T1,0 M ) =
−c1 (T 1,0 ), and deg T 1,0 M = deg K = 2g − 2, and deg T1,0 M = 2 − 2g. From this we
conclude, in particular, that on a compact Riemann surface of genus g > 1, there are
no nonvanishing global holomorphic vector fields, there must be necessarily zeros or other
singularities.
(2) Metrics on T1,0 M are precisely the conformal Riemannian metrics, which were discussed
in Section 3.3, ρ = ρj (dx2j + dyj2 ) = ρj |dzj |2 in Uj . The corresponding area form equals
ωρ = ρj dxj ∧ dyj =
i
ρj dzj ∧ dz j ,
2
and the Chern form equals
c1 (ρ) =
1
1
∂∂ ln ρj = − ∆j (ln ρj )dxj ∧ dyj ,
2πi
4π
where ∆j is the Laplace operator in Uj . Let us represent this as c1 (ρ) =
the function
∆j ln ρj
Kj = −
2ρj
1
2π Kρ ωρ ,
where
in Uj is the classical Gauss curvature of the Riemannian metric ρ (in terms of local
conformal coordinates (xj , yj ) in Uj ). Since any Riemannian metric on an oriented surface
M is a conformal metric of some complex structure J on M (see Section 3.10), and
deg T1,0 M = 2 − 2g is the Euler characteristic, we derived the Gauss-Bonnet formula
Z
1
Kρ ωρ = χ(M ),
2π M
which is valid for any Riemannian metric ρ on any compact orientable surface M .
5.9. Riemann-Roch theorem for vector bundles. The quantities involved in the RiemannRoch theorem can be also defined for vector bundles. From Section 5.6 for L = LD we have
OD = OL and Ω1−D = Ω1L∗ , since (LD )∗ = L−D . It follows that the left-hand side of the RiemannRoch theorem equals dim OL − dim Ω1L∗ (see the proof of the theorem). By the Serre duality for a
compact Riemann surface M , dim Ω1L∗ = dim HL0,1 (dimension over C, sections are global on M ).
Denote by ∂ L the operator ∂ : Λ0L → Λ0,1
L that acts on the vector space of global sections of L
with the values in the space of smooth forms of bidegree (0, 1) on M (with values in L). Recall
0
that HL0,1 = λ0,1
L /∂ΛL . Then,
dim HL0,1 = codim(im ∂ L = ∂Λ0L ) = dim(coker∂ L ).
For an arbitrary continuous linear map A : L1 → L2 between topological vector spaces with a
finite-dimensional kernel and cokernel, the number dim kerA − dim cokerA is called the index of
A, and denoted by ind A.
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Riemann Surfaces
Sect. 5
In our case, ker∂ L = OL . When M is compact, OL and Ω1L∗ are finite dimensional. (From any
sequence of bounded holomorphic functions
in local coordinates Uj one can extract a subsequence
S
converging uniformly on Vj ⋐ Uj ,
Vj = M ; the cover is finite, and the sections in these
coordinate representations are functions. Thus the vector spaces have compact neighbourhoods
of the origin, and therefore are finite-dimensional.) Hence, the left hand-side of the Riemann-Roch
formula equals ind ∂ L . The right hand-side deg D = c1 (L, M ) and thus, the divisor D is replaced
in the formula by its vector bundle. Thus we get
Theorem 5.4 (Riemann-Roch). Let M be a compact Riemann surface of genus g, L - a holomorphic line bundle of M . Then
ind ∂ L = c1 (L, M ) − g + 1.
In particular, for a trivial bundle L = M × C, ∂ L is the usual ∂ : Λ0 → Λ0,1 , c1 (L, M ) = 0
(Exercise 5.17), and therefore, ind ∂ = 1 − g. This, however, also follows from the classical
formulation of the Riemann-Roch theorem for the divisor D = 0.
In this formulation the theorem emphasizes once again the central role of the operator ∂, and
admits various generalizations, for example, to complex manifolds of higher dimensions, and to
vector bundles of arbitrary rank. The latter can be formulated as follows:
Theorem 5.5. Let L be a holomorphic vector bundle of rank r on a compact Riemann surface
M of genus g. Then
ind ∂ L = c1 (L, M ) − r(g − 1).
(Recall that the Chern class c1 (L) of the bundle with the cocycle of transition functions {gkj }
by definition equals the Chern class c1 (det L) ∈ H 2 (M, R)Rof the line bundle with the cocycle of
the transition functions {det gkl }, and clearly, c1 (L, M ) = M c1 (L).)
Let us explain where the rank comes from. If L is a direct (fibrewise) product of r line bundles
1 are isomorphic to direct sums
Lν (i.e., the transition matrices are diagonal), then OL and OL
∗
1
of the spaces OLν and ΩL∗ν . Applying the Riemann-Roch formula and then taking the sum, we
P
verify the identity c1 (L) = c1 (Lν ) .
In general, any holomorphic vector bundle admits a non-zero global meromorphic section (Exercise 5.14), and as a result it has a holomorphic subbundle of rank one, which can be use to
factorize the bundle and reduce the rank by one. Thus, the proof can be done by induction on
the rank r.
5.10. Embeddings into Pn . Holomorphic vector bundles have the advantage that they enrich
the class of non-singular holomorphic objects (at the expense of some technicalities, which are not
that important, especially in higher dimensions). We illustrate that on the example of holomorphic
embeddings of compact Riemann surfaces into complex projective space.
Let us start with a canonical bundle LK = T 1,0 M , holomorphic sections of which are the usual
holomorphic forms. Let α1 , . . . , αg be a basis in Ω1 (M ), and g > 1. In a domain Uj = {p : αj |p 6=
0} the quotients αk /αj are holomorphic functions. Since Uj cover all M , we have a correctly
defined holomorphic map
F : M → Pg−1 ,
for which
Uj ∋ p 7→ F (p) = (α1 : · · · : αg ) := [(α1 /αj (p), . . . , (αg /αj )(p)] ∈ Pg−1 .
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Riemann Surfaces
Sect. 5
If we choose a different basis in Ω1 (M ), then the corresponding map can be obtained from F by
a linear-fractional (linear with respect to homogeneous coordinates) transformation of Pg−1 , i.e.,
essentially the same map.
Suppose that F (p) = F (q) for some p 6= q, and consider the divisor D = [p] + [q]. The subspace
1
Ω−D ⊂ Ω1 (M ) is defined then just by one condition α|p = 0. This means that dim Ω1−D = g − 1,
and by the Riemann-Roch theorem, dim OD = 2. Therefore, there exists a meromorphic function
f on M with simple poles at points p and q, and holomorphic everywhere else. By Corollary 4.2,
f : M → P1 is a two-sheeted map, i.e., M is a hyperelliptic Riemann surface.
The differential of F at p ∈ (U, z) is degenerate, if
(α/dz)′ (p) = 0 for any α ∈ Ω1 (M ).
For such a point p and divisor D = 2[p], the space Ω1−D is defined with one equation α|p = 0, and
therefore, again, dim OD = 2. Hence, there exists a meromorphic function with a single pole of
order two at a point p, i.e., M is again hyperelliptic. As a result, If M is not hyperelliptic, then
the canonical map
M ∋ p 7→ (α1 : · · · : αg )|p ∈ Pg−1
is a holomorphic embedding.
The image of M under the canonical embedding is called the canonical curve in Pg−1 , and is
independent under biholomorphically equivalent realizations of M : if M ′ is biholomorphic to M ,
then the image of M ′ under the canonical embedding can be obtained from the image of M by a
linear-fractional automorphism of Pg−1 . For details on the situation with hyperelliptic surfaces,
see Exercise 5.9.
For a general line bundle of degree deg L > g the space OL of the global holomorphic sections
has dimension ≥ deg L − g + 1 > 1 (Riemann inequality), and if s0 , . . . , sn ∈ OL , then the map
p 7→ (s0 : · · · : sn )|p
is well-defined and holomorphic everywhere except the basis points of the system, i.e., the points
where all sj = 0. Since sections can be multiplied by meromorphic functions, the condition deg L >
g guarantees the existence of two holomorphic sections without common zeros (Exercise 5.20). By
raising the degree of the bundle, we may choose the sections in such a way that p 7→ [s0 , . . . , sn ]|p
will be an embedding (including hyperelliptic surfaces). Thus, any Riemann surface admits a
holomorphic embedding into Pn .
Instead of sections, one can take usual meromorphic functions, thus embedding first M into
(P1 )n , and then using the fact that (P1 )n can be itself embedded into PN with N = 2n + 1. But
our methods also work in higher dimensions.
For n > 3, the projection from a generic point a ∈ Pn to a⊥ ∼
= Pn−1 of any smooth holomorphic
curve S ⊂ Pn is a smooth (without singularities) holomorphic curve in Pn−1 . We can continue
the process till P3 . Holomorphic tangent lines (spheres) to a smooth holomorphic curve S in P3
form a complex analytic set of dimension 2. Thus, the projection P3 → P2 from any point outside
this set gives an immersion of S into P2 . By changing slightly the centre of the projection we
have: any compact Riemann surface admits a holomorphic embedding into P3 and holomorphic
immersion with simple transversal double intersections in P2 .
Not any Riemann surface can be embedded into P2 (see Exercise 5.21).
For non-compact surfaces the situation is similar: by repeating the standard proof of Whitney’s
embedding theorem, and using Runge’s theorem one can prove that any non-compact Riemann
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Riemann Surfaces
Sect. 5
surface admits a proper holomorphic embedding into C3 , and a proper immersion with simple
singularities into C2 .
5.11. Algebraic curves revisited. By Chow’s theorem any complex analytic set (in particular,
any compact holomorphic curve) in Pn is algebraic, i.e., is the zero set of a system of homogeneous
polynomials. For holomorphic curves (images of compact Riemann surfaces) the proof is quite
simple.
Lemma 5.2. Let f : M → P1 be a meromorphic function of degree n on a compact Riemann
surface, and let h be another, arbitrary meromorphic function on M . Then there exists a Weierstrass polynomial P (z, w) = wn + a1 (z)wn−1 + · · · + an (z) with rational coefficients such that
P (f (p), h(p)) ≡ 0 on M \ {f inite set of singular points}.
Proof. By the definition of degree, the number of solutions of the equation f (p) = a counting
multiplicities equals n independently of a ∈ P1 (see Section 4.3). The set of critical points of f
over C (non-poles where df = 0) is finite. Denote by Σ0 its union with the set of poles of f and g,
and let Σ = f −1 (f (Σ0 )). Then f |M \ Σ → C \ f (Σ) is a holomorphic n-sheeted covering, i.e., for
any point a ∈ C \ f (Σ) there exists a connected neighbourhood U ∋ a such that f −1 (U ) consists
of n pairwise disjoint domains Uj such that the restrictions f |Uj → U is biholomorphic. Let
fj−1 : U → Uj be the inverse mappings. Set wj (z) = h ◦ fj−1 (z), z ∈ U , and construct symmetric
sums
n
X
sν (z) =
wj (z)ν , ν ∈ Z+ .
1
As in Section 1.2, the functions sν are holomorphic in U . If Ũ is another such neighbourhood, then
in U ∩ Ũ the functions sν and s̃ν , constructed above, agree. Thus, each sν defines a holomorphic
function on C \ f (Σ). The set f (Σ) is finite. Since h : M → P1 is a continuous mapping, the
functions sν extend to all of C as continuous maps into P1 . Thus by the removable singularity
theorem of one complex variable, the functions sν are meromorphic on P1 , hence, these are rational
functions.
Let P (z, w) be the Weierstrass polynomial of degree n, the roots of which over z ∈ U are
w1 (z), . . . , wn (z). By Vieta’s and Newton’s theorems, the coefficients of P are polynomials of sν ,
and therefore, they are themselves rational functions of z. If z = f (p) ∈ U , then, by construction,
h(p) = h ◦ fj−1 (z) for some j. Hence, P (f, h) ≡ 0 on M \ Σ.
Lemma 5.3. Let F : M → P2 be a nonconstant holomorphic mapping. Then S = F (M ) is an
algebraic curve.
Proof. We may assume that F (M ) does not contain the point [0, 0, 1] ∈ P2 \ C2 (at infinity). Let
z = z1 /z0 , w = z2 /z0 be affine coordinates in C2 = P2 ∩ {z0 6= 0}. Then f = z ◦ F and h = w ◦ F
are meromorphic functions on M , and moreover, f 6=const. By the lemma above, there exists a
Weierstrass polynomial P (z, w) such that S ∩ C2 = ZP . Since [0, 0, 1] 6∈ S, all the coefficients of
P are polynomials in z. It follows that S is the zero set in P2 of a homogeneous polynomial Q
which is obtained by projectivization of P (see Section 1.6).
Corollary 5.6. Let F : M → Pn be a nonconstant holomorphic mapping. Then S = F (M ) is an
algebraic curve.
13
Riemann Surfaces
Sect. 5
Proof. According to the results proved above, by taking various projections π onto two-dimensional
subspaces we obtain homogeneous polynomials Q in the projections which vanish exactly at the
corresponding projections of S. Then Q ◦ π are homogeneous polynomials in Pn , which vanish on
S. The set of common zeros of such polynomials equals S. (In fact, one can show that to define
S we only need n such Q.)
The general result of this section can be formulated as follows.
Theorem 5.6. Any compact Riemann surface is biholomorphically equivalent to some smooth
algebraic curve in P3 .
5.12. Exercises. In Exercises 5.1 - 5.10 assume that M is a compact Riemann surface of genus
g.
5.1 Prove that if D ≥ 0, then there exists α ∈ Ω1 (M ) such that the support of (α) and D are
disjoint.
Let p1 , p2 , . . . be an arbitrary sequence of points on M . Show that there exists α ∈
Ω1 (M ) such that α|pj 6= 0 for any j.
5.2 Give an example of a canonical divisor on P1 .
5.3 Let f : M̃ → M be a holomorphic n-sheeted mapping, Wf is a divisor df (with respect
to local coordinates), i.e., the divisor of the branching of the map f . Prove the RiemannHurwitz formula
1
g̃ = ng + deg Wf − n + 1.
2
5.4 Let f : M → P1 be a meromorphic function, Wf its divisor of the branching. Prove that
KM = f ∗ (KP1 ) + Wf (the formula for canonical divisors). Derive from this the formula
for deg KM .
5.5 How does the Riemann-Roch formula and the Weierstrass problem look like for P1 . Formulate and prove.
5.6 Let A be a fixed divisor of degree g. Prove that any divisor D of degree 0 has the form
D = B − A + (f ), where B ≥ 0, and f ∈ M∗ (M ).
5.7 The curve Sn = {z0n + z1n + z2n = 0} in P2 for n ≥ 4 is not a hyperelliptic Riemann surface,
i.e., there is no two-sheeted holomorphic covering Sn → P1 .
5.8 Let Γ1 , . . . , Γ2g be the basis of integer-valued 1-cycles on M , and α1 , . . . , αg be a basis in
Ω1 (M ). Prove that the vectors
Z
Z
αg , k = 1, . . . , 2g,
α1 , . . . ,
Γk
5.9
5.10
5.11
5.12
Γk
are R-linearly independent.
(Continuation) Does there exist a neighbourhood
of theorigin in Cg that does not contain
R
R
any linear combinations of vectors
Γk α1 , . . . , Γk αg with integer coefficients, except
zero?
Prove that the Jacobi varieties, constructed from different bases Γ1 , . . . , Γ2g , α1 , . . . , αg ,
and Γ′1 , . . . , Γ′2g , α′1 , . . . , α′g are biholomorphically equivalent.
Prove that on a non-compact Riemann surface any holomorphic line bundle is trivial
Let M be a compact Riemann surface of genus g. Prove that the Wronskian is a section
of a line bundle with the transition functions gkj = (dzj /dzk )N , N = g(g + 1)/2.
14
Riemann Surfaces
Sect. 5
5.13 The weight of a Weierstrass point p ∈ M is called the order of zero of the Wronskian W at
p. Prove that the number of Weierstrass points counting weights on a compact Riemann
surface M of genus g equals (g + 1)g(g − 1).
5.14 Prove that any holomorphic vector bundle (of any rank) on any Riemann surface has a
non-zero meromorphic section. A submanifold L1 ⊂ L of a vector bundle π : L → M is
called a subbundle, if π|L1 → M is a vector bundle, and the inclusion L1p → Lp is linear
for all p.
5.15 Prove that any holomorphic vector bundle of rank r ≥ 1 on a Riemann surface has a
holomorphic subbundle of rank 1.
5.16 Let L be a holomorphic vector bundle on a non-compact Riemann surface. Prove that
(a) L has a global nonzero meromorphic section
(b) L has a global nonzero holomorphic section
(c) L has a global nonvanishing holomorphic section
(d) L is trivial (i.e., equivalent to M × C).
5.17 Prove that for L = M × C the Chern class is trivial, c1 (L) = 0.
5.18 Let c1 (L) ∈ H 2 (M, C) be the Chern class of a holomorphic line bundle L on a compact
Riemann surface M , and φ ∈ Λ2 (M ) be a smooth form from the same class as c1 (L).
Prove that there exists a metric h such that φ = c1 (h).
5.19 Let M be a hyperelliptic Riemann surface of genus g > 1. Prove that its image under the
canonical mapping is biholomorphically equivalent to the Riemann sphere, more precisely,
to the closure in Pg−1 of the holomorphic curve {[1, z, . . . , z g−1 ] : z ∈ C}.
5.20 Let L be a holomorphic line bundle over a compact Riemann surface M , deg L > g. Prove
that there exist holomorphic sections s, s̃ without common zeros on M .
5.21 Prove that hyperelliptic curves of genus g > 1 cannot be biholomorphically embedded into
P2 (but can be embedded topologically!)
5.22 Let M be a compact Riemann surface, and A be a finite subset of M . Prove that M \ A
admits a holomorphic embedding into CN as an affine algebraic curve.
Rasul Shafikov, Department of Mathematics, the University of Western Ontario, London, Canada
N6A 5B7
E-mail address: [email protected]
15