Contents
0.1 Assumed Knowledge ...... 5 0.2 Outline ...... 5
Chapter 1 Sheaves and sheaf cohomology 7 1.1 Sheaves ...... 7 1.2 Sheaves associated with functions ...... 12 1.3 Sheaf cohomology ...... 14 1.4 Vanishing theorems ...... 22 n n 1.5 Cohomology of C and P ...... 24
Chapter 2 Riemann surfaces 27 2.1 Properties of Riemann surfaces ...... 28 2.2 Examples of Riemann surfaces ...... 29 2.3 Cohomology of Riemann surfaces ...... 31 2.4 The Riemann-Hurwitz formula ...... 32 2.5 Hyperellipticity ...... 33
Chapter 3 The classical theorems of Abel and Jacobi 35 3.1 Divisors ...... 35 3.2 The Abel-Jacobi map and the Jacobian variety ...... 38 3.3 Line bundles ...... 41 3.4 Pic(S)...... 45
Chapter 4 Linear systems and the Riemann-Roch theorem 51 4.1 The Riemann-Roch theorem ...... 57 4.2 Application and Examples ...... 59
Chapter 5 Complex tori 65 5.1 Cohomology of complex tori ...... 66 5.2 Line bundles on complex tori ...... 67
1 5.3 Theta functions ...... 72
Chapter 6 The Jacobian Variety 77 6.1 Motivation: Abelian integrals ...... 77 6.2 Properties of the Jacobian variety ...... 79 6.3 Riemann’s theorem ...... 83
Chapter 7 The Torelli theorem 91 7.1 Proof of the Torelli theorem ...... 91
Chapter 8 Concluding remarks 98 8.1 The Schottky problem ...... 98
Chapter 9 Background material 100 9.1 Group cohomology ...... 100 9.2 Major theorems ...... 101
2 Acknowledgements
I would like to extend my gratitude to all those who helped in the production of this thesis, and to those who supported me throughout this year. I thank my supervisor, Dr Daniel Chan, for all of his guidance and assistance. Also to my family and friends, especially Lorraine, who put up with me during all this time. Special thanks go to Prof. Tony Dooley for lending me his copy of [Sha74]. Finally, I would like to thank God, who makes all things possible.
3 Introduction
In this thesis, the principal objects of study are Riemann surfaces, and the aim will be to expound the classical Torelli theorem relating Riemann surfaces to their Jacobians, which are central to their study. Riemann’s original definition in his doctoral thesis [Rie51] amounts to saying that a 1 Riemann surface is an n-sheeted branched cover of P . At that time, Riemann surfaces were merely a convenient way to represent multi-valued functions. Klein took up the subject after Riemann and studied Riemann surfaces via differential geometry as objects in their own right. Weyl formalised Klein’s ideas in his famous monograph Die Idee der Riemannschen Fl¨ache [Wey23]. Today we define a Riemann surface as a (compact) connected one dimensional complex manifold. It is interesting to note that the definition of a complex manifold did not appear in the literature until mid 40’s. The phrase komplexe analytische Mannigfaltigkeit 1 first appeared in Teichm¨uller’s [Tei44], and the English version appears in Chern’s [Che46] in 1946. For more on the history of Riemann surfaces, see Remmert’s delightful recount in [Rem98]. The Jacobian of a Riemann surface S is a complex torus, and in fact, is an abelian variety. Its definition is intrinsic to S, and captures much of its information. Torelli’s theorem states that given a Jacobian of a Riemann surface and an additional piece of data, called the principal polarisation, one can recover the Riemann surface up to isomorphism. The proof which we present follows Andreotti’s [And58]. It is interesting to note that Marten published a new proof of the Torelli theorem [Mar63], which uses combinatorial tech- niques together with Abel’s theorem and the Riemann-Roch theorem. Torelli’s original publication on Jacobians can be found here [Tor13]. In [Mum75], Mumford speaks of the “amazing synthesis” of algebra, analysis, and geome- try that is at the heart of the geometry of algebraic curves. This trichotomy is evident in that complex algebraic curves are in a one to one correspondence with Riemann surfaces, each emphasising different methods used to explore the geometry of these objects. The
1This is German for complex analytic manifold.
4 amazing synthesis goes much further; to quote Mumford again [Mum95], algebraic geom- etry is not an “elementary subject” but draws from, and contributes to, many disparate disciplines in mathematics. So the difficulty for any initiate of algebraic geometry lies in the tremendous amount of background which has to be covered, as well as the depth and breadth of the ideas in algebraic geometry which itself has enjoyed a long and illustri- ous history. This is the vindication for the long list of topics assumed. For a history of algebraic geometry, we refer the reader to Dieudonn´e’sarticle, [Die72].
0.1 Assumed Knowledge
We assume knowledge of very basic differential geometry and complex manifolds, an elementary treatment of complex manifolds can be found in chapter 7 of [Che00]. To give an idea of the depth of knowledge assumed, concepts such as K¨ahler manifolds, Hermitian metrics, differentials, tangent and cotangent bundles will be used without comment. Also assumed is a basic understanding of algebraic geometry, where the relevant back- ground can be found in the notes of a course on algebraic geometry taught by Daniel Chan at UNSW in 2004. The course was based on [Sha74], and the notes, edited by the author, can be found at [CC04]. We will not list the concepts assumed, and explicit references to these notes will be made in the thesis. Another very good source of information for the subset of algebraic geometry associated to the thesis material is [Mum95]. The basic concepts algebraic topology, homological algebra, and category theory are also assumed. Again to give some idea of what is assumed, the following concepts will be used without digression; categories, functors, cochain complexes, exact sequences, Poincar´e duality, the Euler characteristic, the Mayer-Vietoris sequence, and simplicial, de Rham, and Dolbeault cohomology. For an exposition of these concepts we refer the reader [Hat02] for algebraic topology, and [Osb00] for homological algebra.
0.2 Outline
This section provides an outline for the development of the material. The chapters should be read in sequence to maintain coherence. In chapter 1, we begin with sheaf theory and their cohomology. The reason for beginning with this technical topic is that the chapters which follow employ extensively the language and techniques of sheaves and sheaf cohomology. The definition of coherent sheaves can be found in [Uen01]. Chapters IX and X of [Mir95] contains a very clear exposition on sheaves. Pages 11-18 of [EH00] contain a basic introduction to sheaf theory.
5 From chapter 2 onwards, the main reference will be [GH78], which is a very comprehensive treatment of algebraic geometry from an analytic perspective. In chapter 2, we introduce Riemann surfaces, and derive some of their selected properties which will be used in proving the Torelli theorem. The differences between hyperelliptic and non-hyperelliptic Riemann surfaces are discussed. Kirwan’s book [Kir92] is elemen- tary in its treatment; [Cle80] contain many interesting examples, but assumes previous knowledge in many areas. The topic of Riemann surfaces are thoroughly developed in both [Mir95] and [FK92]. Chapter 3 and 4 develop more advanced material concerning Riemann surfaces. The Abel and Jacobi theorems are first discussed, then the concepts of divisors, line bundles are introduced. Linear systems are explored in chapter 4, with an emphasis on linear systems on Riemann surfaces, culminating in the Riemann-Roch theorem, which is a formula for the dimension of a linear system on a Riemann surface. Complex Tori are discussed at length in chapter 5, in anticipation to the discussion of the Jacobian variety in chapter 6. The line bundles on complex tori are classified, and the theta functions are obtained from global sections of such line bundles. The references for this chapter are [Pol03] and [GH78]. The next chapter on the Jacobian variety applies the results for complex tori, culminating in Riemann’s theorem. The penultimate chapter gives the proof of the Torelli theorem, using much of material developed above. Finally, we end with some concluding remarks regarding the Torelli theorem.
6 Chapter 1
Sheaves and sheaf cohomology
Algebraic geometry was transformed by Serre in the 1950’s by his introduction of sheaf theoretic techniques [Ser55]. The main reason for using sheaves in this thesis is to access sheaf cohomology, which is, as we shall see, a powerful and concise technique. We seek to emphasis this approach as much as possible. This chapter contains the elementary definitions and theorems of sheaves and their cohomology. For a full treatment of sheaves in the context of algebraic geometry, see [Uen01].
1.1 Sheaves
The machinery of sheaves allows one to organise local information and extract global properties of a topological space, X. A sheaf associates algebraic data to each open set of X, and does so functorially. Let X denote a topological space; we first define a presheaf over X.
Definition 1.1 A presheaf F of abelian groups over X is a contravariant functor from the category of open sets of X, where the morphisms are given by the inclusion maps, to the category of abelian groups Ab, where the morphisms are given by the group homomor- phisms. That is, for every pair of open sets V and U such that V ⊂ U, we have the restric- tion homomorphism ρU,V : F(U) −→ F(V ). Moreover these restriction homomorphisms satisfy
1. ρU,U = idU for all U, and 2. that the following diagram commutes for all open sets W ⊂ V ⊂ U
ρU,V ρV,W F(U) / F(V ) 3/ F(W ) ρU,W
For concision, we will often write ρU,V (σ) = σ|V for σ ∈ F(U).
7 Presheaves of rings or vector spaces 1 can be analogously defined. When we speak about sheaves in general, we will always refer to sheaves of abelian groups. The presheaf organ- ises local information and stipulates they are consistent. The presheaves form a category with the following
Definition 1.2 Let F and G be presheaves over X.A presheaf morphism α : F −→ G is a collection of group morphisms αU : F(U) −→ G(U) for every open set U ⊂ X, such that for every pair U ⊂ V of open sets in X the following diagram commutes
αU F(U) / G(U)
ρU,V ρU,V
αV F(V ) / G(V )
We need an extra patching condition to define a sheaf.
Definition 1.3 A presheaf F on X is a sheaf if X satisfies the sheaf condition: for every open set U ⊂ X, let {Ui}i∈I be an open cover of U. If the collection σi ∈ F(Ui), i ∈ I 2 satisfies σi|Uij = σj|Uij for all i, j ∈ I, then there exists a unique σ ∈ F(U) such that
σ|Ui = σi for all i ∈ I. We call the elements σ ∈ F(U) sections of F over U. If U = X we call σ a global section.
Definition 1.4 Let F and G be sheaves over X.A sheaf morphism α : F −→ G is defined to be the presheaf morphism α : F −→ G.
In particular, let {Ui}i∈I be an open cover of X; then to check that σ = τ where σ, τ ∈
F(X), it suffices to check that σ|Ui = τ|Ui for all i ∈ I. Note that a presheaf is a priori not a sheaf, so the sheaf condition is not vacuous. We give the following example of a presheaf which is not a sheaf.
Example 1.5 Let Z be the presheaf of constant functions on a topological space X, that is, for every open set U ⊂ X,
Z(U) = {f : U −→ Z | f is constant}.
Suppose X has two connected components, X1 and X2. Let {Ui}i∈I be any open cover of X and we take a refinement such that Ui ⊂ X0 or Ui ⊂ X1 for all i ∈ I. Then the 1The notion of sheaves of modules requires more explanation, see [Uen01]. 2 Note that Uij = Ui ∩ Uj . We will keep this notation throughout.
8 collection of sections σi ∈ Z(Ui) where 0 if Ui ⊂ X0 σi = 1 if Ui ⊂ X1
satisfies σi|Uij = σj|Uij for all i, j ∈ I. However, there exists no global section σ such that
σ|Ui = σi, since if such a σ exists
0 if Ui ⊂ X0 σ|Ui = (1.1) 1 if Ui ⊂ X1 contradicting the fact that σ is constant on X. In other words, there is no section which is constant on X and which agrees with the value of σi in each Ui. We see that Z is not a sheaf, and that a possible remedy is the addition of ‘extra’ sections. This is accomplished by allowing locally constant functions. Denote
0 Z(U) = {f : U −→ Z | ∀p ∈ U, ∃ an open set U ⊂ U such that f|U 0 is constant} for each open U ⊂ X 3, to be the presheaf of locally constant functions on X. Then we 0 if x ∈ X0 see that in (1.1), σ(x) = is locally constant, and hence σ ∈ Z(X). 1 if x ∈ X1 Given a presheaf morphism α : F −→ G of presheaves over X, define the kernel of α, cokernel of α, and image of α to be the corresponding presheaves,
ker(α)(U) := ker(αU : F(U) −→ G(U))
im(α)(U) := im(αU : F(U) −→ G(U))
coker(α)(U) := G(U)/ im(αU ) for all open sets U ⊆ X. That the above define presheaves follow from the definition of presheaves.
Proposition 1.6 Let F and Fe be sheaves over X and α : F −→ Fe be a sheaf morphism. Then the presheaf ker(α)(U) is a sheaf.
Proof It suffices to check the sheaf condition. Let {Ui}i∈I be an open cover of U, and
σi ∈ ker(α)(Ui) satisfying σi|Uij = σj|Uij for all i, j ∈ I. Now since F is a sheaf, consider
3This is standard notation, so unfortunately the burden is on the reader to remember that this is the sheaf of locally constant functions with values in Z, not the ring of integers, Z. However, the context should eliminate any ambiguity.
9 σi as elements of F(Ui), so there exists a unique σ ∈ F(U) such that σ|Ui = σi for all i ∈ I. It remains to show that σ ∈ ker(α)(U). Consider the following commutative diagram
αU ker(αU ) / F(U) / F˜(U)
ρ ρ U,Ui U,Ui ρ˜U,Ui αU i ˜ ker(αUi ) / F(Ui) / F(Ui)
for all i ∈ I. Hence αU (σ)|Ui = αUi (σ|Ui ) = αUi (σi) = 0 for all i ∈ I, so by the sheaf condition on Fe, αU (σ) = 0, that is σ ∈ ker(α)(U). 2 However the presheaves im(α) and coker(α) need not be sheaves. To define cokernels in the category of sheaves, we need the sheafification construction. First consider an open set U ⊂ X and an open cover {Ui}i∈I of U.
Definition 1.7 Let F be a presheaf of abelian groups over X, U ⊂ X be any open set, and {Ui}i∈I be an open cover of U. Define
+ Y Y F (U) = ker F(Ui) ⇒ F(Uj ∩ Uk) i (j,k) for all open sets U ⊂ X and all ; where
Y Y F(Ui) ⇒ F(Uj ∩ Uk) i (j,k)
(σj)i∈I 7−→ (σi|Uij − σj|Uij )i,j∈I .
Then the sheafification of F, denoted sheaf(F), is defined as the sheaf F ++ together ϕ with the canonical morphism F −→ sheaf(F).
ϕ Proposition 1.8 The sheafification F −→ sheaf(F) satisfies the following universal prop- erty. Let F be a presheaf, G be a sheaf and α : F −→ G be a presheaf morphism. Then there exists a unique sheaf morphism α˜ such that the following diagram commutes
α F / G . HH v; HH vv HH vv ϕ HH vvα˜ H$ vv sheaf(F)
Example 1.9 Returning to example 1.5, Z = sheaf(Z). Moreover Z(X) is a free abelian group with its number of generators equal to the number of connected components of X.
10 The kernel, cokernel and image of a sheaf morphism α : F −→ G are defined to be the respective sheafifications of the kernel, cokernel, and image of α considered as a presheaf morphism.
Note 1.10 We see that the category of presheaves and the category of sheaves are abelian categories, which roughly means a category where kernels and cokernels are well-defined for any of its morphisms. As a result, exact sequences are well defined in abelian categories.
Definition 1.11 Suppose
αn−1 αn ... −→ Fn−1 −→ Fn −→Fn+1 −→ ...
is a sequence of sheaves over X. Then we say that the sequence is exact at Fn if αn−1 ◦
αn = 0 and ker(αn) = im(αn−1). We say the sequence is exact if it is exact at each Fk.
An important instance of an exact sequence of sheaves is the short exact sequence. For this we need the concept of a zero sheaf. This is simply the assignment 0(U) = 0 for all open sets U.
Example 1.12 Let ϕ ψ 0 −→ F −→G −→H −→ 0 be an exact sequence of sheaves over X. We call this a short exact sequence and we see that ker(ψ) = F and coker(ϕ) = H. In this case, we say F is a subsheaf of G and H is the quotient sheaf of G with F, denoted G/F.
Now given a presheaf F over X we can define a functor from the category of presheaves to the category of abelian groups by the assignment Γ : F 7−→ F(X). This is called the global sections functor. The definition for the global sections functor in the category of sheaves is identical.
Definition 1.13 Suppose A and A0 are abelian categories and
ϕ φ 0 −→ A −→ B −→ C −→ 0 is an exact sequence in A. Then a functor F :A−→A0 is said to be exact if the sequence
F (ϕ) F (φ) 0 −→ F (A) −→ F (B) −→ F (C) −→ 0
11 is exact in A0; and left exact if
F (ϕ) F (φ) 0 −→ F (A) −→ F (B) −→ F (C) is exact in A0.
Note that right exactness of a functor is defined analogously. The important point here is that the categories of sheaves and presheaves have the same notion of morphisms, but not the same notion of cokernels. A consequence of this is the following
Proposition 1.14
1. The global sections functor is an exact functor from the category of presheaves to abelian groups. 2. The global sections functor is a left exact functor from the category of sheaves to the category of abelian groups. In particular, it is not exact.
The first part of the above definition is by the definition of a presheaf. To see the second part, we will produce examples to show that the global sections functor in the category of sheaves is not exact. It turns out that this is the reason why there is the need for a cohomology theory for sheaves. The discussion of cohomology of sheaves continue in section 1.3.
1.2 Sheaves associated with functions
There is often a distinguished class of functions over X which we are interested in.
Definition 1.15 Let M be a complex manifold. The assignment U 7−→ {f : M −→ C | f holomorphic} for every open set U ⊆ M is called the structure sheaf and is denoted
OM , or O when there is no ambiguity.
We have defined the structure sheaf to be the sheaf of holomorphic functions. However, this need not always be the case. For instance, in algebraic geometry over an arbitrary
field K of characteristic 0, one may defines the structure sheaf to be O(U) := {f : U −→ 4 n K | f rational} where U ⊂ V is open and V is a projective variety in P . The following example collects some frequently occurring sheaves in complex geometry.
4See [CC04] for definition.
12 Example 1.16 We have met some of these previously. The following are related to OM as they depend on the analytic structure of M, these are as follows
∗ OM sheaf of nonvanishing holomorphic functions on M
KM sheaf of meromorphic functions on M K∗ sheaf of meromorphic functions on M not identically zero M . Ωk sheaf of holomorphic k-differentials on M Ωp,q sheaf of holomorphic differentials of type (p, q) ∂ O(L) sheaf of sections of holomorphic line bundle L
We shall adopt the convention that Ω0 = O. We also have
C∞ sheaf of smooth functions on M Ap,q sheaf of smooth differentials of type (p, q) . ∂ Ak sheaf of smooth k-differentials
Finally we have the locally constant sheaves which are related to the topological structure of M, these are Z, R, and C for the sheaves of locally constant functions M −→ Z, R, C.
Example 1.17 Let M be a compact complex manifold. A very important short exact sequence is the following ι exp ∗ 0 −→ Z −→O −→O −→ 0, called the exponential sequence. The map ιU : Z(U) −→ O(U) is simply inclusion, 2πif(z) and expU : O(U) −→ O(U) is given by (expU (f))(z) = e for z ∈ U.
Moreover this sequence is exact. Firstly (ιU ◦ expU )(f)(z) = exp(2πif(z)) = 1 since f is a locally constant function taking integer values. Now
∗ ker(exp)(U) = ker(expU : O(U) −→ O (U))
= Z(U) so ker(exp) = Z as sheaves. Finally to show exp is surjective as a sheaf map, we show exp has a local inverse. That is, for every g ∈ O∗, and every p ∈ M, there exists an open neighbourhood of p such that the equation exp(2πif)(z) = g(z) has a solution: namely 1 2πi log(g(z)), which is holomorphic on some neighbourhood of p chosen to not contain any branch cuts of log(g(z)).
13 1.3 Sheaf cohomology
We begin with a sketch of why one studies sheaf cohomology. Firstly, sheaf cohomology replicates important instances of classical cohomology, in particular, we will see that H•(X, ), H•(X, ) and H•(X, Ωp,q) correspond to simplicial, de Rham and Dolbeault Z R ∂ cohomology respectively. As with classical cohomology, one of the aims is to formulate algebraic invariants for topological spaces. Sheaf cohomology, in general, allows one to do so with arbitrary sheaves and in this way generalises classical cohomology theories. The second reason, as alluded to above, is the fact that the global sections functor from the category of sheaves to the category of abelian groups is not exact. Experience shows that exact sequences are a natural and concise way to express certain facts in mathematics. An exact sequence of sheaves, say 0 −→ F −→ G −→ H −→ 0, over X generally correspond to some property holding locally, while 0 −→ F(X) −→ G(X) −→ H(X) −→ 0 correspond to the same property holding globally on X. Hence the obstruction to exactness of the global sections functor correspond to the obstruction to passing from local properties to global properties. We give an example to illustrate this.
Example 1.18 Let X = C−{0}. Applying the global sections functor to the exponential sequence ι exp ∗ 0 −→ Z −→O −→O −→ 0 (1.2) over X, we obtain the left exact sequence
exp ∗ 0 −→ Z(X) −→ O(X) −→O (X) of C-vector spaces. The exponential map in the second sequence is not surjective, since z ∈ O∗(X) is not in the image of exp. We can interpret (1.2) as saying that exp is only locally invertible, but does not have a holomorphic inverse on all of X.
We will briefly sketch the derived functor approach to sheaf cohomology, which measures the obstruction to exactness of the global sections functor. No proofs will be given below, see [Uen01] for details.
Definition 1.19 A sheaf R over X is said to be flasque if the restriction map R(X) −→ R(U) is surjective for all open sets U ⊂ X.A flasque resolution of a sheaf is a sequence
0 −→ G −→ R1 −→ R2 −→ R3 −→ ... (1.3)
such that R1, R2,... are flasque sheaves over X and (1.3) is exact.
14 A flasque resolution exists for any sheaf F and in fact the flasque resolution is canonical. Let G be a sheaf over X, and Γ be the global sections functor. Further let
δ1 δ2 δ3 0 −→ G −→ R1 −→R2 −→R3 −→ ...
• be the canonical choice of flasque resolution for G and denote R := 0 −→ R1 −→ R2 −→ .... Apply the functor Γ to obtain
δ1X δ2X δ3X 0 −→ G(X) −→ R1(X) −→R2(X) −→R3(X) −→ ... which is a cochain complex of abelian groups. We define the i-th cohomology group of X with coefficients in G to be
ker(δ ) Hi(X, G) := Hi(F (R•)) = iX im(δi−1X )
We have skipped most of the details in the above sketch, the point is to see that sheaf cohomology does in fact measure the obstruction to exactness of Γ. The is called the derived functor approach as it is a special case of such a construction in homological algebra (c.f. [Osb00]). The above constitutes the conceptual scaffold, but it is not a computable theory. We will approach sheaf cohomology via Cechˇ cohomology, which is an alternative, and computable way of doing sheaf cohomology5.
Definition 1.20 Let U := {Uα}α∈A be a locally finite cover of X. For every multi-index T ˇ I = {i0, . . . , ik} ⊆ A, denote UI = i∈I Ii. Define the Cech complex to be
C•(U, F) := 0 −→ C0(U, F) −→ C1(U, F) −→ C2(U, F) −→ ...
k k Q where C := C (U, F) := |I|=k+1 F(UI ). k Q • We call an element σ ∈ C (U, F) a k-cochain and write σ = |I|=k+1(σI ) =
(σI )|I|=k+1 with σI ∈ F(UI ). k k+1 Q • The coboundary map, δ : C −→ C is given by δσ = |J|=j+2(δσ)J where
k+1 X i (δσ)J = (−1) σJ−{j } i UJ i=0
and we call τ ∈ δCk+1 a coboundary.
5The Cechˇ cohomology groups agree with the derived functor cohomology groups for a quasi-coherent sheaf over a separated Noetherian scheme.
15 • An element σ ∈ ker(δ) is called a cocycle. • The p-th Cˇech cohomology group of F is the direct limit (see [Osb00] for defi- nition)
Hp(X, F) = limHp(U, F) −→ U
p ker(δ:Cp−→Cp+1) ˇ where H (U, F) = im(δ:Cp−1−→Cp) is the p-th cohomology group of the Cech complex C•(U, F).
The direct limit which appears in the definition of a Cechˇ cohomology group defies com- putation. Leray’s theorem tells us when the open cover U of X is ‘good enough’ such that Hp(U, F) = Hp(X, F).
Theorem 1.21 (Leray’s theorem) Suppose F is a sheaf over X and U = {Ui}i∈I is an ˇ q open cover of X such that for some integer p, the Cech cohomology groups H (Ui1,...,ip ) vanish for all q > 0 and for all i1, . . . , ip ∈ I. Then
H•(U, F) = H•(X, F)
Proposition 1.22 There is a natural isomorphism of vector spaces F(X) ' H0(X, F).
Proof Let U = {Uα}α∈A be an open cover for X. Then
H0(U, F) = ker(δ : C0(U, F) −→ C1(U, F))/{0}
0 Q For σ ∈ C (U, F) = α∈A F(Uα), (δσ)α,β = σβ|Uαβ − σα|Uαβ for any α, β ∈ A. The 0 1 condition σ ∈ ker(δ : C −→ C ) holds iff σβ = σα on Uα ∩ Uβ for all α, β ∈ A. This is equivalent to σ ∈ F(X). Taking the limit, limH0(U, F) = H0(X, F) we have −→ U F(X) ' H0(X, F). 2 Note 1.23 The zeroth Cechˇ cohomology groups agree with the zeroth derived functor sheaf cohomology groups.
Note 1.24 Some authors denote the Cechˇ cohomology groups Hˇ p(X, F), but since most of our discussion will involve Cechˇ cohomology, I will simply denote them Hp(X, F). Classical cohomology groups will be distinguished by the appropriate subscripts. For instance Hp ,Hp,Hp , will denote the de Rham, Dolbeault and simplicial homology DR ∂ simplicial groups respectively, which appear in proposition 1.27.
16 The following basic fact from homological algebra produces a most useful corollary. Its name derives from the shape of the accompanying long exact sequence diagram, as shown below.
Proposition 1.25 (The snake lemma) Let C•,D•,E• be cochain complexes in an abelian category A and suppose the sequence
ψ ϕ 0 −→ C• −→ D• −→ E• −→ 0 is exact. Then this induces the long exact sequence in cohomology
ψ∗ ϕ∗ ... / Hi(C•) / Hi(D•) / Hi(E•) (1.4) c
ψ∗ ϕ∗ BCED / Hi+1(C•) / Hi+1(D•) / Hi+1(E•) / ... GF@A the map c is called a connecting homomorphism.
k • k k+1 Proof Denote Z (A ) := ker(d : A −→ A ) for k ∈ N and similarly for B and C. The exact sequence of cochain complexes 0 −→ A• −→ B• −→ C• −→ 0 is expanded into the two-dimensional complex
O O O
ψ ϕ 0 / Ci+1 / Di+1 / Ei+1 / 0 O O O δ ∂ d ψ ϕ 0 / Ci / Di / Ei / 0 O O O δ ∂ d ψ ϕ 0 / Ci−1 / Di−1 / Ei−1 / 0 O O O
where the rows are exact. We first define the maps ψ∗, ϕ∗ and c in (1.4). The induced maps ψ∗, ϕ∗ in cohomology are given by
ψ∗ ϕ∗ Hi(A•) −→ Hi(B•) and Hi(B•) −→ Hi(C•) [a] 7−→ [ψa][b] 7−→ [ϕb] where [x] ∈ Hi(X•) denotes the cohomology class of x ∈ Zi(X•), X = A, B, C.
17 Let a ∈ Zi(A•); by commutativity of the top left square, ∂ψa = ψδa = ψ0 = 0, we obtain ψa ∈ Zi(B•). Moreover, suppose a ∈ δAi−1, that is a = δa0 for some a0 ∈ Ai−1. Applying ψ and by commutativity of the bottom left square we obtain ψa = ψδa0 = ∂ψa0, so ψa ∈ ∂Bi−1. Hence ψ∗, and similarly ϕ∗, are well-defined maps in cohomology. Let γ ∈ Zi(C•); the connecting homomorphism c is given by
Hi(C•) −→c Hi+1(A•)
[γ] 7−→ [aγ] where aγ is defined below. Let us recall that exactness of rows in the two-dimensional complex above means that ψ is injective, ker(ϕ) = im(ψ), and ϕ is surjective. Also recall that along the columns, the maps δ2, ∂2, d2 are the zero maps. The following diagram will keep track of the various maps and choices in the following paragraphs.
ψ Ai+2 3 0 / 0 . O O
_ ψ _ ϕ aγ / ∂b / 0 O O
_ ϕ _ b / γ ∈ Ci
Since ϕ is surjective, we can choose b ∈ ϕ−1{γ} ⊂ Bi, and since ker(ϕ) = im(ψ), ∂b ∈ −1 im(ψ). Hence ψ {∂b} 6= ∅, in fact, ψ is injective, so there is a unique choice of aγ ∈ ψ−1{∂b}. Moreover, since ∂2b = 0, the top right hand corner is zero since ψ is injective, i+1 • and by commutativity of the top right hand square, δaγ = 0, that is, aγ ∈ Z (A ). We show that choosing a different b0 ∈ ϕ−1{γ} in the bottom row, middle position changes 0 −1 0 aγ by a coboundary. Now b − b ∈ ker(ϕ) = im(ψ), so ψ {b − b } 6= ∅. Choose a ∈ −1 0 ψ {b − b } so aγ changes by a coboundary, namely δa. 0 Finally we show that choosing a different representative γ ∈ [γ] does not change aγ. Let γ0 ∈ Zi(C•) such that γ − γ0 ∈ dCi−1. We use the diagram
ψ Ai+1 3 0 / 0 O O
_ ψ _ ϕ a00γ / ∂b00 / γ − γ0 O O
_ ϕ _ b00 / γ00 ∈ Ci−1
18 to keep track of the arguments. Choose γ00 ∈ d−1{γ − γ0} ⊂ Ci−1 and b00 ∈ ϕ−1{γ00}. Following through the rest of the diagram in much the same manner as above, we obtain aγ−γ0 = 0. This shows that c is a well-defined map in cohomology. Exactness at Hi(B•): Let [a] ∈ Hi(A•) , then (ϕ∗ ◦ ψ∗)[a] = ϕ∗[ψa] = [(ϕ ◦ ψ)(a)] = [0], so ker(ψ∗) ⊃ im(ϕ∗). For the converse, we use the following to keep track of arguments.
ψ Ai+1 3 0 / 0 O O
ϕ _ ψ _ ϕ Ai 3 b / ϕb a / ∂b0 − b / 0 ∈ Ci O
ϕ _ b0 / γ ∈ Ci−1
Suppose [b] ∈ ker(ϕ∗), that is ϕb ∈ dCi−1. So choose γ ∈ d−1{ϕb} ⊂ Ci−1, and since ϕ is surjective, choose b0 ∈ ϕ−1{γ} ⊂ Bi−1. Now ϕ∂b0 = ϕb so ∂b0 −b ∈ ker(ϕ) = im(ψ), hence choose a ∈ ψ−1{∂b0 − b}. Now a is a cocycle, that is δa = 0, since δa = δψ(∂b0 − b) = ψ∂(∂b0 − b) = ψ(∂2b0 − ∂b) = 0 since b is a cocycle. Hence ψ∗[a] = [b], so [b] ∈ im(ϕ∗) and ker(ψ∗) = im(ϕ∗). Exactness at Hi(C•): Let [b] ∈ Hi(B•), then (c ◦ ϕ∗)[b] = c[ϕb], now b ∈ Zi(B•), ∗ so aϕb = 0 by the definition of c above, and ker(c) ⊃ im(ϕ ). For the converse, let i • i−1 −1 [γ] ∈ ker(c) ⊂ H (C ), that is aγ ∈ δA . So let a ∈ δ {aγ}, as follows
ψ Ai+1 3 aγ / ∂b O O
_ _ ϕ a b / γ ∈ Ci where b ∈ ϕ−1{γ}. Now ∂ψa = ∂b so ψa − b ∈ Zi(B•). Moreover ϕ(ψa − b) = 0 + γ, hence ϕ∗[ψa − b] = [γ], and we have proved ker(c) = im(ϕ∗). i+1 • i • ∗ Exactness at H (A ): Let [γ] ∈ H (C ), then (ψ ◦c)[γ] = [ψaγ], but ψaγ is by definition a coboundary, so (ψ∗ ◦ c)[γ] = 0, and ker(ψ∗) ⊃ im(c). Conversely, let [a] ∈ ker(ψ∗) ⊂ Hi+1(A•), that is ψa ∈ ∂Bi. So let b ∈ Bi such that ∂b = ψa then ϕb ∈ Zi(C•) since dϕb = ϕψa = 0. So c[ϕb] = [a]. Thus ker(ψ∗) = im(c). The following diagram sums up the above paragraph.
ψ ϕ Ai+1 3 a / ψa / 0 O O
_ ϕ _ b / γ ∈ Ci
19 This completes the proof of the theorem. 2 Corollary 1.26 The short exact sequence of sheaves
0 −→ F −→ G −→ H −→ 0 on a topological space X induces the following long exact sequence in cohomology,
... / Hi(X, F) / Hi(X, G) / Hi(X, H)
BCED / Hi+1(X, F) / Hi+1(X, G) / Hi+1(X, H) / ... GF@A Proof Simply note that the short exact sequence of sheaves induce an exact sequence of Cechˇ complexes 0 −→ C•(X, F) −→ C•(X, G) −→ C•(X, H) −→ 0 and applying the snake lemma gives the corollary. 2 The style of the proof of proposition 1.25 is typical in homological algebra, it even has a name: diagram chasing. As one can gather from the above proof, cohomology is an extremely concise language. We now return to the complex analytic case, a complex manifold M is clearly also a smooth real manifold. Recall that the ordinary Poincar´elemma states that the groups p HDR(U) = 0 for all p > 0 and U an open convex set in M. In sheaf theoretic language, this says that the de Rham resolution
∞ d 1 d d p d 0 −→ R −→ C −→A −→ ... −→A −→ ... is exact. Similarly, the ∂-Poincar´elemma, which states that Hp,q(V ) = 0 for q > 0 and ∂ V a polycylinder in M, is equivalent to the sheaf sequence
0 −→ Ωp −→ Ap,0 −→A∂ p,1 −→∂ ... −→A∂ p,q −→∂ being exact. This is known as the Dolbeault resolution. We can now construct an appropriate 2-dimensional complex and use diagram chasing to verify the claim that Cechˇ cohomology generalise classical cohomology.
Proposition 1.27 We have the following isomorphisms of cohomology groups
p p 1. (de Rham’s theorem) HDR(M) ' H (M, R) 2. (Dolbeault’s thoerem) Hp,q(M) ' Hq(M, Ωp) ∂
20 p p 3. Hsimplicial(M) ' H (M, Z) for all p, q ∈ Z.
Proof The proof of part 3 can be found in pages 42-43 of [GH78]. Parts 1 and 2 above can be proved by putting the de Rham resolution and the Dolbeault resolution respective in place of F • (bottom row) in the following two-dimensional complex
. . . . O O ∂ ∂
1 d 1 d 0 / C (F1) / C (F2) / ... O O ∂ ∂
0 d 0 d 0 / C (F1) / C (F2) / ... O O ∂ ∂
• d d F : 0 / F1 / F2 / ...
Since this is similar to the proof of proposition 1.25, we will be more sparing with the details. The relevant part of the double complex is
p−1 p−1 0 / C (F1) / C (F2) / O O
p−2 p−2 0 / C (F1) / C (F2) / O
p−3 . C (F2) .. O
1 1 C (Fp−1) / C (Fp+1) O O
0 0 0 C (Fp−1) / C (Fp+1) / C (Fp+1) O O
Fp / Fp+1
21 i−1 Let σ0 ∈ ker(d : Fp −→ Fp+1), and σi ∈ C (Fp−i) such that dσi = ∂σi−1. This is possible due to exactness of the rows. We summarise this as follows.
0 / 0 / 0 O O
_ _ σp−1 / ∂σp−2 / O
σ _ . p−2 ..
∂σ1 / 0 O O
_ _ σ1 / ∂σ0 / 0 O O
_ _ σ0 / d(σ0) 0
p−2 p−1 Now σp−1 ∈ ker(∂ : C (F1) −→ C (F1)). To show the map σ0 7−→ σp−1 induces a well defined map in cohomology
p • p−2 H (F ) −→ H (F1,X), we check that each choice of σi for 0 6 i < p − 1 changes σp−1 by a coboundary. Also we have to show the map is surjective. These are accomplished by tracing through the diagram as in the proof of proposition 1.25, and we will omit these details. 2 Note 1.28 Since no properties peculiar to sheaf cohomology were used, this result holds for any double complex in an abelian category with exact rows.
1.4 Vanishing theorems
In the case of left exact functors, we have the long exact sequence in cohomology, and the best one can hope for is the vanishing of some higher cohomology groups. However, there is still a wealth of information encoded in the long exact cohomology sequence. The first theorem identifies a class of sheaves which has trivial Cechˇ cohomology.
Definition 1.29
22 • Let U := {Ui}i∈I be an open cover for M. A family {fi}i∈I where fi : F(Ui) −→ F(U) is called a partition of unity with respect to the open cover U, if for all
σ ∈ F(U), supp(fiσ) ⊂ Ui and
X fi(σ|Ui ) ≡ σ. i∈I
• A sheaf F over a topological space X is called a fine sheaf if it admits a partition of unity for any open cover of X.
Theorem 1.30 Let F be a fine sheaf on X. Then Hp(X, F) = 0 for all p > 0.
Proof Let U := {Ui}i∈I be an open cover for U, and let {fi}i∈I be a partition of unity with respect to the open cover U. We show that if σ ∈ Ck(U, F) satisfies δσ = 0, that is,
k X t (δσ)i ,...,i = (−1) σ = 0, (1.5) 0 k+1 i0,...,ibt,...,ik+1 t=0 then σ ∈ δCk−1(U, F). Define τ ∈ Ck−1(U, F) by
X (τ)i0,...,ik−1 = fνσν,i0,...,ik−1 ν∈I
The claim is that τ satisfies δτ = σ, we verify this by calculation:
k−1 X t (δτ)i ,...,i = (−1) τ 0 k i0,...,ibt,...ik t=0 k−1 X t X = (−1) fνσ ν,i0,...,ibt,...,ik t=0 ν∈I k−1 X X t = (−1) fνσ ν,i0,...,ibt,...,ik ν∈I t=0 k−1 ! X X t = (−1) fνσ − fνσi ,...,i + fνσi ,...,i ν,i0,...,ibt,...,ik 0 k 0 k ν∈I t=0 by (1.5) X = −(δσ)ν,i0,...,ik + fνσi0,...,ik ν∈I X = fνσi0,...,ik ν∈I
= σi0,...,ik
23 This shows ker(δ : Ck −→ Ck+1) = im(δ : Ck−1 −→ Ck), hence the cohomology groups Hk(U, F) = 0 for all k > 0. Taking the direct limit with respect to U we have Hk(X, F) = 0 for all k > 0. 2 ∞ In particular, the sheaf C over X admits a partition of unity: for any open cover {Ui}i∈I P of X, there exists functions fi : X −→ R such that supp(fi) ⊂ Ui and i∈I fi ≡ 1. This is simply the ordinary partition of unity construction, so C∞ has trivial cohomology. Similarly the k-th cohomology of Ap, Ap,q vanish for k > 0. We will not prove the next ∂ theorem, due to Grothendieck, which deals with the higher cohomology groups of coherent sheaves.
Theorem 1.31 (Grothendieck vanishing theorem) Let F be a coherent sheaf and M is a n p compact submanifold of P . Then H (M, F) = 0 for all p > dimC(M). The hypothesis above has been weakened to avoid having to mention schemes. We do not have the space to develop the theory of coherent sheaves, but we simply note that OS is a coherent sheaf, where S is a Riemann surface. The vanishing theorem will be applied in the case of OS only.
1.5 Cohomology of Cn and Pn We conclude the chapter on sheaves and cohomology by determining some cohomology groups, most of which will be used later on. Computations of cohomology by its definition is laborious, which is another reason why the long exact sequence in cohomology is so useful- we can infer the structure of cohomology groups without having to do explicit computations. 6 n In the case of P , we have the Hodge decomposition (c.f. theorem 9.3), which states that for a compact K¨ahlermanifold, M, the following holds
r M q p H (M, C) ' H (M, Ω ) (1.6) p+q=r Hq(M, Ωp) = Hq(M, Ωp) (1.7)
0 0 for all r, p, q ∈ N. We will denote h (M, C) = dim(H (M, C)) and maintain this conven- tion throughout. This gives the following
Corollary 1.32
6 n n P is compact. Moreover P is K¨ahler via the Study metric, see [Mum95] for more details.
24 p n q C if p = q and p, q 6 n H (P , Ω ) = 0 otherwise
k n In particular, H (P , O) = 0 for all k > 0 and n > 0.
Proof First recall that k n 0 if k is odd h (P , C) = . 1 if k is even
n This can be shown using either by writing P as CW complex or by using the Mayer- Vietoris sequence, but we will omit these details. Hence by (1.6),
k n X p n q h (P , C) = h (P , Ω ) p+q=k
p n q 0 0 This implies h (P , Ω ) = 0 if p + q is odd. Now suppose p 6= q 6 k, then
2k n X p n q 1 = h (P , C) = h (P , Ω ) p+q=2k p0 n q0 q0 n p0 > h (P , Ω ) + h (P , Ω ) by (1.7) p0 n q0 = 2h (P , Ω ).
0 0 p0 n q0 p n p So if p 6= q then h (P , Ω ) = 0. This leaves h (P , Ω ) = 1 for p 6 k. so we have the result. 2 The cohomology of n is easy: by the ∂-Poincar´elemma Hp,q( n) = 0 for q > 0. Putting C ∂ C p = 0 we get
0 = H0,q( n) = Hq( n, O). ∂ C C
Moreover
p n p n 0 = Hsimplicial(C ) ' H (C , Z) for p > 0. Lastly we finish with an important fact about holomorphic functions on compact complex manifolds.
25 Proposition 1.33 Let M be a connected, compact complex manifold, then H0(M, O) =
C. In other words, the only global holomorphic functions are the constant functions.
Proof Suppose f ∈ H0(M, O) and f obtains a maximum at say x ∈ M. Consider an open set U ⊂ M containing x. By the maximum principle f is constant on U. Now f − f(x) vanishes on an open set, so by analytic continuation, f − f(x) vanishes on all of M. 2
26 Chapter 2
Riemann surfaces
A distinguishing feature of complex function theory is that there exist natural functions √ f : C −→ C whose domain of holomorphy is not C. Examples of this include z 7−→ z and z 7−→ log(z). p Consider the function f : C −→ C, z 7−→ z(z − 1)(z − 2). Take two copies of the 1 Riemann sphere, P , and make branch cuts along the intervals [0, 1] and [2, ∞]. Identifying 1 the two copies of P along these cuts, we obtain the following topological picture of the
resulting space, T ,
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2 The above torus, with the infinity point removed, can be considered as a subset of Cx,y satisfying the algebraic equation p(x, y) = 0 where p(x, y) = y2 − x(x − 1)(x − 2). The 1 map f can be considered as the holomorphic map πf : T −→ P given by (x, y) 7−→ y, 1 and via the map πf , T is a two sheeted branched cover of P with ramification points at
0, 1, 2 and ∞. That is, the fibre of the map πf is finite with cardinality 2, except for the points 0, 1, 2, ∞, where it has cardinality 1. In the above example, a Riemann surface was constructed by analytically continuing the complex valued function f. We see that T is 1. a complex manifold of dimension 1, 2. a complex algebraic variety of dimension 1, that is, it is a complex algebraic curve, and 3. a real manifold of dimension 2, that is, a surface 1.
1Here we are using the word dimension in three different ways; the dimension of a complex (resp. real) manifold is the complex (resp. real) dimension of the codomain of any local chart, and the dimension of an algebraic variety is the transcendence degree of its coordinate ring.
27 These three aspects are typical of Riemann surfaces in general and validate what was said in the introduction.
2.1 Properties of Riemann surfaces
We will restrict the definition of an abstract Riemann surface to be compact and con- nected.
Definition 2.1 A Riemann surface is a one-dimensional, connected, compact complex manifold.
There are ‘non-compact Riemann surfaces,’ for instance C, but for the most part of this thesis, we are concerned with the compact case.
Note 2.2 In fact, all Riemann surfaces as defined above can be realised as a n-sheeted 1 branched cover of P , so this is equivalent to Riemann’s original concept. This amounts to the existence of a nonconstant f ∈ K∗(S) with a pole of order n, and the Riemann-Roch theorem (c.f. (4.3)) adequately answers such problems.
n Definition 2.3 Consider Px0,...,xn and the set of common zero loci of homogeneous poly- n nomials p1, . . . , pk ∈ C[x0, . . . , xn]. Denote this set C := V (p1, . . . , pk) ⊆ P , then C is n called a complex algebraic curve if C is a one dimensional submanifold of P .
n More generally, we call a subset X ⊆ P algebraic if X is the common zero loci of some homogeneous polynomials q1, . . . , qj ∈ C[x0, . . . , x0]. We first show that any Riemann surface is algebraic.
Proposition 2.4 Every Riemann surface is a complex algebraic curve.
n Proof By the implicit function theorem, any submanifold of P is an analytic subvariety. n Chow’s theorem (c.f. page 167 of [GH78]) states that any analytic subvariety of P is an n algebraic subvariety of P . Hence if there is an embedding of the Riemann surface S into projective space, then it is algebraic. The Kodaira embedding theorem ensures such an embedding exist. To prove this, we will wait until the end of chapter 3. 2 n Note 2.5 Complex algebraic curves in P are sometimes is referred to as complex pro- jective curves 2.
Invoking the Kodaira embedding theorem is certainly overkill in this case, for a more direct argument, see page 214-215 of [GH78]. The converse to proposition 2.4 is obtained n by noting that an embedded complex algebraic curve inherits the complex structure of P .
2 n Terminology also used to distinguish between algebraic curves in A .
28 Even though Riemann surfaces and complex algebraic curves are essentially equivalent objects, we will refer to a Riemann surface S in general, and use the terminology of curves in conjunction with a particular embedding- we will see that ‘most’ Riemann surfaces possess a canonical embedding. The higher dimensional analogue of proposition 2.4 fails for general complex compact manifolds. In chapter 5, we will encounter examples of complex manifolds which are not algebraic. Finally, we give two fundamental properties of Riemann surfaces; the first is topological and the second is analytic.
Proposition 2.6 Let S be a Riemann surface then 1. S is orientable, and 2. S is a K¨ahlermanifold.
Proof Part 1 is simply due to the fact that all complex manifolds have a natural orienta- tion induced by the complex structure. Recall that a complex manifold M with Hermitian metric ds2 is a K¨ahler manifold (c.f. page 259 of [Che00]) if the associated (1,1)-form ω 2 3 of ds satisfies dω = 0. Now for M = S, dω ∈ A (S), and since dimR(S) = 2 we have dω = 0. 2 Since S is oriented we can assign to S the topological invariant,
−χ(S) + 2 g = = number of ‘handles’ of underlying real manifold of S, 2 called the genus of S. The K¨ahler condition on S facilitates the use of the Hodge decomposition, which is used to decompose cohomology of S- this is done in section 2.3. The next section contains some examples of Riemann surfaces.
2.2 Examples of Riemann surfaces
1 The simplest Riemann surface is the Riemann sphere, which we will denote P . This is the one point compactification of C by adding a point at infinity, which we denote ∞. An 1 atlas for P is {(U0, ϕ0), (U∞, ϕ∞)} where
1 1 {z ∈ P | z 6= 0} = U0 −→ C {z ∈ P | z 6= ∞} = U∞ −→ C and z 7−→ 1/z z 7−→ z
2 0 It has genus zero and can be realised as a conic in P via the invertible map p 7−→ p
29
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given by projecting from (0, 0). The next simplest example is the elliptic curve, E. The etymology of the name elliptic is explained in section 6.1. We start with the complex torus C/Λ where Λ = Z + τZ, =(τ) > 0. We have the classical Weierstrass ℘-function with respect to Λ,
1 X 1 1 ℘(z) = − − z2 (z − λ)2 λ2 λ∈Λ−{0} which is doubly periodic with periods 1, τ. So ℘ is naturally a function on C/Λ. Define the map
2 ϕ : C/Λ −→ P (℘(z): ℘0(z) : 1) if z 6∈ Λ z 7−→ . (0 : 1 : 0) if z ∈ Λ
2 This is an embedding of C/Λ into P . Now the function ℘ satisfies the important identity,
0 2 3 ℘ (z) = 4℘(z) − g2℘(z) − g3
P −4 P −6 where g2 = 60 ω∈Λ−{0} ω and g3 = 140 ω∈Λ−{0} ω . Hence the image ϕ is equal to the subset
2 2 3 2 2 C = {(x : y : z) ∈ P | y z = 4x − g2xz − g3z} ⊂ P
This realises the complex torus as an algebraic curve. We will see more of the ℘-function in later chapters, especially its connection with the elliptic integral (c.f. example 6.3). We will collect some facts about the genus 2 and 3 cases for use later. Let S be a Riemann 0 1 0 1 surface of genus g and let ω1, . . . , ωg be the basis of H (S, Ω ); that h (S, Ω ) = g will be substantiated in example 4.25. First we make a
30 Definition 2.7 Define the canonical map of S to be
0 1 g−1 ιK : S −→ PH (S, Ω ) ' P
p 7−→ (ω1(p): ... : ωg(p))
g−1 The image of S in P is called the canonical curve of S. When ιK is an embedding, this gives a canonical way to study S extrinsically. In example 4.26, we will see that ιK is an embedding iff S is not hyperelliptic. Now genus 2 Riemann surfaces are hyperelliptic, so ιK is not an embedding for these Riemann surfaces. In fact, to embed a genus 2 3 Riemann surface, we need to consider P and use at least three equations [Mum75]. The genus 3 case is the first instance where Riemann surfaces exhibit both hyperelliptic and non-hyperelliptic behaviour. In the non-hyperelliptic case, S can be canonically 2 embedded as a plane curve via ιK : S −→ P . Moreover, the degree of ιK in this case is
2g − 2 = 4, so ιK (S) is a plane quartic. We will need the following fact in the proof of the Torelli theorem.
Proposition 2.8 Every plane quartic has twenty eight bitangents.
This is a classical result which can be obtained via the Pl¨ucker formulas. We do not have the space to prove this, for more on the Pl¨ucker formulas and a proof of proposition 2.8, see pages 277-282 of [GH78]. To prove the Torelli theorem, we only need to know that the plane quartic has a finite number of bitangents.
2.3 Cohomology of Riemann surfaces
n As with P , any Riemann surface S is a compact K¨ahler manifold, so we can apply the Hodge decomposition. The decomposition of cohomology can be summarised by the Hodge diagram, H1(S, Ω1) M qq MMM qqq MM qq MMM qqq MM H1(S, O) H1(S, O) M MM qqq MMM qq MM qqq MM qqq H0(S, O)
n where H (S, C) is isomorphic to direct sum of the entries in the n-th row. In particular, 0 this says H (S, C) can be decomposed into holomorphic and anti-holomorphic forms. The 0 0 bottom row H (S, O) ' H (S, C) reflects the fact that the only holomorphic functions k on S are the constant functions. Moreover H (S, C) = 0 for all k > 2.
31 Since S is a two dimensional manifold, Poincar´eduality says
2 0 H (S, Z) ' H (S, Z)
0 and H (S, Z) ' Z.
2.4 The Riemann-Hurwitz formula
Given a holomorphic map f : S −→ S0 of degree d (that is, f is a d to one map), where S and S0 are Riemann surfaces with genus g and g0 respectively.
Definition 2.9 Let f : S −→ S0 be a holomorphic map and for p ∈ S, let z be a local coordinate around p and w be a local coordinate around f(p). If f can be given locally at p as w = zν(p), for some integer ν(p), then we say ν(p) is the ramification index of f at p. The point p is a branch point if ν(p) > 1. Moreover we define the branch locus of p to be the divisor
X (ν(p) − 1) · p ∈ Div(S) p∈S or its image
X (ν(p) − 1) · f(p) ∈ Div(S0) p∈S
We can see that away from the branch locus, f is a d to 1 covering, and two or more of these sheets come together at the branch locus. The Riemann-Hurwitz formula relates d, g, g0 and the numbers ν(p).
Theorem 2.10 (Riemann-Hurwitz) Let ν denote the ..., and χ be the Euler character- istic, then we have
X χ(S) = dχ(S0) − (ν(q) − 1). q∈S
Proof (Sketch) A triangulation exists on S0 since it is compact. Let T 0 = (V 0,E0,F 0) be a triangulation on S0 such that all the branch points lie on a vertex. Pull this triangulation back to S via f to obtain a triangulation T = (V,E,F ) on S, and we count the numbers
32 of vertices, edges, and faces of T
|E| = d|E0|
|F | = d|F 0| X |V | = d|V 0| − (ν(q) − 1) q∈S and we obtain the Riemann-Hurwitz formula. 2 2.5 Hyperellipticity
We finish this chapter with a brief discussion of the simplest types of Riemann surfaces. We saw that all Riemann surfaces of genus 2 are hyperelliptic, and in fact, there exists hy- perelliptic Riemann surfaces for all genus g > 2. Hyperellipticity can be characterised by the existence of meromorphic functions with two poles, this is equivalent to the following
Definition 2.11 A Riemann surface S is hyperelliptic if it admits a two to one covering 1 map f : S −→ P .
The function f is essentially unique, as we shall see.
Proposition 2.12 Let S be a hyperelliptic Riemann surface with genus S, then f : S −→ 1 P has 2g + 2 branch points.
1 Proof This is a direct application of the Riemann-Hurwitz formula. Since χ(P ) = 2,
X 2 − 2g = 2 · 2 − (ν(p) − 1) p∈S X (ν(p) − 1) = 2g + 2. p∈S
P Also 1 6 ν(p) 6 2, so p∈S(ν(p) − 1) = number of branch points. 2 These branch points actually determine S, we will need this fact when proving the Torelli theorem.
Proposition 2.13 A hyperelliptic Riemann surface S of genus g with two to one map 1 f : S −→ P . Then S is determined completely by the 2g + 2 branch points of f.
1 ∗ 1 Proof Now S −→ Px is of degree 2 and f : C(Px) = C(x) ,→ C(S) is an injective field ∗ homomorphism. On identifying C(x) with its image under f , C(x) is a subfield of C(S),
33 and moreover [C(S): C(x)] = 2. That is, C(S) is a quadratic extension of C(x). Given y ∈ C(S), y 6∈ C(x), it satisfies the quadratic equation
2 y + yf1(x) + f2(x) = 0
2 where f1, f2 are polynomials in x. Completing the square gives y = h(x), so C(S) ' p C(x, h(x)) where for some polynomial h. 0 0 Now C(S) and C(S ) as isomorphic fields iff to S and S birationally equivalent. By a theorem in algebraic geometry, birational curves are isomorphic. We have shown that p every hyperelliptic Riemann surface S of genus g has C(S) ' C(x, h(x)), and since the zeroes of h are precisely the 2g + 2 Weierstrass points of S, these points determine S 2 2 completely. Moreover S is birationally equivalent to the curve {(x, y) ⊂ Cx,y | y = h(x)}.
2 Hyperelliptic Riemann surfaces often behave differently from their non-hyperelliptic rel- g−1 atives. We will see in example 4.27, that the canonical map ιK : S −→ P fails to be an embedding iff S is hyperelliptic. This phenomenon will resurface in chapter 7, when we prove the Torelli theorem. Note that the moduli space of genus g Riemann surfaces has dimension 3g − 3, this can be determined by counting the parameters which define a Riemann surface. However, the hyperelliptic Riemann surfaces of the same genus has a moduli space of dimension 2g − 1. This agrees with the fact that in the case of genus g = 2, all Riemann surfaces are hyperelliptic; and shows that for genus g > 2, ‘most’ Riemann surfaces are non- hyperelliptic. Chapter 2 of Mumford’s book [Mum75], contains a very readable account on moduli spaces of Riemann surfaces.
34 Chapter 3
The classical theorems of Abel and Jacobi
In the context of the Torelli theorem, Abel’s theorem implies Torelli in the genus 1 case; and the Jacobi inversion theorem is needed in section ?? to prove Riemann’s theorem. The aim of this chapter is to present Abel’s and Jacobi’s theorems using the terminology of divisors and line bundles, as well as introducing these essential concepts. Taken together, these theorems give us the following commutative triangle,
Div0(S) / J (S) (3.1) KK u: KK uu KK uu KK uu K% uu Pic0(S) where S is a Riemann surface, Div0(S) is the group of divisors of degree zero, Pic0(S) the connected component of Pic(S) containing the identity, Pic(S) is the group of isomorphism classes of line bundles on S, and J (S) is the Jacobian variety of S. This correspondence is remarkable as it relates three seemingly disparate objects. We will use the theorems to describe the geometry of Pic(S).
3.1 Divisors
The nomenclature in this section have their origins in algebraic number theory, where analogous constructions arose. For an elementary discussion of fractional ideals, the number theoretic analogue of divisors, and the class group, the number theoretic analogue of the Picard group, refer to [Ste79]. Divisors can be thought of as a generalisation of hypersurfaces.
Definition 3.1 A divisor on a compact complex manifold M is a formal finite sum
k X D = niHi i=1
35 where ni ∈ Z and Hi ⊂ M are irreducible hypersurfaces. Let Div(M) be the free abelian group generated by the divisors on M, where the identity element is denoted 0.
In the case of a Riemann surface S, D ∈ Div(S) is simply a formal sum of points D = Pk i=1 nipi, where pi ∈ S. The more general definition above is needed when we discuss divisors on the Jacobian variety. Let M be a compact complex manifold. To every nonzero meromorphic function f ∈ K∗(M) one can associate a divisor (f) as follows. For any hypersurface H ⊂ M define ∗ ordH : K (M) −→ Z by n if f has a zero of order n along H ordH (f) = −n if f has a pole of order n along H 0 otherwise
−1 −1 Note that ordH (fg ) = ordH (f) + ordH (g ) = ordH (f) − ordH (g), so ordH is a group homomorphism. Then define the divisor of f to be
X (f) = ordH (f)H. H hypersurface in M
Of course, not all divisors arise in this way, as the following example shows.
Example 3.2 Let S be a Riemann surface. If D = p where p ∈ S, then D is not the divisor of any nonzero meromorphic function on S. Suppose f ∈ K∗(S) such that p = (f). Then f has no poles, f ∈ H0(S, O) and since S is compact and connected, proposition 1.33 implies f is constant. Now f(p) = 0 implies f = 0, contradicting f ∈ K∗(S).
Proposition 3.3 If D ∈ Div(M) is a divisor of a nonzero meromorphic function, then D is called a principal divisor. The set of principal divisors of M, denoted PDiv(M), is a subgroup of Div(M).
That PDiv(M) is a subgroup of Div(M) follow from the fact that the map (·): K∗(M) −→ Div(M) is a group homomorphism, since
−1 X −1 (fg ) = (ordH (f) + ordH (g ))H H hypersurface in M = (f) − (g) for all f, g ∈ K∗(M).
Definition 3.4
36 P • A divisor D = niHi effective if ni > 0 for all i. We use this to define an 0 0 important partial ordering on Div(M): if D,D be two divisors on M, then D > D iff D − D0 is effective. • Consider the quotient Div(M)/ PDiv(M), then we say that two divisors are linearly equivalent if they are in the same coset of PDiv(M).
Define Pic(S) = Div(S)/ PDiv(S), we will see the connection between this definition and line bundles in proposition 3.30. Effective divisors reappear in the definition of linear systems in chapter 4.
Example 3.5 One can think of linear equivalent divisors as continuous deformations x2−x x of each other. Let M = 2 and f(x , x , x ) = 2 0 1 ∈ K∗(M), then D := Px0,x1,x2 0 1 2 x0x1 1 2 2 V (x2 − x0x1) ∼ V (x0) + V (x1) =: D∞. On the open set {(x0, x1, x2) ∈ P | x2 6= 0}, then for different values of f, we obtain the following
which shows a continuous deformation from D1 to D∞.
This point of view is important in intersection theory, under the premise that intersection numbers should be invariant under continuous deformations. Intersection theory is a topic which we do not have the space to develop. The most important theorem therein is B´ezout’s theorem, c.f. [CC04] Refer to [GH78] or [Sha74] for more details. We now specialise the discussion to a Riemann surface S. We first define Div0(S).
Definition 3.6 There is a natural group homomorphism, Div(S) −→ Z given by
X X npp 7−→ np p∈S p∈S called the degree map.
37 P Note that p∈S np < ∞ since it is a finite sum. The kernel of the degree map is the subgroup Div0(M).
Proposition 3.7 A principal divisor on S has degree 0, hence PDiv(S) is a subgroup of Div0(S).
Proof Note that any f ∈ K∗(S) can be considered as an n-sheeted branched covering of 1 1 P P , f : S −→ P . Then deg s∈f −1(r) ords(f)s = n by definition of branched covers. 1 ∗ Let p = (1 : 0), q = (0 : 1) ∈ P , and we have deg((f)) = deg(f (p − q)) = n − n = 0. 2 This means that linearly equivalent divisors have the same degree: if D,D0 ∈ Div(S) and D = D0 +(f), then deg(D) = deg(D0)+deg((f)) = deg(D0). The converse to this is false. Moreover proposition 3.7 says that we can define Pic0(S) = Div0(S)/ PDiv(S).
Note 3.8 The above proposition depends on Riemann surfaces being compact. Consider a non-compact one-dimensional complex manifold, say C, then let
n Y 1 f(z) = zm z − ak k=1
∗ where m > n and deg(f) 6= 0. Now f ∈ K (C) and deg((f)) 6= 0. However if we consider 1 ˜ ∗ 1 P with atlas {(U0, ϕ0), (U∞, ϕ∞)} and f ∈ K (P ) given by
n m Y 1 f˜∞(z) = z on U∞ z − ak k=1 n n−m Y 1 f˜0(z) = z on U0 1 − zak k=1
−1 −1 −1 −1 and (f) = m · ϕ∞ (0) − (ϕ∞ (a1) + ... + ϕ∞ (an)) + (n − m) · ϕ0 (0) has degree 0.
3.2 The Abel-Jacobi map and the Jacobian variety
The Jacobian variety of a Riemann surface S is introduced at this point to state to the Abel and Jacobi theorems. An intrinsic definition will be given in chapter 6, and it will be shown to agree with the following
0 1 Definition 3.9 Let ω1, . . . , ωg be a basis for H (S, Ω ), δ1, . . . , δ2g be a basis for H1(S, Z) such that the intersection form on H1(S, Z) with respect to these basis has the matrix
−I . I
38 Then define the Jacobian variety of S as the quotient
J (S) = C ZΠ1 + ... + ZΠ2g
R R where Πi = ω1,..., ωg for all i ∈ [1, g]. δi δi
A basis δ1, . . . , δ2g for H1(S, Z) can always be chosen, this will be shown in chapter 6. The implicit claim that J (S) is a variety will also be verified in chapter 6. Now we can define the Abel-Jacobi map, this is the prototype for the map Div0(S) −→
J (S) which appears in (3.1). First pick an arbitrary point, say p0, on S, called the base point, then we have the
Definition 3.10 The map µ : S −→ J (S) given by
Z p Z p p 7−→ ω1,..., ωg (3.2) p0 p0 is called the Abel-Jacobi map. Extending via linearity to divisors, we get a map µ : Div(S) −→ J (S), defined by
k k k ! X X Z pi X Z pi pi 7−→ ω1,..., ωg (3.3) i=1 i=1 p0 i=1 p0
One might notice that in (3.2) and (3.3), the right hand side may not be well defined in g C , owing to the fact that S may not be simply connected. However, we have the
Proposition 3.11 The Abel-Jacobi map is a well-defined map into the Jacobian of S.
An argument for this will be provided in section 6.1. k k Consider the space of effective degree k divisors, Div+(S), we can topologise Div+(S) as follows; denote S × ... × S/ Perm(k) = Sk/ Perm(k) =: S(k) where Perm(k) acts on k times k k (k) S by permuting the k coordinates. Then Div+(S) = S , and so inherits the complex k k structure from S . The Abel-Jacobi map restricted to Div+(S) is
µ(k) : S(k) −→ J (S) g g ! X Z pi X Z pi p1 + ... + pk 7−→ ω1,..., ωg (3.4) i=1 p0 i=1 p0
39 0 1 where ω1, . . . , ωg are a basis for H (S, Ω ). The map µ can be made independent of the 0 base point p0 by restricting to Div (S). In this case, we obtain
µ : Div0(S) −→ J (S) k k k ! X X Z pi X Z pi (pi − qi) = ω1,..., ωg . i=1 i=1 qi i=1 qi
This is the map µ in (3.1). Recall that proposition 3.7 states that principal divisors have degree zero, and that the converse is false. Abel’s theorem give a necessary and sufficient condition for a divisor in Div0(S) to be principal; while the Jacobi inversion theorem says that every point in J (S) corresponds to a linearly equivalent class of divisors of degree 0.
µ Theorem 3.12 The sequence 0 −→ PDiv(S) −→ι Div0(S) −→J (S) −→ 0 of abelian groups is exact, where ι is the inclusion map and µ is the Abel-Jacobi map.
Note 3.13 Abel’s theorem states that ker(µ) = PDiv(S), and the Jacobi inversion the- orem states that µ is surjective.
Corollary 3.14 We have the isomorphism Div0(S)/ PDiv(S) 'J (S).
We will see in example 4.22, that in the genus 1 case, there is an isomorphism S −→ Div0(S)/ PDiv(S), and hence S is isomorphic with its Jacobian; this is an instance of the Torelli theorem. To prove the Abel and Jacobi theorems we require the following. An analytic subvariety of a complex manifold M is defined as the common zero locus of some subset of O(M). We will need the proper mapping theorem, which we state without proof
Theorem 3.15 If f : M −→ N is a holomorphic map between complex manifolds, then if V is an analytic subvariety of M then f(V ) is an analytic subvariety of N.
Now we prove theorem 3.12. Proof
µ 1. We will first show that PDiv(S) −→ Div0(S) −→J (S) is the zero map. Define
1 µ ψf : P −→ Div(S) −→ J (S) . (x : y) 7−→ (xf − y) 7−→ µ((xf − y))
∗ 1 Now ψf is the zero map, since there are no global holomorphic 1 forms on P . This 0 1 1 1 1 ∨ is due to Serre duality (c.f. theorem ??), H (P , Ω ) ' H (P , O) , and corollary
40 1 1 1.32, which gives H (P , O) = 0. This shows ψf is constant, hence µ((f)) = ψf (1 :
0) − ψf (0 : 1) = 0. µ Showing exactness of 0 −→ PDiv(S) −→ Div0(S) −→J (S) will complete the proof of Abel’s theorem. The arguments can be found in pages of 232-235 of [GH78]. 2. We claim that every ξ ∈ J (S) can be written as Pk R pi ω ,..., Pk R pi ω . i=1 p0 1 i=1 p0 g Hence it suffices to show that µ(g) : S(g) −→ J (S) is surjective. The Jacobian (g) matrix for µ near D = p1 + ... + pg with local coordinates z1, . . . , zg is given by
ω1(p1)/dz1 . . . ω1(pg)/dzg (g) . . Dµ = . . . ωg(p1)/dz1 . . . ωg(pg)/dzg
Note that the n-th column are the coordinates of ιK (pn). This matrix is generically full rank, hence by the inverse function theorem there exists an open set U ⊂ S(g) such that µ(g) is a local isomorphism U −→ µ(g)(U). Now by the proper mapping theorem im(µ(g)) is an analytic subvariety of J (S), but im(µ(g)) contains the open set µ(g)(U), so the image of µ(g) must be equal to its codomain.
2 3.3 Line bundles
A vector bundle formalise the idea of a family of vector spaces parameterised by a smooth manifold, M, and which varies smoothly with respect to points on M. The most common example is that of a tangent bundle of a smooth manifold in differential geometry. In complex differential geometry, we replace the smoothness condition with a holomorphic requirement. A holomorphic line bundle is a holomorphic vector bundle where the vector spaces are one dimensional.
Definition 3.16 Let E and X be complex manifolds and π : E −→ X a surjective holomorphic map, satisfying the following properties.
1. There exists a local trivialisation. That is, an open cover {Uα}α∈A of X together with the biholomorphic maps,
−1 ϕα : π (Uα) −→ Uα × V
p 7−→ (π(p), ϕ˜α(p))
where V is a complex vector space.
41 2. Denote ϕ˜α,x :=ϕ ˜α|π−1(x). The functions ϕ˜α satisfy the following properties. −1 (a) The restriction ϕ˜α,x : π (x) −→ V is biholomorphic for all x ∈ Uα. (b) The composition
−1 gαβ(x) :=ϕ ˜α,x ◦ (ϕ ˜β,x) : V −→ V
is a linear isomorphism for all x ∈ Uα ∩Uβ, that is, gαβ(x) ∈ GL(V ). Moreover
the assignment x 7−→ gαβ(x) is holomorphic. We call the family {gαβ}α,β the transition functions. A triple E −→π X satisfying the above is called a holomorphic vector bundle. The rank of the vector bundle is dimC(V ). A holomorphic vector bundle of rank 1 is called a σ holomorphic line bundle. A holomorphic map X −→ E satisfying π ◦ σ = idX is called a holomorphic section. The vector space of all global holomorphic sections is denoted Γ(E) or O(E) 1.
0 Definition 3.17 Let E −→π X and E0 −→π X be holomorphic vector bundles. A mor- phism of holomorphic vector bundles, is a holomorphic map
ϕ : E −→ E0
such that ϕ|X : X −→ X is the identity map and the diagram
ϕ E / E0 @@ } @@ }} π @@ }}π0 @ ~}} B commutes. We say that E and E0 are isomorphic if there exists morphisms ϕ : E −→ E0 0 and ψ : E −→ E such that ϕ ◦ ψ = idE and ψ ◦ ϕ = idE0 .
Definition 3.18 Suppose f is a holomorphic map X −→ Y and suppose E −→ Y is a holomorphic vector bundle on Y . Define the pullback of E, f ∗E −→ X, with f ∗E := {(x, e) ∈ X × E | f(x) = π(e)} and projection f ∗π : f ∗E −→ Y given by f ∗π(x, e) = x.
Now line bundles enjoy the property of being specified completely by their transition functions. Let M be a complex manifold with open cover U := {Uα}α∈A and a family
1More generally, O(E)(U) denotes the sections of E over U.
42 ∗ {gαβ}α,β∈A where gαβ ∈ O (Uαβ), satisfying
gαβ · gβγ · gγα = id|Uαβγ (3.5)
gαβ · gβα = id|Uαβ (3.6) for all α, β, γ ∈ A. Note that these are precisely the identities satisfied by the transition functions defined above.
We can construct a line bundle with the family {gαβ} as transition functions. Consider S the union L := α∈A Uα × C and projections πα : Uα × C −→ Uα for each α, we identify −1 −1 the fibre over z ∈ Uαβ via the bijection πα (z) −→ πβ (z) given by p 7−→ gαβ · p. Then π L −→ X is a holomorphic line bundle with π|Uα = πα for all α ∈ A.
This also tells us how to ‘glue’ sections together, given σα : Uα −→ L and σβ : Uβ −→ L, then on Uαβ, we have σα = gαβ · σβ. In the following, let M be a complex manifold.
Definition 3.19 Define Pic(M) to be the group of isomorphism classes of holomorphic line bundles on M.
Definition 3.20 Let L −→ M and L0 −→ M be two holomorphic line bundles over X 2 with transition functions {gαβ} and {hαβ} given on the same open cover of X respec- tively. Define the dual of L, denoted L∗, to be the line bundle given by the transition functions −1 0 0 {gαβ }. Also define the tensor product of L and L , denoted L ⊗ L to be the line bundle given by the transition functions {gαβ · hαβ}.
Note that the tensor product makes Pic(M) a group. We can characterise Pic(M) as a Cechˇ cohomology group.
Proposition 3.21 There is an isomorphism of groups H1(M, O∗) ' Pic(M).
Proof Let U := {Uα}α∈A be an open cover of M, we have established above that a family ∗ g := {gαβ}α,β, with gαβ ∈ O (Uαβ) satisfying (3.5) and (3.6) determines a line bundle L. We check that the map g 7−→ L is well defined and is an isomorphism. Firstly g ∈ Z1(M, O∗), since
−1 (δg)αβγ = gβγ · gαγ · gβγ by(3.6) = gβγ · gγα · gβγ by(3.5) = id |Uαβγ
2One can always take a refinement of the two open covers of X if they are different.
43 0 −1 If we pick a different representative of [g], say g = {fβ · fα · gαβ}α,β, then this amounts to picking a different trivialisation, and defines the same line bundle L. The definition of tensor product in Pic(M) coincide with the group operation on H1(M, O∗), so the map g 7−→ L is a group homomorphism. The existence of the inverse to g 7−→ L is clear, since L is simply mapped to its transition functions, and choosing a different trivialisation changes the image by a coboundary. 2 We give some examples of holomorphic line bundles.
Example 3.22 Let M be an n-dimensional complex manifold, and T ∗(M) be the cotan- gent bundle. Then K := Vn T ∗(M) is a line bundle, called the canonical bundle of M.
n Example 3.23 Consider a hyperplane H ⊂ Px0,...,xn . This is a codimension one subva- riety as it is given by a linear form α0x0 + ... + αnxn for some α1, . . . , αn ∈ C. The n line bundle corresponding to the divisor class [H] ∈ Pic(P ) is called the hyperplane bundle. The dual of the hyperplane bundle is called the universal bundle.
π Example 3.24 The trivial line bundle L := X × C −→ X over a complex manifold X corresponds to the structure sheaf OX via the identification OX (U) = O(L)(U).
n Example 3.25 We determine Pic(P ). First consider
n deg : Div(P ) −→ Z V (f) 7−→ deg(f)
n where f is a irreducible homogeneous polynomial, and extend deg to all of Div(P ) via lin- n earity. This is actually the explicit form of the chern class map for Pic(P ), whose gen- Pm eral definition will be given in the next section. We see that if deg(D) = i=1 niV (fi) = 0, then
m X ni deg(fi) = 0 i=1 m X ni deg(fi ) = 0 i=1 and after a suitable renumbering of the fj’s,
f n1 . . . f nj g = 1 j ∈ K∗( n) nj+1 nm P fj+1 . . . fn (g) = D.
44 n n So the kernel is PDiv(P ). Now deg is surjective, so we have Pic(P ) ' Z. We will show n Pic(P ) = Z another way using cohomology. Note that in the above proof, we have implicitly assumed that all meromorphic functions n in P are rational functions. For proof of this, we refer the reader to page 168 of [GH78].
n Example 3.26 The above proposition shows that Pic(P ) is generated by one element, n [H], since deg(H) = 1. So given any divisor, D ∈ Pic(P ), there exists some d ∈ Z such that D ∼ dH. This shows that there is no ambiguity in writing
O(D) = O([D]) = O([dH]) = O(dH) = O(d)
n so all the line bundles on P are in the form O(d) for d ∈ Z. In fact, we can be even more explicit,
Proposition 3.27 We have the following isomorphism of vector spaces
0 n H (P , O(D)) ' C[x0, . . . , xn]deg(D)
A proof of the above proposition can be found in pages 164-166 of [GH78].
3.4 Pic(S)
We specialise the discussion to holomorphic line bundles over a Riemann surface S. Hence- forth, holomorphic line bundles will be referred to as simply line bundles, and will be denoted L −→π S. In this section, we will examine the structure of Pic(S). The relationship between line bundles and divisors is best expressed in sheaf theoretic language. In definition ??, Pic(S) is defined as a Cechˇ cohomology group. We now express Div(S) in terms of such a group.
Proposition 3.28 There is an isomorphism of groups ϕ : H0(S, K∗/O∗) −→ Div(S).
0 ∗ ∗ Proof Let σ ∈ H (S, K /O ) given by an open cover {Ui}i∈I of S, and σ|Ui = σi ∈ 0 ∗ ∗ H (Ui, K /O ), satisfying
∗ ∗ σi|Uij O (Uij) = σj|Uij O (Uij) (3.7) as cosets for all i, j ∈ I. We can associate to σ the divisor
X Dσ = ordp(σi)p p∈S
45 where i is chosen such that p ∈ Ui. The value of ordp(σi) does not depend on such a choice, since by (3.7), σi and σj has the same poles and zeroes in Uij. So for p ∈ Uij, σi has a zero (or pole) at p iff σj has a zero (or pole) at p.
Conversely, given any divisor D ∈ Div(S), choose an open cover {Vj}j∈J such that for 0 ∗ each Vj there exist fj ∈ H (Vj, K ) such that fj has poles or zeroes at the corresponding f | i Uij ∗ points in D. This gives, ∈ O (Uij), hence the by the sheaf condition there exists fj |Uij 0 ∗ ∗ an f ∈ H (S, K /O ) such that Df = D. We call f the local defining function for the divisor D. Finally, it is clear that ϕ is a homomorphism. 2 Note 3.29 We can define divisors as elements of H0(S, K∗/O∗), in which case they are called Cartier divisors. Divisors defined as a formal sum of irreducible codimension 1 subvarieties are called Weil divisors. When S is smooth, these definitions are equivalent, as the above isomorphism shows. When singularities are present in S, this is not true.
Consider the exact sequence of sheaves on S,
0 / O∗ / K∗ / K∗/O∗ / 0
This induces a long exact sequence in cohomology, from which we extract the following,
... / H0(S, K∗) / H0(S, K∗/O∗) / H1(S, O∗) / ...
ϕ H0(S, K∗) / Div(S) / Pic(S) where ϕ is the isomorphism of proposition 3.28. The kernel of the map Div(S) −→ Pic(S) is H0(S, K∗) by exactness. We first determine the map Div(S) −→ Pic(S) above explicitly. ∗ Let D ∈ Div(S) with local defining equations fj ∈ K (Vj) with respect to some open cover {Vj}j∈J of S. Then let gij = fi/fj, and the family {gij}i,j∈J satisfy the conditions for transition functions:
g · g = fi · fj = id ij ji fj fi Uij fi fj fk gij · gjk · gki = · · = idU fj fk fi ijk for all i, j, k ∈ J. Denote [D] to be the line bundle defined by {gij}i,j∈J . Now if we choose a different set of local defining equations with respect to the same open cover,3 say
3We can always take a refinement of two different open covers, so we can assume without loss of generality that the local defining functions are on the same open cover.
46 ∗ hj ∈ K (Vj), for D. Then for each j ∈ J, fj = ϕjhj for some nonvanishing holomorphic 0 function ϕj. Now let gij = hi/hj, then
fi ϕihi ϕi 0 gij = = = gij fj ϕjhj ϕj
∗ 0 and since ϕi/ϕj ∈ O (Uij), {gij} and {gij} define the same line bundle. Now the map H0(S, K∗) −→ Div(S) is given by f 7−→ (f) by proposition 3.28. This gives the following
[·] Proposition 3.30 The sequence 0 −→ PDiv(S) −→ Div(S) −→ Pic(S) −→ 0 is exact.
The above proposition shows that linearly equivalent divisors give rise to the same line bundles. Moreover for D,D0 ∈ Div(S), [D + D0] = [D] ⊗ [D0]. This follows from the fact that the transition functions for [D + D0] are obtained by multiplying the transition functions of [D] and [D0].
Definition 3.31 We say a section σ of the line bundle [D] over S is holomorphic iff 0 0 (σ) + D > 0 for some D ∼ D. Next we introduce the chern class map. Let S be a Riemann surface, and consider the exponential sheaf sequence on S
ι exp 0 / Z / O / O∗ / 0
Again this sequence is exact, so it induces a long exact sequence in cohomology,
0 0 exp 0 ∗ 0 / H (S, Z) / H (S, O) / H (S, O ) δ
1 1 1 ∗ BCED / H (S, Z) / H (S, O) / H (S, O ) GF@A c1
2 2 2 ∗ EDBC / H (S, Z) / H (S, O) / H (S, O ) / ... GF@A 2 Since O is a coherent sheaf and dimC(S) = 1, H (S, O) is trivial by Grothendieck’s vanishing theorem (c.f. theorem 1.31). Moreover im(exp) = H0(S, O∗) = ker(δ), hence we can extract the following exact sequence,
1 1 1 ∗ c1 2 0 / H (S, Z) / H (S, O) / H (S, O ) / H (S, Z) / 0
1 ∗ c1 2 Definition 3.32 The connecting homomorphism H (S, O ) −→ H (S, Z) is called the chern class map.
47 Note 3.33 More correctly, this is called the first chern class map, which explains the subscript in c1. But since the higher chern class maps will not be used, we will simply stick to chern class map.
n Example 3.34 We can use the above long exact sequence to determine Pic(P ).
1 n 1 n ∗ 2 n 2 n ... −→ H (P , O) −→ H (P , O ) −→ H (P , Z) −→ H (P , O) −→ ...
1 n 2 n The groups H (P , O) and H (P , O) were determined to be both zero in corollary 1.32, 2 n and also H (P , Z) ' Z. This gives the exact sequence
n 0 −→ Pic(P ) −→ Z −→ 0
n and hence the isomorphism Pic(P ) ' Z.
The above long exact sequence in cohomology is needed to finish off the proof of propo- sition 2.4; every Riemann surface is algebraic. Proof (of proposition 2.4) Let S be a Riemann surface and ds2 be a metric on S with R 2 associated (1, 1)-form ω, normalised such that S ω = 1. Then [ω] ∈ H (S, Z) under the 2 2 2 2 identification H (S, R) ' HDR(S) and the injection H (S, Z) ,→ H (S, R). By the exact sequence 1 ∗ 2 ... −→ H (S, O ) −→ H (S, Z) −→ 0
2 [ω] is a positive form under the identification H (S, Z) ' Z, hence there exists a positive line bundle L with c1(L) = [ω]. By the Kodaira embedding theorem, S can be embedded into projective space. 2 c1 2 Theorem 3.35 The chern class map Pic(S) −→ H (S, Z) for a Riemann surface S co- incides with the degree map deg : Div(S) −→ Z.
For a proof of theorem 3.35, see pages 141-144 of [GH78]. To determine Pic(S) for S a Riemann surface, we need to examine Pic0(S), which is defined as
0 Pic (S) = ker(c1)
Proposition 3.36 We have the following isomorphism of groups,
H1(S, O) Pic0(S) ' H1(S, Z)
48 Proof Denote ϕ : H1(S, O) −→ H1(S, O∗) to be the map in the long exact sequence above. Exactness implies the ker(c1) is isomorphic to im(ϕ). By the first isomorphism theorem
H1(S, O) im(ϕ) ' ker(ϕ)
1 1 1 But ker(ϕ) = im(H (S, Z) −→ H (S, O)) ' H (S, Z) since exactness implies the map, 1 1 H (S, Z) −→ H (S, O), is injective. So we have the required isomorphism. 2 Note 3.37 The above characterisation of Pic0(S) together with the Jacobi inversion theorem says that every point on J (S) corresponds to some line bundle with trivial chern class.
Note 3.38 We can use the above characterisation (c.f. proposition 3.36)
H1(S, O) Pic0(S) ' H1(S, Z) to approach Abel’s theorem another way. The right hand side of the above is actually isomorphic to J (S) via Serre duality (c.f. definition 6.4). So we can form the following sequence Div0(S) ϕ : Div0(S) −→ −→˜ Pic0(S)−→J ˜ (S) PDiv(S) and it can be shown that ϕ agrees with the Abel-Jacobi map.
The map π : Div(S) −→ Pic(S) from theorem 3.28 restricts to a map π : Div0(S) −→ Pic0(S). The Hodge theorem (c.f. theorem 9.3) gives the isomorphism, H1(S, O) ' H0(S, Ω1), so H1(S, O) is naturally a g-dimensional complex vector space. Proposition 3.36 tells us that Pic0(S) for a Riemann surface S is a complex torus and we can pull back the geometry of Pic0(S) to Div0(S)/ PDiv(S) via the map π : Div0(S) −→ Pic0(S). The theorems of Abel and Jacobi can now be summarised by the following commutative diagram µ Div0(S) / J (S) KK u: KK uu KK uu π KK uu µ˜ K% uu Pic0(S)
Proposition 3.39 There is a non-canonical isomorphism of Pic0(S)-sets Picj(S) ' Picj+1(S). Hence Picj(S) ' Pic0(S).
49 Proof We can map Picj(S) −→ Picj+1(S) by D 7−→ D + p for some point p ∈ S. The inverse map is given by D0 7−→ D0 − p for D0 ∈ Picj+1(S). 2 This leads to the following non-canonical characterisation of Pic(S),
[ [ Pic(S) = Picn(S) ' Pic0(S). n∈Z n∈Z
This shows that the moduli space of isomorphism classes of line bundles is indexed by a discrete parameter, as well as a ‘continuous’ parameter. As is typical in problems of moduli, the discrete parameter is usually easier to determine, and as we have seen, a lot more work was required to work out Pic0(S). In section 4.2, example 4.22 we will see Abel’s theorem applied to the classical case of an elliptic curve.
50 Chapter 4
Linear systems and the Riemann-Roch theorem
Linear systems are closely related to divisors and line bundles. Given an effective divisor 1 D on a Riemann surface S, consider the C-vector space
∗ L(D) = {f ∈ K (S) | (f) + D > 0} and one might surmise that we can use the functions in L(D) to map S into projective space, which allows the extrinsic 2 study of S. Denote `(D) = dim(L(D)). The reason why we only consider effective divisors D ∈ Div(S) is due to the following
Proposition 4.1 If deg(D) < 0, then L(D) = 0.
∗ Proof Suppose f ∈ K (S) such that (f) + D > 0, then 0 + deg(D) > 0 contradicting deg(D) < 0. 2 We actually have met the vector space L(D) before. Recall that holomorphic sections σ of O(D) satisfy (σ) + D > 0, hence we obtain the
0 Proposition 4.2 There is a natural isomorphism of C-vector spaces L(D) ' H (S, O(D)).
We can characterise this in terms of divisors. Define, set-theoretically,
0 0 0 |D| = {D ∈ Div(S) | D ∼ D,D > 0} then we have the following
Proposition 4.3 There is a bijection |D| −→ P(L(D)).
Proof Let D0 ∈ |D|, then D0 = D + (f) for some f ∈ K∗(S). The function f is unique 0 up to scalar multiplication, and satisfies (f) + D > 0 since D is effective. So define a 0 map |D| −→ P(L(D)) by D 7−→ [f] where [f] = {λf ∈ P(L(D)) | λ ∈ C}. The inverse is 1This is known as the Riemann-Roch space associated to the divisor D. 2 N That is, we study S via its embedding in P .
51 given by [f] 7−→ D + (f) ∈ |D| and clearly does not depend on choice of representative of [f]. 2 We can now define linear systems for a general compact complex manifold M.
Definition 4.4 Let |D| be the complete linear system associated to the divisor D. The 0 effective divisors corresponding to subspaces of PH (M, O(D)) are called linear systems.
The space |D| contains all effective divisors linearly equivalent to D, hence the name com- plete linear system. The same constructions are valid for any compact complex manifold M, in which case, maps to projective space are even more important- they are candidates N for embedding M into P . If this occurs, then M is algebraic by Chow’s theorem, so we can study M using algebro-geometric techniques.
Example 4.5 Let M be a compact complex manifold, D an effective divisor on M, and 0 3 f0, . . . , fN be a basis for H (M, O(D)). Consider
0 N ιD : M − → PH (M, O(D)) ' P
p 7−→ (f0(p): ... : fN (p)).
The map ϕ is well-defined provided that f0, . . . , fN do not simultaneously vanish at some N point p ∈ M. Let X := {p ∈ M | f0(p) = ... = fN (p) = 0}, then ϕ : M − X −→ P is a well-defined map.
Definition 4.6 The dimension of a complete linear system is defined by
0 0 dim |D| = dim(PH (M, O(D))) = h (M, O(D)) − 1.
Linear systems of dimensions 1, 2, and 3 are respectively known as pencils, nets, and webs.
N In general, for ιD to be an embedding M −→ P , the dimension N of |D| must to be greater than M. Recall in example 4.5, the map ιD is not defined for points where elements of H0(M, O(D)) all vanish. This leads to the following
Definition 4.7 A point p ∈ M is a base point of a linear system W ⊂ |D| if every element of W contains p, that is, for all D ∈ W , D > p. Call the set of all base points of |D| its base locus. A linear system is base point free if its base locus is empty.
We can view this in terms of the sections in H0(M, O(D)).
3The notation − → means a map which is not everywhere defined.
52 0 Example 4.8 Let f1, . . . , fr span a linear subspace W of PH (M, O(D)), then the base locus of the linear system corresponding to W is the set B = {p ∈ M | f0(p) = ... = fr(p) = 0}.
As example 4.5 indicates, the map M− → PW is well defined and holomorphic away from its base locus B. N Given any holomorphic map ϕ : M −→ P , we can pullback the hyperplane divisors N ∗ H ∈ Div(P ) to obtain divisors ϕ H ∈ Div(M), provided that ϕ(M) is not contained in ∗ H. Then the linear system corresponding to the map ϕ is given by {ϕ H} N ∨ . H∈(P ) ,ϕ(M)*H Let ϕ be given by m 7−→ (f0(m): ... : fN (m)), and
X D = − min ordH (fi) H ∈ Div(M). 1 i N H a hypersurface in M 6 6
N ∨ Then if H ∈ P is given by the linear form α0x0 + ... + αN xN , we have
∗ ϕ (H) = (α0f0 + ... + αN fN ) + D.
So the basic correspondence
linear systems |D| holomorphic maps ←→ (4.1) with base locus B ιD : M − B −→ P(L(D)) restricted to base point free linear systems give holomorphic maps M −→ P(L(D)). Note that this is not a priori an embedding. The following proposition gives a condition for
ιD : S −→ P(L(D)) to be an embedding in the case of Riemann surface
Proposition 4.9 Let S be a Riemann surface, and |D| be a base point free linear system.
Then ιD is an embedding iff `(D − p − q) = `(D) − 2 for all p, q ∈ S.
N Proof Let f0, . . . , fN be a basis for L(D), so that ιD : S −→ P . Suppose for some N distinct points p, q ∈ S, ιD(p) = ιD(q). Choose coordinates on P such that ιD(p) = (1 :
0 : ... : 0), then f1(p) = ... = fN (p) = 0. This implies f1, . . . , fN is a basis for L(D − p).
The same argument show that f1, . . . , fN are a basis for L(D − q). So
L(D − p) = L(D − q) = L(D − p − q) (4.2)
The arguments are reversible, so ιD is not one to one iff there exists distinct points p, q ∈ S such that (4.2) holds.
53 Now suppose ιK is one to one. Note that `(D) > `(D − p), with equality iff p is a base point of |D|. Since |D| is base point free, we have `(D − p) = `(D) − 1. Moreover
`(D−p−q) 6 `(D−p)−1, with equality implying (4.2) holds. So `(D−p−q) = `(D)−2. Conversely, if `(D−p−q) = `(D)−2 holds, we must have L(D−p−q) ⊂ L(D−p) ⊂ L(D).
Hence ιD is one to one.
Finally ιD is embedding at p iff there exists f ∈ L(D) vanishing exactly to order 1 at p. That is, there exists f ∈ L(D − p) but f 6∈ L(D − 2p). Now
L(D − 2p) ⊆ L(D − p) so we have `(D − 2p) = `(D − p) − 1 = `(D) − 2. 2 Concerning the generic elements of a linear system away from the base locus, we have Bertini’s theorem. The term generic has a precise meaning in algebraic geometry: a property holds generically on a variety X if it fails to hold on some subvariety of X of strictly lower dimension. An example of this appears in an instance of Bezout’s theorem; m the generic hyperplane H intersects a curve C of degree n in P in n distinct points. This is because H is tangent to C at only a finite number of points, hence this set of points of tangency has dimension 0.
Theorem 4.10 (Bertini) Let W ⊂ |D| be a linear system and B its base locus. Then a generic element D0 ∈ W , D0 6∈ B is smooth.
Bertini’s theorem is the analogue to Sard’s theorem in differential geometry, which states that for a smooth map f between smooth manifolds X and Y , the set {f : X −→ Y | rank(f∗) < dim(Y )} has Lebesgue measure 0. We will not prove Bertini’s theorem, but give an example of how it applies.
n Example 4.11 Let S be a Riemann surface with projective embedding ϕ : S −→ P and ∗ n ∨ ∗ consider the pullback divisors ϕ H where H ∈ (P ) . All such divisors ϕ H have the same degree, and are in fact linearly equivalent, since
ϕ∗H = ϕ∗H0 + (f) where f is the quotient of the linear forms defining H and H0 respectively. The set ∗ n ∨ {ϕ H | H ∈ (P ) } is a complete linear system, since the divisors there correspond to H0(S, O(1)). It is clearly base point free. The divisor ϕ∗H is generic iff H is a generic hyperplane. Since the points of intersection of a generic hyperplane with S are distinct,
54 ∗ so too are the points in ϕ H. In this case, the condition D = p1 +...+pd where p1, . . . , pd are distinct characterises smoothness.
The author admits that the last statement is a bit mysterious, but the notion of smooth- ness cannot be formalised without schemes, which we definitely do not have the space to develop. We will use a similar argument in the proof of the Torelli theorem, so the above italicised statement will be taken as the definition of smoothness in this case. Below are some examples of linear systems arising in algebraic geometry.
Example 4.12
• Suppose deg(D) = d, then there is a natural isomorphism of vector spaces
0 n H (Pz0,...,zn , O(D)) −→ C[z0, . . . , zn]d
This is proposition 3.27, and using the above, we obtain the d-Veronese embed- ding
n N V : P −→ P
(x0 : ... : xn) 7−→ (m0 : ... : mN )
where m0, . . . , mN is the monomial basis of C[z0, . . . , zn]d. This is base point free,
since the equations m0(x) = ... = mN (x) = 0 implies x0 = ... = xn = 0. • The Cremona transformation
2 2 C : P − → P (x : y : z) 7−→ (xy : yz : zx)
0 2 is the map into the subspace of PH (P , O(2)) spanned by xy, yz, zx. The associated linear system has base points where any two of x, y, z are zero, that is, (0 : 0 : 1), (0 : 1 : 0), (1 : 0 : 0).
55 ¢¡¤£¦¥¨§ ©
2 Example 4.13 Denote Hλx+µy+νz ⊂ P to be the divisor corresponding to the line defined by λx + µy + νz = 0. Consider the linear system L corresponding to W = 0 2 span{x, y} 6 PH (P , O(1)). The base locus of L is the point p := Hx ∩ Hy = (0 : 0 : 1). 2 1 The map P −→ PW ' P corresponding to the linear system L is simply projection away from p, and so is not defined at p. Bertini’s theorem is trivial in this case, since the generic divisor Hλx+µy ∈ L is nonsingular, unless λ = µ = 0, that is, at the base point p.
0 2 Example 4.14 Consider the complete linear system corresponding to PH (P , O(2Hx)), and let yx zx yz y2 z2 1, , , , , x2 x2 x2 x2 x2
0 2 be a basis for H (P , O(2Hx)). The linear system |2Hx| is base point free, and any 0 2 D ∈ |2Hx| is given by D = 2Hx + (f) for some f ∈ H (P , O(2Hx)), that is
λ + λ yx + λ zx + λ yz + λ y2 + λ z2 D = 2H + 0 1 2 3 4 5 x x2 2 2 = (λ0 + λ1yx + λ2zx + λ3yz + λ4y + λ5z )
2 = a conic in {(x : y : z) ∈ P | x 6= 0}
In this case, Bertini’s theorem says that the generic conic is nonsingular.
Example 4.15 Recall that the Grassmannian is defined as
n G(r, n) = {V 6 C | dim(V ) = r}.
It is naturally embedded into projective space as follows
^ r n G(r, n) −→ P C
V 7−→ (v1 ∧ ... ∧ vr)
56 where v1, . . . , vr is a basis of the r-dimensional subspace V . Again we can pullback V r n hyperplanes in P ( C ) to obtain a linear system. The Grassmannian will make its appearance again in the proof of the Torelli theorem.
4.1 The Riemann-Roch theorem
This is the principal tool in the study of complete linear systems on a Riemann surface S of genus g. Determining the dimension of H0(S, O(D)) is equivalent to finding the number of linearly independent meromorphic functions satisfying D + (f) > 0, which is not always an easy task. The Riemann-Roch theorem gives a formula for h0(S, O(D)), the caveat is that h0(S, O(K − D)) must also be known.
h0(S, O(D)) − h0(S, O(K − D)) = deg(D) − g + 1 (4.3)
We will see in section 4.2 when one can extract useful information from the above formula. In this section, we will examine some interpretations and a proof of the Riemann-Roch theorem. If we restrict to the case where D is an effective divisor, a geometric interpretation of (4.3) can be given. First we make a
n Definition 4.16 If p1, . . . , pk are points in P , then the linear span of the points p1, . . . , pk is the intersection of all hyperplanes containing p1, . . . , pk. If no hyperplanes contain all n of p1, . . . , pk, then we say the linear span of p1, . . . , pk is P . We can generalise this to the linear span of an effective divisor D ∈ Div(S) where ι : S,→ n n ∨ ∗ P . A hyperplane H ∈ (P ) contains D if ι H > D, then we define the linear span, n ∨ ∗ D, of D to be H ∈ (P ) | ι H > D . If D = p1 + ... + pk where p1, . . . , pk are distinct points, then the linear span of D is precisely the linear span of p1, . . . , pk.
Proposition 4.17 Let D ∈ Div(S) be an effective divisor. Then there is a one to one correspondence between the space of hyperplanes containing D, and |K − D|
g−1∨ ∗ ∗ Proof Let H ∈ P containing D and consider ιK H. Then ιK H ∈ |K − D|, since ∗ ∗ 0 ιK H contains D and by definition ιK H ∈ |K|. Conversely, given any D ∈ |K − D|, 0 0 g−1 ∗ 0 0 0 D + D ∼ K so there exists a hyperplane H in P such that ιK H = D + D, hence H contains D. This gives the one to one correspondence.
g−1∨ 2 Moreover the set of hyperplanes containing D is a linear subspace of P , and its dimension is equal to g − 2 − dim(D) 4. Now applying the Riemann-Roch theorem, we
4 2 Think of lines in P which contain a point p. These lines span a one dimensional space, and the span of p is zero dimensional. Adding these we get 1, which is 1 less than 2.
57 get
h0(K − D) − 1 = g − 2 − dim(D)
h0(D) − (deg(D) − g + 1) − 1 = g − 2 − dim(D)
dim |D| = deg(D) − 1 − dim(D)
The Riemann-Roch theorem in this form relates the dimension of the complete linear system |D| to the geometry of the canonical curve ιK (S). The theorem can also be proven, for effective divisors, purely in its geometric form. For details see pages 248-249 of [GH78]. We can also interpret the Riemann-Roch theorem in the context of sheaves and cohomol- ogy. While not as geometric, this formulation paves the way for a generalisation to higher dimensional varieties 5 and leads to a simple, concise proof using Serre duality.
Definition 4.18 Define the holomorphic Euler characteristic of the line bundle L and M to be the alternating sum
X χ(L) = (−1)php(M, O(L)) (4.4) p∈N
Proposition 4.19 Let L be a line bundle and S be a Riemann surface, then
χ(L) = χ(OS) + c1(L) (4.5) holds and is equivalent to (4.3).
Proof Specialising (4.4) to M = S, we have
χ(L) = h0(S, O(L)) − h1(S, O(L))
= h0(S, O(L)) − h0(S, O(K − L)) (4.6)
0 1 χ(OS) = h (S, O) − h (S, O) = 1 − g (4.7)
Equation (4.6) is due to Serre duality (c.f. theorem 9.5); equation (4.7) holds since the only holomorphic functions on S are the constant ones (c.f. proposition 1.33), and
5This is the Hirzebruch-Riemann-Roch theorem (c.f. page 437 of [GH78])
58 1 0 1 h (S, O) = h (S, Ω ) = g (c.f. example 4.25). By proposition 3.35, c1([D]) = deg(D). Putting these facts together gives equivalence of (4.3) and (4.5).
To prove (4.5), we proceed by induction. First note that if L = OS, then c1(OS) = 0 and (4.5) holds. Now suppose (4.5) is true for L = [D], we show that this implies it is true for L = [D + p] and L = [D − p], hence for all L ∈ Pic(S). The following exact sequence of sheaves over S
ι Resp 0 −→ O(D) −→O(D + p) −→ Cp −→ 0 gives
0 0 0 / H (S, O(D)) / H (S, O(D + p)) / Cp
EDBC / H1(S, O(D)) / H1(S, O(D + p)) / 0 GF@A The alternating sum of dimensions of summands in an exact sequence is zero, so
0 0 1 1 0 = h (S, O(D)) − h (S, O(D + p)) + dim(Cp) − h (S, O(D)) + h (S, O(D + p)) = (h0(S, O(D)) − h1(S, O(D))) + 1 − (h0(S, O(D + p)) − h1(S, O(D + p)))
giving χ([D]) + 1 − χ([D + p]) = 0. By assumption χ([D]) = χ(OS) + c1(D), so
χ([D + p]) = χ([D]) + 1 = χ(OS) + c1(D) + 1 = χ(OS) + c1([D + p]) as required. The proof for L = [D−p] is identical, and we have proved the Riemann-Roch theorem. 2 4.2 Application and Examples
The Riemann-Roch theorem does not always give h0(S, O(D)) easily, due to the appear- ance of the term h0(S, O(K − D)), whose value may not be obvious. However, it turns out that for a generic divisor D = p1 + ... + pd, that is p1, . . . , pd is in general position, the subspace spanned will have dimension g − d. This gives the following
h0(S, O(D)) = d − g + 1 − (g − d) = 1
Also if h0(S, O(K − D)) = 0, which occurs if deg(K − D) < 0, (c.f. proposition 4.1), then
h0(S, O(D)) = d − g + 1
59 So for a generic divisor D,
1 if deg(D) < 2g − 2 h0(S, O(D)) = d − g + 1 if deg(D) > 2g − 2
We emphasise that D 6= K in the above formula, whence h0(S, O(K)) = g as shown below in example 4.25. Applying the Riemann-Roch theorem is somewhat harder when h0(S, O(K − D)) 6= 0, and we call such linear systems |D| special linear systems. Regarding special linear systems, we have Clifford’s theorem,
Theorem 4.20 (Clifford) Suppose D ∈ Div(S) is any special effective divisor such that D 6= 0, D 6= K. Then we have
deg(D) dim |D| 6 2 with equality holding only if S is hyperelliptic.
0 0 Proof Let D1 and D2 be two effective divisors on S. If D1 = D1+(f1) and D2 = D2+(f2) ∗ 0 0 for some f1, f2 ∈ K (S), then D1 + D2 = D1 + D2 + (f1f2). This gives
dim|D1| + dim|D2| 6 dim|D1 + D2| (4.8)
If |D| is a special linear system, then h0(S, O(K − D)) 6= 0, and we can substitute D and K − D into (4.8). By Riemann-Roch, dim |K − D| = dim |D| − (deg(D) − g + 1) so
dim|D| + (dim|D| − (deg(D) − g + 1)) 6 g − 1
2dim|D| 6 deg(D) (4.9)
Equality holds in (4.8) iff |K| = |D1| + |D2| where D1 ∈ |K − D2|. Clearly, then, equality 1 holds if D1 = 0 and D1 = K. Also, let S be hyperelliptic let f : S −→ P be the 1 ∗ 1 double cover to P . Consider the pullback divisor f H for some H ∈ Div(P ), then since 2 deg(H) = deg(f ∗H)
h0(K − π∗H) = g − deg(H)
h0(π∗H) = deg(π∗H) − g + 1 + g − deg(H) deg(f ∗H) dim |π∗H| = deg(H) = 2
60 In fact, these are the only instances where equality in (4.8) holds. Let S be non- g−1 hyperelliptic, and consider the canonical embedding ιK : S −→ P . Recall that every ∗ g−1∨ element in |K| is the pullback ιK H of some hyperplane H ∈ P . Suppose |K| = g−1∨ |D1|+|D2|, and D1 is not 0 or K, so assume deg(D2) 6 g−1. Then H ∈ P contains 0 the points of D2, which are linearly dependent, since h (S, O(K − D2)) > g − deg(D2). This contradicts the fact any g − 1 points of a generic hyperplane section are in general position. Hence if S is non-hyperelliptic equality in (4.9) only holds if D = 0 or D = K.
2 We conclude this chapter with some important applications of the Riemann-Roch theo- rem. The letter g will always denote the genus of S.
Example 4.21 Suppose S has genus 0, then S is isomorphic to the Riemann sphere. Using the Riemann-Roch theorem, we get for any p ∈ Div(S)
h0(S, O(p)) = deg(p) − 0 + 1 + h0(K − p)
= 2 since deg(K − p) = 2 · 0 − 2 − 1 = −3 implies h0(K − p) = 1 by proposition 4.1. Hence 0 1 there must be a nonconstant f ∈ H (S, O(p)), giving the isomorphism f : S −→ P .
Example 4.22 Another application of the Riemann-Roch theorem is to prove the ad- dition law on the elliptic curve. This is a Riemann surface, E, of genus 1. Recall that Abel’s theorem gives the following isomorphism
Pic0(S) 'J (S)
where S is any Riemann surface. On picking an arbitrary base point p0 ∈ S, we aim to show the following isomorphism in the genus 1 case,
ϕ : E −→ Pic0(E)
p 7−→ [p − p0].
Then by Abel’s theorem E ' Pic0(E) 'J (E) and the additive group structure on J (E) pulls back to E via ϕ.
61 Let p, q ∈ E such that p 6= q. If p−q = (f) for some f ∈ K∗(E), then h0(E, O(p−q)) > 0. Now by Riemann-Roch,
h0(E, O(p)) = 1 − 1 + 1 + h0(E, O(K − p)) = 1 (4.10) and by proposition 4.1, deg(K − p) = −1 implies h0(E, O(K − p)) = 0. This means H0(E, O(p)) consists of only the constant meromorphic functions. If σ ∈ H0(E, O(p−q)) then σ ∈ H0(E, O(p)) and σ satisfies σ(q) = 0, so we obtain h0(E, O(p − q)) = 0. Hence p q. Now suppose D ∈ Div1(E). By similar reasoning as (4.10), we have
h0(E, O(D)) = 1
0 so |D| = PH (E, O(D)) '(effective divisors linearly equivalent to D) contains one point, pD. Hence any divisor class, [D], of degree 1 has a unique effective representative, pD ∈ E; and we have an inverse to [·], given by [D] 7−→ pD.
Note 4.23 The argument above using Riemann-Roch fails for higher genus. If S has genus g > 1, then (4.10) becomes
0 0 h (S, O(p)) = 1 − g + 1 + h (E, O(K − p)) 6 0
In fact, a simple dimension count shows, S 6' Pic0(S) for g > 1.
Example 4.24 The third application will be to verify the claim in chapter 2 that all genus 2 Riemann surfaces are hyperelliptic. Since deg(K) = 2, and the canonical map 1 1 ∗ ιK : S −→ P . Pick any point z on P and consider the pullback divisor ιK z. Since ιK is −1 two to one, ιK z = 2p where p ∈ ιK z and 2p ∼ K. So
h0(S, O(2p)) = deg(2p) − g + 1 + h0(S, O(K − 2p))
= 1 + 1.
Hence there exists a nonconstant meromorphic function on S with a double pole at p.
Example 4.25 We can determine deg(K) and h0(S, Ω1) = h0(S, O(K)) (these are equal due to Serre duality and the Hodge theorem H0(S, Ω1) ' H1(S, O) ' H0(S, O(K))). by
62 putting D = K and D = 0 in (4.3). This gives
h0(S, O(K)) = deg(K) − g + 1 + h0(S, O)
h0(S, O) = −g + 1 + h0(S, O(K))
Solving for deg(K) and h0(S, O(K)) simultaneously we obtain
deg(K) = h0(S, O(K)) + g − 1 − h0(S, O)
= h0(S, O) + g − 1 + g − 1 − h0(S, O)
= 2g − 2
h0(S, O(K)) = (2g − 2) − g + 1 + h0(S, O)
= g since h0(S, O) = 1 (c.f. proposition 1.33).
Example 4.26 Recall that a Riemann surface S is hyperelliptic if there exists a mero- morphic function f with a double pole, that is h0(S, O(p + q)) > 1 for all p, q ∈ S. Applying the Riemann-Roch theorem,
h0(S, O(p + q)) = 2 − g + 1 + h0(S, O(K − p − q))
g − 2 < h0(S, O(K − p − q)).
g−1 By proposition 4.9, ιK : S −→ P is not an embedding, note that the converse holds as well.
Example 4.27 In this example, we examine the differences between the canonical curve of a hyperelliptic and non-hyperelliptic Riemann surface. Consider pullback divisors of ∗ g−1g−1 ∗ hyperplanes ιK H ∈ Div(S) where H ∈ P , then ιK H ∼ K so deg(ιK H) = 2g − 2.
In the non-hyperelliptic case ιK is an embedding, so H intersects ιK (S) in 2g − 2 points, counting multiplicity. 1 However in the hyperelliptic case, there is a two to one map f : S −→ P . We claim that the canonical map factors through f, that is, the following commutes
ιK S / g−1 ? P< ?? zz ?? zz f ?? zz ? zz 1 P
63 2 First write the hyperelliptic Riemann surface S as the completion of {(x, y) ∈ Cx,y | 2 1 y = h(x)} for some h ∈ C[x]2g+2, with the two to one covering of P given by π, the projection onto the first coordinate. We can then work out a basis for H0(S, Ω1), firstly 0 1 dy dy/x ∈ H (S, Ω ), and if ω is any holomorphic 1-form on S, we can write ω = p(x) x Lg−1 where p ∈ k=0 C[x]k Hence the following,
dy dy dy , x , . . . , xg−1 x x x is a basis for H0(S, Ω1). With respect to this basis, the canonical map is given by
g−1 ιK : S −→ P (x, y) 7−→ (1 : x : x2 : ... : xg−1).
1 g−1 g−1 It is then clear that ιK = ϕ◦π where ϕ : Px −→ P is given by x 7−→ (1 : x : ... : x ). g−1∨ For any H ∈ P , H intersects ιK (S) at g − 1 points, counting multiplicities; while ∗ the pullback ιK (H) ∈ Div(S) will have degree 2g − 2, since ιK is two to one.
The proof of the Torelli theorem in chapter 7 will make use of both the Riemann-Roch theorem and Clifford’s theorem.
64 Chapter 5
Complex tori
This chapter on complex tori is motivated by the fact that Jacobians are complex tori. A complex torus is defined to be the quotient space V/Λ where V is an n-dimensional complex vector space, and Λ is a lattice spanned by 2n linearly independent vectors in V . It is a K¨ahler manifold with the induced Euclidean metric on V . Complex tori are the simplest examples of compact higher dimensional varieties, although not all complex tori are algebraic.
Definition 5.1 A positive definite Hermitian form H on V such that E := =(H) takes integer values on Λ is called a polarisation of V/Λ. We sometimes also refer to E as the polarisation, since this determines H uniquely.
A consequence of the Kodaira embedding theorem is that any complex torus V/Λ admit- ting a polarisation can be embedded into projective space, since the polarisation guaran- tees the existence of a positive line bundle on V/Λ.
Definition 5.2 A complex torus which admits a polarisation is called an abelian vari- ety.
Note 5.3 The definition of an abelian variety given above is analytic. An algebraic definition can be given, see page 100 of [Pol03]. This makes the definition possible over finite characteristic, and hence is an important construction in number theory.
A polarisation is principal if det(E) = 1 and an abelian variety admitting a principal polarisation is called principally polarised abelian variety. We will write (V/Λ,H) if the polarisation is explicitly given. These are important for two reasons. Firstly the Jacobian is, in fact, principally polarised. We will see why in chapter 6. Moreover there 0 0 1 exists a sublattice Λ of Λ such that C/Λ together with n H for some positive integer n is a principally polarised abelian variety. The existence of a polarisation on V/Λ can be expressed in the following coordinate form, known as the Riemann conditions. First we make a
65 Definition 5.4 Let λ1, . . . , λ2n be a basis for Λ over Z and e1, . . . , en be a basis for V over C. Define the period matrix of V/Λ to be the change of basis matrix Ω ∈ Mn,2n(C), that is, Ω satisfies
λ1 e1 . . Ω . = . . λ2n en
Then the Riemann conditions are given by
Theorem 5.5 The complex torus V/Λ is an abelian variety iff there exists bases given in definition 5.4 such that the period matrix satisfies the following conditions
T Ω = (∆δ,Z) Z = Z =(Z) is positive definite
δ1 .. where ∆δ = . and δ1, . . . , δn are integers satisfying δ1|δ2| ... |δn. δn • • The calculation involves finding conditions relating two bases of H (M, Z) and H (M, C). We will prove this explicitly in the case of the Jacobian variety.
5.1 Cohomology of complex tori
As usual, we will compute the some useful cohomology groups. The first proposition is a simple application of the K¨unneth formula, which relates the cohomology groups of product spaces
m M p q H (X × Y, Z) ' H (X, Z) ⊗ H (Y, Z) p+q=m c.f. chapter 3 of [Hat02].
• Proposition 5.6 We have the canonical isomorphism of cohomology rings H (V/Λ, Z) ' V• 1 H (V/Λ, Z).
Proof Consider the natural map
^ • 1 • α : H (V/Λ, Z) −→ H (V/Λ, Z)
ξ1 ∧ ... ∧ ξn 7−→ ξ1 ^ . . . ^ ξn
66 for all n ∈ N, and extended to other elements via linearity. This is an homomorphism since the cup product is skew-symmetric. The complex torus V/Λ is homeomorphic to (S1)2n, so we apply K¨unneth on the m-th graded piece to obtain
m m 1 2n M i 1 2n−1 i 1 H (V/Λ, Z) ' H ((S ) , Z) ' H 1 ((S ) , Z) ⊗ H 2 (S , Z) i1+i2=m M i 1 i 1 ' H 1 (S , Z) ⊗ ... ⊗ H 2n (S , Z) i1+...+i2n=m 06i1,...,i2n61 M := Hi1,...,i2n
i1+...+i2n=m 06i1,...,i2n61
k 1 since H (S , Z) = 0 for all k > 1, and where the last line is a definition. Similarly
^ m 1 ^ m M i ,...,i H (V/Λ, Z) ' H 1 2n i1+...+i2n=1 06i1,...,i2n61
m 2n V m 1 so h (V/Λ, Z) = m = dim H (V/Λ, Z) . Finally α is surjective since the all cohomology of 2n-tori are cup products of 1-dimensional classes. 2 1 ∨ Now there is a canonical isomorphism H (V/Λ, Z) ' Hom(Λ, Z) := Λ , so for all m ∈ N m Vm ∨ • we have H (V/Λ, Z) ' Λ . The next task is to determine H (V/Λ, O) ∨ Proposition 5.7 We have the isomorphism of cohomology rings H•(V/Λ, O) ' V• V , where V denotes complex conjugation.
Proof See pages 4-5 of [Pol03] for proof. 2 5.2 Line bundles on complex tori
In this section, we classify all line bundles on complex tori. In keeping with the coho- φ mological language, we will show Pic(V/Λ) −→ H1(Λ, O∗(V )) is an isomorphism. Here H1(Λ, O∗(V )) refers to group cohomology, not sheaf cohomology, and is the first cohomol- ogy group of Λ with coefficients in O∗(V ). The necessary definitions for group cohomology are given in section 9.1. ρ Let L −→ V/Λ be any holomorphic line bundle, we recall the pullback bundle π∗L −→ V defined in definition ??, π∗L / L
π∗ρ ρ π V / V/Λ
67 1 1 Recall that H (V, O) = H (V, Z) = 0 (c.f. section 1.5), by the long exact sequence in cohomology H1(V, O∗) = 0, every holomorphic line bundle L −→ V is trivial. So choose ∗ ∗ a global trivialisation ϕ : π L −→ V × C of the pullback π L. We adopt the following −1 ∗ ∗ −1 conventions: for z ∈ V , Lz := ρ (z + Λ), (π L)z := (π ρ) (z), and ϕz := ϕ|{z}. The ∗ ∗ action of Λ on V permutes fibres, for λ ∈ Λ, λ · (π L)z = (π L)z+λ, which induces a map eλ(z): C −→ C via the commutative square,
eλ C / C
ϕz ϕz+λ
∗ λ ∗ (π L)z / (π L)z+λ
∗ Since ϕz is a linear isomorphism for all z ∈ V , eλ(z) ∈ Aut(C) ' C for all λ ∈ V . That ∗ is e :Λ −→ O (V ), and {eλ}λ∈Λ are known as multipliers of the line bundle L −→ V/Λ. ∗ Note that the definition of e depends on the choice of ϕ : π L −→ V × C. For any λ, λ0 ∈ Λ, the following diagrams commute
0 eλ(z) eλ0 (z+λ) eλ0 (z) eλ(z+λ ) C / C /4 C and C / C /4 C
eλ+λ0 (z) eλ+λ0 (z)
Given a representative ε of a class in H1(Λ, O∗(V )), we see that by the 1-cocycle equa- tions de = 1, ε satisfies the conditions above. We can obtain a line bundle on V/Λ by defining the following equivalence relation on V × C, identified with the image of a global ˜ trivialisation of L −→ V . Define (z, `) ∼ (z +λ, ελ(z)·`), then (V ×C)/ ∼ with projection onto the first factor is a line bundle over V/Λ with {ελ}λ∈Λ as multipliers.
φ Theorem 5.8 We have the isomorphism of groups Pic(V/Λ) −→ H1(Λ, O∗(V )).
Proof Note that Λ acts on O∗(V ) by λ · f(z) = f(z + λ). For notations of group cohomology, refer to section 9.1. The map φ : L −→ [e], where {eλ}λ∈Λ are the multipliers of L (with respect to ϕ) and [e] is the equivalence class of e in H1(Λ, O∗(V )), is well- defined. This is because eλ+λ0 (z) = eλ0 (z + λ)eλ(z) is precisely the condition for e ∈ ker(d : C1 −→ C2) by note 9.2. ∗ 0 ∗ Also, suppose ϕ : π L −→ V × C and ϕ : π L −→ V × C are two different global trivial- isations for L over V . Then ϕ0 = f · ϕ for some f ∈ O∗(V ). Hence the multipliers of π∗L −1 with respect to the two different trivialisations are eλ and αλ(z) := f(z + λ)f (z)eλ(z). −1 −1 0 ∗ We conclude α ∈ [e] since αλeλ = f(z + λ)f (z) ∈ dC (Λ, O (V )) again by note 9.2, so φ is well defined.
68 φ 0 φ 0 0 0 Moreover if L 7−→[e] and L 7−→[e ], then L ⊗ L ∈ Pic(V/Λ) has multipliers {eλeλ}λ∈Λ, φ which gives L1 ⊗ L2 7−→[e1 · e2]. Hence φ is a group homomorphism. That φ has an inverse is evident from the discussion immediately preceding the theorem. Denote this inverse β. We check that β is a well defined map in cohomology, that is changing e by a coboundary does not change the line bundle defined by ϕ(e). Let e · ε, 0 ∗ −1 ∗ where ε ∈ dC (Λ, O (V )). So write ελ(z) = f(z + λ)f (z) for some f ∈ O (V ). Then the line bundle defined is (V × C)/ ∼ where V × C is identified with the image of some trivialisation of a line bundle L˜ −→ V and
−1 (z, `) ∼ (z + λ, eλ(z)f(z + λ)f (z) · `).
By discussion above, this amounts to choosing a different global trivialisation L˜ −→ V ×C.
2 The next step is the determination of the chern class of a line bundle specified by {eλ}. First we have
Proposition 5.9 For any non-degenerate, skew-symmetric R-bilinear form E, there ex- ists a basis λ1, . . . , λ2g for Λ such that with respect to this basis, E has the matrix
δ1 0 0 ∆ δ .. where ∆δ = . −∆δ 0 0 δn
Proof This is an easy application of classification theorem of PID modules together with a Gram-Schmidt type argument. Refer to page 304-305 of [GH78] for details. 2 This gives a decomposition, called an isotropic decomposition of the lattice Λ = Λ1⊕Λ2 0 0 0 −1 −1 such that E(λ, λ ) = 0 if λ, λ ∈ Λ1 or λ, λ ∈ Λ2. Let e1 = δ1 λ1, . . . , eg = δg λg be a basis for V . Then we have the following,
E(ei, λj) = 0 and E(ei, λg+j) = δij
Proposition 5.10 Define the multipliers eλ(z) = exp(−2πiE(λ, z)) for all λ ∈ Λ, and denote the line bundle defined by these multipliers LE. Then we have
c1(LE) = E
69 2 V2 ∨ under the identification H (V/Λ, Z) ' Λ . Proof From the proof of lemma 9.1 the coboundary map d : C1(M,G) −→ C2(M,G) where M ∈ G-Mod,
dϕ(g1, g2) = g1 · ϕ(g2) − ϕ(g1 + g2) + ϕ(g1)
1 ∗ for ϕ ∈ C (M,G) and g1, g2 ∈ G. Now for M = O (V ),G = Λ, the abelian group operation in O∗(V ) is written multiplicatively. So we check that e satisfies the 1-cocycle equation de = 1,
λ · eλ2 + eλ1 (de)(λ1, λ2) = eλ1+λ2 exp(−2πiE(λ , λ + z)) exp(−2πiE(λ , z)) = 2 1 1 exp(−2πiE(λ1 + λ2, z)) exp(−2πiE((λ , z) + E(λ , z)) = 2 1 exp(−2πiE(λ1 + λ2, z)) = 1.
Hence e represents a class in H1(Λ, O∗(V )). From the exact sequence
∗ 0 −→ Z −→ O(V ) −→ O (V ) −→ 0 of Λ modules, we extract the following from the long exact sequence
1 ∗ c 2 H (Λ, O (V )) −→ H (Λ, Z).
(recall that the snake lemma holds in any abelian category, in this case Λ-Mod), where c is the connecting homomorphism. Construct the diagram
1 ∗ c1 2 V 2 ∨ H (V/Λ, O ) / H (V/Λ, Z) / Λ qq8 φ qq qqq qqq 1 ∗ c 2 H (Λ, O (V )) / H (Λ, Z)
70 where the two diagonal arrows are natural identifications, and it can be checked that the left hand square commutes. The image of the class represented by e under c is
c(e)(λ1, λ2) = −λ · E(λ2, v) + E(λ1 + λ2, v) − E(λ1, v)
= −E(λ2, v + λ1) + E(λ2, v)
^ 2 ∨ = E(λ1, λ2) ∈ Λ
Hence by commutativity of the above diagram, we have c1(LE) = E. 2 Note 5.11 With the basis given in proposition 5.9, the multipliers become
eλ1 (z) = ... = eλg (z) = 1 and
−2πiz1 −2πizg eλg+1 (z) = e , . . . , eλ2g (z) = e where z = z1e1 + ... + znen. This is the form of the multipliers given in [GH78], and they prove proposition 5.10 by calculations involving the metric and curvature of the line bundle. I have spared the reader from reading a mess of calculations by simplifying the group cohomological argument given in pages 6-7 of [Pol03].
We have constructed line bundles on abelian varieties as quotients of trivial line bundles L −→ V . Furthermore, we have shown how to construct line bundles of any given chern class. To finish this section, we have the
Proposition 5.12 Let τα : V/Λ −→ V/Λ be the translation map τα :[µ] 7−→ [µ + α]. ∗ 1. Let L be a line bundle over V/Λ, then c1(ταL) = c1(L). ∗ 2. If L has multipliers {e : λ 7−→ eλ}λ∈Λ, then for α ∈ V/Λ, ταL has multipliers
{ε : λ 7−→ ελ}, where ελ(z) = eλ(z + α). 0 0 ∗ 0 3. Let L, L be line bundles over V/Λ, then c1L = c1L implies L = τλL . Proof Recall that if two continuous maps between topological spaces are homotopic, then they induce the same map in cohomology. The translation map τα : V/Λ −→ V/Λ for any α ∈ V/Λ is homotopic to the identity, hence the following diagram commutes
∗ τλ Pic(V/Λ) / Pic(V/Λ)
c1 c1 2 id 2 H (V/Λ, Z) / H (V/Λ, Z)
71 giving part 1. Part 2 is clear. For part 3, we show that any line bundle L with c1(L) = 0 has constant multipliers. The following maps between exact sheaf sequences
0 / / O / O∗ / 0 ZO O O id ι1 ι2 ∗ 0 / Z / C / C / 0 induce the following commutative diagram in cohomology
1 exp 1 ∗ c1 2 ... / H (V/Λ, O) / H (V/Λ, O ) / H (V/Λ, Z) / ... . O O O ∗ ∗ ι1 ι2 id
1 1 ∗ 2 ... / H (V/Λ, C) / H (V/Λ, C ) / H (V/Λ, Z) / ... O Hodge
H1(V/Λ, O) ⊕ H0(V/Λ, Ω1)
∗ Under the Hodge decomposition isomorphism, ι1 is projection onto the first factor, hence 1 it is surjective. If e ∈ ker(c1), then e ∈ im(exp). So there is some ξ ∈ H (V/Λ, C) such ∗ ∗ that exp(ι1ξ) = e, and by commutative of the leftmost square, we have e ∈ im(ι2). Hence eλ are constant functions. 2 The last proposition says that the chern class determines a line bundle up to translation.
5.3 Theta functions
π g Given any positive line bundle L −→ C /Λ, the pullback of any global section of of L via g g g C −→ C /Λ is a holomorphic function on C . We call these functions theta functions. Using the multipliers constructed in the previous section, we see that any such function g θ : C −→ C must satisfy the functional equations
θ(z + λj) = exp(−2πiE(λj, z))θ(z) j ∈ [1, g]
0 g The aim is to determine h (V/Λ, O(L)). So let L −→ C /Λ be a line bundle, with
−2πizi V2 ∨ multipliers eλi ≡ 1, eλi+g(z) = e normalised with respect to some given E ∈ Λ . 1 g ∗ We will translate L by µ := 2 (Z11,...,Zgg) ∈ C /Λ and consider τµL. the functional
72 equations become
θ(z + λj) = θ(z) (5.1)
−2πizj −πiZii θ(z + λg+j) = e θ(z) (5.2)
0 0 ∗ for j ∈ [1, g]. Note that h (V/Λ, O(L)) = h (V/Λ, O(τµL)). The translation by µ has the effect of simplifying following proof. We will solve these equations and derive a closed form for the theta functions. The equations (5.1) are periodicity conditions, hence by Fourier analysis, we can write
X −1 −1 θ(z) = α` · exp 2πi`1z1δ1 ... exp 2πi`1z1δ1 g `∈Z X −1 = α` · exp 2πi `, ∆δ z g `∈Z
The equations (5.2) will give recurrence relations for the α`’s,
X −1 −1 θ(z + λg+j) = α` · exp 2πi `, ∆δ z exp 2πi `, ∆δ λg+j g `∈Z by 5.2 X −1 = α` · exp 2πi `, ∆δ z exp (−2πizj) exp (−πiZii) g `∈Z X −1 −1 = α` · exp 2πi `, ∆δ z exp −2πi ∆δξj, ∆δ z exp (−πiZii) g `∈Z X −1 = α` · exp 2πi ` − ∆δξj, ∆δ z exp (−πiZii) g `∈Z X −1 = α`+∆δξj exp (−πiZii) · exp 2πi `, ∆δ z g `∈Z where h·, ·i is the dual pairing on V , and hξj, zii = δij. Comparing the first and last lines give
−1 α` · exp 2πi `, ∆δ z = α`+∆δξj exp (−πiZii) (5.3)
0 0 Let Λ := δ1Z + ... + δgZ be a sublattice of Λ, and identify Λ/Λ with Γ := {γ ∈ Λ | 0 6
γi < δi for all i ∈ [1, g]} ⊂ Λ, then the coefficients {αγ }γ∈Γ determine θ.
Proposition 5.13 The series
X −1 θ(z) = α` · exp 2πi `, ∆δ z (5.4) g `∈Z
73 converges uniformly on any compact subset of C for any choice of α`’s, hence gives a well defined holomorphic function θ : V −→ C.
Proof Let K ⊆ V be compact. We first reorder the summation in (5.4),
X X −1 −1 θ(z) = αγ+` · exp 2πi γ, ∆δ z exp 2πi `, ∆δ z γ∈Γ `∈Λ0 ! X −1 X −1 = exp 2πi γ, ∆δ z αγ+` · exp 2πi `, ∆δ z γ∈Γ `∈Λ0
−1 Let M = supz∈K,γ∈Γ exp 2πi γ, ∆δ z , this exists since K is compact and Γ is finite, then
X −1 |θ(z)| 6 M|Γ| |αγ+`| · exp 2πi `, ∆δ z `∈Λ0
We can solve (5.3), and, omitting the details, we obtain
−1 −1 −1 −1 αγ+` = exp πi ∆δ `,Z∆δ ` + 2πi ∆δ γ,Z∆δ `
0 g 0 or more neatly, let ` ∈ Z such that ` = ∆δ` ,
α 0 = exp (πi h`,Z`i + 2πi hγ,Z`i) γ+∆δ`
Now we can put a bound on the norm of αγ+`,
0 0 −1 0 |α | = α 0 = exp −π ` , =(Z)` − 2π ∆ γ, =(Z)` γ+` γ+∆δ` δ
Since =(Z) is positive definite, so all its eigenvalues are real and positive. Let ρ be the smallest eigenvalue of =(Z), then
0 0 0 0 0 2 ` , =(Z)` > ` , ρ` = ρk` k
Now let P be the largest eigenvalue of =(Z), then
−1 0 −1 0 ∆δ γ, =(Z)` 6 P ∆δ γ, ` 0 0 6 P k` k
74 for some constant P0 > 0 since γ is bounded. So for some constant P00 > 0, we have
00 −1 2 |αγ+`| 6 exp −P k∆δ `k
for all ` ∈ Γ ∩ {z ∈ V | |zi| > R for all i ∈ [1, n]} for some R > 0. Hence the series (5.4) converges on all compact subsets of V . 2 Now since θ(z) is well defined and holomorphic, independently of the coefficients {αγ }γ∈Γ, 0 so we can easily count the dimension of H (S, O(L)). Since |Γ| = δ1 . . . δg, the sections
˜ X −1 θγ(z) = exp 2πi ` + γ, ∆δ z `∈Γ0 span H0(S, O(L)). Hence we have the following
Corollary 5.14 Let L be a positive line bundle, then
0 h (V/Λ, O(L)) = δ1 . . . δg
0 In the special case where c1(L) is a principal polarisation, h (V/Λ, O(L)) = 1 and the vector space H0(V/Λ, O(L)) is spanned by one element, say, θ˜. In the case of a Jacobian, the pullback θ : V −→ C is called the Riemann theta function
X θ(z) = exp (πi h`,Z`i + 2πi h`, zi) n `∈Z and its associated divisor Θ = (θ˜) is called the Riemann theta divisor. It is clear then that θ is an even function
X θ(−z) = exp (πi h`,Z`i + 2πi h`, −zi) n `∈Z X = exp (πi h`,Z`i + 2πi h−`, zi) n `∈Z X = exp πi `0,Z`0 + 2πi `0, z 0 n ` ∈Z = θ(z) (5.5) we shall make use of this fact in the proof of Riemann’s theorem (c.f. theorem 6.14). In fact, the Weierstrass ℘-function in chapters 2 and 6 can be obtained naturally from theta functions (see pages 85-89 of [Cle80]). There are many more applications of theta functions. We will describe one more, recall that the Kodaira embedding theorem gives a
75 k bound k0 such that for all k > k0, the positive line bundle L −→ M gives an embedding of M into projective space. The following Lefschetz embedding theorem uses theta functions to make k0 explicit in the case of complex tori.
Theorem 5.15 (Lefschetz) Let L −→ V/Λ be a positive line bundle and σ1, . . . , σN be 0 k N a basis of H (V/Λ, O(L )). Then for k > 3, the map ϕ : V/Λ −→ P given by p 7−→ N (σ1(p): ... : σN (p)) is an embedding of V/Λ into P .
Proof See pages 321-324 of [GH78] or pages 32-35 (section 3.4) of [Pol03]. 2 We mention the Lefschetz theorem in the spirit of specificity; the Kodaira embedding theorem is a very general existence result, and it is pleasing to know that it can be sharpened in this instance using the intrinsic features of complex tori. As mentioned previously, the theory of theta functions is rich and has wide applications in the study of Riemann surfaces, for example to the Schottky problem, which we will mention in chapter 7, and Riemann’s theorem. A major omission from this section are the theta characteristics, ...References: [Mumford- Tata lectures on Theta]
76 Chapter 6
The Jacobian Variety
The Jacobian variety J (S) is the cornerstone in the study of the Riemann surface S. Torelli’s theorem says that all of the information in S is captured by J (S) and its principal polarisation. Since J (S) is a complex torus, it is ‘linear’ and hence much more accessible to study than S. The construction we give via complex tori is analytic, and a purely algebraic definition can be given. In fact, this approach is the basis of the proof of the Riemann Hypothesis in characteristic p. The Jacobian variety is also directly related to abelian integrals, as we shall see in the following.
6.1 Motivation: Abelian integrals
Historically, such integrals first arose as elliptic integrals, so named for their connection n 2 v2 w2 o with the arc length of an ellipse. Let E := (v, w) ∈ R | a2 + b2 = 1 be an ellipse, then the arc length of E from p to q is given by,
s s Z q dv 2 Z q a4 + (b2 − a2)v2 L := + 1 dv = 2 2 2 dv p dw p a (a − v ) Z q a4 + (b2 − a2)v2 = dv p 2 2 2 4 2 2 2 p a (a − v )(a + (b − a )v )
This can be transformed into a nicer looking integral, let x = a4 + (b2 − a2)v2, then
1 Z q˜ x dx L = · a r p 2 2 4 p˜ 2 x−a4 2 (b − a )(x − a ) x a − b2−a2 1 Z q˜ xdx Z q˜ dx = := x p 2 2 4 2a p˜ x (a b − x)(x − a ) p˜ y on setting y2 = 4a2x a2b2 − x (x − a4). A natural generalisation of this is to consider
Z q I := R(x, y)d x (6.1) p
77 where R is a rational function and x, y satisfies a polynomial equation ρ(x, y) = 0. For deg(ρ) 6 2, I in (6.1) can be expressed in terms of elementary functions. But for deg(ρ) > 2, this is not the case; and such integrals are known as abelian integrals after the Norwegian mathematician Niels Hendrik Abel, who first studied them. Elliptic integrals are then abelian integrals with deg(ρ) = 3 or 4. The following example demonstrates why an elliptic integral is not expressible as elemen- tary functions.
Example 6.1 Consider the elliptic integral
Z q dx E := p p x(x − 1)(x − 2)
2 2 R q dx Let y = x(x − 1)(x − 2) and ρ(x, y) = y − x(x − 1)(x − 2) then E = p y can be 2 2 treated as a line integral on the Riemann surface S := {(x, y) ∈ C | ρ(x, y) = 0} ⊆ C . We saw in the example at the beginning of chapter 2 that S is has topological genus 1. Note that the surface S is not simply connected. So let α, β be a symplectic basis for S and γ, γ0 : [0, 1] −→ S be two paths such that γ(0) = γ0(0) = p, γ(1) = γ0(1) = q and 0 0 γ − γ ∼ nα + mβ for some n, m ∈ Z. Then the difference in evaluating E along γ and γ is
Z dx Z dx Z dx Z dx − = = γ y γ0 y γ−γ0 y nα+mβ y
We show ω := dx/y is a holomorphic differential. Implicitly differentiating ρ, we have
dx 2dy ω = = y 3x2 − 6x + 2 and since 3x2 − 6x + 2 and y are not simultaneously zero 1, ω ∈ H0(S, Ω1). This shows R nα+mβ ω is an element of Λ := ZΠ1 + ZΠ2, where Π1 and Π2 are the periods defined below. The difficulty with E is that it is not a well-defined number- the natural range E is
C/Λ, the Jacobian of S. With essentially the same argument, we can generalise this phenomenon to
R q Proposition 6.2 Let I := p R(x, y)dx be an abelian integral with x and y satisfying ρ(x, y) = 0 for some ρ ∈ C[x, y]. Then the natural range of I is the Jacobian, J (S), of 2 the Riemann surface, S = V (ρ) ⊆ C . 1In general, this depends on ρ having distinct roots.
78 R q dx Example 6.3 Consider the elliptic integral I := 3 . Then we see that q0 4x −g2x−g3
µ : S −→ J (S) Z q dx q 7−→ 3 q0 4x − g2x − g3 is the Abel-Jacobi map. Moreover let ℘ be the Weierstrass ℘-function, then the map
2 C/Λ 'J (S) −→ P z 7−→ (℘(z): ℘0(z) : 1) is the explicit inverse to µ.
6.2 Properties of the Jacobian variety
Much of this section consists of working out some of the constructions of the last chapter in the case of J (S).
Definition 6.4 The Jacobian variety of a Riemann surface S is defined as
H0(S, Ω1)∨ J (S) = H1(S, Z)
1 This definition is intrinsic to S, and note that we clearly have J (S) ' H (S,O) ' Pic0(S) H1(S,Z) 0 1 via Serre duality. Choosing bases ω1, . . . , ωg ∈ H (S, Ω ), and δ1, . . . , δ2g ∈ H1(S, Z) we have the following map
0 1 ∨ g H (S, Ω ) −→ C /Λ
α 7−→ (α(ω1), . . . , α(ωg))
g which realises J (S) explicitly as a complex torus C /Λ, where
Λ = ZΠ1 + ... + ZΠ2g Z Z g Πi = ω1,..., ωg ∈ C . δi δi
This agrees with definition 3.9. We proceed to show J (S) is an abelian variety. First we g choose a nice basis for Λ and C . Proposition 5.9 implies the following,
79 Proposition 6.5 There exists a basis for H1(S, Z), {α1, . . . , αg, β1, . . . , βg}, such that with respect to this basis, the matrix of the intersection form, E, is given by
0 −Ig Ig 0
The basis obtained above is called a symplectic basis. By Poincar´eduality, we can 0 1 R choose a basis ω1, . . . , ωg for H (S, Ω ) dual to α1, . . . , αg, that is, ωj = δij. Then αi clearly we have,
Proposition 6.6 With the bases above, the period matrix of J (S) has the form
Ω := IZ
R where Z = ωj and I is the g × g identity matrix. The columns of Ω, denoted Πi βi i,j for i ∈ [1, 2g] are called the periods of Ω.
As promised, we will verify the Riemann conditions (c.f. theorem 5.5) in the case of J (S), thus showing it is algebraic. First we prove
−1 −1 −1
Proposition 6.7 Let S be a Riemann surface and δ1δ1 δ2δ2 . . . δ2gδ2g be its 4g-polygon
representation (c.f. chapter 3 of [FG01]), as shown¡ below,
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0 1 Let η be a meromorphic 1-form on S with simple poles at p1, . . . , pk, and ω ∈ H (S, Ω ). R R Denote Πi := ω and Ni := η for i ∈ [1, 2g], then we have the δi δi
g k X X Z pi (ΠiNg+i − Πg+iNi) = 2π i respi (η) ω (6.2) i=1 i=1 p0
This is classically known as the reciprocity formula.
80 Proof By choice of the cycle representatives δ1, . . . , δ2g, we can assume without loss of generality that none of the poles of η lie on any δi. Denote the interior of the above polygon P and ∂P be its boundary. Pick a point p0 in P , not a pole of η, and we stipulate that all integrals are taken over paths lying entirely in P , or entirely in ∂P . Consider I(p) := R p ω, this is a holomorphic function on P satisfying dI = ω. We evaluate R ηI p0 ∂P in two different ways to derive (6.2). First we use the residue theorem to obtain
k Z X X Z pi ηI = 2πi resp(ηI) = 2πi respi (η) ω ∂P p∈P i=1 p0
0 −1 Let γ, γ : [0, 1] −→ S parameterise the cycles δi and δi respectively, then η(γ(t)) = η(γ0(1 − t)) for t ∈ [0, 1] so
Z Z Z 1 Z 1 ηI + ηI = (η · I)(γ(t))dt + (η · I)(γ0(t))dt −1 δi δi 0 0 Z 1 = η(γ(t)) I(γ(t)) − I(γ0(1 − t)) dt 0
Now for any t ∈ [0, 1],
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Z γ(t) Z γ0(1−t) Z γ(t) I(γ(t)) − I(γ0(1 − t)) = ω − ω = ω 0 p0 p0 γ (1−t) Z γ0(0) Z Z γ(t) = ω − ω + ω 0 γ (1−t) δg+i γ(1) = −Πg+i
R γ0(0) R γ(t) R R since 0 ω = − ω, giving ηI + −1 ηI = −Πg+iNi. We can similarly derive γ (1−t) γ(1) δi δi R R ηI + −1 ηI = ΠiNg+i, thus we obtain δg+i δg+i
g Z X ηI = (ΠiNg+i − Πg+iNi) ∂P i=1
81 as required. 2 In the genus 1 case, the above reduces to Legendre’s relation Π1N2 − Π2N1 = 2πi, with η(z) = ζ(z)dz where ζ is the Weierstrass ζ-function. We obtain the proof of the Riemann conditions as a corollary.
Corollary 6.8 The period matrix, Ω, of J (S) satisfy the Riemann conditions,
Ω = (I,Z) Z = ZT =(Z) positive definite.
Proof We have seen in proposition 6.6 how to write Ω in the form (I,Z). Let δ1, . . . , δ2g 0 1 be a basis of H1(S, Z) and ω1, . . . , ωg be a basis of H (S, Ω ) giving the period matrix in R the form Ω = (I,Z). Denote Πi,j = ωj for i ∈ [1, 2g] and j ∈ [1, g], then Πi,j = δi,j for δi i, j ∈ [1, g], and substituting this into (6.2), we obtain
g X 0 = (δi,jΠg+i,k − Πg+i,jδi,k) = Πg+j,k − Πg+k,j i=1
Hence Z = ZT . Let I (p) = R p ω and consider the positive definite form (·, ·) = j p0 j R 0 1 i P ωj ∧ ωk on H (S, Ω ),
Z Z i ωj ∧ ωk = i d(Ijωk) P P Z = i Ijωk ∂P g X = i Πi,jΠg+i,k − Πg+i,jΠi,k i=1 = i Πg+j,k − Πg+k,j
= 2=(Πg+k,j)
1 so =(Z) is the matrix of 2 (·, ·) with respect to the chosen basis, hence =(Z) is positive definite. 2 Example 6.9 In the genus 1 case, let E be an elliptic curve. This means that we can write the period matrix as
Ω = (1, τ)
R 0 1 where τ = β ω, α, β symplectic basis for H1(S, Z) and ω a basis of H (S, Ω ) dual to α, and =(τ) > 0. So J (E) is a complex torus C/(Z + τZ)
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We claimed that J (S) has a principal polarisation, this is a consequence of the intersection form on S and the natural isomorphism H1(S, Z) ' Λ.
Proposition 6.10 The intersection form E :Λ × Λ −→ Z induces a unique positive definite Hermitian form H on H0(S, Ω1). Moreover these are related by =(H) = E, and H is a principal polarisation of J (S).
Proof The unique positive definite form on H0(S, Ω1)∨ is given by
H(u, v) = E(iu, v) + iE(u, v).
Also with respect to a symplectic basis,
g −I 0 2g det(E) = (−1) = (−1) = 1
0 I and since det is invariant under basis change, H is a principal polarisation. 2 6.3 Riemann’s theorem
Since J (S) has a principal polarisation given by the intersection form E ∈ V2 Λ∨ (c.f. 0 proposition 6.10), we have by corollary 5.14 h (S, LE) = 1 where LE ∈ Pic(J (S)) has 2 V2 ∨ chern class E under the identification H (J (S), Z) ' Λ . Note that E specifies LE up to translation by proposition ??, so we can associate, up to translation, the divisor
Θ = (θ) to E. Conversely, if a divisor D satisfies c1([D]) = E, then [D] must be a 2 translate of LE. Hence E and Θ up to translation are equivalent data . The divisor Θ is called the Riemann theta divisor.
Denote Θλ = (θ(z −λ)) to be the translated theta divisor, and A·B =intersection number of divisors A and B. First we prove the following lemma
2Some references go as far as saying Θ is the principal polarisation as does [FK92] and [Mum75], we shall refrain from this
83 Lemma 6.11 Suppose µ(S) * Θλ, then µ(S) · Θλ = g, that is, µ(S) and Θλ intersect at g points. Denote these points of intersection z1(λ), . . . , zg(λ) ∈ J (S), then there exists a constant κ ∈ J (S) such that
(z1 (λ) + ... + zg (λ)) + κ = λ (6.3)
Proof First we establish µ(S) · Θλ = g. Let α1, . . . , αg, β1, . . . , βg be a symplectic basis 0 1 for H1(S, Z), and ω1, . . . , ωg be a basis for H (S, Ω ) such that the period matrix of J (S) has the form (I,Z). We will denote Zk to be the k-th column of Z and Zkj to be the −1 −1 −1 −1 k, j-th element of Z. Let P := α1β1α1 β1 . . . αgβgαg βg be the associated 4g-polygon representation, as in proposition 6.7. g The Abel-Jacobi map with respect to a base point z0 lifts to a mapµ ˜ : P −→ C by
Z z Z z µ˜(z) = ω1,..., ωg z0 z0
This is summed by the commutative diagram
µ˜ g P −→ C ↓ ↓ . µ S −→ J (S)
The translated Riemann theta function θλ can be pulled back to P viaµ ˜. Then, after ∗ adjusting α1, β1, . . . , αg, βg such that no zeroes ofµ ˜ θλ lie on the boundary of P , the ∗ number of zeroes ofµ ˜ θλ is equal to the number of points of intersection of µ(S) and Θλ.
Now by continuity of the translatio map θ 7−→ θλ and the residue theorem,
Z ∗ ∗ 1 deg((˜µ θλ)) = deg((˜µ θ)) = d log(θ(˜µ(z)) 2πi ∂P g g 1 X Z Z X Z Z = + + + . 2πi −1 −1 j=1 αj αj j=1 βj βj
R R ∗ • Case + −1 . Let z and z be points on αj such that they are identified on S. αj αj Then we have the following identities
∗ µ˜(z ) =µ ˜(z) + Zj Z θ(˜µ(z∗)) = exp −2πi µ˜ (z) + jj θ(˜µ(z∗)) j 2
84 where the second line is due to the quasiperiodic condition of θ (c.f. (??)). This gives
Z Z Z d log(θ(˜µ(z)) + d log(θ(˜µ(z)) = d log(θ(˜µ(z)) − d log(θ(˜µ(z∗)) −1 αj αj αj Z Zjj = 2πi d µ˜j(z) + αj 2 Z Z z = 2πi d ωj αj z0 = 2πi
R R • Case + −1 . In this case the identities become βj βj
∗ µ˜(z ) =µ ˜(z) − ej θ(˜µ(z∗)) = θ(˜µ(z))
which gives
Z Z Z d log(θ(˜µ(z)) + d log(θ(˜µ(z)) = d log(θ(˜µ(z)) − d log(θ(˜µ(z)) −1 βj βj βj = 0.
Adding everything up we have
g g 1 X X deg((˜µ∗θ )) = 2πi + 0 λ 2πi j=1 j=1 = g.
This gives the first assertion.
To prove (6.3), we use a similar argument. If f has zeroes of orders n1, . . . , nm at z1, . . . , zm, respectively, then, then by the residue theorem
1 Z zd log(f(z)) = n1z1 + ... + nmzm. (6.4) 2πi γ
85 ∗ By above, µ(S) ∩ Θλ are the only points where µ θλ is zero. We consider the i-th com- ponent of µ(S) ∩ Θλ, then by (6.4),
1 Z µ˜i(z1(λ)) + ... +µ ˜i(zg(λ)) = µ˜i(z)d log(θλ(˜µ(z)) 2πi ∂P g g 1 X Z Z X Z Z = + + + (6.5) 2πi −1 −1 j=1 αj αj j=1 βj βj
R R R R where in the last line we have omitted the integrand. The + −1 and + −1 cases αj αj βj βj are dealt with separately. We will denote ξ(z) = log(θλ(˜µ(z)) for concision. R R ∗ • For the + −1 case. Let z and z be points on αj such that they are identified αj αj on S. As before
∗ µ˜i(z ) =µ ˜i(z) + Zij
∗ d ξ(z ) = dξ(z) − 2πiωj(z).
Now
Z Z Z ∗ ∗ µ˜i(z)dξ(z) + µ˜i(z)dξ(z) = µ˜i(z)dξ(z) − µ˜i(z )d ξ(z ) αj αj αj Z = µ˜i(z)dξ(z) − (˜µi(z) + Zij) (d ξ(z) − 2πiωj(z)) αj Z = 2πiωj(z)(˜µi(z) + Zij) + Zijd ξ(z) αj Z t := Aij + Zij d log(θλ(˜µ(z)) s θλ(˜µ(t)) = Aij + Zij log θλ(˜µ(s))
R where Aij = 2πiωj(z)(˜µi(z) + Zij) and s, t are the endpoints of αj. Nowµ ˜(t) = αj
µ˜(s) + ej, so θλ(˜µ(t)) = θλ(˜µ(s)) and the above becomes
θλ(˜µ(t)) Ai∗j + Zijlog = Ai∗j + Zi∗j (Log(1) + 2πi ∗ M) θλ(˜µ(s)) = Ai∗j + Zi∗j2πi ∗ M (6.6)
for some M ∈ Z. Note that none of these depend on λ.
86 R R • For the + −1 case, we have βj βj
∗ µ˜i(z ) =µ ˜i(z) − δij dξ(z∗) = d ξ(z)
where z and z∗ are identified on S. Following the same trail as before,
Z Z Z ∗ ∗ µ˜i(z)dξ(z) + µ˜i(z)dξ(z) = µ˜i(z)dξ(z) − µ˜i(z )d ξ(z ) βj βj βj Z = µ˜i(z)dξ(z) − (˜µi(z) − δij) d ξ(z) βj Z v := δijd log(θλ(˜µ(z))) u θλ(˜µ(v)) = δij log θλ(˜µ(u))
(j) (j) (j) (j) where u and v are the endpoints of βj. Nowµ ˜(v ) =µ ˜(u ) + Zj, so
Z θ (˜µ(v(j))) = exp −2πi µ˜ (u(j)) + ii − λ θ (˜µ(u(j))) λ i 2 i λ
and the above becomes
θλ(˜µ(v)) (j) Zi∗i δijlog = δi∗j −2πi µ˜i(u ) + − λi + 2πi ∗ N (6.7) θλ(˜µ(u)) 2
for some N ∈ Z. Now substitute (6.6) and (6.7) into (6.5),
g g g X 1 X X Zii µ˜ (z (λ)) = A + Z 2πiM + δ −2πi µ˜ (u(j)) + − λ + 2πiM i k 2πi ij ij ij i 2 i k=1 j=1 j=1 g Zii X Aij = −µ˜ (u(i)) − + λ + N + + Z M i 2 i 2πi ij j=1 ∈ λi + κi + Ze1 + ... Zeg + ZZ1 + ... + ZZg
where κi includes all the constant terms not depending on λi. This gives (6.3). 2 Note 6.12 This gives the explicit solution to the Jacobi inversion theorem.
To state Riemann’s theorem, we need to recall the Abel-Jacobi map from (3.4), µ(k) : k (k) Div+(S) −→ J (S). Denote the image of µ , by Wk.
87 Note 6.13 Using this notation, the Jacobi inversion theorem says Wg = J (S).
We can identify Wg−1 with a translate of Θ; this is the content of Riemann’s theorem,
Theorem 6.14 The equation Wg−1 = Θ−κ holds, where κ is defined in lemma 6.11.
Proof First we use the properties of theta functions to show Wg−1 ⊂ Θ−κ. Let D be a (g) generic effective divisor of degree g such that µ(S) * µ (D) + κ =: λ. Now D is generic means we can write D = p1 + ... + pg, where p1, . . . , pg ∈ S are distinct. Applying lemma 6.11, we see that
µ(S) ∩ Θλ = µ(D) = µ(p1) + ... + µ(pg)
Recall that θλ(µ(p1)) = ... = θλ(µ(pg)) = 0, and θ is even (c.f. equation (5.5)), so
θ(µ(p1) + . . . µ(pg−1) + κ) = θ(λ − µ(pg)) = θ(µ(pg) − λ) = θλ(µ(pg)) = 0.
This gives θ−κ(µ(p1) + . . . µ(pg−1)) = 0 for all generic effective divisors D = p1 + ... + ∗ (g−1) pg−1. Thus µ θ−κ vanishes on an open set in S , so is identically zero by analytic continuation. Hence Wg−1 ⊂ Θ−κ. 3 Now Wg−1 is irreducible since it is the image of an irreducible algebraic variety , hence we can write Θ−κ = nWg−1 + Ξ for some Ξ ∈ J (S). The next two lemmas will show that n = 1 and Ξ = 0, thus giving Θ−κ = Wg−1. 2 Lemma 6.15 We have µ(S) · Wg−1 > g and µ(S) · Ξ > 0. Then since
g = µ(S) · Θ−κ = n(µ(S) · Wg−1) + (µ(S) · Ξ)
we conclude µ(S) · Wg−1 = g, µ(S) · Ξ = 0 hence n = 1.
Proof Pick a generic point χ := µ(q1) + ... + µ(qg) ∈ J (S) such that µ(S) * Ξ + χ and
−µ(S) * Wg−1 − χ. Recall that the intersection number of two cycles only depends on their respective homol- ogy classes. Firstly µ(S) is homologous to −µ(S) since the involution J (S) 3 ξ 7−→ −ξ ∈
J (S) on H2(J (S), Z) acts as the identity. Moreover, Wg−1 − ζ is homologous to Wg−1 for a generic choice of ζ ∈ J (S).
3 g−1 g−1 (g−1) S is irreducible so S is irreducible. Hence the image of S −→ Div+ (S) −→ Wg−1 is irreducible. A more detailed argument on why the holomorphic image of an irreducible subvariety is irreducible is given in the proof of the Torelli theorem.
88 The following is the important step
X −µ(pj) = µ(pi) − χ ∈ Wg−1 − χ i6=j for all j ∈ [1, g], hence −µ(S) intersects Wg−1 − χ at (at least)−µ(p1),..., −µ(pg), so
(−µ(S)) · (Wg−1 − χ) > g. Now the intersection number only depends on homology class, and by the discussion in the previous paragraph,
−µ(S) · (Wg−1 − χ) = µ(S) · (Wg−1 − χ)
= µ(S) · Wg−1 > g.
Similarly µ(S) · Ξ = µ(S) · (Ξ + χ). Since µ(S) * Ξ + χ, µ(S) · (Ξ + χ) > 0 and the lemma is proved. 2 Lemma 6.16 The divisor Ξ is zero.
Proof First assume Ξ 6= 0. Suppose µ(p) ∈ Ξλ for some p ∈ S. Then µ(S) ⊂ Ξλ, since otherwise µ(S) · Ξλ > 1 contradicting µ(S) · Ξλ = µ(S) · Ξ = 0. This is true for any λ ∈ J (S).
Now if µ(p0) + µ(q0) ∈ Ξλ for some p0, q0 ∈ S, then
µ(q0) ∈ Ξλ+µ(p0)
so by above, µ(S) ⊂ Ξλ+µ(p0). That is µ(q) ∈ Ξλ+µ(p0) for all q ∈ S. Now
µ(p0) ∈ Ξλ+µ(q)
so µ(S) ⊂ Ξλ+µ(q) for all q ∈ S. These statements imply that for any λ ∈ J (S) if
µ(p0) + µ(q0) ∈ Ξλ for some p0, q0 ∈ S then µ(p) + µ(q) ∈ Ξλ for all p, q ∈ S. That is,
W2 ⊂ Ξλ. Repeating the above argument we have the following, for any λ ∈ J (S) if µ(a1) + ... + µ(an) ∈ Ξλ for some a1, . . . , an ∈ S, then µ(b1) + ... + µ(bn) ∈ Ξλ for all b1, . . . , bn ∈ S that is W2 ⊂ Ξλ.
Now for n = g, the Jacobi inversion theorem states that Wg = J (S), so there exists a1, . . . , ag ∈ S such that µ(a1) + ... + µ(an) ∈ Ξλ for any λ ∈ J (S). By the above
89 argument, this implies J (S) = Wg ⊂ Ξλ, which is a contradiction since Ξλ is codimension
1. Hence Ξ = Ξλ = 0. 2 This completes the proof of theorem 6.14. Riemann’s theorem is significant as it relates two seemingly different objects, Θ−κ which is defined in terms of J (S) and E, and Wg−1 which is defined by the geometry of the Abel-Jacobi map µ and S. This provides the essential link leading to the Torelli theorem.
90 Chapter 7
The Torelli theorem
The Torelli theorem is the theoretical justification of the comment made at the begin- ning of chapter 6, that studying a Riemann surface via its Jacobian incurs no loss of information. The proof relies heavily on Riemann’s theorem (c.f. theorem 6.14). By the discussion in section 6.3, the Riemann theta divisor Θ of a Riemann surface S is specified up to translation by the principal polarisation E; the following proof is a geometric recipe for reconstructing S from Θ. First we introduce some notation. Let X be an analytic variety; we define the singular locus of X, Xsing, to be the union of all singular points of X. Denote Xsm = X − Xsing to be the smooth locus of X.
Theorem 7.1 (Torelli) Let (J (S),H) and (J (S0),H0) be principally polarised Jacobians of S and S0 respectively. If (J (S),H) and (J (S0),H0) are isomorphic as principal po- larised abelian varieties, then S and S0 are isomorphic.
7.1 Proof of the Torelli theorem
This section will be devoted to proving the Torelli theorem.
Definition 7.2 Let M be a complex manifold of dimension n and X be a dimension k analytic subvariety. Define the Gauss map of X to be
sm GX : X −→ G(k, n)
0 0 x 7−→ Tx(X) ⊂ Tx(M)
0 n where Tx(X) is the holomorphic tangent space of X at x ∈ X, identified with C , and n G(k, n) is the Grassmannian of k-planes in C (c.f. example 4.15).
91 Example 7.3 Consider the Abel-Jacobi map µ : S −→ J (S) defined by p 7−→ R p ω ,..., R p ω . p0 1 p0 g g On identification Tp(J (S)) with C , the Gauss map
GS : S −→ G(1, g)
0 0 p 7−→ Tp(S) ⊂ Tp(J (S))
g−1 is simply the canonical map S −→ G(1, g) = P .
sm g−1∨ Lemma 7.4 Consider the Gauss map G :Θ−κ −→ G(g−1, g) = P of the Riemann 2g−2 1 theta divisor. Moreover the generic fibres contains g−1 elements.
sm sm Proof Under the identification Θ−κ = Wg−1, let µ(D) = µ(p1) + ... + µ(pg−1) ∈ Wg−1. g−1∨ The tangent hyperplane at µ(D), G(µ(D)) ∈ P , is simply the hyperplane spanned by the points ιK (p1), . . . , ιK (pg−1). We verify this by computation. Let z0, z1, . . . , zg−1 be local coordinates around the points p0, p1, . . . , pg−1, then
g−1 g−1 ! X Z zk X Z zi µ(D) = µ(p1 + ... + pg−1) = ω1,..., ωg k=1 z0 k=1 z0 g−1 g−1 ! Z zk Z zi ∂ ∂ X ∂ X µ(D) = ω1 ,..., ωg ∂zi ∂zi ∂zi zi=0 k=1 z0 zi=0 k=1 z0 zi=0 = (ω1(pi)/dzi, . . . , ωg(pi)/dzi) =: vi
V g−1 g Under the identification G(g−1, g) ' C , the tangent hyperplane at µ(D) is v1∧...∧ g−1 vg−1 ∈ G(g −1, g). Upon projectivising, we see that vi corresponds to ιK (pi) ∈ P , and g−1 the tangent hyperplane at µ(D) is simply the hyperplane in P spanned by the points
ιK (p1), . . . , ιK (pg−1). From the discussion in chapter 4, these points span a unique hyperplane iff D is regular, that is, if h0(D) = 1 iff h0(K − D) = 1 by Riemann-Roch. Recall that a generic hyperplane intersects µ(S) in 2g − 2 points (c.f. example 4.27). g−1∨ 2g−2 Then for a generic H ∈ P the generic fibre contains g−1 elements. Moreover, each hyperplane intersects µ(S) in a finite number of points, so all fibres of G are finite.
2 We arrive at the proof of Torelli’s theorem. As hinted in section 2.5, hyperelliptic and non-hyperelliptic Riemann surfaces often exhibit different behaviour. Accordingly, the proof will be given in two parts, with the first part covering the non-hyperelliptic case. 0 0 0 Let C := ιK (S),C := ιK (S ) be the canonical curves of S and S respectively; points in
1 sm 2g−2 g−1 ∨ This is another way of saying Θ−κ is a g−1 -sheeted branched cover of (P ) .
92 S will be denoted p1, . . . , pk and their images under ιK in C, ξ1, . . . , ξk. We follow the arguments of [And58], also found in pages 359-362 of [GH78]. Proof of theorem 7.1. Elliptic curve case, g = 1. This is a direct consequence of Abel’s theorem (c.f. example 4.22). g−1∨ Non-hyperelliptic case, g > 3. Let B ⊂ P be the branch locus of G, that is B is the union of the images of the singular points of G. Define,
∨ g−1 ∨ g−1∨ C = {H ∈ (P ) | H is a tangent hyperplane of C} ⊂ P
∨ and call this the hyperplane envelop of C. We will show that C is determined by Θ−κ and J (S), explicitly, B = C∨ where B is the branch locus of G. That is, if two curves, C,C0, have isomorphic Jacobians with the same principal polarisations, then C∨ ' C0 ∨; this is the content of lemma ??. The converse is proved in lemma 7.6; in the non- g−1 hyperelliptic case, the canonical map ιK : S −→ P is an embedding, so this completes the proof of the theorem. 1 Hyperelliptic case, g > 2. The difference here is that the canonical map ιK : S −→ P is not an embedding (c.f. example 4.26), we claim that in this case
∨ ∨ B = C ∪ {p }p is a branch point of ιK
∨ n g−1∨ o g−1∨ where p := H ∈ P | p ∈ H ⊂ P , the dual of p. This is substantiated in lemma 7.7. Now B determines C∨ as well as the ramification points of the two to one 1 map f : S −→ P . By the discussion in section 2.5, this determines S completely. 2 Lemma 7.5 Suppose S is non-hyperelliptic and denote C = ιK (S). The hyperplane envelop, C∨ of C, is equal to the closure of the branch locus B of G. Then since G is ∨ intrinsically defined by J (S) and Θ−κ, so is C .
Proof First define, set theoretically,
n g−1∨ o V = H ∈ P | the intersection of H ∩ C are in not general position
The condition on V is equivalent to the following. There exists g − 1 points out of the 2g − 2 points in H ∩ C which are linearly dependent, that is, whose linear span has
93 g−1∨ dimension less than g − 2. This is a clearly a proper subvariety of P . We wish to show B = C∨ ∩ V c. Consider the maps
(g−1) g−1 µ G g−1∨ S / ∼ −→ Wg−1 −→ P . p1 + ... + pg−1 7−→ µ(p1) + ... + µ(pg−1) 7−→ ιK (p1), . . . , ιK (pg−1)
g−1∨ ∗ If H ∈ P is tangent to C at some point, then the pullback divisor ιK H contains −1 2g−2 multiple points. Hence by the proof of lemma 7.4, |G (H)| < g−1 , hence H is a branch
point of G. This fact is obvious geometrically, consider the following diagram
¡ ¢¤£¦¥¨§ © ¦¥¨¡ !" #$ %¥&
so that an infinitesimal change in the position of P will leave Π stationary, so if z is a ∂ local coordinate for p then ∂z G(z) z=0 = 0. Conversely, suppose H is a hyperplane not tangent to C, so it intersects C at 2g − 2 distinct points, ιK (q1), . . . , ιK (q2g−2), in fact, 0 g−1∨ this is true for any H in some neighbourhood of H in P , hence H is not a singular point of G. Hence
B = C∨ ∩ V c.
We now show C∨ is irreducible. Define 2
g−1∨ I := {(p, H) | H is tangent to C at p} ⊂ C × P and consider I AA π1 ÐÐ A π2 ÐÐ AA ÐÐ AA ÐÐ A C C∨
2I is called the incidence correspondence.
94 where π1 and π2 are the projections onto the first and second factors respectively. First note that π1 and π2 are both surjective and continuous. Now C is irreducible, and the
fibres of π1 are all irreducible, since they are hyperplanes with dimension g−1, we conclude ∨ ∨ that I is irreducible (c.f. [CC04]). Now if C is reducible, write C = ∆1 ∪ ∆2 where
∆1, ∆2 are nonempty closed sets. Then since π2 is surjective and defined on all of I, −1 −1 I = π2 (∆1) ∪ π2 (∆2) is a nontrivial decomposition of I into a union of two closed sets, contradicting irreducibility of I. So C∨ is irreducible. It follows that B = C∨ ∩ V c = C∨, otherwise we have the decomposition C∨ = C∨ ∩ V c∪ (C∨ ∩ V ) into two nonempty closed subsets, contradicting irreducibility of C∨. This gives B = C∨ as required. 2 Lemma 7.6 Suppose C is non-hyperelliptic. Then the hyperplane envelop, C∨, of C determine C up to isomorphism.
Proof Suppose C and C0 are two curves with C∨ = C0 ∨. We claim that there is a well-defined regular bijection
ρ : C −→ C0
0 p 7−→ Tp(C) ∩ C (7.1)
where Tp(C) is the tangent line to C at p. Define the set,
n g−1∨ o Xp := H ∈ P | H contains the tangent line to C at p and consider the linear system, L, obtained by
0 L := {H ∩ C }H∈Xp .
The base locus of L is
\ 0 0 βp := H ∩ C = Tp(C) ∩ C
H∈Xp
∨ since all H ∈ Xp contains Tp(C). Now any H ∈ Xp is tangent to C at p, so H ∈ C . Since C∨ = C0 ∨, H must be tangent to C0 ∨ also. We claim that H is tangent to C0 at
βp. To prove this, recall that Bertini’s theorem (c.f. theorem 4.10) states that the generic element of a linear system away from its base locus is smooth. Applying this to L, we see
95 0 0 that for any H ∈ Xp, H cannot be tangent to C at the points (H ∩ C ) − βp; hence H 0 must be tangent to C at βp. 0 3 For g > 3, we claim that C has no bitangents , so βp is the unique point of tangency of 0 0 0 Tp(C) to C . To see this, suppose ` is a bitangent of C at the points p, q ∈ C . By the geometric version of Riemann-Roch, dim |D| = (deg(D) − 1) − dim(D), where
deg(2p + 2q) dim |2p + 2q| = (4 − 1) − 1 = 2 = 2 and Clifford’s theorem (c.f. theorem 4.20) implies S is hyperelliptic. Hence the map in (7.1) is well-defined and is a regular bijection. 0 For g = 3, we see that Xp = Tp(C), and by proposition 2.8, C only has a finite number of bitangents. So we obtain a rational map
ρ : C − → C0
0 p 7−→ Tp(C) ∩ C
0 defined on the open set consisting of points p, where Tp(C ) is not a bitangent. Now by a theorem in algebraic geometry (c.f. [CC04]), birationally equivalent smooth projective curves are isomorphic, so C ' C0. 2
Lemma 7.7 Suppose S is hyperelliptic, and recall C = ιK (S). Let BιK ⊂ C be the set of g−1 branch points of ιK : S −→ P , then
B = C∨ ∪ {p∨} p∈BιK where B is the branch locus of G.
g−1∨ Proof In the hyperelliptic case, a hyperplane H ∈ P intersects the canonical curve ∗ of S at g − 1 points. So if ιK H contains multiple points, then either H is tangent to S or
H passes through a branch point of ιK . This determines the branch points of ιK and by section 2.5 determines S. 2 A neat picture demonstrating the proof of the Torelli theorem can be drawn for the non-hyperelliptic genus 3 Riemann surface, S. In this case, the canonical map is an 2 embedding, hence we can embed S into P . As with all diagrammatic representations of complex curves, we can only draw a “real” cross section.
3 n n A bitangent to a curve C ⊂ P is a line in P which is tangent to C at two distinct points.
96
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4 x2 2 2 y2 1 The equation of the bottom left curve is given by 7 + y − 1 x + 7 − 1 − 100 = 0.
97 Chapter 8
Concluding remarks
In the proof of the Torelli theorem, we recovered the Riemann surface via its canonical curve. The Jacobian of the Riemann surface S stores its analytic structure, whereas the principal polarisation specifies, up to translation, a divisor in J (S), such that the Riemann surface can be reconstructed from this information. To study Riemann surfaces directly is difficult. Mumford describes in detail in [Mum75] that as the genus of S grows, it becomes increasingly difficult to find explicit descriptions for it. The Weierstrass ℘-function which so neatly does the job in genus 1 has no analogues in higher genera, and as we saw in chapter 2, the number of equations increase as well - 3 three equations are needed already to cut out a genus 2 curve in P . The upshot of Torelli’s theorem is that it is enough to study the Jacobian, with its theta function, in order to study the Riemann surface. For instance, in order to count the number of bitangents of a plane quartic, it is enough to count the so called odd theta characteristics, c.f. pages 150-155 (section 5.2) of [Cle80]. This classification of Riemann surfaces is however not completely satisfactory; for we do not have a description of all the Jacobians of a given dimension. This will be explained in the following section.
8.1 The Schottky problem
From corollary 6.8, we see that the period matrix Ω of J (S) can be given in the form Ω = (I,Z). Define
T Sg := {X ∈ Mg(C) | X = X , =(X) positive definite} then we see that Z ∈ Sg. The space Sg ⊆ Mg(C) is known as the Siegel upper half space. In the case of g = 1, this is simply the upper half plane of C. The information in
Ω determines J (S) completely, so determining which Z ∈ Sg such that (I,Z) is a period
98 matrix is equivalent to identifying all the Jacobians of a given dimension. This is known as the Schottky problem.
In the language of moduli, let Mg be the moduli space of all Riemann surfaces of genus g. Consider the map
Mg −→ Sg S 7−→ Z
associating each Riemann surface of genus g in S ∈ Mg to its period matrix. Torelli’s theorem states that this map is injective, and the Schottky problem is the problem of determining its image in Sg. This is still an open problem; Mumford discusses several approaches to the Schottky problem in chapter 4 of [Mum75].
99 Chapter 9
Background material
The appendix contains the basic definitions of group cohomology and some major theo- rems referred to in the thesis.
9.1 Group cohomology
The point of view of studying G-modules using the fixed point functor ·G, which assigns to a G-module M the abelian group M G := {m ∈ M | gm = m}, leads to group cohomology.
As with the case with sheaf cohomology, exact sequences of G-modules 0 −→ M1 −→ G G G G M2 −→ M3 −→ 0 is carried, under · , to the left exact 0 −→ M1 −→ M2 −→ M3 . A cohomology theory with H0(G, M) = M G will allow us to apply the snake lemma (proposition 1.25), giving the long exact sequence
G G G 0 / M1 / M2 / M3
1 1 1 EDBC / H (M1,G) / H (M2,G) / H (M3,G) / ... @AGF and the cohomology groups are the obstructions for ·G from exactness. Let G be a group and M ∈ G-Mod1, define C0 := C0(G, M) = M, the abelian group Ck := Ck(G, M) = {ϕ : Gm −→ M}, and form the cochain complex C• : 0 −→ 0 d0 1 C −→ C −→ ... with the coboundary map d : Cn −→ Cn+1,
n X j d(ϕ)(g1, . . . , gn+1) = g1ϕ(g2, . . . , gn+1) + (−1) ϕ(g1, . . . , gj−1, gjgj+1, . . . , gn+1) j=1 n+1 +(−1) ϕ(g1, . . . , gn)
Call Hk(M,G) of the complex C• the n-th cohomology group of G with coefficients on M. With this definition, H0(G, M) = M G since dm(g) = gm − m = 0 so m ∈ ker(d : C0 −→ C1) iff m ∈ M G. We will need the following lemma for the proof of theorem 5.8.
1G-Mod denotes the category of (left) G-modules.
100 Lemma 9.1 The kernel of d : C1 −→ C2 is the set of all ϕ : G −→ M satisfying 0 1 g1ϕ(g2) + ϕ(g1) − ϕ(g1g2) = 0 for all g1, g2 ∈ G. The image of d : C −→ C is the set of all ψ : G −→ M satisfying ψ(g) = gm − m for some m ∈ M.
1 Proof Let ϕ ∈ C (G, M), the condition dϕ = 0 implies dϕ(g1, g2) = g1ϕ(g2)−ϕ(g1g2) +
ϕ(g1) = 0, that is, ϕ satisfies ϕ(g1g2) = g1ϕ(g2) + ϕ(g1) for all g1, g2 ∈ G. We have seen above that ψ ∈ dC0(M,G) iff ψ(g) = gm − m for some m ∈ M. 2 Note 9.2 The group operation of M is traditionally written as addition, and the opera- tion of G as multiplication, but in the case of the Λ-module O∗(V ) in theorem 5.8, this is reversed! The above equations become e(λ + λ0) = λ · e(λ0)e(λ) and ε(λ) = (λ · f)f −1 for λ, λ0 ∈ Λ, and e, ε ∈ C1(Λ, O∗(V )), f ∈ O∗(V ).
9.2 Major theorems
We give the statements of the Hodge decomposition theorem for compact K¨ahlermani- folds, the Serre duality theorem, and the Kodaira embedding theorem.
Theorem 9.3 (Hodge) Let M be a compact Kahler manifold, then we have the following
r M q p H (M, C) ' H (M, Ω ) p+q=r Hq(M, Ωp) = Hq(M, Ωp).
Example 9.4 The most basic case of the Hodge decomposition states the following,
1 0 1 1 H (M, C) ' H (M, Ω ) ⊕ H (M, O) ' H1,0(M) ⊕ H0,1(M) ∂ ∂ which is the decomposition of forms into their holomorphic and anti-holomorphic compo- nents.
Theorem 9.5 (Serre) Let M be a connected, compact complex manifold of dimension n. Then the following holds
n n 1. H (M, Ω ) ' C and 2. the pairing Hq(M, Ωp) ⊗ Hn−q(M, Ωn−p) −→ Hn(M, Ωn) is nondegenerate. In particular we have the isomorphism H1(S, O) ' H0(S, Ω1)∨.
Definition 9.6 Let M be a complex manifold. A line bundle is called positive if its 2 chern class can be represented by a positive form in HDR(M).
101 Theorem 9.7 (Kodaira) Let M be a compact complex manifold and L −→ M be a positive line bundle. Then there exists k0 ∈ N such that for all k > k0
0 k N ιLk : M −→ PH (M, O(L )) ' P
N is an embedding of M into P .
102 References
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