These two parts are added as apenddix (application) of the series Riemann-Roch Theorem on Riemann Sruface.

Mittag-Leffler Problem:

{Pī} is a set of discrete points on a Riemann Surface М, near every P¡ there is a principal part. Try to find a meromorphic function fⅰ∈Ш(M) so that in each neighborhood of P¡, f-P¡ is holomorphic.

We want to solve this classcical problem form Sheaf Cohomological point of view. Suppose β is an open cover of M. Without lose of generality, we can suppose that β is fine so that in each {P¡}∩Wɑ there is a gɑ∈Ш(Wɑ). The principal part of gɑ is exact with P¡. Let f(Wα,Wβ)=gβ-gα. So, for every αβγ, f(Wα,Wβ)+f(Wβ,Wγ)=f(Wα,Wγ). We can see now that f∈Z¹(β,θ). If H¹(M,θ)=0, then H¹(β,θ)=0. Therefore, we can find a h∈C°(β,θ), such that f(Wα,Wβ)=h(Wα)-h(Wβ). Form the definition of f, the above equality is gβ-gα= h(Wα)-h(Wβ). SO that on Wα∩Wβ, gα-h(Wα)=gβ-h(Wβ).

We can see that there is an ƒ∈Ш(M), such that f|Wα=gα-h(Wα). This ƒ is the solution of Mittag-Leffler problem.

============================================

We assume that readers have some familiarity of differential form. It does not need too much, for we will use some well-known things.

Since we followed the way complex geometry, we now connect this subject to differential geometry.

Suppose ε is a line bundle, a Hermitian metric on ε is a point-related inner product <,>|x; in which each <,>|x is an inner product on π¯¹(x) (the fiber).

When we said a metric, there must be a metric group on eachπ¯¹(x). As what we always do in linear algebra, this metric can be normalized. So, if we go back to the traditional definition of bundles, we will see that we can select εαβ∈Z1(Uα,Θ*) exactly the transition functions. If we choose e(α) be the section of the line bundle corresponding to the identity element of the fiber over Uα, then let gα=< e(α), e(α)> (for e(α)=εβα e(β)), we will find gα |εαβ|² g¯¹β=1. In this sense, we say we normalized the metric.

If let ★ be the hodge star operator. If T¹, T² are two operators such that T¹, T²: ξ(ε) àξ(ε), we say these two are conjugate if (T¹, )=( , T²). Now we want to find the conjugate operator of ð¯.

Define D’: ξp,q(ε)àξp+1,q (ε);

       If σ=ωe(α), De(α)=θαe(α), define:

       D’σ={ðω+(-1)^p+q ω∧θα};

Why we define like this? For if we investigate the local representation of D (connection, as we do in differential geometry), we can write:

Dσ=(dω) e(α)+(-1)^p (ω∧θα)e(α);

Then, we say ψ=-★D’★ is conjugate to ðˉ.

Define: □=ψðˉ+ ðˉψ=(ψ+ ðˉ)^2, which is self-conjugate and be called Laplacian.   If □φ=0, we say φ is a harmonic form.  Harmonic p,q form is represented by Иp,q, and И=⊕Иp,q; Also we have И(ε)=⊕Иp,q (ε).

The theorem of Hodge is:

If ε is a holomorphic line bundle over a compact Riemann surface, and we have already had a metric of ε, then:

(1). И(ε) is finite dimensional;

(2). There exists a bounded operator G: ξ(ε)à ξ(ε), such that the kernel of the operator KerG=И(ε).  There is a decomposition ξ(ε)= И(ε)⊕□Gξ(ε)= И(ε)⊕G□ξ(ε).

Actually, we can prove that Иp,q (ε)≌Hq(M, Ωp(ε)), where Ωp(ε) denotes the germs of sheaves of holomorphic p-form.

Serre duality Theorem: If (ε) if a holomorphic line bundle over compact Riemann surface M, then there is a isomorphism:

     Hq(M, Ωp(ε))= H1-q(M, Ω1-p(-ε)) for any p,q≧0;

Or equivalently, H¹(M,Θ(ε))=H0(M,Θ(кε¯¹)

Here –ε means the line bundle whose transition functions are εβα∈Z¹(M,Θ*), and ε has transition functions εαβ.

Section II: Sheaf theory

Sheaves are proved to be a very useful tool in the theory of function of several variables. Especially by the cohomology of sheaves, the simplicity of its form enables us to use it to get some very interesting and useful results.

Definition: A sheaf (of abelian groups) over a topological space M is a topological space S, together with a mappingπ: Sà M, such that:

(1). π is a local homeomorphism;

(2). For each point p∈M, the set Sp=π¯(p) has the structure of abelian group;

(3). the group operations are continuous in the topology of S;

The mapπis called the projection, and the set Sp=π¯¹(p) is called the stalk over p. Each stalk is an abelian group, although different stalks may be different groups.

Letπ:Sà M be a sheaf, and U is an open set of M. A section of the sheaf S over U is a continuous map f:UàS such thatπf: UàV is the identity mapping; One should notice that f(p)∈Sp=π¯¹(p) is always right. The set for all sections of S over U is denoted by Γ(U,S). We will notice that for every point s∈S there must be an open set V of S, such thatπ|v:VàU is a homomorphism; so that the inverse (π|v)¯¹: UàV is indeed a section of S over U. Actually, we can topologize S just in this way. If {Uα} is the topological basis of U, then the inverses {π¯(Uα)} is just the topological basis of V.

In a sense, a section of a sheaf can determine the sheaf completely. This observation can be made more precise in the following sense:

Definition: A presheaf (of Abelian group) over a topological space consists of:

(1). A basis {Uα} for the open sets of the topology of M;

(2). A separate abelian group Sp to each open set Uα;

(3). A homomorphism ραβ: SαàSβassociated to each inclusion Uα∈Uβ; such that ραβρβγ=ραγ whenever Uα∈Uβ∈Uγ;

For each sheaf S over M and basis for open sets of the topology for M there is an associated presheaf, which is assigned to each Uα the abelian group Sp=π(Uα,S). The restriction map is easy to see and trivial.

On the contrary, we can construct sheaves from presheaves too. For an arbitrary point p in M, we want to establish a stalk of a sheaf over M from a presheaf. Suppose that there are open sets {Uα} in which p lies. For any two fα∈Sα, fβ∈Sβ, we can define a equivalence class fα≈fβ if and only if ραγfα=ρβγfβ for any Uγ≈∈Uα∩Uβ. The space S=∪Sp and the projection such thatπ(Sp)=p give us a sheaf out of the original presheaf.

We have to notice that given a sheaf S, we get a presheaf of sections of S for some basis {Uα} for the topology of M. The presheaf of sections of S is canonically isomorphic to S itself. However, the sheaf of sections of the associated sheaf to a given presheaf is not always isomorphic to the given presheaf. So, we need the following definition:

Definition: A presheaf { Uα, Sα, ραβ} over a topological space M is called a complete sheaf if, whenever, U’=∪Uβ for a subcollection where {Uβ} is the l basis of {Uα}, the following condition fulfilled:

(1). If there are functions fα∈Sβ, and if for all α, ρβγfυβ=ρυγfυ, then there is an f such that ραγf=fγ.

(2). If f∈Sα such that for each β, ραβ=0, then f=0;

Lemma: A presheaf {Uα, Sα, ραβ} over a topological space M is the presheaf of sections of some sheaf over M if and only if it is complete.

We say the sheaf Θ over M with basis of topology {Uα} is a structural sheaf, if to each set Uα we associated a ring structure ΘUα of functions of holomorphic in Uα. The inclusion relation is the obvious restriction of holomorphic functions. We also call it a sheaf of germs of holomorphic functions on M, and will be denoted by Θ. According to complex analysis, it is easy to see that Θ is isomorphic to the ring С{z} of convergent, complex power series in the variable z.

Note that the presheaf {Uα,ΘUα, ραβ} is a complete presheaf, so with natural isomorphism it is convenient to identify ΘUα with Γ(Uα, Θ). That is, the section of the sheaf Θ over any open set U∈M are identity with the ring ΘU of functions of holomorphic in U.

Because we want to use cohomology of sheaves forth hence, so as the usual cohomology theory, exact sequences are playing very important role in it.  The most basic exact sequences are:

0àΖàΘàΘ*à0, in which е:ΘàΘ*

0àΖàξàξ*à0, in which е:ξàξ* ;

Where Θ* denoted the sheaf of germs of everywhere non-vanishing holomorphic functions and ξ denote the sheaf of germs of infinitely differentiable functions.  We will meet more exact sequences later

Section V: Riemann-Roch theorem

 Recall the exact sequence of sheaves:

               0àΖàΘàΘ*à0, in which е:ΘàΘ*

    So that the associated long exact sequence includes the segment:

          Ｈ¹(M,Z)à Ｈ¹(M,Θ)à Ｈ¹(M, Θ*)àＨ²(M,Z)à Ｈ²(M,Θ)

    SinceＨ²(M,Θ)=0, this exact sequence can be written:

          0àＨ¹(M,Θ)/ Ｈ¹(M,Z)à Ｈ¹(M, Θ*)àＨ²(M,Z)à0

   The characteristic homomorphism is c: Ｈ¹(M, Θ*)àＨ²(M,Z);

   For a line bundle ε∈Ｈ¹(M, Θ*), c(ε) is called the characteristic class or Chern class of the line bundle ε. We have known that the group of line bundles and the group of divisors have a homomorphism relation (isomorphism if modulo the group of principle divisor). Now, the characteristic homomorphism connects three groups together.

   If we assume that a compact Riemann surface is a ;2-dimensional manifold, it is known thatＨ²(M,Z)≌Ζ; so that the Chern class of a line bundle can be considered as an integer, which is rather concise and beautiful. In fact, this identification can be derived as follows. The class c(ε) can be thought an element of the groupＨ²(M,С); for either apply the cohomology homomorphismＨ²(M,Z)à Ｈ²(M,С) derived from the inclusion of the sheaves ΖàС, or use the universal coefficient theorem that Ｈ²(M,С) =Ｈ²(M,Z)⊙С. Under the isomorphismＨ²(M,С)=Γ(M,ξ2)/dΓ(M,ξ1) by De Rham sequence, the cohomology class c(ε) will be represented by the differential form φ(ε)∈Γ(M,ξ²). Under the identification Ｈ²(M,С)=С, ∫∫φ(ε) is belongs to С. Actually, ∫∫φ(ε) is an integer.

Lemma: Let εαβ represent a line bundle as usual, and suppose that {gα} are nowhere vanishing functions defined on the open sets {Uα} satisfying gα |εαβ|² g¯¹β=1. Then φ=1/2πi ðð¯(lg(gα))∈Γ(M,ξ²) is the well defined differential form on M, and c(ε)= ∫∫φ=1/2πi ∫∫ðð¯(lg(gα)).

 More important, we know the Chern class is just the integral of curvature form Θ over M.

Theorem: For any line bundle ε on a compact Riemann surface M, and any non-trivial cross-section f∈Γ(M,Μ*(ε)),

                       c(ε)=∑Vp(f);

Proof: As before, we choose an element εαβ∈Z¹(∪,Θ*) to represent the cohomology group H¹(∪,Θ*).  For the non-trivial cross-section f∈Γ(M,Μ*(ε)) we have fα=εαβ fβ, for each fα∈Γ(Uα,Μ*(ε)). The function fα are infinitely differentiable and nowhere-vanishing in Uα-(∪pi)∩Uα, where {pi} are the set with finitely elements where Vp(f)≠0. So, we can define {gα}: gα=|εαβ|2 gβin Uα∩Uβ, gα=|fα|2 in Uα-∪vi, where {vi} are open sets include pi and “small enough”

 Form this theorem, we can see that the Chern class can be represented by a line bundle, more specifically, by the identification of line bundle with a divisor.

We will use this conclusion to describe the Riemann-Roch Theorem.

Corollary: If ε is a line bundle not identically zero on M such that c(ε)<0, then there is no non-trivial cross-section of the sheaf Θ(ε), or equivalently, Γ(M,Θ(ε))=0;

Now, the preparation for introducing Riemann-Roch theorem is quite enough. From now, we wan to present and discuss this theorem.

Introducing the expression:

     χ(ε)=dimH°(M,Θ(ε))-dimH¹(M,Θ(ε))-c(ε);

and according to Serre duality, we can rewrite the expression as:

     χ(ε)= dimΓ(M,Θ(ε))-dimΓ(M,Θ(кε¯¹)),

where к is the canonical bundle of the surface M. The content of Riemann-Roch theorem is that the expression χ(ε) is independent of choice of the line bundle ε. In order to do this, we have:

Lemma: Let D be a divisor on the compact Riemann surface, and let η=δ*(D) be the line bundle associated to the divisor. Then for any line bundle ε∈H1(M, Θ*),χ(ε)= χ(εη).

Proof:

       Θk(D,ε)={ f∈М|p(ε) ; either f=0 or d(f)≧D near p}

So, we can see Θk(D,ε)≌Θ(εη);

The quotient sheaf S=Θ(ε)/ Θk(D,ε) clearly has the form

                             0   if p≠q;

Sp=

             С  if p=q

     There thus follows the exact sequence of sheaves

                  0àΘ(εη)à Θ(ε)àSà0;

Consider then the associated exact cohomology sequence:

    0àH°(M, Θ(εη))àH°(M, Θ(ε))àH°(M,S)àH¹(M, Θ(εη))àH¹(M, Θ(ε))àH¹(M,S)à………

Since S is a skyscraper sheaf, have stalk С at a single point, it follows readily that H°(M,S)≌С and H¹(M,S)=0. Now in the exact sequence above, the alternating sum of the dimensions of the vector spaces is zero; this is can be rewritten as the equality:

dim H°(M, Θ(εη))- dimH¹(M, Θ(εη))+1=

                                                   dim H°(M, Θ(ε))-dim H¹(M, Θ(ε));

Recall the previous proof of the theorem, given a functions dα corresponding to the divisor D|Uα, there are functions fα=f/ dα∈Мp, where f is a given function f∈Θk(D)|p∈Мp. We can choose 1/ dα as the meromorphic cross-section of the sheaf М*(D). SO, under this convention the total order of the divisor D is thus the negative of the total order of any meromorphic cross-section of its associated line bundle. Here, we see c(η)=-1. Recalling that c(ε)+c(η¯¹)=c(εη¯¹), it follows that χ(εη¯¹)= χ(ε).

If you remember previously that we said on a compact Riemann surface M, H¹(M,М*)=0; equivalently, every line bundle on M has a non-trivial meromorphic cross-section, hen every line bundle is the bundle of a divisor. So, we have the Theorem:

On a compact Riemann surface M, then characteristic χ(ε)=dimH°(M,Θ(ε))-dimH¹(M, Θ(ε))-c(ε) is a constant, independent of the choice of the line bundle ε.

Given this theorem, we set ε=1, the trivial bundle, and note (by using Serre duality) that

      x(1)=dimΓ(M,Θ)-dimΓ(M.Θ1,0)-c(1)=1-dimΓ(M.Θ1,0).

The constant g= dimΓ(M.Θ1,0) is the dimension of the space of abelian differentials on the surface M, is called the genus of the surface M. This constant has a simple explanation as follows. Considering the exact sequence of sheaves:

                   0àСàΘàΘ1,0à0;

The associated exact cohomology sequence has the form:

0àH°(M,С)àH°(M,Θ)à H°(M,Θ1,0)àH¹(M,С)àH¹(M,Θ)à

   H¹(M,Θ1,0)àH²(M,С)à0;

Since H²(M,Θ)=0. Now we have H°(M,C)= H°(M,Θ)=C and H²(M,С)=C. Therefore, recalling that the alternating sum of the dimensions of the terms in a finite exact sequence of vector spaces is zero, it follows that:

dimH°(M,Θ1,0)-dim H¹(M,С)+dim H¹(M,Θ)-dim H¹(M,Θ1,0)+1=0

By definition, dimH°(M,Θ1,0)=g, and by the Serre duality

    dim H°(M,Θ)= dimH°(M,Θ1,0)=g;

    dim H¹(M,Θ1,0)= dimH°(M,Θ)=1;

therefore, dimH¹(M,С)=2g;

Consider the canonical bundle κ, then:

1-g=χ(κ)= dimH¹(M,Θ(κ))-dimH¹(M,Θ(κ));

C(κ)=2g-2.

Theorem: if the fine resolution of the sheaf S is stated as above over a paracompact Hausdorff space M, then

                 Hq(M,S)==(Kernel dq*)/(Image dq-1*);

   Proof: split the exact sequence of resolution

                 0àSàS0àК1à0;

                 ………

0àКiàSiàКi+1à0;

         Then one can get a long exact sequence:

            …àΗq-1(M, S0)à Ηq-1(M, К1)à Ηq(M, S)à Ηq(M, S0)à…

From this we can prove the theorem (details be ignored).

By using above theorem, we can obtain a very important theorem in name of Dolbeault.

Theorem: Let g∈ξ∞(M) and let D be a connected open subset (if M itself is open and connected, this theorem is also true for M) of the complex line C such that D^∈M is compact. Then there exists a function f∈ξ∞(M) such that øf/ðz=g(z) whenever z∈M.

The proof of the theorem is to define the function

f=1/2πi∫f(z)/(z-w)dz∧dž

and prove this f is ξ∞ and øf/ðz=g(z).

Given Dolbeault theorem, consider the fine resolution of sheaves:

0àθàξ∞àξ*∞

And the exact sequence of abelian groups

               0àΘàξ∞-ð¯->ξ*∞à0

So that, we have:

         Ｈ1(M, Θ)=Γ(M,ξ∞)/ ø/ð(Γ(M,ξ∞));   and

         Ｈq(M, Θ)≌0;     for q≥2

Corollary: If M is a connected open subset of C, then

                Ｈq(M, Θ)≌0;     for q≥1

It is very improper if we repeatly saying that cohomology theory of sheaves is a very powerful tool but give no example. We shall see that we can use this tool to solve the traditional Mittag-Leffler Problem.

{Pī} is a set of discrete points on a Riemann Surface М, near every P¡ there is a principal part. Try to find a meromorphic function fi∈М(M) so that in each neighborhood of P¡, f-P¡ is holomorphic.

We want to solve this classical problem form Sheaf cohomological point of view. Suppose β is an open cover of M. Without lose of generality, we can suppose that β is fine so that in each {P¡}∩Wɑ there is a gɑ∈М(Wɑ). The principal part of gα is exact with P¡. Let f(Wα,Wβ)=gβ-gα. So, for every α,β,γ, f(Wα,Wβ)+f(Wβ,Wγ)=f(Wα,Wγ). We can see now that f∈Z¹(β,Θ). If H¹(M,Θ)=0, then H¹(β,Θ)=0. Therefore, we can find a h∈C°(β,Θ), such that f(Wα,Wβ)=h(Wα)-h(Wβ). Form the definition of f, the above equality is gβ-gα= h(Wα)-h(Wβ). So that on Wα∩Wβ, gα-h(Wα)=gβ-h(Wβ).

We can see that there is an ƒ∈М(M), such that f|Wα=gα-h(Wα). This ƒ is the solution of Mittag-Leffler problem.

                           Riemann-Roch Theorem on Riemann Surface    

School of Mathematics, Peking University

                                            Sun Fei

Section Ⅰ: Introduction

I want to discuss basically on Riemann-Roch Theorem, which is probably the most important theorem in the area of complex analysis and a basic tool in algebraic geometry.

Definition:

By a complex manifold M, we mean a connected, T², paracompact topological space with a cooperate covers {Ui,φi}, satisfying:

(1)   each Ui is an open set of M, and ∪Ui = M;

(2)   φi: Ui àСn, andφi: Ui àφi (Ui) is homeomorphism and φi (Ui) is open inСn;

(3)   For each i,j φi (φ¯¹j) is holomorphic;

A complex manifold with dimension one is called a Riemann surface.

I want to talk a little more about complex manifold and Riemann surfaces. We

call f is holomorphic if for each i f(φi) is holomorphicφi (Ui)àС. Moreover, if M and N are two complex manifolds, f: MàN is a map for each i,j such that ψfφ¯: φi(Ui)àψi(Vi) is holomorphic, we say that f is holomorphic map between M and N.

If f:MàN is a mapping between Riemann surfaces, for any p∈M there is a coordinate open set U and coordinate mapping z such that w(p)=0; and for the point q=f(p), there is a coordinate open set V and coordinate mapping w such that w(q)=0. So, we can select proper coordinate open sets and coordinate mappings such that for any point p∈M, we have w=z^n;

The index n is called the order of the mapping f. We can see that the index n can be either positive or negative, depends on the point p. If, plus, we have n-1>0, the point p is called a ramification point of the cover mapping f.

If we want to discuss something about Riemann surface, we can do it several ways in which view points of complex manifolds and algebraic geometry are better known by me. It is true that Riemann surfaces can also be considered form algebraic number theory point of view, however, this way is not what I want to follow. My major concentration is lie on the complex manifolds, which is the main method I use in this article. That is to say, I will follow this way: complex manifolds (Riemann surfaces)à sheaf theoryà cohomology theoryà divisors, line bundles and Riemann-Roch theorem. Although algebraic geometry and algebraic number theory viewpoints are not paid major care, I will still try to talk about these a little. To do this, let me introduce projective spaces and algebraic curves first.

Let us see something interesting first. We have to say that retina is flat or 2-dimensional, but the real world is “at-least” 3-dimensional. Therefore the image formed on our retina can only be 2-dimensional, so our feeling reflected by what we see are “projective”. By a projective (complex) space СРn, we mean the set of one dimensional spaces in Сn+1. Specifically, a projection ∏: Сn+1-{0}àСРn, ∏(Z0,Z1,…,Zn)=[Z0,Z1…Zn]. The open sets in СРn is the image of open sets of Сn+1-{0}. The coordinate covers of СРn are:

        U0={[Z0,Z1,…,Zn] | Z0≠0};

        U1={[Z0,Z1,…,Zn] | Z1≠0};

           ………

        Ui={[Z0,Z1,…,Zn] | Zi≠0};

           ………

        Un={[Z0,Z1,…,Zn] | Zn≠0};  

The coordinate maps are:

            Φi:UiàС, φi([Z0,Z1,…,Zn])=(Z0/Zi,Z1/Zi,…,Zn/Zi);

It is easy to check that СРn is a complex manifold. In a special case, we can see that СР1 is just the Riemann sphere S. In order to generalize this kind of identification, we may try to understand projective spaces from another way.

We have already seen that the definition of projective spaces is to identify each 1-dimensional space in СРn to one point. So, consider the hyperplane Сn+1, select a point B=(0,0,0,…,1) and consider each line pass O and pass through the hyperspace H perpendicular to OB. We can identify each point p in H to the line pB. There are lines pass B but parallel to H. We may think these lines “intersect” H in the infinity. So, we can see that the projective space СРn=СРn-1 + Сn; for each n, we may identify СРn-1 to the infinity “point”. So, each projective space is the union of complex plane and the infinity “point”. Specifically, in СР2, is the union of С2 and infinity, that is, the Riemann sphere S. We can easily see that in this definition the projective spaces are compact.

We will also try to define complex algebraic curves.        

By a algebraic manifolds, we mean a manifolds in СРn defined by a group of homogenous polynomials {Pi(Z1,Z2,…,Zn)}; I concentrate this part of discussion on algebraic curves in СР2, that is, algebraic manifolds defined by a homogenous polynomial of three variables.

First, let us see a very simple example. Consider the homogenous polynomial x³+y³=z³ in СР2; Let us first think the case z≠1 (which means we do not consider what happened in the “infinity” first). The polynomial is u³+v³=1, that is v³=1-u³; Then this function has three solutions, 1, ω,ω’(conjugate). Locally, these define three holomorphic functions of u called branches of the multivalued function v(u). When u travels around {1, ω,ω’} the value of v changes from one branch to another. If we slice the complex plane alone the line segment [1, ω] and [1,ω’], we will find that there are three single-valued holomorphic functions fi (i=1,2,3) defined on the cut plane D satisfying fi³(u)=1-u³. So that the subsets

{[u,v,1]∈C, u∈D}=∪{[u,v,1]∈СР2; u∈D, v=fi(u)};

Of С is the disjoint union of the three copies of the cut plane D. If, go back to the case with “infinity” added, we glued D∪{∞} appropriately, corresponding to the way the multivalued function jumps from one branch fi to another when v cross the cuts. When I mean appropriately, we could do it as the figures show:  

图表 1        cut along {1, ω,ω’}

图表 2

In this way, we can imagine visually that an algebraic manifold defined by a homogenous polynomial is a Riemann surfaces. Actually, by degree-genus formula, we will know that a nonsingular, irreducible homogenous polynomial defined inСР2 is topologically a Riemann surface, and the genus of 2-dimensional surface is determined by the degree of the homogenous polynomial. Follow this way, we can prove the Riemann-Roch theorem, and I will go back to this if I can.