AN ANALYTIC APPROACH TO THE THEOREMS OF RIEMANN-ROCH AND ABEL MICHAELWONG Abstract. Developing foundational notions in the theory of Riemann sur- faces, we prove the Riemann-Roch Theorem and Abel’s Theorem. These no- tions include sheaf cohomology, with particular focus on the zeroth and first cohomology groups, exact cohomology sequences induced by short exact se- quences of sheaves, divisors, the Jacobian variety, and the Abel-Jacobi map. Thegeneralmethodofproofinvolvesbasicalgebraandcomplexanalysis. Contents 1. Introduction 1 2. Sheaves and Sheaf Cohomology 2 3. The Exact Sequence of Sheaves 5 4. The Genus of a Compact Riemann Surface 7 5. The Riemann-Roch Theorem 9 6. A Word on E(0,1)(X) 11 7. The Jacobian Variety and the Abel-Jacobi Map 13 8. Abel’s Theorem 15 9. Closing Remarks 18 Acknowledgments 19 References 19 1. Introduction The theorems of Riemann-Roch and Abel are two fundamental results in alge- braicgeometry. Aspartofthe“amazingsynthesis”whichDavidMumforddescribes in the appendix to The Red Book of Varieties and Schemes, these theorems can be proved analytically when we restrict to algebraic curves over C. In the following exposition, we pursue the analytic viewpoint from the theory of Riemann surfaces, incorporatingsheafcohomology, Jacobianvarieties, andtheAbel-Jacobimap. Our hope is that the method of proof, while drawing only from basic algebra and com- plex analysis, gives a small hint of some “synthesis.” Definition 1.1. A Riemann surface is a pair (X,Σ) where X is a connected man- ifold of real dimension 2 and Σ is a complex structure. We assume basic knowledge of Riemann surfaces, including complex structure, holomorphic and meromorphic functions, differential forms, and integration of dif- ferential forms. Also, elementary facts about meromorphic functions on compact Date:August26,2011. 1 2 MICHAELWONG Riemann surfaces will be used freely, e.g., the number of poles minus the number of zeros of a meromorphic function, counted with multiplicity, is 0. Notation 1.2. For each open U ⊂X, let (1) E(U), O(U), M(U) be the sets of smooth functions (in the R2 sense), holomorphic functions, and meromorphic functions on U, respectively, (2) E1(U),E2(U)E(1,0)(U),andE(0,1)(U)bethesetsofsmooth1-forms,smooth 2-forms, smooth 1-forms of type (1,0), and smooth 1-forms of type (0,1) on U, respectively, (3) and Ω(U) be the set of holomorphic 1-forms on U. Each set is an abelian group under pointwise addition of functions. Additionally, weassumebasicknowledgeaboutholomorphicmapsbetweenRie- mann surfaces. We state briefly the main results we shall need (See [4]). Theorem1.3(LocalNormalForm). Letf :X →Y beanonconstantholomorphic map between two Riemann surfaces. For each x∈X, let (V,ψ) be a chart centered at f(x). Then there exists a chart (U,φ) centered at x and integer m≥1 such that F(z):=ψ◦f ◦φ−1(z)=zm. The integer m is independent of charts and is called the multiplicity of f at x, denoted mult f. x A point x ∈ X such that mult f > 1 is called a ramification point of f, and a x point y ∈ Y that is the image of a ramification point under f is called a branch point of f. A nonconstant holomorphic map f : X → Y between Riemann surfaces is an open map. If in addition X is compact, then f is surjective and finite-to-one. Furthermore, f has only finitely many branch points, y ,y ,...,y . It follows from 1 2 k Theorem 1.3 that f is an n-sheeted covering map over Y −{y ,...,y } for some 1 k n≥1. Finally, given an open covering of a compact Riemman surface, we shall assume the existence of a partition of unity subordinate to the open cover. 2. Sheaves and Sheaf Cohomology In this section, we introduce the notions of sheaves and sheaf cohomology. With aneyetowardtheRiemann-RochTheorem,wefocusontheconstructionofonlythe zeroth and first cohomology groups, but the construction can be easily generalized to the higher-order groups. Definition 2.1. Let X be a topological space. Let F ={F(U)|U ⊂Xopen} be a collection of abelian groups and ρ={ρU :F(U)→F(V)|V ⊂U ⊂Xopen} V be a collection of group homomorphisms with the conditions ρU =id U F(U) and ρV ◦ρU =ρU W V W AN ANALYTIC APPROACH TO THE THEOREMS OF RIEMANN-ROCH AND ABEL 3 for all open W ⊂ V ⊂ U ⊂ X. For f ∈ U, we shall denote ρU(f) as f| . In the V V above notation, a sheaf on X is a pair (F,ρ) satisfying the following sheaf axioms. For any open U ⊂X and any open cover {U } of U, i i∈I (1) if f,g ∈F(U) and f| =g| for all i∈I, then f =g ; Ui Ui (2) if f ∈F(U ) and f =f in F(U ∩U ) for each i,j ∈I , then there exists i i i j i j f ∈F(U) such that f| =f for all i∈I. Ui i For each function listed in 1.2, the collection of abelian groups ranging over all open U ⊂X with the natural restriction maps makes a sheaf on X. The elements of ρ are called restriction homomorphisms in light of such examples. Given a sheaf F on a topological space X, we would like to examine the sheaf structure locally, especially when the group elements are functions and the ho- momorphisms are the natural restriction maps. For x ∈ X, the set I := {U ⊂ X open|x∈U} is a directed set under inclusion, and the sets {F(U)|U ∈I} and {ρU |V ⊂ U and U,V ∈ I} together form a directed system. Then the stalk of F V at x is the direct limit F :=limF(U). x → Definition 2.2. Let F be a sheaf on a topological space X. Let U = {U } be i i∈I an open cover of X. A q-cochain is a tuple (f ) where i0,i1,...,iq (i0,i1,...,iq)∈Iq f ∈F(U ∩U ∩···∩U ). i0,i1,...,iq i0 i1 iq The set of all such q-cochains under addition is an abelian group called the qth- cochain group with respect to U, denoted by Cq(U,F). The cochain groups lend themselves to a chain complex, constructed as follows. Fix a covering U of X. Define δ0 :C0(U,F)→C1(U,F) by δ0(f ) =(f ) i i∈I ij i,j∈I where f =f −f for all i,j ∈I ; here, the operation is understood to be on the ij i j restrictionsoff andf toU ∩U . Inlikemanner,defineδ1 :C1(U,F)→C2(U,F) i j i j by δ1(f ) =(f ) ij i,j∈I ijk i,j,k∈I where f =f −f +f for all i,j,k ∈I. Again, the operations are understood ijk jk ik ij to be on the restrictions to U ∩U ∩U . Clearly, δ0 and δ1 are homomorphisms, i j k and δ1◦δ0 =0. Hence, the sequence 0 (cid:47)(cid:47)C0(U,F) δ0 (cid:47)(cid:47)C1(U,F) δ1 (cid:47)(cid:47)C2(U,F) is a chain complex, and we define the first cohomology group with respect to U to be H1(U,F):=Kerδ1/Imδ0. Similarly, we define the zeroth cohomology group with respect to U to be H0(U,F)=Kerδ0/Im(cid:0)0→C0(U,F)(cid:1)=Kerδ0. For q = 0,1, elements of Zq(U,F) := Kerδq are called q-cocycles. Observe that 0-cocycles satisfy (2.3) f =f , for alli,j ∈I i j and 1-cocycles satisfy (2.4) f =f +f for alli,j,k ∈I. jk ik ij 4 MICHAELWONG Elements of Imδq are called q-coboundaries. If (f ) ∈ Z1(U,F) is also a 0- ij i,j∈I coboundary, then there exists (g )∈C0(U,F), such that i f =g −g for alli,j ∈I ij i j and (f ) is said to split. ij Asdefined,thezerothcohomologygroupispracticallyindependentofthecover. Therelation(2.3)andsheafaxiomIimplythatforevery(f )∈Kerδ0,thereexists i f ∈ F(X) such that f| = f for all i ∈ I. So H0(U,F) ∼= F(X). We therefore Ui i define the zeroth cohomology group of X with coefficients in F to be simply H0(X,F):=F(X). However, the first cohomology group generally depends on the cover. To elimi- natethisdependence,wemakethecollectionofopencoveringsofX intoadirected set and take a direct limit. Let U = {U } be an open cover of X. An open cover i V ={V } of X is a refinement of U, denoted V ≤U, if for every k ∈K, there k k∈K exists an i ∈ I such that V ⊂ U . Given a refinement V, let t : K → I be a map k i sending k to an i such that V ⊂ U . Then define τU : C1(U,F) → C1(V,F) by k i V τU(f )=(g ) where V ij kl g =f | for allk,l∈K. kl t(k),t(l) Vk∩Vl Asiseasilyverified,τU preserves1-cocyclesand0-coboundaries. Hence,τU induces V V a homomorphism τU :H1(U,F)→H1(V,F). V One can demonstrate, in fact, that this map on cohomology with respect to open covers is independent of the function t, is injective, and satisfies two conditions: τU =id U H1(U,F) and τV ◦τU =τU, for allW ≤V ≤U. W V W Then we define the first cohomology group of X with coefficients in the sheaf F to be H1(X,F):=limH1(U,F). → Proposition 2.5. Let X be a compact Riemann surface. Then H1(X,E)=0. Proof. Let U = {U } be an open cover of X. There exists a partition of unity i i∈I {ψ } subordinate to U. Take (f ) to be a representative of an element of i i∈I ij i,j∈I H1(U,E). Then for each j ∈I, ψ f can be extended smoothly to U by giving it j ij i the value 0 outside its support. Let (cid:88) g := ψ f . i j ij j∈I For each x ∈ X, g (x) has only a finite number of summands, so g is indeed an i i element of E(U ). Observe that, on U ∩U , i i j (cid:88) (cid:88) (cid:88) (cid:88) g −g = ψ f − ψ f = ψ (f −f )= ψ f =f . i j k ik k jk k ik jk k ij ij k∈I k∈I k∈I k∈I So (f )∈Imδ0, implying H1(U,E)=0. (cid:3) ij As the next theorem shows, cohomology is the same as cohomology relative to a special open cover, called a Leray cover. AN ANALYTIC APPROACH TO THE THEOREMS OF RIEMANN-ROCH AND ABEL 5 Theorem 2.6 (Leray). Let F be a sheaf on the Riemann surface X. Suppose U ={U } is an open cover of X such that i i∈I H1(U ,F)=0for alli∈I. i Then H1(X,F)∼=H1(U,F). Proof. We show that for any refinement V ={V } of U, the homomorphism k k∈K τU :H1(U,F)→H1(V,F) V isanisomorphism. Thismapisinjectiveasnotedabove,soitremainsonlytoshow surjectivity. Let (f ) ∈Z1(V,F). For each i∈I, W :={U ∩V } is an open cover kl k,l∈K i k k∈K of U . Since H1(U ,F)=0 and the canonical homomorphism i i H1(W,F)→H1(U ,F) i isinjective,wehaveH1(W,F)=0. Sothereexists(g ) ∈C0(W,F)suchthat ik k∈K (2.7) f =g −g onU ∩V ∩V . kl ik il i k l Then for j ∈I, g −g =g −g onU ∩U ∩V ∩V . ik jk il jl i j k l So by sheaf axiom II, for each k ∈K, there exists h ∈F(U ∩U ) such that ij i j (2.8) h =g −g onU ∩U ∩V . ij ik jk i j k We see that (h ) satisfies Equation (2.4) and hence is in Z1(U,F). ij Now, for the refinement V of U, fix a map t : K → I as above, and let m = k g ∈F(V ). Then by Equations (2.7) and (2.8) , on V ∩V , tk,k k k l h −f =g −g −(g −g )=g −g =m −m . tk,tl kl tk,k tl,k tl,k tl,l tk,k tl,l k l Thus, (h ) and (f ) are cohomologous, implying τU is surjective. (cid:3) tk,tl kl V 3. The Exact Sequence of Sheaves In this section, we explain how maps between sheaves on a topological space X inducemapsbetweencohomologieswithcoefficientsinthesesheaves. Thisconcept will be key in proving the Riemann-Roch Theorem. Let (F,ρ) and (G,(cid:37)) be two sheaves on a topological space X. A sheaf homo- morphism α:F →G is a collection of homomorphisms {α :F(U)→G(U)|U ⊂Xopen} U such that for all open V ⊂U ⊂X, the diagram F(U) αU (cid:47)(cid:47)G(U) ρU (cid:37)U V (cid:15)(cid:15) (cid:15)(cid:15) V (cid:47)(cid:47) F(V) G(V) αV commutes. One can easily check that a sheaf homomorphism induces a homomor- phism on stalks, α :F →G for all x∈X. Then a sequence of sheaves x x x α (cid:47)(cid:47) β (cid:47)(cid:47) F G H 6 MICHAELWONG is said to be exact if, for all x∈X, the induced sequence F αx (cid:47)(cid:47)G βx (cid:47)(cid:47)H x x x is exact. In particular, a sequence of sheaves (cid:47)(cid:47) α (cid:47)(cid:47) β (cid:47)(cid:47) (cid:47)(cid:47) 0 F G H 0 is said to be short exact if the induced sequence on stalks is short exact. The following proposition follows quickly from the definitions. (cid:47)(cid:47) α (cid:47)(cid:47) β (cid:47)(cid:47) Proposition 3.1. Suppose 0 F G H is an exact sequence of sheaves. Then for every open subset U ⊂X, the sequence (cid:47)(cid:47) α (cid:47)(cid:47) β (cid:47)(cid:47) 0 F(U) G(U) H(U) is exact. As expected, a sheaf homomorphism α : F → G induces homomorphisms on cohomology, αq :Hq(X,F)→Hq(X,G), q =0,1. The map α0 is just α : F(X) → G(X), while the map α1 is obtained as follows. X The homomorphism α∗ defined by α∗(f ) = (α f ) preserves cocycles and ij Ui∩Uj ij coboundaries. Thus, it induces a homomorphism α∗ :H1(U,F)→H1(U,G). Since α∗ commutes with τU as defined above, it induces a homomorphism V α1 :H1(X,F)→H1(X,G). To see how a short exact sequence of sheaves (cid:47)(cid:47) α (cid:47)(cid:47) β (cid:47)(cid:47) (cid:47)(cid:47) (3.2) 0 F G H 0 induces an exact cohomology sequence, we must define a map δ∗ :H0(X,H)→H1(X,F) calledtheconnecting homomorphism. Leth∈H0(X,H)=H(X). Sinceβ :G → x x H is surjective for each x ∈ X, there exists an open cover U = {U } and a x i i∈I 0-cochain (g )∈C0(U,G) such that i β(g )=h| for everyi∈I. i Ui Thus, β(g −g ) = 0 on U ∩U for all i,j ∈ I. By Proposition 3.1, there exists i j i j (f )∈C1(U,F) such that ij α(f )=g −g for alli,j ∈I. ij i j Soα(f −f +f )=0. AgainbyProposition3.1, thisimpliesf −f +f =0, jk ik ij jk ik ij so (f )∈Z1(U,F). Let δ∗(h) be the cohomology class of (f ) in H1(X,F). One ij ij can verify that δ∗(h) is independent of the choices made, so δ∗ is well-defined. AN ANALYTIC APPROACH TO THE THEOREMS OF RIEMANN-ROCH AND ABEL 7 Theorem 3.3. The short exact sequence of sheaves in (3.2) induces an exact co- homology sequence, 0 (cid:47)(cid:47)H0(X,F) α0 (cid:47)(cid:47)H0(X,G) β0 (cid:47)(cid:47)H0(X,H) δ∗ (cid:15)(cid:15) (cid:111)(cid:111) (cid:111)(cid:111) H1(X,H) H1(X,G) H1(X,F) β1 α1 The proof is tedious but elementary. See [1] for detials. 4. The Genus of a Compact Riemann Surface Appearing in the Riemann-Roch formula, the genus g of a Riemann surface X is defined to be the dimension of H1(X,O) as a C-vector space. The task of this sectionistoshowthatg isfiniteifX iscompact. Wefirstsolvetheinhomogeneous Cauchy-Riemann equation, from which we determine a Leray cover for the sheaf O. Lemma 4.1. Suppose g ∈E(C) has compact support. Then there exists f ∈E(C) such that ∂f =g ∂z¯ Proof. Define f :C→C by 1 (cid:90)(cid:90) g(ζ+z) f(ζ)= dz∧dz¯. 2πi z C Letting z =reiθ, we have 1 (cid:90)(cid:90) g(ζ+reiθ) f(ζ)=− drdθ. π eiθ C Since g has compact support, this integral is finite. Furthermore, we have all the conditions for differentiating under the integral, ∂f 1 (cid:90)(cid:90) ∂g(ζ+reiθ) =− e−iθdrdθ. ∂ζ¯ π C ∂ζ¯ Now,letB ={z ∈C|(cid:15)≤|z|≤R}whereRissufficientlylargesothatB properly (cid:15) (cid:15) contains the support of g. In the original coordinates, ∂f 1 (cid:90)(cid:90) ∂g(ζ+z)1 = lim dz∧dz¯ ∂ζ¯ 2πi(cid:15)→0 B(cid:15) ∂ζ¯ z 1 (cid:90)(cid:90) ∂ (cid:16)g(ζ+z)(cid:17) = lim dz∧dz¯ 2πi(cid:15)→0 B(cid:15) ∂z¯ z 1 (cid:90) g(ζ+z) = lim dz 2πi(cid:15)→0 |z|=(cid:15) z where we have used Stoke’s Theorem to obtain the last integral. This integral is justtheaverageofg overthecircleofradius(cid:15)centeredatζ,sothelimitisg(ζ). (cid:3) A standard compact exhaustion argument generalizes the result in Lemma 4.1 to g without compact support. 8 MICHAELWONG Theorem 4.2. Let 0<R≤∞, and let X ={z ∈C||z|<R}. Suppose g ∈E(X). Then there exists f ∈E(X) such that ∂f =g. ∂z¯ Corollary 4.3. With X as in the theorem, H1(X,O)=0. Proof. Let U ={U } be an open cover of X. Take (f )∈Z1(U,F)⊂Z1(U,E). i i∈I ij By Proposition 2.5, H1(U,E)=0, so there exists (g )∈C0(U,E) such that i f =g −g onU ∩U . ij i j i j Then∂¯g =∂¯g foralli,j ∈I,sobysheafaxiomII,thereexistsh∈E(X)suchthat i j h| =∂¯g . Bythetheorem, thereexistsg ∈E(X)suchthat∂¯g =h. Now, observe thUait∂¯(g −i g| )=0,sog −g| isholomorphicforalliand(g −g| )∈C0(U,O). i Ui i Ui i Ui Moreover, g −g−(g −g)=g −g =f onU ∩U . i j i j ij i j So (f )∈Im(cid:0)C0(U,O) δ1 (cid:47)(cid:47)C1(U,O)(cid:1), implying H1(U,O)=0. (cid:3) ij Corollary 4.4. H1(P1,O)=0. Proof. Let U = C and U = C∗ ∪{∞}. Recall that (U ,φ ) and (U ,φ ) where 1 2 1 1 2 2 φ1 = idC and φ2(z) = 1/z form a complex structure on P1. As is easy to check, H1(U ,O) ∼= H1(φ (U ),O) and φ (U ) = C. Therefore, by Corollary 4.3, U := i i i i i {U ,U } is a Leray covering for O. 1 2 We want to show that (f ) ∈ Z1(U,O) splits. Since f is holomorphic on C∗, ij 12 it has a Laurent expansion at 0 ∞ (cid:88) f (z)= c zn 12 n n=−∞ which converges for all z ∈C∗. Then let ∞ −1 (cid:88) (cid:88) g (z)= c zn onU , g (z)=− c zn onU . 1 n 1 2 n 2 n=0 n=−∞ Clearly, g ∈O(U ) for i=1,2 and f =g −g . (cid:3) i i 12 1 2 We need one more lemma, which we shall take for granted. Given a topological space X and a subset V, we say that a subset U is a relatively compact subset of V, denoted U (cid:98)V, if U¯ is compact and U¯ ⊂V. Lemma 4.5. Suppose X is a Riemann surface. Let U∗ = {U∗} be a finite i i∈I collection of coordinate neighborhoods such that z (U∗) ⊂ C is the unit disk. Let i i W ={W } and U ={U } be collections of open sets such that W (cid:98)U (cid:98)U∗ i i∈I i i∈I i i i for each i∈I. Then the natural restriction map H1(U,O)→H1(W,O) has finite dimensional image Remark 4.6. Note that we do not assume the collections of open sets are covers of X. The lemma refers to the cohomology of (cid:91) (cid:91) X = U , X = W . U i W i i∈I i∈I AN ANALYTIC APPROACH TO THE THEOREMS OF RIEMANN-ROCH AND ABEL 9 The shortest proof of this result of which we are aware involves sophisticated methodsin functionalanalysis. Forfear ofdigressing from the mainpoint, we only refer the reader to [1]. Now, let X be a compact Riemann surface. Let U∗ = {U } be as in the i i∈I lemma, but suppose further that U∗ covers X. It is easy to show that there exist collections of open sets W = {W } and U = {U } as in the lemma but with i i∈I i i∈I the additional properties (1) W and U cover X, and (2) z (U ) is an open disk for all i. i i Then the natural restriction mapping τU :H1(U,O)→H1(W,O) W has finite dimensional image. As noted in Section 2, it is also injective. We have H1(U ,O) ∼= H1(z(U ),O), so by Corollary 4.3, U is a Leray covering. Therefore, i i H1(U,O)∼=H1(X,O). This proves that g =dimH1(X,O)<∞. Remark 4.7. It is a consequence of Serre duality that the genus g of a compact Riemann surface equals dimΩ(X). We will use this fact in the proof of Abel’s Theorem, though we will not develop the theory. See [1] and [4]. 5. The Riemann-Roch Theorem TheRiemann-RochTheoremprovidesawayforcomputingthedimensionofthe spaceofmeromorphicfunctionsonacompactRiemannsurfaceX withrestrictions on poles and zeros. The restrictions are introduced via the notion of a divisor. Definition 5.1. A divisor D on a Riemann surface X is a function D : X → Z such that for any compact K ⊂X, D(x)(cid:54)=0 for only finitely many x∈K. Remark 5.2. IfX iscompact,thenD canalternativelybedefinedasafiniteformal sum of points in X with coefficients in Z, m (cid:88) D = n p i i i=1 where m∈Z+, n ∈Z, and p ∈X for all i. i i We denote by Div(X) the group of divisors on X under addition. Moreover, with X compact and D as in the remark, we define a group homomorphism deg : Div(X)→Z by m (cid:88) degD = n , i i=1 called the degree map. The kernel of the degree map is denoted Div (X). 0 Fortherestofthissection,X willbeacompactRiemannsurface. Eachnonzero meromorphic function f ∈ M(X) determines a divisor. Recall that the order function of f is defined by  0 iff(x)(cid:54)=0  ord (f):= k ifxis a zero of orderk x  −k ifxis a pole of orderk. Since f (cid:54)= 0, the zeros (and poles) of f are isolated. Then because X is compact, there are only finitely many of them. Hence, the order function of f is a divisor, 10 MICHAELWONG written as (f). A divisor D which is the order function of a nonzero meromorphic function f is called a principal divisor, and f is said to be a meromorphic solution of D. The subgroup of all such divisors is denoted Div (X). P Let D ∈ Div(X). We let D be a pointwise lower bound for the order function and thereby place a restriction on the zeros and poles of a nonzero meromorphic function. For every open set U ⊂X, let O (U):={f ∈M(U)−{0}:ord (f)≥−D(x)for allx∈U}. D x With the natural restriction maps, O is a sheaf on X. Note that O is just the D 0 sheaf of holomorphic functions. Theorem5.3(Riemann-Roch). SupposeX isacompactRiemannsurfaceofgenus g,andletD beadivisoronX. ThenthecohomologygroupsHq(X,O )forq =0,1 D are finite dimensional C-vector spaces, and dimH0(X,O )−dimH1(X,O )=1−g+degD. D D Our task is made easier if we first construct auxiliary sheaves. The idea of the proof is to obtain the formula from an exact cohomology sequence induced from a short exact sequence of sheaves. First, let p ∈ X. Let P be the divisor which is 1 at p and 0 elsewhere, and let D(cid:48) =D+P. Then we have the inclusion morphism O (cid:44)→O , i.e., the collection D D(cid:48) of natural inclusion maps O (U)(cid:44)→O(cid:48) (U) for all open U ⊂X. D D Next, for p∈X, let (cid:26) C ifp∈U C (U)= p 0 ifp∈/ U. With the obvious restriction maps, C is a sheaf on X, called the skyscraper sheaf p at p. We see immediately that H0(X,C )=C (X)=C. P p To compute the first cohomology group, let U be any open covering of X. Then there exists a refinement V of U such that precisely one element of V contains p. Therefore, H1(U,C )=H1(V,C )=0 p p implying H1(X,C )=0. p Define a sheaf morphism α : O → C as follows. Fix a chart (V,z) centered D(cid:48) p at p. If p ∈/ U, let α be the zero morphism. If p ∈ U, then each f ∈ O (U) has U D(cid:48) a Laurent expansion around p ∞ (cid:88) c zn. n n=−D(p)−1 So set α (f)=c . As one can easily check, for each x∈X, the sequence of U −D(p)−1 stalks at x 0 (cid:47)(cid:47)OD,x ix (cid:47)(cid:47)OD(cid:48),x αx (cid:47)(cid:47)Cp,x (cid:47)(cid:47)0 is short exact. By Theorem 3.2, the corresponding short exact sequence of sheaves induces an exact cohomology sequence, (5.4) 0→H0(X,O )→H0(X,O )→C→H1(X,O )→H1(X,O )→0. D D(cid:48) D D(cid:48)