aa r X i v : . [ m a t h . QA ] N ov Convergence of Sewing Conformal Blocks B IN G UI Contents H p C , Θ C { B p‚ S X qq bstract In recent work [DGT19b], Damiolini-Gibney-Tarasca showed that for a C -cofinite rational CFT-type vertex operator algebra V , sheaves of conformal blocksare locally free and satisfy the factorization property. In this article, we use analyticmethods to prove that sewing conformal blocks is convergent, solving a conjectureproposed by Zhu [Zhu94] and Huang [Hua16]. Introduction
In conformal field theory (CFT), conformal blocks are linear functionals defined for N -pointed compact Riemann surfaces, together with a vertex operator algebra (VOA) V and V -modules W , . . . , W N . One can sew a (possibly disconnected) N -pointedcompact Riemann surface along pairs of points (with local coordinates) to obtain anew pointed compact Riemann surface with possibly higher genus or more markedpoints. Corresponding to this geometric sewing construction, one can also sew con-formal blocks by taking contractions [Seg88, Vafa87, TK88, TUY89, BFM91, Zhu94,Hua97, Hua05a, Hua05b, NT05, Hua16, DGT19b]. In this article, we prove that for V satisfying natural conditions, sewing conformal blocks is convergent, solving a con-jecture proposed in [Zhu94, Conj. 8.1] and [Hua16, Problem 2.2]. Our result providesa necessary step for the construction of higher genus rational (Euclidean) CFT in thesense of Segal [Seg88]. We hope it will also benefit other approaches to Euclidean CFT,such as the functional analytic one in [Ten17, Ten19a, Ten19b, Ten19c].Conformal blocks were first studied by physicists [BPZ84, FS87, MS89]. In math-ematics, conformal blocks were defined and explored by Tsuchiya-Kanie [TK88] forWeiss-Zumino-Witten (WZW) models and genus Riemann surfaces, and were gen-eralized by Tsuchiya-Ueno-Yamada to stable curves of all genera in [TUY89]. Forminimal models, conformal blocks over any stable curve was studied by Beilinson-Feigein-Mazur [BFM91]. In particular, factorization was proved for these modelsin [TUY89, BFM91]. A definition of conformal blocks for quasi-primarily gener-ated VOAs was given by Zhu in [Zhu94], and was generalized to any (CFT-type)VOA on families of compact Riemann surfaces by Frenkel and Ben-Zvi in [FB04]. In[DGT19a, DGT19b], Damiolini-Gibney-Tarasca defined conformal blocks for VOAs as-sociated to (algebraic) families of stable curves, and showed that when V is C -cofiniteand rational, sheaves of conformal blocks are vector bundles with projectively flatconnections, and proved the factorization property and a (formal) sewing theorem.Their results generalize those of Tsuchiya-Ueno-Yamada [TUY89] for WZW models,those of Nagatomo-Tsuchiya [NT05] for genus curves, and part of the results ofHuang for genus and curves [Hua95, Hua05a, Hua05b]. In [DGT19c], the authorsproved Chern characters of these bundles give cohomological field theories, and called(CFT-type) C -cofinite rational VOAs of CohFT-type. In the conformal net approach toCFT, conformal blocks were defined and investigated by Bartels-Douglas-Henriques[BDH17]. 2 ewing conjecture Conformal blocks are the chiral halves of correlation functions of full CFT. Assumethroughout this article that V is a CFT type vertex operator algebra (VOA). For an N -pointed compact Riemann surface X “ p C ; x , . . . , x N q where x , . . . , x N are distinctpoints of a compact Riemann surface C , one associates to these points V -modules W , . . . , W N . Then a conformal block ψ is a linear functional on W ‚ “ W b ¨ ¨ ¨ b W N “invariant” under the actions of certain global sections related to V . If one has afamily of N -pointed compact Riemann surfaces parametrized by a base manifold B ,then one allows ψ to vary holomorphically over B . The vertex operator Y p¨ , z q can beregarded as a conformal block associated to P with distinct points , z, and modules V , V , V (where V is the contragredient module of V ). More generally, if W , W , W are V -modules, an intertwining operator Y p¨ , z q of type ` W W W ˘ (as defined in [FHL93])corresponds to a conformal block associated to p P , , z, and W , W , W .A fundamental problem in rational CFT is to prove the convergence of sewingconformal blocks. Suppose we have an p N ` q -pointed compact Riemann surface r X “ p r C ; x , . . . , x N , x , x q . Then we can sew r X along the pair of points x , x to obtain another Riemann surfacewith possibly higher genus. More precisely, we choose ξ, ̟ to be local coordinates of r C at x , x . Namely, they are univalent (i.e. holomorphic and injective) functions definedrespectively in neighborhoods U Q x , U Q x satisfying ξ p x q “ , ̟ p x q “ . For each r ą we let D r “ t z P C : | z | ă r u and D ˆ r “ D r ´ t u . We choose r, ρ ą so that theneighborhoods U , U can be chosen to satisfy that ξ p U q “ D r and ̟ p U q “ D ρ , that U X U “ H , and that none of x , . . . , x N is in U or U . Then, for each q P D ˆ rρ , weremove the closed subdiscs of U , U determined respectively by | ξ | ď | q | ρ and | ̟ | ď | q | r ,and glue the remaining part using the relation ξ̟ “ q . Then we obtain an N -pointedcompact Riemann surface X q “ p C q ; x , . . . , x N q which clearly depends on ξ and ̟ .Now, if we associate V -modules W , . . . , W N , M , M (where M is the contragredient(i.e. dual) module of M ) to x , . . . , x N , x , x , and choose a conformal block ψ associatedto r X and these V -modules, then its sewing S ψ is an W ˚‚ “ p W b ¨ ¨ ¨ W N q ˚ -valuedformal series of q defined by sending each w ‚ “ w b ¨ ¨ ¨ b w N P W ‚ to S ψ p w ‚ q “ ψ p w ‚ b q L § b đ q P C t q u where § b đ is the element of the “algebraic completion” of M b M correspondingto the identity element of End C p M q , and L is the zero mode of the Virasoro operators t L n : n P Z u . (Cf. [Seg88, Vafa87, TUY89, Hua97, DGT19b].) The sewing conjecture , asproposed in [Zhu94, Conj. 8.1] and [Hua16, Problem 2.2], says that S ψ p w ‚ q convergesabsolutely to a (possibly) multivalued function on D ˆ rρ . Moreover, for each q P D ˆ rρ , S ψ p¨ , q q defines a conformal block associated to X q and W , . . . , W N . If we sew r C We do not assume C to be connected n pairs of points, and if we let x , . . . , x N and r C and ψ vary and be parametrizedholomorphically by variables τ ‚ “ p τ , . . . , τ m q , then sewing conformal blocks is alsoabsolutely convergent with respect to q , . . . , q n and (locally) uniform with respect to τ ‚ . In this article, we give a complete proof of the sewing conjecture (see Section 13for the main result), which complements the results of [DGT19b] on local freeness andfactorization of sheaves of conformal blocks, and provides a necessary step of con-structing rational conformal field theories on arbitrary (families of) compact Riemannsurfaces. We remark that a sewing theorem (Thm. 8.3.1) was proved in [DGT19b]. Inthat theorem, one treats the (infinitesimal) formal disc Spec C rr q ss instead of the ana-lytic disc D rρ , which is sufficient for application in the algebraic category. In particular,the convergence of sewing is not needed and not proved in [DGT19b]. In the analyticcategory, which is the one we are working in, convergence is necessary. (Also, notethat D rρ is not well-defined in the algebraic category.) See Remark 12.4 for details.Historically, the sewing conjecture was proved in some special cases. Our result isgeneral in the following aspects:(a) We consider any (CFT type) C -cofinite VOA V and any (ordinary) V -module. (b) We consider compact N -pointed Riemann surfaces of all genera.(c) We consider any (analytic) coordinates ξ, ̟ at x , x , and prove the convergenceon D rρ whenever r, ρ satisfy the previously described conditions. (Namely, weprove the convergence not only when q is small.)(d) We consider sewing along several pairs of points, and allow r X and ψ to beparametrized holomorphically by some τ ‚ .To our knowledge, no previous results have covered all these aspects. Nevertheless,even those partial results have played extremely important roles in the development ofa rigorous mathematical theory of conformal field theory. For instance: convergencein the genus case is necessary for the statement of braiding and operator productexpansions (fusion) of intertwining operators [TK88, Hua05a]; convergence of self-sewing a -pointed P (which leads to a -pointed elliptic curve) is necessary for thestatement of modular invariance of VOA characters [Zhu96]; convergence of sewing ageneral N -pointed P is necessary for the proof of Verlinde conjecture and the rigidityand modularity of the tensor category of VOA modules [Hua05b, Hua08a, Hua08b]. History of the proof of convergence
In [TK88], Tsuchiya and Kanie proved for type A WZW models the convergenceof the products of intertwining operators, i.e. the convergence of sewing conformalblocks of such VOAs from (possibly disconnected) genus to genus Riemann sur-faces. Their method applies directly to any WZW model. The local coordinates ξ, ̟ in We expect that our result is applied to C -cofinite and rational VOAs. But we do not assume ratio-nality in our result, so that it can be also applied to any C -cofinite VOA whose rationality is expectedbut not yet proved. z, z ´ at , . Their method is to show that the formal series S ψ satis-fies a differential equation (the Knizhnik–Zamolodchikov equation) with simple poleat q “ , and the coefficients of the differential equation is a (matrix-valued) analyticfunction of q and τ ‚ (parametrizing r X and ψ ). Any result about convergence provedafter [TK88], including ours, follows this pattern. The difficulty is, of course, to findsuch differential equations.In [TUY89], Tsuchiya-Ueno-Yamada showed that for all WZW-models and all com-pact Riemann surfaces, there exist local coordinates ξ, ̟ at x , x (which are called z, w and described in [TUY89, Lemma 6.1.2] and [Ueno97, Lemma 5.3.1]) such that suchdifferential equations exist for small q . This leads immediately to the convergence of S ψ under those conditions, which was later explicitly claimed in [Ueno97, Thm. 5.3.4].Generalizing their result to any ξ, ̟ is not straightforward.In [Zhu96], Zhu proved the convergence for any C -cofinite VOA, for self-sewingan p N ` q -pointed P along , with respect to local coordinates z, z ´ to an N -pointedelliptic curve, assuming that the N modules W , . . . , W N not associated , are thevacuum module V . Its generalization to any V -modules W , . . . , W N is non-trivial andproved by Huang in [Hua05b]. When W , . . . , W N are grading-restricted generalized V -modules, the convergence was proved by Fiordalisi in [Fio16]. Meanwhile, for C -cofinite VOAs, for sewing several pointed P to a pointed P , and assuming the localcoordinates at sewing points are z or z ´ , Huang proved in [Hua05a] the convergenceof sewing conformal blocks for any V -modules. This result, with the help of [Hua98],can be generalized to any local coordinates ξ, ̟ at sewing points. The solution ofsewing conjecture is complete in genus . When W , . . . , W N are grading-restrictedgeneralized V -modules, the convergence is due to Huang-Lepowski-Zhang [HLZ11]. Idea of the proof
Our proof of convergence is motivated by [DGT19b]. It was shown in [DGT19b]that by sewing a conformal block ψ , we get a formal conformal block S ψ as a formalseries of q (Thm. 8.3.1) annihilated by ∇ q B q (Rem. 8.3.3) where ∇ is a connection of thesheaf of conformal blocks on the infinitesimal disk Spec p C rr q ssq defined in [DGT19a].The definition of ∇ is unique up to a projective term. In other words, if we choose ∇ to be an (analytic) connection defined on the analytic disc D rρ , then S ψ is annihilatedby ∇ q B q plus a projective term f which is a priori only a formal power series of q . Akey step of proving the convergence of S ψ is to show that f converges. Then, usinga finiteness theorem 7.3 analogous to [DGT19b, Thm. 8.2.1], we obtain the desireddifferential equation.It turns out that the connection ∇ is determined by a (relative) projective structure P on r X . Moreover, when ξ, ̟ belong to P , the projective term f equals . Thus, for achosen P and the corresponding ∇ , if we assume ξ, ̟ belong to P , then ∇ q B q S ψ “ ,which will provide the differential equation. The vanishing of f is due to that ofthe Schwarzian derivatives between local coordinates belonging to P . In general, onecannot expect that ξ, ̟ belong to the same projective structure. To resolve this issue,we fix a projective structure P , and find an explicit formula of f in terms of P , ξ, ̟ , This observation is due to Liang Kong and Hao Zheng. f is analytic. Outline
To carry out the above ideas, we first define and study some basic properties ofsheaves of conformal blocks on complex curves and analytic families of curves. Thisis achieved in Sections 3 and 6. To prepare for this task, we first review Huang’schange of coordinate formulas [Hua97] in Section 1. This formula is used to definesheaves of VOAs on curves and families of curves in Sections 2 and 5. In Section 4,we follow [TUY89, Sec. 6.1] and give a precise description of how to sew a compactRiemann surface and form a family of curves over the disc D rρ . In fact, we describethe simultaneous sewing for a family of compact Riemann surfaces over a complexmanifold r B , which yield a family of complex curve over B : “ r B ˆ D rρ .In Section 7, we prove a finiteness theorem which will turn the relation p ∇ q B q ` f q S ψ “ into a differential equation with analytic coefficients (provided that f con-verges) and simple poles at q “ . Section 8 recalls some basic facts about Schwarzianderivatives, and Section 9 prepares for the calculation of the projective term f . In Sec-tion 10, we give a proof that S ψ is a formal conformal block using an argument similarto but slightly different from the one in [DGT19b]. Then, in Section 11, we prove theconvergence of sewing conformal blocks associated to a family r X of compact Riemannsurfaces along a pair of (families of) points. This result can be easily generalized tosewing along several pairs of points, which is discussed in Section 13. In particular,our convergence theorem in the most general form is given in Theorem 13.1 of that sec-tion. In Section 12, we show that the sewing map ψ ÞÑ S ψ (defined in a suitable andnatural way) is injective, assuming V is C -cofinite. If V is also rational, then this mapis also bijective due to the factorization property proved in [DGT19b]. Using theseresults, we give an analytic version of the sewing theorem (Thm. 8.3.1) of [DGT19b].We would like to point out that, although many ideas are common in the analyticand algebraic settings, some subtle differences exist which will lead to different strate-gies of the proof. See Remark 7.4 for instance. Thus, we believe it is worthwhile toprovide in our article a detailed account of the analytic theory of conformal blocksand a self-contained proof of convergence. In particular, our proof does not rely onthe factorization or the local freeness of sheaves of conformal blocks. Acknowledgment
I would like to thank Chiara Damiolini, Angela Gibney, Yi-Zhi Huang, Liang Kong,Nicola Tarasca, Chuanhao Wei, Shilin Yu, Hao Zheng for helpful discussions. I amgrateful to Xiaojun Wu for answering my many questions in complex geometry. Sec-tion 12 is motivated by a Mathoverflow question asked by Andr´e Henriques. Theauthor is partially supported by an AMS-Simons travel grant.6
Change of coordinates
Throughout this article, we let N “ t , , , . . . u and Z ` “ t , , , . . . u . Also, C ˆ “ C ´ t u . If W is a vector space and z is a (formal) variable, we define W rr z ss “ " ÿ n P N w n z n : each w n P W * ,W rr z ˘ ss “ " ÿ n P Z w n z n : each w n P W * ,W pp z qq “ ! f p z q : z k f p z q P W rr z ss for some k P Z ) ,W t z u “ ! ÿ n P C w n z n : each w n P W ) . Let V be a vertex operator algebra (VOA for short) in the sense of [FHL93]. Welet and c be respectively the vacuum vector and the conformal vector of V . Foreach u P V , we write the vertex operator as Y p v, z q “ ř n P Z Y p v q n z ´ n ´ where each Y p v q n P End p V q . Then t L n “ Y p c q n ` u are the Virasoro operators with a central charge c P C . We write V “ À n P Z V p n q where V p n q is the eigenspace of L with eigenvalue n .We write wt p v q “ n if v P V p n q .We assume throughout this article that a VOA V is of CFT-type, which means that V p n q “ when n ă , and V p q “ C .By a V -module W , we always assume that it is a (grading-restricted) ordinary V -module. This means that W is a weak V -module, and there exists a finite subset E Ă C such that we have a grading W “ à n P E ` N W p n q where each W p n q is the eigenspace of L with eigenvalue n , which is finite-dimensional.We let Y M p v, z q “ ř n P Z Y M p v q n z ´ n ´ be a vertex operator of this module. The Virasorooperator L n acts on W as Y M p c q n ` .It will be convenient to consider an N -grading which makes W admissible. Namely,consider a grading W “ à n P N W p n q , such that if we define a linear operator r L satisfying r L | W p n q “ n W p n q for each n P N ,then r L ´ L commutes with the action of V . Moreover, we assume that under the r L -grading, each W p n q is finite-dimensional . Such grading is possible. For instance, wemay assume that any two numbers of E do not differ by an integer, and set r L suchthat for any α P E , the restriction of r L to À n P α ` N W p n q (which is a V -submodule of W )equals L ´ α .Since r L ´ L P End V p W q , we have r r L , Y W p v q n s “ r L , Y W p v q n s “ Y W p L v q n ´ p n ` q Y W p v q n , and hence Y W p v q m W p n q Ă W p n ` wt p v q ´ m ´ q . (1.1)7 vector w P W is called r L - (resp. L -) homogeneous with weight n if w P W p n q (resp. w P W p n q ). In this case, we write Ă wt p w q “ n (resp. wt p w q “ n ). Note that the r L -weightsare natural numbers but the L -weights are not necessarily. Convention 1.1.
For the vacuum module V , we choose r L to be L . So V p n q “ V p n q . If W is irreducible (i.e., simple), which implies that W “ À n P α ` N W p n q for some α P C ,we fix the r L -grading such that W p q is non-trivial. If W is semi-simple, i.e., a directsum of irreducible modules, we fix the r L -grading such that the restriction of r L toeach irreducible submodule is the one we have described.We now recall the formula for changing coordinates discovered by Huang [Hua97].To begin with, we let O C , be the stalk of the sheaf of holomorphic functions of C at . Namely, an element in O C , is precisely a formal power series f p z q “ ř n P N a n z n ( a n P C ) converging absolutely in a neighborhood of . We consider the subset G ofall ρ P O C , satisfying ρ p q “ and ρ p q ‰ . Then G becomes a group if we definethe multiplication of two elements ρ , ρ to be their composition ρ ˝ ρ . The identityelement of G is the standard coordinate z of C .For each ρ P G , we can find c , c , c , ¨ ¨ ¨ P C such that ρ p z q “ c ¨ exp ´ ÿ n ą c n z n ` B z ¯ z For instance, if we write ρ p z q “ a z ` a z ` a z ` ¨ ¨ ¨ , (1.2)then one has c “ a ,c c “ a ,c c ` c c “ a . In particular, one has c “ ρ p q . We define U p ρ q P End p W q to be U p ρ q “ ρ p q r L ¨ exp ´ ÿ n ą c n L n ¯ (1.3)Notice a n “ ρ p n q p q{ n ! , we have c “ ρ p q ρ p q ,c “ ρ p q ρ p q ´ ´ ρ p q ρ p q ¯ . (1.4)8 emark 1.2. Considering the action of U p ̺ q on W might be inconvenient since W isnot finite-dimensional. On the other hand, W p n q might not be preserved by U p ρ q .Thus, it would be better to consider W ď n “ À k ď n W p k q which is finite-dimensionaland preserved by U p ̺ q .Since L m W ď n Ă W ď n ´ when m ą , from (1.3) it is easy to see that for any w P W p n q , U p ρ q w “ ρ p q n w mod W n ´ . (1.5)In other words, the action of U p ρ q on W ď n { W ď n ´ is ρ p q n .The following was (essentially) proved in [Hua97] section 4.2: Theorem 1.3.
For each V -module W , U is a representation of G on W . Namely, we have U p ρ ˝ ρ q “ U p ρ q U p ρ q for each ρ , ρ P G . Example 1.4.
It is easy to see that p c z B z q n z “ n ! c n z n ` . Thus exp p c z B z q z “ ř n “ c n z n ` “ z {p ´ c z q . For each ξ P C ˆ , we set γ ξ P G to be γ ξ p z q “ ξ ` z ´ ξ . (1.6)If we set α p z q “ ´ ξ ´ z , then γ ξ p z q “ α {p ´ ξα q “ exp p c α B α qp α q . Thus, by (1.3) andthat U preserves composition, we obtain U p γ ξ q “ e ξL p´ ξ ´ q r L . (1.7)In particular, U p γ q “ e L p´ q r L . (1.8)It is easy to see γ ξ p ξz q “ ξ ´ γ p z q . Thus U p γ ξ q ξ r L “ ξ ´ r L U p γ q . (1.9) Remark 1.5.
Let X be a complex manifold and ρ : X Ñ G , x ÞÑ ρ x a function. We saythat ρ is a holomorphic family of transformations if for any x P X , there exists an opensubset V Ă X containing x and an open U Ă C containing such that p z, y q P U ˆ V ÞÑ ρ y p z q is a holomorphic function on U ˆ V . Then it is clear that the coefficients a , a , . . . in (1.2) depend holomorphically on the parameter x P X . Hence the same is true for c , c , c , . . . . Thus, by the formula (1.3), for any w P W ď n , x P X ÞÑ U p ρ x q w is a W ď n -valued holomorphic function on X . Thus U p ρ q can be regarded as an isomorphism of O X -modules U p ρ q : W ď n b C O X » ÝÑ W ď n b C O X (1.10)sending each W ď n -valued function w to the section x ÞÑ U p ρ x q w p x q . Its inverse is U p ρ ´ q . 9f W is a V -module, then its contragredient module W can be describe using U p γ z q .As a vector space, W “ à n P C W ˚p n q where W ˚p n q is the dual space of W p n q . For each v P V , the vertex operator Y W p v, z q isdefined such that if w P W , w P W , then, using x , y to denote the natural pairing of W and W , we have x Y W p v, z q w , w y “ x w , Y W p U p γ z q v, z ´ q w y , (1.11)recalling that U p γ z q “ e zL p´ z ´ q L . That p W , Y W q satisfies the definition of a V -module follows from [FHL93]. Remark 1.6. If W is semi-simple, then W “ À n P C W ˚p n q “ À n P N W p n q ˚ . Since W isalso a semi-simple V -module, by our Convention 1.1, we must have r L t0 “ r L , i.e., x r L w, w y “ x w, r L w y for each w P W , w P W . In this article, for any complex manifold or complex space X , we let O X denotethe sheaf (of germs) of holomorphic functions of X . So for each open subset U Ă X , O X p U q (written also as O p U q for short) is the algebra of holomorphic functions on U .For every (sheaf of) O X -modules E , recall the usual notation that E x is the stalk of E at x P X . The dual O X -module of E is denoted by E ˚ , or E ´ when E is a line bundle.For two O X -modules E , F , we write their tensor product E b O X F as E b F for short.If Y is a complex submanifold or complex subspace of X , we let E | Y (also written as E | Y ) denote the restriction of E to X , namely, the pullback of E along the inclusionmap Y ã Ñ X . The restriction of a section s of E is denoted by s | X or s | X . In the case Y is a single point t x u , the restriction E | x can be naturally identified with E x { m x E x where m x the ideal of all f P O X,x vanishing at x . If s is a section defined near x , we let s p x q be the restriction s | x . If we consider s p x q as an element of E x { m x E x , then s p x q equals s x ` m x E x , where s x P E x is the germ of s at x .By a complex curve C , we mean either a compact Riemann surface or a (simple)nodal curve. For simplicity, we assume throughout this article that a nodal curve hasonly one (simple) node. We let ω C denote the dualizing sheaf of C , which is the sheafof holomorphic -forms when C is smooth (i.e., a compact Riemann surface). Its dualsheaf is denoted by Θ C “ ω ´ C , which is the (holomorphic) tangent bundle when C issmooth. In the case that C is nodal, O C , ω C , Θ C are described as follows.Assume C has only one simple node. Then C can be obtained by gluing two dis-tinct points y , y of a compact Riemann surface r C (the normalization of C ). The glu-ing map is denoted by ν : r C Ñ C . We identify r C ´ t y , y u with C ´ t x u (where Unless otherwise stated, compact Riemann surfaces are not assumed to be connected. To simply the following discussions, when C is smooth, we let r C be C and ν be the identity map. “ ν p y q “ ν p y q ) via ν . Then O C p U q , ω C p U q , Θ C p U q agree with O r C p U q , ω r C p U q , Θ r C p U q when x ‰ U . If x P U and U is small enough such that ν ´ p U q is a disjointunion of neighborhoods V Q y , V Q y , and that there exist univalent functions ξ P O p V q , ̟ P O p V q satisfying ξ p y q “ ̟ p y q “ , then O C p U q consists of all f P O r C p V Y V q satisfying f p y q “ f p y q ; Θ C p U q is the (free) O C p U q -submodule of Θ r C p V Y V q generated by the tangent fields whose restrictions to V Y V are ξ B ξ , resp. ´ ̟ B ̟ ; (2.1) ω C p U q is the (free) O C p U q -submodule of ω r C p ν ´ p U ´ t x uqq generated by ξ ´ dξ, resp. ´ ̟ ´ d̟. We refer the reader to [ACG11, Chap.X] for basic facts about nodal curves.
Definition of V C Let V be a VOA. The sheaf of VOA V C on a complex curve C is defined when C issmooth by [FB04] and generalized to (simple) nodal curves by [DGT19a]. Let us recallthe definition. V C is defined by the filtration V C “ lim ÝÑ n P N V ď nC , where each V ď nC is a locally free O C -module (i.e., a vector bundle) of rank dim V ď n . Weneed some preparation before we describe V ď nC .We first assume C is a smooth curve or the smooth open subset of a nodal curve.Let U, V be open subsets of C , equipped with univalent functions η P O p U q , µ P O p V q .Define a holomorphic family ̺ p η | µ q : U X V Ñ G as follows. For any p P U X V , η ´ η p p q and µ ´ µ p p q are local coordinates at p . We set ̺ p η | µ q p P G satisfying η ´ η p p q “ ̺ p η | µ q p p µ ´ µ p p qq . (2.2)Let z P O C , be the standard coordinate. Then, by composing both sides of (2.2) with µ ´ p z ` µ p p qq , we find the equivalent formula ̺ p η | µ q p p z q “ η ˝ µ ´ p z ` µ p p qq ´ η p p q , (2.3)which justifies that the family of transformation ̺ p η | µ q is holomorphic. It is also clearthat if η , η , η are three local coordinates, then on their common domain the followingcocycle condition holds: ̺ p η | η q “ ̺ p η | η q ̺ p η | η q . (2.4)By Remark 1.5, for each n P N we have an isomorphism of O U X V -modules U p ̺ p η | µ qq : V ď n b C O U X V » ÝÑ V ď n b C O U X V . V ď nC is defined such that its transition functions are given by U p ̺ p η | µ qq . Thus, for any open subset U Ă C and a univalent η P O p U q , we have atrivilization, i.e., an isomorphism of O U -modules U ̺ p η q : V ď nC | U » ÝÑ V ď n b C O U . (2.5)If V Ă C is also open and µ P O p V q is univalent, then on U X V we have U ̺ p η q U ̺ p µ q ´ “ U p ̺ p η | µ qq . (2.6)From (1.5), we can compute that for any section v of V ď n b C O U X V , U ̺ p η q U ̺ p µ q ´ ¨ v “ pB µ η q n ¨ v mod V ď n ´ b C O U X V . (2.7)By comparing the transition functions, we see that V ď nC { V ď n ´ C is naturally equivalentto V p n q b C Θ b nC (cf. [FB04, Sec. 6.5.9]).We now assume C has a node x . We shall define V ď nC to be an O C -submodule of V ď nC ´t x u as follows. Let V ď nC be equal to V ď nC ´t x u outside x . To describe V ď nC near x ,we choose a neighborhood U of x , and choose y , y , V , V , ξ P O p V q , ̟ P O p V q aspreviously. Then U ´ t x u can be identified with p V ´ t y uq Y p V ´ t y uq via ν . We let V ď nC p U q be the O C p U q -submodule of V ď nC ´t x u p U ´ t x uq generated by U ̺ p ξ q ´ ` ξ L v ˘ ` U ̺ p ̟ q ´ ` ̟ L U p γ q v ˘ p@ v P V ď n q . (2.8)To be more precise, (2.8) defines a section on p V ´ t y uq \ p V ´ t y uq which equals U ̺ p ξ q ´ ` ξ L v ˘ on p V ´ t y uq and U ̺ p ̟ q ´ ` ̟ L U p γ q v ˘ on V . Also, ξ L is an elementof O r C p V ´ t y uq acting on the constant section v P V ď n Ă V ď n b C O r C p V ´ t y uq , and ̟ L U p γ q v is understood in a similar way. By U p γ q “ e L p´ q L and induction on n ,it is easy to see that V ď nC p U q is O C p U q -generated freely by U ̺ p ξ q ´ ` ξ wt p v q v ˘ ` U ̺ p ̟ q ´ ` p´ ̟ q wt p v q v ˘ (2.9)for all v P E where E is any basis of V ď n whose elements are homogenous. By (2.9)and (2.7), it is easy to check that the definition of V ď nC p U q is independent of the choiceof the coordinates ξ, ̟ at y , y .If we compare (2.9) with (2.1), and notice (2.7) and the sentence after that, we seethe following fact (cf.[FB04, Sec.6.5.9] and [DGT19b, Lemma 2.7.1]): Proposition 2.1.
Let C be a complex curve and n P N . Then we have the following isomor-phism of O C -modules: V ď nC { V ď n ´ C » V p n q b C Θ b nC . Under this isomorphism, if U Ă C is open and smooth, and η P O p U q is univalent, then forany v P V p n q , v b B nη is identified with the equivalence class of U ̺ p η q ´ v . vanishing theorem In the remaining part of this section, we use Proposition 2.1 to prove a vanishingtheorem for V ď nC b ω C . By an (analytic) local coordinate η of C at a smooth point x ,we mean a (holomorphic) univalent function η defined on a smooth neighborhood of x satisfying η p x q “ . By an N -pointed complex curve with local coordinates X “ p C ; x , . . . , x N ; η , . . . , η N q we mean a complex curve C together with N distinct smooth points x , x , . . . , x N P C such that each x i is associated with a local coordinate η i at x i . If local coordinatesare not specified, we simply say an N -pointed complex curve X “ p C ; x , . . . , x N q .Unless otherwise stated, we assume that each irreducible component of C (equivalently,each connected component of r C ) contains at least one of x , . . . , x N .We set S X “ x ` x ` ¨ ¨ ¨ ` x N , which can be considered as a divisor both of C and of r C . Note that by Kodaira van-ishing theorem, if C is smooth and connected with genus g , and L is a line bundleon C , then H p C, L b ω C q “ and (by Serre duality) equivalently H p C, L ´ q “ whenever deg L ą . Since deg ω C “ g ´ , we conclude that H p C, L q “ whenever deg L ą g ´ .Recall that r C is the normalization of the complex curve C . We let r g be the largestgenus of the connected components of r C . Let M P t , u be the number of nodes.Notice the following elementary fact: Lemma 2.2.
Choose any integer n ě ´ . Then H p C, Θ b nC p kS X qq “ whenever k ą p n ` qp r g ´ q ` M . Recall that for any locally free E (i.e., E is a vector bundle), E p kS X q » E b O p kS X q is the sheaf of meromorphic sections of E whose only possible poles are at x , . . . , x N and of orders at most k . Proof.
Assume first of all that C is smooth. Then it suffices to assume C is connected.Then r g is the genus g of C . Since deg Θ C “ ´ g , the degree of Θ b nC p kS X q is no lessthan n p ´ g q ` k , which is greater than g ´ if k ą p n ` qp g ´ q . In that case, wehave H p C, Θ b nC p kS X qq “ by the discussion before the lemma.Next, assume C has one node x “ ν p y q “ ν p y q . Choose any k ą p n ` qp r g ´ q ` . To prove the vanishing of H , it suffices, by Serre duality, to prove H p C, ω bp n ` q C p´ kS X qq “ . Note that H p C, ω bp n ` q C p´ kS X qq can be viewed as a sub-space of H p r C, L q where L “ ω bp n ` q r C p´ kS X ` y ` y q . We shall prove that for eachconnected component r C of r C , H p r C , L | r C q “ . This is true since deg p L | r C q ď p n ` q deg p ω r C q ´ k ` ď p n ` qp r g ´ q ´ k ` ă . heorem 2.3. Choose any n P N , and assume k ą n ¨ max t , r g ´ u ` M . Then H p C, V ď nC b ω C p kS X qq “ . Proof.
Since V ď´ is trivial, we have V ď C » V ď C { V ď´ C » O C by Proposition 2.1. Thus,the claim follows from Lemma 2.2. Now, suppose the claim is true for n ´ . We shallprove that it is also true for n . Assume k ą n ¨ max t , r g ´ u ` M . By Proposition 2.1,we have an exact sequence H p C, V ď n ´ C b ω C p kS X qq Ñ H p C, V ď nC b ω C p kS X qqÑ H ` C, V p n q b C Θ bp n ´ q C p kS X q ˘ . The first term vanishes by induction, and the last term vanishes by Lemma 2.2. Thusthe middle term vanishes.
We recall the definition of conformal blocks associated to complex curves in [FB04]and [DGT19a, DGT19b], but rephrase it in the analytic language suitable for our pur-pose in this article.
Action of H p C, V C b ω C p‚ S X qq on W ‚ Let X “ p C ; x , . . . , x N ; η , . . . , η N q be an N -pointed complex curve with local co-ordinates. If W is a V -module, then, by considering V as the subspace of constantsections of V pp z qq , we can extend the linear map V b C W Ñ W pp z qq , v b w ÞÑ Y W p v, z q w uniquely to an C pp z qq -module homomorphism V pp z qq b C W Ñ W pp z qq .Let W , W , . . . , W N be V -modules. Set W ‚ “ W b W b ¨ ¨ ¨ b W N . Convention 3.1. By w P W ‚ , we mean a vector of W b ¨ ¨ ¨ b W N . By w ‚ P W ‚ , we meana vector of the form w b w b ¨ ¨ ¨ b w N , where w P W , . . . , w N P W N .For each O C -module E , we set E p‚ S X q “ lim ÝÑ k P N E p kS X q , whose sections are meromorphic sections of E whose only possible poles are at x , . . . , x N . For each ď i ď N , if v is a section of V C b ω C p‚ S X q defined near x i ,we define a linear action of v on W i as follows. Choose a neighborhood U i of x i onwhich η i is defined. By tensoring with the identity map of ω U i , the map (2.5) inducesnaturally an O U i -module isomorphism (also denoted by U ̺ p η i q ): U ̺ p η i q : V C | U i b ω U i p‚ S X q » ÝÑ V b C ω U i p‚ S X q . Identify U i and η i p U i q via η i . Then η i as a variable equals the standard variable z of C .The action of v on any w i P W i is then v ¨ w i “ Res z “ Y W i p U ̺ p η i q v, z q w i (3.1)14Here the η i in U ̺ p η i q is understood as a coordinate but not a variable. So it is differentfrom z .)Define a linear action of H p C, V C b ω C p‚ S X qq on W ‚ as follows. If v P H p C, V C b ω C p‚ S X qq and w ‚ “ w b ¨ ¨ ¨ b w N P W ‚ , then v ¨ w ‚ “ N ÿ i “ w b w b ¨ ¨ ¨ b p v | U i q ¨ w i b ¨ ¨ ¨ b w N . (3.2) Coordinate-independent definition
Let X “ p C ; x , . . . , x N q be an N -pointed complex curve without specifying localcoordinates, and let W , . . . , W N be V -modules. Define a vector space W X p W ‚ q iso-morphic to W ‚ as follows. W X p W ‚ q is a (possibly infinite rank) vector bundle on the -dimensional manifold t C u (consider as the base manifold of the family C Ñ t C u ).For any choice of local coordinates η ‚ “ p η , . . . , η N q of x , . . . , x N respectively, we havea trivialization U p η ‚ q : W X p W ‚ q » ÝÑ W ‚ (3.3)such that if µ ‚ is another set of local coordinates, then the transition function is U p η ‚ q U p µ ‚ q ´ “ U p η ‚ ˝ µ ´ ‚ q : “ U p η ˝ µ ´ q b U p η ˝ µ ´ q b ¨ ¨ ¨ b U p η N ˝ µ ´ N q . (3.4)If v P H p C, V C b ω C p‚ S X qq and w P W X p W ‚ q , we set v ¨ w “ U p η ‚ q ´ ¨ v ¨ U p η ‚ q ¨ w, (3.5)where the action of v on U p η ‚ q w (which depends on η ‚ ) is defined by (3.1) and (3.2). Theorem 3.2 (Cf. [FB04] Thm. 6.5.4) . The action of H p C, V C b ω C p‚ S X qq on W X p W ‚ q defined by (3.5) is independent of the choice of η ‚ .Proof. We prove this theorem for the case N “ . The general cases can be proved ina similar way. Choose local coordinates η, µ at x “ x defined on a neighborhood U .We identify U with µ p U q via µ . So µ as a coordinate is identified with the standardcoordinate C of C , and η P G . As a variable, µ is identified with the standard one z of C . Also, identify W X p W q (where W “ W “ W ‚ ) with W via U p µ q . So U p µ q “ U p C q “ . Choose any w P W , choose any section v of V C b ω C p‚ S X q defined on U . So U ̺ p µ q v “ u p z q dz for some u “ u p z q P V b C O C p‚ qp U q . Then, by (2.6), U ̺ p η q v “ U p ̺ p η | C qq u p z q dz “ U p ̺ p η | C q z q u p z q dz. Set variable ζ “ η p z q . Then U p η q ´ ¨ v ¨ U p η q ¨ w “ Res ζ “ U p η q ´ Y W ` U ̺ p η q v, ζ ˘ U p η q w ζ in Y W is due to the fact that η (as a variable) equals ζ when U is identifiedwith η p U q via η . (Such identification is needed in the definition of the coordinate-dependent action (3.1).) This expression equals Res z “ U p η q ´ Y W ` U p ̺ p η | C q z q u p z q , η p z q ˘ U p η q w ¨ dz, which by Theorem 3.3 (with α “ η ) equals Res z “ Y W ` u p z q , z ˘ w ¨ dz “ U p µ q ´ ¨ v ¨ U p µ q ¨ w. The proof is complete.In the above proof, we have used the following theorem of Huang [Hua97]; seealso [FB04, Lemma 6.5.6].
Theorem 3.3.
Let W be a V -module. Let U Ă C be a neighborhood of . Let α P O p U q be alocal coordinate at (so α p q “ ). Let C P G be the standard coordinate of C (i.e. the identityelement of G ). Then for any v P V and w P W , we have the following equation of elements in W pp z qq : U p α q Y W p v, z q U p α q ´ ¨ w “ Y W ` U p ̺ p α | C qq v, α p z q ˘ ¨ w. (3.6) Note that U p ̺ p α | C qq v is in V b C O p U q and hence can be regarded as an element of V pp z qq .Of course, (3.6) also holds in an obvious way for any v P V pp z qq . For instance, take α p z q “ λz where λ P C ˆ . Then ̺ p α | C q is constantly λ . It followsthat λ r L Y W p v, z q λ ´ r L “ Y W p λ L v, λz q . (3.7)We now define space of covacua T X p W ‚ q “ W X p W ‚ q H p C, V C b ω C p‚ S X qq ¨ W X p W ‚ q (3.8)(we have omitted Span C in the denominator), whose dual vector space is denoted by T ˚ X p W ‚ q and called space of conformal blocks . Elements of T ˚ X p W ‚ q are called con-formal blocks associated to W ‚ and X . They are the linear functionals of W X p W ‚ q vanishing on the denominator of (3.8). By a (holomorphic) family of compact Riemann surfaces X “ p π : C Ñ B q wemean B , C are complex manifolds, B has finitely many connected components, π is aproper surjective holomorphic submersion, and each fiber C b : “ π ´ p b q (where b P B )is a compact Riemann surface. This is assumed only for simplicity. B is connected, as a family of differentialmanifolds, X is trivial, i.e., X is equivalent to the projection C b ˆ B Ñ B onto the B -component. In particular, all fibers are diffeomorphic.We say that X “ p π : C Ñ B ; ς , . . . , ς N ; η , . . . , η N q is a family of N -pointed compact Riemann surfaces with local coordinates , if(a) π : C Ñ B is a family of compact Riemann surfaces.(b) Each ς i is a section of the family, namely, each ς i : B Ñ C is a holomorphic mapsatisfying π ˝ ς i “ B .(c) Each η i is a local coordinate at ς i p B q , which means that there is an open subset U i Ă C containing ς i p B q such that η i P O p U i q is univalent on each fiber U i,b “ U i X C b of U i . In that case, p η i , π q is a biholomorphic map from U i to an opensubset of C ˆ B . Moreover, we assume the restriction of η i to ς i p B q is .(d) ς i p B q X ς j p B q “ H whenever i ‰ j .(e) For each b P B , every connected component of the fiber C b contains at lease oneof the marked points ς p b q , . . . , ς N p b q .In the case that the local coordinates η , . . . , η N are not assigned (and hence condition(c) is not assumed), we say that X “ p π : C Ñ B ; ς , . . . , ς N q is a family of N -pointedcompact Riemann surfaces. We set S X “ N ÿ i “ ς i p B q to be a divisor of C . Then for each b P B , S X p b q : “ N ÿ i “ ς i p b q is a divisor of C b . The definitions of S X and S X p b q will also apply to the case that X isformed by sewing a smooth family.We say that X is a family of N -pointed complex curves (resp. with local coordi-nates) , if X is either a family of N -pointed compact Riemann surfaces (resp. with localcoordinates) (in that case we say X is a smooth family), or if X is formed by sewing asmooth family r X , whose meaning is explained below. Sewing open discs
We first describe how to sew a pair of open discs D r , D ρ .For any r ą , let D r “ t z P C : | z | ă r u and D ˆ r “ D r ´ t u . If r, ρ ą , we define π r,ρ : D r ˆ D ρ Ñ D rρ , p ξ, ̟ q ÞÑ ξ̟. (4.1)17 π r,ρ is surjective at p ξ, ̟ q whenever ξ ‰ or ̟ ‰ . Denote also by ξ and ̟ thestandard coordinates of D r and D ρ , which can be extended constantly to ξ : D r ˆ D ρ Ñ D r , ̟ : D r ˆ D ρ Ñ D ρ . Set q “ π r,ρ , i.e., q : D r ˆ D ρ Ñ C , q “ ξ̟. Then p ξ, ̟ q , p ξ, q q , p ̟, q q are coordinates of D r ˆ D ρ , D ˆ r ˆ D ρ , D r ˆ D ˆ ρ respectively. Thestandard tangent vectors of the coordinates p ξ, ̟ q , p ξ, q q are related by " B ξ “ B ξ ´ ξ ´ ̟ ¨ B ̟ B q “ ξ ´ B ̟ " B ξ “ B ξ ` ξ ´ q ¨ B q B ̟ “ ξ B q (4.2)The formulae between p ξ, ̟ q , p ̟, q q are similar.It is easy to see that p ξ, q qp D ˆ r ˆ D ρ q (resp. p ̟, q qp D r ˆ D ˆ ρ q ) is precisely the subsetof all p ξ , q q P D r ˆ D rρ (resp. p ̟ , q q P D ρ ˆ D rρ ) satisfying | q | ρ ă | ξ | ă r resp. | q | r ă | ̟ | ă ρ. (4.3)We choose closed subsets E r,ρ Ă D r ˆ D rρ and E r,ρ Ă D ρ ˆ D rρ such that p ξ, q q : D ˆ r ˆ D ρ » ÝÑ D r ˆ D rρ ´ E r,ρ , p ̟, q q : D r ˆ D ˆ ρ » ÝÑ D ρ ˆ D rρ ´ E r,ρ (4.4)are bijective. Sewing a family of compact Riemann surfaces
We discuss how to simultaneously sew a family of compact Riemann surfaces. Thisconstruction is also known as smoothing in the world of algebraic geometry. See[TUY89, Sec. 6.1], [Ueno97, Sec. 5.3], or [ACG11, Sec. XI.3]. Its algebraic version isgiven in [Loo10, Sec. 6] and [DGT19b, Sec. 8.1].Consider a family of p N ` q -pointed compact Riemann surfaces with local coordi-nates r X “ p r π : r C Ñ r B ; ς , . . . , ς N , ς , ς ; η , . . . η N , ξ, ̟ q , (4.5)assuming that for every b P r B , each connected component of r C b “ r π ´ p b q contains oneof ς p b q , . . . , ς N p b q . Choose r, ρ ą and a neighborhood U (resp. U ) of ς p r B q (resp. ς p r B q ) on which ξ (resp. ̟ ) is defined, such that p ξ, r π q : U ÝÑ D r ˆ r B resp. p ̟, r π q : U ÝÑ D ρ ˆ r B (4.6)is a biholomorphic map. We also assume that U and U are disjoint and do not inter-sect ς p r B q , . . . , ς N p r B q . Identify U “ D r ˆ r B resp. U “ D ρ ˆ r B ξ, ̟ (when restricted to the first components) becomethe standard coordinates of D r , D ρ respectively, and r π is the projection onto the r B -component. Set q “ ξ̟ “ π r,ρ : D r ˆ D ρ Ñ D rρ as previously.Set B “ D rρ ˆ r B . (4.7)We now sew the smooth family r X to obtain a family X “ p π : C Ñ B ; ς , . . . , ς N ; η , . . . , η N q of N -pointed complex curves with local coordinates. We first explain how to obtain C and π : C Ñ B . We shall freely switch the orders of Cartesian products. Note that F : “ E r,ρ ˆ r B Ă D r ˆ D rρ ˆ r B p“ U ˆ D rρ q ,F : “ E r,ρ ˆ r B Ă D ρ ˆ D rρ ˆ r B p“ U ˆ D rρ q are subsets of r C ˆ D rρ . They are the subsets we should discard in the sewing process.Then C is obtained by gluing r C ˆ D rρ (with F , F all removed) with W : “ D r ˆ D ρ ˆ r B . (4.8)To be more precise, we define C “ W ğ p r C ˆ D rρ ´ F ´ F q M „ (4.9)where the equivalence „ is described as follows. Consider the following subsets of W : W “ D ˆ r ˆ D ρ ˆ r B , (4.10) W “ D r ˆ D ˆ ρ ˆ r B . (4.11)Then the relation „ identifies W and W respectively via p ξ, q, r B q and p ̟, q, r B q to D r ˆ D rρ ˆ r B ´ F pĂ U ˆ D rρ q , (4.12) D ρ ˆ D rρ ˆ r B ´ F pĂ U ˆ D rρ q (4.13)(recall (4.4)), which are subsets of r C ˆ D rρ ´ F ´ F . (In particular, certain open subsetsof (4.12) and (4.13) are glued together and identified with W X W .)We now define π . It is easy to see that the projection r π ˆ : r C ˆ D rρ Ñ r B ˆ D rρ “ B , (4.14)agrees with π r,ρ ˆ : W “ D r ˆ D ρ ˆ r B Ñ D rρ ˆ r B “ B (4.15)when restricted to W , W . Thus, we have a well-defined surjective holomorphic map π : C Ñ B whose restrictions to r C ˆ D rρ ´ F ´ F and to W are r π b and π r,ρ b respectively.Finally, we extend each ς i : r B Ñ r C constantly to r B ˆ D rρ Ñ r C ˆ D rρ , whose imageis disjoint from F , F . Thus ς i can be regarded as a section ς i : B Ñ C . Likewise, weextend η i constantly over r B to a local coordinate at ς i p B q . We say that the N -points ς , . . . , ς N and the local coordinates η , . . . , η N are constant with respect to sewing .19 short exact sequence The goal of this subsection is to recall a short exact sequence (4.20) (cf. [Ueno97,Eq. (4.2.3)] or [AU07, Eq. (4.6)]) which plays an important role in defining a logarith-mic connection on sheaves of conformal blocks. In Section 11, we will use this exactsequence to find the differential equations that sewn conformal blocks satisfy.Consider the discriminant locus ∆ and the critical locus Σ : ∆ “ t u ˆ r B pĂ D rρ ˆ r B “ B q , Σ “ t u ˆ t u ˆ r B pĂ D r ˆ D ρ ˆ r B “ W q . Then ∆ is the set of all b P B such that the fiber C b “ π ´ p b q is a nodal curve with onenode. Outside ∆ , the fibers are compact Riemann surfaces. Σ is the set of all points of C at which dπ is not surjective, and is also the set of all nodes of the fibers. Note that Σ “ W ´ p W Y W q . Also, Σ is the set of points of C not coming from r C ˆ D rρ ´ F ´ F . The union of nodalfibers is therefore C ∆ “ π ´ p ∆ q . Let Θ B p´ log ∆ q and Θ C p´ log C ∆ q be the sheaves of sections of Θ B and Θ C tangentto ∆ and C ∆ respectively. Then the differential dπ : Θ C Ñ π ˚ Θ B of the map π restrictsto an O C -module homomorphism dπ : Θ C p´ log C ∆ q Ñ π ˚ Θ B p´ log ∆ q (4.16)(the later is short for π ˚ p Θ B p´ log ∆ qq ), which is indeed surjective. To understand themeaning and to see the claimed facts, let us describe the two sheaves and the mor-phism dπ using coordinates.We assume r B is small enough to admit a coordinate τ ‚ “ p τ , . . . , τ m q : r B Ñ C m .Then p q, τ ‚ q is a coordinate of B if we let q be the standard coordinate of D rρ . Then Θ B p´ log ∆ q is an O B -module generated freely by q B q , B τ , . . . , B τ m . Their pullback under π will also be denoted by the same symbols q B q , B τ , . . . , B τ m , forsimplicity. Choose any x P C .Case I. x R Σ . Then x can be regarded as a point p r x, q q of r C ˆ D rρ disjoint from F , F . Choose a neighborhood r U Ă r C of r x together with η P O p r U q univalent on eachfiber of r U . Choose a neighborhood V of q P D rρ such that U : “ r U ˆ V is disjoint from F , F . Write τ ‚ ˝ r π also as τ ‚ for short. Then p η, τ ‚ , q q is a coordinate of U Q x . Note that U X C ∆ is described by q “ . The O V -module Θ C p´ log ∆ q| V is generated freely by B η , q B q , B τ , . . . , B τ m , (4.17)and the morphism dπ in (4.16) sends B η to and keeps the other elements of (4.17).20ase II. x P Σ . Then W “ D r ˆ D ρ ˆ r B is a neighborhood of x , and has coordinate p ξ, ̟, τ ‚ q . Note that W X C ∆ is described by ξ̟ “ . The O W -module Θ C p´ log ∆ q| W isgenerated freely by ξ B ξ , ̟ B ̟ , B τ , . . . , B τ m , (4.18)and the morphism dπ satisfies dπ p ξ B ξ q “ dπ p ̟ B ̟ q “ q B q (4.19)(note that q B q is short for π ˚ p q B q q ) and keeps the other elements of (4.18).It is clear that in both cases, dπ (in (4.16)) is surjective. Thus, by letting Θ C { B be thekernel of dπ , we obtain an exact sequence of O C -modules Ñ Θ C { B Ñ Θ C p´ log C ∆ q dπ ÝÑ π ˚ Θ B p´ log ∆ q Ñ (4.20)In case I resp. case II, Θ C { B | V resp. Θ C { B | W is generated (freely) by B η , resp. ξ B ξ ´ ̟ B ̟ . (4.21)So Θ C { B is locally free of rank , whose dual module is the relative dualizing sheaf ω C { B .Using (4.2), we see that when restricted to W (resp. W ) and under the coordinate p ξ, q, τ ‚ q (resp. p ̟, q, τ ‚ q ), the section ξ B ξ ´ ̟ B ̟ in (4.21) equals ξ B ξ resp. ´ ̟ B ̟ . (4.22)Compare this with (2.1), we see that for each b P B , there are natural equivalences Θ C { B | C b » Θ C b , ω C { B | C b » ω C b . Finally, we remark that when X is a smooth family, dπ : Θ C Ñ π ˚ Θ B is surjective.Thus, (4.20) becomes Ñ Θ C { B Ñ Θ C dπ ÝÑ π ˚ Θ B Ñ where Θ C { B is the kernel of dπ . A theorem of Grauert
Recall that if C is a complex curve and E is a coherent O C -module, then H p p C, E q is always finite-dimensional by (for instance) the direct image theorem of Grauert.Also H p p C, E q vanishes when p ą , which follows for instance from the fact that C can be covered by two Stein open subsets. Thus, the character of E is χ p C, E q “ dim H p C, E q ´ dim H p C, E q .Given a family X “ p π : C Ñ B q of complex curves, recall that we have assumed π is proper when X is a smooth family. If X is formed by sewing r X , then properness stillholds and is not hard to check. Finally, in both smooth and singular cases, the map π is clearly open. So π is flat by [Fis76, Sec. 3.20] (see also [GPR94, Thm. II.2.13] or [BS76,Thm. V.2.13]). If E is locally free (of finite rank), then E is flat over B . Therefore, by atheorem of Grauert [Gra60] (cf. [GPR94, Thm. III.4.7] or [BS76, Thm. III.4.12] or [EP96,Thm. 9.4.8]), we have 21 heorem 4.1. Let X “ p π : C Ñ B q be a family of complex curves. Let E be a locally free O C -module.(a) The function B Ñ Z , b ÞÑ χ p C b , E | C b q “ dim H p C b , E | C b q ´ dim H p C b , E | C b q is locally constant.(b) For any p P N , if the function b ÞÑ dim H p p C b , E | C b q is locally constant, then the O B -module R p π ˚ E is locally free of rank dim H p p C b , E | C b q , and for any b P B , the linear map p R p π ˚ E q b Ñ H p p C b , E | C b q defined by restricting the sections s ÞÑ s | C b induces an isomor-phism of vector spaces p R p π ˚ E q b m b ¨ p R p π ˚ E q b » H p p C b , E | C b q . Recall R π ˚ “ π ˚ , and the (higher) direct image sheaf R p π ˚ E is an O B -module asso-ciated to the presheaf V ÞÑ H p p V, E | V q (for all open V Ă B ). As a consequence of thistheorem, we see R p π ˚ E “ when E is locally free and p ą . Let X “ p π : C Ñ B q be a family of complex curves. Recall that Σ is the criticallocus, which is empty when the family is smooth. Let U, V be open subsets of C ´ Σ ,and let η P O p U q , µ P O p V q be univalent on each fiber of U and V respectively. Then p η, π q and p µ, π q are biholomorphic maps from U resp. V to open subsets of C ˆ B . Foreach p P U X V , we define ̺ p η | µ q p P O C , by ̺ p η | µ q p p z q “ η ˝ p µ, π q ´ ` z ` µ p p q , π p p q ˘ ´ η p p q . (5.1)Then ̺ p η | µ q p is a holomorphic function of z on µ ` p U X V q π p p q ˘ where p U X V q π p p q is thefiber U X V X π ´ p π p p qq . It is easy to check that for each n P N , B nz ̺ p η | µ q p p q “ B nµ η p p q , (5.2)where the partial derivative B µ is defined to be vertical to dπ . From this, we see that ̺ p η | µ q p p q “ and B z ̺ p η | µ q p p q ‰ . So ̺ p η | µ q p is an element of G . We thus obtain afamily of transformations ̺ p η | µ q : U X V Ñ G , p ÞÑ ̺ p η | µ q p , which is clearly holomor-phic. According to Remark 1.5, we have an O U X V -module isomorphism U p ̺ p η | µ qq : V ď n b C O U X V » ÝÑ V ď n b C O U X V . As in Section 2, ̺ p η | µ q is also described by η ´ η p p q ˇˇ p U X V q π p p q “ ̺ p η | µ q p ` µ ´ µ p p q ˇˇ p U X V q π p p q ˘ . (5.3)To see this, one composes both sides of (5.3) with p µ, π q ´ ` z ` µ p p q , π p p q ˘ . Thus, wecan get ̺ p η | µ q p by restricting η, µ to the fiber p U X V q π p p q and then using definition(2.2). Therefore, the cocycle relation (2.4) still holds for holomorphic functions η , η , η univalent on each fiber. 22 efinition of V X Sheaves of VOAs were introduced in [FB04] for algebraic families of smoothcurves, and were generalized in [DGT19a] to algebraic families of stable curves. Sim-ilar to their construction, we now define the sheaf of VOA V X associated to a VOA V and the analytic family X .We set V X “ lim ÝÑ n P N V ď n X where each V ď n X is a locally free O C -module of rank dim V ď n defined as follows. Out-side Σ , V ď n X is a vector bundle with transition function U p ̺ p η | µ qq . Thus, for each open U Ă C ´ Σ and η P O p U q univalent on each fiber, we have an isomorphism of O U -modules (a trivilization) U ̺ p η q : V ď n X | U » ÝÑ V ď n b C O U . (5.4)If V is another open subset of C ´ Σ and µ P O p V q is also univalent on each fiber, thenon U X V we have U ̺ p η q U ̺ p µ q ´ “ U p ̺ p η | µ qq . (5.5)Since (2.7) holds when restricting to each fiber, we again have that for any section v of V ď n b C O U X V , U ̺ p η q U ̺ p µ q ´ ¨ v “ pB µ η q n ¨ v mod V ď n ´ b C O U X V . (5.6)We now assume X is formed by sewing a smooth family r X as in section 4. Let W, W , W , ξ, ̟, q be as in that section. (See the discussion near (4.8).) Then p ξ, q, r B q and p ̟, q, r B q are respectively biholomorphic maps from W and W to complex man-ifolds, and the projection π equals p q, r B q when restricted to W or W . Thus, ξ, ̟ areunivalent on fibers of W , W respectively.We shall define V ď n X | W to be an O W -submodule of V ď n X | W ´ Σ generated (freely) bysome sections on W whose restrictions to W and W are described under the trivi-lizations U ̺ p ξ q and U ̺ p ̟ q respectively. For that purpose, we need to first calculate thetransition function U p ̺ p ̟ | ξ qq : V ď n b C O W X W » ÝÑ V ď n b C O W X W . Lemma 5.1.
Choose any p P W X W . Then we have ̺ p ̟ | ξ q p p z q “ q p p q γ ξ p p q p z q and hence U p ̺ p ̟ | ξ q p q “ q p p q L U p γ ξ p p q q . roof. Choose any x P p W X W q π p p q . Then π p x q “ π p p q and hence q p x q “ q p p q . Since ̟ “ ξ ´ q , we have ̟ p x q ´ ̟ p p q “ q p p qp ξ p x q ´ ´ ξ p p q ´ q . By (5.3), we have ̟ p x q ´ ̟ p p q “ ̺ p ̟ | ξ q p p ξ p x q ´ ξ p p qq . If we compare these two equations and set z “ ξ p x q ´ ξ p p q , we obtain ̺ p ̟ | ξ q p p z q “ ̺ p ̟ | ξ q p p ξ p x q ´ ξ p p qq “ q p p qp ξ p x q ´ ´ ξ p p q ´ q“ q p p q ` p ξ p p q ` z q ´ ´ ξ p p q ´ ˘ “ q p p q γ ξ p p q p z q . We define V ď n X | W to be the O W -submodule of V ď n X ´ Σ | W ´ Σ generated by any sectionon W ´ Σ whose restrictions to W and W are U ̺ p ξ q ´ ` ξ L v ˘ resp. U ̺ p ̟ q ´ ` ̟ L U p γ q v ˘ (5.7)where v P V ď n . Since γ “ γ ´ and hence U p γ q “ U p γ q ´ , this definition is symmetricwith respect to ξ and ̟ . To check that (5.7) is well-defined, we need: Lemma 5.2.
The two sections defined in (5.7) agree on W X W .Proof. Using (1.9) and Lemma 5.1, we check that U ̺ p ̟ q U ̺ p ξ q ´ ξ L v “ U p ̺ p ̟ | ξ qq ξ L v “ q L U p γ ξ q ξ L v “ q L ξ ´ L U p γ q v “ ̟ L U p γ q v. It is easy to see that, if we take v P E where E is a basis of V ď n consisting ofhomogeneous vectors, then V ď n X | W is generated freely by sections defined by (5.7) forall v P E . We have completed the definition of the locally free O C -module V ď n X . Remark 5.3.
Since the vacuum vector is annihilated by L n ( n ě ), we see that is fixed by any transition function U p ̺ p η | µ qq . Thus, we can define unambiguously anelement P V X p C ´ Σ q (the vacuum section ) such that for any open U Ă C ´ Σ andany η P O p U q univalent on each fiber, U ̺ p η q is the vaccum vector (considered as aconstant function). Also, by (5.7), it is clear that P V X p C q . estriction to fibers By comparing the transition functions and looking at the generating sections nearthe nodes, it is easy to see:
Proposition 5.4.
For any n P N and b P B , we have a natural isomorphism of O C b -modules V ď n X | C b » V ď n C b . Let now X “ p π : C Ñ B ; ς , . . . , ς N q be a family of N -pointed complex curves. Theorem 5.5.
Let n P N . Then there exists k P N such that for any k ą k , the O B -module π ˚ ` V ď n X b ω C { B p kS X q ˘ is locally free, and for any b P B there is a natural isomorphism of vectorspaces π ˚ ` V ď n X b ω C { B p kS X q ˘ b m b ¨ π ˚ ` V ď n X b ω C { B p kS X q ˘ b » H ` C b , V ď n C b b ω C b p kS X p b qq ˘ (5.8) defined by restriction of sections. In particular, dim H ` C b , V ď n C b b ω C b p kS X p b qq ˘ is locallyconstant over b . Recall that the left hand side of (5.8) is the fiber π ˚ ` V ď n X b ω C { B p kS X q ˘ˇˇ b , which, byCartan’s Theorem A, is formed by the restrictions of global sections if B is Stein. Proof.
By Theorem 2.3, we can find k P N such that for any k ą k , H ` C b , V ď n C b b ω C b p kS X p b qq ˘ vanishes for any b P B . Since the restriction of ω C { B to C b is ω C b , byProposition 5.4, the restriction of V ď n X b ω C { B p kS X q to C b is naturally equivalent to V ď n C b b ω C b p kS X p b qq . Thus, our theorem follows easily from Grauert’s Theorem 4.1.For any O C -module E , we set E p‚ S X q “ lim ÝÑ k P N E p kS X q , whose sections are meromorphic sections of E whose only possible poles are in ς p B q , . . . , ς N p B q . Corollary 5.6. If B is a Stein manifold, then for any n P N and any b P B , the restrictions to C b of the elements of H ` C , V ď n X b ω C { B p‚ S X q ˘ form the vector space H ` C b , V ď n C b b ω C b p‚ S X p b qq ˘ .Proof. Choose k as in Theorem 5.5. Then by that theorem and Cartan’s Theorem A,we see the claim holds if ‚ S X is replaced by kS X for all k ą k . The original claim thusfollows. The subsheaf V ir c If we compare (5.6) with the transition functions of Θ C { B , and compare (5.7) with(4.22), we immediately see that 25 roposition 5.7. For any n P N , we have the following isomorphism of O C -modules: V ď n X { V ď n ´ X » V p n q b C Θ b n C { B . (5.9) Under this isomorphism, if U Ă C ´ Σ is open and smooth, and η P O p U q is univalent on eachfiber of U , then for any v P V p n q , v b B nη is identified with the equivalence class of U ̺ p η q ´ v . We now assume for simplicity that X is a smooth family, and define an important O C -submodule V ir c of V ď X related to the conformal vector c P V p q . See [FB04, Sec.8.2]. If U is an open subset of C equipped with η P O p U q univalent on each fiber, then V ir c | U is the O U -submodule of V X | U generated (freely) by U ̺ p η q ´ c and the vacuumsection , which is locally free of rank . This definition is independent of the choiceof η . Indeed, if µ : U Ñ C is also univalent on each fiber, then U ̺ p µ q U ̺ p η q ´ c “ U p ̺ p µ | η qq c , which can be calculated using the actions of L n ( n ě ) on c , is an O U -linear combination of c and . Thus, by gluing all such U , we get V ir c .By Proposition 5.7, we have a short exact sequence Ñ V ď X Ñ V ď X λ ÝÑ V p q b C Θ b C { B Ñ where λ is described locally by sending U ´ ̺ p η q v (where v P V p q ) to v ¨ B η and sendingthe submodule V ď X to . Using this description of λ , it is easy to see that the restrictionof λ to the subsheaf V ir c has image c b C Θ b C { B » Θ b C { B , and that its kernel is V ď X “ b C O C » O C . Thus, we obtain an exact sequence Ñ O C Ñ V ir c λ ÝÑ Θ b C { B Ñ . (5.10)If we choose U Ă C and η P O p U q holomorphic on each fiber, then λ : U ̺ p η q ´ c ÞÑ B η , ÞÑ . By tensoring with ω C { B , we get an exact sequence Ñ ω C { B Ñ V ir c b ω C { B λ ÝÑ Θ C { B Ñ (5.11)whose local expression is λ : U ̺ p η q ´ c dη ÞÑ B η , dη ÞÑ . (5.12) Let X “ p π : C Ñ B ; ς , . . . , ς N q be a family of N -pointed complex curves. For each b P B , X b “ p C b ; ς p b q , . . . , ς N p b qq is an N -pointed complex curve. If X is equipped with local coordinates η , . . . , η N at ς p B q , . . . , ς N p B q , then, by restricting to C b , X b also has local coordinates.26et W , . . . , W N be V -modules. Then for each b P B , T ˚ X b p W ‚ q is the space of confor-mal blocks associated to W ‚ and X b . In this section, we shall define conformal blocks φ associated to the family X . φ is a function on B whose value φ p b q at each b P B isan element of T ˚ X b p W ‚ q . Moreover, we require φ is a holomorphic function in a cer-tain sense. Since this property is local, we assume that B is small enough such that X admits local coordinates. This assumption will be dropped in the later half of thissection. Definition of conformal blocks
We need to explain how the vector spaces W X b p W ‚ q (for all b P B ) form an (infiniterank) holomorphic vector bundle W X p W ‚ q .Define W X p W ‚ q to be an infinite rank locally free O B -module as follows. For any lo-cal coordinates η , . . . , η N of X at ς p B q , . . . , ς N p B q respectively, we have a trivialization U p η ‚ q ” U p η q b ¨ ¨ ¨ b U p η N q : W X p W ‚ q » ÝÑ W ‚ b C O B (6.1)such that if µ ‚ is another set of local coordinates, then the transition function U p η ‚ q U p µ ‚ q ´ : W ‚ b C O B » ÝÑ W ‚ b C O B is defined such that for any constant section w ‚ “ w b ¨ ¨ ¨ b w N P W ‚ , U p η ‚ q U p µ ‚ q ´ w ‚ ,as a W ‚ -valued holomorphic function, satisfies ´ U ` η ‚ ˘ U ` µ ‚ ˘ ´ w ‚ ¯ p b q ” ´ U ` η ‚ ˇˇ µ ‚ ˘ ´ w ‚ ¯ p b q“ U ` p η | µ q b ˘ w b U ` p η | µ q b ˘ w b ¨ ¨ ¨ b U ` p η N | µ N q b ˘ w N (6.2)for any b P B . Here, for each ď i ď N , p η i | µ i q b is the element in G satisfying p η i | µ i q b p z q “ η i ˝ p µ i , π q ´ p z, b q . (6.3)If we compare the transition functions (3.4) and (6.2), we see that there is a natural andcoordinate-independent isomorphism of vector spaces W X p W ‚ q| b » W X b p W ‚ q where W X b p W ‚ q is defined near (3.3). We shall identify these two spaces in the follow-ing. Definition 6.1. A conformal block φ associated to W ‚ , X (and defined over B ) is an O B -module homomorphism φ : W X p W ‚ q Ñ O B whose restriction to each fiber W X b p W ‚ q isan element of T ˚ X b p W ‚ q , i.e., vanishes on H ` C b , V C b b ω C b p‚ S X p b qq ˘ ¨ W X b p W ‚ q . The vectorspace of all such φ is denoted by T ˚ X p W ‚ qp B q . Remark 6.2.
The holomorphicity of a conformal block φ as a function on B wouldbe easier to understand by shrinking B and choosing local coordinates η , . . . , η N , andidentifying W X p W ‚ q with W ‚ b C O B via U p η ‚ q . That φ is a morphism W X p W ‚ q Ñ O B means precisely that φ is an p W ‚ q ˚ -valued function on B whose evaluation with anyvector of W ‚ is a holomorphic function on B ; also, φ is determined by and can bereconstructed from the corresponding O p B q -module homomorphism W X p W ‚ qp B q Ñ O p B q . Moreover, φ is in T ˚ X p W ‚ qp B q if and only if for each b P B , φ p b q as a linearfunctional on W ‚ » W X b p W ‚ q is in T ˚ X b p W ‚ q .27 ction of H ` C , V X b ω C { B p‚ S X q ˘ on W X p W ‚ qp B q The above definition of conformal blocks is fiberwise, i.e., the restriction of φ toeach fiber of the family is a conformal block defined in Section 3. We now give aglobal description which relates conformal blocks to the sheaf of VOA V X .Recall there are natural equivalences V X | C b » V C b and ω C { B | C b » ω C b . Thus, we candefine a linear action of H ` C , V X b ω C { B p‚ S X q ˘ on W X p W ‚ qp B q whose restriction to eachfiber H p C b , V C b b ω C b p‚ S X p b qqq acts on W X b p W ‚ q . To see that the image of this action isholomorphic on B , let us write down the action more explicitly.Let z be the standard variable of C . If W is a V -module, we have a linear map Y W : ` V b C O p B qpp z qq ˘ b O p B q p W b C O p B qq Ñ W b C O p B qpp z qq ,v p b, z q b w p b q ÞÑ Y W p v p b, z q , z q w p b q (6.4)which is clearly an O p B qpp z qq -module homomorphism. Note that a section of V b C O C ˆ B on a neighborhood of t u ˆ B can be regarded as an element of V b C O p B qpp z qq by taking series expansion.Now, choose local coordinates η ‚ of X . For each ď i ď N , choose a neighborhood U i of ς i p B q on which η i is defined. Identify W X p W ‚ q with W ‚ b C O B via U p η ‚ q . Bytensoring with the identity map of ω C { B , the map (5.4) induces naturally an O U i -moduleisomorphism (also denoted by U ̺ p η i q ): U ̺ p η i q : V X b ω C { B p‚ S X q ˇˇ U i » ÝÑ V b C ω C { B p‚ S X q ˇˇ U i . Identify U i with p η i , π qp U i q via p η i , π q . Then η i as a variable equals the standard variable z . If v is a section of V X b ω C { B p‚ S X q defined near ς i p B q and w i P W i b C O p B q , then againwe have v ¨ w i “ Res z “ Y W i p U ̺ p η i q v, z q w i , (6.5)which is clearly also an element of W i b C O p B q . If v P H ` C , V X b ω C { B p‚ S X q ˘ , then asin (3.2), v acts on W ‚ b C O p B q by summing up the actions on the tensor components. Another description of conformal blocks
We now give a global description of conformal blocks, followed by an application.We shall not assume X admits local coordinates. Then W X p W ‚ q is still an infinite-ranklocally free O B -module whose local trivializations are given by (6.1). Again, we defineconformal blocks using Definition 6.1. Theorem 6.3.
Let φ : W X p W ‚ q Ñ O B be an O B -module homomorphism. If B is a Steinmanifold, then φ is in T ˚ X p W ‚ qp B q if and only if the evaluation of φ with any element of J p B q “ H ` C , V X b ω C { B p‚ S X q ˘ ¨ W X p W ‚ qp B q (6.6) is the zero function on B . Note that we have omitted
Span C in (6.6).28 roof. The “only if” part is obvious by the definition of conformal blocks and does notrequire B to be Stein. The “if” part follows from Corollary 5.6.For any open subset V Ă B , let C V “ π ´ p V q . Then X restricts to X V : “ p π : C V Ñ V ; ς | V , . . . , ς N | V q . Any set of local coordinates η ‚ of X restricts to one of X V . We also write T ˚ X V p W ‚ qp V q as T ˚ X p W ‚ qp V q . The reason for this notation will be explained shortly. Proposition 6.4.
Let φ : W X p W ‚ q Ñ O B be an O B -module homomorphism. Assume that B is connected and contains a non-empty open subset V such that φ | V : W X p W ‚ q| V Ñ O V is anelement of T ˚ X p W ‚ qp V q . Then φ is an element of T ˚ X p W ‚ qp B q .Proof. We first assume B is also Stein. The evaluation of φ with any element of J p B q is a holomorphic function on B vanishing on V . So it must be constantly . So φ is aconformal block by Theorem 6.3.Now, we do not assume B is Stein. Let A be the set of all b P B such that b hasa neighborhood U satisfying that the restriction φ | U is a conformal block. Then A isopen and non-empty. For any b P B ´ A , let U be a connected Stein neighborhood of b .Then, by the first paragraph, φ | U is a conformal block if U has a non-zero open subset V such that φ | V is a conformal block. Therefore U must be disjoint from A . This showsthat B ´ A is open. So B “ A . Remark 6.5.
It is clear that the collection of all T ˚ X p W ‚ qp V q (where V Ă B ) form asheaf of O B -modules, which we denote by T ˚ X p W ‚ q and call the sheaf of conformalblocks associated to W ‚ and X . Let T X p W ‚ q be the sheaf of O B -modules associated tothe presheaf V ÞÑ W X p W ‚ qp V q H ` C V , V X V b ω C V { V p‚ S X V q ˘ ¨ W X p W ‚ qp V q and call it the sheaf of covacua . Then, by Theorem 6.3, T ˚ X p W ‚ q is the dual O B -moduleof T X p W ‚ q . Moreover, using Theorem 5.5 or Corollary 5.6, it is easy to see that for each b P B , the clearly surjective linear map T X p W ‚ q b m b ¨ T X p W ‚ q b Ñ T X b p W ‚ q (6.7)defined by the restriction W X p W ‚ q b Ñ W X p W ‚ q| b is injective. Thus, the above two vectorspaces have the same dimension, and are clearly equal to that of T ˚ X b p W ‚ q .If V is C -cofinite and rational, then by the main results of [DGT19b], the function b ÞÑ dim T X b p W ‚ q has finite values and is locally constant. By Theorem 7.3 (appliedto any subfamily X V where V is Stein), the O B -module T X p W ‚ q is finitely-generated.Thus, T X p W ‚ q is locally free by an easy consequence of Nakayama’s Lemma. (Cf. forinstance [BS76, Lemma III.1.6].) Thus, T ˚ X p W ‚ q is also locally free, and by (6.7), its Since each fiber C b is a projective variety, and since each V ď n X is locally free and (by comparing thetwo constructions) equivalent to the analytification of the corresponding one in [DGT19a, DGT19b], thealgebraic results of T X b p W ‚ q also hold in the analytic setting. X b can be extended to a conformal blockassociated to a family X V whenever V is Stein.If V is only C -cofinite but X is a smooth family, then our claims about T X p W ‚ q and T ˚ X p W ‚ q and their fibers in the last paragraph are still true since, by [DGT19b], theabove result about dimensions still holds. Recall that V is called C -cofinite if the subspace of V spanned by C p V q : “t Y p u q ´ v : u, v P V u has finite codimension. The following important result is dueto Buhl [Buhl02, Thm. 1]. Theorem 7.1.
Assume V is C -cofinite. Then there exist Q P N and a finite set E of homoge-neous vectors of V satisfying the following condition: For any weak V -module W generated bya vector w , there exists L P N such that W is spanned by elements of the form Y W p v k q ´ n k Y W p v k ´ q ´ n k ´ ¨ ¨ ¨ Y W p v q ´ n w (7.1) where n k ě n k ´ ě ¨ ¨ ¨ ě n ą ´ L and v , v , . . . , v k P E . In addition, for any ď j ď k , if n j ą then n j ą n j ´ ; if n j ď then n j “ n i for at most Q different i . We will fix this E in this section. Buhl’s result will be used in the following form. Corollary 7.2.
Assume that V is C -cofinite. Let W be a finitely-generated V -module. Thenfor any n P N , there exists ν p n q P N such that any r L -homogeneous vector w P W whoseweight Ă wt p w q ą ν p n q is a sum of vectors in Y W p v q ´ l W p Ă wt p w q ´ wt p v q ´ l ` q where v P E and l ą n .Proof. Assume without loss of generality that W is generated by a single r L -homogeneous vector w . Let T be the set of all vectors of the form (7.1) where n k ď n .Then, by the above theorem, T is a finite subset of W . Set ν p n q “ max t Ă wt p w q : w P T u .If w P W is r L -homogeneous with weight Ă wt p w q ą ν p n q , then we can also write w as asum of r L -homogeneous vectors of the form (7.1), but now the n k must be greater than ν since such vector is not in T . This proves that w is a sum of r L -homogeneous vectorsof the form Y W p v q ´ l w where v P E , l ą n , and w P W is r L -homogeneous. The sameis true if W is finitely-generated. By (1.1), we have Ă wt p w q “ Ă wt p w q ´ wt p v q ´ l ` .Let X “ p π : C Ñ B ; ς , . . . , ς N ; η , . . . , η N q be a family of N -pointed complex curveswith local coordinates, and let W , . . . , W N be finitely-generated V -modules. NoticeTheorem 6.3 and recall the definition of J p B q in (6.6). Theorem 7.3.
Let V be C -cofinite, and assume that B is a Stein manifold. Then W X p W ‚ qp B q{ J p B q is a finitely-generated O p B q -module.Proof. Since local coordinates are chosen, we identify W X p W ‚ q with W ‚ b C O B . Let E “ max t wt p v q : v P E u . By Theorem 2.3, there exists k P N such that H p C b , V ď E C b b ω C b p kS X qq “ (7.2)30or any b P B and k ě k . We fix an arbitrary k P N satisfying k ě E ` k .Introduce a weight Ă wt on W ‚ such that Ă wt p w ‚ q “ Ă wt p w q ` Ă wt p w q ` ¨ ¨ ¨ ` Ă wt p w N q when w , . . . , w N are r L -homogeneous. For each n P N , W ď n ‚ (resp. W ‚ p n q ) denotes the(finite-dimensional) subspace spanned by all r L -homogeneous homogeneous vectors w P W ‚ satisfying Ă wt p w q ď n (resp. Ă wt p w q “ n ). We shall prove by induction that forany n ą N ν p k q , any vector of W ‚ p n q (considered as constant sections of W ‚ b C O p B q ) isa (finite) sum of elements of W ď n ´ ‚ b C O p B q mod J p B q . Then the claim of our theoremfollows.Choose any w ‚ “ w b ¨ ¨ ¨ b w N P W ‚ p n q such that w , . . . , w N are r L -homogeneous.Then one of w , . . . , w N must have r L -weight greater than ν p k q . Assume, without lossof generality, that Ă wt p w q ą ν p k q . Then, by Corollary 7.2, w is a sum of non-zero r L -homogeneous vectors of the form Y W p u q ´ l w ˝ where u P E , l ą k , w ˝ P W , and Ă wt p w ˝ q “ Ă wt p w q ´ wt p u q ´ l ` . Thus Ă wt p w q ´ Ă wt p w ˝ q ě l ´ ě k ě E ` k . It suffices to show that each Y W p u q ´ l w ˝ b w b ¨ ¨ ¨ b w N is a sum of elements of W ď n ´ ‚ b C O p B q mod J p B q . Thus, we assume for simplicity that w “ Y W p u q ´ l w ˝ .Then w ‚ “ Y W p u q ´ l w ˝ b w b ¨ ¨ ¨ b w N . Set also w ˝‚ “ w ˝ b w b ¨ ¨ ¨ b w N . Then n ´ Ă wt p w ˝‚ q “ Ă wt p w ‚ q ´ Ă wt p w ˝‚ q ě E ` k . Thus Ă wt p w ˝‚ q ď n ´ E ´ k . (7.3)Consider the short exact sequence of O C -modules Ñ V ď E C b ω C { B p k S X q Ñ V ď E C b ω C { B p lS X q Ñ G Ñ where G is the quotient of the previous two sheaves. By (7.2), Proposition 5.4, andGrauert’s Theorem 4.1, we see that R π ˚ p V ď E C b ω C { B p k S X qq “ , and π ˚ p V ď E C b ω C { B p k S X qq is locally free. Thus, we obtain an exact sequence of O B -modules Ñ π ˚ ` V ď E C b ω C { B p k S X q ˘ Ñ π ˚ ` V ď E C b ω C { B p lS X q ˘ Ñ π ˚ G Ñ . Since B is Stein, by Cartan’s Theorem B, H p B , π ˚ p V ď E C b ω C { B p k S X qqq “ . Thus, thereis an exact sequence Ñ H ` B , π ˚ ` V ď E C b ω C { B p k S X q ˘˘ Ñ H ` B , π ˚ ` V ď E C b ω C { B p lS X q ˘˘ Ñ H ` B , π ˚ G ˘ Ñ . (7.4)Note that H ` B , π ˚ G ˘ is exactly G p C q . Choose mutually disjoint neighborhoods W , . . . , W N of ς p B q , . . . , ς N p B q respectively. For each ď i ď N , identify V ď E C b ω C { B | W i with V ď E b C ω C { B | W i via U ̺ p η i q , and identify η i with the standard coordinate z by iden-tifying W i with p η i , π qp W i q . Define an element υ P G p C q as follows. υ | W is the equiva-lence class represented by uz ´ l dz , and υ | C ´ ς p B q “ . Since the second map in the above31xact sequence is surjective, υ lifts to an element p υ of H ` B , π ˚ ` V ď E C b ω C { B p lS X q ˘˘ ,i.e., of ` V ď E C b ω C { B p lS X q ˘ p C q . Moreover, by the definition of G as a quotient, foreach ď i ď N we have an element v i of V ď E b C O C p k S X qp W i q (and hence of V ď E b C O W i p k ς i p B qqp W i q ) such that p υ | W “ uz ´ l dz ` v dz, p υ | W i “ v i dz p ď i ď N q . Notice that
Res z “ Y p¨ , z q z n dz “ Y p¨q n . It follows that the element p υ ¨ w ˝‚ , which is in J p B q , equals w ‚ ` w △ where w △ “ p v dz q ¨ w ˝ b w b ¨ ¨ ¨ b w i b ¨ ¨ ¨ b w N ` N ÿ i “ w ˝ b w b ¨ ¨ ¨ b p v i dz q ¨ w i b ¨ ¨ ¨ b w N . Thus w ‚ equals ´ w △ mod J p B q . For each ď i ď N , v i has pole at z “ with orderat most k . Thus, by (1.1), the action of v i dz on W i increases the r L -weight by at most E ` k ´ . It follows from (7.3) that w ∆ P W ď n ´ ‚ b C O p B q . The proof is complete. Remark 7.4.
Theorem 7.3 is the complex-analytic analogue of [DGT19b] Thm. 8.2.1,which says that for an algebraic family of complex curves, the sheaf of covacua iscoherent (assuming that V is C -cofinite). The key ideas in our proof are similar totheirs.It is clear that Theorem 7.3 implies T X p W ‚ q is a finitely-generated O B -module.However, Theorem 7.3 does not seem to imply that T X p W ‚ q is analytically coherent.This is different from the algebraic setting in which the sheaves of covacua are al-gebraically coherent since they are quasi-coherent. Nevertheless, one can show that T X p W ‚ q is locally free (if V is also rational) by combining the algebraic results withTheorem 7.3, as explained in Remark 6.5.We remark that certain forms of Theorem 7.3 are well-known in the low genuscases: see [Zhu96, Lemma 4.4.1], [Hua05a, Cor. 1.2], [Hua05b, Cor. 3.4]. Let X “ p π : C Ñ B q be a family of compact Riemann surfaces. Choose an opensubset U Ă C and holomorphic functions η, µ : U Ñ C univalent on each fiber. If f P O p U q and B η f is nowhere zero, we define the (partial) Schwarzian derivative of f over η to be S η f “ B η f B η f ´ ´ B η f B η f ¯ (8.1)where the partial derivative B η is defined with respect to p η, π q , i.e., it is annihilated by dπ and restricts to d { dη on each fiber. Similarly, one can define S µ f .We refer the reader to [Ahl, Gun] for the basic facts about Schwarzian derivatives.The change of variable formula is easy to calculate: S µ f “ pB µ η q S η f ` S µ η, (8.2)32ake f “ µ and notice S µ µ “ , we have S µ η “ ´pB µ η q S η µ. (8.3)Assuming f is also univalent on each fiber, we obtain the cocycle relation. S µ η ¨ dµ “ ´ S η µ ¨ dη , S µ f ¨ dµ ` S f η ¨ df ` S η µ ¨ dη “ . (8.4)The transition functions of V ir c (which is a subsheaf of V X ) defined in Section 5 canbe expressed by Schwarzian derivatives. Note that L c “ c , L c “ , L c “ c , and L n c “ for all n ą . Thus, if ρ “ ρ p z q P G , then using the formula (1.3), we have U p ρ q c “ ρ p q L e c L c “ ρ p q L p c ` c c q “ ρ p q c ` c c where c is the central chargeof V , and c , which is given by (1.4), is S z ρ p q . Replace ρ by ̺ p η | µ q : U Ñ G . Then ρ p n q p q should be replaced by B nµ η . Thus the transition function U p ̺ p η | µ qq is describedby U p ̺ p η | µ qq “ , U p ̺ p η | µ qq c “ pB µ η q c ` c S µ η ¨ . (8.5)We recall some well-known properties of Schwarzian derivatives. See [Hub81]. Proposition 8.1.
The following are true.(1) If the restriction of η to each fiber U b “ U X π ´ p b q (where b P B ) is a M¨obius transfor-mation of µ , i.e., of the form aµ ` bcµ ` d where ad ´ bc ‰ , then S µ η “ .(2) Let Q P O p U q . Then, for each x P U , one can find a neighborhood V Ă U of x and afunction f P O p V q univalent on each fiber V b “ V X π ´ p b q , such that S η f “ Q .(3) If f, g P O p U q are univalent on each fiber, then S η f “ S η g if and only if S f g “ . We remark that the converse of (1) is also true: If f is univalent on each fiber, andif S η f “ , then the restriction of f to each fiber is a M ¨obius transformation of η . Proof. (1) can be verified directly. To prove (2), we identify U with an open subsetof C ˆ B via p η, π q . So η is identified with the standard coordinate z . We choose aneighborhood V Ă U of x of the form D ˆ T where T Ă B is open, and D is an opendisc centered at point p “ η p x q P C . By basic theory of ODE, the differential equation B z h ` Qh { “ . have solutions h , h P O p V q satisfying the initial conditions h p¨ , p q “ , B z h p¨ , p q “ and h p¨ , p q “ , B z h p¨ , p q “ . It is easy to check that f : “ h { h satisfies S z f “ Q , andis defined and satisfies B z f ‰ near t p u ˆ T .(3) follows from (8.2), which says S η g “ pB η f q S f g ` S η f . Definition 8.2.
An open cover p U α , η α q α P A of C , where each open set U α is equippedwith a function η α P O p U α q holomorphic on each fiber, is called a (relative) projectivechart of X , if for any α, β P A , we have S η β η α “ on U α X U β . Two projective charts arecalled equivalent if their union is a projective chart. An equivalence class of projectivecharts is called a (relative) projective structure . Equivalently, a projective structure isa maximal projective chart. L c “ L L ´ “ r L , L ´ s “ L ` c “ c . B is Stein, then X has a projective structure. See Section B. Remark 8.3.
Let P be a projective chart on X . Choose an open subset U Ă C and afiberwisely univalent η P O p U q . One can define an element S η P P O p U q as follows. Choose any p U , µ q P P . Then S η P “ S η µ on U X U . To check that S η P is well defined, suppose there is another p U , ζ q P P . Then S µ ζ “ on U X U . Thus S η µ “ S η ζ on U X U X U by Proposition 8.1-(3). H p C , Θ C { B p‚ S X qq In this section, we fix X “ p π : C Ñ B ; ς , . . . , ς N ; η , . . . , η N q to be a family of N -pointed compact Riemann surfaces with local coordinates. We assume for simplicitythat B is a Stein manifold with coordinates τ ‚ “ p τ , . . . , τ N q . Let W , . . . , W N be V -modules.By Lemma 2.2, there exists k P N such that for any k ą k and b P B , we have H p C b , ω C b p kS X qq “ . Thus R π ˚ ω C { B p kS X q “ (and also π ˚ ω C { B p kS X q is locally free)due to Grauert’s Theorem 4.1. Therefore, (5.11) implies an exact sequence Ñ π ˚ ω C { B p kS X q Ñ π ˚ ` V ir c b ω C { B p kS X q ˘ λ ÝÑ π ˚ Θ C { B p kS X q Ñ . By Cartan’s Theorem B, H p B , π ˚ ω C { B p kS X qq “ . So we have an exact sequence Ñ H ` B , π ˚ ω C { B p kS X q ˘ Ñ H ` B , π ˚ ` V ir c b ω C { B p kS X q ˘˘ λ ÝÑ H ` B , π ˚ Θ C { B p kS X q ˘ Ñ . Take the direct limit over all k ą k , we get an exact sequence Ñ H ` B , π ˚ ω C { B p‚ S X q ˘ Ñ H ` B , π ˚ ` V ir c b ω C { B p‚ S X q ˘˘ λ ÝÑ H ` B , π ˚ Θ C { B p‚ S X q ˘ Ñ . (9.1)According to Section 6, H ` B , π ˚ ` V ir c b ω C { B p‚ S X q ˘˘ acts on W X p W ‚ qp B q whichclearly descends to the trivial action on W X p W ‚ qp B q{ J p B q . We shall use the aboveexact sequence to define an action of H ` B , π ˚ Θ C { B p‚ S X q ˘ “ H p C , Θ C { B p‚ S X qq on W X p W ‚ qp B q{ J p B q , which turns out to be an O p B q -scalar multiplication. This actiondepends on the local coordinates η ‚ .Choose mutually disjoint neighborhoods U , . . . , U N of ς p B q , . . . , ς N p B q on which η , . . . , η N are defined respectively. Write each τ j ˝ π as τ j for short, so that p η i , τ ‚ q isa set of coordinates of U i . Set U “ U Y ¨ ¨ ¨ Y U N . Choose any θ P H p C , Θ C { B p‚ S X qq ,which, in each U i , is expressed as θ | U i “ a i p η i , τ ‚ qB η i . (9.2)Define ν p θ q P ` V ir c b ω C { B p‚ S X q ˘ p U q U ̺ p η i q ν p θ q| U i “ a i p η i , τ ‚ q c dη i . (9.3)The action of θ on T X p W ‚ q is defined to be the action of ν p θ q as in Section 6, namely, isdetermined by ν p θ q ¨ w ‚ “ N ÿ i “ w b ¨ ¨ ¨ b ν p θ q ¨ w i b ¨ ¨ ¨ b w N (9.4)for any w ‚ “ w b ¨ ¨ ¨ b w N P W ‚ , where ν p θ q ¨ w i is described by (6.5). Lemma 9.1.
Assume that p U , η q , . . . , p U N , η N q belong to a projective structure P . Then theaction of ν p θ q on W X p W ‚ qp B q{ J p B q is zero.Proof. By (8.5), the transition function for c b ω C { B between two projective coordinatesis the same as that for Θ C { B , namely, when S µ η “ , B µ changes to B µ η ¨ B η , and c dµ changes to B µ η ¨ c dη , sharing the same transition function B µ η . Thus, as θ is over C , ν p θ q can be extended to a global section of V ir c b ω C { B p‚ S X q on C . Thus, ν p θ q acts triviallysince ν p θ q ¨ W X p W ‚ qp B q Ă J p B q . Proposition 9.2.
Let P be a projective structure of X . Choose θ P H p C , Θ C { B p‚ S X qq whoselocal expression is given by (9.2) . Then the action of ν p θ q on W X p W ‚ qp B q{ J p B q (defined bythe local coordinates η ‚ ) is the O p B q -scalar multiplication by p θ q : “ c N ÿ i “ Res η i “ S η i P ¨ a i p η i , τ ‚ q dη i . (9.5)Note that each S η i P (defined in Remark 8.3) is an element of O p U i q . Proof.
It suffices to prove that the claim is locally true. Thus, we may shrinking B and U , . . . , U N so that for each ď i ď N , there exists a coordinate µ i P O p U i q at ς i p B q suchthat p U i , µ i q P P . Then θ | U i “ a i p η i , τ ‚ q ¨ pB µ i η i q ´ B µ i . Our strategy is to compare the action r ν p θ q of θ defined by the coordinates µ ‚ (whichis trivial by Lemma 9.1) with the one ν p θ q defined by η ‚ . We have U ̺ p µ i q r ν p θ q| U i “ a i p η i , τ ‚ q ¨ pB µ i η i q ´ c dµ i on each U i . Then U ̺ p µ i q r ν p θ q| U i “ a i p η i , τ ‚ q ¨ pB µ i η i q ´ c dη i . By Lemma 9.1, the action of r ν p θ q on W X p W ‚ qp B q{ J p B q is zero. Notice that the actionof r ν p θ q is independent of the choice of local coordinates. (See Theorem 3.2.) By (8.5),we have U ̺ p η i q r ν p θ q| U i “ U p ̺ p η i | µ i qq U ̺ p µ i q r ν p θ q| U i “ a i p η i , τ ‚ q c dη i ` c a i p η i , τ ‚ q ¨ pB µ i η i q ´ S µ i η i ¨ dη i
35y (9.3) and (8.3), we have U ̺ p η i q r ν p θ q| U i “ U ̺ p η i q ν p θ q| U i ´ c a i p η i , τ ‚ q ¨ S η i µ i ¨ dη i “ U ̺ p η i q ν p θ q| U i ´ c a i p η i , τ ‚ q ¨ S η i P ¨ dη i . Since the action of r ν p θ q is zero, the action of ν p θ q equals the sum over i of the actionsof c a i p η i , τ ‚ q ¨ S η i P ¨ dη i , which is exactly the scalar multiplication by (9.5).
10 Sewing conformal blocks
In this and the following sections, we let X “ p π : C Ñ B ; ς , . . . , ς N ; η , . . . , η N q bea family of N -pointed complex curves with local coordinates obtained by sewing thefollowing smooth family r X “ p r π : r C Ñ r B ; ς , . . . , ς N , ς , ς ; η , . . . , η N , ξ, ̟ q . (See Section 4.) Recall that the N -points ς , . . . , ς N and the local coordinates η , . . . , η N of X are constant with respect to sewing, and that each connected component of eachfiber r C b of r X contains at least one ς p b q , . . . , ς N p b q . Choose V -modules W , . . . , W N , M ,which together with the contragredient module M are associated to ς , . . . , ς N , ς , ς respectively. We assume M is semi-simple. Sewing conformal blocks
Note W ‚ b M b M is W b ¨ ¨ ¨ b W N b M b M . Note also p M b M q ˚ can be regardedas the algebraic completion of M b M . Define § b đ P p M b M q ˚ such that for any m P M , m P M , x § b đ , m b m y “ x m , m y . (10.1)Let A P End p M q whose transpose A t P End p M q exists, i.e., x Am, m y “ x m, A t m y (10.2)for any m P M , m P M . Then we have an element A § b đ ” § b A t đ P p M b M q ˚ (10.3)whose value at each m b m is (10.2).More explicitly, for each n P N we choose a basis t m p n, a qu a of the finite-dimensional vector space M p n q . Its dual basis t q m p n, a qu a is a basis of M p n q “ M p n q ˚ satisfying x m p n, a q , q m p n, b qy “ δ a,b . Then we have § b đ “ ÿ n P N ÿ a m p n, a q b q m p n, a q , A § b đ “ ÿ n P N ÿ a A ¨ m p n, a q b q m p n, a q“ § b A t đ “ ÿ n P N ÿ a m p n, a q b A t ¨ q m p n, a q . For each n P N , let P p n q be the projection of M onto M p n q . (Similarly, we let P p n q bethe projection of M onto M p n q for each n P C .) Its transpose, which is the projection of M onto M p n q , is also denoted by P p n q . Then we clearly have P p n q § b đ “ § b P p n q đ “ ÿ a m p n, a q b q m p n, a q P M b M . Recall r L t0 “ r L by Remark 1.6. Define q r L “ ÿ k P N P p n q q n P End p M qrr q ss . Then we have q r L § b đ “ § b q r L đ P p M b M qrr q ss . (10.4)For any ψ P T ˚ r X p W ‚ b M b M qp r B q , we define its (normalized) sewing r S ψ which isan O p B q -module homomorphism r S ψ : W X p W ‚ qp B q “ W ‚ b C O p B q Ñ O p r B qrr q ss , and the (standard) sewing S ψ : W X p W ‚ qp B q “ W ‚ b C O p B q Ñ O p r B qt q u , as follows. Regard ψ as an O p r B q -module homomorphism W ‚ b M b M b C O p r B q Ñ O p r B q . r S ψ is defined such that for any w P W ‚ b C O p B q , r S ψ p w q “ ψ ` w b q r L § b đ ˘ . (10.5) S ψ is defined similarly, except that the normalized energy operator r L is replaced bythe standard one L . When M and hence M are irreducible, r S φ differs from S φ by afactor q λ for some λ P C . Formal conformal blocks
Our goal is to show that r S ψ (and hence S ψ ) is a formal conformal block associatedto X and W ‚ , which means r S ψ vanishes on J p B q (defined by (6.6)). To prove this,we first need: Due to Theorem 6.3, it would be proper to use this definition only when r B and hence B are Stein. emma 10.1. Let R be a unital commutative C -algebra. For any u P V and f P R rr ξ, ̟ ss , thefollowing two elements of p M b M b R qrr q ss (where the tensor products are over C ) are equal: Res ξ “ Y M ` ξ L u, ξ ˘ q r L § b đ ¨ f p ξ, q { ξ q dξξ “ Res ̟ “ q r L § b Y M ` ̟ L U p γ q u, ̟ ˘ đ ¨ f p q { ̟, ̟ q d̟̟ . (10.6) Remark 10.2.
We explain the meaning of the left hand side; the other side can be un-derstood in a similar way. As q r L § b đ is an element of p M b M qrr q ss , Y M ` ξ L u, ξ ˘ q r L § b đ is an element of p M b M qpp ξ qqrr q ss , i.e. it is a formal power series of q whose coefficientsare in p M b M qpp ξ qq . (Note that one cannot switch the order of pp ξ qq and rr q ss .) Identify p M b M qpp ξ qqrr q ss » p M b M b qpp ξ qqrr q ss , which is a subspace of the R pp ξ qqrr q ss -module p M b M b R qpp ξ qqrr q ss . On the other hand, write f p ξ, ̟ q “ ř m,n P N f m,n ξ m ̟ n where each f m,n is in R . Then f p ξ, q { ξ q “ ÿ n ě ÿ k ě´ n f n ` k,n ξ k q n , which shows f p ξ, q { ξ q P R pp ξ qqrr q ss . Thus, the term in the residue on the left hand sideis an element in p M b M b R qpp ξ qqrr q ss dξ, whose residue is in p M b M b R qrr q ss . Proof of Lemma 10.1.
Consider Y M ` ξ L u, ξ ˘ q r L as an element of End p M qrr ξ ˘ , q ss . Since r L t0 “ r L , we have the following relations of elements of End p M qrr ξ ˘ , q ˘ ss : ` Y M ` ξ L u, ξ ˘ q r L ˘ t “ q r L ` Y M ` ξ L u, ξ ˘˘ t (1.11) ùùùù q r L Y M ` U p γ ξ q ξ L u, ξ ´ ˘ (1.9) ùùùù q r L Y M ` ξ ´ L U p γ q u, ξ ´ ˘ (3.7) ùùùù Y M ` p q { ξ q L U p γ q u, q { ξ ˘ q r L . Thus, by (10.3), we have the following equations of elements in p M b M q ˚ rr ξ ˘ , q ˘ ss : Y M ` ξ L u, ξ ˘ q r L § b đ “ § b ` Y M ` ξ L u, ξ ˘ q r L ˘ t đ “ § b Y M ` p q { ξ q L U p γ q u, q { ξ ˘ q r L đ “ q r L § b Y M ` p q { ξ q L U p γ q u, q { ξ ˘ đ . (10.7)Since for each n , P p n q § b đ is in M b M , (10.7) is actually an element in p M b M qrr ξ ˘ , q ˘ ss .Let A p ξ, q q “ Y M ` ξ L u, ξ ˘ q r L § b đ ,B p ̟, q q “ q r L § b Y M ` ̟ L U p γ q u, ̟ ˘ đ , considered as elements of p M b M qrr ξ ˘ , q ˘ ss and p M b M qrr ̟ ˘ , q ˘ ss respectively.Then (10.7) says A p ξ, q q “ B p q { ξ, q q . Let C p ξ, ̟ q P p M b M qrr ξ ˘ , ̟ ˘ ss be A p ξ, ξ̟ q ,which also equals B p ̟, ξ̟ q . Since A p ξ, q q contains only non-negative powers of q , so38oes A p ξ, ξ̟ q for ̟ . Similarly, since B p ̟, q q contains only non-negative powers of q ,so does B p ̟, ξ̟ q for ξ . Therefore C p ξ, ̟ q is an element in p M b M qrr ξ, ̟ ss , where thelatter can be identified with the subspace p M b M b qrr ξ, ̟ ss of the R rr ξ, ̟ ss -module p M b M b R qrr ξ, ̟ ss . Thus D p ξ, ̟ q : “ f p ξ, ̟ q C p ξ, ̟ q is well-defined as an element in p M b M b R qrr ξ, ̟ ss . It is easy to check that Res ξ “ ˆ D p ξ, q { ξ q dξξ ˙ “ Res ̟ “ ˆ D p q { ̟, ̟ q d̟̟ ˙ . (Indeed, they both equal ř n P N D n,n q n if we write D p ξ, ̟ q “ ř m,n P N D m,n ξ m ̟ n .) Thisproves (10.6).Recall B “ r B ˆ D rρ “ D rρ ˆ r B , and the order of Cartesian products will be switchedwhen necessary. Theorem 10.3.
Let ψ P T ˚ r X p W ‚ b M b M qp r B q . Then r S ψ vanishes on J p B q .Proof. Step 1. Note that we have divisors S X “ ř Ni “ ς i p B q and S r X “ ř Ni “ ς i p r B q ` ς p r B q ` ς p r B q of C and r C respectively. Choose any v in H ` C , V X b ω C { B p‚ S X q ˘ . In this first step,we would like to construct a formal power series expansion v “ ÿ n P N v n q n (10.8)where each v n is in H ` r C , V r X b ω r C { r B p‚ S r X q ˘ .First, choose any precompact open subset U of r C disjoint from the double points ς p r B q and ς p r B q . Then one can find small enough positive numbers ǫ ă r, λ ă ρ suchthat U ˆ D ǫλ is an open subset of r C ˆ D rρ ´ F ´ F in (4.9), and hence an open subsetof C . Moreover, by (4.14), the projection π : C Ñ B equals r π ˆ : r C ˆ D rρ Ñ r B ˆ D rρ when restricted to U ˆ D ǫλ . It follows that the section v | U ˆ D ǫλ of V X b ω C { B p‚ S X q canbe regarded as a section of V r X ˆ D rρ b ω r C ˆ D rρ { r B ˆ D rρ p‚ S X q , which, by taking power seriesexpansions at q “ , is in turn an element of ` V r X b ω r C { r B p‚ S r X q ˘ p U qrr q ss . The coefficientbefore q n defines v n | U . This defines the section v n of V r X b ω r C { r B p‚ S r X q on r C ´ ς p r B q ´ ς p r B q satisfying (10.8).We now show that v n has poles of orders at most n at ς p r B q and ς p r B q . Let W, W , W be as described near (4.8). By (5.7) and (4.22), v | W ´ Σ is a sum of those whose restrictionsto W , W under the trivializations U ̺ p ξ q , U ̺ p ̟ q are f p ξ, q { ξ, ¨q ξ L u ¨ dξξ resp. ´ f p q { ̟, ̟, ¨q ̟ L U p γ q u ¨ d̟̟ (10.9)where u P V and f “ f p ξ, ̟, ¨q P O p W q , and the coordinates of r B are suppressed as thedot. (Recall q “ ξ̟ .) In the above two terms, if we take power series expansions of q ,then it is obvious that the coefficients before q n have poles of orders at most n at ξ “ and ̟ “ respectively. This proves the claim.39tep 2. By (4.8), we can regard f p ξ, ̟, ¨q as an element of O p r B qrr ξ, ̟ ss . Thus, byLemma 10.1 (applied to R “ O p r B q ) and the fact that v | W ´ Σ is a (finite) sum of those ofthe form (10.9), we have the following equation of elements in p M b M b O p r B qqrr q ss : ÿ n P N ` v n ¨ q r L § b đ ` q r L § b v n ¨ đ ˘ q n “ (10.10)where the actions of v n on M and M are as in (6.5) using the local coordinates ξ, ̟ of r X .On the other hand, since ψ is conformal block, for each n and each w P W ‚ (consideredas a constant section of W ‚ b C O p B q ), the element A n P O p r B q defined by A n : “ ψ ` v n ¨ w b p q r L § b đ q ˘ ` ψ ` w b p v n ¨ q r L § b đ q ˘ ` ψ ` w b p q r L § b v n ¨ đ q ˘ equals . Here, similarly, the action of v n on w is defined by summing up the com-ponentwise actions described by (6.5) using the local coordinates η ‚ . By (10.10), wehave “ ÿ n P N A n q n “ ÿ n P N ψ ` v n ¨ w b p q r L § b đ q ˘ q n , which is exactly r S ψ p v ¨ w q . This finishes the proof that r S ψ vanishes on J p B q . Remark 10.4.
The algebraic version of Theorem 10.3 (i.e. assuming r X is an algebraicfamily and replacing D rρ with Spec p C rr q ssq ) was proved in [DGT19b, Thm. 8.3.1] andits proof can be easily adapted to the analytic setting. We have provided a completeproof of Theorem 10.3 for the reader’s convenience. We remark that [DGT19b] proveda version of Lemma 10.1. Their proof uses [NT05, Lemma 8.7.1] and is different fromours.In low genus cases, similar versions of Theorem 10.3 were proved in [Zhu96, Prop.4.3.6], [Hua05a, Thm. 1.4], [Hua05b, Prop. 3.6].
11 Convergence of sewing
We continue our discussions and assume the setting in Section 10. Recall that B equals D rρ ˆ r B . Then B is Stein if r B is so. We also assume, unless otherwise stated,that M and hence M are simple. So r S ψ and S ψ differ by a scalar multiplication of q λ .We assume V is C -cofinite, and the V -modules W , . . . , W N are finitely-generated. Weidentify W X p W ‚ q with W ‚ b C O B via U p η ‚ q . Absolute and locally uniform convergence
Definition 11.1.
We say that r S ψ converges absolutely and locally uniformly (a.l.u.) if it sends each element of W X p W ‚ qp B q to an element of O p B q . Theorem 11.2. If r S ψ converges a.l.u., then it is an element of T ˚ X p W ‚ qp B q . roof. We may shrink r B so that r B and hence B are Stein. Assume the a.l.u. conver-gence. Note that, since W X p W ‚ q is generated freely by some global sections, the O p B q -module homomorphism r S ψ : W X p W ‚ qp B q Ñ O p B q can be regarded as an O B -modulehomomorphism r S ψ : W X p W ‚ q Ñ O B . Now, the fact that r S ψ is a conformal blockfollows from Theorems 10.3 and 6.3.Since S ψ is possibly multivalued over q , we need to define its a.l.u. convergencein another way. For each w P W ‚ , considered as a constant section of W X p W ‚ qp B q , wehave the series expansion S ψ p w q “ ÿ n P C S ψ p w q n ¨ q n where S ψ p w q n “ ψ p w b P p n q § b đ q is a holomorphic function on r B . Definition 11.3.
We say that S ψ converges a.l.u. if for any w P W ‚ and any compactsubsets K Ă r B and Q Ă D ˆ rρ , there exists C ą such that ÿ n P C ˇˇ S ψ p w q n p b q ˇˇ ¨ | q n | ď C (11.1)for any b P K and q “ p q , . . . , q M q in Q .It is clear that r S ψ converges a.l.u. if and only if S ψ does. To see this, note that thea.l.u. convergence of r S ψ is clearly equivalent to r S ψ satisfying a similar condition asin Definition 11.3. Convergence and differential equations
As in the proof of Theorem 7.3, for each k P N , W ď k ‚ (resp. W ‚ p k q ) denotes the (finitedimensional) subspace spanned by all r L -homogeneous homogeneous vectors w P W ‚ satisfying Ă wt p w q ď k (resp. Ă wt p w q “ k ). This gives a filtration (resp. grading) of W ‚ .We define S ψ ď k P p W ď k ‚ q ˚ b C O p r B qt q u whose evaluation with each w P W ď k ‚ is S ψ p w q . Theorem 11.4.
Assume r B is a Stein manifold. There exists k P Z ` such that for any k ě k ,there exists Ω P End C ` p W ď k ‚ q ˚ ˘ b C O p B q not depending on M , such that q B q p S ψ ď k q “ Ω ¨ S ψ ď k . (11.2)Using this theorem, it is easy to prove:41 heorem 11.5. r S ψ and S ψ converge a.l.u.. In this theorem, we assume M is only semi-simple. Recall we have assumed that V is C -cofinite and W , . . . , W N are finitely-generated V -modules. Proof.
It suffices to assume B is a Stein open subset of C m and M is simple. One candefine r S ψ ď k P p W ď k ‚ q ˚ b C O p r B qrr q ss in a similar way. Then, as r S ψ ď k and S ψ ď k differby a scalar multiplication by some q λ , r S ψ ď k satisfies a similar differential equation asin (11.2). Thus, by Theorem A.1, r S ψ ď k is holomorphic on B “ D rρ ˆ r B . Since this istrue for any k ě k , we obtain the a.l.u. convergence of r S ψ and hence of S ψ .It follows that r S ψ is a conformal block associated to W ‚ and X . Outside the dis-criminant locus ∆ “ t u ˆ r B , S ψ is also a conformal block. Proof of Theorem 11.4
In this subsection, we assume r B and hence B are Stein manifolds. By TheoremB.2, we are allowed to fix a projective structure P of r X . Recall S X “ ř Ni “ ς i p B q ; set S r X “ ř Ni “ ς i p r B q ` ς p r B q ` ς p r B q . As argued for (9.1), we may use (4.20) to obtain anexact sequence Ñ H ` B , π ˚ Θ C { B p‚ S X q ˘ Ñ H ` B , π ˚ Θ C p´ log C ∆ ` ‚ S X q ˘ dπ ÝÑ H ` B , π ˚ ` π ˚ Θ B p´ log ∆ qp‚ S X q ˘˘ Ñ . (11.3) y “ q B q is a section of Θ B p´ log ∆ q and hence of π ˚ ` π ˚ Θ B p´ log ∆ qp‚ S X q ˘ over B . Thus,we have r y P H ` C , Θ C p´ log C ∆ ` ‚ S X q ˘ satisfying dπ p r y q “ q B q . We let Γ “ ς p r B q Y ς p r B q . Our first step is to take the series expansion ř r y K n q n (as in the proof of Theorem 10.3)of the “vertical part” of r y . Choose any precompact open subset U Ă r C ´ Γ togetherwith a fiberwisely univalent η P O p U q . Then as in that proof, we may find a smallsubdisc D “ D ǫλ of D rρ centered at such that D ˆ U » U ˆ D is an open subset of r C ˆ D rρ ´ F ´ F and hence of C . Extend η constantly (over D ) to a fiberwise univalentfunction on D ˆ U . Then we may write r y | D ˆ U “ h B η ` q B q (11.4)for some h P O p‚ S X qp D ˆ U q . Write h “ ř n P N h n q n where h n P O p‚ S r X qp U q . For each n P N , set an element r y K n P Θ r C { r B p‚ S r X qp U q by r y K n | U “ h n B η . (11.5)42 emma 11.6. The locally defined r y K n is independent of the choice of η , and hence can be ex-tended to an element of H p r C ´ Γ , Θ r C { r B p‚ S r X qq Proof.
Suppose we have another µ P O p U q univalent on each fiber, which is extendedconstantly to D ˆ U . So B q µ “ and hence r y | D ˆ U “ h ¨ B η µ ¨ B µ ` q B q . Note that B η µ isconstant over q . Thus, if we define r y K n | U using µ , then r y K n | U “ h n ¨ B η µ ¨ B µ , which agreeswith (11.5).We shall show that r y K n has poles of finite orders at Γ . For that purpose, we needto describe explicitly r y near the critical locus Σ . Recall the open subsets W, W , W of C described near (4.8) and U , U of r C described near (4.6). Note also q “ ξ̟ . In thefollowing, we let τ ‚ be any biholomorphic map from r B to an open subset of a complexmanifold. If r B is small enough, then τ ‚ can be a set of coordinates of r B . The onlypurpose of introducing τ ‚ is to indicate the dependence of certain functions on thepoints of r B . Thus, p ξ, q, τ ‚ q and p ̟, q, τ ‚ q are respectively biholomorphic maps of W “ D ˆ r ˆ D ρ ˆ r B , W “ D r ˆ D ˆ ρ ˆ r B to complex manifolds. By (4.18), we can find a, b P O pp ξ, ̟, τ ‚ qp W qq such that r y | W “ a p ξ, ̟, τ ‚ q ξ B ξ ` b p ξ, ̟, τ ‚ q ̟ B ̟ . Since dπ p ξ B ξ q “ dπ p ̟ B ̟ q “ q B q by (4.19), we must have a ` b “ . (11.6)This relation, together with (4.2), shows that under the coordinates p ξ, q, τ ‚ q and p ̟, q, τ ‚ q respectively, r y | W “ a p ξ, q { ξ, τ ‚ q ξ B ξ ` q B q , r y | W “ b p q { ̟, ̟, τ ‚ q ̟ B ̟ ` q B q . (11.7) Lemma 11.7.
For each n P N , r y K n has poles of orders at most n ´ at ς p r B q and ς p r B q .Consequently, r y K n is an element of H p r C , Θ r C { r B p‚ S r X qq .Proof. Let us write a p ξ, ̟, τ ‚ q “ ÿ m,n P N a m,n p τ ‚ q ξ m ̟ n , b p ξ, ̟, τ ‚ q “ ÿ m,n P N b m,n p τ ‚ q ξ m ̟ n where a m,n , b m,n P O p τ ‚ p r B qq . Then a p ξ, q { ξ, τ ‚ q “ ÿ n ě ,l ě´ n a l ` n,n p τ ‚ q ξ l q n , b p q { ̟, ̟, τ ‚ q “ ÿ m ě ,l ě´ m b m,l ` m p τ ‚ q ̟ l q m . (11.8)Combine these two relations with (11.5) and (11.7), and take the coefficients before q n .We obtain r y K n ˇˇˇ U ´ ς p r B q “ ÿ l ě´ n a l ` n,n p τ ‚ q ξ l ` B ξ , r y K n ˇˇˇ U ´ ς p r B q “ ÿ l ě´ n b n,l ` n p τ ‚ q ̟ l ` B ̟ , (11.9)which finishes the proof. 43he description (11.9) of r y K n near Γ can be found in [Loo10, Lemma 33]. Next, weshall apply the results of Section 9 to the smooth family r X . In particular, V ir c is definedfor r X and is an O r C -module. We let ν p r y K n q be a section of V ir c b ω r C { r B p‚ S r X q defined on U Y U (near ς p r B q , ς p r B q ) and near ς p r B q , . . . , ς N p r B q as in Section (9), which relies on thelocal coordinates η , . . . , η N , ξ, ̟ of r X . Recall the correspondence B ξ ÞÑ c dξ, B ̟ ÞÑ c d̟ .We calculate the actions of ν p r y K n q on M and on M to be respectively Res ξ “ ÿ l ě´ n a l ` n,n Y M p c , ξ q ξ l ` dξ, Res ̟ “ ÿ l ě´ n b n,l ` n Y M p c , ̟ q ̟ l ` d̟. (11.10)In the following proofs, we will suppress the symbol τ ‚ when necessary.The next lemma is crucial to finding the differential equation (11.2), and was ob-served in [DGT19b, Rem. 8.3.3]. Lemma 11.8.
The following equation of elements of p M b M qrr q ss is true. L q r L § b đ “ ÿ n P N ν p r y K n q q n ` r L § b đ ` ÿ n P N q n ` r L § b ν p r y K n q đ (11.11)As M is assumed to be simple, the equation still holds if r L is replaced by L . Proof.
It is obvious that U p γ q c “ c , ξ L c “ ξ c , ̟ L c “ ̟ c . Notice Remark 10.2. Wehave Y M p ξ L c , ξ q q r L § b đ ¨ a p ξ, q { ξ q dξξ “ ÿ n ě ÿ l ě´ n Y M p c , ξ q q n ` r L § b đ ¨ a l ` n,n ξ l ` dξ as elements of p M b M b O p r B qqpp ξ qqrr q ss dξ . Take Res ξ “ and notice (11.10). Then,the above expression becomes the first summand on the right hand side of (11.11). Asimilar thing could be said about the second summand. Thus, the right hand side of(11.11) equals Res ξ “ Y M p ξ L c , ξ q q r L § b đ ¨ a p ξ, q { ξ q dξξ ` Res ̟ “ q r L § b Y M p ̟ L U p γ q c , ̟ q đ ¨ b p q { ̟, ̟ q d̟̟ . By Lemma 10.1 and that a ` b “ , it equals Res ξ “ Y M p ξ L c , ξ q q r L § b đ ¨ dξξ “ Res ξ “ Y M p c , ξ q q r L § b đ ¨ ξdξ “ Y M p c q q r L § b đ “ L q r L § b đ . Lemma 11.9.
For any w ‚ P W ‚ , we have the following relation of elements of O p r B qt q u . q B q S ψ p w ‚ q “ ÿ n P N ψ p w ‚ b ν p r y K n q q n ` L § b đ q ` ÿ n P N ψ p w ‚ b q n ` L § b ν p r y K n q đ q . roof. We have q B q S ψ p w ‚ q “ q B q ψ p w ‚ b q L § b đ q “ ψ p w ‚ b L q L § b đ q . By the Lemma 11.8 (with r L replaced by L ), the desired equation is proved.As usual, we let ν p r y K n q w ‚ denote ř i w b ¨ ¨ ¨ b ν p r y K n q w i b ¨ ¨ ¨ b w N . For any w ‚ P W ‚ ,define ∇ q B q w ‚ P W ‚ b C O p r B qrr q ss to be ∇ q B q w ‚ “ ´ ÿ n P N q n ν p r y K n q w ‚ . (11.12) Proposition 11.10.
There exists p r y K n q P O p r B q for each n P N , such that for any w ‚ P W ‚ ,we have the following equation of elements of O p r B qt q u : q B q S ψ p w ‚ q “ S ψ p ∇ q B q w ‚ q ` ÿ n P N p r y K n q q n ¨ S ψ p w ‚ q . Proof. p r y K n q is defined by Proposition 9.2. Moreover, by that proposition, we have w ‚ b ν p r y K n q q L § b đ ` w ‚ b q L § b ν p r y K n q đ ` ν p r y K n q w ‚ b q L § b đ “ p r y K n q ¨ w ‚ b q L § b đ . By Lemma 11.9 and relation (11.12), it is easy to prove the desired equation.To prove Theorem 11.4, it remains to check that (11.12) and the projective term ř n p r y K n q q n converge a.l.u.. To treat the first one, we choose mutually disjoint neigh-borhoods U , . . . , U N of ς p r B q , . . . , ς N p r B q on which η , . . . , η N are defined respectively,and assume they are disjoint from U , U . Then D rρ ˆ U , . . . , D rρ ˆ U N are neighbor-hoods of ς p B q , . . . , ς N p B q disjoint from F , F . Write τ ‚ ˝ π and τ ‚ ˝ r π as τ ‚ for simplicity.Then p η i , τ ‚ q is a set of coordinates of U i . Recall (11.4). We may write r y | D rρ ˆ U i “ h i p q, η i , τ ‚ qB η i ` q B q (11.13)where h i p q, η i , τ ‚ q P O C p‚ S X qp D rρ ˆ U i q . Let ν p r y K q be a section of V X b ω C { B p‚ S X q on D rρ ˆ p U Y ¨ ¨ ¨ Y U N q satisfying U p η i q ν p r y K q| D rρ ˆ U i “ h i p q, η i , τ ‚ q c dη i . (11.14)Write h i “ ř n h i,n q n . Then by (11.5), r y K n | U i “ h i,n p η i , τ ‚ qB η i . (11.15)So we have ∇ q B q w ‚ P W ‚ b C O p D rρ ˆ r B q “ W ‚ b C O p B q due to the obvious fact: Lemma 11.11.
We have ∇ q B q w ‚ “ ´ ν p r y K q w ‚ . (11.16)We now prove the convergence of the projective term.45 roposition 11.12. ř n P N p r y K n q q n is an element of O p D rρ ˆ r B q “ O p B q .Proof. Combine (11.9) and (11.15), and apply Proposition 9.2 to the family r X . We obtain p r y K n q “ c ` A n ` B n ` N ÿ i “ C i,n ˘ where A n “ ÿ l ě´ n Res ξ “ S ξ P ¨ a l ` n,n p τ ‚ q ξ l ` dξ,B n “ ÿ l ě´ n Res ̟ “ S ̟ P ¨ b n,l ` n p τ ‚ q ̟ l ` d̟,C i,n “ Res η i “ S η i P ¨ h i,n p η i , τ ‚ q dη i . Notice that S η i P “ S η i P p η i , τ ‚ q , S ̟ P “ S ̟ P p ̟, τ ‚ q , S ξ P “ S ξ P p ξ, τ ‚ q areholomorphic functions on U i , U , U which are identified with their images under p η i , τ ‚ q , p ξ, τ ‚ q , p ̟, τ ‚ q respectively.We have ÿ n ě A n q n “ ÿ n ě ÿ l ě´ n Res ξ “ S ξ P ¨ a l ` n,n p τ ‚ q ξ l ` q n dξ. (11.17)We claim that (11.17) is an element of O p D rρ ˆ r B q . Note that a p ξ, q { ξ, τ ‚ q is definedwhen | q |{ ρ ă | ξ | ă r . Choose any ǫ P p , rρ q . Choose a circle γ surrounding D ǫ { ρ andinside D r . Then, when ξ is on γ , a p ξ, q { ξ, τ ‚ q can be defined whenever | q | ă ǫ . Thus, A : “ ¿ γ S ξ P p ξ, τ ‚ q ¨ a p ξ, q { ξ, τ ‚ q ξdξ is a holomorphic function defined whenever | q | ă ǫ . Recall the first equation of (11.8),and note that the series converges absolutely and uniformly when ξ P γ and | q | ď ǫ ,by the double Laurent series expansion of a p ξ, q { ξ, τ ‚ q . Therefore, the above contourintegral equals ÿ n ě ÿ l ě´ n ¿ γ S ξ P p ξ, τ ‚ q ¨ a l ` n,n p τ ‚ q ξ l ` q n dξ, which clearly equals (11.17) as an element of O p r B qrr q ss . Thus (11.17) is an element of O p D ǫ ˆ r B q whenever ǫ ă rρ , and hence when ǫ “ rρ .A similar argument shows ř B n q n converges a.l.u. to B : “ ¿ γ S ̟ P p ̟, τ ‚ q ¨ b p q { ̟, ̟, τ ‚ q ̟d̟ where γ is any circle in D ρ surrounding . Finally, we compute C i : “ ÿ n ě C i,n q n “ ÿ n ě Res η i “ S η i P p η i , τ ‚ q ¨ h i,n p η i , τ ‚ q q n dη i Res η i “ S η i P p η i , τ ‚ q ¨ h i p q, η i , τ ‚ q dη i which is clearly inside O p D rρ ˆ r B q . The proof is now complete. We summarize that theprojective term equals ÿ n P N p r y K n q q n “ c ` A ` B ` N ÿ i “ C i ˘ . (11.18)We can now finish the Proof of Theorem 11.4.
By Theorem 7.3, we may find k P N such that the vectors in W ď k ‚ , considered as constant sections of W ď k ‚ b C O p B q , generate the O p B q -module W X p W ‚ qp B q{ J p B q . Fix any k ě k . We choose a basis s , s , . . . of W ď k ‚ .By propositions 11.10 and 11.12, for each s i of s , s , . . . , we have the followingequation of elements of O p r B qt q u : q B q S ψ p s i q “ S ψ p ∇ q B q s i q ` g S ψ p s i q where g P O p D rρ ˆ r B q “ O p B q equals (11.18). Since we have ∇ q B q s i P W ‚ b C O p B q “ W X p W ‚ qp B q due to Lemma 11.11, we can find f i,j P O p B q such that ∇ q B q s i equals ř j f i,j s j mod elements of J p B q . Since, by Theorem 10.3, r S ψ and hence S ψ vanishon J p B q , we must have q B q S ψ p s i q “ ÿ j f i,j S ψ p s j q ` g S ψ p s i q . The proof is completed by setting the matrix-valued holomorphic function Ω to be p f i,j ` gδ i,j q i,j . Remark 11.13.
Following [TUY89] or [BK01, Chapter 7] or [DGT19a], one can definelocally a (logarithmic) connection ∇ on W X p W ‚ q as follows. Assume first of all that X admits local coordinates η , . . . , η N , and B is a small enough Stein manifold such that Θ B p´ log ∆ q is O B -generated freely by finitely many global sections y , y , . . . . Theneach y j lifts to an element r y j P H p C , Θ C p´ log C ∆ ` ‚ S X qq , i.e. dπ p r y j q “ y j . As in (11.14),one can define a section r y K j of Θ C { B p‚ S X q (near ς p B q , . . . , ς N p B q ) to be the vertical part of r y j , and define ν p r y K j q as in Section 9 using the local coordinates η ‚ . Then ∇ is determinedby ∇ y j w ‚ “ ´ ν p r y K j q w ‚ for any constant section w ‚ P W ‚ . As argued in [DGT19a], ∇ preserves J p B q . So ∇ descends to a connection of T X p W ‚ q . Its dual connection on T ˚ X p W ‚ q is also denotedby ∇ . Thus, Proposition 11.10 says that for the X obtained by sewing r X , and for any ψ P T ˚ r X p W ‚ b M b M qp r B q , its sewing S ψ P T ˚ X p W ‚ qp B q satisfies ∇ q B q S ψ “ g S ψ (11.19)where g P O p B q is given by (11.18). This observation is the analytic analog of [DGT19b,Rem. 8.3.3]. 47 We continue the study of sewing, but assume that X is formed by sewing a single p N ` q -pointed compact Riemann surface with local coordinates r X “ p r C ; x , . . . , x N , x , x ; η , . . . , η N , ξ, ̟ q . Namely, we assume r B is a single point. As usual, each connected component of r C contains one of x , . . . , x N . So X “ p π : C Ñ D rρ ; x , . . . , x N ; η , . . . , η N q where x , . . . , x N , η , . . . , η N are extended from those of r C and are constant over D rρ .Let E be a complete list of mutually inequivalent simple V -modules. “Complete”means that any simple V -module is equivalent to an object of E . Assume V is C -cofinite. Then by Theorems 11.2 and 11.5, for each q P D ˆ rρ , we can define a linearmap S q : à M P E T ˚ r X p W ‚ b M b M q Ñ T ˚ X q p W ‚ q , (12.1) à M ψ M ÞÑ ÿ M S ψ M p q q where X q “ p C q ; x , . . . , x N ; η , . . . , η N q . Similarly, one can define r S q by replacing S with r S . Notice that ř M r S ψ M p q q “ ř M q λ M S ψ M p q q for some constants λ M dependingonly on M . Thus S q is injective (resp. bijective) if and only if r S q is. Also, S q dependson the argument of q . Theorem 12.1.
Assume V is C -cofinite, and choose any q P D ˆ rρ . Then S q and r S q areinjective linear maps. If V is also rational, then S q and r S q are bijective.Proof. Let us fix q P D ˆ rρ and let q denote a complex variable. Let us prove that S q is injective. Suppose that the finite sum ř M S ψ M p q q equals . We shall prove bycontradiction that ψ M “ for any M P E .Suppose this is not true. Let F be the (finite) subset of all M P E satisfying ψ M ‰ .Then F is not an empty set. We first show that φ : “ ř M S ψ M (which is a multivaluedholomorphic function on D ˆ rρ ) satisfies φ p q q “ for each q P D ˆ rρ . Choose any largeenough k P N . Then, by Theorem 11.4, φ ď k satisfies a linear differential equationon D ˆ rρ of the form B q φ ď k “ q ´ Ω ¨ φ ď k . Moreover, it satisfies the initial condition φ ď k p q q “ . Thus, φ ď k is constantly . So is φ .Consider the V b V -module X : “ À M P F M b M . Define a linear map κ : X Ñ W ˚‚ as follows. If m b m P M b M , then the evaluation of κ p m b m q with any w ‚ P W ‚ is x κ p m b m q , w ‚ y “ ÿ M P F ψ M p w ‚ b m b m q . We claim that
Ker p κ q is a non-zero subspace of X invariant under the action of V b V .If this can be proved, then ker p κ q is a semi-simple V b V -submodule of X , which must48ontain M b M for some M P F . Therefore, ψ M p w ‚ b m b m q “ for any w ‚ P W ‚ and m b m P M b M . Namely, ψ M “ . So M R F , which gives a contradiction.For any n P C , recall P p n q is the projection of M onto its L -weight n subspace. Then φ p w ‚ qp q q “ ÿ M P F ÿ n P C ψ M p w ‚ b P p n q § b đ q q n . Since this multivalued function is always , by Lemma 12.2, any coefficient before q n is . Thus P p n q § b đ (which is an element of M p n q b M n q ) is in ker p κ q for any n . Thus ker p κ q must be non-empty.Suppose now that ř j m j b m j P Ker p κ q where each m j b m j belongs to some M b M .We set ψ r M p w ‚ b m j b m j q “ if M , r M P F and M ‰ r M . Choose any n P N and l P Z . Weshall show that ř j Y p u q l m j b m j P Ker p κ q for any u P V ď n . (Here Y denotes Y M for asuitable M .) Thus Ker p κ q is V b -invariant. A similar argument will show that Ker p κ q is b V -invariant, and hence V b V -invariant.Set divisors D “ x ` ¨ ¨ ¨ ` x N and D “ x ` x . Choose a natural number k ě l such that Y p u q k m j “ Y p u q k m j “ for any j , any k ě k , and any u P V ď n . This ispossible by the lower truncation property. By Serre’s vanishing Theorem, we can find k P N such that H p r C, V ď n r C b ω r C p k D ´ k D qq “ . Thus, the short exact sequence Ñ V ď n r C b ω r C p k D ´ k D q Ñ V ď n r C b ω r C p k D ´ lD q Ñ G Ñ (where G is the quotient of the previous two sheaves) induces another one Ñ H ` r C, V ď n r C b ω r C p k D ´ k D q ˘ Ñ H ` r C, V ď n r C b ω r C p k D ´ lD q ˘ Ñ H p r C, G q Ñ . Choose any u P V ď n . Choose v P H p r C, G q to be U ̺ p ξ q ´ uξ l dξ near x and in r C ´ t x u .Then v has a lift ν in H ` r C, V ď n r C b ω r C p k D ´ lD q ˘ , which must be of the form U ̺ p ξ q ν | U “ uξ l dξ ` ξ k p elements of V ď n b C O p U qq dξ, U ̺ p ̟ q ν | U “ ̟ k p elements of V ď n b C O p U qq d̟, where U Q x , U Q x are open subsets of r C (see (4.6)). It is clear that ν ¨ m j “ Y p u q l m j and ν ¨ m j “ . Thus, as each ψ M vanishes on ν ¨ p W ‚ b M b M q , we have ÿ M P F ÿ j ψ M p w ‚ b Y p u q l m j b m j q “ ´ ÿ M P F ÿ j ψ M pp ν ¨ w ‚ q b m j b m j q“ ´ ÿ j x κ p m j b m j q , ν ¨ w ‚ y “ . So ř j Y p u q l m j b m j P Ker p κ q .We have proved the injectivity when V is C -cofinite. If V is also rational, the surjec-tivity follows by comparing the dimensions of both sides and using the factorizationproperty proved in [DGT19b]. 49s a consequence of this theorem, we see that any C -cofinite V has finitely manyequivalence classes of simple modules. Indeed, we let r X “ p P ; 1 , , ; z ´ , z, z ´ q ,and let W “ V . Then T ˚ r X p V b M b M q is nontrivial since Y M defines a non-zero elementof it. Thus, as the range of S q is finite-dimensional, E is finite.In the proof of Theorem 12.1 we have used the following lemma, which is a gener-alization of [Hua95, Lemma 14.5]. Lemma 12.2.
Let E be a finite subset of C . Choose an element f p z q “ ÿ α P E ` N c α z α of C t z u . Let ǫ ą . Assume that f p z q converges absolutely to on D ˆ ǫ . Namely, for any z P D ˆ r , there is C ą such that ÿ α P E ` N | c α z α | ď C, (12.2) and the infinite sum ř α P E ` N c α z α converges to . Then c α “ for each α . Remark 12.3.
Note that f p z q can be written as z α f p z q`¨ ¨ ¨` z α n f n p z q where f , . . . , f n P C rr z ss , and any two of α , . . . , α n do not differ by an integer. It is easy to see that f p z q converges absolutely on D ˆ ǫ if and only if f , . . . , f n P O p D ǫ q . Indeed, the if part isobvious; the only if part follows from the root test. Moreover, it is clear that f , . . . , f n P O p D ǫ q implies f p z q converges a.l.u. on D ˆ ǫ , i.e., for compact subset K Ă D ˆ ǫ , there is C ą such that (12.2) holds for any z P K . Proof of Lemma 12.2.
Assume that the coefficients of f are not all . We can let r P R bethe smallest number such that c α ‰ for some α P E ` N satisfying Re p α q “ r . Let β “ r ` i s , . . . , β k “ r ` i s k be all the elements of E ` N whose real parts are r . (So s , . . . , s k P R .) Notice s i ‰ s j when i ‰ j . Then z ´ r f p z q “ c β z i s ` ¨ ¨ ¨ ` c β k z i s k ` g p z q where g p z q P C t z u can be written as g p z q “ z γ h p z q ` ¨ ¨ ¨ ` z γ m h m p z q for some h , . . . , h m P C rr z ss , the real parts of γ , . . . , γ m P C are all ą , and any twoof γ , . . . , γ m do not differ by an integer. Since z ´ r f p z q converges absolutely on D ˆ ǫ , wehave h , . . . , h m P O p D ǫ q by Remark 12.3. Let t be a real variable. Then it is easy to seeby induction that for any j P N , lim t Ñ´8 B jt g p e t q “ . Since the j -th derivative of e ´ rt f p e t q over t is constantly , we have lim t Ñ´8 p c β s j ´ e i s t ` ¨ ¨ ¨ ` c β k s j ´ k e i s k t q “ . A “ p s j ´ i q ď i,j ď k P M k ˆ k p C q . Then lim t Ñ´8 p c β e i s t , . . . , c β k e i s k t q ¨ A “ . Since A is a Vandermonde matrix which is invertible, we conclude that c β e i s t , . . . , c β k e i s k t all converge to , which forces c β , . . . , c β k to be . This gives a con-tradiction. Remark 12.4.
In [DGT19b, Thm. 8.3.1], Damiolini-Gibney-Tarasca proved a sewingtheorem for algebraic families of complex curves. Their sewing is algebraic and for-mal, in the sense that they consider infinitesimal disc
Spec p C rr q ssq rather than analyticopen disc D rρ . We now prove an analytic version of sewing theorem. Assume V is C -cofinite and W , . . . , W N are finitely-generated. Let r X be a family of N -pointed com-pact Riemann surfaces with local coordinates and base manifold r B , and let X be sewnfrom r X as in Section 4, whose base manifold is B “ D rρ ˆ r B . Let p : D rρ ˆ r B Ñ r B be the projection onto the second component. Recall by Remark 6.5 that T ˚ X p W ‚ q and T ˚ r X p W ‚ b M b M q (for each M P E ) are sheaves of conformal blocks, whichare O B -modules and O r B -modules respectively. Moreover, as explained in the remark, T ˚ r X p W ‚ b M b M q is a vector bundle (of finite rank). So the pullback p ˚ T ˚ r X p W ‚ b M b M q is also vector bundle over B . We can define a morphism of O B -modules r S : à M P E p ˚ T ˚ r X p W ‚ b M b M q Ñ T ˚ X p W ‚ q sending each p ˚ ψ (where ψ P À M P E T ˚ r X p W ‚ b M b M qp V q and V is an open subset of r B ) to r S q ψ P T ˚ X p W ‚ qp D rρ ˆ V q , considering q as the standard complex variable of D rρ .That this morphism is well-defined is due to the convergence of sewing (Theorem 11.5)and Theorem 11.2. The restriction of r S to each fiber over B ´ ∆ “ D ˆ rρ ˆ r B is injectivedue to Theorem 12.1. Namely, the support of Ker r S (which is an O B -submodule of avector bundle) is inside ∆ “ t u ˆ r B . So it must be empty. Therefore r S is injective. If V is also rational, then r S is an isomorphism by the factorization proved in [DGT19b].
13 Convergence of multiple sewing
In this section, we prove the convergence of sewing conformal blocks along severalpairs of points, which generalizes Theorem 11.5. Let
N, M P Z ` . Let r X “ p r π : r C Ñ r B ; ς , . . . , ς N ; ς , . . . , ς M ; ς , . . . , ς M ; η , . . . , η N ; ξ , . . . , ξ M ; ̟ , . . . , ̟ M q be a family of p N ` M q -pointed compact Riemann surfaces with local coordinates. Soeach η i , ξ j , ̟ j are local coordinates at ς i p r B q , ς j p r B q , ς j p r B q respectively. We assume thatfor every b P r B , each connected component of the fiber r C b “ r π ´ p b q contains one of ς p b q , . . . , ς N p b q .For each ď j ď M we choose r j , ρ j ą and a neighborhood U j (resp. U j ) of ς j p r B q (resp. ς j p r B q ) such that p ξ j , r π q : U j » ÝÑ D r j ˆ r B resp. p ̟ j , r π q : U j » ÝÑ D ρ j ˆ r B (13.1)51s a biholomorphic map. We also assume that these r i and ρ j are small enough suchthat the neighborhoods U , . . . , U M , U , . . . , U M are mutually disjoint and are also dis-joint from ς p r B q , . . . , ς N p r B q .Associate to ς , . . . , ς N finitely-generated V -modules W , . . . , W N . Associateto ς , . . . , ς M semi-simple V -modules M , . . . , M M , whose contragredient modules M , . . . , M M are associated to ς , . . . , ς M . We understand W ‚ b M ‚ b M as W b ¨ ¨ ¨ b W N b M b M b ¨ ¨ ¨ b M M b M M , where the order has be changed so that each M j is next to M j .For any ψ P T ˚ r X p W ‚ b M ‚ b M qp r B q and w P W ‚ , we define S ψ p w q “ ψ ´ w b p q L § b đ q b ¨ ¨ ¨ b p q L M § b M đ q ¯ . (13.2)Here, each § b j đ P p M j b M j q ˚ is defined as in (10.1). r S ψ p w q is defined similarly, exceptthat L is replaced by r L .Let D r ‚ ρ ‚ “ D r ρ ˆ ¨ ¨ ¨ ˆ D r M ρ M and D ˆ r ‚ ρ ‚ “ D ˆ r ρ ˆ ¨ ¨ ¨ ˆ D ˆ r M ρ M . Let q j be thestandard coordinate of D r j ρ j . We say that r S ψ converges a.l.u. if r S ψ p w q P O p r B ˆ D r ‚ ρ ‚ q for each w P W ‚ . This is equivalent to S ψ converging a.l.u., which means that for any w P W ‚ and any compact subsets K Ă r B and Q Ă D ˆ r ‚ ρ ‚ , there exists C ą such that ÿ n ‚ P C M ˇˇˇ ψ ´ w b p P p n q § b đ q b ¨ ¨ ¨ b p P p n M q § b M đ q ¯ p b q ˇˇˇ ¨ | q n ‚ ‚ | ď C for any b P K and q ‚ “ p q , . . . , q M q P Q . Here, n ‚ “ p n , . . . , n M q and q n ‚ ‚ “ q n ¨ ¨ ¨ q n M M . Theorem 13.1.
Let V be C -cofinite. Then S ψ and r S ψ converge a.l.u..Proof. This is the content of Theorem 11.5 when M “ . For general M this followsfrom induction. To simplify discussions, we explain the idea of the proof by assuming M “ . By shrinking r B , we assume r B has a set of coordinates τ ‚ . Since the case for M ´ is proved, we know that for any w P W ‚ , r S ψ p w qp q , q , τ ‚ q is an element of O p D r ρ ˆ r B qrr q ss , and the coefficient before each q n defines a conformal block associated to W ‚ b M b M and the family X over D r ρ ˆ r B formed by sewing r X along ς , ς (due toTheorem 11.2). We now restrict the base manifold of X to D ˆ r ρ ˆ r B . Then X becomesa smooth family. Apply Theorem 11.5 to conformal blocks associated to X , we seethat r S ψ p w qp q , q , τ ‚ q is an element of O p D ˆ r ρ ˆ D r ρ ˆ r B q . Since r S ψ p w qp q , q , τ ‚ q isalso an element of O p r B qrr q , q ss , it must be holomorphic on D r ρ ˆ D r ρ ˆ r B . Thisfinishes the proof. A Differential equations with simple poles and parame-ters
Let V be an open subset of C m with coordinates τ ‚ “ p τ , . . . , τ m q . Let A “ End p C N q b C O p D r ˆ V q . D r is the open disc at with radius r . We let q be the standard variable of D r .Consider the following differential equation with simple pole q B q ψ “ Aψ (A.1)where ψ “ ÿ n P N p ψ n p τ ‚ q q n P C N b C O p V qrr q ss (A.2)is a formal solution of this equation. The following result is well-known although theauthor wasn’t able to find a reference explicitly mentioning that. Thus, we provide aproof for completeness purpose. Theorem A.1.
Suppose that the formal series ψ satisfies (A.1) . Then ψ is an element of C N b C O p D r ˆ V q .Proof. It suffices to prove that ψ is an C N -valued holomorphic function on a neighbor-hood of t u ˆ V Ă D r ˆ V . Then, by basic theory of linear differential equations, thereis a (possibly multivalued) C N -valued holomorphic function on D r ˆ V whose “initialvalue” is given by ψ . (See [Kna] the remark after Thm. B.1.) This function must besingle-valued since ψ is so. So ψ is holomorphic on D r ˆ V .Consider the series expansion of A : A p q, τ ‚ q “ ÿ n P N p A n p τ ‚ q q n , where each p A n is in End p C N q b C O p V q . Then for each n P N , n p ψ n “ n ÿ j “ p A n ´ j p ψ j . Choose any open subset U of V with compact closure, and choose M P N such that k p A p τ ‚ q k ď M whenever τ ‚ P U . (Here k ¨ k is the operator norm.) Then for any n ą M , n ´ p A p τ ‚ q is invertible (with inverse n ´ ř j “ p p A p τ ‚ q{ n q j ). Thus, whenever n ą M , p ψ n “ p n ´ p A q ´ n ´ ÿ j “ p A n ´ j p ψ j . (A.3)Choose any r ă r and set α “ sup p q,τ ‚ qP D r ˆ U k A p q, τ ‚ q k . (A.4)Using the fact that p A n p τ ‚ q “ ű B D r A p q, τ ‚ q q ´ n ´ dq i π , we have k p A n p τ ‚ q k ď αr ´ n (A.5)for all n and all τ ‚ in U . 53hoose β ą such that k p n ´ p A p τ ‚ qq ´ k ď βn ´ for any n ą M and τ ‚ P U . (Such β can be found using the explicit formula of inverse matrix given above.) Set γ ě max t , αβ u . Then, from (A.3) and (A.5), we see that for any n ą M and τ ‚ P U , r n k p ψ n p τ ‚ q k ď γn ´ n ´ ÿ j “ r j k p ψ j p τ ‚ q k . (A.6)By induction, one can show that there exists c ą such that r n k p ψ n p τ ‚ q k ď cγ n for any n P N and τ ‚ P U . Indeed, if this is true for , , , . . . , n ´ where n ą N , thenby (A.6), r n k p ψ n p τ ‚ q k ď γn ´ n ´ ÿ j “ cγ j ď γn ´ n ´ ÿ j “ cγ n ´ “ cγ n . Thus k p ψ n p τ ‚ q k ď cγ n r ´ n for all n and τ ‚ P U . Therefore, if we choose any r P p , γ ´ r q ,then the series ř n k p ψ n p τ ‚ q k ¨| q | n is uniformly bounded by some positive number forall | q | ď r and τ ‚ P U . Since each p ψ n p τ ‚ q is holomorphic over τ ‚ , the series (A.2)must converge uniformly to a holomorphic function on D r ˆ U . This proves ψ isholomorphic on a neighborhood of t u ˆ U , and hence of t u ˆ V by choosing arbitrary U . B Existence of projective structures
Let X “ p π : C Ñ B q be a family of compact Riemann surfaces. In [Hub81, Lemma5] Hubbard showed that X has a projective structure when B is Stein and the connectedcomponents of the fibers have genera ą . In this section, we cover the low genuscases.Let U “ t U α u α P A be a Stein cover of C such that each member U α admits η α P O p U α q univalent on each fiber of U α . Define a ˇCech -cochain σ “ p σ α,β q α,β P A P C p U , ω b C { B q such that σ α,β “ S η α η β ¨ dη α P ω b C { B p U α X U β q . By (8.4), σ is a cocycle and hence can be viewed as an element of H p C , ω b C { B q . The proofof the following Lemma is in [Hub81, Lemma 5]. We recall the proof for the reader’sconvenience. Lemma B.1. X admits a projective structure if and only if σ is the zero element of H p C , ω b C { B q . Note that H p C , ω b C { B q equals H p U , ω b C { B q by Leray’s theorem. An open cover U of a complex manifold X is called Stein if each open set U P U is a Stein manifold.Then any finite intersection of open sets of U is also Stein (see [GR84, Sec. 1.4.4]). roof. “If”: We have a -cochain s “ p s α q α P A P C p U , ω b C { B q such that δs “ σ . By Propo-sition 8.1-(2) and by passing to a finer Stein cover (still denoted by U for simplicity),we can find f α P O p U α q univalent on each fiber, such that S η α f α ¨ dη α “ s α . (B.1)Thus, on U α X U β we have S η α η β ¨ dη β “ σ α,β “ s α ´ s β “ S η α f α ¨ dη α ´ S η β f β ¨ dη β . By (8.4), we have S η α η β ¨ dη β “ S η α f β ¨ dη α ´ S η β f β ¨ dη β . These two imply S η α f α “ S η α f β . Thus, by Proposition 8.1-(3), S f α f β “ . So p U α , f α q α P A is a projective chart.“Only if”: Assume p U α , f α q α P A is a projective chart. Define s “ p s α q α P A using (B.1).Notice S η α f α “ S η α f β . One can reverse the argument in the first paragraph to show δs “ σ . Theorem B.2.
Assume B is a Stein manifold. Then X admits a projective structure.Proof. Assume without loss of generality that the fibers are connected and have genus g . (Recall they are diffeomorphic by Ehresmann’s theorem.)Assume g “ . Each fiber C b is a complex torus and hence obviously has a projectivestructure (e.g. the one from the standard local coordinates of C ). Thus, by LemmaB.1, for each b P B the restriction σ | C b is the zero element of H p C b , ω b C b q . Since b ÞÑ dim H p C b , ω b C b q is constantly (because ω C b » O C b ), by Grauert’s Theorem 4.1, R π ˚ ω b C { B is locally free and each fiber is equivalent to H p C b , ω b C b q . So σ is the zero section of R π ˚ ω b C { B over B . Thus, we may find a Stein cover V “ p V i q i P I of B such that therestriction of σ to C V i “ π ´ p V i q is the zero element of H p C V i , ω b C { B q .Let W iα “ U α X C V i which is Stein (cf. [GR84, Sec. 1.4.4]). Then for each i P I , W i “p W iα q α P A is a Stein cover of C V i . So σ | C V i is the zero element of H p W i , ω b C { B q . Choose t i “ p t iα q α P A (where each t iα P ω b C { B p W iα q ) such that δt i “ σ | C V i . On U α X U β X C V i X C V j we have t iα ´ t iβ “ σ α,β “ t jα ´ t jβ and hence t iα ´ t jα “ t iβ ´ t jβ . Therefore, we have awell-defined element u “ p u i,j q i,j P I of H p V , π ˚ ω b C { B q such that u i,j , which is an elementof p π ˚ ω b C { B qp V i X V j q “ ω b C { B p C V i X C V j q , equals t iα ´ t jα when restricted to C V i X C V j X U α for any α . Since B is Stein, by Cartan’s Theorem B, H p V , π ˚ ω b C { B q is trivial. So thereexists v “ p v i q i P I (where each v i P ω b C { B p C V i q ) such that v i ´ v j “ u i,j on C V i X C V j . So v i ´ v j “ t iα ´ t jα on C V i X C V j X U α . So there is a well-defined s α P ω b C { B p U α q whichequals t iα ´ v i on U α X C V i for each i . Let s “ p s α q α P A . One checks easily δs “ σ . Thus,by Lemma B.1, X has a projective structure.Assume g ą . Then H p C b , ω b C b q is trivial (since deg ω C b “ g ´ ą ). A similarargument shows that σ is a coboundary. Assume g “ . Then b ÞÑ dim H p C b , ω b C b q is still constant since any fiber C b is biholomorphic to P . Moreover, C b clearly has aprojective structure. 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