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ON THE JUMPING PHENOMENON OF dimCHq(Xt,Et) 6 KWOKWAICHANANDYAT-HINSUEN 1 0 2 Abstract. Let X be a compact complex manifold and E be a holomorphic vector bundle on X. Given a deformation (X,E) of the pair (X,E) over a n smallpolydiskBcenteredattheorigin,westudythejumpingphenomenonof a J the cohomology groups dimCHq(Xt,Et)near t=0. We show that there are precisely twocohomological obstructions to the stabilityof dimCHq(Xt,Et), 5 which can be expressed explicitly interms of the Maurer-Cartan element as- 2 sociatedtothedeformation. ThisgeneralizestheresultsofX.Ye[7,8]. ] G D . Contents h t a 1. Introduction 1 m Acknowledgment 3 [ 2. Deformations of Pairs 3 3. An acyclic resolution for E 4 1 4. Obstructions 7 v 2 Appendix A. Convergence 13 7 References 17 4 6 0 . 1 1. Introduction 0 6 Let X be a compact complex manifold and π : X → B be a small deformation 1 of X = π−1(0) over a small polydisk B centered at the origin in some complex : v vector space. Suppose that F is a coherent sheaf on X which is flat over B. Then i X the sheaf F can be viewed as a deformation of the sheaf F|X on X. It is known r byGrauert’sdirectimagetheoremthatthedimensiondimCHq(Xt,Ft)isanupper a semi-continuousfunctionint. Moreover,wehavethe followingcharacterizationfor when the dimension dimCHq(Xt,Et) is locally constant, also due to Grauert. Theorem 1.1 (Grauert [2]). Let π : X → B be a flat proper holomorphic map between complex analytic space X,B with B being reduced and connected. Suppose that F is a coherent sheaf on X that is flat over B. Let k(t) := O /m be the B,t t residue field at t ∈ B and F be the pullback of F to X . Then the following are t t equivalent: (a) The function t7→dimCHq(Xt,Ft) is locally constant in t∈B. Date:January26,2016. 1 2 K.CHANANDY.-H.SUEN (b) The sheaf Rqπ F is locally free and the natural map ∗ Rqπ F ⊗k(t)→Hq(X ,F ) ∗ t t is an isomorphism. Nevertheless, condition (b) in the above theorem is not easy to check in general evenwhenF islocallyfree. In[7,8],X.Yestudiedthejumpingphenomenonofthe dimensions dimCHq(X,•) under small deformations of X, where • = ΩpX,TX. He found two obstructions oq , oq−1 and proved that the dimension of Hq(X,•) n,n−1 n,n−1 does not jump if and only if oq ≡0, oq−1 ≡0 for all n,m≥1. n,n−1 m,m−1 In this note, we generalize Ye’s results to a much more general setting, namely, when X is a compact complex manifold and E is an arbitrary holomorphic vector bundle on X. Let (X,E) be a small deformation of (X,E) over a polydisk B centeredatthe origin. We assumethat E is flatoverB viathe properholomorphic submersion π : X → B. Let E := E| . We are interested in characterizing when t Xt the dimension dimCHq(Xt,Et) stays constant near t=0. Following[7,8],weformulatethejumpingphenomenonofdimCHq(Xt,Et)asan extensionproblem, namely,whether wecanextend a nonzeroelementin Hq(X,E) to one in a nearby fiber Hq(X ,E ). In general, such extensions may not exist, and t t thereareobstructionstothisextensionproblem. In[7,8],Yegaveexplicitformulae fortheseobstructionsinthecaseswhenE =Ωp andE =T . Wewillseethat X/B X/B his formulae can be generalized to the above general setting. While Ye [7, 8] employed an algebraic approachto study the extension problem by applying a version of Grauert’s direct image theorem which states that Rqπ E ∗ is a quotient of two locally free sheaves of finite ranks over B, here we adapt a differential-geometric approach, following [4, 1]. We will formulate the problem directly as extending E-valued differential forms over B, which means that, in contrary to [7, 8], we are going to work with sheaves of infinite rank. A key step is to get an explicit description of Rqπ E by using an ∗ acyclicresolution(D•,D¯•)ofthesheafE constructedfromthedifferentialoperators D¯• studied in [4, 1] (see Section 3). The operators D¯• capture the holomorphic structures of the deformed pairs {(X ,E )} (see [1] and also Section 2). Then t t t∈B more or less the same strategy as in Ye’s proofs will work. An advantage of this geometric approach is that the computation of the obstructions becomes much simpler and more transparent, as compared to the Cˇech calculations in [7, 8]. Our main result is as follows (see Section 4, in particular, Theorem 4.11 and Equations (1) & (2) for the details): Theorem 1.2. Let {(A(t),ϕ(t))} be the family of Maurer-Cartan elements as- t∈B sociated to the small deformation (X,E) of (X,E). We define the n-th order obstruction maps Oi : Hi((π D•) ⊗O /mn) → Hi+1((π D•) ⊗O /mn), n,n−1 ∗ 0 B,0 0 ∗ 0 B,0 0 where i=q,q−1, by n−1 Oi ([α ])= tn−1 (ϕn−jy∇+An−j)αj . n,n−1 n−1  n−1 j=0 X   Thenthefunctiont7→dimCHq(Xt,Et)islocallyconstantifandonlyifOmq ,m−1 ≡0 and Oq−1 ≡0 for all m,n≥1. n,n−1 ON THE JUMPING PHENOMENON OF dimCHq(Xt,Et) 3 Acknowledgment WearegratefultoXuanmingYeandRobertLazarsfeldforvarioususefuldiscus- sionsviaemails. WewouldalsoliketothankConanLeungforencouragement. The workofthe firstnamedauthordescribedinthis paperwassubstantially supported by grants from the Research Grants Council of the Hong Kong Special Adminis- trative Region, China (Project No. CUHK404412 & CUHK400213). 2. Deformations of Pairs In this section, we briefly review the deformationtheory of a pair (X,E), where X is a compact complex manifold and E is a holomorphic vector bundle on X, following the exposition in [1] (cf. [4]), and recall severaluseful facts. Definition 2.1. LetB bea small polydisk in somecomplex vector space containing the origin. A deformation of (X,E) consists of a surjective proper submersion π : X → B between complex manifolds X and B, together with a holomorphic vector bundle E on X, such that π−1(0)=X and E|π−1(0) =E. Given such a deformation of (X,E), we put X :=π−1(t) and E :=E| . Since t t Xt wehaveassumedthatBisapolydisk,itiscontractibleandthuswehaveX ∼=X×B assmoothmanifoldsandE ∼=E×B assmoothvectorbundles. Letϕ(t)∈Ω0,1(T ) X be the family of Maurer-Cartanelements which corresponds to the family X →B. In[1], we considereda holomorphicfamily of differentialoperatorsD¯q :Ω0,q(E)→ t Ω0,q+1(E) defined locally by D¯ α ⊗e (t) := (∂¯+ϕ(t)y∂)α ⊗e (t), t j j j j   j j X X   where {e (t)} are local holomorphic frames of E . j t By choosing a Hermitian metric on E and the Chern connection ∇, one can express D¯ as t D¯q =∂¯ +ϕ(t)y∇+A(t), t E for some A(t) ∈ Ω0,1(End(E)). Then (A(t),ϕ(t)) ∈ Ω0,1(A(E)) is the family of Maurer-Cartanelements which corresponds to the deformation (X,E), namely, we have 1 ∂¯ (A(t),ϕ(t))+ [(A(t),ϕ(t)),(A(t),ϕ(t))] =0 A(E) 2 fort∈B;hereA(E)istheAtiyahextensionofE. Thisfamilyofoperatorssatisfies the integrability condition D¯qD¯q−1 =0 (which is equivalent to the Maurer-Cartan t t equation). Another important feature ofthe operatorD¯ , whichwill be useful later,is that t its cohomology computes the Dolbeault cohomology of (X ,E ). t t Proposition 2.2 ([1], Proposition 3.13). For each fixed t∈B, we have Hq(X ,E )∼=Hq((π D•) ⊗k(t))∼=Hq Ω0,•(E),D¯ , t t ∗ t t for any q ≥0. (cid:0) (cid:1) 4 K.CHANANDY.-H.SUEN 3. An acyclic resolution for E From now on, for the purpose of simplifying computations and formulae, we will assume that the base B of the deformation is complex 1-dimensional. In this section, we will construct an acyclic resolution of the sheaf E in order to get an explicit description of the direct image sheaf Rqπ E. ∗ Let π : X → X be the projection (not necessary holomorphic) of X onto X. X Consider the sheaf of O -modules Dq over X defined by X Dq :U 7→{α∈Γ (U,π∗ Ω0,p⊗E):∂¯ α=0}, smooth X X B whose pushforward by π carries a natural O -module structure. Since D¯ varies B t holomorphically in the variable t, it induces a sheaf map D¯q :Dq →Dq+1 for each q ≥ 0. We further define Dq,• to be the sheaf of smooth sections of π∗Ω0,• ⊗ B B π∗ Ω0,p ⊗ E. Clearly, D• ⊂ D•,0 as O -submodules. Let ∂¯ be the Dolbeault X X X B operator on the base B. Theen ∂¯ gives a complex (Dq,•,∂¯•) for each q. B B e Lemma3.1. Foreachp,q ≥0thesheafDq,p isfineandthecomplex(π Dq,•,π ∂¯•) e ∗ ∗ B has no higher cohomology sheaves, that is, e e Hp(π Dq,•)=0 ∗ for all p≥1. e Proof. Finenessisclear,forwecanapplyapartitionofunitytoconcludethatDq,p has no higher direct images. Toprovethat(π Dq,•,π ∂¯•)hasnohighercohomology,werecallthatHp(π Deq,•) ∗ ∗ B ∗ is the sheafification of e W 7→Hp(Γ(π−1(W),Dq,•)). e It suffices to prove that Hp(Γ(π−1(W),Dq,•)) = 0 for any polydisk W ⊂ B and e all p ≥ 1. Let α ∈ Γ(π−1(W),Dq,p) and {U } be a locally finite open covering of i X ⊂ π−1(W) by coordinates charts. Leet α be the restriction of α on U ×W. i i Write e α = α (z,z¯,t,t¯)dt¯J ⊗dz¯I =: α ⊗dz¯I. i IJ,i I,i I,J I X X Then ∂¯ α=0 simply means for each I, B 0=∂¯ α dt¯J =∂¯ α . B IJ,i B I,i ! J X Hence, for fixed z, we can apply the Dolbeault lemma on W to conclude that α =∂¯ β , I,i B I,i for some β ∈Ω0,p−1(W). Since α varies smoothly in z and z¯, we see from the I,i B I,i proof of the Dolbeault-Grothendieck lemma that β can be chosen to be smooth I,i in z as well. Let {ψ } be a partition of unity on X subordinate to the covering i {U }. Define i β := ψ β ⊗dz¯I. i I,i I,i X ON THE JUMPING PHENOMENON OF dimCHq(Xt,Et) 5 Then β ∈Γ(π−1(W),Dq,p−1) and ∂¯ β = ψ (e∂¯ β )⊗dz¯I = ψ α ⊗dz¯I = ψ α=α. B i B I,j i I,i i ! I,i I,i i X X X We have the first equality simply because {ψ } are all independent of t and t¯, i and the second last equality follows from the fact that α is a global section on π−1(W). (cid:3) Lemma3.2. Foreachq ≥0,thesheafDq isacyclicwithrespectivetotheleft-exact functor π . ∗ Proof. Lemma3.1showsthat(Dq,•,∂¯•)isafineresolutionofDq andsoRqπ Dq ∼= B ∗ Hp(π Dq,•)=0 for all q ≥1. (cid:3) ∗ e Proposition 3.3. The complex of sheaves (D•,D¯•) is an acyclic resolution of E e with respective to the left-exact functor π . In particular, we have ∗ Rqπ E ∼=Hq(π D•) ∗ ∗ as O -modules. B Proof. ByLemma3.2, Rpπ Dq =0forallp≥1. Itremainstoprovethatitdefines ∗ a resolution of E. We need to show that for any point (p,t) ∈ X ∼= X ×B, the sequence of stalks 0→E →D0 →D1 →··· (p,t) (p,t) (p,t) is exact. The exactness of 0→E →D0 →D1 (p,t) (p,t) (p,t) follows from the fact that D¯0 and ∂¯ share the same kernel. For the remaining E exactness, we will focus on the case t=0; same argumentworks for general t∈B. Recall that D¯ is locally defined by t D¯ α ⊗e (t) := (∂¯+ϕ(t)y∂)α ⊗e (t), t j j j j   j j X X   so it suffices to prove the exactness for the case E =O . X We wouldliketo firstworkoverC[[t]] insteadofC{t}. LetU ⊂X be a polydisk and denote Ω0,•(U){t}:=Ω0,•(U)⊗CC{t} and Ω0,•(U)[[t]]:=Ω0,•(U)⊗CC[[t]]= Ω0,•(U){t}⊗C{t}C[[t]]. The Maurer-Cartan element ϕ(t) is gauge equivalent to 0 on U. Hence ∂¯+ϕ(t)y∂ =ev(t)∂¯e−v(t), for some v(t)∈Ω0(T )[[t]] and ev(t) acts on Ω0,q(U)[[t]] by U ∞ (v(t)y∂)n ev(t)α(t)= α(t). n! n=0 X Hence we can apply the Dolbeault-Grothendieck lemma with analytic parameter (the t-variable) to conclude that (Ω0,•(U)[[t]],D¯•) is an exact complex. t Now, as C[[t]] is a flat-C{t} module (because C[[t]] is torsion free and C{t} is a PID), we have Hq(Ω0,•(U)[[t]])=Hq(Ω0,•(U){t}⊗C[[t]])∼=Hq(Ω0,•(U){t})⊗C[[t]]. 6 K.CHANANDY.-H.SUEN ButwehaveshownthatHq(Ω0,•(U)[[t]])=0. Therefore,Hq(Ω0,•(U){t})⊗C[[t]]= 0. If we can show that Hq(Ω0,•(U){t}) is torsion free, we see that Hq(Ω0,•(U){t}) vanishes. Assumingthis,weconcludethateveryD¯ -closed(0,q)-formvaluedpower t series on U is locally exact. Now, for any D¯ -closed element α ∈ Dq , we can represent it by a D¯ - (p,0) (p,0) t closed element α(t) ∈ Ω0,q(U){t}, for some polydisk U ⊂ X. The vanishing of Hq(Ω0,•(U){t}) showsthat α(t)=D¯ β(t) for some β(t)∈Ω0,q−1(U){t}. This β(t) t defines an element β ∈ Dq−1 such that D¯q−1β = α. This proves the exactness of (p,0) the complex (D•,D¯•). To completethe proofofthe proposition,weneedto provethatHq(Ω0,•(U){t}) is a torsion free C{t}-module for q > 1. That is, if [α(t)] ∈ Hq(Ω0,•(U){t}) is a nonzeroelement, thenf(t)·[α(t)] is nonzeroforall f(t)∈C{t}−{0}. Since f(t) is invertibleiff(0)6=0,wemayassumef(t)∈(tN)forsomeN ≥1. We mayassume N is chosen such that f(t) = tNg(t) with g(0) 6= 0. Again, we can invert g(t), so we can further assume f(t)=tN. Then the vanishing of f(t)·[α(t)]=[f(t)·α(t)] means tNα(t)=D¯ β(t)=(∂¯+ϕ(t)y∂)β(t), t for some β(t) ∈ Ω0,q−1(U){t}. Since both α(t) and β(t) is holomorphic in t, the equation shows that β(t) is in fact D¯ -closed up to order N −1. t We first prove the following Lemma3.4. Forany∂¯-closedβ ∈Ω0,q−1(U),q >1,thereexistsβ(t)∈Ω0,q−1(U){t} such that β(0)=β and D¯ β(t)=0. t Proof of Lemma 3.4. Since β is ∂¯-closed on the polydisk U, it must be ∂¯-exact. Write β =∂¯α for some α∈Ω0,q−2(U). Define β(t):=β+ϕ(t)y∂α∈Ω0,q−1(U){t}. Then β(0)=β. Since D¯2 =0, we have D¯ β(t)=0. (cid:3) t t With this lemma in hand, we see that α(t) is D¯ -exact and this proves that t Hq(Ω0,•(U){t}) is torsion free. Since β is ∂¯-closed, we can choose β (t)∈Ω0,q−1(U){t} such that 0 1 β (0)=β and D¯ β (t)=0. 1 0 t 1 Then we have β(t)−β (t) tN−1α(t)=D¯ 1 =D¯ γ (t). t t 1 t (cid:18) (cid:19) If N =1, we aredone. Otherwise,we see that γ (0) is ∂¯-closed. Hence we canfind 1 β (t) such that 2 β (0)=γ (0) and D¯ β (t)=0. 2 1 t 2 Hence γ (t)−β (t) tN−2α(t)=D¯ 1 2 . t t (cid:18) (cid:19) Repeating this process, we will arrive at the conclusion that α(t)=D¯ γ (t), t N for some γ (t)∈Ω0,q−1(U){t}. This completes the proof of the proposition. (cid:3) N ON THE JUMPING PHENOMENON OF dimCHq(Xt,Et) 7 4. Obstructions In this section, we will find out explicitly the obstruction maps for extending a given element of Hq(X,E). In [7, 8], the author used the Grauert direct image theoremto obtaina complex of locally free O -modules of finite ranks to compute B theobstructionmaps,whileweusetheinfinite-dimensionalcomplexofO -modules B (π D•,D¯•). We are going to see that more or less the same strategy of proofs in ∗ [7, 8] will work in our infinite-dimensional setting as well. We give most details of the proofs in order to be more self-contained. Recall that Proposition 3.3 gives an isomorphism of O -modules: B Rqπ E ∼=Hq(π D•). ∗ ∗ Together with Proposition 2.2, we see that it is equivalent to work with the sheaf Hq(π D•) and the cohomology group Hq((π D•) ⊗ k(0)). Tensoring the stalk ∗ ∗ 0 (π D•) with O /mn+1 over O , we obtain a complex ∗ 0 B,0 0 B,0 ((π D•) ⊗O /mn+1,D¯•). ∗ 0 B,0 0 n Givenα∈ker(∂¯q),andsupposethatwehavealocalextensionα ∈Γ(U,π Dq) E n−1 ∗ of α such that jn−1(D¯qα )(t)=0, 0 n−1 wedefinetheobstructionmapOq :Hq((π D•) ⊗O /mn)→Hq+1((π D•) ⊗ n,n−1 ∗ 0 B,0 0 ∗ 0 O /mn) by B,0 0 (1) Oq [jn−1(α )(t)]:=[tn−1·(jn(D¯qα )(t)/tn)] n,n−1 0 n−1 0 n−1 Remark4.1. The(n−1)-stjetcanbeviewedasanelementin(π Dq) ⊗O /mn. ∗ 0 B,0 0 The map Oq factors through a map n,n−1 Oq :Hq((π D•) ⊗O /mn)→Hq+1((π D•) ⊗O /m ), n ∗ 0 B,0 0 ∗ 0 B,0 0 given by Oq[jn−1(α )(t)]:=[jq(D¯qα )(t)/tn]. n 0 n−1 0 n−1 This is well-defined because the cohomology class of jq(dqα )(t)/tn only depends 0 n−1 on the cohomology class of the (n−1)-st jet jn−1(α )(t). 0 n−1 For later use, we define Oq [jn−1(α )(t)]:=[ti·(jq(D¯qα )(t)/tn)], n,i 0 n−1 0 n−1 for i≥0 and n≥1 The following proposition characterizes when an extension exists up to order n≥1. Proposition 4.2. For a fixed n≥1, the following are equivalent: (1) For any local section α around t = 0 such that jn−1(D¯qα )(t) = 0, n−1 0 n−1 there exists a local section α around t = 0 such that j0(α −α ) = 0 n 0 n n−1 and jn(D¯qα )(t)=0. 0 n (2) For any c ∈ Hq((π Dq) ⊗O /mn), there exists c ∈ Hq((π Dq) ⊗ n−1 ∗ 0 B,0 0 n ∗ 0 O /mn+1) such that c | =c | ∈Hq((π Dq) ⊗k(0)). B,0 0 n t=0 n−1 t=0 ∗ 0 (3) For any local section α around t = 0 such that jn−1(D¯qα )(t) = 0, n−1 0 n−1 Oq [jn−1(α )(t)]=0. n,n−1 0 n−1 8 K.CHANANDY.-H.SUEN Proof. We shall prove that (1)⇔(2) and (1)⇔(3). For (1) ⇒ (2) : Let c ∈ Hq((π D•) ⊗O /mn) and α be a local sec- n−1 ∗ 0 B,0 0 n−1 tion around t = 0 such that jn−1(α )(t) ∈ ker(D¯q ) represents the class c . 0 n−1 n−1 n−1 Then jn−1(D¯qα )(t) = 0. By assumption, we can extend α to a local sec- 0 n−1 n−1 tion α around t = 0 such that j0(α − α )(t) = 0 and jn(D¯qα )(t) = 0. n 0 n n−1 0 n Then D¯q(jn(α )(t)) = 0 ∈ (π Dq+1) ⊗ O /mn+1. Set c := [jn(α )(t)] ∈ n 0 n ∗ 0 B,0 0 n 0 n Hq((π D•) ⊗O /mn+1). Sincej0(α −α )=0,wehavec | =[j0(α )(t)]= ∗ 0 B,0 0 0 n n−1 n t=0 0 n−1 c | =0. n−1 t=0 For (2) ⇒ (1) : Let α be such that jn−1(D¯qα )(t) = 0. Extend c := n−1 0 n−1 n−1 [jn−1(α )(t)] to a classc ∈Hq((π D•) ⊗O /mn+1). Let α be localsection 0 n−1 n ∗ 0 B,0 0 n around t = 0 such that jn(α )(t) represents the class c . Then jn(D¯qα )(t) = 0. 0 n n 0 n Since c | =c | , we have n t=0 n−1 t=0 j0(α −α )=D¯q−1γ, 0 n n−1 0 for some γ ∈(π Dq−1) ⊗k(0). Choose any representative γ′ of γ and define ∗ 0 α′ :=α −D¯q−1γ′. n n Then jn(D¯qα′ )(t)=jn(D¯qα )(t)=0 and j0(α′ −α )=0. 0 n 0 n 0 n n−1 For (1)⇒(3): Let γ :=α −α . Then n−1 n D¯qγ =jn(D¯q(α −α ))(t)=tn·(jn(D¯qα )(t)/tn), n 0 n−1 n 0 n−1 since jn−1(D¯qα )(t)=jn(dqα )(t)=0. By assumption, j0(γ)(t)=0, so γ =tβ 0 n−1 0 n 0 for some local section β around t=0. Hence D¯q jn−1(β)(t)=tn−1·(jn(D¯qα )(t)/tn), n−1 0 0 n−1 which means that Oq [jn−1(α )(t)]=0. n,n−1 0 n−1 For (3) ⇒ (1) : The vanishing of Oq [jn−1(α )(t)] gives an element β ∈ n,n−1 0 n−1 (π Dq) ⊗O /mn such that ∗ 0 B,0 0 tn−1·(jn(D¯qα )(t)/tn)=D¯qβ. 0 n−1 n Let β′ be a local section around t = 0 representing the germ β and set α := n α −tβ′. Then n−1 jn(D¯qα )(t)=jn(D¯qα )(t)−t·jn−1(D¯qβ′)(t)=tn·(jn(D¯qα )(t)/tn)−t·D¯qβ =0. 0 n 0 n−1 0 0 n−1 n Hence α defines an n-th order extension of α. (cid:3) n Therefore, if Oq ≡ 0 for all n ≥ 1, then by (1) above we obtain a formal n,n−1 elementα(t)suchthatD¯ α(t)=0. InAppendix A,wewillshowthatafteragauge t fixing, α(t) is analytic in a neighborhood around 0∈B. Remark 4.3. The radius of convergence of each extension α(t) may be different as α=α(0) varies. However, since Hq(X,E) is finite dimensional, we can simply choose a basis, for instance, consisting of harmonic forms with respective to a cer- tain hermitian metric. Then we obtain a minimum radius of convergence, uniform in all [α]∈Hq(X,E). Next we shall demonstrate that there is another obstruction for an extension to be nonzero. ON THE JUMPING PHENOMENON OF dimCHq(Xt,Et) 9 Proposition4.4. Anon-exactelementβ ∈ker(∂¯q)admitsalocalextensionβ(t)∈ E Γ(U,π Dq) such that β(t) is exact for t 6= 0 if and only if there exist n ≥ 1 and ∗ [jn−1(α )(t)]∈Hq−1((π D•) ⊗O /mn) such that 0 n−1 ∗ 0 B,0 0 Oq−1[jn−1(α )(t)]=[β]. n 0 n−1 Proof. Suppose that Oq−1[jn−1(α )(t)]=[β]. Then n 0 n−1 β =jn(D¯q−1α )(t)/tn+∂¯q−1γ 0 n−1 E for some γ ∈Ω0,q−1(E). Define β(t) by β(t):=D¯q−1(α (t)/tn)+D¯q−1γ(t), t6=0, n−1 where γ(t) is any extension of γ. Clearly β(t) can be extended through the origin by setting β(0)=β. Then β(t) is a D¯q−1-exact class and equals β at t=0. Hence β(t) serves as an extension of β which is D¯q−1-exact for t6=0. Conversely, if β(t) is an extension of β such that β(t)=D¯q−1γ(t) for t 6= 0. Then γ(t) can be chosen to be meromorphic in t with pole order n ≥ 1 at t=0. Let α (t):=tnγ(t). Then α (t) is holomorphic in t and n−1 n−1 Oq−1[jn−1(α )(t)]=[jn(D¯q−1(tnγ(t))/tn]=[jn(tnβ(t))/tn]=[β]. n 0 n−1 0 0 This completes the proof. (cid:3) Proposition 4.5. Let [jn−1(α )(t)] ∈ Hq−1((π D•) ⊗ O /mn) such that 0 n−1 ∗ 0 B,0 0 Onq−1[j0n−1(αn−1)(t)]6=0. Thenthereexistn′ ≤nand[j0n′−1(αn′−1)(t)]∈Hq−1((π∗D•)0⊗ O /mn′) such that B,0 0 Onq−,n1′−1[j0n−1(αn−1)(t)]=Onq−′,n1′−1[j0n′−1(αn′−1)(t)]6=0. Proof. IfOnq−,n1−1[j0n−1(αn−1)(t)]6=0,wecansimplytaken′ =nandαn′−1 =αn−1. Otherwise, there exists α′ such that 1 D¯q−1α′ =Oq−1 [jn−1(α )(t)]. n−1 1 n,n−1 0 n−1 Then we have Oq−1 [α′]=Oq−1 [jn−1(α )(t)]. n−1,n−2 1 n,n−2 0 n−1 Since Oq−1[jn−1(α )(t)]6=0, we finally arrive at some n′ such that n 0 n−1 Onq−,n1′−1[j0n−1(αn−1)(t)]=Onq−′,n1′−1[j0n′−1(αn′−1)(t)]6=0. (cid:3) These two propositions together prove the following Corollary 4.6. Every local extension of every non-exact element β ∈ ker(∂¯q) is E non-exact if and only if Oq−1 ≡0 for all n≥1. n,n−1 Proof. For a fixed non-exact β ∈ ker(∂¯q), if any extension of β is non-exact, then E [β]∈/ Im(Oq−1 ) for all n≥1. Hence Oq−1 ≡0. n,n−1 n,n−1 Conversely,ifthereisanextensionofβ suchthatitisexactfort6=0,thenthere exist n≥1 and [jn−1(α )(t)]∈Hq−1((π D•) ⊗O /mn) such that 0 n−1 ∗ 0 B,0 0 Oq−1[jn−1(α )(t)]=[β]6=0. n 0 n−1 10 K.CHANANDY.-H.SUEN But we canalso choosen′ ≤n and[j0n′−1(αn′−1)(t)]∈Hq−1((π∗D•)0⊗OB,0/mn0′) such that Onq−,n1′−1[j0n−1(αn−1)(t)]=Onq−′,n1′−1[j0n′−1(αn′−1)(t)]6=0. This proves the corollary. (cid:3) Lemma 4.7. For each q ≥0, π Dq is a flat O -module. ∗ B Proof. This follows from the fact that (π Dq) is torsionfree and O ∼=C{x−t} ∗ t B,t is a PID for every t∈B. (cid:3) Wewillneedthefollowingfactinhomologicalalgebra,whoseproofcanbefound, e.g. in [3]. Proposition 4.8. Let A be a Noetherian ring and C• be a finite cochain complex of flat A-modules whose cohomology Hi(C•) is finitely generated for all i. Then thereexistsacochain complex offinitelygeneratedflatA-modules K• andacochain map C• → K•, which is a quasi-isomorphism. Moreover, for any A-module M, the natural map C•⊗M → K•⊗M is a quasi-isomorphism. Furthermore, if the dimension dim Hq(K•⊗k(p)) k(p) is locally constant in p ∈ Spec(A), then for i = q,q−1, the δ-functors Ti(M) := Hi(K•⊗M) commute with base change. We apply this proposition to the case A = O , C• = (π D•) to prove the B,0 ∗ 0 following Proposition 4.9. If dim Hq((π D•) ⊗k(t)) is locally constant around 0∈B, k(t) ∗ t then the canonical map Hq((π D•) )⊗k(0)→Hq((π D•) ⊗k(0)) ∗ 0 ∗ 0 is an isomorphism. Proof. Since (π D•) is a flat O -module, using Proposition 4.8, we obtain a ∗ 0 B,0 complex of finitely generated flat O -modules K• such that B,0 H•((π D•) ⊗M)∼=H•(K•⊗M) ∗ 0 for any O -module M. We claim that the dimension B,0 dim Hq(K•⊗k(p)) k(p) is locally constant in p∈Spec(O ). B,0 First ofall, since dim Hq((π D•) ⊗k(t)) is locally constant,by Theorem1.1 k(t) ∗ t and Proposition 3.3, the sheaf Hq(π D•) ∼= Rqπ E is a locally free O -module. ∗ ∗ B Hence Hq((π D•) )⊗k(0)∼=(Rqπ E) ⊗k(0)∼=Hq(X,E)∼=Hq((π D•) ⊗k(0)). ∗ 0 ∗ 0 ∗ 0 In particular dim Hq(K•⊗k(0))=dim Hq((π D•) ⊗k(0)) k(0) k(0) ∗ 0 =dim Hq((π D•) )⊗k(0) k(0) ∗ 0 =dim Hq(K•)⊗k(0). k(0)