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1 Extension of 2-forms and symplectic varieties 0 0 2 Yoshinori Namikawa n a J 7 Introduction 1 In this paper we shall prove two theorems (Stability Theorem,Local Torelli ] G Theorem) for symplectic varieties. Let us recall the notion of a symplectic singularity. Let X be a good rep- A resentative of a normal singularity. Then the singularity is symplectic if the . h regular locus U of X admits an everywhere non-degenerate holomorphic closed at 2-formω whereω extendstoaregularformonY foraresolutionofsingularities m Y → X. Similarly we say that a normal compact Kaehler space Z is a symplectic variety ifthe regularlocus V ofZ admits anon-degenerateholomorphic [ closed 2-form ω where ω extends to a regular form on Z˜, where Z˜ → Z is a 3 resolution of singularities of Z. When Z has a resolution π :Z˜ →Z such that v (Z˜,π∗ω) is a symplectic manifold, we call Z has a symplectic resolution. 4 1 Examples 1 (i) This is one of examples of symplectic singularities studied in [Be 1]. For 0 detailssee[Be1]andthereferencesthere. LetQ⊂Pn−1 beageneralquadratic 1 0 hypersurface. IdentifyapointoftheGrassmannianGr(2,n)withalineinPn−1. 0 LetGr (2,n)be the subvarietyofGr(2,n) correspondingto the lines ofPn−1 iso / contained in Q. It is checked that dimGr (2,n)=dimGr(2,n)−3=2n−7. h iso t Embed Griso(2,n) into P1/2n(n−1)−1 by the Plu¨cker embedding Gr(2,n) → a P1/2n(n−1)−1. Nowconsider the cone X overGr (2,n). Then the germ(X,0) m iso at the vertexis a symplectic singularityof dimension 2n−6. The X is actually v: obtained as the closure O¯min of the minimal nilpotent orbit Omin of the Lie i algebra Lie(SO(n)), and O has the Kostant-Kirillov symplectic 2-form. X min (ii)LetA:=C2l/ΓbeanAbelianvarietyofdimension2l. Let(z ,z ,...,z ,z ) r 1 2 2l−1 2l a be the standard coordinates of C2l. Then Z/2Z acts on A by z → −z i i (i=1,...,2l). The quotient Z of A by the action becomes a symplectic variety ofdimension2l. Asymplectic2-formis,forexample,givenbyΣ dz ∧dz . 1≤i≤l i l+i The Z has singularities, and Z has no symplectic resolution when l>1. (iii) These are symplectic varieties studied by O’Grady [O]. Let S be a polarized K3 surface. Let c be an even number with c ≥ 4. Denote by M 0,c the moduli space of rank 2 semi-stable torsion free sheaves with c = 0 and 1 c = c. M becomes a projective symplectic variety of dim = 4c−6. The 2 0,c singularlocus Σ has dimension2c. Moreover,O’Grady showedthat M has a 0,4 1 symplecticresolution,howeverM hasQ-factorialterminalsingularitieswhen 0,c c≥6 (cf. section 3 of the e-print version of [O]: alg-geom/9708009). Therefore M have no symplectic resolution when c≥6. 0,c A symplectic singularity / variety will play an important role in the gener- alized Bogomolov decomposition conjecture (cf. [Kata], [Mo]): Conjecture: Let Y be a smooth projective variety over C with Kodaira di- mension0. Then thereisafiniteetalecoverY′ →Y suchthatY′ isbirationally equivalent to Y ×Y ×Y , where Y is an Abelian variety, Y is a symplectic 1 2 3 1 2 variety, and Y is a Calabi-Yau variety. 3 In this conjecture we hope that it is possible to replace Y and Y by their 2 3 birationalmodelswithonlyQ-factorialterminalsingularitiesrespectively. Main results are these. Theorem 7(Stability Theorem): Let (Z,ω) be a projective symplectic variety. Let g :Z →∆ bea projective flat morphism from Z toa 1-dimensional unit disc ∆ with g−1(0)=Z. Then ω extends sideways in theflat family sothat it gives a symplectic 2-form ω on each fiber Z for t ∈ ∆ with a sufficiently t t ǫ small ǫ. In the above, the result should also hold for a (non-projective) symplectic variety (Z,ω) and for a proper flat morphism g. But two ingredients remained unproved in the general case (cf. Remark below Theorem 7). Let Z be a symplectic variety. Put Σ := Sing(Z) and U := Z \ Σ. Let π :Z →S betheKuranishifamilyofZ,whichis,bydefinition,asemi-universal flat deformation of Z with π−1(0) = Z for the reference point 0 ∈ S. When codim(Σ⊂Z)≥4,S issmoothby[Na1,Theorem2.4]. Z isnotprojectiveover S. But we canshow that everymember of the Kuranishifamily is a symplectic variety (cf. Theorem 7’). Define U to be the locus in Z where π is a smooth map and let π :U →S be the restriction of π to U. Then we have Theorem 8(Local Torelli Theorem): Assume that Z is a Q-factorial projective symplectic variety. Assume h1(Z,O ) = 0, h0(U,Ω2) = 1, dimZ = Z U 2l≥4 and Codim(Σ⊂Z)≥4. Then the following hold. (1) R2π (π−1O ) is a free O module of finite rank. Let H be the image of ∗ S S the composite R2π C→R2π C→R2π (π−1O ). Then H is a local system on ∗ ∗ ∗ S S with H =H2(U ,C) for s∈S. s s (2) The restriction map H2(Z,C) → H2(U,C) is an isomorphism. Take a resolution ν : Z˜ → Z in such a way that ν−1(U) ∼= U. For α ∈ H2(U,C) we take a lift α˜ ∈ H2(Z˜,C) by the composite H2(U,C) ∼= H2(Z,C) → H2(Z˜,C). Choose ω ∈ H0(U,Ω2) = C. This ω extends to a holomorphic 2-form on Z˜. U Normalize ω in such a way that (ωω)l =1. Then one can define a quadratic Z˜ form q :H2(U,C)→C as R 2 q(α):=l/2 (ωω)l−1α˜2+(1−l)( ωlωl−1α˜)( ωl−1ωlα˜). Z Z Z Z˜ Z˜ Z˜ This form is independent of the choice of ν :Z˜ →Z. (3)Put H :=H2(U,C). Then there exists a trivialization of the local system H: H∼=H ×S. Let D :={x∈P(H);q(x)=0,q(x+x)>0}. Then one has a period map p:S →D and p is a local isomorphism. Note that when Z is a symplectic variety with terminal singularities, the condition codim(Σ⊂Z)≥4 is always satisfied ([Na 2]). Thestabilitytheoremwillbeprovedbyusingthe followingtheoremandthe factthataprojectivevarietywithrationalsingularitieshasDuBoissingularities (cf. [Ko]): Theorem 4.. Let X be a Stein open subset of a complex algebraic variety. Assume that X has only rational Gorenstein singularities. Let Σ be the singular locus of X and let f : Y → X be a resolution of singularities such that f|Y\f−1(Σ) : Y \f−1(Σ) ∼= X \Σ. Then f∗Ω2Y ∼= i∗Ω2U where U := X \Σ and i:U →X is a natural injection. The same result was obtained by D. van Straten and Steenbrink [S-S] for an arbitrary isolated normal singularity with dim ≥ 4, and later Flenner [Fl] proveditforarbitrarynormalsingularitywithCodim(Σ⊂X)≥4. ByTheorem 4 X is a symplectic singularity if and only if X is rational Gorenstein and the regular part of X admits an everywhere non-degenerate 2-form (cf. Theorem 6). Thisfactisoftenusefultodeterminethatcertainkindsofsingularitiesgiven by G.I.T. quotient are symplectic (Example 6’). The local Torelli theorem has been proved for non-singular symplectic varieties by Beauville [Be 2, Theoreme 5]. In our singular case, it is based on the Hodge decomposition H2(U,C) = H0(U,Ω2)⊕H1(U,Ω1)⊕H2(U,O ). We U U U need the conditionthat codim(Σ⊂Z)≥4 to have this decomposition(cf. [Oh, Na 1, Lemma 2.5]). We shall prove that R2π C is a constant sheaf around ∗ 0 ∈ S by using the Q-factoriality of Z. When Z is not Q-factorial, the state- ment for H in (1) does not hold as it stands because H2(Z,C) → H2(U,C) is not surjective. We have formulated the local Torelli theorem for a projective symplectic variety, but it is possible to get similar statements for a general (non- projective) symplectic variety. For a non-projective symplectic variety, Q- factoriality should be replaced by a certain condition which is equivalent to the Q-factoriality in the projective case (cf. Remark (2) on the final page). ThefollowingproblemwouldbeofinterestinviewofGlobalTorelliProblem. Problem. LetZ beaQ-factorial projective symplecticvariety withterminal singularities. Assume that Z is smoothable by a suitable flat deformation. Is Z then non-singular? 3 InthefirstsectionweshallproveTheorem4. Othertwotheoremsareproved in the second section. Notation. Let F be a coherent sheaf on a normal crossing variety D. Assume that F is a locally free sheaf on the regular locus of D. Then Fˆ = F/(torsion) by definition. Here (torsion) means the subsheaf of the sections whose support are contained in the singular locus of D. §1. Extension properties for Rational Gorenstein Singularities Proposition 1. Let X be a Stein open subset of a complex algebraic variety. Assume that X has only rational Gorenstein singularities. Let Σ be the singular locus of X and let f :Y →X be a resolution of singularities such that f|Y\f−1(Σ) :Y \f−1(Σ)∼=X\Σ and D :=f−1(Σ) is a simple normal crossing divisor. Then f Ω2(logD) ∼= i Ω2 where U := X \Σ and i : U → X is a ∗ Y ∗ U natural injection. Proof. Let ω ∈ H0(U,Ω2). ω is a meromorphic 2-form on Y. In fact, U Coker[f Ω2 → i Ω2] is a torsion sheaf whose support is contained in Σ. Hence ∗ ∗ U φω is an element of Γ(X,f Ω2) for a suitable holomorphic function φ on X. ∗ Y Since(X,p)∼=(R.D.P)×(Cn−2,0)forallp∈X outsidecertaincodimension 3 (in X) locus Σ ⊂ Σ ([Re]), it is clear that f Ω2(logD) ∼= i Ω2 at such 0 ∗ Y ∗ U points p. Let F be an irreducible component of D with f(F) ⊂ Σ . Put 0 k := dimΣ −dimf(F). We shall prove that ω has at worst log pole along F 0 by the induction on k. (a) k =0: (a-1): Put l := codim(Σ ⊂ X). Note that l ≥ 3. Take a general l dimen- 0 sional complete intersection H := H ∩...∩H . Let p ∈ H ∩f(F). Since 1 n−l H is general, p ∈ f(F) is a smooth point. Replace X by a suitable small open neighborhoodofp. ThenH∩f(F)={p}. H hasauniquedissidentpointpand othersingularitiesarelocallyisomorphicto(R.D.P.)×(Cl−2,0). Byperturbing H we can define a flat holomorphic map g : X →∆n−l such that the fiber X 0 over 0 ∈ ∆n−l coincides with H. We may assume that g has a section passing through p and each fiber g−1(t) intersects f(F) only in this section. The map f :Y →X givesasimultaneousresolutionofX (t∈∆n−l). Since H is general t and X is sufficiently small, D := D∩Y are normal crossing divisors of Y for t t t all t ∈ ∆n−l. Let D′ be the union of irreducible components of D which are mapped in this section. D′ → ∆n−l is a proper map. We put π = g◦f. We often write ∆ for ∆n−l. (a-2): We shall prove the following. Claim. By replacing ∆n−l by a smaller disc and by restricting everything (e.g. X, Y, D, D′ ...) over the new disc, we have a subset K ⊂ Y which contains D′ and which is proper over ∆n−l with the following property: The ω is mapped to zero by the comoposition of the maps 4 H0(U,Ω2)∼=H0(Y\D′,Ω2 (logD))→H0(Y\K,Ω2 (logD))→H1(Y,Ω2(logD)). U Y\D′ Y\K K Y Ifthe claimis verified,thenω| extends to alogarithmic2-formonY. It Y\K is clear that its restriction to U is ω. Proof of Claim. We shall first prove that R1πΩ2(logD) = 0. There is a ! Y filtration π∗Ω2 ⊂G ⊂Ω2(logD) which yields exact sequences ∆ Y 0→G →Ω2(logD)→Ω2 (logD)→0 Y Y/∆ 0→π∗Ω2 →G →π∗Ω1 ⊗Ω1 (logD)→0. ∆ ∆ Y/∆ BytheexactsequencesitsufficestoprovethatR1πO =R1πΩ1 (logD)= ! Y ! Y/∆ R1πΩ2 (logD)=0. ! Y/∆ We shall use the relative duality theorem due to Ramis and Ruget [R- R] to prove these facts. Before applying the relative duality we note that Riπ Ωl−2 (logD)(−D) = Riπ Ωl−1 (logD)(−D) = Riπ Ωl (logD)(−D) = ∗ Y/∆ ∗ Y/∆ ∗ Y/∆ 0fori≥l−1. Toprovethese,weonlyhavetocheckthatRiπ Ωl−2(logD )(−D )= ∗ Yt t t Riπ Ωl−1(logD )(−D ) = Riπ ω = 0 for i ≥ l −1 and for t ∈ ∆, for ex- ∗ Yt t t ∗ Yt ample, by using [B-S, VI, Cor. 4.5, (i)]). Since l ≥ 3, these follow from a vanishing theorem in [St] except for the vanishing of R2π Ω1 (logD )(−D ). ∗ Yt t t But, by the same argument as the proof of [Na-St, Theorem (1.1)] we see that R2π Ω1 (logD )(−D )=0. ∗ Yt t t Now the relative duality says that RπRHom(Y;Ωl−j (logD)(−D)⊗π∗Ωn−l,ω· ) ! Y/∆ ∆ Y ∼=RHomtop(∆,Rπ (Ωl−j (logD)(−D)⊗π∗Ωn−l),ω· ) ∗ Y/∆ ∆ ∆ for j =0,1,2. WehaveR1−nπRHom(Y;Ωl−j (logD)(−D)⊗π∗Ωn−l,ω· )∼=R1πΩj (logD). ! Y/∆ ∆ Y ! Y/∆ ThereforewehavetoprovethatExtop1−n(∆,Rπ (Ωl−j (logD)(−D)⊗π∗Ωn−l),ω· )= ∗ Y/∆ ∆ ∆ 0. ChooseaboundedcomplexofFNfreeO modulesL·representingRπ Ωn−j(logD)(−D) ∆ ∗ Y/∆ (cf. [R-R]). Since Riπ Ωn−j(logD)(−D)=0 for i≥l−1, we have Hi(L·)=0 ∗ Y/∆ fori≥l−1. LetQ:=Ker[Ll−2 →Ll−1]. ThenRHomtop(∆,Rπ (Ωl−j (logD)(−D)⊗ ∗ Y/∆ π∗Ωn−l),ω· )isrepresentedbythecomplexHomtop(...→Ll−3 →Q→0...,Ωn−l[n− ∆ ∆ ∆ l]). HenceweknowthatExtop1−n(∆,Rπ (Ωl−j (logD)(−D)⊗π∗Ωn−l),ω· )= ∗ Y/∆ ∆ ∆ 0. 5 Wearenowinapositiontojustifytheclaim. Letj :Y \D′ →Y. Theω de- finesanelementof(π j j∗Ω2(logD)) . Byacoboundarymap(π j j∗Ω2(logD)) → ∗ ∗ Y 0 ∗ ∗ Y 0 (R1Γπ,D′Ω2Y(logD))0,itdefinestheobstructionclassob(ω)∈(R1Γπ,D′Ω2Y(logD))0. Bytheobservationabove,ob(ω)issenttozerobythenaturalmap(R1Γπ,D′Ω2Y(logD))0 → (R1πΩ2(logD)) . Therefore there is a small disc ∆ ⊂ ∆ and a subset ! Y 0 ǫ K ⊂ π−1(∆ǫ)(= Yǫ) which contains (π|D′)−1(∆ǫ)(= Dǫ′) and which is proper over∆ ,suchthatob(ω)isalreadyzeroinH1(Y ,Ω2 (logD′)). Thisisnothing ǫ K ǫ Yǫ ǫ but our claim. (b) k: general (b-1): Takeagenerall+k dimensionalcompleteintersectionH :=H ∩...∩ 1 H . Let p∈H∩f(F). p∈f(F) is a smooth point. Replace X by a small n−k−l openneighborhoodofp. ThenH∩f(F)={p}. ByperturbingH,wecandefine a flat holomorphic map g : X → ∆n−k−l with g−1(0) = H. We may assume thatg hasasectionpassingthroughpandeachfiberg−1(t)intersectsf(F)only in this section. The map f :Y →X gives a simultaneous resolution of X (t∈ t ∆n−l−k). SinceH isgeneralandX issufficientlysmall,D :=D∩Y arenormal t t crossing divisors of Y for all t ∈ ∆n−l−k. Let D′ be the union of irreducible t components of D which are mapped in the section. Then D′ → ∆n−k−l is a proper map. We put π =g◦f. We often write ∆ for ∆n−k−l. By an induction hypothesis we have an isomorphism H0(Y \D′,Ω2 (logD))∼=H0(U,Ω2). Y\D′ U (b-2): We shall prove the following Claim. By replacing ∆n−l by a smaller disc and by restricting everything (e.g. X, Y, D, D′ ...) over the new disc, we have a subset K ⊂ Y which contains D′ and which is proper over ∆n−l with the following property: The ω is mapped to zero by the comoposition of the maps H0(U,Ω2)∼=H0(Y\D′,Ω2 (logD))→H0(Y\K,Ω2 (logD))→H1(Y,Ω2(logD)). U Y\D′ Y\K K Y Ifthe claimis verified,thenω| extends to alogarithmic2-formonY. It Y\K is clear that its restriction to U is ω. The proof of the claim is similar to the claim in (a-2). When we ap- ply the relative duality we need the vanishings: Riπ Ωl+k−2(logD )(−D ) = ∗ Yt t t Riπ Ωl+k−1(logD )(−D ) = Riπ ω = 0 for i ≥ l +k − 1 and for t ∈ ∆. ∗ Yt t t ∗ Yt Q.E.D. Lemma2. Letp∈X beaSteinopen neighborhood ofapoint pofacomplex algebraic variety. Assume that X is a rational singularity of dimX ≥ 3. Let f : Y → X be a resolution of singularities of X such that E := f−1(p) is a simple normal crossing divisor. Then H0(Y,Ωi ) → H0(Y,Ωi (logE)) are Y Y isomorphisms for i=1,2. Proof. By the assumptionwe cantakea complete algebraicvarietyZ which 6 containsX asanopenset. Wemayassumethatf isobtainedfromaresolution Z˜ →Z. Set V :=Z˜\E. Recall that the natural exact sequence →Hj(Z˜,C)→Hj(V,C)→Hj+1(Z˜,C)→ E is obtained from the following exact sequence of the complexes by taking hy- percohomology 0→Ω· →Ω· (logE)→Ω· (logE)/Ω· →0. Z˜ Z˜ Z˜ Z˜ Introduce the stupid filtrations F· (cf. [De]) on three complexes and take Hj(Gri ) of the sequence of complexes. Then we have F →Hj−i(Ωi )→Hj−i(Ωi (logE))→Hj−i(Ωi (logE)/Ωi )→. Z˜ Z˜ Z˜ Z˜ We know that this exact sequence coincides with the exact sequence →Gri (Hj(Z˜,C)→Gri (Hj(V,C)→Gri (Hj+1(Z˜,C))→ F F F E whichcomesfromthemixedHodgestructures. Inparticular,themapHj−i(Ωi (logE)/Ωi )→ Z˜ Z˜ Hj−i+1(Ωi ) is interpreted as the map Gri (Hj+1(Z˜,C))→Gri (Hj+1(Z˜,C)). Z˜ F E F WenextconsiderthenaturalmapofmixedHodgestructures: Hj+1(Z˜,C)→ Hj+1(E,C). We have Gri (Hj+1(E,C)) ∼= Hj−i+1(Ωî ) (cf. [Fr]), (see Intro- F E duction, for the notation Ωî ). Note that Ωî is isomorphic to the cokernel of E E the injection Ωi (logE)(−E)→Ωi . Therefore the composed map Y Y Hj−i(Ωi (logE)/Ωi )→Hj−i+1(Ωi )→Hj−i+1(Ωi /Ωi (logE)(−E)) Z˜ Z˜ Z˜ Y Y is interpreted as the map Gri (Hj+1(Z˜,C))→Gri (Hj+1(E,C). F E F By the isomorphisms H·(Y,C) ∼= H·(E,C), H·(Y) have natural mixed Hodge structures. We put U′ :=Y \E. (i=2): We shallfirstprovethatthenaturalmapβ :H3(Y,C)→H3(Y,C) E is an injection. Byalocalcohomologyexactsequenceitsufficestoshowthatα:H2(Y,C)→ H2(U′,C)isasurjection. BecauseX hasonlyrationalsingularity,H1(Y,O∗)⊗ Y C∼=H2(Y,C). On the other hand, one can prove that H1(U′,O∗ ) ⊗ C → H2(U′,C) U′ is a surjection; in fact, since H·(U′,C) ∼= H·(Y,Rj Ω· ) ∼= H·(Y,j Ω· ) ∼= ∗ U′ ∗ U′ H·(Y,Ω· (logE))wherej :U′ →Y istheimmersion[De],andsinceH·(U′,C)∼= Y H·(U′,Ω· ), we have a commutative diagram of Hodge spectral sequences U′ 7 Hq(Y,Ωp(logE)) >Hp+q(U′,C) Y (1)     Hq(Uy′,Ωp ) >Hp+yq(U′,C) U′ Let F· (resp. F·) be the filtration of Hp+q(U′,C) by the first (resp. sec- 1 2 ond) spectral sequence. In particular, when p+q = 2, we have a surjection Gr0 H2(U′,C) → Gr0 H2(U′,C). Because X has only rational singularity, F1 F2 H2(Y,O ) = 0, hence Gr0 H2(U′,C) = 0. Therefore Gr0 H2(U′,C) = 0, Y F1 F2 which implies that the natural map H2(U′,C) → H2(U′,OU′) is the zero map. It is now clear that H2(U′,Z) → H2(U′,OU′) is also the zero map, and hence H1(U′,O∗ )→H2(U′,Z) is a surjection. Therefore one has a surjection U′ H1(U′,O∗ )⊗C→H2(U′,C). U′ It is enough to prove that H1(Y,O∗) → H1(U′,O∗ ) is surjective in order Y U′ to prove that α is surjective. Put f0 := f|U′ : U′ → U, where U := X \{p}. Let D be irreducible exceptional divisors of f where f(D ) 6= {p}. Let L ∈ i i Pic(U′). Since dimX ≥3 and U ⊂X is 1-concave at p, there exists a reflexive coherent sheaf F on X of rank 1 such that F| ∼= (f0L)∗∗ by [S, Theorem 5], U ∗ where ∗∗ means the double dual. By taking the double dual of both sides of the natural map (f0)∗f0L → L we get an injection ((f0)∗f0L)∗∗ → L, hence ∗ ∗ L ∼= ((f0)∗f∗0L)∗∗ ⊗OU′(ΣaiDi) with some ai ≥ 0. On the other hand, we have an injection ((f0)∗f∗0L)∗∗ →(f∗f∗F)∗∗|U′ and M :=(f∗f∗F)∗∗ ∈Pic(Y). We have ((f0)∗f∗0L)∗∗ ∼= M|U′ ⊗OU′(Σ(−bi)Di) with some bi > 0. Therefore L∼=M ⊗OY(Σ(ai−bi)Di)|U′. We are now in a position to prove the lemma for (i = 2). Let us consider the exact sequence 0→Ω2/Ω2(logE)(−E)→Ω2(logE)/Ω2(logE)(−E)→Ω2(logE)/Ω2 →0. Y Y Y Y Y Fromthissequencewehaveamapδ :H0(E,Ω2(logE)/Ω2)→H1(E,Ω2/Ω2(logE)(−E)). Y Y Y Y The map β is a morphism of mixed Hodge structures and δ can be interpreted as the map Gr2(H3(Y,C) → Gr2(H3(Y,C). We already proved that β is an F E F injection. Hence δ is also an injection by the strict compatibility of the filtrations F. Note that δ is factorized as H0(E,Ω2(logE)/Ω2) →γ H1(Y,Ω2) → Y Y Y H1(E,Ω2/Ω2(logE)(−E)) where γ is the last map in the following exact se- Y Y quence H0(Y,Ω2)→τ H0(Y,Ω2(logE))→H0(E,Ω2(logE)/Ω2)→H1(Y,Ω2). Y Y Y Y Since δ is injective, γ is also injective. Hence τ is surjective by the exact sequence. 8 (i=1): We shallfirstprovethatthenaturalmapβ :H2(Y,C)→H2(Y,C) E is an injection. By a local cohomology exact sequence it suffices to show that α : H1(Y,C) → H1(U′,C) is a surjection. Since X has rational singularities, the sequence H0(Y,O )→H0(Y,O∗)→H1(Y,Z)→0 Y Y is exact. By the same argument as (i = 2), H1(U′,Z) → H1(U′,OU′) is the zero map because X has rational singularities. Hence the sequence H0(U′,OU′)→H0(U′,OU∗′)→H1(U′,Z)→0 is also exact. The restriction map H0(Y,O∗) → H0(U′,O∗ ) is an iso- Y U′ morphism because it is factorized as H0(Y,O∗) ∼= H0(X,O∗ ) ∼= H0(X \ Y X {p},OX∗\{p})∼=H0(U′,OU∗′). SimilarlytherestrictionmapH0(Y,OY)→H0(U′,OU′) is also an isomorphism. Hence the restriction H1(Y,Z) → H1(U′,Z) is an isomorphism by the exact sequences above, and α is an isomorphism. Let us consider the exact sequence 0→Ω1/Ω1(logE)(−E)→Ω1(logE)/Ω1(logE)(−E)→Ω1(logE)/Ω1 →0. Y Y Y Y Y Fromthissequencewehaveamapδ :H0(E,Ω1(logE)/Ω1)→H1(E,Ω1/Ω1(logE)(−E)). Y Y Y Y The map β is a morphism of mixed Hodge structures and δ can be interpreted as the map Gr1(H2(Y,C) → Gr1(H2(Y,C). We already proved that β is an F E F injection. Hence δ is also an injection by the strict compatibility of the filtrations F. Note that δ is factorized as H0(E,Ω1(logE)/Ω1) →γ H1(Y,Ω1) → Y Y Y H1(E,Ω1/Ω1(logE)(−E)) where γ is the last map in the following exact se- Y Y quence H0(Y,Ω1)→τ H0(Y,Ω1(logE))→H0(E,Ω1(logE)/Ω1)→H1(Y,Ω1). Y Y Y Y Since δ is injective, γ is also injective. Hence τ is surjective by the exact sequence. Q.E.D. Remark. In the proof of Lemma 2 the map H0(E,Ωi /Ω1(logE)(−E))→ Y H0(E,Ωi (logE)/Ωi (logE)(−E)) is surjective for i = 1,2 because we have Y Y proved that δ is injective. Proposition3. Let X beaStein open subsetof acomplex algebraic variety. Assumethat X has only rational Gorenstein singularities. Let Σ be the singular locus of X and let f : Y → X be a resolution of singularities such that D := f−1(Σ)isasimplenormalcrossingdivisor andsuchthatf| :Y \D∼=X\Σ. Y\D Then f Ω2 ∼=f Ω2(logD). ∗ Y ∗ Y 9 Proof. Since (X,p) ∼= (R.D.P.)×(Cn−2,0) for all p ∈ X outside certain codimension 3 (in X) locus Σ ⊂Σ ([Re]), it is clear that f Ω2 ∼=f Ω2(logD) 0 ∗ Y ∗ Y at such points p. Let ω ∈ H0(Y,Ω2(logD)). Let F be an irreducible component of D with Y f(F)⊂Σ . Putk :=dimΣ −dimf(F). Weshallprovethatω isregularalong 0 0 F by the induction on k. (a) k =0: (a-1): Put l := codim(Σ ⊂ X). Note that l ≥ 3. Take a general l dimen- 0 sional complete intersection H := H ∩H ∩...∩H . Let p ∈ H ∩f(F). 1 2 n−l p ∈ f(F) is a smooth point. Replace X by a small open neighborhood of p. Then H ∩f(F) = {p}. Moreover, H˜ := f−1(H) is a resolution of singularities ofH. SinceX hascanonicalsingularities,H hasalsocanonicalsingularities. H has a unique dissident point p and other singular points are locally isomorphic to (R.D.P.) × (Cl−2,0). By perturbing H we can define a flat holomorphic map g : X → ∆n−l with g−1(0) = H. We may assume that g has a section passing through p and each fiber X :=g−1(t) intersects f(F) only in this sec- t tion. Denote by p ∈ X this intersection point. By definition p = p. The t t 0 map f : Y → X gives a simultaneous resolution of X for t ∈ ∆n−l. Let D′ t be the union of irreducible components of D which are mapped in this section. Since H is general and X is sufficiently small, every irreducible component of D′ is mapped onto the section. D′ → ∆n−l is a proper map and every fiber D′ is a normal crossing variety. Note that f−1(p ) = D′. We put π = g◦f. t t t t We often write ∆ for ∆n−l. There are filtrations (π|D′)∗Ω2∆ ⊂ F ⊂ Ωˆ2D′ and π∗Ω2 ⊂G ⊂Ω2(logD′) which yield the following exact sequences ∆ Y 0→F →Ωˆ2 →Ωˆ2 →0, D′ D′/∆ 0→(π|D′)∗Ω2∆ →F →(π|D′)∗Ω1∆⊗Ωˆ1D′/∆ →0, 0→G →Ω2(logD′)→Ω2 (logD′)→0, Y Y/∆ 0→π∗Ω2 →G →π∗Ω1 ⊗Ω1 (logD′)→0. ∆ ∆ Y/∆ (a-2): Let us consider the exact sequence 0→Ω2/Ω2(logD′)(−D′)→Ω2(logD′)/Ω2(logD′)(−D′)→Ω2(logD′)/Ω2 →0. Y Y Y Y Y We shall prove the following. Claim. ThemapH0(D′,Ω2/Ω2(logD′)(−D′))→H0(D′,Ω2(logD′)/Ω2(logD′)(−D′)) Y Y Y is surjective. Proof. Note that Ω2/Ω2(logD′)(−D′) ∼= Ωˆ2 . By the exact sequences Y Y D′ above we have two commutative diagrams with exact columns 10