aa r X i v : . [ m a t h . AG ] M a y UNOBSTRUCTEDNESS OF DEFORMATIONSOF WEAK FANO MANIFOLDS
TARO SANO
Dedicated to Professor Miles Reid on the occasion of his 65th birthday.
Abstract.
We prove that a weak Fano manifold has unobstructed deformations. For ageneral variety, we investigate conditions under which a variety is necessarily obstructed.
Contents
1. Introduction 12. Proof of theorem 23. The surface case 6Acknowledgments 7References 71.
Introduction
We consider algebraic varieties over an algebraically closed field k of characteristic zero.The Kuranishi space of a smooth projective variety has bad singularities in general. Evenin the surface case, Vakil [18] exhibited several examples of smooth projective surfaces ofgeneral type with arbitrarily singular Kuranishi spaces.On the other hand, in some nice situations, the Kuranishi space is smooth. A famousresult is that the Kuranishi space of a Calabi-Yau manifold is smooth. The Kuranishi spaceof a Fano manifold X is also smooth since H ( X, Θ X ) = 0 by the Kodaira–Nakano vanishingtheorem, where Θ X is the tangent sheaf of X .In this paper, we look for several nice projective manifolds with smooth Kuranishi space.A smooth projective variety X is called a weak Fano manifold if the anticanonical divisor − K X is nef and big. The following is the main theorem of this paper. Theorem 1.1.
Deformations of a weak Fano manifold are unobstructed.
Previously, Ran proved the unobstructedness for a weak Fano manifold with a smoothanticanonical element ([15, Corollary 3]). Minagawa’s argument in [13] implies the unob-structedness when |− K X | contains a smooth element. However these assumptions are notsatisfied for a general weak Fano manifold as explained in Example 2.9. We prove it for thegeneral case.We use the T -lifting technique developed by Ran, Kawamata, Deligne and Fantechi-Manetti. Another approach is dealt with by Buchweitz–Flenner in [1].The following more general result implies Theorem 1.1. Mathematics Subject Classification.
Primary 14D15, 14J45; Secondary 14B07.
Key words and phrases. deformation theory, Kuranishi space, weak Fano manifolds.
Theorem 1.2.
Let X be a smooth projective variety. Assume that H ( X, O X ) = 0 and thereexists a positive integer m and a smooth divisor D ∈ |− mK X | such that H ( D, N D/X ) = 0 .Then deformations of X are unobstructed. We sketch the proof of Theorem 1.2. Instead of proving the unobstructedness directly,we first prove the unobstructedness for the pair of a weak Fano manifold X and a smoothelement D of |− mK X | for a sufficiently large integer m in Theorem 2.2. Next we show thatthe unobstructedness for ( X, D ) implies the unobstructedness for X .We also show that the Kuranishi space of a smooth projective surface is smooth if theKodaira dimension of the surface is negative or 0 in Theorem 3.2. It seems to be known toexperts but we give a proof for the convenience of the reader.2. Proof of theorem
Fix an algebraically closed field k of characteristic zero. Let Art k be the category ofArtinian local k -algebras with residue field k and Sets the category of sets. For a propervariety X over k and an effective Cartier divisor D on X , let Def ( X,D ) : Art k → Sets bethe functor sending A ∈ Art k to the set of equivalence classes of proper flat morphisms f : X A → Spec A together with effective Cartier divisors D A ⊂ X A and marking isomor-phisms φ : X A ⊗ A k → X such that φ ( D A ⊗ A k ) = D . This is the pair version of thedeformation functor Def X defined in [10]. We see that Def ( X,D ) is a deformation functor inthe sense of Fantechi–Manetti ([4, Introduction]).We need the following lemma. Lemma 2.1.
Let Z be a smooth proper variety over k and ∆ ⊂ Z a smooth divisor.Set A n := k [ t ] / ( t n +1 ) for a non-negative integer n . Let Z n → Spec A n and ∆ n ⊂ Z n bedeformations of Z and ∆ . Let Ω • Z n /A n (log ∆ n ) be the de Rham complex of Z n /A n withlogarithmic poles along ∆ n (cf. [8, (7.1.1)] ). Then we have the following: (i) the hypercohomology group H k ( Z n , Ω • Z n /A n (log ∆ n )) is a free A n -module for all k ; (ii) the spectral sequence (1) E p,q := H q ( Z n , Ω pZ n /A n (log ∆ n )) ⇒ H p + q ( Z n , Ω • Z n /A n (log ∆ n )) degenerates at E ; (iii) the cohomology group H q ( Z n , Ω pZ n /A n (log ∆ n )) is a free A n -module and commuteswith base change for any p and q .Proof. We can prove this by the same argument as in [2, Th´eor`eme 5.5]. We give a prooffor the convenience of the reader. We can assume that k = C by the Lefschetz principle.(i) Set U := Z \ ∆. Let ι : U ֒ → Z be the open immersion. We see that the complexΩ • Z n /A n (log ∆ n ) is quasi-isomorphic to ι ∗ Ω • U n /A n by a standard argument as in [14, Proposi-tion 4.3], where U n → Spec A n is a deformation of U which is induced by Z n → Spec A n .We have an isomorphism H k ( Z n , ι ∗ Ω • U n /A n ) ≃ H k ( U n , Ω • U n /A n ) since we have R i ι ∗ Ω jU n /A n = 0for i > j . Moreover we have H p + q ( U n , Ω • U n /A n ) ≃ H p + q ( U, A n ), where the latter isthe singular cohomology on U with coefficient A n since Ω • U n /A n is a resolution of the sheaf A n,U , where A n,U is a constant sheaf on U associated to A n (See [2, Lemme 5.3]). Hence weobtain (i) since we have H p + q ( Z n , Ω • Z n /A n (log ∆ n )) ≃ H p + q ( U, A n ) ≃ H p + q ( U, C ) ⊗ A n . EFORMATIONS OF WEAK FANO MANIFOLDS 3
Moreover we obtain the equalitydim C H p + q ( Z n , Ω • Z n /A n (log ∆ n )) = dim C ( A n ) · dim C H p + q ( Z, Ω • Z (log ∆)) . (ii) By the argument as in [2, (5.5.5)], we see that(2) dim C H q ( Z, Ω pZ n /A n (log ∆ n )) ≤ dim C ( A n ) · dim C H q ( Z, Ω pZ (log ∆))and equality holds if and only if H q ( Z, Ω pZ n /A n (log ∆ n )) is a free A n -module. By the spectralsequence (1), we have(3) X p + q = k dim C H q ( Z n , Ω pZ n /A n (log ∆ n )) ≥ dim C H k ( Z n , Ω • Z n /A n (log ∆ n )) . By the two inequalities (2), (3) and (i), we obtain(4) dim C ( A n ) · X p + q = k dim C H q ( Z, Ω pZ (log ∆)) ≥ dim C ( A n ) · dim C H k ( Z, Ω • Z (log ∆)) . We have equality in the inequality (4) since the spectral sequence (1) degenerates at E when n = 0 by [3, Corollaire (3.2.13)(ii)]. Hence we have equality in (3) and obtain (ii).(iii) This follows from (i) and (ii). (cid:3) To prove Theorem 1.2, we prove the following theorem on unobstructedness of deforma-tions of a pair.
Theorem 2.2.
Let X be a smooth proper variety such that H ( X, O X ) = 0 . Assume thatthere exists a smooth divisor D ∈ |− mK X | for some positive integer m . Then deformationsof ( X, D ) are unobstructed, that is, Def ( X,D ) is a smooth functor.Proof. Set A n := k [ t ] / ( t n +1 ) and B n := k [ x, y ] / ( x n +1 , y ) ≃ A n ⊗ k A . For [( X n , D n ) , φ ] ∈ Def ( X,D ) ( A n ), let T (( X n , D n ) /A n ) be the set of isomorphism classes of pairs (( Y n , E n ) , ψ n )consisting of deformations ( Y n , E n ) of ( X n , D n ) over B n and marking isomorphisms ψ n : Y n ⊗ B n A n → X n such that ψ n ( E n ⊗ B n A n ) = D n , where we use a ring homomorphism B n → A n given by x t and y
0. Then we see the following.
Claim . We have(5) T (( X n , D n ) /A n ) ≃ H ( X n , Θ X n /A n ( − log D n )) , where Θ X n /A n ( − log D n ) is the dual of Ω X n /A n (log D n ). Proof.
We can prove this by a standard argument (cf. [17, Proposition 3.4.17]) using B n = A n ⊗ k A . (cid:3) Hence, by [4, Theorem A], it is enough to show that the natural homomorphism γ n : H ( X n , Θ X n /A n ( − log D n )) → H ( X n − , Θ X n − /A n − ( − log D n − ))is surjective for the above X n , D n and for X n − := X n ⊗ A n A n − , D n − := D n ⊗ A n A n − .Note that we have a perfect pairingΩ X n /A n (log D n ) × Ω d − X n /A n (log D n ) → O X n ( K X n /A n + D n ) ≃ ω ⊗ − mX n /A n , where we set d := dim X . We have O X n ( K X n /A n + D n ) ≃ ω ⊗ − mX n /A n since we have H ( X, O X ) =0 (See [6, Theorem 6.4(b)], for example.). Thus we see that H ( X n , Θ X n /A n ( − log D n )) ≃ H ( X n , Ω d − X n /A n (log D n ) ⊗ ω ⊗ m − X n /A n ) . TARO SANO
Let π n : Z n := Spec m − M i =0 O X n ( iK X n /A n ) → X n be the ramified covering defined by a section σ D n ∈ H ( X n , − mK X n /A n ) which correspondsto D n . We have an isomorphism π ∗ n Ω X n /A n (log D n ) ≃ Ω Z n /A n (log ∆ n )for some divisor ∆ n ∈ |− π ∗ n K X n /A n | . Hence we see that( π n ) ∗ Ω d − Z n /A n (log ∆ n ) ≃ m − M i =0 Ω d − X n /A n (log D n )( iK X n /A n )and Ω d − X n /A n (log D n ) ⊗ ω ⊗ m − X n /A n is one of the direct summands.Hence it is enough to show that the natural restriction homomorphism r n : H ( Z n , Ω d − Z n /A n (log ∆ n )) → H ( Z n − , Ω d − Z n − /A n − (log ∆ n − ))is surjective, where we set Z n − := Z n ⊗ A n A n − and ∆ n − := ∆ n ⊗ A n A n − , since γ n is aneigenpart of r n . By Lemma 2.1(iii), we see the required surjectivity. This completes theproof of Theorem 2.2. (cid:3) Remark . Iacono [7] proved Theorem 2.2 when m = 1 without the assumption H ( X, O X ) =0 in [7, Corollary 4.5] as a consequence of the analysis of DGLA. Remark . We can remove the assumption H ( X, O X ) = 0 when m = 1 by a similarargument as in [15, Corollary 2]. In that case, we see that O X n ( K X n /A n + D n ) ≃ O X n sincewe have H ( X n , K X n /A n + D n ) ≃ A n by Claim 2.1, with X n , D n as in the proof of Theorem2.2.We do not know whether we can remove the assumption H ( X, O X ) = 0 in Theorem 2.2when m is arbitrary.Theorem 2.2 implies Theorem 1.2 as follows. Proof of Theorem 1.2.
Since H ( D, N D/X ) = 0, we see that the forgetful morphismDef ( X,D ) → Def X between functors is smooth. Since Def ( X,D ) is smooth by Theorem 2.2, we see that Def X isalso smooth. (cid:3) Theorem 1.2 implies Theorem 1.1 as follows.
Proof of Theorem 1.1.
Let X be a weak Fano manifold of dimension d . By the base pointfree theorem, we can take a sufficiently large integer m such that − mK X is base point freeand contains a smooth element D ∈ |− mK X | . We have H ( D, N D/X ) = 0 since there is anexact sequence H ( X, O X ( D )) → H ( D, N D/X ) → H ( X, O X )and both outer terms are zero by the Kawamata–Viehweg vanishing theorem. Hence The-orem 1.2 implies Theorem 1.1. (cid:3) Remark . We can prove the following theorem by the same argument as Theorem 1.1.
EFORMATIONS OF WEAK FANO MANIFOLDS 5
Theorem . Let X be a complex manifold whose anticanonical bundle is nef and big. Thendeformations of X are unobstructed. Actually we see that such a complex manifold is Moishezon since there is a big divisor on X . Hence we can show Lemma 2.1 and the base-point free theorem in this setting. Usingthese, we can show Theorem 2.7 in the same way as Theorem 1.1. Example 2.8.
We give an example of a weak Fano manifold such that H ( X, Θ X ) = 0,where Θ X is the tangent sheaf.Let f : X → P (1 , , ,
3) be the blow-up of the singular point p of the weighted projectivespace. We can check that X ≃ P P ( O P ⊕ O P ( − f is the anticanonical morphism of X . Hence − K X = f ∗ ( − K P (1 , , , ) and this is nef and big. Set E := O P ⊕ O P ( − P P ( E ) → P , we see that h ( X, Θ X ) = h ( X, Θ X/ P ) = h ( P , E ⊗ E ∗ ) = 1 . Hence H ( X, Θ X ) = 0.Thus we need a technique such as T -lifting for the proof of Theorem 1.2. Example 2.9.
We give an example of a Fano manifold such that neither of the linearsystems |− K X | and |− K X | contain smooth elements. Our example is a modification of anexample in [11, Example 3.2 (3)].Let X := X d ⊂ P (1 , . . . , , , d ) = P (1 n , , d ) be a weighted hypersurface of degree 5 d and dimension n . Assume that d n − d = 2. (For example, d = 6 , n = 21.) The latter condition implies that − K X = O X (2). We see that the baselocus of |− K X | and |− K X | consists of a point p := H ∩ . . . ∩ H n ∩ X d , where H , . . . , H n are degree 1 hyperplanes of the first n coordinates of P (1 n , , d ). We see that every elementof |− K X | has multiplicity 2 at the base point p and hence is singular. We also see that everyelement of |− K X | has multiplicity 4 at the base point p and hence is singular. Example 2.10.
We give an example of a smooth projective variety such that Def X is notsmooth and − K X is big.Let C ⊂ P be a smooth curve with an obstructed embedded deformation which lies in acubic surface as in [6, Theorem 13.1]. Let µ : X → P be the blow-up of P along C . Then X has an obstructed deformation. See [6, Example 13.1.1]. Note that − K X = µ ∗ O P (4) − E where E := µ − ( C ) and C is contained in a cubic surface S ⊂ P . Let ˜ S ⊂ X be the stricttransform of S . Then we see that − K X is big since ˜ S + | µ ∗ O P (1) | ⊂ |− K X | . Example 2.11.
We give an example of X and D ∈ |− K X | such that Def ( X,D ) is smoothbut Def X is not smooth.Let C ⊂ P be a smooth curve in a quartic surface S such that the Hilbert scheme ofcurves in P is singular at the point corresponding to C (cf. [6, Exercise 13.2]). Let X → P be the blow-up of P along C . Then X has an obstructed deformation. However the stricttransform D := ˜ S ∈ |− K X | of S is smooth and H ( X, O X ) = 0. Hence Def ( X,D ) is smoothby Theorem 2.2. Example 2.12.
We give an example of X with an obstructed deformation such that − K X is nef.Set X := T m × P where T m is a complex torus of dimension m ≥
2. Then X has anobstructed deformation ([12, p.436–441]). Note that − K X is nef. It is actually semiample.It is natural to ask the following question: TARO SANO
Problem 2.13.
Let X be a smooth projective variety such that − K X is nef and H ( X, O X ) =0 . Is the Kuranishi space of X smooth? The surface case
The following lemma states that smoothness of the Kuranishi space is preserved underthe blow-up at a point.
Lemma 3.1.
Let S be a smooth projective variety and ν : T → S the blow-up at a point p ∈ S . Then the functor Def S is smooth if and only if the functor Def T is smooth.Proof. Let Def ( S,p ) be the functor of deformations of a closed immersion { p } ⊂ S and Def ( T,E ) the functor as in Section 2, where E := ν − ( p ). We can define a natural transformation ν ∗ : Def ( T,E ) → Def ( S,p ) as follows: given A ∈ Art k and a deformation ( T , E ) of ( T, E ) over A , we see that ν ∗ O T is a sheaf of flat A -algebras by [19, Corollary 0.4.4] since we have R ν ∗ O T = 0. We alsosee that ν ∗ O T ( − E ) is a sheaf of flat A -modules by [19, Corollary 0.4.4] since we have R ν ∗ O T ( − E ) = 0 by a direct calculation. Hence we can define a deformation ( S , p ) of( S, p ) over A by sheaves O S := ν ∗ O T , I p := ν ∗ O T ( − E ) and obtain a natural transformation ν ∗ .We can also define a natural transformation ν ∗ : Def ( S,p ) → Def ( T,E ) as follows: given a deformation ( S , p ) of ( S, p ) over A ∈ Art k , we define a deformation T of T as the blow-up of S along p . We can also define a deformation E of E by the inverseimage ideal sheaf ν − I p · O T , where I p is the ideal sheaf of p ⊂ S .We see that ν ∗ and ν ∗ are inverse to each other. Hence we have Def ( T,E ) ≃ Def ( S,p ) asfunctors.We have forgetful morphisms of functors F T : Def ( T,E ) → Def T and F S : Def ( S,p ) → Def S .We see that F T and F S are smooth since we have H ( E, N E/T ) ≃ H ( P d − , O P d − ( − H ( N p/S ) = 0, where we set d := dim S .Thus we have a diagram Def ( T,E ) ∼ / / F T (cid:15) (cid:15) Def ( S,p ) F S (cid:15) (cid:15) Def T Def S , where F T and F S are smooth. Hence we see the required equivalence. (cid:3) By this lemma, we see that a smooth projective surface has unobstructed deformations ifand only if its relatively minimal model has unobstructed deformations.Using Lemma 3.1, we can prove the following:
Theorem 3.2.
Let X be a smooth projective surface with non-positive Kodaira dimension.Then the deformations of X are unobstructed.Proof. By Lemma 3.1, we can assume that X does not contain a − X is negative, it is known that X ≃ P or X ≃ P C ( E ) forsome projective curve C and a rank 2 vector bundle E on C . In these cases, we see that H ( X, Θ X ) = 0 by the Euler sequence or the argument in [16, p.204]. EFORMATIONS OF WEAK FANO MANIFOLDS 7
If the Kodaira dimension of X is zero, it is a K (cid:3) Remark . Kas [9] gave an example of a smooth projective surface of Kodaira dimension1 with an obstructed deformation.
Acknowledgments
The author would like to thank Professors Osamu Fujino, Yoshinori Gongyo, AndreasH¨oring, Donatella Iacono, Yujiro Kawamata, Marco Manetti, Tatsuhiro Minagawa, Yoshi-nori Namikawa and Hirokazu Nasu for valuable conversations. He thanks the referee forconstructive comments. He thanks Professor Miles Reid and Michael Selig for improvingthe presentation of the manuscript and valuable conversations. He is partially supported byWarwick Postgraduate Research Scholarship.
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TARO SANO
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