Deformations of the tangent bundle of projective manifolds
aa r X i v : . [ m a t h . AG ] J u l DEFORMATIONS OF THE TANGENT BUNDLE OFPROJECTIVE MANIFOLDS
THOMAS PETERNELL
Abstract.
We investigate when the tangent bundle of a projective manifoldhas a non-trivial first order (or positive-dimensional) deformation. This leadsto a new conjectural characterization of the complex projective space.
Contents
1. Introduction 12. Some general results 33. Surfaces 64. Threefolds 114.1. Fano threefolds 114.2. P -bundles 134.3. P -bundles 144.4. Blow-ups 165. Calabi-Yau threefolds: birational morphisms and flops 196. Higher Dimensions 31References 321. Introduction
In the early 1990’s, physicists asked to compute the dimension of the space H ( X, T X ⊗ Ω X ) for a Calabi-Yau threefold X ; see [DGKM89], [EH90]. Theycomputed the dimension for some special complete intersections. A few years laterthe problem was taken up by D.Huybrechts [Huy95]; he proved in particular that H ( X, T X ⊗ Ω X ) = 0 for three-dimensional Calabi-Yau complete intersections inprojective spaces. In terms of deformation theory, this space parametrizes firstorder deformations of the holomorphic tangent bundle T X .In this paper we propose to systematically study deformations of the tangentbundle T X of any compact complex manifold X , up to deformations of the form T X ⊗ L , where L is a deformation of the trivial line bundle O X . To be precise, weintroduce the following notation. Let X be a compact complex manifold. We say that T X has a genuine first order deformation if T X has a deformation E over the double point D which is not induced by a deformation of O X , i.e., E 6≃ p ∗ T X ⊗ L with L adeformation of O X over D ; here p : X × D → X denotes the projection. Mathematics Subject Classification.
Key words and phrases. tangent bundle, deformation, Calabi-Yau threefold. o have a genuine first oder deformation is equivalent to saying that(1.1.1) h ( X, T X ⊗ Ω X ) > q ( X ) := h ( X, O X ) . Note that T X has a non-trivial first order deformation if and only if the morphism π : P ( T X ) → X has a first order deformation with fixed target X which is nottrivial, i.e., not constant.In the same way we define genuine deformations of T X over a positive-dimen-sional parameter space. The obstructions to lifting a first order deformation in thesense of (1) ly in the space(1.1.2) H ( X, T X ⊗ Ω X ) /H ( X, O X ) ≃ H ( P ( T X ) , T P ( T X ) /X ) . Thus T X has a non-obstructed genuine deformation provided(1.1.3) h ( X, T X ⊗ Ω X ) − h ( X, T X ⊗ Ω X ) − q ( X ) + h ( X, O X ) > . Let E nd ( T X ) denote the sheaf of traceless endomorphisms of T X ,hence T X ⊗ Ω X ≃ E nd ( T X ) ⊕ O X and a fortiori H q ( X, T X ⊗ Ω X ) = H q ( X, E nd ( T X )) = H q ( X, E nd ( T X )) ⊕ H q ( X, O X ))for all q . Then Equation (1.1.1) is equivalent to h ( X, E nd ( T X )) = 0 . whereas Equation (1.1.3) is equivalent to h ( X, E nd ( T X )) − h ( X, E nd ( T X )) > . The theme of this paper is now the following
Let X be a projective (compact K¨ahler, or simply compact)manifold. Give necessary and sufficient conditions such that T X has a genuinefirst-order deformation resp. a genuine non-obstructed deformation.In dimension 2 we show Let X be a compact K¨ahler surface. Then T X does not havea genuine first order deformation if and only if either X ≃ P or if X is a ballquotient (so c ( X ) = 3 c ( X ) ) with H ( X, S Ω X ) = 0 . The case that X is of general type has already been shown in [CWY03]. Itwould certainly be interesting to classify the two-dimensional ball quotients X with H ( X, S Ω X ) = 0.In dimension larger than one however, things get much more complicated, evenin dimension 3. Of course, the tangent bundle of the complex projective space isrigid in any dimension. In dimension 3 we prove - among other things - Let X be a three-dimensional compact complex manifold. Then T X has a genuine first order deformation provided one of the following conditionsis satisfied.(1) X is a Fano threefold, different from P .(2) Suppose X is rationally connected and that c ( X ) ≤ .(3) X is the blow-up of the smooth threefold Y in a point or a smooth curve C .Suppose that h ( Y, T Y ⊗ Ω Y ) = 1 , e.g., T Y is stable for some polarization,and in case of the blow-up of C , additionally that − K Y · C ≥ . X is a P -bundle over a smooth curve.(5) X = P ( F ) with a semi-stable rank two bundle over a smooth K¨ahler surface S with H ( S, O S ) = 0 .(6) X is a Calabi-Yau threefold and ϕ : X → Y the contraction of an irreducibledivisor E to a point or a curve with Y projective or ϕ : X → Y a smallcontraction, with a few exceptions, given in Theorem 5.12 and Theorem5.16. It is also easy to see that the tangent bundles of smooth hypersurfaces X ⊂ P n +1 of degree d ≥ n ≥ X is a ball quotient of dimension at least three, then Siu [Siu91] has shown that T X has no genuine deformation. Further, if X is a product X = X × X with X j ballquotients with q ( X j ) = 0, then T X again has no genuine first order deformation.This procedure can also be iterated.All these considerations lead to the following Suppose X is a compact K¨ahler manifold. Then T X has nogenuine first order deformation if and only if X is one of the following. • X ≃ P n • X is a ball quotient of dimension at least three • X is a twodimensional ball quotient such that H ( X, E nd ( T X )) = 0 • X is a product (with at least two factors) X = Π j X j with X j ball quotients such that q ( X j ) = 0 and H ( X j , E nd ( T X j )) = 0 forall j . Acknowledgements
Many thanks go to Ulrike Peternell for carefully readingthe first four parts of the paper and for suggesting many improvements. Also I wouldlike to thank Alan Huckleberry and Stefan Nemirovski for very helpful discussionsand Jun-Muk Hwang for pointing out the references [CWY03] and [Siu91]. Finally,I would like to thank the referee for pointing out an error in section 5 and manyother very valuable suggestions.2.
Some general results
In this section we prove some general results with a focus on blow-ups.
Let X be a compact complex manifold. Let E ⊂ X be a smoothirreducible divisor. Assume that(1) H ( E, T X | E ) = 0 . (2) H ( X, Ω X ⊗ T X ) = H ( X, Ω X (log E ) ⊗ T X ) , Then H ( X, Ω X ⊗ T X ) = 0 . Proof.
The residue sequence tensorized with T X reads0 → Ω X ⊗ T X → Ω X (log E ) ⊗ T X → T X | E → . Taking cohomology and using our assumption (2) we obtain an injection H ( E, T X | E ) → H ( X, T X ⊗ Ω X )Hence assumption (1) gives the claim. (cid:3) In the same way, we have .2. Proposition. Let X be a compact complex manifold. Let E ⊂ X be a smoothirreducible divisor. Assume that(1) H ( E, T X | E ) = 0 . (2) h ( X, E nd ( T X )) = h ( X, (Ω X (log E ) ⊗ T X ) / O X ) .Then H ( X, E nd ( T X ) = 0 .Proof. Dividing by the trivial summand of Ω X ⊗ T X , the residue sequence inducesan exact sequence0 → E nd ( T X ) → R := (Ω X (log E ) ⊗ T X ) / O X → T X | E → . Then we conclude as before. (cid:3)
Since 0 = H ( E, T E ) ⊂ H ( E, T X | E ), we obtain Let X be a compact complex manifold. Let E ⊂ X be a smoothirreducible divisor. Assume that(1) H ( E, T E ) = 0 , (2) H ( X, Ω X ⊗ T X ) = H ( X, Ω X (log E ) ⊗ T X ) .Then H ( X, T X ⊗ Ω X ) = 0 . Let X be a compact complex manifold. Let E ⊂ X be a smoothirreducible divisor. Assume that(1) H ( E, T E ) = 0 , (2) h ( X, E nd ( T X )) = h ( X, Ω X (log E ) ⊗ T X / O X ) .Then H ( X, E nd ( T X )) = 0 . The condition H ( X, Ω X ⊗ T X ) = H ( X, Ω X (log E ) ⊗ T X ) , certainly holds provided dim H ( X, Ω X (log E ) ⊗ T X ) = 1 . Let X be a compact complex n -dimensional manifold, x ∈ X and π : ˆ X → X the blow up of X at x . Then(1) There is an exact sequence (2.6.1) 0 → π ∗ ( T ˆ X ⊗ Ω X ) → T X ⊗ Ω X → Q → with a sheaf Q supported on x of length n − .(2) h ( ˆ X, T ˆ X ⊗ Ω X ) ≥ h ( X, T X ⊗ Ω X ) with strict inequality if h ( X, T X ⊗ Ω X ) < n ;(3) h ( ˆ X, T ˆ X ⊗ Ω X ) ≤ h ( X, T X ⊗ Ω X ) ;(4) h ( ˆ X, E nd ( T ˆ X )) ≥ h ( X, E nd ( T X )) with strict inequality if h ( X, T X ⊗ Ω X ) < n ;(5) h ( ˆ X, E nd ( T ˆ X )) ≤ h ( X, E nd ( T X )) .Proof. Set E = π − ( x ). We clearly have an exact sequence0 → π ∗ ( T ˆ X ⊗ Ω X ) → T X ⊗ Ω X → Q → Q which is zero outside x . Tenorize the exact sequence0 → π ∗ (Ω X ) → Ω X → Ω X/X → y T ˆ X , take π ∗ and use R π ∗ ( T ˆ X ) = 0 to obtain the exact sequence0 → π ∗ ( T ˆ X ) ⊗ Ω X → π ∗ ( T ˆ X ⊗ Ω X ) → π ∗ ( T ˆ X | E ⊗ Ω E ) → . Further, we have an exact sequence0 → π ∗ ( T ˆ X ) ⊗ Ω X → T X ⊗ Ω X → C nx ⊗ Ω X → . Then we obtain the following commutative diagram(2.6.2) 0 (cid:15) (cid:15) (cid:15) (cid:15) / / π ∗ ( T ˆ X ) ⊗ Ω X (cid:15) (cid:15) / / π ∗ ( T ˆ X ⊗ Ω X ) (cid:15) (cid:15) / / π ∗ ( T ˆ X ⊗ Ω E ) / / T X ⊗ Ω X = / / (cid:15) (cid:15) T X ⊗ Ω X (cid:15) (cid:15) / / ker λ / / C nx ⊗ Ω X λ / / (cid:15) (cid:15) Q (cid:15) (cid:15) / /
00 0A diagram chase shows thatker λ ≃ π ∗ ( T ˆ X | E ⊗ Ω E ) , yielding an exact sequence0 → π ∗ ( T ˆ X | E ⊗ Ω E ) → C nx ⊗ Ω X → Q → . Observe finally(2.6.3) π ∗ ( T ˆ X | E ⊗ Ω E ) ≃ C x . In fact, use the exact sequence0 → T E ⊗ Ω E → T ˆ X | E ⊗ Ω E → N E ⊗ Ω E → h ( E, T ˆ X | E ⊗ Ω E ) = 1and H ( E, N ∗⊗ µE ⊗ T ˆ X | E ⊗ Ω E ) = 0for all µ ≥
1, which yields (2.6.3). Then Assertion (1) follows by (2.6.3).Assertions (2) and (3) then follow immediately by taking cohomology and usingthe Leray spectral sequence. Finally, (4) and (5) are immediate from (2) and (3),since H q ( ˆ X, O ˆ X ) = H q ( X, O X ) for all q . (cid:3) Let X be a compact complex n -dimensional manifold, x ∈ X and π : ˆ X → X the blow up of X at x . Suppose that T X has a genuine first order (resp.genuine non-obstructed) deformation. Suppose further that h ( X, T X ⊗ Ω X ) < n .Then T ˆ X has a genuine first order (genuine non-obstructed) deformation. f T X is simple, we can say more: Let X be a compact complex manifold such that H ( X, Ω X ⊗ T X ) ≃ C ; e.g., T X is stable with respect to a Gauduchon metric. Let either π : ˆ X → X be theblow-up of a point in X or dim X ≥ and π : ˆ X → X be the blow-up of a smoothrational curve C ⊂ X . Then H ( ˆ X, E nd ( T ˆ X )) = 0 . Proof.
We restrict ourselves to the case of the blow-up of a curve; the point blow-upbeing completely analogous. We verify the conditions of Corollary 2.4, applied tothe exceptional divisor E = π − ( C ) and use again the notation R = (Ω X (log E ) ⊗ T ˆ X ) / O ˆ X . Since E = P ( N ∗ C/X ), the relative Euler sequence and the splitting of N C/X yields h ( E, T
E/C ) = h ( C, N ∗ C/X ⊗ N C/X ) − ≥ , thus H ( E, T E ) = 0. As to the second condition, we first observe the followingchain of inclusions and equations H ( ˆ X \ E, R ) = H ( ˆ X \ E, E nd ( T ˆ X ) ) = H ( X \ C, E nd ( T X )) == H ( X, E nd ( T X )) , the last equation coming from Riemann’s extension theorem.By our assumption, H ( X, E nd ( T X )) = 0, hence H ( ˆ X \ E, R ) = 0. Now R istorsion free; in fact, otherwise O X would not be saturated in Ω X (log E ) ⊗ T X ,thus O X → Ω X (log E ) ⊗ T X would vanish along E , which is clearly not the case.Consequently, H ( X, R ) → H ( X \ E, R ) is injective, hence we conclude. (cid:3) Theorem 2.8 remains true for curves C of genus g ≥ , provided H ( T E ) = 0 . If g ≥ , this is equivalent to h ( N C ⊗ N ∗ C ) ≥ , i.e., N C is not simple. In case g = 1 , we might also have h ( N C ⊗ N ∗ C ) = 1 andthe vector field on C lifts to E. Instead of assuming T X to be simple in Theorem 2.8, it sufficesto assume that H ( ˆ X, T ˆ X ⊗ Ω X ) = H ( X, T X ⊗ Ω X ) . Surfaces
We start by some general calculations.
Let X be a smooth compact complex surface. Then(1) χ ( X, T X ⊗ Ω X ) = (cid:0) c ( X ) − c ( X ) (cid:1) (2) h ( X, T X ⊗ Ω X ) = (cid:0) − c ( X ) + 11 c ( X ) (cid:1) + h ( X, T X ⊗ Ω X ) + h ( X, (Ω X ) ⊗ ) (3) h ( X, T X ⊗ Ω X ) = (cid:0) c ( X ) − c ( X )) + q ( X ) + h ( X, E nd ( T X )) + h ( X, S Ω X ) . h ( X, E nd ( T X )) − h ( X, E nd ( T X )) = (3 c ( X ) − c ( X )) + h ( X, T X ⊗ Ω X ) − . (5) If c ( X ) < c ( X ) , then h ( X, E nd ( T X )) − h ( X, E nd ( T X )) > .Proof. (1) follows from Riemann-Roch, since c ( T X ⊗ Ω X ) = 0 and c ( T X ⊗ Ω X ) =4 c ( X ) − c ( X ) . (2) is a consequence of (1), using H ( X, T X ⊗ Ω X ) ≃ H ( X, T X ⊗ Ω X ⊗ K X ) = H ( X, (Ω X ) ⊗ ) . For (3), we apply (2), observe that(Ω X ) ⊗ ≃ S Ω X ⊕ K X and use h ( X, K X ) = h ( X, O X ) = χ ( X, O X ) − q ( X ) = 112 (cid:0) c ( X ) + c ( X ) (cid:1) − q ( X ) . As to (4), we have, using (1), h ( X, End ( T X )) − h ( X, End ( T X )) = h ( X, T X ⊗ Ω X ) − h ( X, T X ⊗ Ω X ) − q ( X )++ h ( X, O X ) = − χ ( X, T X ⊗ Ω X ) + h ( X, T X ⊗ Ω X ) + χ ( X, O X ) − c ( X ) − c ( X ) + h ( X, T X ⊗ Ω X ) − . This yields claim (4), and (5) follows from (4). (cid:3)
Let X be a compact complex surface. If c ( X ) < c ( X ) , then T X has a genuine non-obstructed deformation. Let X be a compact complex K¨ahler surface. Then the followingassertions are equivalent.(1) T X has a genuine first order deformation.(2) X P and X is not a ball quotient with H ( X, S Ω X ) = 0 . As already mentioned, surfaces of general type have been treated in [CWY03].
Proof.
First that if X = P or if X is a ball quotient with H ( X, S Ω X ) = 0, thenby Proposition 3.1, T X has no genuine first order deformations. Hence Assertion(1) implies Assertion (2).To prove the converse, we note first that by Proposition 2.6, we may assume X to beminimal and by Corollary 3.2 that c ( X ) ≥ c ( X ). The Miyaoka-Yau inequalityand surface classification gives c ( X ) = 3 c ( X ), unless X is a ruled surface over acurve of genus at least two. More specifically, again by classification, X is one ofthe following(1) X = P ;(2) X is a ball quotient;(3) κ ( X ) = 1 and c ( X ) = 0;(4) X is a torus or hyperelliptic;(5) X is a ruled surface over a curve B of genus g = g ( B ) ≥ ase (4) is immediately settled by Proposition 3.1(3).(3) Assume that κ ( X ) = 1 and c ( X ) = 0. Let f : X → B be the Iitaka fibra-tion. Suppose T X does not have a genuine first order deformation, then we have h ( X, T X ⊗ Ω X ) = q ( X ). Hence H ( X, S Ω X ) = 0 by Proposition 3.1(3). Hence q ( X ) = 0, so that χ ( X, O X ) >
0, contradicting c ( X ) = c ( X ) = 0.(5) Let π : X → B denote a ruling over the curve B . Since χ ( X, T X ⊗ Ω X ) = 4( g − g ≥
2. By Proposition 3.1(2), h ( X, T X ⊗ Ω X ) = 4(1 − g ) + h ( X, T X ⊗ Ω X ) + h ( X, (Ω X ) ⊗ ) ≥≥ − g ) + h ( X, T X ⊗ Ω X ) + h ( B, K B ) = 1 − g + h ( X, T X ⊗ Ω X ) . We will now prove that(3.3.1) h ( X, T X ⊗ Ω X ) > g − . This yields h ( X, T X ⊗ Ω X ) > q ( X ) = g, which was to be proved. In order to show (3.3.1), we consider the subbundles T X ⊗ π ∗ (Ω B ) ⊂ T X ⊗ Ω X and T X/B ⊗ π ∗ (Ω B ) ⊂ T X ⊗ π ∗ (Ω B ) . Write X = P ( E ) with a rank two bundle E on B . Then T X/B ≃ O P ( E ) (2) ⊗ π ∗ (det E ∗ ) . Hence h ( T X/B ⊗ π ∗ (Ω B )) = h ( B, S E ⊗ det E ∗ ⊗ Ω B ) ≥ χ ( S E ⊗ det E ∗ ⊗ Ω B ) = 3( g − . The last equation is Riemann-Roch for the rank three bundle S ( E ) ⊗ det E ∗ , re-calling that c ( S ( E ) ⊗ det E ∗ ) = 0. Thus h ( X, T X ⊗ Ω X ) > h ( T X ⊗ π ∗ (Ω B )) ≥ h ( T X/B ⊗ π ∗ (Ω B )) ≥ g − , proving (3.3.1). The strictness of the first equality comes from the fact that theidentity map in H ( X, T X ⊗ Ω X ) is not induced by an element of H ( X, T
X/B ⊗ π ∗ (Ω B ).In summary, if T X does not have a genuine deformation, then X is either P or aball quotient, and in the latter case, necessarily H ( X, S Ω X ) = 0 by Proposition3.1(3). (cid:3) There are non-K¨ahler surfaces whose tangent bundles have nogenuine first order deformations. For example, let X be an Inoue surface of type S + N , [Ino74]. These are exactly the surfaces with κ ( X ) = −∞ , having no curvesand c ( X ) = c ( X ) = 0. Moreover h ( X, T X ) = h ( X, T X ) = 1 , [Ino74, Prop.3]. Let v ∈ H ( X, T X ) be a non-zero vector field. Then v has nozeroes and induces an exact sequence0 → O X → T X → O X ( − K X ) → . rom Riemann-Roch, we have χ (Ω X ) = 0, hence H ( X, Ω X ) (since H ( X, Ω X ) = H ( X, Ω X ) = 0). Thus taking cohomology of the preceeding exact sequence, ten-sorized by Ω X , we obtain0 = H ( X, Ω X ) → H ( X, T X ⊗ Ω X ) → H ( X, T X ) ≃ C , and therefore h ( X, T X ⊗ Ω X ) = 1 = h ( X, O X ) . Theorem 3.3 has the following partial strengthening
Let X be a compact K¨ahler surface. Assume that X is neither P , nor a ball quotient nor of the form P ( E ) with E a stable locally free sheaf of ranktwo over an elliptic curve. Then T X has a genuine non-obstructed deformation.Proof. By Proposition 2.6, we may assume X minimal. What remains to be provedis the following. Suppose X is one of the following.(1) κ ( X ) = 1 and c ( X ) = 0(2) X is a torus or hyperelliptic(3) X is a ruled surface over a curve B of genus g = g ( B ) ≥
1, but not ofthe form P ( E ) with E a stable locally free sheaf of rank two over an ellipticcurve.Then T X has a non-trivial non-obstructed deformation.(1) Assume first that κ ( X ) = 1. By Proposition 3.1(4), it suffices to show that h ( X, T X ⊗ Ω X ) ≥ . To do this, consider the Iitaka fibration f : X → B . Since c ( X ) = 0, the onlysingular fiber of f are multiples m i F i of elliptic curves F i ; write D = P ( m i − F i .The elliptic bundle formula now reads K X = f ∗ ( K B ⊗ L ) ⊗ O X ( D )with a torsion line bundle L . Further, there is an exact sequence0 → f ∗ ( K B ) ⊗ O X ( D ) → Ω X → K X/B ⊗ O X ( − D ) → , and therefore an inclusion T X ⊗ f ∗ K B ⊗ O X ( D ) ⊂ T X ⊗ Ω X . Thus it suffices to show H ( X, T X ⊗ f ∗ K B ⊗ O X ( D )) = 0 . Indeed, a non-zero element in the space is a morphism Ω X → f ∗ K B ⊗ O X ( D ).Composing with the inclusion f ∗ K B ⊗ O X ( D ) → Ω X yields a morphism Ω X → Ω X which is not a multiple of id. Dualizing the last exact sequence yields an inclusion f ∗ ( K B ⊗ L ∗ ) ⊗ O X ( D ) ≃ K ∗ X/B ⊗ O X (2 D ) ⊗ f ∗ ( K B ) → T X ⊗ f ∗ K B ⊗ O X ( D ) . Now H ( X, f ∗ ( K B ⊗ L ∗ ) ⊗ O X ( D )) = H ( B, K B ⊗ L ∗ ) = 0 , unless g = 1 and L not trivial, we are done except for this special case. Here weperform a finite ´etale base change ˜ B → B to trivialize L and set ˜ X = X × B ˜ B withprojection ˜ f : ˜ X → ˜ B . Then the associated line bundle ˜ L and therefore h ( ˜ X, T ˜ X ⊗ Ω X ) ≥ . hus there exists a morphism λ : Ω X → Ω X , which is not a multiple of the identity. Let µ : ˜ X → X be the projection andconsider µ ∗ ( λ ) : µ ∗ (Ω X ) → µ ∗ (Ω X ) . Via the decomposition µ ∗ (Ω X ) = µ ∗ µ ∗ (Ω X ) = Ω X ⊗ µ ∗ ( O ˜ X ) = Ω X ⊗ − s M j =0 L j , there exists a number k and a non-zero morphism ψ : Ω X → Ω X ⊗ L k . We aim to prove that k = 0; hence we obtain an endomorphism of Ω X which is nota multiple of the identity and conclude. Using the cotangent sequence, which nowreads 0 → O X ( D ) ⊗ Ω X → K X ⊗ O X ( − D ) → , the morphism ψ induces by composition a morphism ψ : O X ( D ) → K X ⊗ O X ( − D ) ⊗ f ∗ ( L k ) = f ∗ ( L ⊗ L k ) . If ψ = 0, then k = − D = 0, a contradiction. Therefore ψ and ψ induces anonvanishing map O X ( D ) → O X ( D ) ⊗ f ∗ ( L k ) , hence k = 0.(2) If X is a torus or hyperelliptic, then h ( X, T X ⊗ Ω X ) = h ( X, Ω X ⊗ Ω X ) = h ( X, S Ω X ) + h ( X, K X ) ≥ , hence we conclude again by Proposition 3.1(4).(3) Finally, let p : X → B be a ruled surface over a curve B of genus g = g ( B ) ≥ → T X/B → T X → p ∗ ( T B ) → ζ ∈ H ( X, T
T/B ⊗ p ∗ ( T ∗ B )) ≃ H ( X, − K X ). Now H ( X, − K X ) = 0, unless g = 1 and X = P ( E ) with E a stable locally free sheafof rank two on B . This is a direct consequence of the structure results of ruledsurfaces, [Har77, chap. V.2]. The latter case ruled out by assumption, we candeform the extension class ζ and obtain a deformation F of T X over X × ∆. Every F t sits in an exact sequence0 → T X/B → F t → p ∗ ( T B ) → p ∗ ( T B ) → . Since there are no non-trivial maps T X/B → p ∗ ( T B ), the sheaves F t are differentfrom T X for t = 0, and we obtain a non-trivial positive-dimensional deformation of T X .It remains to treat the case X = P ( E ) with E stable over the elliptic curve B . (cid:3) .6. Remark. Assume that X = P ( E ) with E a stable locally free sheaf of ranktwo over an elliptic curve or that X is a ball quotient with H ( X, S Ω X ) = 0. Then T X has a genuine first order deformation and one might suspect that a suitable suchdeformation is not obstructed. Then Theorem 3.5 could be stated as follows: T X has a genuine non-obstructed deformation if and only if X is neither P nor a ballquotient with H ( X, S Ω X ) = 0.4. Threefolds
The Riemann-Roch formula gives, using c ( T X ⊗ Ω X ) = − c ( X ) + 6 c ( X ), Let X be a -dimensional compact complex manifold. Then χ ( X, T X ⊗ Ω X ) = c ( X ) − c ( X ) c ( X ) . Fano threefolds.4.2. Proposition.
Let X be a Fano threefold. Then T X has a genuine non-obstructed deformation unless X = P .Proof. Since χ ( X, O X ) = 1, we have c ( X ) c ( X ) = 24 by Riemann-Roch. Hence χ ( X, T X ⊗ Ω X ) = c ( X ) − . By the classification of Fano threefolds, c ( X ) ≤
62, unless X = P . Notice furtherthat H ( X, T X ⊗ Ω X ) = H ( X, Ω X ⊗ Ω X ) = 0 , e.g., since X is rationally connected.Hence h ( X, T X ⊗ Ω X ) − h ( X, T X ⊗ Ω X ) > , unless X = P . Since q ( X ) = 0, this proves the claim. (cid:3) The arguments actually show more (having in mind that h ( X, T X ⊗ Ω X ) = 0) Let X be a smooth threefold with χ ( X, O X ) ≥ and H ( X, Ω X ) = H ( X, Ω X ) = H ( X, Ω X ⊗ Ω X ) = 0 . Assume that c ( X ) ≤ . Then T X has a genuine non-obstructed deformation. If X is rationally connected, then the first two conditions in Corollary 4.3 aresatisified, hence we obtain Let X be a smooth rationally connected threefold. Assume that c ( X ) ≤ . Then T X has a non-trivial non-obstructed deformation. In view of Proposition 4.2 it is natural to consider the case that X is ”weakFano”, i.e., − K X is big and nef. In that case, c ( X ) ≤
72 by [Pro05, Thm.1.5].In fact, for a suitable positive integer m the line bundle − mK X is spanned byglobal sections and defines a birational morphism ϕ : X → Y to a Fano Gorensteinvariety Y with at most canonical singularities such that − K X = ϕ ∗ ( − K Y ) . By Pro05, Thm.1.5], ( − K Y ) ≤
72, hence c ( X ) = ( − K X ) ≤
72. The bound 72is sharp; actually ( − K Y ) = 72 if and only if Y is either the weighted projectivespace P (3 , , ,
1) or P (6 , , , . Thus we cannot conclude directly that T X has afirst order or non-obstructed deformation. It should however be possible to classifiyall X in the range 64 ≤ c ( X ) ≤
72 and treat this cases by hand. We give oneexample, namely X = P ( O P ⊕ O P (3)) . In this case Y = P (3 , , , . For simplicity, we consider only first order deforma-tions.
Let X be the weak Fano threefold P ( O P ⊕ O P (3)) . Then T X has a genuine first order deformation. We prepare the proof by the following
Let π : X → S be a P -bundle over the smooth compact surface S .Assume that h ( S, T S ⊗ Ω S ) = 1 . Then h ( X, T X ⊗ Ω X ) = 1 + h ( X, T
X/S ⊗ Ω X ) = 1 + h ( X, − K X/S ⊗ Ω X ) . Proof.
The sequence0 → T X/S ⊗ Ω X → T X ⊗ Ω X → π ∗ ( T S ) ⊗ Ω X → → H ( X, T
X/S ⊗ Ω X ) → H ( X, T X ⊗ Ω X ) α → H ( X, π ∗ ( T S ) ⊗ Ω X ) . The sequence0 → π ∗ ( T S ) ⊗ π ∗ (Ω S ) → π ∗ ( T S ) ⊗ Ω X → π ∗ ( T S ) ⊗ K X/S → H ( X, π ∗ ( T S ) ⊗ Ω X ) = H ( S, T S ⊗ Ω S ) ≃ C . Hence it suffices to observe that α = 0. This is however clear: id : Ω X → Ω X yieldsvia α a non-zero morphism π ∗ (Ω S ) → Ω X . (cid:3) Proof of Proposition 4.5
By Proposition 2.1, applied to the exceptional section E := P ( O P ) ≃ P in X , it suffices to show that(4.6.1) H ( X, T X ⊗ Ω X ) = H ( X, T X ⊗ Ω X (log E )) . We use the exact sequence0 → T X/S ⊗ Ω X (log E ) → T X ⊗ Ω X (log E ) → π ∗ ( T S ) ⊗ Ω X (log E ) → . Using Lemma 4.6, things come down to show H ( X, T
X/S ⊗ Ω X ) = H ( X, T
X/S ⊗ Ω X (log E ))and h ( X, π ∗ ( T S ) ⊗ Ω X (log E )) = 1 . The first equation is seen by taking cohomology of the exact sequence0 → T X/S ⊗ Ω X → T X/S ⊗ Ω X (log E ) → T X/S | E → H ( E, T
X/S | E ) = H ( E, − K X/S | E ) = H ( E, N
E/X ) = 0 . he second equation follows from the observation π ∗ (Ω X ) = π ∗ (Ω X (log E )) , which is seen either by restricting to the fibers of π or by noticing that, taking π ∗ , the induced morphism O S → R π ∗ (Ω X ) is injective. Thus Equation (4.6.1) isshown and the proof of Proposition 4.5 is complete.In Subsection 4.3 we come back to P -bundles over surfaces in general. How thelater results do not yield Proposition 4.5.4.2. P -bundles. We start to study threefolds carrying a projective bundle struc-ture by studying P -bundles. Let π : X → C be a P -bundle over the smooth projective curve C . Then T X has a genuine first order deformation.Proof. Write X = P ( F ) with a locally free sheaf F of rank three on C . Let g bethe genus of C . If g = 0, then ( − K X ) = 54, hence we conclude by Corollary 4.4.To compute ( − K X ) , just use the formula − K X = O P ( F ) (3) ⊗ π ∗ (det F ∗ ) ⊗ O C ( − K C )) , see e.g. [Har77, Ex.III.8.4].Thus we will assume from now on that g ≥
1. In this case χ ( X, E nd ( T X )) ≥ → T X/C ⊗ Ω X → T X ⊗ Ω X → π ∗ ( T C ) ⊗ Ω X → H ( X, T X ⊗ Ω X ) → H ( X, π ∗ ( T C ) ⊗ Ω X ) = H ( X, π ∗ ( T C ⊗ Ω C )) ≃ C is surjective, leads to an exact sequence0 → H ( X, T
X/C ⊗ Ω X ) → H ( X, T X ⊗ Ω X ) → H ( X, π ∗ ( T C ) ⊗ Ω X ) →→ H ( X, T
X/C ⊗ Ω X ) . We will now show(1) h ( X, T
X/C ⊗ Ω X ) ≥ g ;(2) H ( X, π ∗ ( T C ) ⊗ Ω X ) = 0;(3) H ( X, T
X/C ⊗ Ω X ) = 0.Then the exact sequence yields h ( X, T X ⊗ Ω X ) > g = h ( X, O X ) , which was to be proved. Proof of (1).
Write L := O P ( F ) (1) and tensor the relative Euler sequence0 → O X → π ∗ ( F ∗ ) ⊗ L → T X/C → X to obtain an exact sequence H ( X, T
X/C ⊗ Ω X ) → H ( X, Ω X ) → H ( X, π ∗ ( F ∗ ) ⊗ L ⊗ Ω X ) . Now H ( X, π ∗ ( F ∗ ) ⊗ L ⊗ Ω X ) = 0 via the Leray spectral sequence. Further h ( X, Ω X ) = g >
0: use Hodge decomposition and H , = 0 to obtain h ( X, Ω X ) = b ( X )2 = b ( C )2 = g ( C ) . ence (1) follows. Proof of (2).
Since π ∗ (Ω X ) = Ω C , we have h ( X, π ∗ ( T C ) ⊗ Ω X ) ≥ h ( C, T C ⊗ Ω C ) = g, proving (2). Proof of (3).
This follows again by the Leray spectral sequence. (cid:3) P -bundles. In many cases the non-rigidity of the tangent bundle of a P -bundle over a surface S can be established as follows. For simplicity, we assumethat q ( X ) = H ( X, O X ) = H ( S, O S ) = 0. Let π : X → S be a P − bundle over the smooth compactsurface S with q ( S ) = 0 . If H ( X, T X ⊗ π ∗ (Ω S )) = H ( S, π ∗ ( T X ) ⊗ Ω S ) = 0 , then T X has a genuine first order deformation.Proof. We use the exact sequence(4.8.1) 0 → T X ⊗ π ∗ (Ω S ) → T X ⊗ Ω X → T X ⊗ K X/S → . From the exact sequence(4.8.2) 0 → O X = − K X/S ⊗ K X/S → T X ⊗ K X/S → π ∗ ( T S ) ⊗ K X/S → , we deduce h ( X, T X ⊗ K X/S ) = 1 . Now H ( X, T X ⊗ Ω X ) → H ( X, T X ⊗ K X/S )does not vanish: id : T X → T X induces a non-zero morphism − K X/S → T X . Henceby Sequence (4.8.1), H ( X, T X ⊗ π ∗ (Ω S ) = H ( S, π ∗ ( T X ) ⊗ Ω S )injects into H ( X, T X ⊗ Ω X ). (cid:3) Let π : X → S be a P − bundle over the smooth compact surface S with q ( S ) = 0 . If h ( S, T S ⊗ Ω S ) = 1 and if h ( X, T
X/S ⊗ π ∗ (Ω S )) ≥ , then T X has a genuine first order deformation.Proof. This is immediate, taking cohomology of0 → T X/S ⊗ π ∗ (Ω S ) → T X ⊗ π ∗ (Ω S ) → π ∗ ( T S ⊗ Ω S ) → . (cid:3) With a little more care and a slighty stronger assumption on h ( X, T
X/S ⊗ π ∗ (Ω S )), but without assumption on q ( S ), we obtain non-obstructed deformations: .10. Proposition. Let π : X → S be a P -bundle. Assume that h ( S, T S ⊗ Ω S ) = 1 and that h ( X, T
X/S ⊗ π ∗ (Ω S )) ≥ . Then T X has a genuine non-obstructed deformation.Proof. We consider the tangent bundle sequence0 → T X/S −→ T X α −→ π ∗ ( T S ) → . Since dim Ext ( π ∗ ( T S ) , T X/S ) = h ( X, T
X/S ⊗ π ∗ (Ω S )) ≥ , we obtain an at least three-dimensional family of extensions0 → T X/S β −→ E t −→ π ∗ ( T S ) → , and it suffices to show that E t T X for general t . Assume to the contrary that E t ≃ T X . Then the composed map α ◦ β : T X/S → π ∗ ( T S ) must vanish (restrict tofibers of π ). Hence β induces a morphism T X/S → T X/S which must be a multipleof the identity map. Thus we have an induced map π ∗ ( T S ) → π ∗ ( T S ), which byassumption is another multiple of the identity. Hence the space of extension istwodimensional, contradicting our dimension assumption. (cid:3) We now give a criterion for the nonvanishing of H ( X, T
X/S ⊗ π ∗ (Ω S )). Weassume for simplicity that the P -bundle X → S is actually of the form X = P ( F )with a locally free sheaf F of rank two on S ; this can always arranged by passingto a finite ´etale cover of S , see [Ele82]. Let F be a locally free sheaf of rank two on the smooth compactcomplex surface S with q ( S ) = 0 and set X = P ( F ) . Assume that h ( S, T S ⊗ Ω S ) = 1 and that c ( F ) − c ( F ) + 12 ( K S − c ( S )) ≤ − . Then T X has a genuine first order deformation. This happens e.g., when F is ω -semistable for some K¨ahler form ω , so that c ( F ) ≤ c ( F ) .Proof. Notice that π ∗ ( T X/S ) = S ( F ) ⊗ det F ∗ , thus h ( S, S ( F ) ⊗ det F ∗ ⊗ Ω S ) = h ( X, T
X/S ⊗ π ∗ (Ω S ) . Hence it suffices to show that χ ( S, S ( F ) ⊗ det F ∗ ⊗ Ω S ) ≤ − , which is equivalent by Riemann-Roch to our assumption.Notice finally that since q ( S ) = 0, then K S ≤ c ( S ) and c ( S ) = χ top ( S ) ≥ (cid:3) In the same manner, we have .12. Corollary. Let F be a locally free sheaf of rank two on the smooth compactcomplex surface S and set X = P ( F ) . Assume that h ( S, T S ⊗ Ω S ) = 1 and that c ( F ) − c ( F ) + 12 ( K S − c ( S )) ≤ − . Then T X has a genuine unobstructed deformation. The equation H ( X, T X ⊗ π ∗ (Ω S )) = 0 also holds under theassumptions H ( S, T S ⊗ Ω S ) = 0and h ( S, S ( F ∗ ) ⊗ det F ⊗ Ω S ) = 0 , again by taking cohomology of the exact sequence0 → T X/S ⊗ π ∗ (Ω ) S ) → T X ⊗ π ∗ (Ω S ) → π ∗ ( T S ⊗ Ω S ) → . In our setting X = P ( F ), we have χ ( X, E nd ( T X )) = 2 c ( F ) − c ( F ) + 23 (cid:0) K S − c ( S ) (cid:1) . So if this number is negative and if H ( X, E nd ( T X ))) = 0 , then H ( X, E nd ( T X )) = 0. The last vanishing amounts via Serre duality to h ( X, Ω X ⊗ Ω X ) ≤ h ( X, K X )) . Blow-ups.
We conclude the section by studying blow-ups. First we recollectour knowledge on blow-ups of points.
Let Y be a smooth compact complex threefold and π : X → Y be the blow-up at y with exceptional divisor E . Then T X has a genuine first orderdeformation provided one of the following holds.(1) Y is rationally connected with c ( Y ) ≤ ;(2) Y is smooth threefold with c ( Y ) ≤ and Y is uniruled.If the MRC fibration has two-dimensional (smooth) image S , suppose fur-ther that χ ( S, O S ) = 0 and that H ( S, Ω S ⊗ K S ) = 0 ;(3) T Y has a genuine first order deformation and h ( Y, T Y ⊗ Ω Y ) < ;(4) h ( Y, T Y ⊗ Ω Y ) = 1 , i.e., T Y is simple.In the cases (1) and (2) T X has even a genuine non-obstructed deformation; thesame being true in case (3) provided T Y has a non-obstructed genuine deformation.Proof. Assertion (1) is settled by Corollary 4.4; (3) by Corollary 2.7 and (4) byTheorem 2.8. Thus only (2) needs to be proven. We calculate χ ( X, E nd ( T X )) = χ ( X, T X ⊗ Ω X ) − χ ( X, O X ) == c ( X ) − χ ( X, O X ) = c ( X ) − χ ( S, O S ) ≤ − χ ( S, O S ) . Hence by our assumption χ ( S, O S ) ≥ χ ( X, E nd ( T X )) < h ( X, E nd ( T X )) − h ( X, E nd ( T X )) + h ( X, E nd ( T X )) > . Hence it suffices to show that(4.15.1) H ( X, T X ⊗ Ω X ) = 0 . y Serre duality, H ( X, T X ⊗ Ω X ) = H ( X, T X ⊗ Ω X ⊗ O X ( K X )) = H ( X, Ω X ⊗ Ω X ) . We may assume that Y is not rationally connected. Y being uniruled, we considerthe MRC fibration f : X S , [Kol96], with S not uniruled and smooth. Ifdim S = 2, Proposition 4.17 shows that H ( X, Ω X ⊗ Ω X ) = H ( S, Ω S ⊗ K S ) . If dim S = 1, then S is a smooth curve of genus at least two, and f is actually amorphism. But then, restricting to a general fiber F , a rational surface, a directcalculation shows that H ( X, Ω X ⊗ Ω X ) = 0 , proving (4.15.1). (cid:3) In the proof of the next proposition we will use the following
Let X be a normal complex algebraic variety. Then Ω [ q ] X := ( q ^ Ω X ) ∗∗ denotes the sheaf of reflexive q -forms. If X has canonical singularites and if π :ˆ X → X is a desingularization, then by [GKKP11] , Ω [ q ] X = π ∗ (Ω q ˆ X ) . Let X be a smooth projective threefold. Suppose that X isuniruled with MRC fibration f : X S to the smooth projective surface S . Thenthe pull-back f ∗ : H ( S, Ω S ⊗ K S ) → H ( X, Ω X ⊗ Ω X ) is an isomorphism.Proof. Note that dim h ( X, Ω X ⊗ Ω X ) is a birational invariant of smooth projectivemanifolds. Hence we may assume that f is a morphism. Clearly, f ∗ is injective.Running a relative MMP, we obtain a factorization X X ′ → S, where X X ′ is a sequence of relative contractions and flips, and where f ′ : X ′ → S is a Mori fiber space.We proceed with the following observations. If Y and Z are normal projectivevarieties with terminal singularities and if ϕ : Y → Z is a divisorial contraction,then ϕ ∗ ((Ω Y ⊗ Ω Y ) ∗∗ ) ⊂ (Ω Z ⊗ Ω Z ) ∗∗ . Moreover, if ϕ : Y Z is a flip, then H ( Y, (Ω Y ⊗ Ω Y ) ∗∗ ) = H ( Z, (Ω Z ⊗ Ω Z ) ∗∗ ) . Hence it suffices to consider f ′ : X ′ → S , which is equidimensional and a conicbundle outside a finite set of S , i.e., there is a finite set A ⊂ S such that f ′ : X ′ \ ( f ′ ) − ( A ) → S \ A is a conic bundle, and we need to prove that( f ′ ) ∗ : H ( S, Ω S ⊗ Ω S ) → H ( X ′ , (Ω X ′ ⊗ Ω X ′ ) ∗∗ ) s surjective. Since f ′ is a submersion outside a set of dimension at most one, Ω X ′ /S is torsion free. We will use the exact sequences0 → f ′∗ (Ω S ) → Ω X ′ → Ω X ′ /S → → f ′∗ (Ω S ) → Ω [2] X ′ → S → , where S is a torsion free sheaf with S| X ′ = ( f ′∗ (Ω S ) ⊗ Ω X ′ /S ) | X ′ and where X ′ is the regular locus of X ′ and f ′ . Now we observe that H ( X ′ , (cid:0) Ω X ′ /S ⊗ Ω [2] X ′ (cid:1) ∗∗ ) = 0and H ( X ′ , S ⊗ ( f ′ ) ∗ (Ω S )) = 0 . Indeed, both sheaves in question are negative on the general fiber of f ′ . Then theassertion follows, tensoring the first exact sequence with Ω [2] X ′ , and then the secondwith ( f ′ ) ∗ (Ω S ) and computing on X ′ . (cid:3) Proposition 4.15 suggests to proceed by induction on the Picard number ρ ( X ),performing an MMP. In this context, we notice that in a similar way as in Propo-sition 4.15, it possible to treat the other contraction of extremal rays on threefoldswhich contract a divisor E to a point. This opens a way to reduce the problem toMori fiber spaces and threefolds with nef canonical bundles (possibly singular).Blowing up curves is more complicated; we restrict ourselves to first order de-formations. Let Y be a smooth projective threefold such that h ( Y, T Y ⊗ Ω Y ) = 1 . Let π : X → Y be the blow-up of a smooth curve C ⊂ Y . Assume that (4.18.1) − K Y · C ≥ . Then H ( X, E nd ( T X )) = 0 .Proof. It suffices to show that H ( Y, π ∗ ( E nd ( T X )) = 0 . To prove this, we consider the canonical exact sequence0 → π ∗ ( E nd ( T X )) → E nd ( T Y ) → Q → , where Q is a coherent sheaf supported on C . Then things comes down to show that(4.18.2) H ( Y, Q ) = 0 . The sheaf Q appears as well in the exact sequence0 → π ∗ ( T X ⊗ Ω X ) → T Y ⊗ Ω Y → Q → , and we shall work with this sequence. Set E = π − ( C ). The normal bundle of C in Y will simply be denoted N C . Using the exact sequences0 → π ∗ ( T X ) ⊗ Ω Y → π ∗ ( T X ⊗ Ω X ) → π ∗ ( T X | E ⊗ Ω E/C ) → → π ∗ ( T X ) ⊗ Ω Y → T Y ⊗ Ω Y → N C ⊗ Ω Y → , diagram chase yields an exact sequence0 → π ∗ ( T X | E ⊗ Ω E/C ) → N C ⊗ Ω Y | C → Q → . Since h ( C, π ∗ ( T X | E ⊗ Ω E/C )) = h ( E, T X | E ⊗ Ω E/C ) == h ( E, T E ⊗ Ω E/C ) = h ( E, T
E/C ⊗ Ω E/C ) = h ( E, O E ) = 1 , Equation (4.18.2) comes therefore down to show that(4.18.3) h ( C, N C ⊗ Ω Y | C ) ≥ . This follows the stronger inequality(4.18.4) χ ( C, N C ⊗ Ω Y | C ) ≥ , which by Riemann-Roch is equivalent to6(1 − g ) + c ( N C ⊗ Ω Y | C ) ≥ . This is just our assumption via the adjunction formula. (cid:3)
Let Y be a Fano threefold such that h ( Y, T Y ⊗ Ω Y ) = 1 . Let π : X → Y be the blow-up of a smooth curve C ⊂ Y . Then H ( X, E nd ( T X )) = 0 . Proof.
It remains to treat the case that − K Y · C = 1. But then C is a smoothrational curve, and Theorem 2.8 applies. In fact, if − K Y · C = 1, then Y musthave index one. In almost all cases, − K Y is spanned, hence defines a morphism ϕ : Y → P N such that ϕ ∗ ( O P N )(1) = O Y ( − K Y ). Hence ϕ ( C ) must be a line ℓ and C → ℓ is an isomorphism. There are only two exceptional cases, [IP99, p.49] whichcan be checked by hand. (cid:3) Keeping track of the H -term in χ ( N C ⊗ Ω Y | C ) and using Serre duality, theproof of Theorem 4.18 actually shows the slightly stronger Let Y be a smooth projective threefold such that h ( Y, T Y ⊗ Ω Y ) = 1 . Let π : X → Y be the blow-up of a smooth curve C ⊂ Y . If − K Y · C + h ( N C ⊗ Ω Y | C ) ≥ , then H ( X, E nd ( T X )) = 0 .Proof. It suffices to note that in case g = 0, Theorem 2.8 applies. (cid:3) Calabi-Yau threefolds: birational morphisms and flops
We are now turning to Calabi-Yau manifolds, mostly in dimension three. To beprecise, a Calabi-Yau manifold is a simply connected projective manifold X with K X ≃ O X . Therefore T X has a genuine first order deformation if and only if H ( X, T X ⊗ Ω X ) = 0. .1. Definition. Let X be a Calabi-Yau manifold and ϕ : X → Y be a birationalmorphism to a normal projective variety Y . Then ϕ is said to be primitive ifthe relative Picard number ρ ( X/Y ) = 1 . A primitive contraction is divisorial if theexceptional locus E of ϕ has codimension . Then automatically E is an irreducibledivisor. We first collect a few known results on divisorial contractions in dimension three.
Let X be a Calabi-Yau threefold, ϕ : X → Y a primitivedivisorial birational map contracting the irreducible divisor E . Then the followingholds.(1) K Y ≃ O Y and Y has canonical singularities; we say that Y is a weakCalabi-Yau variety.(2) If dim ϕ ( E ) = 0 , then − K E is ample, so E is a (possibly singular) del Pezzosurface.(3) If dim ϕ ( E ) = 1 , then C := ϕ ( E ) is a smooth curve. The map ϕ | E definesa conic bundle structure on E .Proof. We refer to the fundamental papers of Wilson [Wil92], [Wil93], [Wil94],[Wil97], [Wil99]. (cid:3)
A primitive contraction might also be a small birational contraction, in whichcase the exceptional locus is a finite union of smooth rational curves. This casewill be treated at the end of this section. Or we have dim Y = 1 or 2; then ϕ is a K3-fibration, an abelian fibration or an elliptic fibration. These cases willbe treated in a different paper. Note also that it is expected that a Calabi-Yauthreefold X with ρ ( X ) ≥ N E ( X ) is locallyrational polyhedral and that nef line bundles on Calabi-Yau manifolds should besemiample; see e.g. the above cited papers of Wilson or the survey [LOP18].Recall that, given be a normal complex algebraic variety X , thenΩ [ q ] X := ( q ^ Ω X ) ∗∗ is the sheaf of reflexive q -forms. Further, if X has canonical singularites and if π : ˆ X → X is a desingularization, then by [GKKP11],Ω [ q ] X = π ∗ (Ω q ˆ X ) . Let X be a Calabi-Yau manifold, ϕ : X → Y a primitive birationalmorphism to a normal projective variety, contracting an irreducible divisor E . Thenthe following holds.(1) The reflexive cotangent sheaf Ω [1] Y is H -polystable for any ample divisor H on X. (2) If Ω [1] Y is not H -stable for some ample divisor H , then there exists a quasi-´etale cover η : Z → Y such that Z decomposes into a product an abelianvariety and (possibly singular) Calabi-Yau varieties and irreducible sym-plectic varieties (in the sense of [GKP16] ) (possibly Z is just an abelianvariety or there is no abelian factor).(3) Assume n = 3 and that Ω [1] Y is not H -stable for some ample divisor H .Then dim ϕ ( E ) = 1 and there exists a quasi-´etale cover η : Z → Y such hat either Z is an abelian threefold or Z = B × S with Z an elliptic curveand S a K3-surface (possibly with rational double points).Proof. Assertion (1) has been shown by Guenancia [Gue16], while Assertion (2)is the main result in [HP19]. Note that either Z is abelian or Z has a productdecomposition with at least two factor, since Calabi-Yau or irreducible symplecticvarieties have stable tangent sheaves (even after quasi-´etale cover).So it remains to prove (3). By (2), it is clear that Z is either abelian or a product B × S with B an elliptic curve and S a K3-surface. So suppose that dim ϕ ( E ) = 0,i.e., Y has a single singular points y , a case which we will rule out. Note that Z can have at most finitely many singular point, hence if Z = C × S , hence S is smooth. Thus Z always smooth. We consider the orbifold Euler characteristic e orb ( Y ), see e.g. [Bla96, 2.14]. By [Bla96, Cor. p.26], e orb ( Z ) = deg( η ) · e orb ( Y ) . Now e orb ( Z ) = e top ( Z ) = 0. On the other hand, e orb ( Y ) = e top ( Y ) − (1 − y ) , where G y is the group attached to the quotient singularity y , which is absurdsince e top ( Y ) is an integer. (cid:3) The case n = 3 and E is contracted to a curve is more complicated; here wesimply state what is needed later. Let X be a Calabi-Yau threefold, ϕ : X → Y a primitive birationalmorphism to a normal projective variety, contracting an irreducible divisor E tothe smooth curve C . If C ≃ P (or, more generally, the genus g ( C ) is even) and if E is a P -bundle over C , then Ω [1] Y is H -stable for any ample divisor H .Proof. We argue as in the previous proof, part (3). Still we have e orb ( C × S ) = 0,even if S is singular. This is immediate from the definition of the orbifold Eulernumber, since C is a elliptic curve. Since E is a P -bundle, Y is locally of the form∆ × ( A ) with ∆ a small disc in C . Thus for all y ∈ C , the group G y has order 2and by definition of the orbifold Euler number, e orb ( Y ) = e top ( Y ) − (1 − e top ( C )) . Since C ≃ P by assumption, we conclude that e top ( Y ) = 1 , hence b ( Y ) is odd. This is impossible: either use the Leray spectral sequence todeduce that H ( X, C ) = H ( Y, C ). But b ( X ) is even due to Hodge decompositionon X . (cid:3) If C is an elliptic curve in the setting of Lemma 5.4, then weconclude that χ top ( Y ) = 0, hence χ top ( X ) = 4. Let X be a Calabi-Yau manifold, ϕ : X → Y a primitivedivisorial birational map contracting the irreducible divisor E . Suppose that Ω [1] Y is H -stable for some ample divisor H . Then dim H ( X, Ω X ⊗ T X ) = dim H ( X, O X ( E ) ⊗ Ω X ⊗ T X ) = 1 . roof. The equation dim H ( X, Ω X ⊗ T X ) = 1 follows from the stability of T X . Toobtain the second equation, observe that any non-zero section of Ω X ( E ) ⊗ T X canbe seen as a non-zero morphism λ : Ω X ( − E ) → Ω X . Taking direct images, this gives a morphism ϕ ∗ ( λ ) : ϕ ∗ (Ω X ( − E )) → ϕ ∗ (Ω X ) . Since ϕ ∗ (Ω X ( − E )) ⊂ Ω [1] Y , with equality outside S = ϕ ( E ), the map ϕ ∗ ( λ ) gives amorphism µ ′ : (Ω [1] Y ) | Y \ S → (Ω [1] Y ) | Y \ S . Since S has codimension at least 2 in Y, and since Ω [1] Y is reflexive, µ ′ extends to amorphism µ : Ω [1] Y → Ω [1] Y . By construction, ϕ ∗ ( λ ) = µ ◦ ι , where ι : ϕ ∗ (Ω X ( − E )) → Ω [1] Y is the inclusion map. Now Ω [1] Y is stable for any ample line bundle on Y by Lemma5.3. In particular, Ω [1] Y is simple, thus there exists a complex number c = 0 suchthat µ = c id . It follows that λ | X \ E = c id Ω X \ E , and therefore λ = c id Ω X ◦ κ, where κ : Ω X ( − E ) → Ω X is the inclusion map. This proves the assertion. (cid:3) With n = dim X , suppose in the setting of Proposition 5.6 addi-tionally that H n − ( E, Ω X ⊗ T X | E ) = 0 . Then H ( X, T X ⊗ Ω X ) = 0 . Proof.
We show equivalently that H n − ( X, Ω X ⊗ T X ) = 0 . Consider the cohomologysequence H n − ( X, Ω X ⊗ T X ) → H n − ( E, Ω X ⊗ T X | E ) →→ H n ( X, O X ( − E ) ⊗ Ω X ⊗ T X ) → H n ( X, Ω X ⊗ T X ) → . From Proposition 5.6 and Serre duality, we know thatdim H n ( X, O X ( − E ) ⊗ Ω X ⊗ T X ) = dim H n ( X, Ω X ⊗ T X ) . This yields the claim. (cid:3)
Let X be a Calabi-Yau manifold, ϕ : X → Y a primitive divisorialbirational map contracting the irreducible divisor E . If H ( E, T E ) = 0 , and if Ω [1] Y is H -stable for some ample divisor H , then H ( X, Ω X ⊗ T X ) = 0 . roof. We aim to apply Proposition 5.7 and verify that H n − ( E, Ω X ⊗ T X ) | E ) = H ( E, T X ⊗ Ω X ⊗ K E ) = 0 . In fact, T X ⊗ Ω X | E ⊗ K E contains - via the (co)tangent sequence and the adjunctionformula - the subsheaf T E ⊗ N ∗ E ⊗ K E ≃ T E , hence the nonvanishing follows from our assumption H ( E, T E ) = 0. (cid:3) In case E is smooth, Theorem 5.8 also follows from Corollary 2.3. Suppose that dim X = 3. The condition H ( E, T E ) = 0 holds inthe following cases E smooth).(1) E be a del Pezzo surface and K E ≥ . (2) E is a rational ruled surface.(3) E is a ruled surface over an elliptic curve, and the vector field on the ellipticcurve lifts to E .(4) E is a ruled surface over C , and E = P ( V ) with a rank 2-vector bundle V such that h ( V ∗ ⊗ V ) ≥
2, since by the relative Euler sequence, the relativevector fields are computed by h ( E, T
E/C ) = h ( C, V ∗ ⊗ V ) − H ( E, T E ) = 0, things get more involved, we restrict ourselves to dimensionthree. The key is the following Let X be a Calabi-Yau threefold, ϕ : X → Y be a birationalmorphism to a normal compact complex (Moishezon) space Y , whose exceptionallocus is a smooth rational curve C . Then H ( X, T X ⊗ Ω X ) = 0 .Proof. We argue by contradiction and assume to the contrary that H ( X, T X ⊗ Ω X ) = 0 . By a theorem of Laufer [Lau81, Thm.4.1], the normal bundle N C = N C/X has thefollowing form(1) N C = O C ( − ⊕ O C ( − N C = O C ⊕ O C ( − N C = O C (1) ⊕ O C ( − . Moreover, y = ϕ ( C ) is a hypersurface singularity.We claim that(5.10.1) h ( Y, R ϕ ∗ ( T X ⊗ Ω X )) ≥ . To prove Claim (5.10.1), we use the inequality h ( Y, R ϕ ∗ ( T X ⊗ Ω X )) ≥ h ( C, T X ⊗ Ω X | C ) . In fact, all cohomology classes in H ( C, T X ⊗ Ω X | C ) extend to all infinitesimalneighborhoods, since H ( C, ( N ∗ C ) ⊗ k ⊗ T X ⊗ Ω X | C ) = 0 for all k . Since h ( C, T X ⊗ Ω X | C ) = 5in Case (2) and h ( C, T X ⊗ Ω X | C ) = 7 n Case (3), we need only to consider Case (1). In this case, h ( C, T X ⊗ Ω X | C ) = 4 . Here we need to consider the second infinitesimal neighborhood C , defined by theideal I C , and use the inequality h ( Y, R ϕ ∗ ( T X ⊗ Ω X )) ≥ h ( C , T X ⊗ Ω X | C ) . The right hand side appears in the cohomology sequence0 → H ( C, N ∗ C ⊗ T X ⊗ Ω X | C ) → H ( C , T X ⊗ Ω X | C ) α → H ( C, T X ⊗ Ω X ) →→ H ( N ∗ C ⊗ T X ⊗ Ω X ) → H ( C , T X ⊗ Ω X ) | C ) . By [Lau81, Thm.3.2], a sufficiently small neighborhood of C ⊂ X is biholomorphicto a small neighborhood of the zero section of the normal bundle N C , hence α issurjective. Since h ( N ∗ C ⊗ T X ⊗ Ω X ) = 4 , Claim (5.10.1) also holds in Case (1).Since ϕ is small, the sheaf ϕ ∗ ( T X ⊗ Ω X ) is reflexive, hence ϕ ∗ ( T X ⊗ Ω X ) = ( T Y ⊗ Ω [1] Y ) ∗∗ =: F . Since we assume H ( X, T X ⊗ Ω X ) = 0, the Leray spectral sequence yields H ( Y, F ) = 0;further, the edge morphism µ : E , → E , is injective, hence by Claim (5.10.1),(5.10.2) h ( Y, F ) ≥ h ( Y, R ϕ ∗ ( T X ⊗ Ω X )) ≥ . We now compute H ( Y, F ) in a different way to obtain a contradiction. By Serreduality, H ( Y, F ) ≃ Ext ( F , O Y ) . We will use the Grothendieck spectral sequence, with E -terms E p,q = H p ( Y, E xt q ( F , O Y )) , converging to Ext p + q ( F , O Y ). Notice that H p ( Y, E xt ( F , O Y )) = H p ( Y, H om ( F , O Y )) = H p ( Y, F ) , since F ≃ F ∗ . Thus, introducing the edge morphism δ : E , → E , , the spectralsequence together with the vanishing E , = 0 yields H ( Y, F ) ≃ ker δ. Since Ext ( F , O Y ) = H ( Y, F ) = 0, necessarily E , = 0, i.e., E , = im δ , so themorphism δ is surjective. Since E , = H ( Y, F ) ≃ ker δ , we obtain(5.10.3) 2 h ( Y, F ) = dim E , = h ( Y, E xt ( F , O Y )) . The sheaf E xt ( F , O Y ) being supported on y , we need to compute its length at y . Recalling that y is a hypersurface singularity, Ω Y = Ω [1] Y by a theorem of Kunz[Kun86, Cor. 9.8], hence F = H om (Ω Y , Ω Y ) . or our local computation, we may assume Y itself to be a hypersurface in C . Weconsider the cotangent sequence0 → N ∗ Y/ C → Ω C | Y → Ω Y → , which after possibly shrinking Y reads0 → O Y → O ⊕ Y → Ω Y → . Tensoring by T Y gives(5.10.4) 0 → T Y → T ⊕ Y → T Y ⊗ Ω Y → . The sheaf T Y ⊗ Ω Y is clearly torsion free, as seen directly from the exact sequence(5.10.4), but possibly not reflexive. To see the difference, introduce the quotient R = F / ( T Y ⊗ Ω Y ) which is supported on y . Dualizing the resulting exact sequence0 → T Y ⊗ Ω X → F → R → E xt ( R , O Y ) → E xt ( F , O Y ) → E xt ( T Y ⊗ Ω Y , O Y ) . Since R is supported on y , we have E xt ( R , O Y ) = 0, and thus it suffices toestimate h ( Y, E xt ( T Y ⊗ Ω Y , O Y ). Dualizing the exact sequence (5.10.4) yieldsthe exact sequence(Ω Y ) ⊕ α → Ω Y → E xt ( T Y ⊗ Ω Y , O Y ) → E xt (Ω Y , O Y ) ⊕ . Since α is simply the mapid ⊗ ( df ) ∗ : Ω Y ⊗ O ⊕ Y → Ω Y ⊗ O Y = Ω Y , where f is the equation for Y ⊂ C , it followscoker α = O ⊕ y . Dualizing the cotangent sequence and using the same argument gives also E xt (Ω Y , O Y ) = O y . Hence in total, h ( Y, E xt ( T Y ⊗ Ω Y , O Y )) ≤ , and consequently by Equation (5.10.3), h ( Y, F ) = 12 h ( Y, E xt ( F , O Y )) ≤ . This contradicts Inequality (5.10.2), completing the proof of Proposition 5.10. (cid:3)
Let X be a Calabi-Yau threefold and C ⊂ X be a smooth rationalcurve with normal bundle N C . Assume either that N C = O C ( − ⊕ O C ( − or that N C = O C ⊕ O C ( − and that C is an isolated curve in the sense of [Rei83] , i.e., C does not move in X . Then H ( X, T X ⊗ Ω X ) = 0 .Proof. By Proposition 5.10, it suffices to prove that C is contractible. In the firstcase this is Grauert’s criterion [Gra62]; in the second case we apply a theorem ofReid [Rei83, Cor. 5.6]. (cid:3) We now apply Corollary 5.11 to compute H ( X, T X ⊗ Ω X ). .12. Theorem. Let ϕ : X → Y be a primitive contraction of the Calabi-Yauthreefold with exceptional divisor E . Assume that dim ϕ ( E ) = 0 and that one ofthe following conditions holds.(1) E is smooth;(2) K E ≥ ;(3) E is normal, rational and contains a smooth contractible rational curve,e.g., E carries a birational contraction of an extremal ray;(4) E is normal and irrational;(5) E is non-normal.Then H ( X, Ω X ⊗ T X ) = 0 . Proof. (1) If E is smooth and if K E ≥
6, the claim follows from Theorem 5.8,combined with Remark 5.9, since E is a del Pezzo surface. If E is smooth with K E ≤
7, then we may choose a ( − − curve C ⊂ E . Then, using the normalbundle sequence for C ⊂ E ⊂ X , it is immediate that C has normal bundle N C/X = O C ( − ⊕ O C ( − U = U ⊂ X of E := E and a deformation π : U → ∆over the unit disc and a divisor
E ⊂ U such that X = π − (0), such that E ∩ X = E and such that - after possibly shrinkling ∆ - the divisor E t = E ∩ X t is smooth.Since the normal bundle N E/X is negative, so does N E t /U t , hence E t is contractible.Thus, we obtain a family φ t : U t → V t contracting fiberwise the divisor E t . Noticethat(5.12.1) K U t ≃ O U t for all t. In fact, we may choose U t such that E t is a deformation retract of U t . Hence the restriction H ( U t , Z ) → H ( E t , Z )is an isomorphism. Since H q ( E t , O E t ) = 0, the restrictionPic( U t ) → Pic( E t )is an isomorphism, too. Since K X | E ≃ O E , it follows that K U t | E t ≡ , hence K U t | E t ≃ O E t . Hence Equation (5.12.1) follows.We assume now that K E ≥
6. Using the inclusion N ∗ E t /U t ⊂ Ω U t , and observing N ∗ E t /U t ≃ O E t ( − K E t ), we have an inclusion H ( E t , T E t ) = H ( E t , T E t ⊗ N ∗ E t /U t ⊗ O E t ( K E t )) →→ H ( E t , T U t ⊗ Ω U t | E t ⊗ O E t ( K Et )) . Since K E t = K E ≥
6, we conclude Observe that, using the conormal sheaf sequenceand the triviality of K U t that H ( E t , T U t ⊗ Ω U t | E t ⊗ O E t ( K E t )) = = 0 . y semi-continuity, H ( E, T U ⊗ Ω U | E ⊗ O E ( K E )) = 0 . By Serre duality, H ( E, T X ⊗ Ω X | E ) = 0 , and we conclude by Lemma 5.7.(3) Suppose now that K E ≤ E is normal and rational (but singular). Thenby [HW81], E is either a rational surface with only ADE singularities or an ellipticcone; hence in our case, the first alternative holds. Let C ⊂ E be a smoothcontractible rational curve; C is a Q -divisor in E , but possibly not Cartier. Theconormal sheaf N ∗ C/E is of the form N ∗ C/E = O C ( a ) ⊕ T with a ≥ T a torsion sheaf, supported on C ∩ Sing( E ). Consider the conormalsheaf sequence 0 → N ∗ E/X | C → N ∗ C/X → N ∗ C/E → . Since N ∗ E/X | C = − K E | C is ample, either a = 1 and N ∗ C/X = O C (1) ⊕ O C (1),or a = 0, T is supported on one point with one-dimensional stalk and N ∗ C/X = O C (2) ⊕ O C . In both cases we conclude by Corollary 5.11.(4) If E is normal with a non-rational singularity, then, as already mentioned, E isan elliptic cone. In this case, H ( E, T E ) = 0, and we conclude by Theorem 5.8.(5) Suppose finally that E is non-normal. In case y is not a hypersurface singu-larity, K E = 7 by [Gro97a, Thm. 5.2]. This case is settled by (2). Thus we mayassume that y is a hypersurface singularity. Then1 ≤ K E ≤ H ( E, T E ) = 0. Then we con-clude by Theorem 5.8. Actually, we have more informations on E , [Gro97a, p.201]and [Wil97, p.620]. In fact, if K E = 1, then E is a hypersurface in the weightedprojective space P (3 , , ,
1) with explicit equation, in case K E the surface E is ahypersurface in P (2 , , ,
1) and if K E = 3, then E is a cubic. The normalization e E in the last two cases are Hirzebruch surfaces, while the normalization in the firstcase is P .We give two examples. First assume that K E = 3; so E is a cubic surface in P .The nonnormal locus N (with the structure given by the conductor ideal) is a linein E . Then the tangent sheaf sequence reads0 → T E → T P | E → I N ⊗ N E/ P → T E → T P | E is contained in I N ⊗ N E/ P ).Now h ( E, T P | E ) = 15 and h ( E, N E/ P ) = h ( E, O E (3)) = 19. Since the image ofthe restriction map H ( E, O E (3)) → H ( N, O N (3)) is surjective, it has dimension5, hence h ( E, I N ⊗ N E/ P ) = 14, hence h ( E, T E ) = 0.Second, consider the case K E = 1, hence e E ≃ P . Let η : e E → E be thenormalization map; the non-normal locus N of E is a line. We consider the subcasewhen the preimage e N is a smooth conic; it might also be a line pair or a doubleline. The degree of η | e N → N is two. Then, H ( e E, T e E ) = H ( e N , T e E | e N ) and the atter contains H ( e N , T e N ). Thus there are three vectors fields on e E , tangent to e N and one of them is invariant under the double cover e N → N (by considering T e N → η ∗ ( T N )). (cid:3) The remaning open case in Theorem 5.12 is the following: E is normal, but singular, rational with only ADE-singularities containing no con-tractible smooth rational curve and K E ≤
5. Then necessarily ρ ( E ) ≤
2, otherwise E carries a birational contraction of an extremal ray. The way to treat this opencase would be to show that there is a global deformation X t of X (not only a localdeformation of a neighborhood of E ), such that E deforms to a smooth del Pezzosurface E t and then to conclude again by semicontinuity.We now turn to the case that E is contracted to a curve C . We already knowthat T X has a first order deformation if E is smooth with H ( E, T E ) = 0, e.g., if C ≃ P . The case that C is an elliptic curve is easy as well, provided c ( X ) = 4: Let X be a Calabi-Yau threefold, ϕ : X → Y be a primitivecontraction with smooth exceptional divisor E such that C = ϕ ( E ) is a curve. If g ( C ) = 1 and if c ( X ) = 4 , then H ( X, Ω X ⊗ T X ) = 0 . Proof.
By Lemma 5.7 and Remark 5.5, it suffices to show H ( E, Ω X ⊗ T X | E ) = 0 . By Serre duality, this is equivalent to H ( E, Ω X ⊗ T X | E ⊗ K E ) = 0 . Using the tangent bundle sequence, it suffices to show that H ( E, Ω X ⊗ T E ⊗ K E ) = H ( E, Ω X ⊗ Ω E ) = 0 . By the cotangent sequence, we obtain an exact sequence H ( E, T E ) → H ( E, Ω X ⊗ Ω E ) →→ H ( E, Ω E ⊗ Ω E ) δ → H ( E, N ∗ E ⊗ Ω E ) ≃ H ( E, T E ) . Since we may assume H ( E, T E ) = 0 by Theorem 5.8 and since clearly H ( E, T E ) =0 , Riemann-Roch shows, using g ( C ) = 1 , that H ( E, T E ) = 0 . Since H ( E, Ω E ⊗ Ω E ) = 0 , our claim follows. (cid:3) If g ( C ) ≥ , these simple arguments do no longer work (pro-vided Ω [1] Y is H -stable). The difficulty is that dim H ( E, T E ) = 6( g − , assuming H ( E, T E ) = 0, whereas dim H ( E, Ω E ⊗ Ω E ) = 3( g − . One would need to show that the connecting map δ : H ( E, Ω E ⊗ Ω E ) → H ( E, N ∗ E ⊗ Ω E ) ≃ H ( E, T E ) s not injective. Actually, this statement can still be sharpened. In fact, assumingas always that H ( E, T E ) = 0, then the sequence0 → T E/C → T E → ϕ ∗ ( T C ) → H ( E, Ω X | E ⊗ ϕ ∗ ( T C )) = 0 . This comes down to show that the canonical morphism ǫ : H ( E, Ω E ⊗ ϕ ∗ ( T C )) → H ( E, N ∗ E ⊗ Ω E )is not injective. Notice also that both vector spaces have the same dimension3( g ( C ) − g ( C ) ≥ E is singular by a moresophisticated method. Let X be a Calabi-Yau threefold, ϕ : X → Y a primitivedivisorial contraction contracting the exceptional divisor E to a curve C . Then H ( X, T X ⊗ Ω X ) = 0 unless (possibly) E is one of the following surfaces(1) E is a normal singular surface and all singular fibers are double lines;(2) E is a non-normal surface, C ≃ P , but the normalization of E is irrational.Proof. Recall that p := ϕ | E : E → C is a conic bundle over the smooth curve C . We will also use the finer classification of E , due to Wilson [Wil92], [Wil93],[Wil97]. In fact, consider a singular fiber E c of p which is a line pair E c = C ∪ C (with C = C ). Then there are three possibilities: • E is smooth along C ∪ C ; • E is normal along along C ∪ C and has an A n -singularity at the intersec-tion point C ∩ C and is smooth elsewhere; • E is non-normal, the normalisation is a P -bundle over a smooth curve ˜ C which is a double cover over C and unramified over c . The singular locusof E meets C exactly in the intersection point C ∩ C .(A) Assume first that the general fiber of p is irreducible, but p has a reduciblefiber E c , and fix an irreducible component B ≃ P . Then the conormal sheafsequence 0 → N ∗ E/X | B → N ∗ B/X → N ∗ B/E → E · B = − N ∗ B/X is either O B (1) ⊕O B (1)or O B (2) ⊕ O B . In both cases B is contractible; in the first case by Grauert’scriterion [Gra62], in the second case we apply again [Rei83, Cor. 5.6], using ourassumption that the general fiber of p is irreducible, hence B does not move. Wealso use the sequence0 → N ∗ E c /E | B ≃ O B → N ∗ B/E → N ∗ B/E c → N ∗ B/E ≃ O B (1) or that N ∗ B/E ≃ O B ⊕ O x , where O x is asheaf supported on the singularity of E c with one-dimensional stalk at x . Once weknow that B is contractible, we conclude by Proposition 5.10.(B) Next we consider the case g ( C ) ≥
1. By [Gro97b, 1.2,1.3], [Wil97, p.631 ff]there exists a flat family π : X → ∆of Calabi-Yau threefolds X t over the unit disc with the following properties X = π − (0) ≃ X ;(2) there is a relative crepant contraction Φ : X → Y over ∆ with Φ | X = ϕ ;(3) ϕ t = Φ | X t : X t → Y t is small;(4) ϕ t contracts the deformations of the finitely many fibers of E → C whichdeform to X t .By Theorem 5.18, H ( X t , T X t ⊗ Ω X t ) = 0, hence H ( X, T X ⊗ Ω X ) = 0 by semi-continuity.(C) We next consider the case that C ≃ P and that all fibers of p : E → C areirreducible. If E is smooth, p is a P -bundle, and we are done by Lemma 5.4,Theorem 5.8 in connection with Remark 5.9. If E is singular, then E is normal andby (A), we may assume that the only singular fibers are double lines. This case isruled out by assumption.(D) Assume finally that the general fiber of p is reducible. Then there is a doublecover ˜ C → C such that the fiber product ˜ E → ˜ C is a P -bundle. If g ( C ) >
0, thenby [Wil97, p.635] and [Gro97b, p.294], the general deformation X t of X carries asmall contraction. Hence H ( X t , T X t ⊗ Ω X t ) = 0by Theorem 5.18, and we conclude by semicontinuity.If g ( ˜ C ) = 0, then by [Wil97, p.635], the non-normal locus of E is a ( − , − g ( ˜ C ) > g ( C ) = 0 isruled out by assumption. (cid:3) The exceptional case (b) might be ruled out as follows. Consideragain a general deformation X t of X . Then E deforms to a rational surface which isa conic bundle over P ; see [Wil97, p.635]. One might expect that not all singularfibers are double lines, hence H ( X t , T X t ⊗ Ω X t ) = 0 and we conclude again bysemicontinuity. Thus the difficulty is that in all deformation E t ⊂ X t , we land inthe exceptional case (1).We finally consider small contractions ϕ : X → Y . Let X be a Calabi-Yau threefold and ϕ : X → Y be a smallcontraction. Then H ( X, T X ⊗ Ω X ) = 0 .Proof. By [Lau81], [Pin83], [Mor85, Thm.5.5], see also [Fri86], any fiber F of ϕ hasthe form F = ∪ C j with smooth rational curves C j . All C j have normal bundle N C/X = O C ( a ) ⊕ O C ( b )with ( a, b ) = ( − , − , (0 , − , (1 , − . Moreover, there is at most one curve with( a, b ) = (1 , − C j whose normal bundle is not of type (1 , −
3) - unless F = C is irreducible withnormal bundle of type (1 , − (cid:3) Any small contraction ϕ : X → Y gives rise to a flop h : X X + with a(smooth) Calabi-Yau threefold X + , [Kol89, 2.4]. We finally relate the deformationsof T X to those of T X + . .19. Proposition. Let h : X X + be a flop of the Calabi-Yau threefold X ,induced by the small contraction ϕ : X → Y. Then H ( X, T X ⊗ Ω X ) ≃ H ( X, T X + ⊗ Ω X + ) . Moreover, every positive-dimensional deformation of T X over (a germ of ) an irre-ducible reduced complex space S induces canonically a positive-dimensional defor-mation of T X + over S. Proof.
In order to show the first claim, it suffices to show that H X, T X ⊗ Ω X ) ≃ H ( X, T X + ⊗ Ω X + ) . The flop being induced by a small contraction ϕ : X → Y , we let ϕ + : X + → Y denote the associated flopped small morphism. The Leray spectral sequence gives H ( X, T X ⊗ Ω X ) ≃ H ( Y, ϕ + ∗ ( T X ⊗ Ω X ))and H ( X + , T X + ⊗ Ω X + ) ≃ H ( Y, ϕ + ∗ ( T X + ⊗ Ω X + )) . The sheaves ϕ ∗ ( T X ⊗ Ω X ) and ϕ + ∗ ( T X + ⊗ Ω X + ) are reflexive and isomorphic outsidea finite set. Hence they are isomorphic on all of Y , and the claim follows.As to the second claim, let E be a flat deformation of T X over X × S. The locally freesheaf E induces a coherent sheaf E + over X + × S , such that ( E + ) | X + ×{ } ≃ T X + . Inparticular, ( E + ) | X + ×{ } is locally free and so does ( E + ) | X + ×{ s } for small s . Hence E + is flat over S. (cid:3) Let X and X ′ be birationally equivalent Calabi-Yau threefolds.Then H ( X, T X ⊗ Ω X ) ≃ H ( X, T X ′ ⊗ Ω X ′ ) . Moreover, every positive-dimensional deformation of T X induces canonically a posi-tive-dimensional deformation of T X ′ . Proof.
It suffices to remark that any birational map between (smooth) Calabi-Yauthreefolds is a sequence of flops, [Kol89]. (cid:3) Higher Dimensions
We finish the paper with some results in higher dimensions: hypersurfaces inprojective space, and products, the latter being important for the correct set-up ofQuestion 1.6.
Let X ⊂ P n +1 , n ≥ , be a smooth hypersurface of degree d ≥ .Then T X has a genuine first order deformation.Proof. Since H ( X, O X ) = 0, it suffices to show that H ( X, T X ⊗ Ω X ) = 0. Weuse the cohomology sequence0 → H ( X, T X ⊗ Ω X ) → H ( X, T P n +1 | X ⊗ Ω X ) → H ( X, Ω X ( d )) →→ H ( X, T X ⊗ Ω X ) . The Euler sequence in combination with the vanishing H ( X, Ω X (1)) = 0 (use thecotangent sequence for X ⊂ P n +1 ) yields h ( X, T P n +1 | X ⊗ Ω X ) = 1 . hus H ( X, Ω X ( d )) ⊂ H ( X, T X ⊗ Ω X )and it suffices to observe that H ( X, Ω X ( d )) = 0, which is clear since Ω P n +1 (2) isgenerated by global sections. (cid:3) Concerning products, we first consider the case of two factors.
Let X = X × X be a projective manifold with dim X j ≥ .Then H ( X, E nd ( T X )) = 0 if and only if the following conditions hold for j = 1 , .(1) H ( X j , T X j ) = 0 ;(2) q ( X j ) = 0 ;(3) H ( X j , T X j ⊗ Ω X j ) = H ( X j , E nd ( T X j )) = 0 .In particular, dim X j ≥ for j = 1 , .Proof. Let p j : X → X j denote the projections. Then T X ⊗ Ω X ≃ (cid:0) p ∗ ( T X ⊗ Ω X ) (cid:1) ⊕ (cid:0) p ∗ T X ⊗ p ∗ Ω X (cid:1) ⊕⊕ (cid:0) p ∗ T X ⊗ p ∗ Ω X (cid:1) ⊕ (cid:0) p ∗ ( T X ⊗ Ω X ) (cid:1) . Using the K¨unneth formula, a direct computation shows that the conditions (1),(2)and (3) are equivalent to h ( X, T X ⊗ Ω X ) = q ( X ), i.e., h ( X, E nd ( T X )) = 0. Inparticular, dim X j = 1 is impossible. (cid:3) Inductively, we obtain
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Thomas Peternell, Mathematisches Institut, Universit¨at Bayreuth, 95440 Bayreuth,Germany
E-mail address : [email protected]@uni-bayreuth.de