Bott vanishing for algebraic surfaces

Abstract

For William Fulton on his eightieth birthdayA smooth projective variety X over a field is said to satisfy Bott vanishing if H j (X, Ω i X ⊗ L) = 0 for all ample line bundles L, all i ≥ 0, and all j > 0. Bott proved this when X is projective space. Danilov and Steenbrink extended Bott vanishing to all smooth projective toric varieties; proofs can be found in [4,7,28,15]. What does Bott vanishing mean? It does not have a clear geometric interpretation in terms of the classification of algebraic varieties. But it is useful when it holds, as a sort of preprocessing step, since the vanishing of higher cohomology lets us compute the spaces of sections of various important vector bundles. Bott vanishing includes Kodaira vanishing as a special case (where i equals n := dim X), but it says much more.For example, any Fano variety that satisfies Bott vanishing must be rigid, since H 1 (X, T X) = H 1 (X, Ω n−1 X ⊗ K * X ) = 0 for X Fano. So Bott vanishing holds for only finitely many smooth complex Fano varieties in each dimension. Even among rigid Fano varieties, Bott vanishing fails for quadrics of dimension at least 3 and for Grassmannians other than projective space [7, section 4]. As a result, Achinger, Witaszek, and Zdanowicz asked whether a rationally connected variety that satisfies Bott vanishing must be a toric variety [1, after Theorem 4].In this paper, we exhibit several new classes of varieties that satisfy Bott vanishing. First, we answer Achinger-Witaszek-Zdanowicz's question: there are non-toric rationally connected varieties that satisfy Bott vanishing, since Bott vanishing holds for the quintic del Pezzo surface (Theorem 2.1). Over an algebraically closed field, a quintic del Pezzo surface is isomorphic to the moduli space M 0,5 of 5-pointed stable curves of genus zero. It is the only rigid del Pezzo surface that is not toric: del Pezzo surfaces of degree at least 5 are rigid, and those of degree at least 6 are toric. (The quintic del Pezzo surface also does not have a lift of the Frobenius endomorphism from Z/p to Z/p 2 , a property known to imply Bott vanishing [7], [1, Proposition 7.1.4].) In view of this example, there is a good hope of finding more Fano or rationally connected varieties that satisfy Bott vanishing.We also consider varieties that are not rationally connected, with most of the paper devoted to K3 surfaces. Bott vanishing holds for abelian varieties over any field: it reduces to Kodaira vanishing, since the tangent bundle is trivial. On the other hand, Riemann-Roch shows that Bott vanishing fails for all K3 surfaces of degree less than 20 (Theorem 3.1). But recent work of Ciliberto-Dedieu-Sernesi and Feyzbakhsh [9,14] implies: Bott vanishing holds for all K3 surfaces of degree 20 or at least 24 with Picard number 1 (Theorem 3.2). Version 2 of this paper on the arXiv gave a more elementary proof, not using Feyzbakhsh's work on Mukai's program (reconstructing a K3 surface from a curve), but here we give a short proof using her work. Surprisingly, Bott vanishing fails in degree 22.More stron...

Remarks on the positivity of the cotangent bundle of a K3 surface

“…As explained in the above reference, by Kodaira vanishing, one need only treat the case p = 1. The problem is essentially solved in [Tot20], but let us show how one can use our above results to easily obtain Bott vanishing for K3 surfaces (S, O S (1)) of Picard number one with O S (1) 2 > 75. Indeed, using the upper bound for the nef slope α n,2t from Proposition 3.2, we obtain that if O S (1) 2 > 75, the divisor D = 3L + H is ample on E = P(Ω 1 S ).…”

Bott vanishing for elliptic surfaces

“…Totaro gives an answer in [16] for Achinger-Witaszek-Zdanowicz's question [1,after Theorem 4] that Bott vanishing holds for the quintic del Pezzo surfaces which are non-toric rationally connected varieties. Sebastián Torres generalizes this example by giving in [17] that Bott vanishing holds for every stable GIT quotient of (P 1 ) n by the action of P GL 2 , over an algebraically closed field of characteristic zero.…”

“…Sebastián Torres generalizes this example by giving in [17] that Bott vanishing holds for every stable GIT quotient of (P 1 ) n by the action of P GL 2 , over an algebraically closed field of characteristic zero. In [16], there are also investigations for irrationally connected varieties. Bott vanishing fails for all K3 surfaces of degree less than 20 or at least 24 with Picard number 1 [16,Theorem 3.2], which is obtained by using recent work of Ciliberto-Dedieu-Sernesi and Feyzbakhsh [7,9].…”

“…In [16], there are also investigations for irrationally connected varieties. Bott vanishing fails for all K3 surfaces of degree less than 20 or at least 24 with Picard number 1 [16,Theorem 3.2], which is obtained by using recent work of Ciliberto-Dedieu-Sernesi and Feyzbakhsh [7,9].…”

“…The geometric properties of a K3 surface X leading to this failure is the existence of elliptic curves of low degree on X or a (possibly singular) Fano 3−fold containing the X as a hyperplane section. Theorems 5.1, 6.1 and 6.2 in [16] give geometric interpretation for this failure: the existence of certain special type of singular fibers of the elliptic fibration plays an important role in whether H 1 (X, Ω 1 X ⊗ A) is zero for an ample line bundle A which is the key point for Bott Vanishing. These results inspire the research in this paper.…”