Gaussian maps and generic vanishing I: Subvarieties of abelian varieties

doi:10.1017/CBO9781107416000.019

18 - Gaussian maps and generic vanishing I: Subvarieties of abelian varieties

Published online by Cambridge University Press: 05 January 2015

G. Pareschi

Edited by

Christopher D. Hacon ,

Mircea Mustaţă and

Mihnea Popa

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G. Pareschi: Affiliation:
Università di Roma
Christopher D. Hacon: Affiliation:
University of Utah
Mircea Mustaţă: Affiliation:
University of Michigan, Ann Arbor
Mihnea Popa: Affiliation:
University of Illinois, Chicago

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Abstract

The aim of this paper is to present an approach to Green–Lazarsfeld'sgeneric vanishing combining Gaussian maps and the Fourier–Mukai transform associated with the Poincaré line bundle. As an application, we prove the generic vanishing theorem for all normal Cohen–Macaulay subvarieties of abelian varieties over an algebraically closed field.

Dedicated to my teacher, Rob Lazarsfeld, on the occasion of his 60th birthday

1 Introduction

We work with irreducible projective varieties on an algebraically closed field of any characteristic, henceforth called varieties. The contents of this paper are:

(1) A general criterion expressing the vanishing of the higher cohomology of aline bundle on a Cohen–Macaulay variety in terms of certain first-order conditions on hyperplane sections (Theorem 2). Such conditions involve Gaussian maps and the criterion is a generalization of well-known results on hyperplane sections of K3 and abelian surfaces.

(2) Using a relative version of the above, we prove the vanishing of higher direct images of Poincaré line bundles of normal Cohen–Macaulay subvarieties of abelian varieties (Theorem 5). As is well known, this is equivalent to Green–Lazarsfeld's generic vanishing, a condition satisfied by all irregular compact Kahler manifolds [5]. This implies in turn a Kodaira-type vanishing for line bundles which are restrictions to normal Cohen–Macaulay subvarieties of abelian varieties, of ample line bundles on the abelian variety (Corollary 6).

Concerning point (2), it should be mentioned that at present we are not able to extend this approach efficiently to the general generic vanishing theorem (GVT), i.e., for varieties mapping to abelian varieties, even for smooth projective varieties over the complex numbers (where it is well known by the work of Green and Lazarsfeld). This will be the object of further research. However, concerning possible extensions of the general GVT to singular varieties and/or to positive characteristic, one should keep in mind the work of Hacon and Kovacs [8] where – by exploiting the relation between GVT and the Grauert–Riemenschneider vanishing theorem – they show examples of failure of the GVT for mildly singular varieties (over C) and even smooth varieties (in characteristic p > 0) of dimension ≥ 3, with a (separable) generically finite map to an abelian variety.

Type: Chapter
Information: Recent Advances in Algebraic Geometry
A Volume in Honor of Rob Lazarsfeld’s 60th Birthday
, pp. 370 - 404

DOI: https://doi.org/10.1017/CBO9781107416000.019 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2015

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