GV-sheaves, Fourier-Mukai transform, and generic vanishing

@article{Pareschi2006GVsheavesFT,
  title={GV-sheaves, Fourier-Mukai transform, and generic vanishing},
  author={Giuseppe Pareschi and Mihnea Cristian Popa},
  journal={American Journal of Mathematics},
  year={2006},
  volume={133},
  pages={235 - 271}
}
We prove a formal criterion for generic vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon, but in the context of an arbitrary Fourier-Mukai correspondence. For smooth projective varieties we apply this to deduce a Kodaira-type generic vanishing theorem for adjoint bundles associated to nef line bundles, and in fact a more general generic Nadel-type vanishing theorem for multiplier ideal sheaves. Still in the context of the Picard variety, the same method… 
View PDF on arXiv
93 Citations
Highly Influential Citations
8
Background Citations
14
Methods Citations
6
Results Citations
2

93 Citations

Gaussian maps and generic vanishing I: subvarieties of abelian varieties

We present an approach to Green-Lazarsfeld's generic vanishing combining gaussian maps and the Fourier-Mukai transform associated to the Poincar\`e line bundle. As an application we prove the Generic…

Strong generic vanishing and a higher-dimensional Castelnuovo–de Franchis inequality

We extend to manifolds of arbitrary dimension the Castelnuovo-de Franchis inequality for surfaces. The proof is based on the theory of Generic Vanishing and higher regularity, and on the…

Injectivity theorem for pseudo-effective line bundles and its applications

We formulate and establish a generalization of Kollár’s injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár’s torsion-freeness,…

Generic vanishing and minimal cohomology classes on abelian varieties

We establish a—and conjecture further—relationship between the existence of subvarieties representing minimal cohomology classes on principally polarized abelian varieties, and the generic vanishing…

Generic vanishing and the geometry of irregular varieties in positive characteristic

Vanishing theorems play a paramount role in modern birational geometry. Over fields of characteristic zero, the classical vanishing theorems of Kodaira, Nakano and KawamataViehweg show that under…

Cohomological rank functions on abelian varieties

Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of $\mathbb Q$-twisted (complexes of) coherent sheaves on abelian…

Generic vanishing filtration and perverse objects in derived categories of coherent sheaves

This is a mostly expository paper, intended to explain a very natural relationship between two a priori distinct notions appearing in the literature: Generic Vanishing in the context of vanishing…

Derived equivalence and non-vanishing loci

The purpose of this note is to propose and motivate a conjectu re on the behavior of cohomological support loci for topologically trivial line bundles under derived equivalence, to verify it in the…

Regularity on Abelian Varieties III: Relationship with Generic Vanishing and Applications

We describe the relationship between the notions of M-regular sheaf and GV-sheaf in the case of abelian varieties. The former is a natural strengthening of the latter, and we provide an algebraic…

Fourier-Mukai Transforms and Bridgeland Stability Conditions on Abelian Threefolds

We show that the construction of Bayer, Bertram, Macri and Toda gives rise to a Bridgeland stability condition on a principally polarized abelian threefold with Picard rank one by establishing their…
...

References

SHOWING 1-10 OF 63 REFERENCES

Generic vanishing and minimal cohomology classes on abelian varieties

We establish a—and conjecture further—relationship between the existence of subvarieties representing minimal cohomology classes on principally polarized abelian varieties, and the generic vanishing…

Fourier-Mukai transforms for K3 and elliptic fibrations

Given a non-singular variety with a K3 fibration π : X → S we construct dual fibrations π : Y → S by replacing each fibre Xs of π by a two-dimensional moduli space of stable sheaves on Xs. In certain…

Lectures on Vanishing Theorems

1 Kodaira's vanishing theorem, a general discussion.- 2 Logarithmic de Rham complexes.- 3 Integral parts of Q-divisors and coverings.- 4 Vanishing theorems, the formal set-up.- 5 Vanishing theorems…

Regularity on Abelian Varieties III: Relationship with Generic Vanishing and Applications

We describe the relationship between the notions of M-regular sheaf and GV-sheaf in the case of abelian varieties. The former is a natural strengthening of the latter, and we provide an algebraic…

Singularities of theta divisors and the birational geometry of irregular varieties

The purpose of this paper is to show how the generic vanishing theorems of M. Green and the second author can be used to settle several questions and conjectures concerning the geometry of irregular…

Regularity on abelian varieties I

We introduce the notion of Mukai regularity ($M$-regularity) for coherent sheaves on abelian varieties. The definition is based on the Fourier-Mukai transform, and in a special case depending on the…

A conjecture of Beauville and Catanese revisited

Abstract.A theorem of Green, Lazarsfeld and Simpson (formerly a conjecture of Beauville and Catanese) states that certain naturally defined subvarieties of the Picard variety of a smooth projective…

Higher obstructions to deforming cohomology groups of line bundles

Our purpose is to study the cohomological properties of topologically trivial holomorphic line bundles on a compact Kahler manifold. (See [GLl, GL2] for our prior work on this topic.) Let M be a…

A ∞ -structures, Brill-noether Loci and the Fourier-mukai Transform

The goal of this paper is to show how the techniques of A∞-categories can be applied to the study of variation of cohomology spaces of coherent sheaves under deformations. Our starting point is the…

Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville

Introduction 389 w Notation and conventions 392 w Deformations of cohomology groups 392 w Generic vanishing criteria for topologically trivial line bundles 397 w A Nakano-type generic vanishing…
...