Abstract

We prove a few cases of a conjecture on the invariance of cohomological support loci under derived equivalence by establishing a concrete connection with the related problem of the invariance of Hodge numbers. We use the main case in order to study the derived behavior of fibrations over curves.

Dedicated to Rob Lazarsfeld on the occasion of his 60th birthday, with warmth and gratitude

1 Introduction

This paper is concerned with the following conjecture made in [11] on the behavior of the non-vanishing loci for the cohomology of deformations of the canonical bundle under derived equivalence. We recall that given a smooth projective X these loci, more commonly called cohomological support loci, are the closed algebraic subsets of the Picard variety defined as

Vi(ωX) := {α | Hi(X, ωX ⊗ α) ≠ 0} ⊆ Pic0(X).

All varieties we consider are defined over the complex numbers. We denote by D(X) the bounded derived category of coherent sheaves Db(Coh(X)).

Conjecture 1 ([11]) Let X and Y be smooth projective varieties with D(X) ≃ D(Y) as triangulated categories. Then

Vi(ωX)0 ≃ Vi(ωY)0 for all i ≥ 0,

where Vi(ωX)0 denotes the union of the irreducible components of Vi(ωX) passing through the origin, and similarly for Y.

We refer to [10] and [11] for a general discussion of this conjecture and its applications, and of the cases in which it has been known to hold (recovered below as well). The main point of this paper is to relate Conjecture 1 directly to part of the well-known problem of the invariance of Hodge numbers under derived equivalence; we state only the special case we need.

Conjecture 2Let X and Y be smooth projective varieties withD(X) ≃ D(Y).

Then

h0,i(X) = h0,i(Y) for all i ≥ 0.

Our main result is the following:

Theorem 3Conjecture 2 implies Conjecture 1. More precisely, Conjecture 1 for a given i is implied by Conjecture 2 for n − i, where n = dim X.