We study the behavior of cohomological support loci of the canonical bundle
under derived equivalence of smooth projective varieties. This is achieved by
investigating the derived invariance of a generalized version of Hochschild
homology. Furthermore, using techniques coming from birational geometry, we
establish the derived invariance of the Albanese dimension for varieties having
non-negative Kodaira dimension. We apply our machinery to study the derived
invariance of the holomorphic Euler characteristic and of certain Hodge numbers
for special classes of varieties. Further applications concern the behavior of
particular types of fibrations under derived equivalence.Comment: Small fixes and substantial expository improvements. 22 page
Based on work of R. Lazarsfeld and M. Popa, we use the derivative complex associated to the bundle of the holomorphic p-forms to provide inequalities for all the Hodge numbers of a special class of irregular compact Kähler manifolds. For 3-folds and 4-folds we give an asymptotic bound for all the Hodge numbers in terms of the irregularity. As a byproduct, via the BGG correspondence, we also bound the regularity of the exterior cohomology modules of bundles of holomorphic p-forms.
We prove that any Fourier-Mukai partner of an abelian surface over an algebraically closed field of positive characteristic is isomorphic to a moduli space of Gieseker-stable sheaves. We apply this fact to show that the Fourier-Mukai set of canonical covers of hyperelliptic and Enriques surfaces over an algebraically closed field of characteristic greater than three is trivial. These results extend to positive characteristic earlier results of Bridgeland-Maciocia and Sosna.
Dedicated to Rob Lazarsfeld on the occasion of his sixtieth birthday, with warmth and gratitude.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.
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