GV-sheaves, Fourier-Mukai transform, and Generic Vanishing

Details GV-sheaves, Fourier-Mukai transform, and Generic Vanishing GV-sheaves, Fourier-Mukai transform, and Generic Vanishing

2006-08-04

arxiv.org/abs/math/0608127v4

Mathematics

Algebraic Geometry

28 pages; many more corrections and improved statements with respect to previous versions; especially fixed some inaccuracies

Scientific paper

We use homological methods to establish a formal criterion for Generic Vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon and the first author, 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 of the form $K_X + L$ with $L$ a nef line bundle, 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 generates various other generic vanishing results, by reduction to standard vanishing theorems. We further use the formal criterion in order to address examples related to generic vanishing on higher rank moduli spaces (on curves and on some threefold Calabi-Yau fiber spaces).

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