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Published online by Cambridge University Press:  17 May 2013

Department of Mathematics, Statistics & Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607,
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794,


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We extend most of the results of generic vanishing theory to bundles of holomorphic forms and rank-one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin associated with irregular varieties. Our main tools are Saito’s mixed Hodge modules, the Fourier–Mukai transform for $\mathscr{D}$-modules on abelian varieties introduced by Laumon and Rothstein, and Simpson’s harmonic theory for flat bundles. In the process, we also discover two natural categories of perverse coherent sheaves.

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