Abstract
AbstractWe 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.
Highlights
The results in this paper can be divided into two groups: (1) vanishing and dimension results, which we prove for arbitrary mixed Hodge modules, and where results from Saito’s theory are crucially needed; (2) linearity results, which we prove only for certain Hodge modules, but where harmonic theory for flat line bundles suffices for the proofs
The graded pieces grFk DRA(M) of the de Rham complex associated with a filtered D-module (M, F) underlying a mixed Hodge module form a class of GV-objects which is closed under Grothendieck duality
Recall that Laumon [19] and Rothstein [30] have extended the Fourier–Mukai transform to D-modules. Their Fourier–Mukai transform takes bounded complexes of coherent algebraic D-modules on A to bounded complexes of algebraic coherent sheaves on A ; we briefly describe it following the presentation in [19, Section 3], which is more convenient for our purpose
Summary
In order to obtain a generic Nakano-type vanishing statement similar to (D), or statements for cohomological support loci of rank-one local systems, we apply Theorem 1.1 to the direct image of the trivial Hodge module on an irregular variety under the Albanese map. Note that this t-structure is different from the dual standard t-structure that appears in the generic vanishing theory of topologically trivial line bundles [28]. Simple examples show that this result is optimal; see Section 5.2
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