Generic vanishing theory via mixed Hodge modules (Q2849813)

From the abstract: ``We extend the dimension and strong linearity 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 to irregular varieties. Our main tools are Saito's mixed Hodge modules, the Fourier-Mukai transform for D-modules on abelian varieties introduced by Laumon and Rothstein, and Simpson's harmonic theory for flat bundles. In the process, we discover two natural categories of perverse coherent sheaves, one on the Picard variety, and the other on the moduli space of Higgs line bundles or that of line bundles with flat connection.NEWLINENEWLINEThe motivations and the aims of work under review have been conveniently surveyed by the second author inside the introduction to his recent preprint [``Holonomic \( {\mathcal {D} } \)-modules on abelian varieties'', \url{arXiv:1307.1937}].NEWLINENEWLINEThe author writes: In a sequence of papers on the {generic vanishing theorem} \textit{M. Green} and \textit{R. Lazarsfeld} [Invent. Math. 90, 389--407 (1987; Zbl 0659.14007)], \textit{D. Arapura} [Bull. Am. Math. Soc., New Ser. 26, No. 2, 310--314 (1992; Zbl 0759.14016)] and \textit{C. Simpson} [J. Ann. Sci. soc. Norm. Super. (4) 26, No. 3, 361--401 (1993; Zbl 0798.14005)], analyzed the loci NEWLINE\[NEWLINE S_m^{p,q}(X) = \{ L \in \mathrm{Pic}^0(X) \,\, | \,\, \dim H^q \bigl( X, \Omega _X^p \otimes L \bigr) \geq m \} \subseteq \mathrm{Pic}^0(X), NEWLINE\]NEWLINE for \(X\) a projective (or compact Kähler) complex manifold. Among other things, they showed that each irreducible component of \(S_m^{p,q}(X)\) is a translate of a subtorus by a point of finite order; and they obtained bounds on the codimension in the most interesting cases (\(p=0\) and \(p = \dim X\)). These bounds imply for example that when the Albanese mapping of \(X\) is generically finite over its image, all higher cohomology groups of \(\omega_X \otimes L\) vanish for a generic line bundle \(L \in \mathrm{Pic}^0(X)\). \textit{C. D. Hacon} [J. Reine Angew. Math. 575, 173--187 (2004; Zbl 1137.14012)] pointed out that the codimension bounds can be interpreted as properties of certain coherent sheaves on abelian varieties, and then reproved them using the Fourier-Mukai transform. His method applies particularly well to those coherent sheaves that occur in the Hodge filtration of a mixed Hodge module; based on this observation, Popa and I generalized all of the results connected with the generic vanishing theorem (in the projective case) to Hodge modules of geometric origin on abelian varieties. As a by-product, we also obtained results about certain regular holonomic \({\mathcal {D} }\)-modules on abelian varieties, namely those that can be realized as direct images of structure sheaves of smooth projective varieties with nontrivial first Betti number.