ON THE IRREGULAR HODGE FILTRATION OF EXPONENTIALLY TWISTED MIXED HODGE MODULES

Abstract

Given a mixed Hodge module$\mathcal{N}$and a meromorphic function$f$on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module$\mathcal{N}\otimes \mathcal{E}^{f}$, which extends the construction of Esnaultet al.($E_{1}$-degeneration of the irregular Hodge filtration (with an appendix by Saito),J. reine angew. Math.(2015), doi:10.1515/crelle-2014-0118). We show the strictness of the push-forward filtered${\mathcal{D}}$-module through any projective morphism${\it\pi}:X\rightarrow Y$, by using the theory of mixed twistor${\mathcal{D}}$-modules of Mochizuki. We consider the example of the rescaling of a regular function$f$, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss–Manin systems of$f$in terms of the Harder–Narasimhan filtration of the Kontsevich bundles associated with$f$.