Description

We introduce the category of irregular mixed Hodge modules consisting of possibly irregular holonomic D-modules which can be endowed in a canonical way with a filtration, called the irregular Hodge filtration. Mixed Hodge modules with their Hodge filtration naturally belong to this category, as well as their twist by the exponential of any meromorphic function. This category is stable by various standard functors, which produce many more filtered objects. The irregular Hodge filtration satisfies the E₁-degeneration property with respect to any projective morphism. This generalizes some results previously obtained by H. Esnault, J.-D. Yu and the author. We also show that, modulo a condition on eigenvalues of monodromies, any rigid irreducible holonomic D-module on the complex projective line underlies an irregular pure Hodge module. In a chapter written jointly with Jeng-Daw Yu, we make explicit the case of irregular mixed Hodge structures, for which we prove in particular a Thom-Sebastiani formula.