Irregular Hodge theory

Abstract

We introduce a category of possibly irregular holonomic D-modules which can be endowed in a canonical way with an irregular Hodge filtration. Mixed Hodge modules with their Hodge filtration naturally belong to this category, as well as their twist by $\exp \phi$ for any meromorphic function $\phi$. This category is stable by various standard functors, which produce many more filtered objects. The irregular Hodge filtration satisfies the $E_1$-degeneration property by a projective morphism. This generalizes some results proved by Esnault-Sabbah-Yu arxiv:1302.4537 and Sabbah-Yu arxiv:1406.1339. We also show that those rigid irreducible holonomic D-modules on the complex projective line whose local formal monodromies have eigenvalues of absolute value one, are equipped with such an irregular Hodge filtration in a canonical way, up to a shift of the filtration. 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.

V3: 69 pages. An error in Section 7.b corrected and Section 7.b rewritten. Appendix B added. Various improvements. V4: 126 pages, Major revision: (1) title changed; (2) A simplification suggested by T. Mochizuki; (3) Chapter 3 added, written in collaboration with Jeng-Daw Yu; (4) various other improvements; V5: Final version to be published in Mem. SMF vol. 156

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