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arXiv · 1511.00176

Irregular Hodge theory

Claude Sabbah
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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 $\expφ$ for any meromorphic function $φ$. 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.

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Claude Sabbah. 2018-06-20. Irregular Hodge theory. https://doi.org/10.24033/msmf.464

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