Takahiro Saito (RIMS, Kyoto University)

Title: A description of monodromic mixed Hodge modules

Abstract: The irregular Hodge filtration is expected to be a ``Hodge filtration'' of a twisted de Rham cohomology of a holomorphic function that appears as a mirror of a Fano manifold. In general, the irregular Hodge filtrations are complicated and it is difficult to deal with them. Note that the twisted de Rham cohomology can be expressed in terms of the Fourier-Laplace transform of a certain regular D-module. One reason that makes the irregular Hodge filtration difficult is that the Fourier-Laplace transform is irregular in general. I try to make everything clear in the simple case: ``monodromic".

For a complex manifold X, regular monodromic D-modules on X\times C have the important property that ``their Fourier-Laplace transforms are regular again". For a mixed Hodge module whose underlying D-module is monodromic (which is called a monodromic mixed Hodge module), it is natural to expect that its Hodge filtration and the irregular Hodge filtration of its Fourier-Laplace transform are ``easy". In fact, in my papers: [1] and [2], I clarified the structure of the Hodge filtrations of monodromic mixed Hodge modules, described the irregular Hodge filtrations of the Fourier-Laplace transforms concretely, and proved that they have good properties. I think my result is a good step toward understanding general (irregular) Hodge filtrations.

In this talk, in the first half, I will introduce the theory of mixed Hodge modules and the way from classical Hodge theory to it. In the second half, I will explain my results on the description of monodromic mixed Hodge modules.

[1] Takahiro Saito, A description of monodromic mixed Hodge modules, J. Reine Angew. Math. 786 (2022), 107–153. MR4434749

[2] Takahiro Saito, The Hodge filtrations of monodromic mixed Hodge modules and the irregular Hodge filtrations, 2022. arXiv:2204.13381.