GEOMETRY AND TOPOLOGY OF EXTERNAL AND SYMMETRIC PRODUCTS OF VARIETIES

We give a brief overview of recent developments on the calculation of generating series for invariants of external products of suitable coefficients (e.g., constructible or coherent sheaves, or mixed Hodge modules) on complex quasi-projective varieties.

Kodaira–Saito vanishing via Higgs bundles in positive characteristic

The goal of this paper is to give a new proof of a special case of the Kodaira–Saito vanishing theorem for a variation of Hodge structure on the complement of a divisor with normal crossings. The proof does not use the theory of mixed Hodge modules, but instead reduces it to a more general vanishing theorem for semistable nilpotent Higgs bundles, which is then proved by using some facts about Higgs bundles in positive characteristic.

Thom–Sebastiani Theorems for Filtered 𝒟-Modules and for Multiplier Ideals

We give a proof of the Thom–Sebastiani type theorem for holonomic filtered D-modules satisfying certain good conditions (including Hodge modules) by using algebraic partial microlocalization. By a well-known relation between multiplier ideals and V-filtrations of Kashiwara and Malgrange, the argument in the proof implies also a Thom–Sebastiani type theorem for multiplier ideals, which cannot be deduced from an already known proof of the Thom–Sebastiani theorem for mixed Hodge modules (since the latter gives only the information of graded pieces of multiplier ideals). We also sketch a more elementary proof of the Thom–Sebastiani type theorem for multiplier ideals (as communicated to us by M. Mustaţǎ), which seems to be known to specialists, although it does not seem to be stated explicitly in the literature.

ON THE IRREGULAR HODGE FILTRATION OF EXPONENTIALLY TWISTED MIXED HODGE MODULES

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$f$.