Constructible Sheaves and Filtered D-modules

Speaker: Bhargav Bhatt
Affiliation: University of Michigan
03/29/22

The Riemann-Hilbert correspondence relates constructible sheaves to D-modules on complex algebraic varieties. The theory of mixed Hodge modules equips the D-module attached to a constructible sheaf of geometric origin with a natural "Hodge" filtration; the associated graded of this filtration plays a central role in many applications of this theory in algebraic geometry.

In this talk, I will explain why the Hodge filtration is in fact automatic in the p-adic world: one can functorially attach a filtered D-module to any constructible sheaf on an algebraic variety over a p-adic field. I will also illustrate how one might use this functor by deducing Kollár's vanishing theorem for higher direct images of the canonical bundle from the BBDG decomposition theorem. (Joint work in progress with Jacob Lurie.)