Harvard-MIT Algebraic Geometry Seminar: Intersection Cohomology of p-adic Shimura Varieties

In this talk, we investigate the relationship of the intersection cohomology of the minimal compactification of a Shimura variety with a structure coming from the incarnation of the Shimura variety as a p-adic manifold. Namely, we study how the intersection cohomology interacts with Mantovan’s filtration, or the filtration coming from excision with respect to certain locally closed strata of the Shimura variety defined by isogeny classes of p-divisble groups. We accomplish this by describing the intersection cohomology as a certain Hecke operator applied to a sheaf on Bun_{G}, the moduli stack of G-bundles of the Fargues-Fontaine curve, extending results of this form for the compactly supported and regular cohomology. The aforementioned filtration can then be understood via excision applied to the Harder-Narasimhan stratification of this sheaf on Bun_{G}. While the motivation for this construction stems from number theory and the categorical local Langlands program, we employ various interesting technical tools in the realm of perfect/rigid analytic algebraic geometry to carry it out and we will try to emphasize some of these tools in this talk. This is joint with Ana Caraiani and Mingjia Zhang.

https://researchseminars.org/talk/harvard-mit-ag-seminar/135/