Intersection Cohomology of Igusa Stacks

Abstract

We study the intersection cohomology of minimally compactified Shimura varieties of PEL type AC using Igusa stacks and the work of Fargues-Scholze. More precisely, we construct a sheaf on the moduli stack of $G$-bundles on the Fargues-Fontaine curve, which recovers this intersection cohomology after applying a Hecke operator in the sense of geometric Langlands. We show that this sheaf has several desirable properties; for example, it is Verdier self-dual and perverse. This leads to several applications to intersection cohomology, including a version of the Mantovan product formula, as well as torsion-vanishing and Eichler-Shimura relations. Along the way, we investigate the interaction between Baily-Borel and Newton stratifications on minimally compactified Igusa stacks, and we study perverse t-structures on stratified v-stacks.