On the generic part of the cohomology of compact unitary Shimura varieties

TL;DR

This paper proves that the cohomology of certain compact unitary Shimura varieties is concentrated in the middle degree and free of torsion under generic conditions, using geometric analysis of the Hodge-Tate period map.

Contribution

It establishes the concentration and torsion-freeness of cohomology for compact unitary Shimura varieties under generic conditions, and develops foundational geometric results on the Hodge-Tate period map.

Findings

Cohomology is concentrated in the middle degree.

Cohomology is torsion-free after localization.

Comparison of fibers of the Hodge-Tate period map with Igusa varieties.

Abstract

The goal of this paper is to show that the cohomology of compact unitary Shimura varieties is concentrated in the middle degree and torsion-free, after localizing at a maximal ideal of the Hecke algebra satisfying a suitable genericity assumption. Along the way, we establish various foundational results on the geometry of the Hodge-Tate period map. In particular, we compare the fibres of the Hodge-Tate period map with Igusa varieties.