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.
"On the generic part of the cohomology of compact unitary Shimura varieties." Ann. of Math. (2) 186 (3) 649 - 766, November 2017. https://doi.org/10.4007/annals.2017.186.3.1