  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.
