The goal of this paper is to show that the cohomology of compac
t
unitary Shimura varieties is concentrated in the middle deg
ree and torsion-free,
after localizing at a maximal ideal of the Hecke algebra sati
sfying 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.