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The goal of this paper is to show that the cohomology of compact unitaryShimura varieties is concentrated in the middle degree and torsion-free, afterlocalizing at a maximal ideal of the Hecke algebra satisfying a suitablegenericity assumption. Along the way, we establish various foundational resultson the geometry of the Hodge-Tate period map. In particular, we compare thefibres of the Hodge-Tate period map with Igusa varieties.

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