  We prove that the generic part of the mod l cohomology of Shimura varieties
associated to quasi-split unitary groups of even dimension is concentrated
above the middle degree, extending our previous work to a non-compact case. The
result applies even to Eisenstein cohomology classes coming from the locally
symmetric space of the general linear group, and has been used in joint work
with Allen, Calegari, Gee, Helm, Le Hung, Newton, Taylor and Thorne to get good
control on these classes and deduce potential automorphy theorems without any
self-duality hypothesis. Our main geometric result is a computation of the
fibers of the Hodge-Tate period map on compactified Shimura varieties, in terms
of similarly compactified Igusa varieties.
