  We show that the mod p cohomology of a smooth projective variety with
semistable reduction over K, a finite extension of Qp, embeds into the
reduction modulo p of a semistable Galois representation with Hodge-Tate
weights in the expected range (at least after semisimplifying, in the case of
the cohomological degree > 1). We prove refinements with descent data, and we
apply these results to the cohomology of unitary Shimura varieties, deducing
vanishing results and applications to the weight part of Serre's conjecture.
