  We consider integral models of Hilbert modular varieties with Iwahori level
structure at primes over p, first proving a Kodaira-Spencer isomorphism that
gives a concise description of their dualizing sheaves. We then analyze fibres
of the degeneracy maps to Hilbert modular varieties of level prime to p and
deduce the vanishing of higher direct images of structure and dualizing
sheaves, generalizing prior work with Kassaei and Sasaki (for p unramified in
the totally real field F). We apply the vanishing results to prove flatness of
the finite morphisms in the resulting Stein factorizations, and combine them
with the Kodaira-Spencer isomorphism to simplify and generalize the
construction of Hecke operators at primes over p on Hilbert modular forms
(integrally and mod p). The vanishing results also provide an ingredient for
forthcoming work with Sasaki, associating Galois representations to mod p
Hilbert modular eigenforms of non-paritious weight (for p ramified in F).
