  We prove a relative Kawamata Viehweg vanishing type theorem for birational
morphisms. We use this to prove a Grauert Riemenschneider theorem over log
canonical threefolds without zero dimensional log canonical centers, in residue
characteristic p greater than five. In large enough residue characteristics, we
prove a Grauert Riemenschneider theorem over threefold log canonical
singularities with standard coefficients. These vanishing theorems can also be
used to study the depth of log canonical singularities at non log canonical
centers, as well as the singularities of the log canonical centers themselves.
The former simplifies joint work with F. Bernasconi and Z. Patakfalvi, and the
latter appears in joint work with Q. Posva.
