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

On a vanishing theorem for birational morphisms of threefolds in positive and mixed characteristics