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

TL;DR

This paper establishes vanishing theorems for birational morphisms of threefolds in positive and mixed characteristics, extending classical results to singular varieties in algebraic geometry.

Contribution

It proves a relative Kawamata Viehweg vanishing theorem and a Grauert Riemenschneider theorem for threefolds with log canonical singularities in positive characteristic.

Findings

Proves vanishing theorems for threefolds in characteristic p > 5.

Extends Grauert Riemenschneider theorem to singular threefolds in positive characteristic.

Applications to the study of singularities and their depths.

Abstract

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.