The failure of Kodaira vanishing for Fano varieties, and terminal singularities that are not Cohen-Macaulay

TL;DR

This paper demonstrates the failure of Kodaira vanishing on smooth Fano varieties in positive characteristic and constructs the first examples of terminal singularities that are not Cohen-Macaulay, including a 3-dimensional example in characteristic 2.

Contribution

It provides the first known examples of terminal singularities that are not Cohen-Macaulay and shows Kodaira vanishing can fail in positive characteristic.

Findings

Kodaira vanishing fails on smooth Fano varieties in characteristic p > 0

Constructs the first terminal singularities that are not Cohen-Macaulay

Provides a 3-dimensional terminal singularity in characteristic 2 that is not Cohen-Macaulay

Abstract

We show that the Kodaira vanishing theorem can fail on smooth Fano varieties of any characteristic p > 0. Taking cones over some of these varieties, we give the first examples of terminal singularities which are not Cohen-Macaulay. By a different method, we construct a terminal singularity of dimension 3 (the lowest possible) in characteristic 2 which is not Cohen-Macaulay.