Kawamata-Viehweg vanishing fails for log del Pezzo surfaces in characteristic 3

TL;DR

This paper demonstrates that the Kawamata-Viehweg vanishing theorem does not hold for certain log del Pezzo surfaces in characteristic three, revealing limitations of vanishing theorems in positive characteristic.

Contribution

It constructs explicit counterexamples of klt del Pezzo surfaces in characteristic three where Kawamata-Viehweg vanishing fails, and shows related non-Cohen-Macaulay singularities.

Findings

Kawamata-Viehweg vanishing fails in characteristic 3 for specific surfaces

Existence of non-Cohen-Macaulay Kawamata log terminal threefold singularities in characteristic 3

Counterexamples challenge the general applicability of vanishing theorems in positive characteristic

Abstract

We construct a klt del Pezzo surface in characteristic three violating the Kawamata-Viehweg vanishing theorem. As a consequence we show that there exists a Kawamata log terminal threefold singularity which is not Cohen-Macaulay in characteristic three.