Divisorial Cohomology Vanishing on Toric Varieties

TL;DR

This paper advances the understanding of cohomology vanishing for divisorial sheaves on toric varieties, providing refined theorems, new vanishing results, and criteria for Cohen-Macaulayness, with implications for algebraic geometry.

Contribution

It introduces a refined Kawamata-Viehweg type theorem, a new vanishing theorem for specific divisors, and a criterion for maximal Cohen-Macaulay divisorial sheaves on toric varieties.

Findings

Refined Kawamata-Viehweg vanishing theorem for toric varieties

New vanishing theorem for divisors with nef inverse and small Kodaira dimension

Criterion for divisorial sheaves to be maximal Cohen-Macaulay

Abstract

This work discusses combinatorial and arithmetic aspects of cohomology vanishing for divisorial sheaves on toric varieties. We obtain a refined variant of the Kawamata-Viehweg theorem which is slightly stronger. Moreover, we prove a new vanishing theorem related to divisors whose inverse is nef and has small Kodaira dimension. Finally, we give a new criterion for divisorial sheaves for being maximal Cohen-Macaulay.