Vanishing Theorems on Toric Varieties

TL;DR

This paper presents a characteristic-free approach to vanishing theorems on toric varieties using Cox's sheaf description, and applies it to prove a strong form of Fujita's conjecture for toric varieties.

Contribution

It introduces a new characteristic-free method for vanishing theorems on toric varieties and extends Cox's correspondence to all toric varieties, not just simplicial ones.

Findings

Established a characteristic-free vanishing theorem for toric varieties.

Proved a strong form of Fujita's conjecture for toric varieties.

Showed that every sheaf on a toric variety corresponds to a module over the homogeneous coordinate ring.

Abstract

We use Cox's description for sheaves on toric varieties and results about the local cohomology with respect to monomial ideals to give a characteristic free approach to vanishing results on arbitrary toric varieties. As an application, we give a proof of a strong form of Fujita's conjecture in the case of toric varieties. We also prove that every sheaf on a toric variety corresponds to a module over the homogeneous coordinate ring, generalizing Cox's result for the simplicial case.