Computing Cohomology on Toric Varieties

TL;DR

This paper reviews a new technique for computing line bundle cohomology on toric varieties, highlighting a vanishing theorem and the use of Alexander duality to facilitate the algorithm.

Contribution

It introduces a vanishing theorem based on Stanley-Reisner ideals and emphasizes the role of Alexander duality in the computation of cohomology groups.

Findings

The technique efficiently computes cohomology dimensions.

The vanishing theorem simplifies the calculation process.

Alexander duality is central to the proof methods.

Abstract

In these notes a recently developed technique for the computation of line bundle-valued sheaf cohomology group dimensions on toric varieties is reviewed. The key result is a vanishing theorem for the contributing components which depends on the structure of the Stanley-Reisner ideal generators. A particular focus is placed on the (simplicial) Alexander duality that provides a central tool for the two known proofs of the algorithm.