Cohomology of Line Bundles: Proof of the Algorithm

TL;DR

This paper proves an algorithm for computing line bundle cohomology on toric varieties and introduces a combinatorial Serre duality for Betti numbers, enhancing computational methods in algebraic geometry.

Contribution

It provides a rigorous proof of a previously conjectured algorithm and proposes a new combinatorial duality principle for Betti numbers.

Findings

Proof of the conjectured cohomology algorithm.

Introduction of a combinatorial Serre duality for Betti numbers.

Enhanced understanding of line bundle cohomology on toric varieties.

Abstract

We present a proof of the algorithm for computing line bundle valued cohomology classes over toric varieties conjectured by R.~Blumenhagen, B.~Jurke and the authors (arXiv:1003.5217) and suggest a kind of Serre duality for combinatorial Betti numbers that we observed when computing examples.