# Displaying the cohomology of toric line bundles

## TL;DR

This paper provides a concise and direct proof of a simplified formula for computing the cohomology of toric line bundles, replacing complex topological methods with a set-theoretic difference approach.

## Contribution

It offers a straightforward proof of a known cohomology formula for toric line bundles, simplifying the computational process.

## Key findings

- Simplified cohomology calculation using set-theoretic difference.
- Provides a direct proof of the formula from prior work.
- Enhances understanding of toric line bundle cohomology.

## Abstract

There is a standard method to calculate the cohomology of torus-invariant sheaves $L$ on a toric variety via the simplicial cohomology of associated subsets $V(L)$ of the space $N_{\mathbb R}$ of 1-parameter subgroups of the torus. For a line bundle $L$ represented by a formal difference $\Delta^+-\Delta^-$ of polyhedra in the character space $M_{\mathbb R}$, [ABKW18] contains a simpler formula for the cohomology of $L$, replacing $V(L)$ by the set-theoretic difference $\Delta^- \setminus \Delta^+$. Here, we provide a short and direct proof of this formula.