# Index Formulae for Line Bundle Cohomology on Complex Surfaces

## TL;DR

This paper develops explicit index formulae for line bundle cohomology on complex surfaces, including del Pezzo, Hirzebruch, and toric surfaces, providing a unified approach and new theoretical insights.

## Contribution

It introduces closed-form index expressions for line bundle cohomology on various complex surfaces and an algorithm to compute cohomology in terms of an index.

## Key findings

- Derived explicit index formulae for del Pezzo and Hirzebruch surfaces.
- Provided an algorithm for expressing cohomology of line bundles on toric surfaces.
- Proved theorems relating effective and nef line bundles preserving cohomology.

## Abstract

We conjecture and prove closed-form index expressions for the cohomology dimensions of line bundles on del Pezzo and Hirzebruch surfaces. Further, for all compact toric surfaces we provide a simple algorithm which allows expression of any line bundle cohomology in terms of an index. These formulae follow from general theorems we prove for a wider class of surfaces. In particular, we construct a map that takes any effective line bundle to a nef line bundle while preserving the zeroth cohomology dimension. For complex surfaces, these results explain the appearance of piecewise polynomial equations for cohomology and they are a first step towards understanding similar formulae recently obtained for Calabi-Yau three-folds.