# Machine Learning Line Bundle Cohomology

## TL;DR

This paper explores machine learning methods to compute line bundle cohomology on complex surfaces and Calabi-Yau three-folds, focusing on region-based polynomial formulas and rigid divisor identification.

## Contribution

It introduces neural networks that identify cohomology regions and polynomials, advancing computational techniques in algebraic geometry.

## Key findings

- Networks successfully identify cohomology regions and polynomials.
- Method improves conjecture generation for cohomology formulas.
- Applicable to complex surfaces and Calabi-Yau three-folds.

## Abstract

We investigate different approaches to machine learning of line bundle cohomology on complex surfaces as well as on Calabi-Yau three-folds. Standard function learning based on simple fully connected networks with logistic sigmoids is reviewed and its main features and shortcomings are discussed. It has been observed recently that line bundle cohomology can be described by dividing the Picard lattice into certain regions in each of which the cohomology dimension is described by a polynomial formula. Based on this structure, we set up a network capable of identifying the regions and their associated polynomials, thereby effectively generating a conjecture for the correct cohomology formula. For complex surfaces, we also set up a network which learns certain rigid divisors which appear in a recently discovered master formula for cohomology dimensions.