Formulae for Line Bundle Cohomology on Calabi-Yau Threefolds

TL;DR

This paper derives explicit, polynomial-based formulas for computing the cohomology groups of line bundles on various Calabi-Yau threefolds, facilitating easier calculations and potential automation.

Contribution

It provides the first closed-form, region-based cohomology formulas for line bundles on Calabi-Yau threefolds, including non-simply connected cases, verified through computational methods.

Findings

Cohomology ranks expressed as degree-three polynomials within specific regions.

Formulas verified computationally for multiple Calabi-Yau examples.

Potential for machine learning to extend systematic cohomology calculations.

Abstract

We present closed form expressions for the ranks of all cohomology groups of holomorphic line bundles on several Calabi-Yau threefolds realised as complete intersections in products of projective spaces. The formulae have been obtained by systematising and extrapolating concrete calculations and they have been checked computationally. Although the intermediate calculations often involve laborious computations of ranks of Leray maps in the Koszul spectral sequence, the final results for cohomology follow a simple pattern. The space of line bundles can be divided into several different regions, and in each such region the ranks of all cohomology groups can be expressed as polynomials in the line bundle integers of degree at most three. The number of regions increases and case distinctions become more complicated for manifolds with a larger Picard number. We also find explicit cohomology formulae for several non-simply connected Calabi-Yau threefolds realised as quotients by freely acting discrete symmetries. More cases may be systematically handled by machine learning algorithms.