The cohomology of spherical vector bundles on K3 surfaces

TL;DR

This paper develops an algorithm to compute the cohomology of spherical vector bundles on K3 surfaces using Mukai vectors, with simplifications in many cases, and relates it to weak Brill-Noether conditions when the Picard rank is one.

Contribution

It introduces a new algorithm for cohomology computation of spherical vector bundles on K3 surfaces based on Mukai vectors, including significant simplifications and applications to Brill-Noether theory.

Findings

Algorithm for cohomology computation using Mukai vectors

Simplifications of the algorithm in many cases

Numerical condition for weak Brill-Noether when Picard rank is one

Abstract

We find an algorithm to compute the cohomology groups of spherical vector bundles on complex projective K3 surfaces, in terms of their Mukai vectors. In many good cases, we give significant simplifications of the algorithm. As an application, when the Picard rank is one, we show a numerical condition that is equivalent to weak Brill-Noether for a spherical vector bundle.