Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic

TL;DR

This paper introduces an iterative numerical method to solve the hermitian Yang-Mills equation on stable holomorphic vector bundles, demonstrating its effectiveness through explicit constructions on complex manifolds like P^2 and the Fermat quintic.

Contribution

It presents a novel iterative approach for numerically solving the hermitian Yang-Mills equation on complex manifolds, including the Fermat quintic, with practical examples.

Findings

Successfully constructed hermitian Einstein metrics on tangent bundles.

Found hermitian Yang-Mills connections on stable rank three bundles on the Fermat quintic.

Validated the iterative method's effectiveness for complex geometric problems.

Abstract

We develop an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson. As illustrations, we construct numerically the hermitian Einstein metrics on the tangent bundle and a rank three vector bundle on P^2. In addition, we find a hermitian Yang-Mills connection on a stable rank three vector bundle on the Fermat quintic.