Some numerical results in complex differential geometry

TL;DR

This paper presents numerical methods for approximating special Kahler metrics on complex projective manifolds, exemplified by detailed results on a specific K3 surface, linking geometric invariant theory and line bundle asymptotics.

Contribution

It introduces numerical procedures for computing distinguished Kahler metrics, connecting geometric invariant theory with line bundle asymptotics, and demonstrates these methods on a K3 surface.

Findings

Numerical approximations of Calabi-Yau metrics achieved

Procedures linked to geometric invariant theory

Detailed results for a specific K3 surface

Abstract

The first part of this paper discusses general procedures for finding numerical approximations to distinguished Kahler metrics, such as Calabi-Yau metrics, on complex projective manifolds. These procedures are closely related to ideas from Geometric Invariant Theory, and to the asymptotics of high powers of positive line bundles. In the core of the paper these ideas are illustrated by detailed numerical results for a particular K3 surface.