Numerical Algorithms for Finding Balanced Metrics on Vector Bundles

TL;DR

This paper proves that a dynamical system designed to find balanced metrics on vector bundles always converges to such a metric if it exists, extending previous results from polarized manifolds to vector bundles.

Contribution

It confirms the conjecture that the dynamical system converges to a balanced metric on vector bundles when such a metric exists, generalizing earlier work on polarized manifolds.

Findings

The dynamical system converges to a balanced metric on vector bundles.

Existence of a balanced metric guarantees convergence of the algorithm.

Extends Donaldson's framework from manifolds to vector bundles.

Abstract

In \cite{D3}, Donaldson defines a dynamical system on the space of Fubini-Study metrics on a polarized compact K\"ahler manifold. Sano proved that if there exists a balanced metric for the polarization, then this dynamical system always converges to the balanced metric (\cite{S}). In \cite{DKLR}, Douglas, et. al., conjecture that the same holds in the case of vector bundles. In this paper, we give an affirmative answer to their conjecture.