Quot-scheme limit of Fubini-Study metrics and its applications to balanced metrics

TL;DR

This paper extends the theory of balanced metrics and Gieseker stability to singular holomorphic vector bundles using Quot-scheme limits of Fubini-Study metrics, connecting to broader geometric stability concepts.

Contribution

It generalizes the equivalence between balanced metrics and Gieseker stability to singular bundles and relates Bergman 1-parameter subgroups to subgeodesics in hermitian metric space.

Findings

Balanced metrics exist iff Gieseker stability for singular bundles

Bergman 1-parameter subgroups form subgeodesics

Techniques relate to Yau-Tian-Donaldson conjecture

Abstract

We present some results that complement our prequels [arXiv:1809.08425,arXiv:1907.05770] on holomorphic vector bundles. We apply the method of the Quot-scheme limit of Fubini-Study metrics developed therein to provide a generalisation to the singular case of the result originally obtained by X.W. Wang for the smooth case, which states that the existence of balanced metrics is equivalent to the Gieseker stability of the vector bundle. We also prove that the Bergman 1-parameter subgroups form subgeodesics in the space of hermitian metrics. This paper also contains a review of techniques developed in [arXiv:1809.08425,arXiv:1907.05770] and how they correspond to their counterparts developed in the study of the Yau-Tian-Donaldson conjecture.