Perturbations of Vector Bundle whose Curvature Form Solves a Polynomial Equation

TL;DR

This paper studies how small perturbations affect solutions of geometric PDEs on holomorphic Hermitian vector bundles, establishing a local correspondence between solutions and stability conditions using advanced geometric invariant theory techniques.

Contribution

It introduces a local Kobayashi-Hitchin correspondence for perturbed bundles, linking solutions to local polystability, and develops new local stability and filtration results.

Findings

Existence of solutions characterized by local polystability.

Continuity and uniqueness of solutions under perturbations.

A local version of the Kempf-Ness theorem and filtrations.

Abstract

We investigate the behaviour of local perturbations of a wide class of geometric PDEs on holomorphic Hermitian vector bundles over a compact complex manifold. Our main goal is to study the existence of solutions near an initial solution under small deformations of both the holomorphic structure of the bundle and the parameters of the equation. Inspired by techniques from geometric invariant theory and the moment map framework, under suitable assumptions on the initial solution, we establish a local Kobayashi-Hitchin correspondence. A perturbed bundle admits a solution to the equation if and only if it satisfies a local polystability condition. We also show additional results, such as continuity and uniqueness of solutions when they exist, and a local version of the Kempf-Ness theorem. We also provide a local version of the Jordan-H\"older and Harder-Narasimhan filtrations.