Canonical metrics on families of vector bundles

TL;DR

This paper develops a new geometric PDE for families of holomorphic vector bundles, extending Hermite--Einstein metrics, and proves existence and uniqueness results for these canonical metrics on complex manifolds.

Contribution

It introduces a family Hermite--Einstein equation as a canonical metric concept and proves existence on product manifolds and long-term existence of a related parabolic flow.

Findings

Constructed Hermite--Einstein metrics in adiabatic classes.

Proved the parabolic flow admits a unique smooth solution for all time.

Established the Dirichlet problem has a unique solution.

Abstract

We introduce a geometric partial differential equation for families of holomorphic vector bundles, generalising the theory of Hermite--Einstein metrics. We consider families of holomorphic vector bundles which each admit Hermite--Einstein metrics, together with a first order deformation. On such families, we define the family Hermite--Einstein equation for Hermitian metrics, which we view as a notion of a canonical metric in this setting. We prove two main results concerning family Hermite--Einstein metrics. Firstly, we construct Hermite--Einstein metrics in adiabatic classes on product manifolds, assuming the existence of a family Hermite--Einstein metric. Secondly, we prove that the associated parabolic flow admits a unique smooth solution for all time, and use this to show that the Dirichlet problem always admits a unique solution.