Quot-scheme limit of Fubini-Study metrics and Donaldson's functional for vector bundles

TL;DR

This paper introduces the Quot-scheme limit of Fubini-Study metrics to connect slope stability of vector bundles with the asymptotic behavior of Donaldson's functional, providing new insights and proofs in complex differential geometry.

Contribution

It defines the Quot-scheme limit of Fubini-Study metrics and links it to slope stability, offering explicit estimates and a new proof of Hermitian-Einstein metrics implying slope stability.

Findings

Donaldson's functional is coercive on Fubini-Study metrics if the bundle is slope stable

Established a direct link between slope stability and the asymptotic behavior of Donaldson's functional

Provided a new proof that Hermitian-Einstein metrics imply slope stability

Abstract

For a holomorphic vector bundle $E$ over a polarised K\"ahler manifold, we establish a direct link between the slope stability of $E$ and the asymptotic behaviour of Donaldson's functional, by defining the Quot-scheme limit of Fubini-Study metrics. In particular, we provide an explicit estimate which proves that Donaldson's functional is coercive on the set of Fubini-Study metrics if $E$ is slope stable, and give a new proof of Hermitian-Einstein metrics implying slope stability.