Constructing stable Hilbert bundles via Diophantine approximation

TL;DR

This paper constructs Hermitian--Einstein metrics on Hilbert bundles over complex curves of positive genus, using Diophantine approximation and stability conditions to analyze their geometric and analytic properties.

Contribution

It introduces a novel approach combining Diophantine approximation with stability conditions to construct and analyze Hilbert bundles with Hermitian--Einstein metrics.

Findings

Construction of Hilbert bundles with Hermitian--Einstein metrics on complex curves

Introduction of well-approximating sequences of stable bundles

Application of Diophantine approximation to bound Hermitian-Einstein metrics

Abstract

On any complex smooth projective curve with positive genus, we construct Hilbert bundles that admit Hermitian--Einstein metrics. Our main constructive step is by investigating the arithmetic property of the upper half plane in Bridgeland's definition of stability conditions and its homological countparts. The main analytic ingredient in our proof is a notion called a well-approximating sequence of stable bundles. This notion helps us to apply the Diophantine approximation to Donaldson's functional and bound the $L^\infty$ norm of Hermitian-Einstein metrics. We further study the continuous structures, smooth structures, and holomorphic structures on such Hilbert bundles. We hope that this construction can shed some new light on the geometric background of quantum field theory.