Asymptotic Hodge Theory of Vector Bundles

TL;DR

This paper develops new filtrations on vector bundles over smooth projective varieties using asymptotic analysis of the Dolbeault Dirac operator, linking geometric structures with Hodge theory.

Contribution

It introduces novel filtrations based on large k asymptotics that measure the failure of bundles to admit holomorphic structures and explores their compatibility with Hodge filtrations.

Findings

Filtrations effectively quantify the deviation from holomorphicity.

Compatibility established between these filtrations and classical Hodge filtrations.

Provides new tools for understanding vector bundle structures in algebraic geometry.

Abstract

We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the bundle to admit a holomorphic structure. We study compatibility under the Chern isomorphism of these filtrations with the Hodge filtration on cohomology.