Geometric singularities and Hodge theory

TL;DR

This paper studies the stability of the kernel of Hodge-Dirac operators under rough geometric conditions, demonstrating isomorphism with classical cohomology even with non-smooth metrics.

Contribution

It establishes the invariance of Hodge-Dirac kernels under perturbations of non-smooth geometric data, extending classical results to rough Riemannian metrics.

Findings

Kernel of Hodge-Dirac operator remains isomorphic under uniform perturbations.

Non-smooth metrics induce kernels isomorphic to classical cohomology.

Results apply to both compact and non-compact manifolds.

Abstract

We consider smooth vector bundles over smooth manifolds equipped with non-smooth geometric data. For nilpotent differential operators acting on these bundles, we show that the kernels of induced Hodge-Dirac-type operators remain isomorphic under uniform perturbations of the geometric data. We consider applications of this to the Hodge-Dirac operator on differential forms induced by so-called rough Riemannian metrics, which can be of only measurable coefficient in regularity, on both compact and non-compact settings. As a consequence, we show that the kernel of the associated non-smooth Hodge-Dirac operator with respect to a rough Riemannian metric remains isomorphic to smooth and singular cohomology when the underlying manifold is compact.