Geometric singularities and Hodge theory

Lashi Bandara; Georges Habib

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arXiv · 2407.01170

Geometric singularities and Hodge theory

Lashi Bandara·Georges Habib

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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.

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Lashi Bandara, Georges Habib. 2024-07-01. Geometric singularities and Hodge theory. https://arxiv.org/abs/2407.01170

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