$L^p$-cohomology, heat semigroup and stratified spaces

TL;DR

This paper investigates the relationship between $L^p$ and $L^2$ cohomology on incomplete Riemannian manifolds and applies these findings to stratified spaces and singular varieties, revealing new connections in geometric analysis.

Contribution

It establishes conditions under which $L^p$-cohomology injects into or surjects onto $L^2$-cohomology and applies these results to stratified pseudomanifolds and complex varieties with isolated singularities.

Findings

Injective and surjective maps between $L^p$ and $L^2$ cohomology groups under certain conditions

Applications to curvature and intersection cohomology of stratified spaces

New insights into the topology of singular algebraic varieties

Abstract

Let $(M,g)$ be an incomplete Riemannian manifold of finite volume and let $2\leq p<\infty$. In the first part of this paper we prove that under certain assumptions the inclusion of the space of $L^p$-differential forms into that of $L^2$-differential forms gives rise to an injective/surjective map between the corresponding $L^p$ and $L^2$ cohomology groups. Then in the second part we provide various applications of these results to the curvature and the intersection cohomology of compact Thom-Mather stratified pseudomanifolds and complex projective varieties with only isolated singularities.