Sobolev spaces and Bochner Laplacian on complex projective varieties and stratified pseudomanifolds

TL;DR

This paper extends known spectral and Sobolev space results from the regular parts of complex projective varieties to sections and Schrödinger operators on stratified pseudomanifolds with edge metrics.

Contribution

It generalizes previous results on Sobolev spaces and heat operators to sections and Schrödinger operators on more complex geometric structures.

Findings

Extension of Sobolev space results to sections and Schrödinger operators.

Proved compactness of certain embeddings and trace class properties of heat operators.

Applicable to stratified pseudomanifolds with iterated edge metrics.

Abstract

Let $V\subset \mathbb{C}\mathbb{P}^n$ be an irreducible complex projective variety of complex dimension $v$ and let $g$ be the K\"ahler metric on $\reg(V)$, the regular part of $V$, induced by the Fubini Study metric of $\mathbb{C}\mathbb{P}^n$. In this setting Li and Tian proved that $W^{1,2}_0(\reg(V),g)=W^{1,2}(\reg(V),g)$, that the natural inclusion $W^{1,2}(\reg(V),g)\hookrightarrow L^2(\reg(V),g)$ is a compact operator and that the heat operator associated to the Friedrich extension of the scalar Laplacian $\Delta_0:C^{\infty}_c(\reg(V))\rightarrow C^{\infty}_c(\reg(V))$, that is $e^{-t\Delta_0^{\mathcal{F}}}:L^2(\reg(V),g)\rightarrow L^2(\reg(V),g)$, is a trace class operator. The goal of this paper is to provide an extension of the above result to the case of Sobolev spaces of sections and symmetric Schr\"odinger type operators with potential bounded from below where the underling riemannian manifold is the regular part of a complex projective variety endowed with the Fubini-Study metric or the regular part of a stratified pseudomanifold endowed an iterated edge metric.