Sobolev Spaces and Bochner Laplacian on Complex Projective Varieties and Stratified Pseudomanifolds

Abstract

Let $$V\subset {\mathbb {C}}{\mathbb {P}}^n$$ be an irreducible complex projective variety of complex dimension v and let g be the Kähler metric on $${{\mathrm{reg}}}(V)$$ , the regular part of V, induced by the Fubini–Study metric of $${\mathbb {C}}{\mathbb {P}}^n$$ . In (J Am Math Soc 8:857–877, 1995) Li and Tian proved that $$W^{1,2}_0({{\mathrm{reg}}}(V),g)=W^{1,2}({{\mathrm{reg}}}(V),g)$$ , that the natural inclusion $$W^{1,2}({{\mathrm{reg}}}(V),g)\hookrightarrow L^2({{\mathrm{reg}}}(V),g)$$ is a compact operator and that the heat operator associated with the Friedrich extension of the scalar Laplacian $$\Delta _0:C^{\infty }_c({{\mathrm{reg}}}(V))\rightarrow C^{\infty }_c({{\mathrm{reg}}}(V))$$ , that is, $$e^{-t\Delta _0^{{\mathcal {F}}}}:L^2({{\mathrm{reg}}}(V),g)\rightarrow L^2({{\mathrm{reg}}}(V),g)$$ , 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 Schrödinger type operators with potential bounded from below where the underlying 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 with an iterated edge metric.