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

Let V⊂ CPn be an irreducible complex projective variety of complex dimension v and let g be the Kähler metric on reg (V) , the regular part of V, induced by the Fubini–Study metric of CPn. In (J Am Math Soc 8:857–877, 1995) Li and Tian proved that W01,2(reg(V),g)=W1,2(reg(V),g), that the natural inclusion W1 , 2(reg (V) , g) ↪ L2(reg (V) , g) is a compact operator and that the heat operator associated with the Friedrich extension of the scalar Laplacian Δ0:Cc∞(reg(V))→Cc∞(reg(V)), that is, e-tΔ0F:L2(reg(V),g)→L2(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.