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

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• 作者：Francesco Bei
• 关键词：Projective variety ; Fubini–Study metric ; Stratified pseudomanifold ; Iterated edge metric ; Sobolev space ; Bochner Laplacian ; Heat kernel
• 刊名：The Journal of Geometric Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：27
• 期：1
• 页码：746-796
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Differential Geometry; Convex and Discrete Geometry; Fourier Analysis; Abstract Harmonic Analysis; Dynamical Systems and Ergodic Theory; Global Analysis and Analysis on Manifolds;
• 出版者：Springer US
• ISSN：1559-002X
• 卷排序：27

文摘

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.