Demailly’s asymptotic Morse inequalities: a heat equation proof. (English) Zbl 0649.58030

From the introduction: “J. P. Demailly [C. R. Acad. Sci., Paris, Ser. I 301, 119-122 (1985; Zbl 0595.58014); Ann. Inst. Fourier 35, 189- 229 (1985; Zbl 0575.58013)] proved remarkable asymptotic Morse inequalities for the \({\bar \partial}\) complex constructed over the line bundle \(E^{\otimes k}\) as \(k\uparrow \uparrow +\infty\), where E is a holomorphic hermitian line bundle. The inequalities of Demailly give asymptotic bounds on the Morse sums of the Betti numbers of \({\bar \partial}\) on \(E^{\otimes k}\) in terms of certain integrals of the curvature of E. The analysis of Demailly is based on the beautiful remark that in the formula for (\({\bar \partial}+{\bar \partial}^*)^ 2\) on \(E^{\otimes k}\), the metric of E plays formally the role of the Morse function. In this paper, we give a heat equation proof of Demailly’s inequalities.”
Reviewer: G.Warnecke


58J10 Differential complexes
58J35 Heat and other parabolic equation methods for PDEs on manifolds
57R70 Critical points and critical submanifolds in differential topology
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