Holomorphic Morse Inequalities on Non-compact Manifolds
We start by the L 2 Hodge theory on non-compact Hermitian manifolds in Section 3.1. In Section 3.2, we prove holomorphic Morse inequalities for the L 2-cohomology in a quite general context, namely, when the fundamental estimate (3.2.2) holds. This gives a fairly general method which may be applied in many situations. The main idea, going back to Witten, is to show that the spectral spaces of the Laplacian, corresponding to small eigenvalues, inject in the spectral spaces of the Laplacian with Dirichlet boundary conditions on a smooth relatively domain, The asymptotic of the latter operator is calculated in Theorem 3.2.9. For a compact manifold we recover of course Theorem 1.7.1.
KeywordsLine Bundle Fundamental Estimate Hermitian Form Spectral Space Hodge Theory
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3.7 Bibliographic notes
- A. Andreotti and G. Tomassini, Some remarks on pseudoconcave manifolds, Essays on Topology and Related Topics dedicated to G. de Rham (A. Haefinger and R. Narasimhan, eds.), Springer-Verlag, 1970, pp. 85–104.Google Scholar
- G. Fischer, Complex analytic geometry, Lect. Notes, vol. 538, Springer-Verlag, 1976. p. 180.Google Scholar
- L. Schwartz, Théorie des distributions, Actualités Sci. Ind., no. 1091, Hermann & Cie., Paris, 1950.Google Scholar