Holomorphic Morse Inequalities on Non-compact Manifolds

Part of the Progress in Mathematics book series (PM, volume 254)


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.


Line Bundle Fundamental Estimate Hermitian Form Spectral Space Hodge Theory 
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