Holomorphic Morse inequalities on covering manifolds

Details Holomorphic Morse inequalities on covering manifolds Holomorphic Morse inequalities on covering manifolds

1998-02-23

arxiv.org/abs/math/9802110v2

Mathematics

Complex Variables

9 pages, Latex2e; see also preprint MS-98-003 at http://www.maths.ed.ac.uk/ preprints/; corrected typos, a reference added

Scientific paper

The goal of this paper is to generalize Demailly's asymptotic holomorphic Morse inequalities to the case of a covering manifold of a compact manifold. We shall obtain estimates which involve Atiyah's ``normalized dimension'' of the square integrable harmonic spaces. The techniques used are those of Shubin who gave a proof for the usual Morse inequalities in the presence of a group action relying on Witten ideas. As a consequence we obtain estimates for the dimension of the square integrable holomorphic sections of the pull-back of a line bundle on the base manifold under some mild hypothesis for the curvature.

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