Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 11 (2016), No. 3, pp. 579-602 DOI: http://dx.doi.org/10.21915/BIMAS.2016304 EXISTENCE OF HOLOMORPHIC SECTIONS AND PERTURBATION OF POSITIVE LINE BUNDLES OVER q–CONCAVE MANIFOLDS GEORGE MARINESCU To Gennadi Henkin, in memoriam Universit¨at zu K¨oln, Mathematisches Institut, Weyertal 86-90, 50931 K¨oln, Germany. Institute of Mathematics ‘Simion Stoilow’, Romanian Academy, Bucharest, Romania. E-mail:
[email protected] Abstract By using holomorphic Morse inequalities we prove that sufficiently small deformations of a pseudoconcave domain in a projective manifold is Moishezon. 1. Introduction Let X be a compact complex manifold with strongly pseudoconcave boundary. The question of projectively embedding X has been studied in [1], [4], [5], [11], [13]. In particular the generalization of Kodaira’s embedding theorem would give an intrinsic characterisation of projective pseudoconcave manifolds in terms of positive line bundles. Our model is the case when X admits a positively embedded (i.e. with positive normal bundle) smooth compact divisor Z. By a rigidity theorem of Griffiths [13] we infer that global sections in high tensor powers of the the associated bundle [Z] embed a small neighbourhood of Z in the projective space. In particular X has a maximal number of independent meromorphic functions. Received May 15, 2016. AMS Subject Classification: 32L10, 32F10, 32G05. Key words and phrases: Deformation of complex structure, q-concave manifold, Moishezon man- ifold, holomorphic Morse inequalities. 579 580 GEORGE MARINESCU [September We will be concerned in the sequel with general positive line bundles.