Limits of Projective Manifolds Under Holomorphic Deformations

Details Limits of Projective Manifolds Under Holomorphic Deformations Limits of Projective Manifolds Under Holomorphic Deformations

2009-10-11

arxiv.org/abs/0910.2032v1

Mathematics

Algebraic Geometry

38 pages

Scientific paper

We prove that if in a complex analytic family of compact complex manifolds all the fibres, except one, are supposed to be projective, then the remaining (limit) fibre must be Moishezon. The proof is based on the so-called singular Morse inequalities for integral cohomology classes that we obtained in a previous work. The strategy, originating in the work of J.-P. Demailly, consists in using the Aubin-Calabi-Yau theorem to construct K\"ahler forms on non-limit fibres in a certain integral De Rham cohomology 2-class and in showing that this family of forms is bounded in mass in a suitable sense. By weak compactness, a subsequence of K\"ahler forms converges weakly to a $(1, 1)$-current that may have wild singularities and is defined on the limit fibre. The singular Morse inequalities are then used on the limit fibre to produce a K\"ahler current in the same integral cohomology class. The existence of a K\"ahler current with integral cohomology class is known to characterise Moishezon manifolds.

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