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A rigidity theorem for $${\mathbb{P}}_ 3({\mathbb{C}})$$. (English) Zbl 0573.32027
In complex analytic geometry there is a basic problem: Let X be a complex manifold homeomorphic to $${\mathbb{P}}^ n$$. Is then X biholomorphic to $${\mathbb{P}}^ n?$$ For a long time it has been known that it is so for $$n=1$$. For larger n it is true if X is assumed to be projective (shown by Hirzebruch and Kodaira, and Yau).
The author describes how to show it for $$n=2$$ and proves the following theorem: Let a complex manifold X be bimeromorphic to a Kähler manifold (e.g. let X be Moishezon) and homeomorphic to $${\mathbb{P}}^ 3$$, then X is projective (hence is biholomorphic to $${\mathbb{P}}^ 3)$$. The author asserts that any global deformation of $${\mathbb{P}}^ 3$$ is $${\mathbb{P}}^ 3$$. He also proves that if a complex manifold X is bimeromorphic to a Kähler manifold and has $$h^{0,2}=0$$, then X is Moishezon. (In general it is very difficult to show that on X homeomorphic to $${\mathbb{P}}^ 3$$ there are any non-constant meromorphic functions.) The main tool is Mori’s theory of projective threefolds whose canonical bundle is not numerically effective [S. Mori, Ann. Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)].
Reviewer: K.Dabrowski

##### MSC:
 32J99 Compact analytic spaces 14J30 $$3$$-folds 32H99 Holomorphic mappings and correspondences 32G05 Deformations of complex structures 32G13 Complex-analytic moduli problems
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