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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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