Kähler manifolds with almost non-negative bisectional curvature.
type: Item
instance of: scholarly article
title: Kähler manifolds with almost non-negative bisectional curvature.
zbMATH Open document ID: 1041.53043
published in: The Asian Journal of Mathematics
publication date: 2002-01-01T00:00:00Z
review text: This paper presents a generalization of the Mok uniformization theorem asserting that a simply connected compact Kähler manifold with non-negative bisectional curvature is biholomorhic to the product of a certain number of projective spaces and irreducible Kähler Hermitian symmetric spaces of rank at least \(2\). For this generalization, the author introduces the following notion: A Hermitian manifold \((M, g)\) has almost nonnegative sectional curvature if for any positive constant \(\varepsilon\), the bisectional curvature \(H\) satisfies the inequality \(H\cdot \text{diam}(M, g)^2 \geq -\varepsilon\). A main theorem states that for a simply connected \(n\)-dimensional Kähler manifold \(M\) with sectional curvature \(K\leq \Lambda\), there exists a universal positive constant \(\varepsilon (n,\Lambda)\), depending only on the dimension \(n\) and on \(\Lambda\), such that if the bisectional curvature \(H\) and the diameter of \(M\) satisfy \(H\cdot \text{diam}(M)^2 \geq -\varepsilon(n,\Lambda)\), then \(M\) is diffeomorphic to a product \(M_1\times\cdots\times M_k\), where each \(M_i\) is a \(C^{1,\alpha}\)-Kähler manifold with second Betti number \(b_2(M_i)=1\) for any prescribed real number \(\alpha\in (0,1)\). Furthermore, if \(M\) is Kähler-Einstein, then \(M_i\) are all biholomorphic to irreducible Kähler Hermitian symmetric spaces.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1925765
zbMATH Keywords: Kähler manifold
zbMATH Keywords: bisectional curvature
zbMATH Keywords: Kähler Hermitian symmetric spaces
author: Fuquan Fang
MaRDI profile type: Publication
full work available at URL: ajm.2002.v6.n3.a1
OpenAlex ID: W124907631
DOI: 10.4310/AJM.2002.V6.N3.A1
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description: scientific article; zbMATH DE number 1925765