Kähler manifolds with almost non-negative bisectional curvature.
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.
scientific article; zbMATH DE number 1925765
scientific article