Kähler manifolds with almost non-negative bisectional curvature

Abstract

The main purpose of this paper is to study the topology of Kahler manifolds with almost non-negative bisectional curvature.Among others we prove that for simply connected ndimensional Kahler manifolds M of sectional curvature K < A, there exists a universal positive constant e(n, A), depending only on the dimension n and A, such that if the bisectional curvature H and the diameter of M satisfy, H • diam 2 (M) > -e(n, A), then M is diffeomorphic to the product Mi x • • • X Mfc, where each Mi is a simply connected C 1 ' a -Kahler manifold with second Betti number 62(Mi) = 1 for any prescribed real number o; G (0,1).Furthermore, if M is Kahler-Einstein, then Mi are all biholomorphic to irreducible Kahler Hermitian symmetric spaces.In the non-simply connected case, we prove that M is a holomorphic fiber bundle over the Jacobian J(M). Introduction.Let M be a compact complex manifold.We say M has almost nonnegative bisectional curvature, if for any positive constant 5, there is a Hermitian metric g on M whose bisectional curvature H satisfies that H • diam(Mg) 2 > -e.Besides Hermitian manifolds of non-negative bisectional curvature, there are many examples of complex manifolds of almost non-negative bisectional curvature but do not admit any Hermitian metric of non-negative bisectional curvature (c.f.Section 1.)When the Hermitian manifold is Kahlerian, the uniformization theorem of Mok [Mo] (generalized Prankel conjecture, compare Siu-Yau [SY]) asserts that a simply connected compact Kahler manifold M with non-negative bisectional curvature is biholomorphic to the product of P(C) mi x • • • x P(C) mfc xNx'-xNi where Ni, 1 < i < I, are irreducible Kahler Hermitian symmetric spaces of rank at least 2. The Mok theorem depends on an earlier decomposition theorem of Howard-Smyth-Wu [HSW], Mori's celebrated work [Mo] and Hamilton's heat equation technique.A natural question is whether one can extend the Mok theorem and the Howard-Smyth-Wu theorem to Kahler manifold of almost non-negative bisectional curvature.In this paper we will prove, among others, for simply connected n-dimensional Kahler manifold M with sectional curvature K < A, there exists a universal positive constant £(n, A), depending only on the dimension n and A, such that if the bisectional curvature H and the diameter of M satisfy, H • diam 2 (M) > -e(n, A), then M is diffeomorphic to the product Mi x • • • x M&, where each Mi is a simply connected C 1 ' a -Kahler manifold with second Betti number 62(Mi) = 1 for any prescribed real number a € (0,1).Furthermore, if M is Kahler-Einstein, then Mi are all biholomorphic to irreducible Kahler Hermitian symmetric spaces.In the non-simply connected case, we prove that M is a holomorphic fiber bundle over the Jacobian J(M).Now we start to state our main results.For convenience, let M(n, A) denote the set of all n-dimensional Kahler manifolds so that K < A.