Kähler manifolds with almost non-negative bisectional curvature

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(M) > — e(n, A), then M is diffeomorphic to the product Mi x • • • X Mfc, where each Mi is a simply connected C'-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). 0. 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) > —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) x • • • x P(C) 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-SmythWu [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 HowardSmyth-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(M) > —e(n, A), then M is diffeomorphic to the product Mi x • • • x M&, where each Mi is a simply connected C'-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. * Received November 15, 2001; accepted for publication April 23, 2002. t Nankai Institute of Mathematics, Nankai University, Tianjin 300071, P.R. C. (ffang@nankai.edu.cn). Institute de Matematica, Universidade Federal Fluminense, Niteroi 24005, RJ, Brasil (fuquan@mat.uff.br). Supported by NSFC Grant 19741002, RFDP, CNPq and the Qiu-Shi Foundation.