A note on K$\ddot{a}$hler manifolds with almost nonnegative bisectional curvature

Abstract

In this note we prove the following result: There is a positive constant $\epsilon(n,\Lambda)$ such that if $M^n$ is a simply connected compact K$\ddot{a}$hler manifold with sectional curvature bounded from above by $\Lambda$, diameter bounded from above by 1, and with holomorphic bisectional curvature $H \geq -\epsilon(n,\Lambda)$, then $M^n$ is diffeomorphic to the product $M_1\times ... \times M_k$, where each $M_i$ is either a complex projective space or an irreducible K$\ddot{a}$hler-Hermitian symmetric space of rank $\geq 2$. This resolves a conjecture of F. Fang under the additional upper bound restrictions on sectional curvature and diameter.