A note on Kähler manifolds with almost nonnegative bisectional curvature

Abstract

In this note, we prove the following result. There is a positive constant ε(n, Λ) such that if M n is a simply connected compact Kähler manifold with sectional curvature bounded from above by Λ, diameter bounded from above by 1, and with holomorphic bisectional curvature H ≥ −ε(n, Λ), then M n is diffeomorphic to the product M 1 × ⋯ × M k , where each M i is either a complex projective space or an irreducible Kähler–Hermitian symmetric space of rank ≥ 2. This resolves a conjecture of Fang under the additional upper bound restrictions on sectional curvature and diameter.