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

TL;DR

This paper proves that simply connected compact Kähler manifolds with almost nonnegative bisectional curvature, bounded sectional curvature, and diameter are diffeomorphic to products of well-understood symmetric spaces, confirming a conjecture under certain bounds.

Contribution

It establishes a classification result for Kähler manifolds with near nonnegative bisectional curvature under curvature and diameter bounds, confirming Fang's conjecture.

Findings

Manifolds are diffeomorphic to products of projective spaces or symmetric spaces.

Provides explicit bounds for curvature and diameter ensuring the classification.

Resolves Fang's conjecture with additional geometric restrictions.

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.