Diameter Rigidity for Kähler manifolds with positive bisectional curvature

Abstract

Let $M^n$ be a compact Kähler manifold with bisectional curvature bounded from below by $1$. If $diam(M)=\pi /\sqrt{2}$ and $vol(M)>vol(\mathbb{C}{\mathbb{P}}^n)/2^n$, we prove that $M$ is biholomorphically isometric to $\mathbb{C}{\mathbb{P}}^n$ with the standard Fubini-Study metric.