# Diameter Rigidity for K\"ahler manifolds with positive bisectional curvature

## TL;DR

This paper proves a rigidity result for compact K"ahler manifolds with positive bisectional curvature, showing that under certain diameter and volume conditions, the manifold must be biholomorphically isometric to complex projective space with the Fubini-Study metric.

## Contribution

It establishes a diameter rigidity theorem for K"ahler manifolds with positive bisectional curvature, characterizing the complex projective space uniquely under specific geometric constraints.

## Key findings

- Manifolds with given curvature bounds and diameter are isometric to complex projective space.
- Volume conditions ensure the manifold's isometry class matches that of n.
- The result extends classical rigidity theorems to the K"ahler setting.

## Abstract

Let $M^n$ be a compact K\"ahler 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.