On the simply connectedness of nonnegatively curved Kähler manifolds and applications

Abstract:

We study complete noncompact long-time solutions $(M, g(t))$ to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. $R_{i\bar {\jmath }} \ge cRg_{i\bar {\jmath }}$ at $(p,t)$ for all $t$ for some $c>0$, then there always exists a local gradient Kähler-Ricci soliton limit around $p$ after possibly rescaling $g(t)$ along some sequence $t_i \to \infty$. We will show as an immediate corollary that the injectivity radius of $g(t)$ along $t_i$ is uniformly bounded from below along $t_i$, and thus $M$ must in fact be simply connected. Additional results concerning the uniformization of $M$ and fixed points of the holomorphic isometry group will also be established. We will then consider removing the condition of positive Ricci curvature for $(M, g(t))$. Combining our results with Cao’s splitting for Kähler-Ricci flow (2004) and techniques of Ni and Tam (2003), we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point $p \in M$, then $M$ has a special holomorphic fiber bundle structure. We will treat as special cases, complete Kähler manifolds with nonnegative holomorphic bisectional curvature and average quadratic curvature decay as well as the case of steady gradient Kähler-Ricci solitons.