On the Simply Connectedness of Non-negatively Curved Kähler Manifolds and Applications

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. Ri̄ ≥ cRgi̄ 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 ti → ∞. We will show as an immediate corollary that the injectivity radius of g(t) along ti is uniformly bounded from below along ti, 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 for (M, g(t)). Combining our results with Cao’s splitting for Kähler-Ricci flow [7] and techniques of Ni-Tam [25], we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point p ∈ M , then M has a special holomorphic fiber bundle structure. We will treat as special cases, complete Kähler manifolds with non-negative holomorphic bisectional curvature and average quadratic curvature decay aswell as the case of steady gradient Kähler-Ricci solitons.