On the tangent cone of Kähler manifolds with Ricci curvature lower bound

Abstract

Let $X$ be the Gromov-Hausdorff limit of a sequence of pointed complete Kähler manifolds $(M_i^n,p_i)$ satisfying $Ric(M_i)\ge -(n-1)$ and the volume is noncollapsed. We prove that, there exists a Lie group isomorphic to $\mathbb{R}$, acting isometrically, on the tangent cone at each point of $X$. Moreover, the action is locally free on the cross section. This generalizes the metric cone theorem of Cheeger-Colding to the Kähler case. We also discuss some applications to complete Kähler manifolds with nonnegative bisectional curvature.

16pages