On the tangent cone of K\"ahler manifolds with Ricci curvature lower bound

TL;DR

This paper extends the metric cone theorem to K"ahler manifolds with Ricci curvature lower bounds, showing tangent cones admit a Lie group action and exploring implications for manifolds with nonnegative bisectional curvature.

Contribution

It proves the existence of an $ $-action on tangent cones of K"ahler limit spaces, generalizing Cheeger-Colding's metric cone theorem to the K"ahler setting.

Findings

Existence of an $ $-action on tangent cones at each point.

The action is locally free on the cross section.

Applications to K"ahler manifolds with nonnegative bisectional curvature.

Abstract

Let $X$ be the Gromov-Hausdorff limit of a sequence of pointed complete K\"ahler manifolds $(M^n_i, p_i)$ satisfying $Ric(M_i)\geq -(n-1)$ 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 K\"ahler case. We also discuss some applications to complete K\"ahler manifolds with nonnegative bisectional curvature.