### Abstract

Let X be the Gromov–Hausdorff limit of a sequence of pointed complete Kähler manifolds (Min,pi) satisfying Ric(M_{i}) ≥ - (n- 1) and the volume is noncollapsed. We prove that, there exists a Lie group isomorphic to 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.

Original language | English (US) |
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Pages (from-to) | 649-667 |

Number of pages | 19 |

Journal | Mathematische Annalen |

Volume | 370 |

Issue number | 1-2 |

DOIs | |

State | Published - Feb 1 2018 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Liu, G. (2018). On the tangent cone of Kähler manifolds with Ricci curvature lower bound.

*Mathematische Annalen*,*370*(1-2), 649-667. https://doi.org/10.1007/s00208-017-1536-0