Mathematische Annalen

, Volume 370, Issue 1–2, pp 649–667 | Cite as

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

  • Gang LiuEmail author


Let X be the Gromov–Hausdorff limit of a sequence of pointed complete Kähler manifolds \((M^n_i, p_i)\) 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.



The author thanks Professor Gang Tian for his encouragement and interest on this work during the visit to UC Berkeley on March 31, 2014. The author also thanks Professors John Lott, Xiaochun Rong and Jiaping Wang for valuable discussions. Special thanks also go to the anonymous referee for correcting some inaccuracies.


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA

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