Metric rigidity of Kähler manifolds with lower Ricci bounds and almost maximal volume

Datar, Ved; Seshadri, Harish; Song, Jian

doi:10.1090/proc/15473

Metric rigidity of Kähler manifolds with lower Ricci bounds and almost maximal volume
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by Ved Datar, Harish Seshadri and Jian Song PDF

Proc. Amer. Math. Soc. 149 (2021), 3569-3574 Request permission

Abstract:

In this short note we prove that a Kähler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results on holomorphic rigidity of such Kähler manifolds (see Gang Liu [Asian J. Math. 18 (2014), 69–99]) with the structure theorem of Tian-Wang (see Gang Tian and Bing Wang [J. Amer. Math. Soc 28 (2015), 1169–1209]) for almost Einstein manifolds. This can be regarded as the complex analog of the result on Colding on the shape of Riemannian manifolds with almost maximal volume.

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