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Proceedings of the American Mathematical Society

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


Authors: Ved Datar, Harish Seshadri and Jian Song
Journal: Proc. Amer. Math. Soc. 149 (2021), 3569-3574
MSC (2020): Primary 53C24, 53C55
DOI: https://doi.org/10.1090/proc/15473
Published electronically: May 18, 2021
MathSciNet review: 4273157
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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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Additional Information

Ved Datar
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
MR Author ID: 1138557
Email: vvdatar@iisc.ac.in

Harish Seshadri
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
MR Author ID: 712201
Email: harish@iisc.ac.in

Jian Song
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
MR Author ID: 746741
Email: jiansong@math.rutgers.edu

Received by editor(s): October 30, 2020
Published electronically: May 18, 2021
Additional Notes: Research supported in part by National Science Foundation grant DMS-1711439, UGC (Govt. of India) grant no. F.510/25/CAS- II/2018(SAP-I), and the Infosys Young investigator award.
Communicated by: Jiaping Wang
Article copyright: © Copyright 2021 American Mathematical Society