## 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.## References

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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
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 3569-3574 - MSC (2020): Primary 53C24, 53C55
- DOI: https://doi.org/10.1090/proc/15473
- MathSciNet review: 4273157