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Second-order estimates for collapsed limits of Ricci-flat Kähler metrics

Part of: Global differential geometry Differential operators in several variables Surfaces and higher-dimensional varieties Complex manifolds

Published online by Cambridge University Press:  05 January 2023

Kyle Broder Open the ORCID record for Kyle Broder [Opens in a new window]
Kyle Broder*
Affiliation:
School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD 4067, Australia
*
e-mail: kyle.broder@anu.edu.au
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Abstract

We show that the singularities of the twisted Kähler–Einstein metric arising as the longtime solution of the Kähler–Ricci flow or in the collapsed limit of Ricci-flat Kähler metrics are intimately related to the holomorphic sectional curvature of reference conical geometry. This provides an alternative proof of the second-order estimate obtained by Gross, Tosatti, and Zhang (2020, Preprint, arXiv:1911.07315) with explicit constants appearing in the divisorial pole.

Keywords

Calabi–Yau manifoldsKähler metricscollapsed geometryKähler–Ricci flowholomorphic sectional curvatureSchwarz lemma

MSC classification

Primary: 32Q25: Calabi-Yau theory 32Q20: Kähler-Einstein manifolds 32W20: Complex Monge-Ampère operators 14J32: Calabi-Yau manifolds 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 3 , September 2023 , pp. 912 - 926
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The author was partially supported by an Australian Government Research Training Program (RTP) Scholarship and funding from the Australian Government through the Australian Research Council’s Discovery Projects funding scheme (project DP220102530).

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