Kähler–Einstein metrics with prescribed singularities on Fano manifolds

Antonio Trusiani

doi:10.1515/crelle-2022-0047

Article

Kähler–Einstein metrics with prescribed singularities on Fano manifolds

Antonio Trusiani

Published/Copyright: September 29, 2022

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2022 Issue 793

Abstract

Given a Fano manifold ( X , ω ) , we develop a variational approach to characterize analytically the existence of Kähler–Einstein metrics with prescribed singularities, assuming that these singularities can be approximated algebraically. Moreover, we define a function α ω on the set of prescribed singularities which generalizes Tian’s α-invariant, showing that its upper lever set { α ω ⁢ ( ⋅ ) > n n + 1 } produces a subset of the Kähler–Einstein locus, i.e. of the locus given by all prescribed singularities that admit Kähler–Einstein metrics. In particular, we prove that many K-stable manifolds admit all possible Kähler–Einstein metrics with prescribed singularities. Conversely, we show that enough positivity of the α-invariant function at nontrivial prescribed singularities (or other conditions) implies the existence of genuine Kähler–Einstein metrics. Finally, through a continuity method we also prove the strong continuity of Kähler–Einstein metrics on curves of totally ordered prescribed singularities when the relative automorphism groups are discrete.

Funding statement: The author is supported by a postdoctoral grant of the Knut and Alice Wallenberg Foundation.

Acknowledgements

I would like to thank my PhD advisors Stefano Trapani and David Witt Nyström for their comments.

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Received: 2021-02-14

Revised: 2022-04-26

Published Online: 2022-09-29

Published in Print: 2022-12-01

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https://doi.org/10.1515/crelle-2022-0047