A lifting principle for canonical stability indices of varieties of general type

Meng Chen; Hexu Liu

doi:10.1515/crelle-2024-0054

Article

A lifting principle for canonical stability indices of varieties of general type

Meng Chen and Hexu Liu

Published/Copyright: August 6, 2024

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2024 Issue 816

Abstract

For any integer n > 0 , the 𝑛-th canonical stability index r n is defined to be the smallest positive integer so that the r n -canonical map Φ r n is stably birational onto its image for all smooth projective 𝑛-folds of general type. We prove the lifting principle for { r n } as follows: r n is equal to the maximum of the set of those canonical stability indices of smooth projective ( n + 1 ) -folds with sufficiently large canonical volumes. Equivalently, there exists a constant V ⁢ ( n ) > 0 such that, for any smooth projective 𝑛-fold 𝑋 with the canonical volume vol ⁢ ( X ) > V ⁢ ( n ) , the pluricanonical map φ m , X is birational onto the image for all m ≥ r n − 1 .

Funding source: National Key Research and Development Program of China

Award Identifier / Grant number: 2020YFA0713200

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12121001

Award Identifier / Grant number: 12071078

Funding statement: Project supported by National Key Research and Development Program of China (#2020YFA0713200), NSFC for Innovative Research Groups (#12121001) and NSFC grant (#12071078).

Acknowledgements

The authors are grateful for fruitful discussions with Zhi Jiang and Chen Jiang during the preparation of this paper. Especially Chen Jiang pointed out to us an imprecise application of the adjunction in Section 2.7 in an earlier version of this paper. The first author is a member of LMNS, Fudan University. The second author would like to thank Jianshi Yan, Yu Zou, Minzhe Zhu, Mengchu Li, Wentao Chang for useful discussions and help in study.

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Received: 2024-01-09

Revised: 2024-06-25

Published Online: 2024-08-06

Published in Print: 2024-11-01

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https://doi.org/10.1515/crelle-2024-0054