Valuative stability of polarised varieties

Dervan, R. and Legendre, E. (2023) Valuative stability of polarised varieties. Mathematische Annalen, 385(1-2), pp. 357-391. (doi: 10.1007/s00208-021-02313-4)

Abstract

Fujita and Li have given a characterisation of K-stability of a Fano variety in terms of quantities associated to valuations, which has been essential to all recent progress in the area. We introduce a notion of valuative stability for arbitrary polarised varieties, and show that it is equivalent to K-stability with respect to test configurations with integral central fibre. The numerical invariant governing valuative stability is modelled on Fujita’s β-invariant, but includes a term involving the derivative of the volume. We give several examples of valuatively stable and unstable varieties, including the toric case. We also discuss the role that the δ-invariant plays in the study of valuative stability and K-stability of polarised varieties.

• Item Type: Articles
• Additional Information: RD was funded by a Royal Society University Research Fellowship. EL held a visiting position at Churchill College in Cambridge, she was also supported by CNRS (IEA-International Emerging Actions grant number 295351) and a CIMI mobility grant.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Dervan, Dr Ruadhaí
• Authors: Dervan, R., and Legendre, E.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Mathematische Annalen
• Publisher: Springer
• ISSN: 0025-5831
• ISSN (Online): 1432-1807
• Published Online: 06 January 2022
• Copyright Holders: Copyright © 2022 The Authors
• First Published: First published in Mathematische Annalen 385(1-2): 357-391
• Publisher Policy: Reproduced under a Creative Commons License