Valuative stability of polarised varieties

Authors
Dervan, Ruadhai 
Eveline, Legendre 

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Type
Article
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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 beta 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 delta invariant plays in the study of valuative stability and K-stability of polarised varieties.

Publication Date
2023-02
Online Publication Date
2022-01-06
Acceptance Date
2021-09-21
Keywords
4902 Mathematical Physics, 4904 Pure Mathematics, 49 Mathematical Sciences
Journal Title
Mathematische Annalen
Journal ISSN
0025-5831
Volume Title
Publisher
Springer
Rights
Publisher's own licence
Sponsorship
Royal Society (URF\R1\201041)
Royal Society University Research Fellowship