Valuative stability of polarised varieties

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.

Automated summary

The article introduces the concept of valuative stability and its relationship with K stability, and provides several examples. Additionally, the article discusses the role of the delta-invariant in the study of valuative stability and K stability.