Valuative Invariants with Higher Moments

Abstract

In this article we introduce a family of valuative invariants defined in terms of the p-th moment of the expected vanishing order. These invariants lie between \(\alpha \) and \(\delta \)-invariants. They vary continuously in the big cone and semi-continuously in families. Most importantly, they give sufficient conditions for K-stability of Fano varieties, which generalizes the \(\alpha \) and \(\delta \)-criterions in the literature. They are also related to the \(d_p\)-geometry of maximal geodesic rays.