Valuative invariants with higher moments

TL;DR

This paper introduces a new family of valuative invariants based on higher moments, which can detect K-stability of Fano varieties and generalize existing criteria like alpha and delta invariants.

Contribution

It defines a novel family of invariants using the $p$-th moment of expected vanishing order, extending the tools for studying K-stability.

Findings

Invariants vary continuously in the big cone.

They can detect K-stability of Fano varieties.

Related to $d_p$-geometry of geodesic rays.

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 can detect the 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.