# Thresholds, valuations, and K-stability

## TL;DR

This paper investigates valuation-based invariants like the log canonical and stability thresholds on complex projective varieties, providing new characterizations of K-stability and explicit formulas in the toric case.

## Contribution

It introduces the stability threshold as a generalization of Fujita and Odaka's notion, linking it to K-stability and volume bounds, and proves the attainment of infima for ample line bundles.

## Key findings

- The stability threshold characterizes K-semistability and uniform K-stability.
- Infima of the thresholds are attained when L is ample.
- Explicit formulas are obtained for toric varieties in terms of moment polytopes.

## Abstract

Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter generalizes a notion by Fujita and Odaka, and can be used to characterize when a Q-Fano variety is K-semistable or uniformly K-stable. It can also be used to generalize volume bounds due to Fujita and Liu. The two thresholds can be written as infima of certain functionals on the space of valuations on X. When L is ample, we prove that these infima are attained. In the toric case, toric valuations acheive these infima, and we obtain simple expressions for the two thresholds in terms of the moment polytope of L.