# Some criteria for uniform K-stability

## TL;DR

This paper establishes criteria for uniform K-stability of log Fano pairs, linking it to the positivity of the beta-invariant and exploring its relation to destabilization conjectures and K-stability equivalence.

## Contribution

It provides new criteria for uniform K-stability based on the beta-invariant and investigates its connection with destabilization and K-stability conjectures.

## Key findings

- Uniform K-stability is equivalent to the beta-invariant having a positive lower bound.
- The paper explores the relationship between optimal destabilization and K-stability conjectures.
- Provides criteria that can be used to verify uniform K-stability in practice.

## Abstract

We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to $\beta$-invariant having a positive lower bound. Then we study the relation between optimal destabilization conjecture and the conjectural equivalence between uniform K-stability and K-stability in the twisted setting.