# K-semistability of optimal degenerations

## TL;DR

This paper explores the relationship between K-stability and degenerations of polarized varieties, showing that if a destabilising degeneration exists, its limit is K-semistable, advancing understanding of stability conditions in algebraic geometry.

## Contribution

It proves that the limit of a destabilising degeneration of a K-unstable variety is necessarily K-semistable, clarifying the structure of destabilisations.

## Key findings

- Existence of a destabilising degeneration implies the limit is K-semistable.
- Provides a link between destabilisation and K-semistability in algebraic geometry.
- Enhances understanding of the stability landscape for polarized varieties.

## Abstract

K-polystability of a polarised variety is an algebro-geometric notion conjecturally equivalent to the existence of a constant scalar curvature K\"ahler metric. When a variety is K-unstable, it is expected to admit a "most destabilising" degeneration. In this note we show that if such a degeneration exists, then the limiting scheme is itself relatively K-semistable.