# K-stability of cubic threefolds

## TL;DR

This paper establishes that the moduli space of cubic threefolds defined by K-stability matches the GIT moduli space, providing explicit stability criteria and implications for Kähler-Einstein metrics.

## Contribution

It proves the equivalence of K-stability and GIT stability for cubic threefolds and introduces a new volume estimate for three-dimensional singularities.

## Key findings

- K-stability coincides with GIT stability for cubic threefolds
- All smooth cubic threefolds admit Kähler-Einstein metrics
- Provides explicit classification of singular KE cubic threefolds

## Abstract

We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit K\"ahler-Einstein metric as well as provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension three of the volumes of kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of three dimensional canonical and terminal singularities, which was established during the study of the explicit three dimensional minimal model program.