K-stability for varieties with a big anticanonical class

TL;DR

This paper extends algebraic K-stability theory to certain pairs with big anticanonical classes, showing that K-semistability ensures a klt anticanonical model with equivalent stability.

Contribution

It introduces a framework for K-stability of pairs with big anticanonical classes and demonstrates their stability properties align with their klt anticanonical models.

Findings

K-semistability implies the existence of a klt anticanonical model

The stability property of the model matches that of the original pair

Extension of K-stability theory to a broader class of varieties

Abstract

We extend the algebraic K-stability theory to projective klt pairs with a big anticanonical class. While in general such a pair could behave pathologically, it is observed in this note that K-semistability condition will force them to have a klt anticanonical model, whose stability property is the same as the original pair.