Log K-stability of GIT-stable divisors on Fano varieties

TL;DR

This paper establishes a connection between GIT stability of divisors on Fano varieties and their K-stability, providing criteria that relate these two notions under specific conditions.

Contribution

It proves that GIT stability of divisors on a K-polystable Fano variety is equivalent to K-stability of the pair with a scaled divisor, with a universal constant depending only on the variety and the integer l.

Findings

GIT stability characterized by K-stability of pairs with scaled divisors

Existence of a universal constant c_1 for stability equivalence

Applicable to divisors in the linear system |-lK_X| on Fano varieties

Abstract

For a given K-polystable Fano variety $X$ and a natural number $l$ such that $(X, \frac{1}{l} B)$ is log canonical for some $B\in |-lK_X|$, we show that there exists a rational number $0<c_1<1$ depending only on $X$ and $l$, such that $D\in |-lK_X|$ is GIT-(semi/poly)stable under the action of Aut(X) if and only if the pair $(X, \frac{\epsilon}{l}D)$ is K-(semi/poly)stable for some rational $0<\epsilon<c_1$.