Log K-stability of GIT-stable divisors 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 $0<\epsilon<c_1$.