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 $B\in |-l{K}_X|$, we show that there exists a rational number $0<c_1<1$ depending only on $X$ and $l$, such that $D\in |-l{K}_X|$ 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 $0<\epsilon <c_1$.

arXiv commentFinal version. Title changed, to appear in Math. Res. Lett

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