On the Shape of the K-semistable Domain and Wall Crossing for K-stability

Abstract

Fixing two positive integers d and k, a positive number v, and a positive integer I, we prove that the K-semistable domain of the log pair $(X, \sum _{j=1}^kD_j)$ is a rational polytope lying in the k-dimensional simplex $\overline{\Delta ^k}$, where X is a Fano variety of dimension d, $D_j\sim _{\mathbb {Q}} -K_X$, $(-K_X)^d=v$, $I(K_X+D_j)\sim 0$, and $(X, \sum _{j=1}^kc_jD_j)$ is a K-semistable log Fano pair for some $c_j\in [0,1)\cap {\mathbb {Q}}$. Moreover, we show that there are only finitely many polytopes that may appear as the K-semistable domains for such log pairs. Based on this, we establish a wall crossing theory for K-moduli with multiple boundaries.

Keywords

Log Fano pair / K-stability / K-semistable domain / K-moduli / Wall crossing / 14J45

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