On the shape of the K-semistable domain and wall crossing for K-stability

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 $(X,{\sum }_{j=1}^k{D}_j)$ is a rational polytope lying in the $k$-dimensional simplex $\overline{{\mathrm{\Delta }}^k}$, 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 $(X,{\sum }_{j=1}^k{c}_j{D}_j)$ 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 which 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.