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 \((X, \sum _{j=1}^kD_j)\) $(X,{\sum }_{j=1}^k{D}_j)$ is a rational polytope lying in the k-dimensional simplex \(\overline{\Delta ^k}\) $\overline{{\mathrm{\Delta }}^k}$ , where X is a Fano variety of dimension d, \(D_j\sim _{\mathbb {Q}} -K_X\) $D_j{\sim }_{\mathbb{Q}}-K_X$ , \((-K_X)^d=v\) ${(-K_X)}^d=v$ , \(I(K_X+D_j)\sim 0\) $I(K_X+D_j)\sim 0$ , and \((X, \sum _{j=1}^kc_jD_j)\) $(X,{\sum }_{j=1}^k{c}_j{D}_j)$ is a K-semistable log Fano pair for some \(c_j\in [0,1)\cap {\mathbb {Q}}\) $c_j\in [0,1)\cap \mathbb{Q}$ . 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.