Two results regarding the variation of K-moduli

Abstract

In this note, we prove two results regarding the variation of K-moduli. The first one reveals the relationship between the chamber decomposition for K-semistable domains and the variation of GIT. The second one presents the relationship between the K-moduli generically parametrizing K-semistable smooth Fano complete intersections of the form $S_1\cap ...\cap S_k$ and the K-moduli generically parametrizing K-semistable log Fano manifolds of the form $({\mathbb{P}}^n,{\sum }_{j=1}^k{x}_j{S}_j)$, where $x_j\in (0,1)\cap \mathbb{Q}$ and $S_j\subset {\mathbb{P}}^n$ is a hypersurface of degree $d_j$ for each $1\le j\le k$.

v4, 23 pages, title changed and largely revised, several arguments rewritten and a cubic-hyperplane example added. Two authors added