Two results regarding the variation of K-moduli

TL;DR

This paper explores two theoretical results on K-moduli, linking chamber decompositions with GIT variation and relating K-moduli of smooth Fano complete intersections to log Fano manifolds.

Contribution

It establishes new connections between chamber decompositions, GIT variation, and different types of K-moduli parametrizations.

Findings

Chamber decomposition relates to GIT variation.

K-moduli of Fano intersections links to log Fano manifolds.

Provides theoretical insights into K-moduli structure.

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 $(\mathbb{P}^n, \sum_{j=1}^kx_jS_j)$, 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 $1\leq j\leq k$.