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

## TL;DR

This paper proves that the K-semistable domain for certain Fano varieties with multiple divisors is a rational polytope within a simplex, and introduces a wall crossing theory for K-moduli, revealing finiteness and structural properties.

## Contribution

It establishes the rational polytope structure of K-semistable domains and develops a wall crossing theory for K-moduli with multiple boundary divisors.

## Key findings

- K-semistable domain is a rational polytope in the simplex
- Finiteness of possible K-semistable polytopes
- Development of a wall crossing theory for K-moduli

## 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 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.