Non-proportional wall crossing for K-stability

In this paper, we present a general wall crossing theory for K-stability and K-moduli of log Fano pairs whose boundary divisors can be non-proportional to the anti-canonical divisor. Along the way, we prove that there are only finitely many K-semistable domains associated to the fibers of a log bounded family of couples. Under the additional assumption of volume bounded from below, we show that K-semistable domains are semi-algebraic sets (although not necessarily polytopes). As a consequence, we obtain a finite semi-algebraic chamber decomposition for wall crossing of K-moduli spaces. In the case of one boundary divisor, this decomposition is an expected finite interval chamber decomposition. As an application of the theory, we prove a comparison theorem between GIT-stability and K-stability in non-proportional setting when the coefficient of the boundary is sufficiently small.

💡 Research Summary

This paper develops a comprehensive wall‑crossing theory for K‑stability and K‑moduli of log Fano pairs in which the boundary divisors are not required to be proportional to the anticanonical divisor. The authors begin by fixing integers (d) (dimension), (k) (number of boundary components) and a rational polytope (P\subset