Nonproportional wall crossing for K‐stability

Abstract 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 nonproportional to the anticanonical 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 semialgebraic sets (although not necessarily polytopes). As a consequence, we obtain a finite semialgebraic 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 geometric invariant theory (GIT)‐stability and K‐stability in nonproportional setting when the coefficient of the boundary is sufficiently small.