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Non-proportional wall crossing for K-stability

Yuchen Liu, Chuyu Zhou

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 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.

Non-proportional 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 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.

Paper Structure

This paper contains 21 sections, 53 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Wall crossing of K-moduli
  3. Properties of the K-semistable domain
  4. Constructible properties
  5. GIT-stability vs K-stability
  6. Conjectures
  7. The relation with LZ24
  8. Organization and convention
  9. K-stability of Fano varieties and boundedness
  10. Constructible properties
  11. Algebraic log Fano locus
  12. Finiteness of K-semistable domains
  13. Wall crossing for K-semistability
  14. Boundedness of couples
  15. On the shape of the K-semistable domain
  16. ...and 6 more sections

Key Result

Theorem 1.1

(Theorem thm: fcd, Theorem thm: kps fcd) Suppose $\widetilde{\mathcal{G}}\subset \mathcal{G}$ is a subset which is log bounded and satisfies the volume condition $(\clubsuit)$. Then there exists a decomposition of $P$ into finite disjoint subsets (depending only on $\widetilde{\mathcal{G}}$), denote

Theorems & Definitions (117)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Proposition 1.10
  • ...and 107 more