Log K-stability of GIT-stable divisors on Fano varieties

TL;DR

This work establishes a precise link between GIT-stability of divisors in $|-lK_X|$ and K-stability of the corresponding small-perturbation log Fano pairs on a K-polystable Fano variety $X$, proving the existence of a universal $0<c_1<1$ such that $D$ is GIT-stable iff $(X, \frac{\epsilon}{l}D)$ is K-stable for all $0<\epsilon<c_1$. Using the CM-line bundle as the bridge between GIT weights and Futaki invariants, along with boundedness of Fano degenerations and a degeneration analysis showing the central fiber must be $X$ for small $\epsilon$, the authors construct an isomorphism between the corresponding K-moduli and GIT moduli spaces for these small perturbations, generalizing prior ADL19 results to arbitrary K-polystable Fano varieties. The results provide a practical criterion to identify GIT-stable divisors via K-stability, clarifying the relationship between K-stability and GIT-stability in moduli problems and enabling concrete moduli-space comparisons and constructions. The approach highlights the role of stability invariants, degenerations, and CM-line bundles in unifying two prominent moduli theories in algebraic geometry.

Abstract

For a given K-polystable Fano variety $X$ and a natural number $l$ such that $(X, \frac{1}{l} B)$ is log canonical for some $B\in |-lK_X|$, we show that there exists a rational number $0<c_1<1$ depending only on $X$ and $l$, such that $D\in |-lK_X|$ is GIT-(semi/poly)stable under the action of Aut(X) if and only if the pair $(X, \fracε{l}D)$ is K-(semi/poly)stable for some rational $0<ε<c_1$.

Theorems & Definitions (21)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Definition 2.5
• Definition 2.6
• Remark 2.7
• Example 2.8
• Definition 3.1