Optimal destabilizing centers and equivariant K-stability (Q2238047)

The notion of K-stability is introduced as an algebraic condition for the existence of Kähler-Einstein metrics on Fano varieties. K-stability of a Fano variety \(X\) is defined by positivity of Futaki invariant of test configurations of \(X\), which are \(\mathbb{G}_m\)-equivariant polarized one parameter degenerations of \(X\). One of the main challenges in the theory of K-stability is to find an effective way of checking K-stability for an explicit Fano variety. If the Fano variety \(X\) is equipped with an action of an algebraic group \(G\), then it is expected that to test K-stability it suffices to check those test configurations that are \(G\)-equivariant, known as the \(G\)-equivariant K-stability. This expectation was proved in cases when \(X\) is a smooth Fano manifold and \(G\) reductive by \textit{V. Datar} and \textit{G. Székelyhidi} [Geom. Funct. Anal. 26, No. 4, 975--1010 (2016; Zbl 1359.32019)], when \(G\) is an algebraic torus by \textit{C. Li} et al. [J. Am. Math. Soc. 34, No. 4, 1175--1214 (2021; Zbl 1475.14062)], or when \(G\) is a finite group by \textit{Y. Liu} and \textit{Z. Zhu} [Int. J. Math. 33, No. 1, Article ID 2250007, 21 p. (2022; Zbl 1487.14089)]. Note that the approaches of Datar-Székelyhidi and Liu-Zhu heavily relies on analytic methods from solutions of the Yau-Tian-Donaldson conjecture. The main theorem of this paper verifies this expectation in full generality using purely algebraic methods, showing the following result. Theorem 1.1. Let \(X\) be a Fano variety with an action of an algebraic group \(G\). Then \begin{itemize} \item[1.] If \(X\) is \(G\)-equivariantly K-semistable, then \(X\) is K-semistable. \item[2.] If \(X\) is \(G\)-equivariantly K-polystable and \(G\) is reductive, then \(X\) is K-polystable. \end{itemize} To prove Theorem 1.1, the author proceed in two steps. The first step addresses the uniqueness of the minimal optimal destabilizing center. The stability threshold \(\delta(X)\) of a Fano variety \(X\), introduced by \textit{K. Fujita} and \textit{Y. Odaka} [Tohoku Math. J. (2) 70, No. 4, 511--521 (2018; Zbl 1422.14047)], measures the singularity of an average anti-canonical \(\mathbb{Q}\)-divisor on \(X\). An equivalent definition by \textit{H. Blum} and \textit{M. Jonsson} [Adv. Math. 365, Article ID 107062, 57 p. (2020; Zbl 1441.14137)] gives that \[ \delta(X)=\min_{v} \frac{A_X(v)}{S(v)}, \] where \(v\) runs over all real valuations of \(K(X)\). According to Fujita-Odaka and Blum-Jonsson, \(X\) is K-semistable if and only if \(\delta(X)\geq 1\). If \(\delta(X)<1\), an optimal destabilizing center \(Z\subset X\) is the center of a valuation \(v\) achieving the minimum in the above equality. In Theorem 1.5, the author shows that if \(X\) is K-unstable, then there exists a unique minimal optimal destabilizing center of \(X\). Then in the second step, the author shows that the stability threshold around the minimal optimal destabilizing center can be computed asympotically by a \(G\)-equivariant prime divisor. The proofs of these results cleverly combine the connectedness theorem of Kollár-Shokurov, compatible divisors explored in an earlier joint work of the author with \textit{H. Ahmadinezhad} [``K-stability of Fano varieties via admissible flags'', Preprint, \url{arXiv:2003.13788}], and induction on dimension from inversion of adjunction.