K-stable valuations and Calabi-Yau metrics on affine spherical varieties

TL;DR

The paper develops a comprehensive framework connecting K-stability with Calabi–Yau metrics on affine spherical varieties. It defines and analyzes polarized spherical log cones, proves that the existence of log Calabi–Yau cone metrics is equivalent to K-stability, and establishes the existence and uniqueness of a $G$-equivariant K-stable degeneration to a horospherical cone. Using Donaldson–Sun theory, it shows that the valuation induced by a complete Calabi–Yau metric on a $G$-spherical manifold is $G$-invariant and that the asymptotic cone is uniquely determined up to $G$-equivariant isomorphism, with the negative valuation corresponding to the Reeb valuation on the limit cone. The results yield new nonexistence and rigidity statements (e.g., no $G_2$-horospherical CY metric) and give a detailed description of K-stable valuations on smooth affine spherical varieties and spherical modules. Together, these contributions advance the understanding of the Yau–Tian–Donaldson correspondence in the noncompact, highly symmetric setting and illuminate the geometry of asymptotic cones for Calabi–Yau metrics.

Abstract

After providing an explicit K-stability condition for a $\mathbb{Q}$-Gorenstein log spherical cone, we prove the existence and uniqueness of an equivariant K-stable degeneration of the cone, and deduce uniqueness of the asymptotic cone of a given complete $K$-invariant Calabi-Yau metric in the trivial class of an affine $G$-spherical manifold, $K$ being the maximal compact subgroup of $G$. Next, we prove that the valuation induced by $K$-invariant Calabi-Yau metrics on affine $G$-spherical manifolds is in fact $G$-invariant. As an application, we point out an affine smoothing of a Calabi-Yau cone that does not admit any $K$-invariant Calabi-Yau metrics asymptotic to the cone. Another corollary is that on $\mathbb{C}^3$, there are no other complete Calabi-Yau metrics with maximal volume growth and spherical symmetry other than the standard flat metric and the Li-Conlon-Rochon-Székelyhidi metrics with horospherical asymptotic cone. This answers the question whether there is a nontrivial asymptotic cone with smooth cross section on $\mathbb{C}^{3}$ raised by Conlon-Rochon when the symmetry is spherical.

Figures (4)

• Figure 1: Degeneration of $(W,\xi)$ along a valuation $\nu$ in the Futaki vanishing locus $F$ on $\mathcal{V}_W$.
• Figure 2: Colored cone $(\mathcal{C}_Y, \mathcal{D}_Y)$ of a symmetric rank one conical embedding of $G/H \times \mathbb{C}^{*}$ where $G = \operatorname{SL}_2$, $H = N_{SL_2}(T)$. Here the lattice of $G/H$ is generated by the unique restricted root $\alpha$ ($= 2 \widehat{\alpha}$, $\widehat{\alpha}$ being the unique root of $\operatorname{SL}_2$). There is a unique color $d_{\alpha}$ of $G/H$ with $\rho(d_{\alpha}) = \alpha^{\vee}$. The polytope $Q_X$ is then $\left\{ t \alpha^{\vee}2, \left|t\right| \leq 1\right\}$, and $\mathcal{C}_Y^{\vee}$ is the cone over $Q_X$ in $\mathbb{Z}( \alpha, \eta)_{\mathbb{R}}$. Note also that $\varpi = \alpha$ and $\Delta_X = \left\{t \alpha, 0 \leq t \leq 2\right\}$.
• Figure 3: Restricted root system of $G_2$ symmetric spaces.
• Figure 4: Colored cone $(\mathcal{C}, \mathcal{D})$ of the symmetric manifold $\mathbb{C}^3$ with open orbit $\operatorname{SO}_3/ \operatorname{SO}_2 \times \mathbb{C}^{*}$. The K-stable valuations correspond to the vectors of coordinates $(1,-1)$ and $(0,-1)$ in the lattice generated by $(\alpha^{\vee}/2, \chi)$.

Theorems & Definitions (88)

• Theorem 1: Prop. \ref{['theorem_kstability_spherical_cone']}
• Remark 1.1
• Remark 1.2
• Theorem 2: Prop. \ref{['prop_stable_degeneration_equals_equivariant_stable_degeneration']}
• Definition 1.3
• Theorem 3: Prop. \ref{['prop_kstable_valuation_invariant']} and \ref{['prop_asymptotic_cone_spherical']}
• Corollary 4
• Example 1.4
• Corollary 5: Prop. \ref{['proposition_calabiyaumetrics_c3']}
• Theorem 2.1