Luna-Vust Data in Spherical Varieties

• Luna-Vust data are combinatorial invariants that classify equivariant embeddings of spherical homogeneous spaces using lattices, cones, and colored divisors.
• They transform complex geometric operations, such as Cox ring analysis and real form classification, into explicit combinatorial procedures.
• Applications include determining smoothness, factoriality, and the structure of spherical varieties, enabling precise classification and analysis.

Luna-Vust data constitute the foundational combinatorial invariants classifying equivariant embeddings of spherical homogeneous spaces for a connected reductive group $G$ over an algebraically closed field of characteristic zero. For a normal irreducible $G$-variety $X$ containing a dense $B$-orbit for a fixed Borel subgroup $B \subset G$ and maximal torus $T \subset B$, these invariants distill all relevant geometric and representation-theoretic structure into a collection of lattices, cones, and marked divisors. The Luna-Vust theory provides a dictionary to translate geometric operations—such as passage to the spectrum of the Cox ring or the study of real forms—into purely combinatorial transformations, enabling explicit classification and structure theorems for spherical varieties (Gagliardi, 2016, Moulin, 2023).

1. Canonical Luna-Vust Invariants

Given $X$ as above, the Luna-Vust data are encoded in the tuple $(M, N, V, \Delta, \rho, \varsigma, \Sigma)$ where:

• Weight lattice $M$: Consists of weights of $B$-semi-invariant rational functions in the function field $k(X)$, i.e., $M \subset X^*(B)$. Its dual is $$N := \operatorname{Hom}_{\mathbb{Z}}(M, \mathbb{Z})$$, with natural pairing $$\langle \cdot, \cdot \rangle : N \times M \rightarrow \mathbb{Z}$$.
• Valuation cone $V$: The set of $G$-invariant discrete valuations $\nu : k(X)^* \rightarrow \mathbb{Q}$, injected into $N_{\mathbb{Q}}$ via $\langle \iota(\nu), \chi \rangle := \nu(f_\chi)$ for $f_\chi$ any $B$-semi-invariant of weight $\chi$. $V$ forms a strictly convex, finitely generated polyhedral cone. Its dual cone in $M_{\mathbb{Q}}$ is the tail cone $T := -V^{\vee}$.
• Set of colors $\Delta$: The $B$-invariant prime divisors on $X$, equipped with two structure maps: $\rho:\Delta \rightarrow N$, where $$\langle \rho(D), \chi \rangle = \operatorname{ord}_D(f_\chi)$$, and $\varsigma:\Delta \rightarrow \mathcal{P}(S)$, with $S$ the set of simple roots, and $\varsigma(D)$ records simple roots $\alpha$ with $P_\alpha \cdot D \ne D$.
• Spherical roots $\Sigma$: The unique minimal set of primitive elements in $M$ generating the tail cone $T$.

This tuple suffices to classify $G$-equivariant embeddings of $G/H$, fully encoding the interaction of orbits, divisors, and invariant functions (Gagliardi, 2016).

2. Luna-Vust Data for Spectra of Cox Rings

Given the Cox ring $$R(X) = \bigoplus_{[D] \in \operatorname{Cl}(X)} \Gamma(X, \mathcal{O}_X(D))$$ and the affine spectrum $\overline{X} := \operatorname{Spec} R(X)$, Brion’s construction furnishes a natural action by an enlarged group $\widehat{G}$ that is reductive and commutes with the grading torus $$\mathbb{T} = \operatorname{Spec} k[\operatorname{Cl}(X)]$$. The Luna-Vust data for $\overline{X}$, denoted by the corresponding barred objects, transform as follows (Gagliardi, 2016):

• The new weight lattice $\overline{M}$ is freely generated by weights $e_D$ associated to canonical sections $1_D$ for each $D \in \Delta$, i.e., $$\overline{M} = \bigoplus_{D \in \Delta} \mathbb{Z} \cdot e_D$$, and the monoid $\overline{M}^+$ is generated by the $e_D$.
• The pullback $\pi^* : M \rightarrow \overline{M}$ is determined by $$\pi^*(\chi) = \sum_{D \in \Delta} \langle \rho(D), \chi \rangle \, e_D$$.
• Dualizing, $\pi_* : \overline{N} \rightarrow N$ satisfies $\pi_*(e_D^*) = \rho(D)$, with $(e_D^*)$ dual to $(e_D)$.
• The valuation cone $\overline{V}$ is given by $\pi_*^{-1}(V) \subset \overline{N}_{\mathbb{Q}}$, with $\pi^*(T) = \overline{T}$.
• Colors split according to the set $$\Delta^{S} := \{ D \in \Delta : \varsigma(D) = \{ \alpha \}, 2\alpha \in \Sigma^{2a} \}$$, with $D \in \Delta^S$ mapping to two distinct colors in $\overline{\Delta}$, otherwise to a unique one.

Every $D' \in \overline{\Delta}$ arises exactly once via this splitting mechanism. This explicit behavior determines the transformation of combinatorial invariants under Cox ring iteration.

3. Divisor Class Group of the Cox Spectrum

The divisor class group $\operatorname{Cl}(\overline{X})$ is governed by the Luna-Vust formalism: select a subset $\overline{\Delta}_1 \subset \overline{\Delta}$ so that the $\rho$-images form a basis of $\overline{N}$; remaining divisors generate $\operatorname{Cl}(\overline{X})$ freely. For $X$ spherical, $\overline{\Delta}_1$ consists of those $D' \in \overline{\Delta}$ associated to $D \in \Delta \setminus \Delta^S$, yielding:

$$\operatorname{Cl}(\overline{X}) \cong \mathbb{Z}^{|\Delta^S|},$$

and establishing that $R(X)$ is factorial if and only if $\Delta^S = \varnothing$ (Gagliardi, 2016).

4. Spherical Skeletons and Combinatorial Determination

The spherical skeleton $RS_X = (R, S, T, \Delta, c, \varsigma)$ serves as a minimal combinatorial package for spherical varieties:

• $R$ is the root system of $G$ with $S$ its simple roots,
• $T$ the tail cone in $M_{\mathbb{Q}}$,
• $\Delta$ an abstract finite set with structure maps $$c:\Delta \to (\mathrm{Span}T)^* \subset M_{\mathbb{Q}}^*$$, $c(D) = \rho(D)|_{\mathrm{Span}T}$, and $\varsigma: \Delta \rightarrow \mathcal{P}(S)$.

It follows that two varieties with isomorphic skeletons, in the sense that their root systems and maps $c, \varsigma$ are compatible, have equivariantly isomorphic Cox spectra and thus isomorphic (non-graded) Cox rings [(Gagliardi, 2016), Thm. 3.6]. This gives a combinatorial classification in terms of skeleton data, independently of explicit geometric realizations.

5. Smoothness Criteria and Factorial Affine Reduction

For any spherical $X$ one defines an invariant $\{ X \}$ using a canonical global section $s \in \Gamma(X, \omega_X^\vee)$ whose divisor is

$$\operatorname{div} s = \sum_{D \in \Delta} m_D D, \quad m_D \in \mathbb{Z}_{>0},$$

and

$$Q^* := \bigcap_{D \in \Delta} \{ v \in \operatorname{Span}T : \langle c(D), v \rangle \geq -m_D \}.$$

Then

$$\{ X \} := \sup_{v \in Q^* \cap T} \sum_{D \in \Delta} (m_D - 1 + \langle c(D), v \rangle).$$

The conjectures state:

• If $X$ is complete spherical then $\{ X \} \le \dim X - \operatorname{rank} X$, with equality if and only if $X$ is toric.
• For $X$ factorial affine with a $G$-fixed point, the same inequality holds with equality precisely when $X \cong \mathbb{A}^n$.

These can be reduced to the factorial case: any spherical skeleton $R$ can be modified to a factorial skeleton $R'$ with $\{ R \} \le \{ R' \}$ and dimension is not increased. Thus, to check the conjecture, only factorial skeletons need to be considered [(Gagliardi, 2016), Sec. 5].

6. Luna–Vust Data in the Classification of Real Forms

Luna-Vust invariants underlie the classification of real forms of spherical varieties, as demonstrated for minimal smooth complete $\mathrm{SL}_2$-threefolds (Moulin, 2023). The key combinatorial objects—weight and valuation lattices, colored cones, and associated fans—translate under real forms via the induced Galois action, which either preserves or permutes the skeleton data. For $\mathrm{SL}_2$-varieties of complexity one, the one-dimensional nature of the valuation group $N \cong \mathbb{Z}$ enables explicit enumeration of colored fans and detection of real structures by analysis of Galois-invariance. Rationality and non-emptiness of the real locus are direct consequences of how the involution acts on the colored skeleton.

Table: Principal Luna-Vust Invariants for a Spherical Variety

Invariant	Description	Transformation under Cox Spectrum
$M$	Weight lattice of $B$-semi-invariants	$$\overline{M} = \bigoplus_{D \in \Delta} \mathbb{Z}\cdot e_D$$
$N$	Dual lattice, $N = \operatorname{Hom}_{\mathbb{Z}}(M, \mathbb{Z})$	$$\overline{N} = \operatorname{Hom}_{\mathbb{Z}}(\overline{M}, \mathbb{Z})$$
$V$	Cone of $G$-invariant valuations	$\overline{V} = \pi_*^{-1}(V)$
$\Delta$	Set of colors ($B$-invariant prime divisors)	Colors split per $\Delta^S$
$\rho$	Map $\Delta \rightarrow N$, via order of vanishing	$$\overline{\rho}: \overline{\Delta} \to \overline{N}, \overline{\rho}(D') = e_D^*$$
$\varsigma$	Map recording associated simple roots	$\overline{\varsigma} = \varsigma$ under splitting

Further Context and Significance

Luna-Vust data are the central language for spherical embeddings, Cox ring structure, and the combinatorial classification of equivariant varieties. Their transformation under various constructions (Cox ring, real form, reduction to affine factorial case) enables a unified framework subsuming both algebraic and geometric invariants, culminating in explicit algorithms for classification, determination of smoothness, and analysis of automorphisms and real loci (Gagliardi, 2016, Moulin, 2023).