Spherical Homogeneous Varieties

• Spherical homogeneous varieties are normal G-homogeneous spaces with an open dense Borel orbit, generalizing toric and symmetric varieties.
• They are characterized by combinatorial invariants such as the weight lattice, valuation cone, and colors that encode their equivariant embeddings.
• Applications include equivariant compactifications, harmonic analysis, and representation theory, demonstrating their pivotal role in modern algebraic geometry.

A spherical homogeneous variety is a normal G-homogeneous space $X = G/H$ for a connected reductive algebraic group $G$ (over an algebraically closed field, or more generally over an arbitrary base), such that a Borel subgroup $B \subseteq G$ possesses an open dense orbit in $X$—that is, $B$ acts on $G/H$ with an open dense orbit. This fundamental class of varieties exhibits rich combinatorial and geometric structures, generalizing both toric and symmetric varieties, and serves as the basic object for the Luna–Vust theory of equivariant embeddings. Spherical homogeneous varieties are central in invariant theory, representation theory, and algebraic geometry.

1. Definition and Main Characterizations

Let $G$ be a connected reductive group, $H \subseteq G$ a closed (possibly non-reduced) subgroup, $B \subseteq G$ a Borel subgroup. The quotient $X^0 = G/H$ is called a spherical homogeneous variety if any of the following equivalent conditions holds:

• $B$ acts on $X^0$ with a dense open orbit ($B \cdot x_0$ is open in $G/H$ for $x_0 = eH$).
• $X^0$ contains only finitely many $B$-orbits.
• The field of rational functions $k(X^0)$ contains a free abelian group $\mathcal{X}(X^0)$ of $B$-eigenfunctions of full rank (i.e., $$k(X^0)^{(B)} = \bigoplus_{\chi \in \mathcal{X}(X^0)} k \cdot f_\chi$$ is a multiplicity-free $B$-module).

Over arbitrary bases, the notion of sphericity extends to so-called spherical spaces: a $G$-space $X \to S$ over a base scheme $S$ is spherical if every geometric fiber $X_{\bar{s}}$ is a spherical $G_{\bar{s}}$-variety, i.e., normal, connected, separated, and $B_{\bar{s}}$-homogeneous with a dense open $B_{\bar{s}}$-orbit (Wedhorn, 2015).

2. Combinatorial Invariants: Weight Lattice, Valuation Cone, and Colors

The Luna–Vust theory classifies spherical homogeneous varieties and their equivariant embeddings through discrete combinatorial data:

• Weight lattice $\mathcal{X}(X^0)$: the group of $B$-weights for nonzero $B$-eigenfunctions in $k(X^0)$, a free abelian group of rank $r = \mathrm{rk}_G(X^0)$. Explicitly,

$$\mathcal{X}(X^0) = \{\chi \in X^*(T) \mid \exists\, f \in k(X^0)^\times,\, b\cdot f = \chi(b)f \ \forall b \in B\}.$$

• Valuation cone $V(X^0)$: the convex rational polyhedral cone in $$\mathrm{Hom}_\mathbb{Z}(\mathcal{X}(X^0), \mathbb{Q})$$ spanned by the images of all $G$-invariant discrete valuations of $k(X^0)$. The cone $V(X^0)$ is strictly convex and of full rank (Tange, 2016, Knop, 2013, Timashev, 11 Nov 2025).
• Colors $\mathcal{D}(X^0)$: the finite set of $B$-stable, non-$G$-stable prime divisors in $X^0$ ("colors"). Each color $D$ defines a vector $$\rho_D \in \mathrm{Hom}_\mathbb{Z}(\mathcal{X}(X^0), \mathbb{Q})$$ given by evaluating $B$-eigenfunctions along $D$ (Tange, 2016, Wedhorn, 2015, Kaveh et al., 2016).

These invariants fully encode the geometry of spherical homogeneous varieties and control their equivariant completions.

3. Classification of Equivariant Embeddings: Colored Fans

All normal $G$-varieties $X \supset X^0$ that contain a fixed spherical homogeneous open orbit $X^0 = G/H$ as an open dense $G$-orbit and are themselves spherical are classified by colored fans:

• A colored cone is a pair $(C,F)$, where $C \subset V(X^0)$ is a strictly convex rational cone generated by valuations corresponding to closures of $G$-stable divisors and by $\rho_D$ for colors $D$ in a subset $F \subset \mathcal{D}(X^0)$, subject to compatibility conditions.
• The set $F$ records colors whose closures meet the closed $G$-orbit of the embedding.
• A general $G/H$-embedding is classified by a finite collection of compatible colored cones—i.e., a colored fan—covering a subcone of $V(X^0)$.

Over arbitrary fields (with Galois group $\Gamma$), spherical embeddings are classified by $\Gamma$-invariant colored fans (Wedhorn, 2015, Tange, 2016, Knop, 2013).

4. Structure Theory: Spherical Roots, Little Weyl Group, and Valuation Polyhedral Geometry

The spherical roots $$\Sigma(X^0) \subset \mathcal{X}(X^0) \otimes \mathbb{Q}$$ are the primitive elements corresponding to codimension-one faces ("walls") of the valuation cone $V(X^0)$. These determine the combinatorial structure of $V(X^0)$:

• The little Weyl group $W_{X^0}$ is the finite reflection group generated by orthogonal reflections $s_\sigma$ associated to each $\sigma \in \Sigma(X^0)$ (Knop, 2013).
• The valuation cone is a fundamental domain for this Weyl group: for $\mathrm{char}\,k \neq 2$, $V(X^0)$ is a single Weyl chamber for $W_{X^0}$ (Knop, 2013).
• The root system $R_{X^0} = W_{X^0} \cdot \Sigma(X^0)$ governs the distinguishing features of the G-orbit structure and its embeddings.

Dihedral angle constraints and localizations at colors and valuations further refine the structure, including a color-root compatibility dictated by the behavior of simple roots moving colors (Knop, 2013).

5. Geometric, Topological, and Cohomological Properties

Spherical homogeneous varieties and their embeddings manifest strong geometric and cohomological features:

• Frobenius Splittings: Every smooth toroidal embedding under mild conditions admits a Frobenius splitting compatible with all $G$-stable subvarieties and, under further hypotheses, a $B$-canonical splitting (Tange, 2016).
• Cohomology Vanishing: For projective spherical varieties and semi-ample line bundles, higher cohomology vanishes ($H^i(Y, \mathcal L) = 0$ for $i > 0$), and restriction maps $H^0(X, \mathcal L) \to H^0(Y, \mathcal L)$ are surjective if $X$ is simple or $Y$ is irreducible (Tange, 2016).
• Resolutions: Any normal $G/H$-embedding admits a $G$-equivariant rational resolution by a smooth quasi-projective toroidal embedding (Tange, 2016).

Topologically, the orbit space $X/K$ by a maximal compact $K \subset G$ is canonically homeomorphic to the valuation cone $V_{X^0}$, with the natural orbit-type stratification corresponding to the face stratification of $V_{X^0}$ (Timashev, 11 Nov 2025).

6. Examples, Closure Properties, and Applications

Examples:

• Toric varieties ($G = T$, $H = \{e\}$) are spherical with $V_X = \mathbb{R}^n$ and colors trivial.
• Flag varieties ($G/P$) are spherical; $V_X$ is a point, and the Borel acts transitively on the dense Bruhat cell (Achinger et al., 2013).
• Reductive group embeddings, symmetric spaces $G/G^\theta$, and various products of flag varieties are all spherical in appropriate settings (Tange, 2016, Achinger et al., 2013).

Closure properties:

• The class of spherical homogeneous varieties is stable under parabolic induction and includes symmetric spaces for $\mathrm{char}\,k \neq 2$ (Tange, 2016).
• Products such as $G/P_1 \times G/P_2$ are spherical in precisely determined cases (minuscule flag varieties, Stembridge classification) and have well-controlled Borel orbit closure properties, including normality and Cohen–Macaulayness for simply-laced groups (Achinger et al., 2013).

Applications extend to harmonic analysis on homogeneous spaces, equivariant compactifications, tropical and moment geometry, and multiplicity-free representation theory (Knop et al., 2013, Kaveh et al., 2016, Kaveh et al., 2015).

7. Real, Relative, and Arithmetic Aspects

In the real algebraic setting, a real spherical variety is a normal real $G$-variety on which a minimal parabolic subgroup $P$ has an open orbit. The local structure theorem establishes a canonical isomorphism for the open $P$-orbit as a fiber bundle $Q \times_L S$, factoring the geometry into parabolic, Levi, and "elementary spherical" components. This enables explicit classification of real orbits and their combinatorial invariants (Knop et al., 2013, Cupit-Foutou et al., 2019).

Under field extensions, sphericity is an open and closed condition in algebraic families; the set of points where the fiber is a spherical homogeneous space is both open and closed in the base (Wedhorn, 2015). Galois descent and classification over arbitrary fields is governed by colored fans invariant under the Galois group, with subtle examples even in the presence of inseparable subgroups (Wedhorn, 2015).

In summary, spherical homogeneous varieties comprise a robust class with a rich combinatorial, geometric, and arithmetic framework, underpinning much of the structure theory for reductive group actions on algebraic and analytic varieties.