Affine Complexity-Zero Horospherical Varieties

• Affine complexity-zero horospherical varieties are normal affine algebraic varieties characterized by a reductive group action with an open Borel orbit and a combinatorial structure arising from dominant weights and rational cones.
• They are classified via rational polyhedral cones, colored cone theory, and the geometric roles of Demazure roots, which directly inform their orbit structure and valuation properties.
• The analysis of these varieties reveals flexibility and infinite transitivity in their automorphism groups, bridging invariant theory and convex geometry for broader applications.

An affine complexity-zero horospherical variety is a normal affine algebraic variety $X$ endowed with an action of a connected reductive algebraic group $G$ such that $G$ acts with an open orbit isomorphic to $G/H$ for a closed subgroup $H \subset G$ containing a maximal unipotent subgroup $U \subset G$, and the action is of complexity zero (i.e., the transcendence degree of the field of $B$-invariant rational functions is zero, or equivalently, $B$ has an open orbit where $B$ is a Borel subgroup of $G$). Affine complexity-zero horospherical varieties generalize toric varieties and play a central role in the structure theory of spherical varieties and algebraic transformation groups. Their geometry, automorphism groups, and flexibility properties are governed by a combinatorial description that interlaces invariant theory, valuation theory, and convex geometry (Gaifullin et al., 22 Nov 2025, Gaifullin et al., 2018, Avdeev et al., 2023).

1. Structural and Combinatorial Classification

Let $\mathbb{k}$ be an algebraically closed field of characteristic zero. Suppose $G \simeq G' \times T$ for $G'$ semisimple and $T$ a torus, with $B \simeq B' \times T$ a Borel subgroup, and $X$ an affine complexity-zero horospherical $G$-variety. By the Luna–Vust theory, every such $X$ arises as $X = \operatorname{Spec} S_{\mathcal{F}}$, where

$$S_{\mathcal{F}} := \bigoplus_{\lambda\in \mathcal{F}} S_\lambda \subseteq \mathbb{k}[G]$$

for a finitely generated subsemigroup $\mathcal{F} \subset X^+(B)$ of dominant weights. The algebra $S_{\mathcal{F}}$ is $B$-graded, $G$-stable, and $X$ inherits a combinatorial structure controlled by the weight lattice $M = \mathbb{Z}\mathcal{F}$, its dual $N = \operatorname{Hom}(M, \mathbb{Z})$, and the associated rational cones

$$\sigma^\vee := \mathbb{Q}_{\ge 0} \mathcal{F} \subset M_\mathbb{Q}, \quad \sigma \subset N_\mathbb{Q}.$$

Faces of $\sigma$ correspond bijectively (Popov–Vinberg) to $G$-orbits in $X$, with extremal rays $\rho \preceq \sigma$ of dimension one representing closures $D_\rho$ of codimension-one $G$-orbits (Gaifullin et al., 22 Nov 2025).

For normal $X$, the Luna–Vust colored cone classification further refines the structure. Each normal affine complexity-zero horospherical $G$-variety corresponds to a strictly convex rational polyhedral cone $C \subset N_\mathbb{Q}$, decorated with a finite set of "colors" $\mathcal{F}_X$—$B$-stable but not $G$-stable prime divisors—subject to compatibility conditions. The coordinate ring is then

$$\mathbb{k}[X] = \bigoplus_{\lambda \in M \cap C^\vee} S_\lambda,$$

where $C^\vee$ is the dual cone and $S_\lambda$ is the weight space of highest weight $\lambda$ (Gaifullin et al., 2018).

2. Orbit Structure, Valuations, and Demazure Roots

Each codimension-one $G$-orbit closure $D_\rho$ corresponds to an extremal ray $\rho$ of $\sigma$. The valuations associated to $B$-orbits and colors translate geometric features of $X$ to faces and rays of $\sigma$. The rich geometry of affine complexity-zero horospherical varieties is encoded via Demazure roots: for each ray $\rho_i = \mathbb{Q}_{\ge 0} v_i$, the set of Demazure roots is

$$\mathcal{R}_{\rho_i}(\sigma) := \{ e \in M \mid \langle e, v_i \rangle = -1, \ \langle e, v_j \rangle \ge 0\text{ for all }j \ne i\}.$$

A Demazure root $e \in \mathcal{R}_{\rho_i}(\sigma)$ yields a homogeneous locally nilpotent derivation (LND) on $\mathbb{k}[X]$ whose kernel cuts out exactly $D_{\rho_i}$. The existence of such LNDs depends on the "almost saturation" condition: there exists a saturation point $p \in \hat\rho \cap \mathcal{F}$ in the dual face; this is combinatorial in nature and critical for geometric properties such as regularity and the existence of automorphisms (Gaifullin et al., 22 Nov 2025, Avdeev et al., 2023).

3. Regularity Cone and Smooth Locus

The smooth locus $X^{\mathrm{reg}}$ is governed by the regularity cone $\gamma(X) \subset N_\mathbb{Q}$, defined as the cone generated by all significant rays (i.e., rays $\rho \preceq \sigma$ for which the associated $G$-orbit has codimension one and is smooth) together with a canonical subcone $\theta$ corresponding to the torus factor, dual to the fundamental Weyl chamber. The coordinate ring of the regular locus is

$$\mathbb{k}[X^{\mathrm{reg}}] = \bigoplus_{\lambda \in \gamma^\vee \cap M} S_\lambda,$$

with regularity determined by vanishing order along nonregular codimension-one orbits. The structure of $\gamma$ explicitly controls both singularities and transitive group actions by automorphisms derived from LNDs (Gaifullin et al., 22 Nov 2025).

4. Flexibility: Group Actions and Infinite Transitivity

A variety $X$ is called flexible if the subgroup $\operatorname{SAut}(X)$ generated by all one-parameter unipotent automorphisms ($\mathbb{G}_a$-subgroups, corresponding to LNDs) acts transitively (equivalently, infinitely transitively) on $X^{\mathrm{reg}}$. For affine complexity-zero horospherical $G$-varieties, the main result is the flexibility criterion (Gaifullin et al., 22 Nov 2025): $$X \text{ is flexible} \iff \gamma(X) \text{ is not contained in any hyperplane of } N_\mathbb{Q} \iff \forall f \notin \mathbb{k},~V(f)\cap X^{\mathrm{reg}}\neq\varnothing.$$ The proof proceeds by exhibiting, for each significant ray, a Demazure-root LND generating a $\mathbb{G}_a$-subgroup whose orbits span tangent directions at generic points, including those coming from both the semisimple and toric factors, and showing that the union of these flows acts transitively on $X^{\mathrm{reg}}$. The absence of nontrivial invertible regular functions on $X^{\mathrm{reg}}$ (i.e., $ML(X) = \mathbb{k}$) is both a consequence and a technical ingredient of flexibility (Gaifullin et al., 22 Nov 2025, Gaifullin et al., 2018).

Infinite transitivity follows as a corollary: on the smooth locus of any normal affine complexity-zero horospherical $G$-variety of dimension at least $2$, $\operatorname{SAut}(X)$ acts infinitely transitively (Gaifullin et al., 2018).

5. Automorphism Groups and Root Subgroups

The automorphism group $\operatorname{Aut}^0(X)$ contains $G$ and all $\mathbb{G}_a$-subgroups corresponding to LNDs described via Demazure roots. For horospherical $X$, all $B$-root subgroups (i.e. $\mathbb{G}_a$-subgroups normalized by a Borel $B$) decompose into horizontal (non-$B$-orbit-preserving; parameterized by Demazure roots in the cone) and vertical (preserving the open $B$-orbit; parameterized combinatorially by roots shifted by the highest root and constrained by inequalities on colors) (Avdeev et al., 2023).

For the case $G$ of semisimple rank $1$ (essentially $SL_2$-type) and $X$ complete, all $B$-root subgroups (both horizontal and vertical) are classified combinatorially, leading to an explicit splitting of the automorphism Lie algebra as

$$\mathfrak{a} = \operatorname{Lie}(G) \oplus \bigoplus_{\alpha} L_\alpha \oplus \bigoplus_{\beta} K_\beta,$$

with $L_\alpha$ and $K_\beta$ the $G$-modules generated by the vector fields from horizontal and vertical root subgroups, respectively. The bracket relations generalize the Demazure commutator pattern from toric to horospherical cases (Avdeev et al., 2023).

6. Special and Limiting Cases

Several familiar varieties arise as special or limiting cases:

• Semisimple case ($T$ trivial): The regularity cone $\gamma$ is full-dimensional; all such varieties are flexible (Gaifullin et al., 22 Nov 2025).
• Toric case ($G = T$): The flexibility criterion coincides with the Boldyrev–Gaifullin condition for possibly non-normal toric varieties; flexibility is determined by the presence of enough significant rays (Gaifullin et al., 22 Nov 2025).
• Normal case ($\mathcal{F}$ saturated): All codimension-one orbits are regular, so $\gamma = \sigma$ and flexibility reduces to $\sigma$ not being contained in any hyperplane, equivalent to the unit group of $k[X]$ being $k^\times$ (Gaifullin et al., 22 Nov 2025, Gaifullin et al., 2018).

Explicit examples demonstrate both flexible and non-flexible affine complexity-zero horospherical varieties, depending on the configuration of generators of the semigroup $\mathcal{F}$ and the geometry of the associated cones.

7. Research Directions and Connections

The theory of affine complexity-zero horospherical varieties interconnects invariant theory, automorphism group structure, and convex geometry. Flexibility results generalize earlier work on normal horospherical varieties and toric varieties (Gaifullin et al., 22 Nov 2025, Gaifullin et al., 2018). The combinatorics of Demazure roots and colored cones encode both the existence of large automorphism groups and geometric features such as singularities and orbit structure. Classification projects extend further to spherical varieties and to non-normal embeddings.

Recent work provides a complete description of all $B$-root subgroups, both standard (arising from Demazure roots) and nonstandard, yielding explicit structure theorems for $\operatorname{Aut}^0(X)$ that parallel the classical toric situation but with new phenomena arising from the nature of horospherical stabilizers and color data (Avdeev et al., 2023). This suggests further potential in generalized invariant-theoretic constructions and in the study of dynamical and transitivity properties of large automorphism groups in algebraic geometry.