Automorphism groups of toroidal horospherical varieties

Abstract: We establish a structure theorem for the connected automorphism groups of smooth complete toroidal horospherical varieties, that is, toric fibrations over rational homogeneous spaces. A key ingredient is an extension of the notion of Demazure roots from toric varieties to toroidal horospherical varieties. In particular, we provide a criterion for the reductivity of the connected automorphism groups of such varieties. As an application, we prove the K-unstability of certain $\mathbb{P}<sup>1$-bundles over rational homogeneous spaces.

• The paper extends Demazure roots to smooth complete toroidal horospherical varieties and uses them to describe the connected automorphism group through explicit Levi and unipotent components.
• The authors derive a dimension formula in which each admissible root contributes an irreducible G-module, and prove that the automorphism group is reductive exactly when every B⁺-root is semisimple.
• The results characterize reductivity for decomposable projective bundles and produce new K-unstable Fano bundles, including examples over bases of Fano index one.

Setting and motivation

The paper by Barban, Hwang, and Kwon studies the connected automorphism group $\mathrm{Aut}^0(X)$ of a smooth complete toroidal horospherical variety $X$, that is, a smooth equivariant toric bundle $X \to G/P$ over a rational homogeneous space. Horospherical varieties interpolate between two classical classes with sharply contrasting behavior of automorphism groups: for a rational homogeneous space, $\mathrm{Aut}^0$ is semisimple and hence reductive; for a complete toric variety, $\mathrm{Aut}^0$ is described combinatorially by Demazure roots and is generally non-reductive (2602.07879). Prior to this work, a description of the unipotent part of $\mathrm{Aut}^0(X)$ was largely missing, and no criterion for reductivity existed in this generality.

The main technical contribution is an extension of Demazure roots from toric varieties to toroidal horospherical varieties — which the authors claim is the first such extension to smooth complete varieties — together with a structure theorem giving a Levi decomposition of $\mathrm{Aut}^0(X)$ in terms of these generalized roots.

The key combinatorial object: $B^+$-roots

A horospherical subgroup $H \subset G$ has normalizer a parabolic $P = N_G(H)$, and $X$0 fibers equivariantly over $X$1 with fiber the torus $X$2. A toroidal horospherical variety is, by Luna–Vust theory, the parabolic induction $X$3 for an $X$4-toric variety $X$5. The relevant data are thus: the Demazure root set $X$6 of the toric fiber, and the color map $X$7 sending each $X$8-color (preimage of a Schubert divisor) to its induced valuation functional on the weight lattice.

The central definition introduces:

• $X$9: Demazure roots $X \to G/P$0 of the fiber with $X \to G/P$1;
• semisimple roots $X \to G/P$2: those with $X \to G/P$3 also satisfying this condition;
• unipotent roots: the complement.

A lemma of independent interest identifies dominance with the color pairing condition: $X \to G/P$4 is $X \to G/P$5-dominant if and only if $X \to G/P$6, via Timashev's identification of color pairings with coroot pairings.

Lifting theorem and Lie algebra decomposition

The lifting criterion states that for a Demazure root $X \to G/P$7 of the fiber, the associated $X \to G/P$8-normalized $X \to G/P$9-action on $\mathrm{Aut}^0$0 extends to a $\mathrm{Aut}^0$1-normalized $\mathrm{Aut}^0$2-action on $\mathrm{Aut}^0$3 preserving every fiber of $\mathrm{Aut}^0$4 if and only if $\mathrm{Aut}^0$5. Semisimple roots yield $\mathrm{Aut}^0$6-normalized actions.

Computationally, the authors use the relative tangent sequence and Blanchard's lemma, reducing to $\mathrm{Aut}^0$7, then apply Grauert's theorem and Borel–Weil–Bott. Since $\mathrm{Aut}^0$8 has fiber $\mathrm{Aut}^0$9 by Demazure's computation on the toric fiber, they obtain as $\mathrm{Aut}^0$0-modules:

$\mathrm{Aut}^0$1

where $\mathrm{Aut}^0$2 and $\mathrm{Aut}^0$3 denotes the irreducible $\mathrm{Aut}^0$4-module of highest weight $\mathrm{Aut}^0$5. This yields the dimension formula:

$\mathrm{Aut}^0$6

Note the structural contrast with the toric formula: each root contributes an entire irreducible $\mathrm{Aut}^0$7-module rather than a one-dimensional weight space, reflecting the conjugation action of $\mathrm{Aut}^0$8.

Levi decomposition and reductivity criterion

The structure theorem establishes that $\mathrm{Aut}^0$9, with Lie algebra $\mathrm{Aut}^0(X)$0 as a multiplicity-free $\mathrm{Aut}^0(X)$1-module, generated by the root subgroups $\mathrm{Aut}^0(X)$2 and their $\mathrm{Aut}^0(X)$3-conjugates. A Levi subgroup of $\mathrm{Aut}^0(X)$4 is generated by the torus $\mathrm{Aut}^0(X)$5 and the subgroups $\mathrm{Aut}^0(X)$6 for semisimple $\mathrm{Aut}^0(X)$7; adjoining the image of $\mathrm{Aut}^0(X)$8 gives a connected reductive group that is a Levi subgroup of $\mathrm{Aut}^0(X)$9 whenever $\mathrm{Aut}^0(X)$0 is surjective.

This surjectivity hypothesis fails only for the three classical exceptions ($\mathrm{Aut}^0(X)$1 short-root quadrics, $\mathrm{Aut}^0(X)$2 Lagrangian Grassmannians, $\mathrm{Aut}^0(X)$3), and even there it is removable: replacing $\mathrm{Aut}^0(X)$4 by the group of completely regular automorphisms $\mathrm{Aut}^0(X)$5 preserves the horospherical-toroidal structure and colors. Consequently, without any hypothesis, a Levi subgroup of $\mathrm{Aut}^0(X)$6 is generated by $\mathrm{Aut}^0(X)$7 and the $\mathrm{Aut}^0(X)$8.

The immediate corollary is a clean reductivity criterion: $\mathrm{Aut}^0(X)$9 is reductive if and only if every $B^+$0-root of $B^+$1 is semisimple, i.e., no unipotent roots exist. This directly generalizes Nill's characterization for toric varieties and fills the gap noted in the literature regarding the unipotent radical.

Application to projective bundles and K-stability

For a decomposable projective bundle $B^+$2 over a rational homogeneous space $B^+$3, the fiber is $B^+$4 with Demazure roots $B^+$5. Applying the criterion:

Root set	Elements	Condition
$B^+$6	$B^+$7 ($B^+$8)	$B^+$9 nef
Semisimple roots	$H \subset G$0	$H \subset G$1

Hence $H \subset G$2 is reductive if and only if for all $H \subset G$3 with $H \subset G$4, the bundle $H \subset G$5 is not nef. For $H \subset G$6 and $H \subset G$7, reductivity forces $H \subset G$8. In the toric case $H \subset G$9, reductivity holds exactly when either all twisting degrees vanish or some pair satisfies $P = N_G(H)$0; notably, in dimension at least 3 Fano examples with reductive automorphism group beyond $P = N_G(H)$1 exist, e.g. $P = N_G(H)$2.

Combining the Matsushima obstruction (K-polystable implies reductive automorphism group), Delcroix's result that K-polystability equals K-semistability for horospherical Fano varieties, and Debarre's Fano criterion for projective bundles, the paper derives: for $P = N_G(H)$3 rational homogeneous and $P = N_G(H)$4 nontrivial nef with $P = N_G(H)$5 ample, the Fano $P = N_G(H)$6-bundle $P = N_G(H)$7 is K-unstable. This strengthens results of Zhang–Zhou, who required Fano index at least 2 of the base and specific twistings. An explicit index-1 example is given: $P = N_G(H)$8 with $P = N_G(H)$9 yields a smooth K-unstable Fano threefold-bundle.

Limitations and open questions

Two caveats bear on the strength of the results. First, the analysis is restricted to smooth complete toroidal horospherical varieties; singular or non-toroidal spherical cases are not covered. Second, while Pezzini settled the construction of Levi subgroups for general toroidal spherical varieties, the description of the unipotent radical via $X$00-actions remains open outside the horospherical case treated here and the wonderful case (where $X$01 is semisimple). The authors pose explicitly: how to construct a Levi subgroup of $X$02 and describe $X$03 geometrically for an arbitrary smooth complete spherical variety — noting that a full answer would yield an effective reductivity criterion in that generality. Additionally, the extension of Demazure roots given here relies on smoothness of $X$04 through its use of Borel–Weil–Bott and the tangent sheaf computation, so a singular version would require different techniques.

Conclusion

The paper provides a complete, computable description of the connected automorphism group of a smooth complete toroidal horospherical variety: dimension formula, explicit generators for the unipotent radical and a Levi factor, all governed by the interplay between Demazure roots of the toric fiber and the color map of the base. The resulting reductivity criterion yields new concrete families of K-unstable Fano $X$05-bundles, including examples over bases of Fano index 1 previously out of reach.