Toric varieties modulo reflections

TL;DR

The paper shows that when a finite reflection group $W$ preserves a lattice $M$ and a full-dimensional $M$-rational polytope $P$, the toric quotient $X_P / W$ is canonically isomorphic to the toric variety $X_{P \cap D}$, with $D$ a fundamental domain for $W$; this unifies and extends Lorenz’s affine result and recovers several prior toric-quotient findings. It also establishes a parallel statement for affine and projective settings via invariant semigroup algebras, proving $\mathbb{Z}[D \cap S] \cong \mathbb{Z}[S]^W$ for saturated affine semigroups $S$, which yields $X / W \cong \mathrm{Spec}\, \mathbb{Z}[D \cap S]$ and in the projective case $X_P / W \cong X_{D \cap P}$. Beyond the algebraic results, the authors analyze quotients on real points, proving contractibility for real permutohedral quotients $X_P^{\mathbb{R}} / W$ when $P$ is a permutohedron of an irreducible crystallographic root system, and formulate a conjecture that such real quotients are wedge-sum homotopy types of spheres with Euler characteristic $2 - \ell$, where $\ell$ counts $P$-vertices in $D$. The approach combines invariant-theoretic isomorphisms, fundamental-domain combinatorics, and topological gluing arguments to connect toric geometry with the topology of real loci.

Abstract

Let $W$ be a finite group generated by reflections of a lattice $M$. If a lattice polytope $P \subset M \otimes_{\mathbb Z}\mathbb R$ is preserved by $W$, then we show that the quotient of the projective toric variety $X_P$ by $W$ is isomorphic to the toric variety $X_{P \cap D}$, where $D$ is a fundamental domain for the action of $W$. This answers a question of Horiguchi-Masuda-Shareshian-Song, and recovers results of Blume, of Song, of the second author, and of Gui-Hu-Liu. We also study quotients of real toric varieties, proving that $X_P^{\mathbb R} / W$ is contractible when $P$ is a permutohedron.

Figures (2)

• Figure 1: The lattice points in the shaded cone are $D \cap S$, and the bold lattice points marked with coefficients represent $\Psi(\chi^{u})$ for $u = e_1 + 3e_2$.
• Figure 2: Three possible shapes of the quotient polygon $P \cap D$. The blue arrows indicate the fundamental weights $\lambda_1,\lambda_2$ that define the fundamental domain $D$.

Theorems & Definitions (60)

• Theorem 1.1
• Theorem \ref{lorenz}$'$
• Theorem \ref{lorenz}$'$
• Theorem \ref{main}$'$
• Corollary \ref{main}$'$
• Theorem \ref{main}$'$
• Theorem \ref{main}$'$
• Corollary \ref{main}$'$
• Conjecture \ref{main}$'$
• Remark \ref{main}$'$