[Paper Review] Automorphism group of a toric variety

M. T Sancho, J. P. Moreno+1 more|arXiv (Cornell University)|Sep 24, 2018

Algebraic Geometry and Number Theory2 references4 citations

TL;DR

This paper fully determines the automorphism group of a complete toric variety over an algebraically closed field of characteristic zero. It shows that the identity component $\mathrm{Aut}_k^0X$ is a semidirect product of a unipotent radical (a product of additive groups indexed by divisor classes) and a reductive quotient of a product of general linear groups, with explicit action rules. The finite quotient $\mathrm{Aut}_kX / \mathrm{Aut}_k^0X$ is isomorphic to the group of automorphisms of the lattice preserving the fan, modulo symmetric groups on linear equivalence classes of rays.

ABSTRACT

We calculate the automorphism group of a complete toric variety $X$ with torus $T_M$. We prove that the radical unipotent of $Aut_k^0X$ is a semidirect product of additive groups, the reductive part is a quotient of a product of lineal groups and we give the action of linear groups on the additive groups. We also prove that $Aut_kX/Aut_k^0X$ is a quotient of the group of automorphisms of $M$ leaving invariant the fan.

Motivation & Objective

• To fully determine the automorphism group of a complete toric variety with a given torus action.
• To decompose the identity component $\mathrm{Aut}_k^0X$ into a semidirect product of a unipotent radical and a reductive subgroup.
• To describe the action of the reductive group on the unipotent radical via irreducible representations.
• To characterize the finite quotient $\mathrm{Aut}_kX / \mathrm{Aut}_k^0X$ as a quotient of the group of lattice automorphisms preserving the fan.

Proposed method

• Identify $\mathrm{Aut}_k^0X$ with the quotient of the graded automorphism group of the Cox ring.
• Construct the unipotent radical as a semidirect product of additive groups $V_F$, indexed by linear equivalence classes $F$ of rays in the fan.
• Define reductive subgroups $\mathrm{GL}_F$ acting on each $V_F$, with the action encoded via root systems.
• Use the partial order on the set $W$ of linear equivalence classes of rays to define the semidirect product structure of the unipotent radical.
• Prove that the reductive part of $\mathrm{Aut}_k^0X$ is a quotient of the product $\prod_F \mathrm{GL}_F$, with the action of $\mathrm{GL}_F$ on $V_F$ given by irreducible representations.
• Show that the finite group $\mathrm{Aut}_kX / \mathrm{Aut}_k^0X$ is isomorphic to $\mathrm{Aut}_\Delta M / (S_1 \times \cdots \times S_k)$, where $\mathrm{Aut}_\Delta M$ is the group of lattice automorphisms preserving the fan and $S_i$ is the symmetric group on each linear equivalence class.

Experimental results

Research questions

• RQ1 What is the structure of the identity component $\mathrm{Aut}_k^0X$ of the automorphism group of a complete toric variety?
• RQ2 How is the unipotent radical of $\mathrm{Aut}_k^0X$ constructed from the rays and divisor classes of the fan?
• RQ3 What is the action of the reductive subgroup on the unipotent radical, and how is it realized as a representation?
• RQ4 How is the finite quotient $\mathrm{Aut}_kX / \mathrm{Aut}_k^0X$ related to the automorphisms of the lattice $M$ preserving the fan?
• RQ5 What is the role of symmetric groups $S_i$ in the structure of the full automorphism group?

Key findings

• The unipotent radical of $\mathrm{Aut}_k^0X$ is isomorphic to a semidirect product of additive groups $V_F$, indexed by linear equivalence classes $F$ of rays, ordered by a partial order on $W$.
• The reductive part of $\mathrm{Aut}_k^0X$ is isomorphic to a quotient of the product $\prod_F \mathrm{GL}_F$, where $\mathrm{GL}_F$ acts on the corresponding $V_F$.
• The action of $\mathrm{GL}_F$ on $V_F$ is given by an irreducible representation, explicitly determined by the root system associated to the class $F$.
• The group $\mathrm{Aut}_kX / \mathrm{Aut}_k^0X$ is isomorphic to $\mathrm{Aut}_\Delta M / (S_1 \times \cdots \times S_k)$, where $\mathrm{Aut}_\Delta M$ is the group of lattice automorphisms preserving the fan.
• The symmetric group $S_i$ on each linear equivalence class $F_i$ is realized as a subgroup of $\mathrm{Aut}_k^0X$ via permutations of the corresponding variables in the Cox ring.
• The automorphism group $\mathrm{Aut}_kX$ is completely determined by the fan and the lattice $M$, with the finite part arising from combinatorial symmetries of the fan.