Galois actions on analytifications and tropicalizations

TL;DR

This paper develops a framework connecting Galois actions with tropicalizations of algebraic varieties, enabling the recovery of Galois representations from combinatorial data and showing that tropicalizations encode all arithmetic information of the variety.

Contribution

It introduces Galois-equivariant tropicalizations, linking Galois actions to tropical geometry, and proves they capture the full arithmetic structure of varieties over Henselian fields.

Findings

Galois-equivariant tropicalizations are Galois-equivariant maps.

They can recover Galois orbits of points faithfully.

The Berkovich analytification is the inverse limit of all such tropicalizations.

Abstract

This paper initiates a research program that seeks to recover algebro-geometric Galois representations from combinatorial data. We study tropicalizations equipped with symmetries coming from the Galois-action present on the lattice of $1$-parameter subgroups inside ambient Galois-twisted toric varieties. Over a Henselian field, the resulting tropicalization maps become Galois-equivariant. We call their images Galois-equivariant tropicalizations, and use them to construct a large supply of Galois representations in the tropical cellular cohomology groups of Itenberg, Katzarkov, Mikhalkin, and Zharkov. We also prove two results which say that under minimal hypotheses on a variety $X_{0}$ over a Henselian field $K_{0}$, Galois-equivariant tropicalizations carry all of the arithmetic structure of $X_{0}$. Namely: (1) The Galois-orbit of any point of $X_{0}$ valued in the separable closure of $K_{0}$ is reproduced faithfully as a Galois-set inside some Galois-equivariant tropicalization of our variety. (2) The Berkovich analytification of $X_{0}$ over the separable closure of $K_{0}$, equipped with its canonical Galois-action, is the inverse limit of all Galois-equivariant tropicalizations of our variety.

Equations

$$\text{trop}(x)\operatorname{\overset{{}_{\text{def}}}{=}}\big{(}-\log|x_{1}|,\dots,-\log|x_{r}|\big{)}\ \in\ {\mathbb{R}}^{r}.$$

$$\text{trop}:X_{0}^{\text{an}}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\!\text{Trop}(X_{0}).$$

$$G\stackrel{{}_{\text{def}}}{=}\text{Gal}(K/K_0)$$

$$X\stackrel{{}_{\text{def}}}{=}{X}_{0,K}.$$

$$|a|_{L}\ \operatorname{\overset{{}_{\text{def}}}{=}}\ \Big{|}\prod_{h\in H}h(a)\Big{|}^{1/d},$$

$$\text{trop}:X(K)⟶\text{Trop}(X).$$

$$G\text{rotatebox}-90.0\circlearrowright M.$$

$$\left\{\!\!\begin{array}[]{rl}\mbox{{\bf objects:}}&\!\!\!\mbox{algebraic $K_{0}$-tori}\\ \mbox{{\bf morphisms:}}&\!\!\!\mbox{algebraic $K_{0}$-group}\\ &\!\!\!\mbox{homomorphisms}\end{array}\!\!\right\}\ \xrightarrow{\ \ T_{0}\ \mapsto\ M(T)\ \ }\ \left\{\!\!\begin{array}[]{rl}\mbox{{\bf objects:}}&\!\!\!\mbox{finite rank lattices with}\\ &\!\!\!\mbox{continuous $G$-action}\\ \mbox{{\bf morphisms:}}&\!\!\!\mbox{$G$-equivariant}\\ &\!\!\mbox{homomorphisms}\end{array}\!\!\right\}.$$

$$T_{0}(M)\ \ \ \operatorname{\overset{{}_{\text{def}}}{=}}\ \ \ T(M)\big{/}G.$$

$$\text{Trop}(X,ı)\subset N_{\mathbb{R}}\stackrel{{}_{\text{def}}}{=}{\text{Hom}}_{\mathbb{Z}}(M,\mathbb{R}).$$

$$\text{trop}:X^{\text{an}}\stackrel{{}_{\text{def}}}{=}(X_{0,K}{)}^{\text{an}}⟶N_{\mathbb{R}}$$

$$M\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\!{K}[M]\xrightarrow{\ \ |-|_{x}\ \ }{\mathbb{R}}_{\geq 0}\xrightarrow{\ -\log\ }{\mathbb{R}}.$$

$${\text{Trop}}_G(X,ı),$$

$$\text{trop}:{X}^{\text{an}}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\!\text{Trop}_{G}({X},\imath)$$

$$\ \ G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}x\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\!{X}^{\text{an}}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\!\text{Trop}_{G}({X},\imath)\ \ ?$$

$$X$$

$${X}\ \!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 6.4706pt\raise 5.35417pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.77084pt\hbox{$\scriptstyle{\imath\times\imath^{\prime}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\!\ Y_{\Sigma}\times_{{K}}Y_{\Sigma^{\prime}}\xrightarrow{\ \sim\ }Y_{\Sigma\times\Sigma^{\prime}}$$

$$\underset{{\mathcal{S}}_G}{\underleftarrow{\lim }}{\text{Trop}}_G(X,ı)$$

$$\pi :X^{\text{an}}⟶\underset{{\mathcal{S}}_G}{\underleftarrow{\lim }}{\text{Trop}}_G(X,ı).$$

$$G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}x\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\!{X}^{\text{an}}\xrightarrow{\ \text{trop}\ }\text{Trop}_{G}({X},\imath)$$

$$G\ \ \raisebox{8.0pt}{\rotatebox{-90.0}{$\circlearrowright$}}\ \ H^{p,q}_{\text{trop}}\big{(}\ \!\text{Trop}({X},\imath),F\ \!\big{)}\ \ \ \ \mbox{and}\ \ \ \ G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \ \raisebox{8.0pt}{\rotatebox{-90.0}{$\circlearrowright$}}\ \ H_{p,q}^{\text{trop}}\big{(}\ \!\text{Trop}({X},\imath),F\ \!\big{)}\ \ \ \mbox{for each}\ \ \ p,q\geq 0$$

$$Y\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$Y_{0,L}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$Y_{0,K}^{\prime }\cong Y.$$

$${\text{Aut}}_K(T)\cong T(K)⋊{\text{GL}}_{\mathbb{Z}}(M{)}^{\text{smallersmallersmallersmallersmallerop}}.$$

$$g^{\ast }:Y_{\Sigma }⟶Y_{\Sigma }$$

$$(y_{\mbox{{\smaller\smaller\smaller\smaller\smaller$(-)$}}},\varphi_{\mbox{{\smaller\smaller\smaller\smaller\smaller$(-)$}}})\ \!:\ G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\longrightarrow T({K})\rtimes\text{Aut}(\Sigma),$$

$$\text{Aut}(\Sigma)\ \operatorname{\overset{{}_{\text{def}}}{=}}\ \big{\{}\psi\in\text{GL}_{{\mathbb{Z}}}(N)\ \!:\ \Sigma\mbox{ is invariant under }N_{{\mathbb{R}}}\xrightarrow{\ \psi\ }N_{{\mathbb{R}}}\big{\}}.$$

$$H^{1}\big{(}G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}},\text{Aut}_{{K}}(Y)\big{)},$$

$$\displaystyle\left.\left\{\!\!\begin{array}[]{c}\mbox{arithmetic toric $K_{0}$-varieties $T^{\prime}_{0}\ \raisebox{8.0pt}{\rotatebox{-90.0}{$\circlearrowright$}}\ Y_{0}$}\\ \mbox{admitting ${K}$-variety isomorphisms}\\ \mbox{$T^{\prime}\cong T$ and $Y\cong Y_{\Sigma}$ s.t. the diagram}\\ \begin{aligned} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 20.23076pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\&\crcr}}}\ignorespaces{\hbox{\kern-17.86745pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{T^{\prime}\times_{K}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 46.25299pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-7.94444pt\raise-19.7622pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\wr}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.6911pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 46.25299pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 53.26688pt\raise-19.7622pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\wr}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 53.26688pt\raise-29.6911pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-20.23076pt\raise-39.52441pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{T\times_{K}Y_{\Sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 44.23076pt\raise-39.52441pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 44.23076pt\raise-39.52441pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{Y_{\Sigma}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{aligned}\mbox{\ \ commutes}\end{array}\!\!\right\}\right/\mbox{iso.}\ \ \ \xrightarrow{\ \ \ \mbox{$\sim$}\ \ \ }\ \ \ H^{1}\big{(}G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}},{T}({K})\!\rtimes\!\text{Aut}(\Sigma)\big{)}.$$

Full text

1. 1 Introduction
2. 2 Galois-equivariant tropicalization
3. 3 An extensive supply of Galois-equivariant toric embeddings
4. 4 Proofs of the Main Theorems

1. Introduction

1.1. Tropicalization: frontiers and confines.

Let $K_{0}$ be a non-Archimedean field. Fix an algebraic $r$-dimensional $K_{0}$-torus $T_{0}=\text{Spec}_{\ \!}K_{0}[{\mathbb{Z}}^{r}]$. Each closed subvariety $X_{0}\subset T_{0}$ has an associated polyhedral subspace $\text{Trop}(X_{0})\subset{\mathbb{R}}^{r}$ called the tropicalization of $X_{0}$, and comes equipped with a map on $K_{0}$-rational points $\text{trop}:X_{0}(K_{0})\longrightarrow\text{Trop}(X_{0})$ that takes each $K_{0}$-rational $r$-tuple $x=(x_{1},\dots,x_{r})$ in $X_{0}$ to its coordinatewise valuation

[TABLE]

The map extends to a surjective map on the Berkovich analytification

[TABLE]

In this way, tropicalization is an operation that collapses large swaths of information in the algebraic variety $X_{0}$, reducing the variety to a far more tractable object. The beauty of tropicalization lies in the fact that despite eliminating so much information, it preserves important algebro-geometric features of $X_{0}$. The guiding example of this sort is G. Mikhalkin’s enumerative result [22, §3, Theorem 1] [23, §7.1, Theorem 1]. Motivated by ideas of M. Kontsevich [20], Mikhalkin showed that Gromov-Witten invariants in \mathbb{P}^{2}_{\mbox{{\smaller\smaller\smaller\smaller\smaller{\mathbb{C}}}}} can be recovered as counts of tropical curves. In a similar vein, B. Osserman, S. Payne, and J. Rabinoff showed that intersection numbers between algebraic varieties can be recovered via intersection numbers of their tropicalizations [27] [24] [25].

Given this situation, one is inclined to ask if there are other important geometric features of algebraic varieties that are preserved under tropicalization. Having fixed a separable closure $K\operatorname{\overset{{}_{\text{def}}}{=}}K^{\text{sep}}_{0}$, one obvious candidate is the canonical action of the absolute Galois group

[TABLE]

on the variety

[TABLE]

Unfortunately, there is an immediate obstacle to the recovery of Galois actions in tropicalizations. Indeed, if $K_{0}$ is Henselian, then there exists a unique absolute value on ${K}$ extending the absolute value on $K_{0}$ [12, Lemma 4.1.1]. Given a finite Galois extension $L/K_{0}$ with Galois group $H\operatorname{\overset{{}_{\text{def}}}{=}}\text{Gal}(L/K_{0})$, the restriction of this absolute value to $L$ is given by

[TABLE]

where $d=\#H$. From the form of $|-|_{L}$ it follows that the $G$-action on $L$-rational points of $X$ becomes trivial under the tropicalization map

[TABLE]

As we range over all Galois extensions of $K_{0}$, we see that (1) collapses all Galois orbits, and one can show that the same is true for the Berkovich analytification of ${X}$; all Galois orbits collapse under tropicalization.

This negative observation has lead to a pervasive idea that Galois actions on algebraic varieties are forever lost to tropical geometers. But this is not the case. The tool needed for tropicalizing algebraic varieties in a manner that is sensitive to their Galois structures is already present in the classification of non-split tori [7, Chapter III, §8.12], which plays a central role in the Borel-Chevalley theory of linear algebraic groups, completed in the late ‘50s and early ‘60s [9] [8] [6].

1.2. Galois actions on tropicalizations

Recall that a (not necessarily split) algebraic torus over $K_{0}$ is any algebraic $K_{0}$-group $T_{0}$ such that for some finite separable extension $L/K_{0}$, the algebraic $L$-group $T_{0,L}$ is isomorphic to the split algebraic torus $\mathbb{G}^{r}_{\text{m},L}=\text{Spec}_{\ \!}L[{\mathbb{Z}}^{r}]$ for some $r$. Since $T\operatorname{\overset{{}_{\text{def}}}{=}}T_{0,K}$ comes with a canonical $G$-action, its character lattice M=M(T)\operatorname{\overset{{}_{\text{def}}}{=}}\text{Hom}_{\mbox{{\smaller\smaller\smaller\smaller\smaller{K}\text{-}\mathbb{Grp}}}}({T},\mathbb{G}_{\text{m},\mbox{{\smaller\smaller\smaller\smaller\smaller{K}}}}) comes equipped with a continuous $G$-action

[TABLE]

This gives rise to a functor

[TABLE]

The fundamental classification result for algebraic tori over $K_{0}$ says that the functor (2) is an equivalence of categories [7, §III.8.12]. The quasi-inverse $T_{0}(-)$ takes a finite rank lattice $M$ with Galois-action G\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ M, forms the split torus ${T}(M)\operatorname{\overset{{}_{\text{def}}}{=}}\text{Spec}_{\ \!}{K}[M]$ with twisted $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action[1][1][1]Throughout the paper, $G$-action is synonymous with left $G$-action, and $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action is synonymous with right $G$-action. Maps equivariant with respect to either $G$- or $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-actions are called $G$-equivariant. Given $g\in G$, we write “$g(-)$” to denote its application with respect to a $G$-action, and “$g^{\ast}(-)$” to denote its application with respect to a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action. The general rule of thumb throughout the paper is: $G$-actions occur on spaces of functions, $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-actions occur on underlying topological spaces. coming from the Galois-action on $M$, and then produces an algebraic torus $T_{0}(M)$ over $K_{0}$ via Galois descent:

[TABLE]

If $X_{0}$ is a $K_{0}$-variety, then we can ask for closed embeddings of $X_{0}$ into (not necessarily split) algebraic tori over $K_{0}$, or what is the same, $G$-equivariant closed embeddings of ${X}=X_{0,{K}}$ into $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted tori ${T}(M)=\text{Spec}_{\ \!}{K}[M]$. Given such an embedding \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!{T}(M), the tropicalization of ${X}$ with respect to $\imath$ is a polyhedral subspace that we denote

[TABLE]

The $G$-action on $M$ puts extra structure on $\text{Trop}({X},\imath)$. First, it induces a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action on $N_{{\mathbb{R}}}$. Recall that the tropicalization map

[TABLE]

takes each multiplicative seminorm $|-|_{x}:{K}[M]\longrightarrow{\mathbb{R}}_{\geq 0}$, constituting a point of ${X}^{\text{an}}$, to the composite

[TABLE]

When $K_{0}$ is Henselian, the fact that $\imath$ is $G$-equivariant implies that the subspace $\text{Trop}({X},\imath)\subset N_{{\mathbb{R}}}$ is $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-invariant and that the map (3) is $G$-equivariant. For details, see Lemmas 2.2.5 and 2.2.5. To highlight the fact that $\text{Trop}(X,\imath)$ comes equipped with a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action, we denote it

[TABLE]

and refer to it as a Galois-equivariant or $G$-equivariant tropicalization of $X$. We refer to the resulting $G$-equivariant map

[TABLE]

as the corresponding $G$-equivariant tropicalization map.

Remark 1.2.1**.**

See Examples 2.2.3, 2.2.7, and 2.3.4 below for pictures of Galois-equivariant tropicalizations.

Although described above only for Galois-invariant closed embeddings of $X$ into Galois-twisted tori, our construction of Galois-equivariant tropicalizations extends to all Galois-equivariant closed embeddings of $X$ into Galois-twisted toric varieties. One way these closed embeddings arise naturally is from closed subvarieties of arithmetic toric $K_{0}$-varieties in the sense of E. J. Elizondo, P. Lima-Filho, F. Sottile, and Z. Teitler [11, §3]. We develop the extended theory of Galois-equivariant tropicalization in §2.

1.3. First questions: faithfulness.

Given the existence of Galois-equivariant tropicalizations, one would like to understand just how faithfully they reproduce the Galois-action on a given algebraic variety. Because tropicalization collapses so much of the information inside an algebraic variety, there are various concrete questions one can ask.

Question 1**.**

Assume that $K_{0}$ is Henselian. Given a point $x\in{X}^{\text{an}}$ with finite $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-orbit, for instance a point $x\in X({K})$, does there exist a $G$-equivariant tropicalization $\text{Trop}_{G}({X},\imath)$ such that the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-set $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}x\subset X^{\text{an}}$ maps bijectively onto its image under the $G$-equivariant composite map

[TABLE]

Remark 1.3.1**.**

We provide an affirmative answer to Question 1 in Theorem C below under minimal hypotheses on $X_{0}$ when $K_{0}$ is a perfect field, or when $X$ is quasiprojective. This suggests that one should ask Question 1 with “point” replaced by other geometric structures on ${X}$. We plan to address variants of Question 1 in a future paper. One concrete variant appears in Question 3 below. Let us first offer a global version of Question 1.

Question 2**.**

Can we reconstruct the Berkovich analytification ${X}^{\text{an}}$ together with its natural $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action from Galois-equivariant tropicalizations of ${X}$?

Remark 1.3.2**.**

We provide an affirmative answer to Question 2 in Theorem B below, again under minimal hypotheses on $X_{0}$ when $K_{0}$ is a perfect field, or when $X$ is quasiprojective.

Question 3**.**

Fix a prime $\ell$. Can we realize the $\ell$-adic representations G\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ H^{n}({X}_{\text{\'{e}t}},{\mathbb{Q}}_{\ell}) using representations of $G$ on appropriately defined tropical cohomology groups H^{n}_{\text{trop}}\big{(}\text{Trop}({X},\imath),{\mathbb{Q}}_{\ell}\big{)}?

Remark 1.3.3**.**

We do not know if Question 3 has an affirmative answer. However, we show in Theorem D and Remark 3.2.3 below that the tools we develop to answer Questions 1 and 2 allow us to produce a large supply of representations of $G$ in the tropical cellular cohomology groups of I. Itenberg, L. Katzarkov, G. Mikhalkin, and I. Zharkov [17]. We plan to address Question 3 further in a future paper.

1.4. Main results

Fix a $K_{0}$-variety $X_{0}$. Let $X=X_{0,{K}}$ with its canonical $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action over $\text{Spec}_{\ \!}K_{0}$. Let $\mathbb{TEmb}_{G}({X})$ denote the category where an object is any $G$-equivariant closed embedding into a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric variety \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} (see Definition 2.1.5), and where a morphism from one $G$-equivariant closed embedding \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} to another \imath^{\prime}:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma^{\prime}} is any morphism of $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric varieties $f_{\varphi}:Y_{\Sigma}\longrightarrow Y_{\Sigma^{\prime}}$ (see Definition 2.1.5) fitting into a commutative diagram

[TABLE]

Definition 1.4.1**.**

A system of $G$-equivariant toric embeddings of ${X}$, denoted $\mathcal{S}_{G}$, is any subcategory of $\mathbb{TEmb}_{G}({X})$.

• ($\boldsymbol{\Pi}$)

We say that $\mathcal{S}_{G}$ contains all products if given any pair of $G$-equivariant closed embeddings \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} and \imath^{\prime}:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma^{\prime}} in $\mathcal{S}_{G}$, the composite $G$-equivariant embedding

[TABLE]

is an object in $\mathcal{S}_{G}$, and if each of the $G$-equivariant projections $Y_{\Sigma}\xleftarrow{\ \text{pr}_{1}\ }Y_{\Sigma\times\Sigma^{\prime}}\xrightarrow{\ \text{pr}_{2}}Y_{\Sigma^{\prime}}$ is a morphism in $\mathcal{S}_{G}$.

• ($\boldsymbol{\star}$)

We say that $\mathcal{S}_{G}$ satisfies condition ($\star$) if there exists an affine open cover $\{U_{1},\dots,U_{r}\}$ of ${X}$ such that for each $1\leq i\leq r$ and each regular function $f$ on $U_{i}$, there exists a $G$-equivariant closed embedding \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} such that $U_{i}$ is the inverse image of a torus invariant open subset of $Y_{\Sigma}$ and $f$ is the pullback of a monomial on the dense torus in $Y_{\Sigma}$.

Construction 1.4.2**.**

The category of topological spaces equipped with a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action, i.e., the category of $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-spaces, admits inverse limits. If $\mathcal{S}_{G}$ is a system of $G$-equivariant toric embeddings of ${X}$, then we can form the inverse limit

[TABLE]

in the category of $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-spaces. It comes with a continuous $G$-equivariant map

[TABLE]

Remark 1.4.3**.**

In [29], J. Włodarczyk shows that if $K_{0}$ is algebraically closed and $X_{0}$ is normal, then $X_{0}$ admits a closed embedding into a toric $K_{0}$-variety if and only if $X_{0}$ is an A2-variety, i.e., $X_{0}$ satisfies

• **(A2)**​​​​​

every pair of points $x,y\in X_{0}$ lies in some affine open subset of $X_{0}$.

Włodarczyk’s result can be understood as a generalization of the Chevalley-Kleiman criterion [19, §IV.2, Theorem 3] [15, §I.9, Theorem 9.1]. In [13, §4], P. Gross, S. Payne, and the present author use techniques developed by Włodarczyk in [29, §4] to show that (if one ignores $G$-action throughout), then for any system of closed toric embeddings satisfying the non-$G$-equivariant versions of Conditions ($\Pi$) and ($\star$), the map (4) is a homeomorphism [13, Theorem 1.1]. Using details of Włodarczyk’s result, this allows one to show that for any variety over an algebraically closed non-Archimedean field, admitting a closed embedding into at least one toric variety over an algebraically closed non-Archimedean field, there are abundant of examples of systems of toric embeddings for which the non-equivariant version of (4) is a homeomorphism [13, Theorem 1.2].

Our first two results in the present paper reproduce this theorem in the Galois-equivariant setting.

Theorem A**.**

For any system $\mathcal{S}_{G}$ of $G$-equivariant toric embeddings satisfying Conditions ($\Pi$) and ($\star$) of Definition 1.4.1, the map (4) is a $G$-equivariant homeomorphism.

Theorem B**.**

Suppose that $K_{0}$ is Henselian and that $X_{0}$ is a $K_{0}$-variety such that either $X$ is quasiprojective or $K_{0}$ is perfect and ${X}$ admits a closed embedding into at least one toric variety. Then ${X}$ admits systems of $G$-equivariant toric embeddings satisfying Conditions ($\Pi$) and ($\star$). The category $\mathbb{TEmb}_{G}({X})$ of all $G$-equivariant toric embeddings is one such system.

Remark 1.4.4**.**

Theorems A and B together imply that the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-space given by ${X}^{\text{an}}$ with its $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action can be reconstructed from Galois equivariant tropicalizations of ${X}$, giving an affirmative answer to Question 2.

Because the canonical continuous surjection {X}^{\text{an}}\!\!\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!{X} used in formulating the universal property of the Berkovich analytification [5, Theorem 3.4.1] is $G$-equivariant, Theorem B implies that the underlying topological space of the ${K}$-variety ${X}$ with its $G$-action is encoded, in its entirety, in the totality of Galois equivariant tropicalizations of ${X}$. Succinctly: all Galois-action on a torically embeddable variety over a Henselian field is tropical.

Our third main result gives an affirmative answer to Question 1.

Theorem C**.**

Suppose that $K_{0}$ is Henselian and that $X_{0}$ is a $K_{0}$-variety such that either $X$ is quasiprojective or $K_{0}$ is perfect and ${X}$ admits a closed embedding into at least one toric variety. Then for each point $x\in X^{\text{an}}$ with finite $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-orbit, there exists a $G$-equivariant closed embedding \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} into a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric variety $Y_{\Sigma}$ such that the composite

[TABLE]

is a $G$-equivariant isomorphism of the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-set $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}x$ onto its image in $\text{Trop}_{G}({X},\imath)$.

Remark 1.4.5**.**

Theorem C says that a finite analytic Galois-orbit in an algebraic variety can always be recovered in a single Galois-equivariant tropicalization. Our last main result provides the beginning of an answer to the corresponding question for a different kind of finite geometric structure on a variety:

Theorem D**.**

Suppose that $K_{0}$ is Henselian and that ${X}$ is projective. Let $F$ be any field of characteristic-[math]. Then for each $G$-equivariant closed embedding \imath:X\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma}, we obtain an induced representation

[TABLE]

on the tropical cohomology and homology groups of Itenberg, Katzarkov, Mikhalkin, and Zharkov.

Remark 1.4.6**.**

Because the proofs of Theorems B and C provide us with a large supply of $G$-equivariant closed embeddings of $X$, Theorem D provides us with a large supply of nontrivial, Artin Galois-representations in $F$-vector spaces.

1.5. Outline of the paper

In §2 we review the theories of Galois-twisted and arithmetic toric varieties, we use them to define extended Galois-equivariant tropicalizations, and we prove Theorem D. In §3 we show how to use either a lemma of Payne or algorithms of Włodarczyk or Berchtold and Hausen to construct a large supply of Galois-equivariant toric embeddings. In §4 we prove Theorems A, B, and C.

1.6. Acknowledgements

First, I thank the organizers of the Georgia Algebraic Geometry Symposium 2015, where the early ideas for this paper took shape. Thank you to Efrat Bank, Max Hully, and Dhruv Ranganathan for listening to and commenting on these ideas, and to Kristin Shaw and Frank Sottile for helpful conversations. Alexander Duncan played a crucial role by alerting me to his paper [10] and to the theory of twisted toric varieties. I thank him and Sam Payne for their enthusiasm and for detailed comments on drafts of the paper. Finally, I thank the organizers of the Tropical Geometry Seminar at l’Institut Math matiques de Jussieu, namely Erwan Brugallé, Penka Georgieva, and Ilia Itenberg, for a speaking opportunity that greatly improved the paper.

Research for the paper was conducted at University of Michigan, at l’Institut des Hautes Études Scientifiques, and at l’Institut Henri Poincaré. I thank all three institutions for their hospitality. Support for the author came from NSF RTG grant DMS-0943832 and from Laboratoire d’Excellence CARMIN.

2. Galois-equivariant tropicalization

Having defined Galois-equivariant tropicalizations of closed subvarieties of Galois-twisted tori in §1.2 of the Introduction, we now extend this construction to provide Galois-equivariant tropicalizations of closed Galois-invariant subvarieties of all Galois-twisted toric varieties.

2.1. Toric varieties over non-closed fields

We begin by providing a brief review the theory of toric varieties twisted by a Galois action, as developed by E. J. Elizondo, P. Lima-Filho, F. Sottile, and Z. Teitler in [11], and by A. Duncan in [10]. This theory has a long gestation period that precedes [11] and [10]. A thorough account of this development appears in [11, §1].

Fix a field $K_{0}$, not necessarily equipped with a non-Archimedean absolute value. Fix a separable closure ${K}=K^{\text{sep}}_{0}$. If $Y_{0}$ is a $K_{0}$-variety, with $Y\operatorname{\overset{{}_{\text{def}}}{=}}Y_{0,{K}}$, then the $G$-action G\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ {K} induces the canonical $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y, wherein the action of each $g\in G$ is given by the Cartesian diagram

[TABLE]

A twisted $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action on $Y$ is any $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y induced by the action H^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{0,L} of some finite quotient $H=G/Z\cong\text{Gal}(L/K_{0})$ with fixed field $L={K}^{Z}$, such that for each $h\in H$ the diagram

[TABLE]

commutes. For any twisted $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action on $Y$, the diagram (5) commutes for each $g\in G$, although the diagram is not Cartesian in general. A $K_{0}$-form of $Y$ is any $K_{0}$-variety $Y^{\prime}_{0}$ that admits an isomorphism of ${K}$-varieties

[TABLE]

Definition 2.1.1** ([11, §3]).**

An arithmetic toric variety over $K_{0}$ is a triple $(Y_{0},T_{0},\mu)$ consisting of:

• (i)

a $K_{0}$-variety $Y_{0}$;

• (ii)

a (not-necessarily split) $K_{0}$-torus $T_{0}$;[2][2][2]See §1.2

• (iii)

a faithful $T_{0}$-action \mu:T_{0}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{0} with dense orbit in $Y_{0}$.

We often write an arithmetic toric variety as simply $Y_{0}$, leaving $T_{0}$ and $\mu$ implicit.

Remark 2.1.2**.**

Fix a split $K_{0}$-torus $T_{0}=\text{Spec}_{\ \!}K_{0}[M]$ with character lattice $M$. Define $T\operatorname{\overset{{}_{\text{def}}}{=}}T_{0,{K}}$. Equip $M$ with the trivial $G$-action. Then the automorphism group of $T$ as a ${K}$-scheme is

[TABLE]

It comes with a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action that restricts to the canonical action on the first factor $T({K})$ and to the trivial action on the second factor $\text{GL}_{{\mathbb{Z}}}(M)^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$.

Let T_{0}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{0} be an arithmetic toric variety over $K_{0}$. Then ${Y}=Y_{0,{K}}$ is isomorphic to a toric ${K}$-variety $Y_{\Sigma}$ for some fan $\Sigma$ in $N_{{\mathbb{R}}}$. For each $g\in G$, the resulting morphism of $K_{0}$-schemes

[TABLE]

is $T$-equivariant. Up to translation by $T({K})$, it is a toric morphism, which is to say that up to translation by some element $y_{g}\in T({K})$, it comes from a ${\mathbb{Z}}$-integral automorphism \varphi_{g}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ N_{{\mathbb{R}}}\operatorname{\overset{{}_{\text{def}}}{=}}\text{Hom}_{{\mathbb{Z}}}(M,{\mathbb{R}}) that leaves the fan $\Sigma$ invariant. In this way, the twisted $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{\Sigma} gives rise to a map of sets

[TABLE]

where

[TABLE]

Equip $\text{Aut}(\Sigma)$ with the trivial $G$-action, ${T}({K})$ with its canonical $G$-action, and ${T}({K})\rtimes\text{Aut}(\Sigma)$ with the resulting product $G$-action.

Proposition 2.1.3** ([28, §III.1, Proposition 5], [11, Lemma 3.1 & Theorem 3.2]).**

(i). The set of $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted actions on a ${K}$-variety $Y$, up to $G$-equivariant automorphisms, is in natural bijection with the non-Abelian Galois cohomology set

[TABLE]

where $\text{Aut}_{{K}}(Y)$ denotes the group of automorphisms of $Y$ over ${K}$. If $Y$ is quasiprojective, then the set (7) is also in bijection with the set of twisted $K_{0}$-forms of $Y$.

(ii). The map (6) is a $1$-cocycle on $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$. If $Y$ is a quasiprojective toric ${K}$-variety with fan $\Sigma$, then the construction of the $1$-cocycle (6) gives rise to a bijection

[TABLE]

Corollary 2.1.4**.**

(i). Given any lattice $N$ and any projective fan $\Sigma$ in $N_{{\mathbb{R}}}$, each group homomorphism

[TABLE]

determines a $1$-cocyle (6) with $y_{g}$ the identity in $T({K})$ for all $g\in G$. It therefore defines a toric ${K}$-variety $Y_{\Sigma}$ equipped with a twisted $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{\Sigma}. The representation of $G$ in finite rank ${\mathbb{Z}}$-lattices

[TABLE]

is exactly the image of the corresponding $K_{0}$-twisted form of $T$ under the equivalence (2).

(ii). When $Y$ is quasiprojective, each homomorphism (8) has a uniquely associated arithmetic toric variety \mu_{\varphi}:T_{0}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{0} over $K_{0}$. $\square$

Definition 2.1.5**.**

A $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric variety over ${K}$ is any toric ${K}$-variety $Y_{\Sigma}$ equipped with a twisted $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action arising from a homomorphism (8) as in Corollary 2.1.4.(i). If G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{\Sigma} and G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{\Sigma^{\prime}} are $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric varieties, then a morphism $Y_{\Sigma}\longrightarrow Y_{\Sigma^{\prime}}$ of $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric varieties is any $G$-equivariant morphism of toric ${K}$-varieties induced by a $G$-equivariant morphism of fans $\Sigma\longrightarrow\Sigma^{\prime}$.

Remark 2.1.6**.**

We can construct the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric variety G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{\Sigma} in Corollary 2.1.4 explicitly. Observe that for each element $g\in G$ and each cone $\sigma\in\Sigma$, the fact that the actions g\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ M and g^{\ast}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ N are adjoint implies that

[TABLE]

Thus for each $g\in G$, the automorphism $g:M\xrightarrow{\ \sim\ }M$ restricts to an isomorphism of semigroups

[TABLE]

Letting $g$ act on ${K}$, this induces a $K_{0}$-algebra (!) isomorphism

[TABLE]

hence a commutative diagram of $K_{0}$-scheme morphisms

[TABLE]

It is easy to see that for a given $g\in G$, the diagrams (11) commute with the inclusions of toric affine opens induced by inclusions of cones $\tau\subset\sigma$, and thus glue to produce $Y_{\Sigma}$ with twisted $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action.

Example 2.1.7**.**

Brauer-Severi variety arising as an arithmetic toric variety. Let $K_{0}=\mathbb{C}(\!(t)\!)$, so that $K=\mathbb{C}(\!(t)\!)^{\text{sep}}\cong{\mathbb{C}}(\!(t^{{\mathbb{Q}}})\!)$, the field of Puiseux series, and G=\text{Gal}\big{(}{\mathbb{C}}(\!(t^{{\mathbb{Q}}})\!)\big{/}{\mathbb{C}}(\!(t)\!)\big{)}\cong\widehat{{\mathbb{Z}}}. Let Y=\mathbb{P}^{2}_{\mbox{{\smaller\smaller\smaller\smaller\smaller\mathbb{C}(!(t)!)}}}. Define $N\operatorname{\overset{{}_{\text{def}}}{=}}{\mathbb{Z}}^{2}$, and let $\Sigma\subset$ be the standard fan in ${\mathbb{R}}^{2}$ that realizes \mathbb{P}^{2}_{\mbox{{\smaller\smaller\smaller\smaller\smaller{\mathbb{C}}(!(t^{{\mathbb{Q}}})!)}}} as a toric ${\mathbb{C}}(\!(t^{{\mathbb{Q}}})\!)$-variety, with rays generated by the vectors $e_{1}=(1,0)$, $e_{2}=(0,1)$, and $(-1,-1)$. Consider the 3-periodic matrix

[TABLE]

acting on $N_{{\mathbb{R}}}$. As pictured in Figure 1, the fan $\Sigma$ is invariant with respect to the action of this matrix. The matrix provides a cyclic permutation of the cones and rays in $\Sigma$.

Fix an isomorphism

[TABLE]

Let ${\mathbb{Z}}/3{\mathbb{Z}}\longrightarrow\text{Aut}(\Sigma)$ be the group homomorphism taking $1\mapsto\left(\begin{smallmatrix}0&-1\\ 1&-1\end{smallmatrix}\right)$, and consider the composite

[TABLE]

By Corollary 2.1.4.(ii), the homomorphism $\varphi$ determines an arithmetic toric variety \mu:T_{0}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{0} over ${\mathbb{C}}(\!(t)\!)$. The ${\mathbb{C}}(\!(t)\!)$-variety $Y_{0}$ is a $2$-dimensional Brauer-Severi variety [1] that comes equipped with the faithful, dense action of a non-split $2$-dimensional torus $T_{0}$.

2.2. Galois equivariant extended tropicalizations of twisted toric varieties

Equip $K_{0}$ with the structure of a (rank-$1$) non-Archimedean field, i.e., equip $K_{0}$ with a non-Archimedean absolute value

[TABLE]

Fix a separable closure $K=K^{\text{sep}}_{0}$, and let $|-|:{K}\longrightarrow{\mathbb{R}}_{\geq 0}$ be a non-Archimedean valuation extending our chosen valuation on $K_{0}$. When $K_{0}$ is Henselian, for instance when $K_{0}$ is complete with respect to its valuation, the extension of $|-|$ to ${K}$ is unique [12, Lemma 4.1.1]. Define

[TABLE]

Fix a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric variety G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{\Sigma} determined by a group homomorphism $\varphi:G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\longrightarrow\text{Aut}(\Sigma)$ for some fan $\Sigma\subset N_{{\mathbb{R}}}$ as in Corollary 2.1.4.(i). Let us temporarily ignore the Galois-structure on $Y_{\Sigma}$, interpreting $Y_{\Sigma}$ as a toric ${K}$-variety without $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action. S. Payne [26, §3] explains how to associate a Hausdorff space to $Y_{\Sigma}$, called the extended tropicalization of $Y_{\Sigma}$. In detail, each cone $\sigma\in\Sigma$ has a corresponding topological space

[TABLE]

where $S_{\sigma}$ is the semigroup $S_{\sigma}\operatorname{\overset{{}_{\text{def}}}{=}}\sigma^{\vee}\cap M$ in $M_{{\mathbb{R}}}$, and where ${\mathbb{R}}\sqcup\{\infty\}$ is the ordered semigroup with $\infty>r$ for all $r\in{\mathbb{R}}$, equipped with its order topology. Inclusions of cones $\tau\subset\sigma$ in $\Sigma$ induce open embeddings of topological spaces

[TABLE]

The extended tropicalization of $Y_{\Sigma}$, denoted $\text{Trop}(Y_{\Sigma})$, is the topological space obtained by gluing along these inclusions:

[TABLE]

It contains $N_{{\mathbb{R}}}=\text{Trop}(Y_{\{0\}})$ as a dense open subset. The extended tropicalization comes with a continuous surjective tropicalization map

[TABLE]

On each $Y_{\sigma}=\text{Spec}_{\ \!}{K}[S_{\sigma}]$, the map (13) takes a point $y\in Y^{\text{an}}_{\sigma}$, given by a multiplicative seminorm $|-|_{y}:{K}[S_{\sigma}]\longrightarrow{\mathbb{R}}_{\geq 0}$, and sends it to the semigroup homomorphism

[TABLE]

Lemma 2.2.1**.**

The $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ N_{{\mathbb{R}}} extends to a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action

[TABLE]

If $K_{0}$ is Henselian, then the twisted $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{\Sigma} induces a $G^{op}$-action

[TABLE]

and the tropicalization map (13) is $G$-equivariant with respect to the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-actions (14) and (15).

Proof.

For $g\in G$, the semigroup homomorphism (9) induces a homeomorphism of topological spaces

[TABLE]

for each $\sigma\in\Sigma$. Because the semigroup homomorphisms (9) commute with the morphisms dual to inclusions of cones $\tau\subset\sigma$, the homeomorphisms (16) glue to give an automorphism

[TABLE]

Taking these maps for all $g\in G$, we obtain a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action on $\text{Trop}(Y_{\Sigma})$.

For each $\sigma\in\Sigma$, a point $y\in Y^{\text{an}}_{\sigma}$ is given by a multiplicative seminorm $|-|_{y}:{K}[S_{\sigma}]\longrightarrow{\mathbb{R}}_{\geq 0}$ who’s restriction to ${K}\subset{K}[S_{\sigma}]$ is our fixed absolute value $|-|$ on ${K}$. If $K_{0}$ is Henselian, then for each $g\in G$, the $K_{0}$-algebra isomorphism (10) fits into a commutative diagram

[TABLE]

In particular, the composite multiplicative seminorm |-|_{g^{\ast}(y)}\operatorname{\overset{{}_{\text{def}}}{=}}\big{|}g(-)\big{|}_{v} defines a point $g^{\ast}(y)\in Y^{\text{an}}_{g^{\ast}(\sigma)}$. The resulting homeomorphisms $g^{\ast}:Y^{\text{an}}_{\sigma}\xrightarrow{\ \sim\ }Y^{\text{an}}_{g^{\ast}(\sigma)}$ glue to give an automorphism g^{\ast}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y^{\text{an}}_{\Sigma}, and together these give a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action (15). Finally, commutativity of the diagrams

[TABLE]

implies that the tropicalization map (13) is equivariant with respect to the actions constructed. ∎

Definition 2.2.2**.**

If $Y_{\Sigma}$ is a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric ${K}$-variety, then the $G$-equivariant extended tropicalization of $Y_{\Sigma}$ is the extended tropicalization $\text{Trop}(Y_{\Sigma})$ equipped with the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action constructed in the proof of Lemma 2.2.1. We denote it

[TABLE]

When $K_{0}$ is Henselian, we refer to (13) as the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-equivariant tropicalizaiton map.

Example 2.2.3**.**

Let $K_{0}={\mathbb{C}}(\!(t)\!)$, with $Y_{0}$ the $2$-dimensional Brauer-Severi ${\mathbb{C}}(\!(t)\!)$-variety realized as an arithmetic toric variety \mu:T_{0}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{0} in Example 2.1.7. The compactifications $\overline{\sigma_{i}}$ of each of the three $2$-dimensional cones $\sigma_{i}$ in $\Sigma$ fit together to form $\text{Trop}(Y_{\Sigma})$, which appears in Figure 2. The ${\mathbb{Z}}/3{\mathbb{Z}}$-action {\mathbb{Z}}/3{\mathbb{Z}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ N_{{\mathbb{R}}}, cyclically permuting the cones $\sigma_{i}$, $1\leq i\leq 3$, extends to a ${\mathbb{Z}}/3{\mathbb{Z}}$-action on the whole space $\text{Trop}(Y_{\Sigma})$. It cyclically permutes the three compactified cones $\overline{\sigma_{i}}$ as depicted in Figure 2. Precomposing with G\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!{\mathbb{Z}}/3{\mathbb{Z}}, we get a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action on $\text{Trop}(Y_{\Sigma})$.

2.2.4**.**

Extended tropicalizations of $\boldsymbol{K}$-varieties with canonical $\boldsymbol{G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}}$-action. Let G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{\Sigma} be a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric variety over ${K}$. Let $X_{0}$ be a $K_{0}$-variety. Equip $X\operatorname{\overset{{}_{\text{def}}}{=}}X_{0,{K}}$ with its canonical $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action. Fix an affine open cover $\{\text{Spec}_{\ \!}A_{i}\hookrightarrow X_{0}\}_{i\in I}$. If $K_{0}$ is Henselian, then for each $g\in G$ and each point $x\in(\text{Spec}_{\ \!}A_{i,{K}})^{\text{an}}$, we have have a commutative diagram

[TABLE]

and thus an induced canonical $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ X^{\text{an}}. Each $G$-equivariant closed embedding

[TABLE]

induces a $G$-equivariant closed embedding of topological spaces \imath^{\text{an}}:{X}^{\text{an}}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y^{\text{an}}_{\Sigma}. The (extended) tropicalization of ${X}$ (with respect to $\imath$), denoted $\text{Trop}({X},\imath)$, is the image

[TABLE]

The restriction of (13) to $X^{\text{an}}$ is a continuous surjective map

[TABLE]

that we call the tropicalization map on $X^{\text{an}}$ (with respect to $\imath$).

Lemma 2.2.5**.**

If $K_{0}$ is Henselian, then the tropicalization $\text{Trop}({X},\imath)$ is a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-invariant subspace of $\text{Trop}(Y_{\Sigma})$ under the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action in Lemma 2.2.1, and the tropicalization map (19) is $G$-equivariant.

Proof.

For each $\sigma\in\Sigma$, let $\mathfrak{a}_{\sigma}\subset{K}[S_{\sigma}]$ denote the ideal cutting out $\imath(X)\cap Y_{\sigma}$. A point $v$ in $\text{Trop}(X,\imath)\cap\text{Trop}(Y_{\sigma})$ is given by a semigroup homomorphism

[TABLE]

that admits at least one factorization

[TABLE]

for some $y\in Y^{\text{an}}_{\sigma}$, such that $|\mathfrak{a}_{\sigma}|_{y}=\{0\}$. The assumption that $\imath$ is $G$-equivariant implies that the isomorphism (10) satisfies $g(\mathfrak{a}_{g^{\ast\!}(\sigma)})=\mathfrak{a}_{\sigma}$. Thus if $K_{0}$ is Henselian, commutativity of the diagrams (17) and (18) implies that $g^{\ast}(v)$ lies inside $\text{Trop}({X},\imath)\cap\text{Trop}(Y_{g^{\ast\!}(\sigma)})$. That (19) is $G$-equivariant follows from commutativity of the diagram (18). ∎

Definition 2.2.6**.**

If \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} is a $G$-equivariant closed embedding into a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric ${K}$-variety, then the $G$-equivariant tropicalization of ${X}$ (with respect to $\imath$) is the extended tropicalization $\text{Trop}({X},\imath)$ equipped with the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action provided by Lemma (2.2.5). We denote it

[TABLE]

and refer to the map (19) as its $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-equivariant tropicalizaiton map.

Example 2.2.7**.**

A Galois-equivariant tropicalization of $\boldsymbol{\mathbb{P}^{1}}$. Let $K_{0}={\mathbb{C}}(\!(t)\!)$ with absolute Galois group G=\text{Gal}\big{(}{\mathbb{C}}(\!(t^{{\mathbb{Q}}})\!)\big{/}{\mathbb{C}}(\!(t)\!)\big{)} as in Example 2.1.7. Let $X_{0}$ be the projective line over ${\mathbb{C}}(\!(t)\!)$:

[TABLE]

Consider the surjective homomorphism G\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!\text{Gal}\big{(}{\mathbb{C}}(\!(t^{\frac{1}{2}})\!)\big{/}{\mathbb{C}}(\!(t)\!)\big{)}\cong{\mathbb{Z}}/2{\mathbb{Z}}. Define $M\operatorname{\overset{{}_{\text{def}}}{=}}{\mathbb{Z}}u\oplus{\mathbb{Z}}v\oplus{\mathbb{Z}}w$ with $G$-action factoring trough the ${\mathbb{Z}}/2{\mathbb{Z}}$-action that exchanges $u$ and $v$. Note that the plane in $N_{{\mathbb{R}}}$ dual to $u-v\in M$ is the maximal ${\mathbb{Z}}/2{\mathbb{Z}}$-fixed subspace of $N_{{\mathbb{R}}}$. Let $\Sigma$ be the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-invariant fan in $N_{{\mathbb{R}}}$ with rays spanned by the vectors $e_{1}$, $e_{2}$, $e_{3}$, and $-(e_{1}+e_{2}+e_{3})$. It determines the toric ${\mathbb{C}}(\!(t^{{\mathbb{Q}}})\!)$-variety Y_{\Sigma}\cong\mathbb{P}^{3}_{\mbox{{\smaller\smaller\smaller\smaller\smaller{\mathbb{C}}(!(t^{{\mathbb{Q}}})!)}}} with twisted $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action. Give {X}=X_{0,\mbox{{\smaller\smaller\smaller\smaller\smaller{\mathbb{C}}(!(t^{{\mathbb{Q}}})!)}}} the canonical $G$-action. Then we have a $G$-equivariant closed embedding

[TABLE]

hence a $G$-equivariant tropicalization $\text{trop}:{X}^{\text{an}}\longrightarrow\text{Trop}_{G}({X},\imath)$. See Figure 3 for a depiction.

The action G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ {X}^{\text{an}} is quite complex, but we can describe pieces of it. Let B_{r}\big{(}\pm t^{\frac{1}{2}}\big{)}^{\circ} denote the open analytic disks of radius $r=e^{-\frac{1}{2}}$ in ${X}^{\text{an}}$, centered at $\pm t^{\frac{1}{2}}$ (red in Figure 3). Then

[TABLE]

This implies that every element $g\in G$ mapping to $1$ under G\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!{\mathbb{Z}}/2{\mathbb{Z}} induces a homeomorphism

[TABLE]

The open analytic disk $B_{1}(1)^{\circ}$ (blue in Figure 3) and the open analytic domain ${X}^{\text{an}}\backslash B_{1}(1)$ (purple in Figure 3) are each invariant under the full Galois group $G$. All points of ${X}^{\text{an}}$ lying in the dense torus, except those in the open disks B_{r}\big{(}\pm t^{\frac{1}{2}}\big{)}^{\!\circ}, map into the ${\mathbb{Z}}/2{\mathbb{Z}}$-fixed plane in $N_{{\mathbb{R}}}$. The $G$-action on $N_{{\mathbb{R}}}$ exchanges the images of the open disks B_{r}\big{(}\pm t^{\frac{1}{2}}\big{)}^{\!\circ}.

Corollary 2.2.8**.**

Assume that $K_{0}$ is Henselian and equip $X^{\text{an}}$ with its canonical $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action. Then for any system $\mathcal{S}_{G}$ of $G$-equivariant toric embeddings, the continuous map (4) is $G$-equivariant. $\square$

2.3. The polyhedral structure on a Galois equivariant tropicalization

For the remainder of §2.3, assume that $K_{0}$ is Henselian. Denote the valuation ring, maximal ideal, and residue field of ${K}=K^{\text{sep}}_{0}$

[TABLE]

Suppose given a $K_{0}$-variety $X_{0}$ and a closed embedding \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} of $X=X_{0,{K}}$ into a toric ${K}$-variety $Y_{\Sigma}$. Equip $Y_{\Sigma}$ with a twisted $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{\Sigma}.

As in the proof of Lemma 2.2.5, for each $\sigma\in\Sigma$, let $\mathfrak{a}_{\sigma}\subset{K}[M]$ denote the ideal cutting out $\imath(X)\cap Y_{\sigma}$. Each point $v\in N_{{\mathbb{R}}}$ determines a $K^{\circ}$-model $\mathscr{X}^{v}$ of $X$ glued from the spectra of the $K^{\circ}$-algebras {K}[S_{\sigma}]^{v}\big{/}\mathfrak{a}^{v}_{\sigma}, where

[TABLE]

and

[TABLE]

Because each ${K}[S_{\sigma}]^{v}$ is a subring of ${K}[S_{\sigma}]$, it makes sense to say that the special fibers \mathscr{X}^{v_{1}}_{\mbox{{\smaller\smaller\smaller\smaller\smaller\widetilde{K}}}} and \mathscr{X}^{v_{2}}_{\mbox{{\smaller\smaller\smaller\smaller\smaller\widetilde{K}}}} are “equal” or “not equal” for distinct $v_{1},v_{2}\in N_{{\mathbb{R}}}$. For any closed embedding \imath:X\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!\mathbb{P}^{n}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smallerK}}}, where \mathbb{P}^{n}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smallerK}}} has its standard toric structure, the Gröbner complex on $\text{Trop}(Y_{\Sigma})$ is the closure of the complex in $N_{{\mathbb{R}}}$ whose cells consist of vectors $v$ with fixed $\mathscr{X}^{v}$ [21, §2.5] [14, proof of Theorem 10.14].

Define a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted \mathbb{P}^{n}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smallerK}}} to be any $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric ${K}$-variety with underlying ${K}$-variety Y_{\Sigma}\cong\mathbb{P}^{n}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smallerK}}}.

Lemma 2.3.1**.**

If $K_{0}$ is Henselian and \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} is a $G$-equivariant closed embedding into a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted \mathbb{P}^{n}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smallerK}}}, the Gröbner complex in $\text{Trop}(Y_{\Sigma})$ induced by $\imath$ is invariant under the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action on $\text{Trop}(Y_{\Sigma})$.

Proof.

If $Y_{\Sigma}$ is a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric variety and the closed embedding \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} is $G$-equivariant, then for each $\sigma\in\Sigma$, each $u\in S_{\sigma}$, each $v\in\text{Trop}(Y_{\Sigma})$, and each $g\in G$, invariance of the absolute value on ${K}$ under $g$ implies that

[TABLE]

Thus the automorphism g\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ {K}[M] restricts to an isomorphism

[TABLE]

and for any pair of points $v_{1},v_{2}\in N_{{\mathbb{R}}}$, we have ${K}[S_{\sigma}]^{v_{1}}={K}[S_{\sigma}]^{v_{2}}$ inside ${K}[S_{\sigma}]$ if and only if ${K}[S_{g^{\ast\!}(\sigma)}]^{g^{\ast\!}(v_{1})}={K}[S_{g^{\ast\!}(\sigma)}]^{g^{\ast\!}(v_{2})}$ inside ${K}[S_{g^{\ast\!}(\sigma)}]$. Because $g(\mathfrak{a}_{g^{\ast\!}(\sigma)})=\mathfrak{a}_{\sigma}$, this implies that $v_{1}$ and $v_{2}$ lie in the same Gröbner cell in $N_{{\mathbb{R}}}$ if an only if $g^{\ast\!}(v_{1})$ and $g^{\ast\!}(v_{2})$ lie in the same Gröbner cell in $N_{{\mathbb{R}}}$. ∎

Corollary 2.3.2**.**

If $K_{0}$ is Henselian and \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} is a $G$-equivariant closed embedding into a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted \mathbb{P}^{n}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smallerK}}}, then $\text{Trop}_{G}({X},\imath)$ admits a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-invariant polyhedral decomposition.

Proof.

This follows immediately from Lemma 2.3.1 and [14, Theorem 10.14]. ∎

Corollary 2.3.3**.**

If $K_{0}$ is Henselian and \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} is a $G$-equivariant closed embedding into a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted \mathbb{P}^{n}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smallerK}}}, then there exists a $G$-invariant polyhedral decomposition of $\text{Trop}_{G}(X,\imath)$ that refines the polyhedral decomposition $\{\overline{\ \!\sigma\ \!}\}_{\sigma\in\Sigma}$.

Proof.

Let $\mathcal{P}$ be the Gröbner complex on $\text{Trop}({X},\imath)$. Then the set \mathcal{Q}\operatorname{\overset{{}_{\text{def}}}{=}}\big{\{}\ \!P\cap\overline{\ \!\sigma\ \!}\ \!:\ \!P\in\mathcal{P}\ \text{and}\ \overline{\ \!\sigma\ \!}\in\Sigma\ \!\big{\}} is a polyhedral decomposition with the required property. ∎

Proof of Theorem D.

By Corollary 2.3.2, $\text{Trop}({X},\imath)$ admits a polyhedral decomposition that is $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-invariant. For each polyhedral cell $\Delta$ in this polyhedral decomposition and for each integer $p\geq 0$, every element $g\in G$ induces isomorphisms

[TABLE]

between the terms $\mathcal{F}^{p}(\Delta)$ and $\mathcal{F}(g^{\ast}\!\Delta)$ in the cochain group C^{\text{dim}_{\ \!}\Delta}\big{(}\text{Trop}({X},\imath),\mathcal{F}^{p}\big{)} as defined in [17, Definition 16 on]. Thus for each integer $q\geq 0$, the cochain group C^{q}\big{(}\text{Trop}({X},\imath),\mathcal{F}^{p}\big{)} picks up a $G$-action. This action (21) commutes with the restriction morphisms “$\iota^{\ast}$” defined in [17, Equation (8)]. Thus the coboundary morphisms [17, §2.3, after Equation (8)] on C^{q}\big{(}\text{Trop}({X},\imath),\mathcal{F}^{p}\big{)} are $G$-equivariant, and we obtain a $G$-action on each tropical cellular cohomology group:

[TABLE]

The representation on tropical homology arises in the same way after dualizing $\mathcal{F}^{p}(\Delta)\mapsto\mathcal{F}_{p}(\Delta)$. ∎

Example 2.3.4**.**

A Galois representation in tropical cellular cohomology. Let $K_{0}={\mathbb{C}}(\!(t)\!)$ with separable closure $K={\mathbb{C}}(\!(t^{{\mathbb{Q}}})\!)$ and Galois group G=\text{Gal}\big{(}{\mathbb{C}}(\!(t^{{\mathbb{Q}}})\!)\big{/}{\mathbb{C}}(\!(t)\!)\big{)}. Let $Y_{0}$ be the Brauer-Severi ${\mathbb{C}}(\!(t)\!)$-variety constructed in Example 2.1.7. If we give $M=\text{Hom}_{{\mathbb{Z}}}(N,{\mathbb{Z}})$ the basis dual to the standard basis on $N={\mathbb{Z}}^{2}$, then the action of the matrix $\left(\begin{smallmatrix}0&-1\\ 1&-1\end{smallmatrix}\right)$ on $N$ in Example 2.1.7 is dual to the $3$-periodic action of its transpose $\left(\begin{smallmatrix}\ 0\!&\ 1\!\\ \!-1&\!-1\end{smallmatrix}\right)$ on $M$. This action induces a twisted ${\mathbb{Z}}/3{\mathbb{Z}}$-action, and thus a $G$-action, on ${\mathbb{C}}(\!(t^{\frac{1}{3}})\!)[M]={\mathbb{C}}(\!(t^{\frac{1}{3}})\!)[x^{\pm 1},y^{\pm 1}]$. Consider the Laurent polynomial

[TABLE]

in ${\mathbb{C}}(\!(t^{\frac{1}{3}})\!)[x^{\pm 1},y^{\pm 1}]$. It cuts out a smooth, degree-$6$, genus-$10$ curve $X$ inside Y_{\Sigma}\cong\mathbb{P}^{2}_{\mbox{{\smaller\smaller\smaller\smaller\smaller{\mathbb{C}}(!(t^{{\mathbb{Q}}})!)}}}. Each of the nine trinomials in parentheses in (22) constitutes an orbit under the ${\mathbb{Z}}/3{\mathbb{Z}}$-action on ${\mathbb{C}}(\!(t^{\frac{1}{3}})\!)[x^{\pm 1},y^{\pm 1}]$, making $X$ a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-invariant subvariety of the twisted toric ${\mathbb{C}}(\!(t^{{\mathbb{Q}}})\!)$-variety G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{\Sigma}. It arises as the pullback of a ${\mathbb{C}}(\!(t)\!)$-curve $X_{0}$ inside the resulting Brauer-Severi ${\mathbb{C}}(\!(t)\!)$-variety $Y_{0}$.

The $G$-equivariant tropicalization of ${X}$ is pictured in Figure 4 below. Note the apparent ${\mathbb{Z}}/3{\mathbb{Z}}$-symmetry.

In the language of [17, Definitions 7, 10, & 11], $\text{Trop}(X,\imath)$ is a smooth regular projective ${\mathbb{Q}}$-tropical variety inside \text{Trop}(\mathbb{P}^{2}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smaller{\mathbb{C}}(!(t^{{\mathbb{Q}}})!)}}})\cong{\mathbb{T}}\mathbb{P}^{2}. The dimensions of its nonzero tropical cellular cohomology groups H^{p,q}_{\text{trop}}\big{(}\text{Trop}(X,\imath),{\mathbb{Q}}\big{)} sit in a symmetric Hodge diamond

[TABLE]

The Galois group $G^{op}$ acts trivially on the degree-$(0,0)$ and -$(1,1)$ homology groups. Its action on the degree-$(1,0)$ and -$(0,1)$ groups factors through a ${\mathbb{Z}}/3{\mathbb{Z}}$-action that permutes the elements of a basis in each case. In degree-$(0,1)$, these basis elements can be taken as the classes of the $1$-cycles labeled “1” through “10” in Figure 4, say with counter-clockwise orientations. With respect to this basis, the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action on H_{0,1}^{\text{trop}}\big{(}\text{Trop}(X,\imath),{\mathbb{Q}}\big{)} factors through the ${\mathbb{Z}}/3{\mathbb{Z}}$-action generated by the action of the $10\times 10$-matrix

[TABLE]

Under the isomorphism provided by [17, Theorem 1], the ${\mathbb{Z}}/3{\mathbb{Z}}$-actions on tropical cohomology groups’ ${\mathbb{Q}}$-linear duals H_{0,1}^{\text{trop}}\big{(}\text{Trop}(X,\imath),{\mathbb{Q}}\big{)}^{\ast} and H_{1,0}^{\text{trop}}\big{(}\text{Trop}(X,\imath),{\mathbb{Q}}\big{)}^{\ast} induce $G$-representations on the respective weight-[math] and -$1$ associated graded components of the weight filtration for the limit mixed Hodge structure of the nearby fiber associated to the family of projective varieties cut out by (22) over the punctured unit disk inside $\text{Spec}_{\ \!}{\mathbb{C}}[t]$.

3. An extensive supply of Galois-equivariant toric embeddings

We explain how to construct an extensive supply of Galois-equivariant closed embeddings. Throughout §3, let $X$ be a variety over $K=K^{\text{sep}}_{0}$, without any assumption that $K_{0}$ is non-Archimedean.

3.1. Algorithms for constructing non-equivariant toric embeddings

In order to prove Theorems B and C, we need to construct Galois-equivariant toric embeddings X\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} that realize pre-specified rational functions on ${X}$ as pullbacks of characters on the dense torus in $Y_{\Sigma}$. To this end, we provide three constructions of toric embeddings in the non-equivariant setting: an embedding for projective varieties, J. Włodarczyk’s algorithm, and the algorithm of F. Berchtold and J. Hausen. Włodarczyk’s algorithm is sufficient when $K_{0}$ is a perfect field, but Berchtold and Hausen’s algorithm, which lacks some of the generality of Włodarczyk’s, provides a far more explicit picture of what is happening and provides the user with tools that will be crucial (we believe) to applications.

3.1.1**.**

Payne’s lemma for projective varieties.* *In [26], S. Payne shows how to construct systems of closed embeddings of a given quasiprojective variety into quasiprojective toric varieties such that the inverse limit of all tropicalizations in these systems is homeomorphic to the Berkovich analytification. A key lemma in the course of his proof is the following:

Lemma 3.1.2** ([26*, Lemma 4.3])*.

Assume that ${X}$ is a projective ${K}$-variety. Given an ample effective divisor $E$ on ${X}$ with complement $U\subset{X}$, given a closed subvariety contained in the support of $E$, and given any set $f_{1},\dots,f_{r}$ of generators of the coordinate ring of $U$, there exists a closed toric embedding

[TABLE]

such that U=\imath^{-1}\big{(}D(y_{0})\big{)}, such that $V$ is the preimage of some coordinate linear subspace in \mathbb{P}^{n}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smaller{K}}}}, and such that each $f_{i}$ is the pullback $f_{i}=\imath^{\#}(y_{i})$. $\square$

3.1.3**.**

Włodarczyk’s algorithm for A2-varieties.* *Recall from Remark 1.4.3 that an A2-variety is a variety in which every pair of points lies in some affine open subset of the variety.

Let $K^{\text{alg}}\supset K$ denote an algebraic closure of $K$ (and thus an algebraic closure of $K_{0}$). Recall that $K_{0}$ is a perfect field if the inclusion ${K}=K^{\text{sep}}_{0}\subset K^{\text{alg}}_{0}=K^{\text{alg}}$ is an isomorphism. In this case $X_{K^{\text{alg}}}=X$. All fields are perfect in characteristic [math].

Suppose that ${X}_{K^{\text{alg}}}$ is a normal A2-variety over $K^{\text{alg}}$. In his seminal 1993 paper [29], J. Włodarczyk details an algorithm that takes as input any finite collection $\mathfrak{F}$ of nonzero rational functions on $X_{K^{\text{alg}}}$. The algorithm enlarges the collection $\mathfrak{F}$ through step-by-step adjustments of a corresponding set of divisors on $X_{K^{\text{alg}}}$. The final enlarged collection $\mathfrak{F}^{\prime}$ satisfies properties that make it a generating set for the mutliplicative group of restrictions of characters under a closed embedding of $X_{K^{\text{alg}}}$ into a toric $K^{\text{alg}}$-variety $Y_{\Sigma,K^{\text{alg}}}$, and one can reconstruct $Y_{\Sigma,K^{\text{alg}}}$ from $\mathfrak{F}^{\prime}$.

Remark 3.1.4**.**

Włodarczyk works over the algebraic closure $K^{\text{alg}}$ because it is important in his algorithm that all classes in the quotient K^{\text{alg}}(X)^{\times}\big{/}(K^{\text{alg}})^{\times} be torsion free [29, bottom of p. 710]. If we take $K_{0}$ to be the non-perfect field $K_{0}={\mathbb{F}}_{\!p}(\!(t)\!)$ for instance, then on the $K={\mathbb{F}}_{\!p}(\!(t)\!)^{\text{sep}}$-variety X=\text{Spec}_{\ \!}{\mathbb{F}}_{\!q}(\!(t)\!)^{\text{sep}}[x]\big{/}(x^{p}-t) we have a non-trivial class [x]\in{\mathbb{F}}_{\!p}(\!(t)\!)^{\text{sep}}(X)^{\times}\big{/}\big{(}{\mathbb{F}}_{\!p}(\!(t)\!)^{\text{sep}}\big{)}^{\times} with $[x]^{p}=[1]$.

Using Włodarczyk’s algorithm, P. Gross, S. Payne, and the present author show the following:

Theorem 3.1.5** ([13*, Theorem 4.2])*.

Assume that ${X}_{K^{\text{alg}}}$ is a normal A2-variety. Given affine open subvarieties $U_{1},\dots,U_{r}\subset X_{K^{\text{alg}}}$ and any finite collection $\mathfrak{F}=\mathfrak{F}_{1}\sqcup\cdots\sqcup\mathfrak{F}_{r}$ of rational functions such that each $f\in\mathfrak{F}_{i}$ is regular on $U_{i}$, there exists a closed embedding

[TABLE]

into a toric ${K^{\text{alg}}}$-variety $Y_{\Sigma,K^{\text{alg}}}$ such that for each $1\leq i\leq r$ we have $U_{i}=\imath^{-1}(Y_{\sigma_{i},K^{\text{alg}}})$ for some $\sigma_{i}\in\Sigma$, and such that each $f\in\mathfrak{F}_{i}$ is the pullback $\imath^{\#}(\chi^{u})$ of a character $\chi^{u}\in S_{\sigma_{i}}$. $\square$

3.1.6**.**

Berchtold and Hausen’s algorithm for “A2-Mori dream spaces” in characteristic 0.* In the last decade and a half, F. Berchtold and J. Hausen [3] [4] [16] have clarified large parts of the geometry behind Włodarczyk’s original algorithm using the language of Cox rings. See Hausen’s book with I. Arzhantsev, U. Derenthal, and A. Laface [2, §1-§3] for a detailed exposition. Berchtold and Hausen employ combinatorial objects called “bunches of cones,” dual to fans, inside the Cox ring of ${X}$. The clarity that Berchtold and Hausen’s construction affords comes at a cost: one must assume that $\text{char}_{\ \!}K=0$ (thus that $K_{0}$ is perfect) and that ${X}$ satisfies an A2 version of the Mori dream space hypotheses. Specifically, $X$ must be an irreducible factorial A2*-variety over $K^{\text{alg}}={K}$ with H^{0}({X},\mathscr{O}^{\times}_{\!\mbox{{\smaller\smaller\smaller\smaller\smaller{X}}}})=K^{\times} such that the ideal class group and Cox ring of ${X}$ are finitely generated. These conditions hold, for instance, for any $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted proper (not-necessarily projective) toric ${K}$-variety in characteristic [math]. Using a finite set of generators $\mathfrak{F}$ of ${X}$’s Cox ring $R_{{X}}$, Berchtold and Hausen construct a characteristic space over ${X}$ that comes equipped with a canonical closed embedding into a toric ${K}$-variety $Y_{\widehat{\Sigma}}$. Taking a quotient of both $Y_{\widehat{\Sigma}}$ and the characteristic space lying inside it, one obtains a closed embedding \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} into a toric ${K}$-variety $Y_{\Sigma}$.

In Theorem 3.1.7 below, we prove an analogue of Theorem 3.1.5 that uses Berchtold and Hausen’s algorithm in place of Włodarczyk’s algorithm. The proof makes heavy use of notation and terminology from [2]. It is the only place in the present text where we make use of this notation, and readers interested only in the proofs of Theorems A, B, and C may want to read the statement of Theorem 3.1.7 and then skip ahead to §3.2.

Theorem 3.1.7**.**

Assume that $\text{char}_{\ \!}K_{0}=0$, that ${X}$ is an A2-variety over ${K}$ with H^{0}({X},\mathscr{O}^{\times}_{\!\mbox{{\smaller\smaller\smaller\smaller\smaller{X}}}})=K^{\times}, and that the ideal class group $\text{Cl}(X)$ and Cox ring $R_{X}$ of $X$ are finitely generated. Then there exists a finite affine open cover $\mathscr{U}=\{U_{1},\dots,U_{r}\}$ of ${X}$, consisting of images of standard affine opens in $\text{Spec}_{\ \!}R_{X}$, and satisfying the following property: For any function $f$ regular on some $U_{i}$, we can construct a closed embedding

[TABLE]

into a toric ${K}$-variety $Y_{\Sigma^{\prime}}$ such that for each $1\leq i\leq r$, the affine open $U_{i}$ is the inverse image of a torus invariant open in $Y_{\Sigma^{\prime}}$, and such that some $K^{\times}$-multiple of $f$ is the pullback of a character on the dense torus in $Y_{\Sigma^{\prime}}$.

Proof.

By [2, Theorem 1.5.3.7], we can choose an ordered generating set $\mathfrak{F}=(f_{1},\dots,f_{r})$ of the Cox ring $R_{{X}}$ satisfying the hypotheses of [2, §3.2.1]. By [2, Theorem 3.2.1.9.(ii)] and the A2 hypothesis, we can find a maximal $\mathfrak{F}$-bunch $\Phi$. Via [2, Construction 3.2.1.3], this provides us with a characteristic space over ${X}$ and an affine open cover $\mathscr{U}=\{U_{\gamma_{I}}\}_{I\in\mathscr{I}}$ of ${X}$ indexed by the set $\mathscr{I}$ of relevant faces of $\Phi$. Using [2, Construction 3.2.5.3 & Proposition 3.2.5.4], we obtain a closed embedding into a toric variety

[TABLE]

such that each $U_{\gamma_{I}}$ is of the form $\imath^{-1}(Y_{\sigma})$ for some $\sigma\in\Sigma$.

We want to check that for each function $f$ regular on some $U_{\gamma_{I}}$, there exists another closed toric embedding \imath^{\prime}:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma^{\prime}} such that each $U_{\gamma_{I}}$ is the inverse image of a torus-invariant affine open, and such that some $K^{\times}$-multiple of $f$ is the pullback of a character. Fix one such $U_{\gamma_{I}}$ and fix a regular function $f$ on $U_{\gamma_{I}}$. By construction, we can interpret $f$ as a degree-[math] element

[TABLE]

for some $n\geq 0$ and for any choice of $u=(u_{1},\dots,u_{r})\in\gamma^{\circ}_{I}$, where $f_{0}$ is homogeneous of degree $n\ \!Q(u)\in\text{Cl}({X})$. By [2, Theorem 1.5.3.7], we can assume that $f_{0}$ is $\text{Cl}({X})$-prime.

If $f_{0}$ is associated to one of the functions in $\mathfrak{F}$, then since H^{0}({X},\mathscr{O}^{\times}_{\!\mbox{{\smaller\smaller\smaller\smaller\smaller{X}}}})=K^{\times}, after multiplying a scalar in $K^{\times}$ we have that $f_{0}\in\mathfrak{F}$. In this case, the fact that $f$ is degree-[math] implies that it is the pullback under $\imath_{\Phi}$ of a character that is regular on the affine toric subvariety $Y_{P(\gamma^{\ast}_{I})}$.

If $f_{0}$ is not associated to any function in $\mathfrak{F}$, then by the previous arguments, the ordered set of homogeneous elements $\mathfrak{F}^{\prime}\ =\ (f_{0},f_{1},\dots,f_{r})$ admits an $\mathfrak{F}^{\prime}$-bunch $\Phi^{\prime}$ that provides us with a new closed toric embedding \imath_{\Phi^{\prime\prime}}:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma^{\prime\prime}} for which $f$ becomes the pullback of a monomial. Then the product toric embedding

[TABLE]

satisfies the two requisite properties. ∎

3.2. From non-equivariant to equivariant toric embeddings

Assume now and for the remainder of the text that $X$ arises as $X=X_{0,{K}}$ for some $K_{0}$-variety $X_{0}$ and $K=K^{\text{sep}}_{0}$. Equip $X$ with its canonical $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action. Since every toric variety has a canonical ${\mathbb{Z}}$-model, the toric ${K}$-varieties produced by Lemma 3.1.2 when $X$ is projective, by Theorem 3.1.3 when $K_{0}$ is perfect, and by Theorem 3.1.6 when $\text{char}_{\ \!}K_{0}=0$ come at least with a canonical $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-actions on $X$ and $Y_{\Sigma}$. However, because each algorithm constructs closed embeddings over ${K}$, not $K_{0}$, the embeddings may not be $G$-equivariant with respect to the canonical $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action. We get $G$-equivariant toric embeddings by running a second construction reminiscent of the construction of an induced representation.

Theorem 3.2.1**.**

For each closed embedding \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} into a toric ${K}$-variety $Y_{\Sigma}$ associated to a fan $\Sigma\subset N_{{\mathbb{R}}}$ without $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action, there exists a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-equivariant closed embedding \jmath:X\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma^{\prime}} into a $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric variety $Y_{\Sigma^{\prime}}$ and a (non-$G$-equivariant) surjective morphism of toric ${K}$ varieties Y_{\Sigma^{\prime}}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} that fits into a commutative diagram of $K$-varieties

[TABLE]

Proof.

The toric ${K}$-variety $Y_{\Sigma}$ is of the form ${K}\otimes_{K_{0}}Y_{\Sigma,K_{0}}$, where $Y_{\Sigma,K_{0}}$ denotes the toric $K_{0}$-variety associated to $\Sigma$. Equip $Y_{\Sigma}$ with its resulting canonical $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action. Let $M$ denote the lattice of characters on the dense torus in $Y_{\Sigma}$. Define $\mathfrak{F}$ to be the multiplicative group

[TABLE]

consisting of rational functions on ${X}$. The canonical $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ {X} induces a $G$-action on the multiplicative semigroup ${K}({X})$ of rational functions. Because $M$ is a finitely generated free group, $\mathfrak{F}$ is a finitely generated multiplicative subsemigroup of $K(X)$. Hence the $G$-orbit $G\mathfrak{F}$ is itself a finitely generated multiplicative subsemigroup of ${K}({X})$. Consequently, there exists an open normal subgroup $Z\lhd G$ fixing $\mathfrak{F}$, i.e., such that $Zf=f$ for all $f\in\mathfrak{F}$. If we define

[TABLE]

then the natural $G$-action on the orbit $G\mathfrak{F}$ factors through an $H$-action H\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ G\mathfrak{F}. Let $L\operatorname{\overset{{}_{\text{def}}}{=}}{K}^{Z}$ denote the fixed field of $Z$, so that $H=\text{Gal}(L/K_{0})$.

If $Y_{\Sigma,L}$ denotes the toric $L$-variety associated to the fan $\Sigma$, then our construction of $L$ implies that the closed embedding \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} over $K$ is actually the pullback of a closed embedding

[TABLE]

Equip $X_{0,L}$ and $Y_{\Sigma,L}$ with their canonical $H^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-actions. For each $h\in H$, let \imath^{h}_{L}:X_{0,L}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma,L} denote the pullback morphism fitting into the Cartesian diagram

[TABLE]

Define $N^{H}\operatorname{\overset{{}_{\text{def}}}{=}}\prod_{H}N$ to be $\#H$-fold Cartesian power of $N$ with factors indexed by $H$. Likewise, define $\Sigma^{H}\operatorname{\overset{{}_{\text{def}}}{=}}\prod_{H}\Sigma$ to be the fan in $N^{H}_{{\mathbb{R}}}$ obtained as the $\#H$-fold Cartesian power of the fan $\Sigma$ with factors indexed by $H$. Let $Y_{\Sigma^{H}\!\!,\ \!L}$ be the associated toric $L$-variety. Then we have a closed embedding

[TABLE]

The cocharacter lattice $N^{H}$ comes with a natural $H^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action that permutes the factors of $N^{H}$, and the fan $\Sigma^{H}$ is invariant under the induced $H^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action H^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ N^{H}_{{\mathbb{R}}}. This $H^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action on $N^{H}$ induces a twisted $H^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action H^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}\ \raisebox{8.0pt}{\rotatebox{-90.0}{\circlearrowright}}\ Y_{\Sigma^{H}\!\!,\ \!L}, which is to say that the diagram

[TABLE]

commutes for each $h\in H$.

We claim that the closed embedding (25) is $H$-equivariant, i.e., that each diagram

[TABLE]

commutes. Since $Y_{\Sigma^{H}\!\!,\ \!L}=\prod_{H}Y_{\Sigma,L}$, we can check this by verifying the universal property of each of the two composites in (26). To this end, observe that for each $\ell\in H$, we have commutative diagrams

[TABLE]

and

[TABLE]

Because the diagram (24) is Cartesian, the two composites are equal.

Finally, define $Y_{\Sigma^{H}}$ to be the pullback of $Y_{\Sigma^{H}\!\!,\ \!L}$ to ${K}$, equipped with the resulting twisted $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-action, and define \jmath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma^{H}} to be the pullback of $\jmath_{L}$ to ${K}$. Then $G$-equivariance of $\jmath$ follows immediately from $H$-equivariance of $\jmath_{L}$. ∎

Corollary 3.2.2**.**

Assume that ${X}$ is projective. Then for each integer $n>0$ and each non-equivariant closed embedding \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!\mathbb{P}^{n}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smallerK}}}, where \mathbb{P}^{n}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smallerK}}} is equipped with its standard toric structure, there exists a $G$-equivariant closed embedding

[TABLE]

that factors as

[TABLE]

where $\jmath$ is a $G$-equivariant closed embedding constructed from $\imath$ as in the proof of Proposition 3.2.1, where Seg is the $\#H$-fold Segre embedding, and where \mathbb{P}^{m}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smaller{K}}}} has the $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted action induced by the $H$-action on the factors of \prod_{H}\mathbb{P}^{m}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smaller{K}}}}. $\square$

Remark 3.2.3**.**

A large supply of G-representations. Assume that ${X}$ is projective. Then by Lemma 3.1.2, we can construct a large system of closed toric embeddings \imath:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!\mathbb{P}^{n}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smaller{K}}}}. Each embedding realizes a given set of functions on the complement of an effective ample divisor on ${X}$ as the pullbacks of coordinate linear functions on \mathbb{P}^{n}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smaller{K}}}}. By Corollary 3.2.2, we can turn any given choice of $\imath$ into a $G$-equivariant closed embedding \jmath^{\prime}:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!\mathbb{P}^{m}_{\!\!\mbox{{\smaller\smaller\smaller\smaller\smaller{K}}}}. Thus we have an extensive supply of $G$-equivariant closed embeddings of ${X}$ into $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-twisted toric projective spaces. If we assume that $K_{0}$ is Henselian, then Theorem D turns this into a extensive supply of $G$-representations in tropical cellular cohomology groups.

4. Proofs of the Main Theorems

4.1. Criterion for a G-equivariant homeomorphism: Theorem A

We begin by proving Theorem A, which says that Conditions ($\Pi$) and ($\star$) together provide a criterion for checking if a system of $G$-equivariant toric varieties has enough closed embeddings to give an affirmative answer to Question 2. The proof amounts to an appeal, through cofinality, to the results of [13].

Proof of Theorem A.

Let $\mathcal{S}_{G}$ be a system of $G$-equivariant toric embeddings of ${X}$ satisfying Conditions ($\Pi$) and ($\star$) in Definition 1.4.1. There is a forgetful functor

[TABLE]

that takes each $G$-equivariant closed toric embedding in $\mathcal{S}_{G}$ to its underlying closed toric embedding. Note that this forgetful functor (27) will not be fully faithful, since a given toric ${K}$-variety $Y_{\Sigma}$ can have several distinct twisted $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-actions. Let $\mathcal{S}$ denote the system of toric embeddings of ${X}$ given by the image of the forgetful functor (27). The fact that $\mathcal{S}_{G}$ satisfies Conditions ($\Pi$) and ($\star$) of Definition 1.4.1 implies that $\mathcal{S}$ satisfies the hypotheses of [13, Theorem 1.1]. Thus the map

[TABLE]

is a homeomorphism of topological spaces (without $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-actions). The functor

[TABLE]

is final (or what is also called “co-cofinal” in [18, Definition 2.5.1.(ii)]). By [18, Proposition 2.5.2.(ii)], this implies that the forgetful functor induces a homeomorphism of the underlying topological spaces

[TABLE]

Hence the map (4) is a homeomorphism. Finally, the map (4) is $G$-equivariant by Corollary 2.2.8. ∎

4.2. Proofs of Theorems B and C

Theorem A provides us with a criterion for verifying whether or not a given system $\mathcal{S}_{G}$ of $G$-equivariant toric embeddings induces a $G$-equivariant homeomorphism (4). In order to construct $G$-equivariant systems of toric embeddings satisfying this criterion, we employ any one of the algorithms of §3.1.

Proof of Theorem B.

Fix an affine open cover $\{U_{1},\dots,U_{r}\}$ provided by Lemma 3.1.2 if $X$ is quasiprojective, or as in Theorem 3.1.5 if $K_{0}$ is perfect and $X$ admits at least one closed embedding into a toric ${K}$-variety (using the fact that any ambient toric variety is normal and A2), or as in Theorem 3.1.7 if $\text{char}_{\ \!}K_{0}=0$ and $X$ is an A2-variety over ${K}$ with H^{0}({X},\mathscr{O}^{\times}_{\!\mbox{{\smaller\smaller\smaller\smaller\smaller{X}}}})=K^{\times} and the ideal class group and Cox ring of ${X}$ are finitely generated. Let $\mathfrak{F}$ denote the set of all rational functions on ${X}$ regular on at least one of the affine opens $U_{i}$. For each $f\in\mathfrak{F}$ and each choice of open $U_{i}$ with $f\in{K}[U_{i}]$, construct a non-equivariant toric embedding \imath_{f}:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma_{f}} as in Lemma 3.1.2, Theorem 3.1.5, or Theorem 3.1.7, respectively. Use $\imath_{f}$ and Theorem 3.2.1 to build a $G$-equivariant closed toric embedding

[TABLE]

Note that the existence of the projection Y_{\Sigma^{H}_{f}}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma} fitting into an instance of the commutative diagram (23) implies that $U_{i}$ is the inverse image, under $\jmath_{f}$, of a torus-invariant open subscheme in $Y_{\Sigma^{H}_{f}}$, and that $f$ is the inverse image $f=\jmath^{\#}_{f}(a\chi^{u})$ of a monomial on the dense torus in $Y_{\Sigma^{H}_{f}}$.

Let $\mathbb{Sep}_{G}(X;\mathfrak{F})$ denote the discrete subcategory of $\mathbb{TEmb}_{G}({X})$ with objects consisting of all the $G$-equivariant closed embeddings (28) that we’ve constructed as $(f,U_{i})$ ranges over $f\in\mathfrak{F}$ and $U_{i}$ such that $f\in{K}[U_{i}]$. Then $\mathbb{Sep}_{G}(X;\mathfrak{F})$ satisfies Condition ($\star$) of Definition 1.4.1. Any system $\mathcal{S}_{G}$ of $G$-equivariant toric embeddings containing $\mathbb{Sep}_{G}(X;\mathfrak{F})$ and satisfying Condition ($\Pi$) of Definition 1.4.1 satisfies the theorem, with $\mathbb{TEmb}_{G}({X})$ being an example of one such system. ∎

Proof of Theorem C.

Let $x$ be a point in $X^{\text{an}}$ with finite $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}$-orbit $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}x$. For each pair of distinct points $y$ and $z$ in $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}x$, [13, Proposition 3.2] shows that one of the closed embeddings \imath_{yz}:{X}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces\!Y_{\Sigma_{yz}} provided by Lemma 3.1.2, Theorem 3.1.5, or Theorem 3.1.7 satisfies

[TABLE]

Applying Theorem 3.2.1, we obtain a $G$-equivariant toric embedding

[TABLE]

such that $\text{trop}(x^{\prime})\neq\text{trop}(x^{\prime\prime})$ inside $\text{Trop}_{G}({X},\jmath_{yz})$. Let $\text{S}^{2}_{\ \!}G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}x$ denote the set of all unordered pairs of distinct points in $G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}x$. Then we can form a new $G$-equivariant closed toric embedding as the product

[TABLE]

Each projection

[TABLE]

induces a morphism of $G$-equivariant tropicalizations $\text{Trop}_{G}(\prod_{\{y,z\}}Y_{\Sigma_{yz}})\longrightarrow\text{Trop}_{G}(Y_{\Sigma_{yz}})$ that restricts to a morphism of $G$-equivariant tropicalizations $\text{Trop}_{G}({X},\jmath)\longrightarrow\text{Trop}_{G}({X},\jmath_{yz})$. Because each pair of distinct points $y,z\in G^{\textsf{{\smaller\smaller\smaller\smaller\smaller op}}}x$ is separated in its corresponding tropicalization $\text{Trop}({X},\jmath_{yz})$, we conclude that the $G$-equivariant composite

[TABLE]

is a bijection onto its image. ∎