# Real phase structures on tropical varieties and patchworks in higher
codimension

Johannes Rau  Johannes Rau, Departamento de Matemáticas, Universidad de los
Andes, Carrera 1 # 18A - 12, 111711 Bogotá, Colombia<EMAIL_ADDRESS>,
Arthur Renaudineau  Arthur Renaudineau, Univ. Lille, CNRS, UMR 8524 -
Laboratoire Paul Painlevé, F-59000 Lille, France.  arthur.renaudineau@univ-
lille.fr and  Kris Shaw  Kris Shaw, University of Oslo, Oslo, Norway.
<EMAIL_ADDRESS>
###### Abstract.

This paper generalises the homeomorphism theorem behind Viro’s combinatorial
patchworking of hypersurfaces in toric varieties to arbitrary codimension
using tropical geometry. We first define the patchwork of a polyhedral space
equipped with a real phase structure. When the polyhedral subspace is
tropically non-singular, we show that the patchwork is a topological manifold.
When a non-singular tropical variety appears as a tropical limit of a real
analytic family, we show that the real part of a fibre of the family near the
tropical limit is homeomorphic to the patchwork.

Finally we extend the spectral sequence introduced by the last two authors in
the case of hypersurfaces to non-singular tropical varieties with real phase
structures. As a corollary, we obtain bounds on the Betti numbers of the
patchwork in terms of the dimensions of the tropical homology groups with
coefficients modulo two.

###### Contents

  1. 1 Introduction
  2. 2 Real phase structures
    1. 2.1 Tropical toric varieties
    2. 2.2 Polyhedral complexes
    3. 2.3 Real phase structures on polyhedral subspaces
    4. 2.4 Tropical patchworking
  3. 3 Non-singular tropical subvarieties and their patchworks
    1. 3.1 Tropically non-singular polyhedral subspaces
    2. 3.2 Patchworks of non-singular subvarieties are topological manifolds
  4. 4 Patchworking close to the tropical limit
    1. 4.1 Non-singular tropical limits
    2. 4.2 Real tropical limits and patchworking
    3. 4.3 Reduction to the unimodular case
    4. 4.4 Regular CW complexes and pairs
    5. 4.5 The local cases
    6. 4.6 Semi-stable degenerations — classical side
    7. 4.7 Semi-stable degenerations — tropical side
    8. 4.8 Comparison to other real tropicalizations
  5. 5 A spectral sequence for the homology of a patchwork
    1. 5.1 Celluar cosheaves
    2. 5.2 Tropical homology
    3. 5.3 The sign cosheaf
    4. 5.4 The filtration of the sign cosheaf
    5. 5.5 The spectral sequence
    6. 5.6 Hirzebruch polynomials

##  1\. Introduction

Viro’s patchworking method is a powerful method to study the topological types
of real algebraic hypersurfaces in toric varieties [Vir84]. A particular case
of the method is _combinatorial_ patchworking which, from some combinatorial
data, produces two objects: a polyhedral space and a patchworking polynomial.
Viro’s theorem provides a homeomorphism between the polyhedral space and the
real part of a hypersurface defined by the polynomial. It was later
generalized to complete intersections, see [Stu94] and [Bih02]. This paper
aims to generalise the homeomorphism theorem of combinatorial patchworking
beyond the case of hypersurfaces and complete intersections.

To generalise the theorem to arbitrary codimension, we begin by considering
rational polyhedral spaces and tropical varieties equipped with real phase
structures, which were defined in the case of polyhedral fans in [RRS22]. For
a rational polyhedral subspace $X$ in ${\mathbb{R}}^{n}$, a real phase
structure is an assignment of an affine subspace of the vector space
${\mathbb{Z}}_{2}^{n}$, where ${\mathbb{Z}}_{2}:={\mathbb{Z}}/2{\mathbb{Z}}$
to each facet of $X$.

The assignment of a real phase structure is then used to describe the
_patchwork_ of a tropical variety, which should be thought of as its real
part, see Definition 2.12. Our first result is a description of the structure
of the patchwork when we start from a non-singular tropical subvariety.

######  Theorem 1.1.

The patchwork of a non-singular tropical subvariety equipped with a real phase
structure in a tropical toric variety is a topological manifold.

We then show that a patchwork describes, up to homeomorphism, fibers of real
analytic families with non-singular tropical limits. Let us describe our
setup: Let ${\mathbf{X}}\subset({\mathbb{C}}^{*})^{n}\times{\mathcal{D}}^{*}$
be a real analytic family of algebraic varieties over the punctured disc
${\mathcal{D}}^{*}$. That is, locally ${\mathbf{X}}$ can be described by
polynomials whose coefficients are Laurent series convergent on
${\mathcal{D}}^{*}$. Moreover, the family is compatible with complex
conjugations on $({\mathbb{C}}^{*})^{n}$ and ${\mathcal{D}}^{*}$. To such a
family, we can associate a tropical limit
$\operatorname{Trop}({\mathbf{X}})=X\subset{\mathbb{R}}^{n}$, by taking the
image of ${\mathbf{X}}$ under the valuation map, see Section 4.1. Let us
assume that $\operatorname{Trop}({\mathbf{X}})$ is tropically non-singular,
see Section 4.1 for definition. Then additionally the family ${\mathbf{X}}$
induces a real phase structure ${\mathcal{E}}$ on
$\operatorname{Trop}({\mathbf{X}})$, see Definition 4.4. We relate this to
pre-existing notions of tropicalising real varieties in Remark 4.8.

For any pointed unimodular fan $\Sigma$, we can consider the compactification
of ${\mathbf{X}}\subset({\mathbb{C}}^{*})^{n}\times{\mathcal{D}}^{*}$ in
${\mathbb{C}}\Sigma\times{\mathcal{D}}^{*}$. Consider a subdivision
${\mathcal{X}}$ of $X$ and the set $\operatorname{RecCone}({\mathcal{X}})$ of
all recession cones for polyhedron in ${\mathcal{X}}$, see Section 2.2. The
second main theorem is that, under the assumption that
$\operatorname{RecCone}({\mathcal{X}})\cup\Sigma$ forms a fan, the patchwork
$\mathfrak{R}(\overline{X},{\mathcal{E}}({\mathbf{X}}))$ of $X$ (see
Definition 2.12) in the real tropical toric variety
$\mathfrak{R}{\mathbb{T}}\Sigma$ of $\Sigma$ (see Equation 2.3) gives the
homeomorphism type of
${\mathbb{R}}\overline{{\mathbf{X}}}_{t}\subset{\mathbb{R}}\Sigma$ for
sufficiently small positive $t$.

######  Theorem 1.2 (Patchworking for non-singular tropical limits).

Let ${\mathbf{X}}\subset({\mathbb{C}}^{*})^{n}\times{\mathcal{D}}^{*}$ be a
real analytic family with non-singular tropical limit
$X=\operatorname{Trop}({\mathbf{X}})$ and associated real phase structure
${\mathcal{E}}={\mathcal{E}}({\mathbf{X}})$. Let ${\mathcal{X}}$ be a
subdivision of $X$ and $\Sigma$ a pointed unimodular fan such that
$\operatorname{RecCone}({\mathcal{X}})\cup\Sigma$ is a fan.

Then for sufficiently small positive $t\in{\mathcal{D}}^{*}\cap{\mathbb{R}}$
the pairs $({\mathbb{R}}\Sigma,{\mathbb{R}}\overline{{\mathbf{X}}}_{t})$ and
$(\mathfrak{R}{\mathbb{T}}\Sigma,\mathfrak{R}(\overline{X},{\mathcal{E}}))$
are homeomorphic. Moreover, the homeomorphism can be chosen to respect the
stratification of ${\mathbb{R}}\Sigma$ and $\mathfrak{R}{\mathbb{T}}\Sigma$ by
torus orbits.

We expect that the assumption that
$\operatorname{RecCone}({\mathcal{X}})\cup\Sigma$ forms a fan can be weakened.

We give a rough outline of the proof strategy here. The first step is the
reduction to a variant of the above theorem in which we assume the existence
of a unimodular subdivision ${\mathcal{P}}$ of ${\mathbb{R}}^{n}$ compatible
with $\operatorname{Trop}({\mathbf{X}})$ in 4.7. Under the unimodular
assumption, the strategy is then to view both topological pairs from 1.2 as
regular CW pairs in the sense of Definition 4.11. In order to do this, we must
prove a sequence of lemmas in Section 4.5 establishing topological properties
of closures of real linear spaces and their patchworks in toric varieties. We
then use these lemmas and [Rau23] to give a description of
$({\mathbb{R}}\Sigma,{\mathbb{R}}\overline{{\mathbf{X}}}_{t})$ as a CW pair
for small and positive $t$, see 4.19.

We then aim to compare the CW description of
$({\mathbb{R}}\Sigma,{\mathbb{R}}\overline{{\mathbf{X}}}_{t})$ to the
patchwork by introducing some intermediate CW pairs. Figure 1 shows the spaces
involved in the steps we now outline and the relations between them. First we
introduce the notion of tropical semi-stable degeneration and of tropical
special fibre arising from the subdivision ${\mathcal{P}}$, see Definition
4.20. We consider the pair of tropical special fibers, denoted by
$({\mathbb{T}}{\mathcal{P}}_{\infty},\overline{X}_{\infty})$. We prove that
this pair is homeomorphic to the pair $({\mathbb{T}}\Sigma,\overline{X})$, by
using an adaptation of cubical subdivisions which we call the _bounded cubical
subdivision_ , see Proposition 4.30. The operation of taking this subdivision
is denoted $BC$ in Figure 1. We then pass to the patchwork of the tropical
positive special fiber
$(\mathfrak{R}{\mathbb{T}}{\mathcal{P}}^{+}_{\infty},\mathfrak{R}(\overline{X},{\mathcal{E}})^{+}_{\infty})$,
which, it follows, is homeomorphic to the pair
$(\mathfrak{R}{\mathbb{T}}\Sigma,\mathfrak{R}(\overline{X},{\mathcal{E}}))$.

${({\mathbb{T}}{\mathcal{P}}_{\infty},\overline{X}_{\infty})}$${(\mathfrak{R}{\mathbb{T}}{\mathcal{P}}^{+}_{\infty},\mathfrak{R}(\overline{X},{\mathcal{E}})^{+}_{\infty})}$${({\mathbb{T}}\Sigma,\overline{X})}$${(\mathfrak{R}{\mathbb{T}}\Sigma,\mathfrak{R}(\overline{X},{\mathcal{E}}))}$${({\mathbb{R}}\Sigma,{\mathbb{R}}\overline{{\mathbf{X}}}_{t})}$$\scriptstyle{patch}$$\scriptstyle{BC}$$\scriptstyle{patch}$$\scriptstyle{BC}$$\scriptstyle{cub}$
Figure 1. The CW complexes involved in the proof of 1.2

We finish our comparison of
$(\mathfrak{R}{\mathbb{T}}\Sigma,\mathfrak{R}(\overline{X},{\mathcal{E}}))$
and $({\mathbb{R}}\Sigma,{\mathbb{R}}\overline{{\mathbf{X}}}_{t})$ by showing
that the tropical positive special fiber
$(\mathfrak{R}{\mathbb{T}}{\mathcal{P}}^{+}_{\infty},\mathfrak{R}(\overline{X},{\mathcal{E}})^{+}_{\infty})$
is also a cubical subdivision (denoted $cub$ in Figure 1) of the CW pair
structure that we define on
$({\mathbb{R}}\Sigma,{\mathbb{R}}\overline{{\mathbf{X}}}_{t})$ in 4.19.
Lastly, we use that CW pairs with isomorphic face posets are homeomorphic.

We say that a real algebraic variety is _close to a non-singular limit_ if it
is isomorphic to a fibre $\overline{{\mathbf{X}}}_{t}$ for which the
assumptions of 1.2 hold. The homeomorphism from 1.2 implies that statements
about the topology of $\mathfrak{R}(\overline{X},{\mathcal{E}})$ translate
into statements about the topology of real algebraic varieties close to a non-
singular tropical limit.

Next, we adapt the arguments from [RS23] to bound the Betti numbers of the
patchwork in terms of the dimensions of the tropical homology groups. For a
tropical manifold $X$, we denote the tropical homology groups with
${\mathbb{Z}}_{2}$ coefficients by $H_{q}(X;{\mathcal{F}}_{p})$. We let
$H_{q}^{BM}(X;{\mathcal{F}}_{p})$ denote the Borel-Moore variants of these
homology groups. See Section 5.2 for the definition of tropical homology.

We define the (Borel-Moore) tropical signature of $X$ as

$\sigma^{\diamond}(X):=\sum_{p,q}(-1)^{q}H_{q}^{\diamond}(X;{\mathcal{F}}_{p}),$

where $\diamond$ is either empty, denoting usual homology, or $\diamond$ is
BM, denoting Borel-Moore homology. Notice that $\sigma^{\diamond}(X)$ is also
equal to
$\sum_{p,q}(-1)^{q}H_{q}^{\diamond}(X;{\mathcal{F}}_{p}^{{\mathbb{R}}})$ where
$H_{q}^{\diamond}(X;{\mathcal{F}}_{p}^{{\mathbb{R}}})$ denotes tropical
homology with real coefficients.

######  Theorem 1.3.

If $X$ is a $d$-dimensional non-singular tropical subvariety equipped with a
real phase structure ${\mathcal{E}}$, then

$b_{q}(\mathfrak{R}(X,{\mathcal{E}}))\leq\sum_{p=0}^{d}\dim
H_{q}(X;{\mathcal{F}}_{p})$

and

$b^{BM}_{q}(\mathfrak{R}(X,{\mathcal{E}}))\leq\sum_{p=0}^{d}\dim
H^{BM}_{q}(X;{\mathcal{F}}_{p}).$

Moreover, the (Borel-Moore) Euler characteristic of
$\mathfrak{R}(X,{\mathcal{E}})$ is equal to the (Borel-Moore) tropical
signature of $X$, namely,

$\displaystyle\chi(\mathfrak{R}({\mathcal{X}},{\mathcal{E}}))=\sigma(X),$
$\displaystyle\chi^{\text{BM}}(\mathfrak{R}({\mathcal{X}},{\mathcal{E}}))=\sigma^{\text{BM}}(X).$

We then combine Theorems 1.2 and 1.3 to obtain the following two corollaries.

######  Corollary 1.4.

Under the same hypothesis as in Theorem 1.2, for sufficiently small
$t\in{\mathcal{D}}^{*}\cap{\mathbb{R}}$ we have

$b_{q}({\mathbb{R}}\overline{{\mathbf{X}}}_{t})\leq\sum_{p=0}^{d}\dim
H_{q}(\overline{X};{\mathcal{F}}_{p})\quad\text{ and }\quad
b^{BM}_{q}({\mathbb{R}}\overline{{\mathbf{X}}}_{t})\leq\sum_{p=0}^{d}\dim
H^{BM}_{q}(\overline{X};{\mathcal{F}}_{p}).$

We define the _signature_ of a generic fibre $\overline{{\mathbf{X}}}_{t}$ as

$\sigma_{c}(\overline{{\mathbf{X}}}_{t}):=\sum_{p,q}(-1)^{p}e_{c}^{p,q}(\overline{{\mathbf{X}}}_{t})$

where $e_{c}^{p,q}(\overline{{\mathbf{X}}}_{t})$ is defined via the mixed
Hodge structure on $\overline{{\mathbf{X}}}_{t}$ by
$e_{c}^{p,q}(\overline{{\mathbf{X}}}_{t}):=\sum_{k}(-1)^{k}h^{p,q}(H_{c}^{k}(\overline{{\mathbf{X}}}_{t})).$
See for instance [DK86].

######  Corollary 1.5.

Let ${\mathbf{X}}\subset{\mathbb{C}}\Sigma\times{\mathcal{D}}^{*}$ be a real
meromorphic family with non-singular tropical limit satisfying the same
hypothesis as in Theorem 1.2. Then for sufficiently small
$t\in{\mathcal{D}}^{*}\cap{\mathbb{R}}$ we have

$\chi^{\text{BM}}({\mathbb{R}}\overline{{\mathbf{X}}}_{t})=\sigma_{c}(\overline{{\mathbf{X}}}_{t}).$

This equality was already proved in the case of codimension one by Bertrand
[Ber10] (see also [Arn17]), and by Brugallé in the same situation as above by
utilising a different point of view [Bru21].

In a recent paper [AM22], the authors generalized the ideas of [RS23] beyond
the case of tropical varieties. They proved that the same bounds still apply
for any totally real semi-stable degeneration satisfying strong homological
properties. Essentially what they require is that the open strata of the
special fiber are all maximal in the sense of the Smith-Thom inequality, the
real parts only have non-trivial homology in degree $0$, and that the
cohomology of the complex open strata satisfy a strong condition on their
mixed Hodge structures. These properties are satisfied in our context and so,
up to a comparison of tropical homology and the complexes obtained by Ambrosi
and Manzaroli, this eventually gives another proof of Corollary 1.4.

We would also like to mention the work of Brugallé, López de Medrano and the
first author [BLdMR22]. This work transfers tropical homology, real phase
structures, and the Betti number bounds of 1.3 to the original framework of
Viro’s patchworking method, namely the framework of (unimodular)
triangulations of lattice polytopes. An overlap with the present work is given
if the triangulation is assumed to be convex. Then the objects of study in
[BLdMR22] correspond to self-intersections of a tropical hypersurface
associated to the triangulation (and the correspondence can supposedly
extended to complete intersections using mixed subdivisions). So while
[BLdMR22] also covers non-convex triangulations which do not have a tropical
counterpart, the present approach also treats non-complete intersections which
seem to have no counterpart on the triangulation side.

## Acknowledgements

The authors are grateful to Erwan Brugallé, Ilia Itenberg, Oleg Viro for
helpful conversations and comments.

This research was supported in part by the Trond Mohn Foundation project
“Algebraic and Topological Cycles in Complex and Tropical Geometries”. We also
acknowledge the support of the Centre for Advanced Study (CAS) in Oslo,
Norway, which funded and hosted the Young CAS research project Real Structures
in Discrete, Algebraic, Symplectic, and Tropical Geometries (REACTIONS) during
the 2021-23 academic years.

Johannes Rau was supported by the FAPA project “Matroids in tropical geometry”
from the Facultad de Ciencias, Universidad de los Andes, Colombia.

Arthur Renaudineau acknowledges support from the Labex CEMPI
(ANR-11-LABX-0007-01), and from ANR, project ANR-22-CE40-0014.

##  2\. Real phase structures

###  2.1. Tropical toric varieties

Given a polyhedron $\sigma\subset{\mathbb{R}}^{n}$ we denote by $T(\sigma)$
its tangent space, that is, the vector subspace of ${\mathbb{R}}^{n}$
generated by all vectors $u-v$, $u,v\in\sigma$. We are often interested in the
quotient space ${\mathbb{R}}^{n}/T(\sigma)$ for which we hence use the
shorthand ${\mathbb{R}}(\sigma)$. If $\sigma$ is a rational polyhedron, i-e
the supporting hyperplanes are defined over ${\mathbb{Q}}$, we denote by
$T_{\mathbb{Z}}(\sigma)=T(\sigma)\cap{\mathbb{Z}}^{n}$ its _tangent lattice_ ,
and set ${\mathbb{Z}}(\sigma)={\mathbb{Z}}^{n}/T_{{\mathbb{Z}}}(\sigma)$.
Finally, for a ${\mathbb{Z}}$-module $G$, we set
$T_{G}(\sigma)=T_{\mathbb{Z}}(\sigma)\otimes G$ and
$G(\sigma)=G^{n}/T_{G}(\sigma)$.

Let $\Sigma$ be a rational polyhedral pointed fan in ${\mathbb{R}}^{n}$. Given
$\rho\in\Sigma$, we denote the dual cone by

$\rho^{\vee}=\\{\lambda\in({\mathbb{R}}^{n})^{\vee}:\lambda|_{\rho}\geq
0\\}\subset({\mathbb{R}}^{n})^{\vee}.$

We denote by
${\mathbb{T}}U_{\rho}=\operatorname{Hom}(\rho^{\vee}\cap({\mathbb{Z}}^{n})^{\vee},{\mathbb{T}})$
the space of semigroup homomorphisms, where
${\mathbb{T}}={\mathbb{R}}\cup\left\\{-\infty\right\\}$ endowed with the
addition is the _tropical_ semigroup. We equip it with the coarsest topology
such that for all $\lambda\in\rho^{\vee}\cap({\mathbb{Z}}^{n})^{\vee}$ the
evaluation map ${\mathbb{T}}U_{\rho}\to{\mathbb{T}}$,
$\alpha\mapsto\alpha(\lambda)$, is continuous. If $\rho\subset\eta$, we have
by restriction a continuous map ${\mathbb{T}}U_{\rho}\to{\mathbb{T}}U_{\eta}$.

######  Definition 2.1.

[MR, Chapter 3] [Pay09, Section 3] The tropical toric variety associated to
the fan $\Sigma$ is the direct limit

${\mathbb{T}}\Sigma:=\varinjlim_{\rho\in\Sigma}{\mathbb{T}}U_{\rho}.$

We call ${\mathbb{T}}\Sigma$ _non-singular_ if $\Sigma$ is unimodular (and
hence simplicial).

The space ${\mathbb{T}}\Sigma$ is compact if and only if the corresponding fan
is complete. The action of ${\mathbb{R}}^{n}$ on itself extends to a group
action on ${\mathbb{T}}\Sigma$ whose orbits are given by
${\mathbb{T}}{\mathcal{O}}_{\rho}={\mathbb{R}}(\rho)\cong\operatorname{Hom}(\rho^{\perp},{\mathbb{T}})\subset{\mathbb{T}}U_{\rho}$
(the inclusion is given by extension by $-\infty$). Notice that
$\dim{\mathbb{T}}{\mathcal{O}}_{\rho}=n-\dim\rho$. Given a point
$y\in{\mathbb{T}}{\mathcal{O}}_{\rho}$, we call $\rho$ the _sedentarity_ of
$y$. The _order of sedentarity_ of $y$, denoted by $\operatorname{sed}(y)$, is
set to be $\dim(\rho)$.

######  Example 2.2.

Tropical affine space ${\mathbb{T}}^{n}$ is the tropical toric variety
associated to the fan that consists of the negative orthant in
${\mathbb{R}}^{n}$ and all its faces. Given a subset $A$ of
$E=\\{1,\dots,n\\}$, we denote by $\rho_{A}$ the cone generated by
$\\{-e_{i},i\in A\\}$, where the $e_{i}$ are the vectors of the standard basis
of ${\mathbb{R}}^{n}$. Then the torus orbit
${\mathbb{R}}^{n}_{A}:={\mathbb{T}}{\mathcal{O}}_{\rho_{A}}$ is equal to
${\mathbb{R}}^{E\setminus A}$, while its closure in ${\mathbb{T}}^{n}$,
denoted by ${\mathbb{T}}^{n}_{A}$, can be identified with
${\mathbb{T}}^{E\setminus A}$.
Similarly, we set $e_{0}=-e_{1}-\dots-e_{n}$ and for any _proper_ subset
$A\subset\\{0,\dots,n\\}$ we let $\sigma_{A}$ be the cone generated by
$\\{-e_{i},i\in A\\}$. The collection of these cones forms a complete fan
defining the tropical toric variety ${{\mathbb{T}}{\mathbb{P}}}^{n}$. We
denote the closure of ${\mathbb{T}}{\mathcal{O}}_{\rho_{A}}$ in
${{\mathbb{T}}{\mathbb{P}}}^{n}$ by ${{\mathbb{T}}{\mathbb{P}}}^{n}_{A}$. It
can be identified with a projective space ${{\mathbb{T}}{\mathbb{P}}}^{n-|A|}$
(whose homogeneous coordinates are labelled by $E\setminus A$).

######  Definition 2.3.

Given two fans $\Sigma$ in ${\mathbb{R}}^{n}$ and $\Sigma^{\prime}$ in
${\mathbb{R}}^{m}$, a _morphism of fans_ $A\colon\Sigma\to\Sigma^{\prime}$ is
a ${\mathbb{Z}}$-linear map $A\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ such
that $A(\sigma)\in\Sigma^{\prime}$ for all $\rho\in\Sigma$.

Let ${\mathbb{T}}\Sigma$ and ${\mathbb{T}}\Sigma^{\prime}$ denote the tropical
toric varieties associated to $\Sigma$ and $\Sigma^{\prime}$. A _(toric)
morphism_ $\psi\colon{\mathbb{T}}\Sigma\to{\mathbb{T}}\Sigma^{\prime}$ is a
continuous map that sends ${\mathbb{R}}^{n}\subset{\mathbb{T}}\Sigma$ to
${\mathbb{R}}^{m}\subset{\mathbb{T}}\Sigma^{\prime}$ and such that the
restriction ${\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ is of the form $x\mapsto
Ax+v$ where $A$ is a morphism of fans from $\Sigma$ to $\Sigma^{\prime}$.

We note that given $x\mapsto Ax+v$, there is a unique continuous extension
$\psi\colon{\mathbb{T}}\Sigma\to{\mathbb{T}}\Sigma^{\prime}$. Moreover, this
extension sends the torus orbit ${\mathbb{T}}{\mathcal{O}}_{\rho}$ to the
torus orbit ${\mathbb{T}}{\mathcal{O}}_{A(\rho)}$ and the restriction of
$\psi$ to these orbits is given by the induced map

${\mathbb{R}}(\rho)\to{\mathbb{R}}(A(\rho)),[x]\mapsto[Ax+v].$

######  Example 2.4.

Let ${\mathbb{T}}\Sigma$ be a tropical toric variety. The open subsets
${\mathbb{T}}U_{\rho}$ are toric varieties themselves (their defining fan
consists of $\rho$ and all its faces). If $\rho$ is a unimodular cone
generated by the first $k$ vectors of the ${\mathbb{Z}}^{n}$-basis
$v_{1},\dots,v_{n}$, then

$\rho^{\vee}\cap({\mathbb{Z}}^{n})^{\vee}=\langle
v_{1}^{*},\dots,v_{k}^{*},\pm v_{k+1}^{*},\dots,\pm
v_{n}^{*}\rangle\cong{\mathbb{N}}^{k}\times{\mathbb{Z}}^{n-k}$

as a cone, and hence
${\mathbb{T}}U_{\rho}\cong{\mathbb{T}}^{k}\times{\mathbb{R}}^{n-k}$. Hence if
${\mathbb{T}}\Sigma$ is non-singular, it is covered by toric open subsets
isomorphic to ${\mathbb{T}}^{k}\times{\mathbb{R}}^{n-k}$ (for varying $k$).

######  Example 2.5.

The only automorphisms of ${\mathbb{T}}^{n}$ are coordinate permutations
composed with translations, so
$\text{Aut}({\mathbb{T}}^{n})=\text{Sym}_{n}\ltimes{\mathbb{R}}^{n}$. This
follows from the fact that (fan) automorphisms of the negative orthant of
${\mathbb{R}}^{n}$ must permute its rays.

######  Remark 2.6.

We focused here on tropical toric varieties, but the given construction of
toric varieties makes sense for any semigroup $S$ (the given description of
torus orbits requires additionally that $S$ contains a single non-invertible
element). In particular, we will later use the cases
$S={\mathbb{C}},{\mathbb{R}},{\mathbb{R}}_{\geq}=[0,\infty)$ (all with respect
to multiplication) and (briefly) a few more. The toric variety of the fan
$\Sigma$ over the semigroup $S$ will be denoted by $S\Sigma$.

###  2.2. Polyhedral complexes

A rational polyhedron in a tropical toric variety ${\mathbb{T}}\Sigma$ is the
closure in ${\mathbb{T}}\Sigma$ of a rational polyhedron in some stratum
${\mathbb{T}}{\mathcal{O}}_{\rho}$. For a polyhedron $\sigma$ in
${\mathbb{T}}\Sigma$, the intersection
$\sigma\cap{\mathbb{T}}{\mathcal{O}}_{\eta}$, if non-empty, is a polyhedron in
${\mathbb{T}}{\mathcal{O}}_{\eta}$. A _face_ $\tau$ of $\sigma$ is the closure
of a face of any of these intersections (hence $\tau$ is a polyhedron in
${\mathbb{T}}\Sigma$). The _relative interior_ $\operatorname{relint}(\sigma)$
is the set of points in $\sigma$ not contained in a proper face. We set
$\operatorname{sed}(\sigma):=\operatorname{sed}(y)$ for an arbitrary
$y\in\operatorname{relint}(\sigma)$. Therefore, we have
$\operatorname{sed}(\sigma)=\dim(\rho)$ referring to $\rho$ from above.

A _rational polyhedral complex_ $\mathcal{X}$ in ${\mathbb{T}}\Sigma$ is a
collection of rational polyhedra in ${\mathbb{T}}\Sigma$ such that

  * •

if $\sigma\in{\mathcal{X}}$ and $\tau\subset\sigma$ is a face, then
$\tau\in{\mathcal{X}}$,

  * •

if $\sigma_{1},\sigma_{2}\in{\mathcal{X}}$ and
$\sigma_{1}\cap\sigma_{2}\neq\emptyset$, then $\sigma_{1}\cap\sigma_{2}$ is a
face of $\sigma_{1}$ and $\sigma_{2}$.

We refer to the elements $\sigma\in{\mathcal{X}}$ as _faces_ of
${\mathcal{X}}$. The maximal faces of ${\mathcal{X}}$ are called _facets_. The
_support_ $|{\mathcal{X}}|$ of ${\mathcal{X}}$ is the union of all the faces
of ${\mathcal{X}}$. Given a polyhedral complex ${\mathcal{X}}$, a
_subdivision_ of ${\mathcal{X}}$ is a polyhedral complex
${\mathcal{X}}^{\prime}$ with the same support and such that any face of
${\mathcal{X}}^{\prime}$ is contained in some face of ${\mathcal{X}}$. A
_rational polyhedral subspace_ $X$ of ${\mathbb{T}}\Sigma$ is a subspace which
is the support of a rational polyhedral complex $\mathcal{X}$. Note that two
polyhedral complexes have the same support if and only if they have a common
subdivision. Hence, alternatively we can think of a polyhedral subspace as an
equivalence class of polyhedral complexes under the equivalence relation
induced by subdivision. We allow ourselves to call $\mathcal{X}$ a subdivision
of the polyhedral space $|{\mathcal{X}}|$.

A rational polyhedral subspace $X\subset{\mathbb{T}}\Sigma$ is called of _pure
sedentarity $\rho\in\Sigma$_ if
$X=\overline{X\cap{\mathbb{T}}{\mathcal{O}}_{\rho}}$. A polyhedral complex is
of _pure sedentarity $\rho\in\Sigma$_ if $|{\mathcal{X}}|$ is of pure
sedentarity $\rho$. A polyhedral complex ${\mathcal{X}}$ is _of pure dimension
$d$_ if all facets have dimension $d$. Clearly, the notion is preserved under
subdivision and hence extends to polyhedral spaces $X$. In the following,
polyhedral complexes ${\mathcal{X}}$ and polyhedral spaces $X$ are _always_
assumed to be rational, of pure dimension, and of pure sedentarity.

Let ${\mathcal{X}}$ be a polyhedral complex in ${\mathbb{R}}^{n}$ and
$\tau\in{\mathcal{X}}$. By $\operatorname{star}_{\tau}{\mathcal{X}}$ we denote
the fan in ${\mathbb{R}}(\tau)$ whose cones are given by

$\displaystyle\frac{\sigma+T(\tau)}{T(\tau)},$
$\displaystyle\tau\subseteq\sigma\in{\mathcal{X}}.$

If ${\mathcal{X}}$ is a polyhedral complex in a tropical toric variety and
$\tau\in{\mathcal{X}}$ of sedentarity $\rho$, we set
$\operatorname{star}_{\tau}{\mathcal{X}}=\operatorname{star}_{\tau\cap{\mathbb{T}}{\mathcal{O}}_{\rho}}({\mathcal{X}}\cap{\mathbb{T}}{\mathcal{O}}_{\rho})$.
We note that ${\mathbb{T}}\operatorname{star}_{\rho}\Sigma$ is naturally
embedded in ${\mathbb{T}}\Sigma$ as the closure of the torus orbit
${\mathbb{T}}{\mathcal{O}}_{\rho}$.

Given a polyhedron $\sigma\subset{\mathbb{R}}^{n}$, its _recession cone_ is
defined by

$\operatorname{RecCone}(\sigma)=\\{v\in{\mathbb{R}}^{n}:\lambda
v+p\in\sigma\;\forall\;p\in\sigma,\lambda\in{\mathbb{R}}_{\geq}\\}.$

If $\sigma\subset{\mathbb{T}}\Sigma$ is a polyhedron of sedentarity $\rho$, we
set
$\operatorname{RecCone}(\sigma)=\operatorname{RecCone}(\sigma\cap{\mathbb{T}}{\mathcal{O}}_{\rho})$.
Given a polyhedral complex ${\mathcal{X}}$, we set
$\operatorname{RecCone}({\mathcal{X}})=\\{\operatorname{RecCone}(\sigma):\sigma\in{\mathcal{X}}\\}$.

Let us assume that ${\mathcal{X}}$ is a polyhedral complex in
${\mathbb{R}}^{n}$ such that $\Sigma=\operatorname{RecCone}({\mathcal{X}})$ is
a fan in ${\mathbb{R}}^{n}$. Let us additionally assume that ${\mathcal{X}}$
contains a vertex, or equivalently, that $\Sigma$ is pointed. We will now
define a polyhedral complex $\overline{{\mathcal{X}}}$ in ${\mathbb{T}}\Sigma$
with the property that $|\overline{{\mathcal{X}}}|=\overline{|{\mathcal{X}}|}$
and $\overline{{\mathcal{X}}}\cap{\mathbb{R}}^{n}={\mathcal{X}}$. The
construction is sometimes called the canonical compactification of
${\mathcal{X}}$, cf. [KSW20], [AP20, Section 2.4]. We set

$\overline{{\mathcal{X}}}:=\\{\overline{\sigma}\cap{\mathbb{T}}\operatorname{star}_{\rho}\Sigma:\sigma\in{\mathcal{X}},\rho\in\Sigma\\}.$
(2.1)

Note that, for example by [MR, Proposition 3.2.4], the intersection
$\overline{\sigma}\cap{\mathbb{T}}\operatorname{star}_{\rho}\Sigma$ is non-
emtpy if and only if $\rho\subset\operatorname{RecCone}(\sigma)$, and
moreover, in this case
$\overline{\sigma}\cap{\mathbb{T}}\operatorname{star}_{\rho}\Sigma$ is the
closure in ${\mathbb{T}}\operatorname{star}_{\rho}\Sigma$ of

$\sigma/T(\rho)\subset{\mathbb{R}}(\rho)={\mathbb{T}}{\mathcal{O}}_{\rho}.$

For further reference, we note that the labelling of faces of
$\overline{{\mathcal{X}}}$ by pairs $(\sigma,\rho)$ provides an isomorphism
between the poset $\overline{{\mathcal{X}}}$ and the subposet of
${\mathcal{X}}\times\Sigma$ given by

$\\{(\sigma,\rho)\in{\mathcal{X}}\times\Sigma:\rho\subset\operatorname{RecCone}(\sigma)\\},$
(2.2)

equipped with the product order, that is,

$(\sigma^{\prime},\rho^{\prime})\leq(\sigma,\rho)\Longleftrightarrow\sigma^{\prime}\leq\sigma\text{
in }{\mathcal{X}},\rho\leq\rho^{\prime}\text{ in }\Sigma.$

We note at this point that throughout the following, given a polyhedral
complex ${\mathcal{X}}$ we regard it as a poset whose partial order is given
by inclusion. We will sometimes refer to this as the _face poset_ of
${\mathcal{X}}$ to emphasize that we forget all geometric information and only
consider the poset structure.

###  2.3. Real phase structures on polyhedral subspaces

Given a polyhedron $\sigma$ in ${\mathbb{T}}\Sigma$ of sedentarity $\rho$, we
define its _tangent space_ by
$T(\sigma):=T(\sigma\cap{\mathbb{T}}{\mathcal{O}}_{\rho})\subset{\mathbb{T}}{\mathcal{O}}_{\rho}={\mathbb{R}}(\rho).$
Its integer and binary versions are

$\displaystyle T_{\mathbb{Z}}(\sigma)$
$\displaystyle:=T(\sigma)\cap{\mathbb{Z}}(\rho)\subset{\mathbb{Z}}(\rho),$
$\displaystyle T_{{\mathbb{Z}}_{2}}(\sigma)$
$\displaystyle:=T_{\mathbb{Z}}(\sigma)\otimes{\mathbb{Z}}_{2}\subset{\mathbb{Z}}_{2}(\rho).$

For a vector space $V$ over ${\mathbb{Z}}_{2}:={\mathbb{Z}}/2{\mathbb{Z}}$, we
denote by $\text{Aff}_{d}(V)$ the set of all affine subspaces in $V$ of
dimension $d$.

######  Definition 2.7.

A collection of subsets $S_{1},\dots,S_{k}\subset V$ is called an _even
covering_ if every element in $V$ is contained in an even number of the sets.
Equivalently,

$S_{1}\triangle\dots\triangle S_{k}=\emptyset,$

where $S\triangle T:=(S\cup T)\setminus(S\cap T)$ is the symmetric difference.

######  Definition 2.8.

Let $\mathcal{X}$ be a polyhedral complex of pure dimension $d$ and
sedentarity $\rho$ contained in a tropical toric variety ${\mathbb{T}}\Sigma$.
A _real phase structure_ on $\mathcal{X}$ is a map

$\mathcal{E}\colon\operatorname{Facets}({\mathcal{X}})\to\text{Aff}_{d}({\mathbb{Z}}_{2}(\rho))$

satisfying

  1. (1)

for every facet $\sigma$ of $\mathcal{X}$, the set $\mathcal{E}(\sigma)$ is an
affine subspace of ${\mathbb{Z}}_{2}(\rho)$ parallel to $\sigma$, in formulas,
$T({\mathcal{E}}(\sigma))=T_{{\mathbb{Z}}_{2}}(\sigma)$;

  2. (2)

for every codimension one face $\tau$ of $\mathcal{X}$ with facets
$\sigma_{1},\dots,\sigma_{k}$ adjacent to it, the sets
${\mathcal{E}}(\sigma_{1}),\dots,{\mathcal{E}}(\sigma_{k})$ are an even
covering,

A real phase structure on a polyhedral subspace $X\subset{\mathbb{T}}\Sigma$
is a real phase structure on some subdivision $\mathcal{X}$ of $X$.

######  Remark 2.9.

If ${\mathcal{X}}^{\prime}$ is a subdivision of $\mathcal{X}$, there is a
bijection between the set of real phase structures on $\mathcal{X}$ and those
on $\mathcal{X}^{\prime}$. This bijection is given by
${\mathcal{E}}^{\prime}(\sigma^{\prime})={\mathcal{E}}(\sigma)$ if $\sigma$ is
the face of ${\mathcal{X}}$ containing $\sigma^{\prime}$ in its relative
interior. It is obvious that this assignment is injective, and condition 2) in
Definition 2.8 for a real phase structure on ${\mathcal{X}}^{\prime}$ ensures
that it is surjective. In fact, if a codimension one face $\tau$ is adjacent
to exactly two facets $\sigma_{1}$ and $\sigma_{2}$, then it follows from
condition 2) that ${\mathcal{E}}(\sigma_{1})={\mathcal{E}}(\sigma_{2})$.
Therefore, a real phase structure on a polyhedral subspace $X$ is a real phase
structure on any polyhedral complex ${\mathcal{X}}$ such that
$|{\mathcal{X}}|=X$.

###  2.4. Tropical patchworking

Following [GKZ08, Chapter 11], we now describe how to obtain a space
homeomorphic to the real part of a complex toric variety by glueing together
multiple symmetric copies of the tropical toric variety. Let $\Sigma$ be a
pointed polyhedral fan in ${\mathbb{R}}^{n}$ defining a tropical toric variety
${\mathbb{T}}\Sigma$. For every $\varepsilon\in{\mathbb{Z}}_{2}^{n}$, we
denote by ${\mathbb{T}}\Sigma(\varepsilon)$ a copy of ${\mathbb{T}}\Sigma$
indexed by $\varepsilon$. We then define

$\mathfrak{R}{\mathbb{T}}\Sigma:=\bigsqcup_{\varepsilon\in{\mathbb{Z}}_{2}^{n}}{\mathbb{T}}\Sigma(\varepsilon)/\sim,$
(2.3)

where $\sim$ identifies strata
${\mathbb{T}}\operatorname{star}_{\rho}\Sigma(\varepsilon)$ and
${\mathbb{T}}\operatorname{star}_{\rho}\Sigma(\varepsilon^{\prime})$ if and
only if $\varepsilon+\varepsilon^{\prime}$ is in $T_{{\mathbb{Z}}_{2}}(\rho)$.

We can also give a more precise description of
$\mathfrak{R}{\mathbb{T}}\Sigma$ as a direct limit of topological spaces.
Consider the poset

$Q(\Sigma):=\\{(\rho,\varepsilon):\rho\in\Sigma,\varepsilon\in{\mathbb{Z}}_{2}(\rho)\\}$

with partial order given by

$(\rho,\varepsilon)\leq(\eta,\delta):\Longleftrightarrow\eta\subset\rho,\varepsilon=\pi(\delta),$

where $\pi\colon{\mathbb{Z}}_{2}(\eta)\to{\mathbb{Z}}_{2}(\rho)$ denotes the
canonical projection. We construct a direct system of topological spaces over
$Q(\Sigma)$ by assigning to $(\rho,\varepsilon)$ a copy of the toric variety
${\mathbb{T}}\operatorname{star}_{\rho}\Sigma$, denoted by
${\mathbb{T}}\operatorname{star}_{\rho}\Sigma(\varepsilon)$. For
$(\rho,\varepsilon)\leq(\eta,\delta)$, the corresponding map is given by the
inclusion
${\mathbb{T}}\operatorname{star}_{\rho}\Sigma\subset{\mathbb{T}}\operatorname{star}_{\eta}\Sigma$.
Then $\mathfrak{R}{\mathbb{T}}\Sigma$ is equal to the limit of this direct
system. In particular, we note that $\mathfrak{R}{\mathbb{T}}\Sigma$ contains
$2^{n-\dim\rho}$ copies of ${\mathbb{T}}\operatorname{star}_{\rho}\Sigma$ (and
hence of ${\mathbb{T}}{\mathcal{O}}_{\rho}$) labelled by
$\varepsilon\in{\mathbb{Z}}_{2}(\rho)$.

######  Remark 2.10.

The construction can also be understood using the formalism of toric varieties
over arbitrary semigroups, cf. Remark 2.6. Consider the semigroup
$S={\mathbb{T}}\times{\mathbb{Z}}_{2}$. The toric variety $S\Sigma$ is the
disjoint union
$\bigsqcup_{\varepsilon\in{\mathbb{Z}}_{2}^{n}}{\mathbb{T}}\Sigma(\varepsilon)$
appearing in Equation 2.3. The quotient of $S$ under
$\\{-\infty\\}\times{\mathbb{Z}}_{2}$ is called the semigroup (more often,
semifield or hyperfield) of _real_ tropical numbers which we denote by
$\mathfrak{R}{\mathbb{T}}$, see [Vir11]. As a set,
$\mathfrak{R}{\mathbb{T}}={\mathbb{T}}\cup_{-\infty}{\mathbb{T}}$ is a union
of two copies of ${\mathbb{T}}$ along $-\infty$. Then
$\mathfrak{R}{\mathbb{T}}\Sigma$ is in fact the toric variety associated to
the semigroup $\mathfrak{R}{\mathbb{T}}$, and the glueing described in
Equation 2.3 is coming from the map $S\Sigma\to\mathfrak{R}{\mathbb{T}}\Sigma$
induced by the quotient homomorphism $S\to\mathfrak{R}{\mathbb{T}}$.

To explain the connection to the real toric variety ${\mathbb{R}}\Sigma$,
first note that $\log\colon({\mathbb{R}}_{\geq},\cdot)\to({\mathbb{T}},+)$
(with $\log(0)=-\infty$) is an isomorphism of semigroups. Moreover,
${\mathbb{R}}$ can be described as the semigroup quotient of
${\mathbb{R}}_{\geq}\times{\mathbb{Z}}_{2}$ modulo
$\\{0\\}\times{\mathbb{Z}}_{2}$. Hence we get an extended semigroup
homomorphism
$\log^{\mathfrak{R}}\colon{\mathbb{R}}\to\mathfrak{R}{\mathbb{T}}$. These
semigroup homomorphisms yield homeomorphisms
$\operatorname{Log}\colon{\mathbb{R}}_{\geq}\Sigma\to{\mathbb{T}}\Sigma$ and
$\operatorname{Log}^{\mathfrak{R}}\colon{\mathbb{R}}\Sigma\to\mathfrak{R}{\mathbb{T}}\Sigma$.
We also note that this construction is a tropical reformulation of for example
[GKZ08, Theorem 11.5.4].

######  Example 2.11.

We consider the tropical affine space ${\mathbb{T}}\Sigma={\mathbb{T}}^{n}$.
In continuation of the notation introduced Example 2.2, given $A\subset
E=\\{1,\dots,n\\}$ and $\varepsilon\in{\mathbb{Z}}_{2}^{E\setminus A}$, we
denote by ${\mathbb{T}}^{n}_{A}(\varepsilon)$ the corresponding copy of
${\mathbb{T}}^{n}_{A}$ in $\mathfrak{R}{\mathbb{T}}^{n}$. Instead of tuples
$(A,\varepsilon)$, we can alternatively label the strata of
$\mathfrak{R}{\mathbb{T}}^{n}$ by the set of _signed vectors_
$C\in\\{0,+1,-1\\}^{n}$. The connection is given by setting
$A=E\setminus\text{supp}(C)$ and $C|_{\text{supp}(C)}=(-1)^{\varepsilon}$. The
stratification agrees with the ordinary sign stratification of
${\mathbb{R}}^{n}$ under the homeomorphism
$\operatorname{Log}^{\mathfrak{R}}\colon{\mathbb{R}}^{n}\to\mathfrak{R}{\mathbb{T}}^{n}$.
For $i=1,\dots,n$, we use the shorthand
$H_{i}:=\bigcup_{\varepsilon\in{\mathbb{Z}}_{2}^{E\setminus\\{i\\}}}{\mathbb{T}}^{n}_{\\{i\\}}(\varepsilon)\subset\mathfrak{R}{\mathbb{T}}^{n}$.
Under $\operatorname{Log}^{\mathfrak{R}}$, this subset corresponds to the
coordinate hyperplane $\\{x_{i}=0\\}\subset{\mathbb{R}}^{n}$.

Given a polyhedron $\sigma$ in
${\mathbb{T}}\operatorname{star}_{\rho}\Sigma\subset{\mathbb{T}}\Sigma$ and
$\varepsilon\in{\mathbb{Z}}_{2}(\rho)$, we let $\sigma(\varepsilon)$ denote
the copy of $\sigma$ in
${\mathbb{T}}\operatorname{star}_{\rho}\Sigma(\varepsilon)\subset\mathfrak{R}{\mathbb{T}}\Sigma$.
We extend the notion of polyhedral complex to $\mathfrak{R}{\mathbb{T}}\Sigma$
be declaring the sets $\sigma(\varepsilon)$ to be the polyhedra of
$\mathfrak{R}{\mathbb{T}}\Sigma$.

Let ${\mathcal{X}}$ be a polyhedral complex of sedentarity $\rho$ in
${\mathbb{T}}\Sigma$ together with a real phase structure $\mathcal{E}$. Given
an arbitrary face $\tau\in{\mathcal{X}}$ of sedentarity $\eta$, we set

${\mathcal{E}}(\tau)=\bigcup_{\begin{subarray}{c}\sigma\in\operatorname{Facets}({\mathcal{X}})\\\
\tau\subset\sigma\end{subarray}}\pi({\mathcal{E}}(\sigma)),$

where $\pi$ denotes the canonical projection
$\pi\colon{\mathbb{Z}}_{2}(\rho)\to{\mathbb{Z}}_{2}(\eta)$. Note that in
general ${\mathcal{E}}(\tau)$ is not an affine subspace of
${\mathbb{Z}}_{2}(\eta)$.

######  Definition 2.12.

Let ${\mathcal{X}}$ be a polyhedral complex in ${\mathbb{T}}\Sigma$ together
with a real phase structure $\mathcal{E}$. The _patchwork_
$({\mathcal{X}},{\mathcal{E}})$ is the polyhedral complex
$\mathfrak{R}({\mathcal{X}},\mathcal{E})$ in $\mathfrak{R}{\mathbb{T}}\Sigma$
given by

$\mathfrak{R}({\mathcal{X}},\mathcal{E})=\\{\sigma(\varepsilon):\sigma\in{\mathcal{X}},\varepsilon\in{\mathcal{E}}(\sigma)\\}.$

Let $X$ be a polyhedral subspace in ${\mathbb{T}}\Sigma$ together with a real
phase structure ${\mathcal{E}}$. The _patchwork_ of $X$ with respect to
${\mathcal{E}}$ is the polyhedral subspace

$\mathfrak{R}(X,\mathcal{E}):=|\mathfrak{R}({\mathcal{X}},\mathcal{E})|\;\subset\mathfrak{R}{\mathbb{T}}\Sigma$

for some subdivision ${\mathcal{X}}$ of $X$.

If the real phase structure ${\mathcal{E}}$ is clear from the context, we
allow ourselves to write $\mathfrak{R}{\mathcal{X}}$ and $\mathfrak{R}X$
instead of $\mathfrak{R}({\mathcal{X}},\mathcal{E})$ and
$\mathfrak{R}(X,\mathcal{E})$.

######  Remark 2.13.

Let ${\mathcal{X}}^{\prime}$ be a subdivision of ${\mathcal{X}}$ with induced
real phase structure ${\mathcal{E}}^{\prime}$ as explained in Remark 2.9. Then
clearly $\mathfrak{R}({\mathcal{X}}^{\prime},{\mathcal{E}}^{\prime})$ is a
subdivision of $\mathfrak{R}({\mathcal{X}},{\mathcal{E}})$. In particular,
$|\mathfrak{R}({\mathcal{X}},{\mathcal{E}})|=|\mathfrak{R}({\mathcal{X}}^{\prime},{\mathcal{E}}^{\prime})|$
and hence $\mathfrak{R}(X,\mathcal{E})$ is well-defined.

######  Proposition 2.14.

Let ${\mathcal{X}}$ be a polyhedral complex in ${\mathbb{T}}\Sigma$ together
with a real phase structure $\mathcal{E}$. The polyhedral complex
$\mathfrak{R}({\mathcal{X}},{\mathcal{E}})\subset\mathfrak{R}{\mathbb{T}}\Sigma$
represents a closed cellular chain in the homology of
$\mathfrak{R}{\mathbb{T}}\Sigma$ over ${\mathbb{Z}}_{2}$.

Note that the associated homology class
$[\mathfrak{R}({\mathcal{X}},{\mathcal{E}})]\in
H^{BM}_{d}(\mathfrak{R}{\mathbb{T}}\Sigma,{\mathbb{Z}}_{2})$ is invariant
under subdivisions of ${\mathcal{X}}$ by Remark 2.13. In particular, we can
associate a well-defined homology class $[\mathfrak{R}(X,{\mathcal{E}})]\in
H^{BM}_{d}(\mathfrak{R}{\mathbb{T}}\Sigma,{\mathbb{Z}}_{2})$ to any polyhedral
subspace $X$ with real phase structure ${\mathcal{E}}$.

###### Proof.

To see that $\partial\mathfrak{R}({\mathcal{X}},{\mathcal{E}})=0$ we need to
verify that every codimension one face of
$\mathfrak{R}({\mathcal{X}},{\mathcal{E}})$ is contained in an even number of
facets of $\mathfrak{R}({\mathcal{X}},{\mathcal{E}})$. If ${\mathcal{X}}$ is
of sedentarity $\rho$, we can replace ${\mathbb{T}}\Sigma$ by
${\mathbb{T}}\operatorname{star}_{\rho}\Sigma$, hence it suffices to prove the
statement for sedentarity $0$.

Let $\tau(\varepsilon)$ be a face of
$\mathfrak{R}({\mathcal{X}},{\mathcal{E}})$ of codimension one. We first
consider the case $\operatorname{sed}(\tau)=0$. Then the facets of
$\mathfrak{R}({\mathcal{X}},{\mathcal{E}})$ containing $\tau(\varepsilon)$ are
in bijection with the facets $\sigma$ of ${\mathcal{X}}$ containing $\tau$ and
such that $\varepsilon\in{\mathcal{E}}(\sigma)$. By Condition 2) of Definition
2.8, the number of such facets is even, which proves the claim in this case.

Now suppose $\tau$ is of sedentarity $\rho$. Let $\sigma$ be a facet of
${\mathcal{X}}$ containing $\tau$. This implies that
$\operatorname{RecCone}(\sigma)$ intersects the relative interior of $\rho$
and moreover the intersection
$\rho^{\prime}:=\operatorname{RecCone}(\sigma)\cap\rho$ is a one-dimensional
cone, see [MR, Proposition 3.2.6], [OR13, Lemma 3.9]. Let
$v\in{\mathbb{Z}}_{2}^{n}$ be the generator of
$T_{{\mathbb{Z}}_{2}}(\rho^{\prime})\cong{\mathbb{Z}}_{2}$. Since $v\in
T_{{\mathbb{Z}}_{2}}(\rho)$, we have $\pi(v)=0\in{\mathbb{Z}}_{2}(\rho)$.
Since $v\in T_{{\mathbb{Z}}_{2}}(\sigma)$, by Condition 1) of Definition 2.8
the vector space parallel to ${\mathcal{E}}(\sigma)$ contains $v$. Hence the
assignment $\sigma(\varepsilon^{\prime})\mapsto\sigma(\varepsilon^{\prime}+v)$
defines a fixed-point-free involution on the set of facets of
$\mathfrak{R}({\mathcal{X}},{\mathcal{E}})$ containing $\tau(\varepsilon)$.
This proves that there is an even number of such facets, which proves the
claim. ∎

######  Remark 2.15.

A tropical variety is commonly defined as a weighted $n$-dimensional rational
polyhedral complex ${\mathcal{X}}$ that satisfies the so-called balancing
condition. The balancing condition can be reformulated in terms of tropical
homology. From a weighted polyhedral complex ${\mathcal{X}}$ one can construct
its fundamental chain in tropical homology. The balancing condition on
${\mathcal{X}}$ is then equivalent to this fundamental chain being closed, see
[MZ14, Proposition 4.3].

In analogy to this picture, we propose to think of Conditions 1) and 2) in
Definition 2.8 as a real balancing condition in tropical geometry. Indeed,
given an arbitrary map
$\mathcal{E}\colon\operatorname{Facets}(\mathcal{X})\to\operatorname{Pow}({\mathbb{Z}}_{2}(\rho))$,
we may still construct $\mathfrak{R}({\mathcal{X}},{\mathcal{E}})$ as a
(cellular) Borel-Moore chain in $\mathfrak{R}{\mathbb{T}}\Sigma$. Then
Proposition 2.14 states that this chain is closed if ${\mathcal{E}}$ is a real
phase structure. Moreover, the inverse is true at least for certain natural
special cases, for example when ${\mathcal{X}}\cap{\mathbb{R}}^{n}=\Sigma$.
More precisely, the sedentarity $0$ part of the proof of Proposition 2.14 can
be inverted to show that Condition 2) is equivalent to
$\mathfrak{R}({\mathcal{X}},{\mathcal{E}})$ being closed in sedentarity $0$.
Moreover, assuming ${\mathcal{X}}\cap{\mathbb{R}}^{n}=\Sigma$, given
$\rho\in\Sigma$ one can extend the argument in the proof to show that
$\mathfrak{R}({\mathcal{X}},{\mathcal{E}})$ is closed at higher sedentarity
faces of $\overline{\rho}$ if and only if ${\mathcal{E}}(\sigma)$ is a union
of affine subspace satisfying Condition 1). Allowing unions may be interesting
when treating tropical varieties with weight functions which are not equal to
one. Here we do not treat this generality for the sake of simplicity.

##  3\. Non-singular tropical subvarieties and their patchworks

###  3.1. Tropically non-singular polyhedral subspaces

The local models for tropically non-singular polyhedral spaces are matroid
fans. We recall them for convenience. Given a loopfree matroid $M$ on the base
set $E$, we denote by $\Sigma_{M}$ the _affine_ matroid fan in
${\mathbb{R}}^{E}$ and by
${\mathbb{P}}\Sigma_{M}=\Sigma_{M}/\langle(1,\dots,1)\rangle$ the _projective_
matroid fan in ${\mathbb{R}}^{E}/\langle(1,\dots,1)\rangle$. We now describe
how to construct both of these fans following Ardila and Klivans [AK06]. Fix
the vectors $v_{i}=-e_{i}$ where $\\{e_{1},\dots,e_{n}\\}$ is the standard
basis of ${\mathbb{R}}^{E}$ for $E=\\{1,\dots,n\\}$ and set $v_{I}=\sum_{i\in
I}v_{i}$ for any subset $I\subset E$. For a chain of flats

${\mathcal{F}}=\\{\emptyset\subsetneq F_{1}\subsetneq
F_{2}\subsetneq\dots\subsetneq F_{k}\subsetneq E\\}$

in the lattice of flats ${\mathcal{L}}$, define the $k+1$-dimensional cone

$\sigma_{{\mathcal{F}}}=\langle v_{F_{1}},\dots,v_{F_{k}},\pm
v_{E}\rangle_{\geq 0}.$

The affine matroid fan $\Sigma_{M}$ is the collection of all such cones
ranging over the chains in ${\mathcal{L}}$. In particular, the top dimensional
faces of $\Sigma_{M}$ are in one to one correspondence with the maximal chains
in the lattice of flats of $M$. The projective matroid fan
${\mathbb{P}}\Sigma_{M}$ is the image of $\Sigma_{M}$ in the quotient
${\mathbb{R}}^{E}/(1,\dots,1)$.

If a matroid $M$ has loops $L=\operatorname{cl}(\emptyset)$, then we set
$\Sigma_{M}:=\Sigma_{M/L}\subset{\mathbb{R}}^{E\setminus L}$ and
${\mathbb{P}}\Sigma_{M}:={\mathbb{P}}\Sigma_{M/L}\subset{\mathbb{R}}^{E\setminus
L}/(1,\dots,1)$. We will often assume $E=\\{0,\dots,n\\}$. Note that by
Example 2.2 we can consider ${\mathbb{R}}^{E\setminus
L}={\mathbb{T}}{\mathcal{O}}_{\rho_{L}}$ as a torus orbit of
${\mathbb{T}}^{n+1}$, and ${\mathbb{R}}^{E\setminus
L}/(1,\dots,1)={\mathbb{T}}{\mathcal{O}}_{\rho_{L}}$ as a torus orbit of
${{\mathbb{T}}{\mathbb{P}}}^{n}$ (excluding the trivial case $E=L$). In this
sense, in the following we will regard $\Sigma_{M}$ and
${\mathbb{P}}\Sigma_{M}$ as subsets of ${\mathbb{T}}^{n+1}$ and
${{\mathbb{T}}{\mathbb{P}}}^{n}$, respectively.

######  Definition 3.1.

[MR, Chapter 6] Let $\Sigma$ be a fan and ${\mathbb{T}}\Sigma$ the associated
tropical toric variety. Let $X\subset{\mathbb{T}}\Sigma$ be a polyhedral
subspace and consider $p\in X$ of sedentarity $\rho\in\Sigma$ and
$\operatorname{sed}(p)=\dim(\rho)=k$.

Then $X$ is _non-singular_ at $p$ if there exist

  * •

a toric isomorphism
$\psi\colon{\mathbb{T}}U_{\rho}\to{\mathbb{T}}^{k}\times{\mathbb{R}}^{n-k}={\mathbb{T}}U_{\rho_{\\{1,\dots,k\\}}}\subset{{\mathbb{T}}{\mathbb{P}}}^{n}$,

  * •

an open neighbourhood $p\in U\subset{\mathbb{T}}U_{\rho}$,

  * •

a matroid $M$ on $E=\\{0,1,\dots,n\\}$,

such that $\psi(p)=(-\infty,0)$ and $\psi(X\cap
U)=\overline{{\mathbb{P}}\Sigma}_{M}\cap\psi(U)$.

The polyhedral subspace $X$ is a _non-singular tropical subvariety_ of $Y$ if
it is non-singular at all its points.

###  3.2. Patchworks of non-singular subvarieties are topological manifolds

Our next goal is to prove 1.1 from the introduction. We point the reader to
[RRS22] for the background on the connection between oriented matroids and
real phase structures.

Theorem 1.1. Let $X\subset{\mathbb{T}}\Sigma$ be a non-singular tropical
subvariety of a tropical toric variety ${\mathbb{T}}\Sigma$. Let
${\mathcal{E}}$ be a real phase structure on $X$. Then the patchwork
$\mathfrak{R}X$ is a topological manifold.

By [RRS22, Theorem 1.1], we proved that matroidal fans equipped with real
phase structures are equivalent to oriented matroids. In Proposition 3.4, we
show that the patchwork of a matroid fan with a real phase structure is a
topological manifold with determined topology. This is essentially a corollary
of the Folkman-Lawrence topological representation theorem for oriented
matroids [FL78]. We note that similar results on patchworking of matroids were
obtained in [AKW06, Theorem 3.4] and in [CLY22, Theorem A]. We start with two
examples describing the faces of the simplicial complexes
$\mathfrak{R}\overline{\Sigma}_{M}$ and
$\mathfrak{R}\overline{{\mathbb{P}}\Sigma}_{M}$.

######  Example 3.2.

Let ${\mathcal{M}}$ be an oriented matroid with underlying matroid $M$ and let
${\mathcal{E}}$ be the associated real phase structure on $\Sigma_{M}$ [RRS22,
Definition 7]. Hence ${\mathcal{E}}$ is also a real phase structure on
$\overline{\Sigma}_{M}\subset{\mathbb{T}}^{n}$. By combining Example 2.11 with
the definition of ${\mathcal{E}}$, we see that the faces of the polyhedral
complex $\mathfrak{R}\overline{\Sigma}_{M}\subset\mathfrak{R}{\mathbb{T}}^{n}$
are labelled by the chains of covectors $0<Z_{1}<\dots<Z_{l}$ from
${\mathcal{M}}$. The final covector $Z_{l}$ determines the stratum of
$\mathfrak{R}{\mathbb{T}}^{n}$ that contains the face. The associated chain of
flats $F_{i}=\phi(Z_{l-i})$ determines the corresponding face of
$\overline{\Sigma}_{M}$. The map $\phi$ here denotes the forgetful map from
oriented matroids to matroids sending a covector $X$ to the flat
$\mathrm{Supp}(X)^{c}$, see for example [RRS22, Section 3.1]. In particular,
the intersection $H_{i}\cap\mathfrak{R}\overline{\Sigma}_{M}$ is the
subcomplex corresponding to all chains such that $i\in\phi(Z_{l})$.

######  Example 3.3.

Let ${\mathcal{M}}$ be an oriented matroid with underlying matroid $M$ and let
${\mathcal{E}}$ be the associated real phase structure on
${\mathbb{P}}\Sigma_{M}\subset{\mathbb{R}}^{n+1}/(1,\dots,1)\simeq{\mathbb{R}}^{n}$.
Again, we denote by the same letter the induced real phase structure on
$\overline{{\mathbb{P}}\Sigma}_{M}\subset{{\mathbb{T}}{\mathbb{P}}}^{n}$.

Pick $\varepsilon\in{\mathbb{Z}}_{2}^{n+1}$ such that $T=(-1)^{\varepsilon}$
is a tope of ${\mathcal{M}}$. We denote by
${{\mathbb{T}}{\mathbb{P}}}^{n}(\varepsilon)\subset\mathfrak{R}{\mathbb{T}}{\mathbb{P}}^{n}$
the copy of ${{\mathbb{T}}{\mathbb{P}}}^{n}$ in
$\mathfrak{R}{\mathbb{T}}{\mathbb{P}}^{n}$ obtained as the projection of
${\mathbb{T}}^{n+1}(\varepsilon)$. We are interested in the polyhedral complex
$\mathfrak{R}\overline{{\mathbb{P}}\Sigma}\cap\partial{{\mathbb{T}}{\mathbb{P}}}^{n}(\varepsilon)$.
By Examples 2.2 and 3.2, it is a simplicial complex whose faces are labelled
by chains of covectors $0<Z_{1}<\dots<Z_{l}<T$ or, in other words, chains of
flats $\emptyset\subsetneq F_{1}\subsetneq\dots\subsetneq F_{l}\subsetneq E$
which are touching $T$. The lattice of flats touching $\varepsilon$ is known
as the _Las Vergnas lattice_ of ${\mathcal{M}}$ with respect to $T$. We denote
it by ${\mathcal{F}}_{lv}^{{\mathcal{M}}}(T)$. Let us also remark that by
[AKW06, Theorem 3.4], the simplicial complex
$\mathfrak{R}\overline{{\mathbb{P}}\Sigma}_{M}\cap\partial{\mathbb{T}}{\mathbb{P}}^{n}(0)$
is equal to the _positive Bergman complex_ from _loc. cit._ , though rather
than intersection with a standard sphere, we intersect with
$\partial{\mathbb{T}}{\mathbb{P}}^{n}(0)$ which is homeomorphic to a sphere.

The simplicial complexes mentioned in the previous examples have been studied
by Folkman and Lawrence in the context of the topological representation
theorem for oriented matroids. We translate the results to our setup in the
following proposition. To state the results in the conventional way, it is
useful to intersect with a sphere
$S^{n-1}\subset\mathfrak{R}{\mathbb{T}}^{n}$. A natural choice for $S^{n-1}$
is given by taking, in each stratum ${\mathbb{T}}^{n}(\varepsilon)$ of
$\mathfrak{R}{\mathbb{T}}^{n}$, the symmetric copy of the boundary of the
negative orthant, that is

$S(\varepsilon)=\\{x\in{\mathbb{T}}^{n}(\varepsilon)\ |\ \forall\ i\ x_{i}\leq
0\text{ and }\exists\ i:\ x_{i}=0\\}.$

The union of all of the $S(\varepsilon)$ in $\mathfrak{R}{\mathbb{T}}^{n}$ is
a polyhedral sphere of dimension $n-1$, see Figure 2. In fact, it is the
boundary of a cube.

$\bullet$$(0,0)\subset{\mathbb{T}}^{2}((0,0))$$\\{0\\}\times\left[-\infty,0\right]$

Figure 2. On the left is a polyhedral sphere (drawn in red) in
$\mathfrak{R}{\mathbb{T}}^{2}$. On the right is a patchwork of
$\mathfrak{R}\overline{{\mathbb{P}}\Sigma}_{M}$ (drawn in red) in
${\mathbb{R}}{\mathbb{P}}^{2}$ for the uniform matroid $U_{2,3}$. The real
projective plane is obtained from the square by identifying antipodal points
on the boundary.

######  Proposition 3.4.

Let $M$ be a matroid of rank $d$ on $n$ elements and let ${\mathcal{E}}$ be a
real phase structure on $\Sigma_{M}$.

  1. (1)

The complexes $\mathfrak{R}\overline{\Sigma}_{M}\cap S^{n-1}$,
$\mathfrak{R}\overline{{\mathbb{P}}\Sigma}_{M}$ and
$\mathfrak{R}\overline{\Sigma}_{M}$ are topological manifolds and homeomorphic
to $S^{d-1}$, ${\mathbb{R}}{\mathbb{P}}^{d-1}$ and ${\mathbb{R}}^{d}$,
respectively.

  2. (2)

For a sign vector
$\varepsilon\in\cup_{\sigma\in\Sigma_{M}}{\mathcal{E}}(\sigma)$, the complexes
$\mathfrak{R}\overline{{\mathbb{P}}\Sigma}_{M}\cap\partial{\mathbb{T}}{\mathbb{P}}^{n}(\varepsilon)$
and
$\mathfrak{R}\overline{{\mathbb{P}}\Sigma}_{M}\cap{\mathbb{T}}{\mathbb{P}}^{n}(\varepsilon)$
are topological manifolds (with boundary in the second case) and homeomorphic
to $S^{d-2}$ and $D^{d-1}$, respectively.

  3. (3)

The subcomplexes $H_{i}\cap\mathfrak{R}\overline{\Sigma}_{M}\cap
S^{n-1}\subset\mathfrak{R}\overline{\Sigma}_{M}\cap S^{n-1}$, $i=1,\dots,n$,
form a pseudosphere arrangement. The associated oriented matroid is equal to
${\mathcal{M}}_{\mathcal{E}}$.

###### Proof.

Let ${\mathcal{M}}={\mathcal{M}}_{\mathcal{E}}$ denote the oriented matroid
constructed in [RRS22, Section 3.3]. By [RRS22, Theorem 1.1] and Example 3.2,
the simplicial complex $\mathfrak{R}\overline{\Sigma}_{M}\cap
S^{n-1}\subset\mathfrak{R}{\mathbb{T}}^{n}$ is a geometric realization of the
order complex of the poset ${\mathcal{C}}$ of covectors of ${\mathcal{M}}$.
Analogously, by Theorem [RRS22, Theorem 1.1] and Example 3.3, given
$\varepsilon$ such that $T=(-1)^{\varepsilon}$ is tope of ${\mathcal{M}}$, the
simplicial complex
$\mathfrak{R}\overline{{\mathbb{P}}\Sigma}_{M}\cap\partial{\mathbb{T}}{\mathbb{P}}^{n}(\varepsilon)$
is the order complex of the poset
${\mathcal{F}}_{lv}^{{\mathcal{M}}}(T)\setminus\\{0,T\\}$. The statements then
follow from the Folkman-Lawrence topological representation theorem [FL78].
See also [BLVS+99, Theorems 4.3.3, 4.3.5, 5.2.1]. ∎

###### Proof of 1.1.

Fix a point $p\in X$. Let
$\psi\colon{\mathbb{T}}U_{\rho}\to{\mathbb{T}}^{k}\times{\mathbb{R}}^{n-k}$,
$p\in U\subset{\mathbb{T}}U_{\rho}$ and $M$ be the objects guaranteed by the
definition of $X$ being non-singular at $p$. Since
$(-\infty,0)\in\overline{{\mathbb{P}}\Sigma}_{M}$, the set $F=\\{1,\dots,k\\}$
is a flat of $M$. After replacing $M$ by $F\oplus M/F$ (and possibly shrinking
$U$), we may assume that $v_{F}$ is contained in the lineality space of
${\mathbb{P}}\Sigma_{M}$. In particular, any facet of ${\mathbb{P}}\Sigma_{M}$
(in the coarse sense) intersects $\psi(U)$. It follows that $\psi$ induces a
real phase structure from $X\cap U$ on ${\mathbb{P}}\Sigma_{M}$.

Given an open subset of a toric variety $U\subset{\mathbb{T}}\Sigma$, we
denote by $\mathfrak{R}U\subset\mathfrak{R}{\mathbb{T}}\Sigma$ the open subset
obtained by copying $U$ in each stratum
${\mathbb{T}}\Sigma(\varepsilon)\subset\mathfrak{R}Y$. We consider the open
subset $\mathfrak{R}X\cap\mathfrak{R}U\subset\mathfrak{R}X$. By definition,
under $\psi$ (or rather, its extension to
$\mathfrak{R}U_{\rho}\to\mathfrak{R}({\mathbb{T}}^{k}\times{\mathbb{R}}^{n-k})$)
it is homeomorphic to
$\mathfrak{R}\overline{{\mathbb{P}}\Sigma}_{M}\cap\psi(\mathfrak{R}U)$. Since
$\mathfrak{R}\overline{{\mathbb{P}}\Sigma}_{M}$ is a topological manifold by
Corollary 3.4, it follows that $\mathfrak{R}X$ is a topological manifold at
all points in $\mathfrak{R}X\cap\mathfrak{R}U$. Since $p$ was arbitrary, these
sets cover $\mathfrak{R}X$, hence the claim follows. ∎

##  4\. Patchworking close to the tropical limit

###  4.1. Non-singular tropical limits

We denote by ${\mathbb{K}}={\mathbb{C}}((t))$ the field of formal Laurent
series with complex coefficients, and by ${\mathbb{K}}_{\mathbb{R}}$ the
subfield of series with real coefficients. As algebraic closure we fix the
field of Puiseux series $\bar{{\mathbb{K}}}={\mathbb{C}}\\{t\\}$. Throughout
the following, we fix an algebraic variety
${\mathbf{X}}\subset({\mathbb{K}}^{*})^{n}$ defined over the subring in
${\mathbb{K}}$ of series that converge for $0<|t|<1$. We can alternatively
regard ${\mathbf{X}}$ as a subset of
$({\mathbb{C}}^{*})^{n}\times{\mathcal{D}}^{*}$ describing a family of
algebraic varieties in $({\mathbb{C}}^{*})^{n}$ parametrized by
${\mathcal{D}}^{*}$, the punctured unit disc in ${\mathbb{C}}$. From now on,
we refer to such ${\mathbf{X}}$ as an _analytic family over
${\mathcal{D}}^{*}$_. For $t\in{\mathcal{D}}^{*}$, we denote the fibre by
${\mathbf{X}}_{t}={\mathbf{X}}\cap(({\mathbb{C}}^{*})^{n}\times\\{t\\})$. We
define the tropical limit
$\operatorname{Trop}({\mathbf{X}})\subset{\mathbb{R}}^{n}$ as the closure of
$-\text{val}({\mathbf{X}}\otimes\bar{{\mathbb{K}}})$, where

$\text{val}\colon(\bar{{\mathbb{K}}}^{*})^{n}\to{\mathbb{R}}^{n}$

is the coordinate-wise valuation map. The tropicalization
$\operatorname{Trop}({\mathbf{X}})$ is a tropical subvariety (that is, a
weighted polyhedral subspace that satisfies the balancing condition). We set
$X=\operatorname{Trop}({\mathbf{X}})$.

For each $p\in X$, we have an associated initial scheme
$\operatorname{in}_{p}{\mathbf{X}}\subset({\mathbb{C}}^{*})^{n}$. We have
$d:=\dim_{\mathbb{K}}{\mathbf{X}}=\dim_{\mathbb{C}}\operatorname{in}_{p}{\mathbf{X}}$.
For a generic point $p\in X$, $\operatorname{in}_{p}{\mathbf{X}}$ is invariant
under the action of an $d$-dimensional subtorus ${\mathbf{T}}_{p}$ of
$({\mathbb{C}}^{*})^{n}$. We denote the quotient under this action by
${\mathbf{X}}_{p}=\operatorname{in}_{p}{\mathbf{X}}/{\mathbf{T}}_{p}$. Hence
${\mathbf{X}}_{p}$ is a $0$-dimensional scheme. The _weight_ of a generic
point $p$ is defined as
$\omega(p)=\deg({\mathbf{X}}_{p})=\dim_{\mathbb{C}}K[{\mathbf{X}}_{p}]$. In
particular, the weight of $p$ is $1$ if and only if ${\mathbf{X}}_{p}$ is a
reduced point if and only if $\operatorname{in}_{p}{\mathbf{X}}$ is reduced
and irreducible. Throughout the following, and by slight abuse of notation, we
say that _${\mathbf{X}}$ has non-singular tropical limit_, or just
_$X=\operatorname{Trop}({\mathbf{X}})$ is non-singular_, if both of the
following conditions hold:

  * •

the scheme $\operatorname{in}_{p}{\mathbf{X}}$ is reduced and irreducible for
generic $p$ (weight $1$),

  * •

the polyhedral subspace $X$ is non-singular in the sense of Definition 3.1.

We first collect a few consequences of ${\mathbf{X}}$ having non-singular
tropical limit. A subscheme $\textbf{L}\subset({\mathbb{C}}^{*})^{n}$ is
called _linear_ if its closure $\overline{\textbf{L}}\subset{\mathbb{P}}^{n}$
under the standard inclusion of $({\mathbb{C}}^{*})^{n}$ to ${\mathbb{P}}^{n}$
is a linear subspace. In particular, this implies that L is reduced and
irreducible. Then $\overline{\textbf{L}}\backslash\textbf{L}$ is the
complement of an arrangement of $n+1$ (not necessarily distinct) hyperplanes.
We denote the associated matroid by $M_{\textbf{L}}$.

######  Lemma 4.1.

[KP11, Proposition 4.2] Let $\textbf{L}\subset({\mathbb{C}}^{*})^{n}$ be a
subscheme with non-singular tropical limit
$\operatorname{Trop}(\textbf{L})={\mathbb{P}}\Sigma_{M}$, where $M$ is a
matroid. Then L is linear and $M_{\textbf{L}}=M$.

A polyhedral subdivision ${\mathcal{X}}$ of
$X=\operatorname{Trop}({\mathbf{X}})$ is called _tropical_ if it satisfies the
following two conditions.

  1. (1)

Fix $\sigma\in{\mathcal{X}}$. Then for any $p,q$ in the relative interior of
$\sigma$ we have
$\operatorname{in}_{p}{\mathbf{X}}=\operatorname{in}_{q}{\mathbf{X}}$. We set
$\operatorname{in}_{\sigma}{\mathbf{X}}=\operatorname{in}_{p}{\mathbf{X}}$ for
one such $p$.

  2. (2)

The scheme $\operatorname{in}_{\sigma}{\mathbf{X}}$ is invariant under the
action of the subtorus
${\mathbf{T}}_{\sigma}=T(\sigma)\otimes{\mathbb{C}}^{*}\subset({\mathbb{C}}^{*})^{n}$.
We denote the quotient by
${\mathbf{X}}_{\sigma}=\operatorname{in}_{\sigma}{\mathbf{X}}/{\mathbf{T}}_{\sigma}$.
We have $\dim_{\mathbb{C}}{\mathbf{X}}_{\sigma}=d-\dim\sigma$.

It is shown for example in [Spe05, Theorem 2.2.1] that tropical subdivisions
exist for any analytic family ${\mathbf{X}}$. Moreover, we have the following.

######  Proposition 4.2.

If ${\mathbf{X}}$ has non-singular tropical limit
$X=\operatorname{Trop}({\mathbf{X}})$, then any subdivision ${\mathcal{X}}$ of
$X$ is tropical.

###### Proof.

This is a result known more generally for so-called schön subvarieties, see
[Tev07, Definition 1.3, Theorem 1.4] for the “constant coefficient” case and
[LQ11, Definition 7.1, Theorem 7.12] for the general case. By [Hac08, Lemma
2.7] and [Spe05, Theorem 2.4.2], the variety ${\mathbf{X}}$ is schön if and
only of $\operatorname{in}_{p}{\mathbf{X}}$ is non-singular for all $p\in X$.
To finish the proof we show the latter property (see also [KS12, Section 2,
Propositions 2.6 and 2.7] for a summary of the previous discussion). By
[Spe05, Proposition 2.2.3], the tropical limit of
$\operatorname{in}_{p}{\mathbf{X}}$ for $p\in X$ is
$\operatorname{Trop}(\operatorname{in}_{p}{\mathbf{X}})=\operatorname{star}_{p}X$.
By our assumptions, $\operatorname{star}_{p}X$ is isomorphic to a matroid fan
${\mathbb{P}}\Sigma_{M}$ via a change of coordinates in
${\text{GL}}(n,{\mathbb{Z}})$. Hence by Lemma 4.1,
$\operatorname{in}_{p}{\mathbf{X}}$ is isomorphic to a linear subvariety via a
change of coordinates in ${\text{GL}}(n,{\mathbb{Z}})$. In particular,
$\operatorname{in}_{p}{\mathbf{X}}$ is non-singular. ∎

######  Remark 4.3.

In the situation of Theorem 1.2 and Corollary 1.4, the tropical subvarieties
appearing are those whose vertices in ${\mathbb{R}}^{N}$ have rational
coordinates, and have been called ${\mathbb{Q}}$-rational tropical varieties
in [IKMZ19]. This is because $\text{val}(\bar{{\mathbb{K}}})={\mathbb{Q}}$.
These have also been called rational tropical varieties (we use ${\mathbb{Q}}$
to avoid confusion with rational varieties in algebraic geometry or with
rational polyhedral complexes). We do not know of an example of a tropical
subvariety in a tropical toric variety which cannot be deformed to a
${\mathbb{Q}}$-tropical variety.

###  4.2. Real tropical limits and patchworking

Let ${\mathbf{X}}$ be an analytic family over ${\mathcal{D}}^{*}$. From now
on, we additionally assume that ${\mathbf{X}}$ is _real_ , that is, defined
over the subfield ${\mathbb{K}}_{\mathbb{R}}={\mathbb{R}}((t))$ of real
Laurent series. Equivalently,
${\mathbf{X}}\subset({\mathbb{C}}^{*})^{n}\times{\mathcal{D}}^{*}$ is
invariant under the canonical conjugation on
$({\mathbb{C}}^{*})^{n}\times{\mathcal{D}}^{*}$. In particular, for any
$t\in{\mathcal{D}}^{*}\cap{\mathbb{R}}$,
${\mathbf{X}}_{t}\subset({\mathbb{C}}^{*})^{n}$ is a real variety (invariant
under conjugation on $({\mathbb{C}}^{*})^{n}$). Moreover, all initial
varieties $\operatorname{in}_{p}{\mathbf{X}}\subset({\mathbb{C}}^{*})^{n}$ are
real. We want to use the real structure on the analytic family to define a
real phase structure on $X=\operatorname{Trop}({\mathbf{X}})$.

######  Definition 4.4.

Let ${\mathbf{X}}$ be a real analytic family over ${\mathcal{D}}^{*}$,
$X=\operatorname{Trop}({\mathbf{X}})$ and ${\mathcal{X}}$ a tropical
subdivision of $X$. We define the map
${\mathcal{E}}_{\mathbf{X}}\colon\operatorname{Facets}({\mathcal{X}})\to{\operatorname{Pow}}({\mathbb{Z}}_{2}^{n})$
as follows: The set ${\mathcal{E}}_{\mathbf{X}}(\sigma)$ is the set of
orthants of $({\mathbb{R}}^{*})^{n}$ meeting
${\mathbb{R}}\operatorname{in}_{\sigma}{\mathbf{X}}$,

${\mathcal{E}}_{\mathbf{X}}(\sigma):=\left\\{\varepsilon\in{\mathbb{Z}}_{2}^{n}:(-1)^{\varepsilon}\cdot{\mathbb{R}}\operatorname{in}_{\sigma}{\mathbf{X}}\cap{\mathbb{R}}_{>}^{n}\neq\emptyset\right\\}.$

For general ${\mathbf{X}}$, this definition might not produce a real phase
structure in the sense of Definition 2.8 since ${\mathcal{E}}(\sigma)$ could
be empty or a union of several affine subspaces. However, our first goal is to
show that if ${\mathbf{X}}$ has non-singular tropical limit $X$ then
${\mathcal{E}}_{\mathbf{X}}$ is a real phase structure on $X$.

Recall that the projective matroid fan $\mathbb{P}\Sigma_{M}$ of $M$ is a
projection of the affine matroid fan $\Sigma_{M}$ of $M$ along the direction
$(1,\dots,1)$. Given an oriented matroid ${\mathcal{M}}$ over $M$, the real
phase structure ${\mathcal{E}}_{{\mathcal{M}}}$ on $\Sigma_{M}$ from [RRS22,
Definition 7] can also be projected along this direction to produce a real
phase structure on $\mathbb{P}\Sigma_{M}$. Given a real linear subvariety
$\textbf{L}\subset({\mathbb{C}}^{*})^{n}$, we denote by
${\mathcal{M}}_{\textbf{L}}$ the oriented matroid associated to the (cone over
the) real hyperplane arrangement
${\mathbb{R}}\overline{\textbf{L}}\setminus{\mathbb{R}}\textbf{L}$. The next
lemma compares the real phase structure on
$\operatorname{Trop}(\textbf{L})={\mathbb{P}}\Sigma_{M_{\textbf{L}}}$ from
Definition 4.4 with the one coming from ${\mathcal{M}}_{\textbf{L}}$.

######  Lemma 4.5.

Let $\textbf{L}\subset({\mathbb{C}}^{*})^{n}$ be a real and linear subvariety
of the torus. Then the real phase structure ${\mathcal{E}}_{\textbf{L}}$ from
Definition 4.4 on $\operatorname{Trop}(\textbf{L})={\mathbb{P}}\Sigma_{M_{L}}$
coincides with the projection to ${\mathbb{P}}\Sigma_{M_{L}}$ of the real
phase structure ${\mathcal{E}}_{{\mathcal{M}}_{\textbf{L}}}$ from [RRS22,
Definition 7].

###### Proof.

For $\textbf{L}\subset({\mathbb{C}}^{*})^{n}$ linear and real, we denote by
$C\textbf{L}$ the cone over L, that is, the preimage of L under
$({\mathbb{C}}^{*})^{n+1}\to({\mathbb{C}}^{*})^{n}$,
$(z_{0},\dots,z_{n})\mapsto(z_{1}/z_{0},\dots,z_{n}/z_{0})$. We denote by
$\overline{C\textbf{L}}$ the closure of $C\textbf{L}$ in ${\mathbb{C}}^{n+1}$.
Then ${\mathbb{R}}\overline{C\textbf{L}}\backslash{\mathbb{R}}C\textbf{L}$
defines a central real hyperplane arrangement which yields an oriented matroid
${\mathcal{M}}_{\textbf{L}}$. We set $M=M_{\textbf{L}}$ and
${\mathcal{M}}={\mathcal{M}}_{\textbf{L}}$. A facet $\sigma$ of $\Sigma_{M}$
is labelled by a complete flag of flats
${\mathcal{F}}=\\{\emptyset=F_{0}\subsetneq\dots\subsetneq F_{k}=E\\}$ in $M$.
It is straightforward to check that the initial variety
$\operatorname{in}_{\sigma}C\textbf{L}$ is another (central) linear subvariety
whose associated oriented matroid is
${\mathcal{M}}_{\mathcal{F}}:=\bigoplus_{j=1}^{k}{\mathcal{M}}|F_{j}/F_{j-1}$.
Hence, by definition, $\varepsilon\to(-1)^{\varepsilon}$ gives a bijection
from ${\mathcal{E}}_{C\textbf{L}}(\sigma)$ to the tope vectors of
${\mathcal{M}}_{\mathcal{F}}$. However, it is explained in the proof of
[RRS22, Lemma 3.1] that the same statement is true for
${\mathcal{E}}_{{\mathcal{M}}}(\sigma)$ as defined in [RRS22, Definition 7]. ∎

######  Proposition 4.6.

Let ${\mathbf{X}}$ be a real analytic family over ${\mathcal{D}}^{*}$ with
non-singular tropical limit $X=\operatorname{Trop}({\mathbf{X}})$. Then the
map ${\mathcal{E}}_{\mathbf{X}}$ is a real phase structure on $X$.

###### Proof.

For a facet $\sigma\in{\mathcal{X}}$, the associated initial scheme
$\operatorname{in}_{\sigma}{\mathbf{X}}$ is reduced and irreducible. Hence, by
Condition (2) of a tropical subdivision,
$\operatorname{in}_{\sigma}{\mathbf{X}}$ is a translation of the subtorus
${\mathbf{T}}_{\sigma}$. It follows that ${\mathcal{E}}(\sigma)$ is an affine
space in ${\mathbb{Z}}_{2}^{n}$ tangent to $T_{{\mathbb{Z}}_{2}}(\sigma)$,
which proves Condition (1) of a real phase structure. To check Condition (2),
up to change of coordinates (as in the proof of Proposition 4.2) and using
Lemma 4.1, we can assume that ${\mathbf{X}}$ is real and linear. Condition (2)
then follows from Lemma 4.5 and [RRS22, Theorem 1.1]. ∎

Our aim in the remainder of this section is to prove the following theorem
from the introduction.

Theorem 1.2 (Patchworking for non-singular tropical limits). Let
${\mathbf{X}}$ be a real analytic family over ${\mathcal{D}}^{*}$ with non-
singular tropical limit $X=\operatorname{Trop}({\mathbf{X}})$ and associated
real phase structure ${\mathcal{E}}={\mathcal{E}}_{\mathbf{X}}$. Let
${\mathcal{X}}$ be a subdivision of $X$ and $\Sigma$ a pointed unimodular fan
such that $\operatorname{RecCone}({\mathcal{X}})\cup\Sigma$ is a fan.

Then for sufficiently small and positive
$t\in{\mathcal{D}}^{*}\cap{\mathbb{R}}$ the pairs
$({\mathbb{R}}\Sigma,{\mathbb{R}}\overline{{\mathbf{X}}}_{t})$ and
$(\mathfrak{R}{\mathbb{T}}\Sigma,\mathfrak{R}(\overline{X},{\mathcal{E}}))$
are homeomorphic. Moreover, the homeomorphism can be chosen to respect the
stratification of ${\mathbb{R}}\Sigma$ and $\mathfrak{R}{\mathbb{T}}\Sigma$ by
torus orbits.

Here, the closures are taken with respect to the partial compactifications
$({\mathbb{C}}^{*})^{n}\subset{\mathbb{C}}\Sigma$ and
${\mathbb{R}}^{n}\subset{\mathbb{T}}\Sigma$.

###  4.3. Reduction to the unimodular case

The first step in the proof of 1.2 is to construct out of the given data a
toric degeneration of the variety ${\mathbb{C}}\Sigma$ such that the closure
of ${\mathbf{X}}$ in this family is a semi-stable degeneration. To do so, we
follow closely [NS06] and [HK12].

Given a polyhedron $\sigma\in{\mathbb{R}}^{n}$, we denote by $C(\sigma)$ the
polyhedral cone in ${\mathbb{R}}^{n+1}$ obtained as the closure of the cone
over $\sigma\times\\{1\\}$. Given a polyhedral complex ${\mathcal{X}}$ in
${\mathbb{R}}^{n}$ such that $\Sigma=\operatorname{RecCone}({\mathcal{X}})$ is
a fan, we denote by $C({\mathcal{X}})$ the fan in
${\mathbb{R}}^{n}\times{\mathbb{R}}_{\geq}$ given by the cones $C(\sigma)$,
$\sigma\in{\mathcal{X}}$, and $\rho\times\\{0\\}$, $\rho\in\Sigma$.

We say $\sigma$ is _unimodular_ if $C(\sigma)$ is unimodular. In particular,
the polyhedron $\sigma$ must have ${\mathbb{Q}}$-coordinates. We call $\sigma$
_strongly unimodular_ if there exist $x_{0}\in{\mathbb{Z}}^{n}$ and
$e_{1},\dots,e_{k},f_{1},\dots,f_{l}\in{\mathbb{Z}}^{n}$ a part of a lattice
basis such that $\sigma=\kappa+\rho$,

$\displaystyle\kappa=\text{Conv}(x_{0},x_{0}+e_{1},\dots,x_{0}+e_{k}),\text{
and }$ $\displaystyle\rho=\langle f_{1},\dots,f_{l}\rangle_{\geq 0}.$ (4.1)

This is equivalent to the condition that $C(\sigma)$ is unimodular and that
the primitive generators of the rays of $C(\sigma)$ have last coordinate equal
to $x_{n+1}=0$ or $x_{n+1}=1$. In particular, the vertices of $\sigma$ must
have ${\mathbb{Z}}$-coordinates. A polyhedral complex ${\mathcal{P}}$ in
${\mathbb{R}}^{n}$ is called strongly unimodular if all its polyhedra are
strongly unimodular. For convenience, we also require that ${\mathcal{P}}$
contains a $0$-dimensional cell. If ${\mathcal{P}}$ is a strongly unimodular
subdivision of ${\mathbb{R}}^{n}$, then
$\Sigma=\operatorname{RecCone}({\mathcal{P}})$ is a pointed unimodular
complete fan of ${\mathbb{R}}^{n}$ [BGS11]. Our additional requirement that
${\mathcal{P}}$ contains a $0$-dimensional cell is made to ensure that
$\Sigma$ is pointed. Moreover, $C({\mathcal{P}})$ is a pointed unimodular fan
with support ${\mathbb{R}}^{n}\times{\mathbb{R}}_{\geq}$. We will deduce the
general Patchworking Theorem 1.2 from the following version.

######  Theorem 4.7 (Unimodular Patchworking).

Let ${\mathbf{X}}$ be a real analytic family with non-singular tropical limit
$X=\operatorname{Trop}({\mathbf{X}})$ and associated real phase structure
${\mathcal{E}}={\mathcal{E}}({\mathbf{X}})$. Let ${\mathcal{P}}$ be a strongly
unimodular subdivision of ${\mathbb{R}}^{n}$ such that

${\mathcal{X}}=\\{\sigma\in{\mathcal{P}}:\sigma\subset X\\}$

is a subdivision of $X$. Set $\Sigma=\operatorname{RecCone}({\mathcal{P}})$.

Then for sufficiently small and positive
$t\in{\mathcal{D}}^{*}\cap{\mathbb{R}}$ the pairs
$({\mathbb{R}}\Sigma,{\mathbb{R}}\overline{{\mathbf{X}}}_{t})$ and
$(\mathfrak{R}{\mathbb{T}}\Sigma,\mathfrak{R}(\overline{X},{\mathcal{E}}))$
are homeomorphic. Moreover, the homeomorphism can be chosen to respect the
stratification of ${\mathbb{R}}\Sigma$ and $\mathfrak{R}{\mathbb{T}}\Sigma$ by
torus orbits.

The reduction from 1.2 to 4.7 is based on the following lemma.

######  Lemma 4.8.

Let $\Sigma$ be a pointed fan in ${\mathbb{R}}^{n}$ and ${\mathcal{X}}$ a
polyhedral complex in ${\mathbb{R}}^{n}$ such that
$\operatorname{RecCone}({\mathcal{X}})\cup\Sigma$ is a fan.

  1. (1)

There exists a complete subdivision ${\mathcal{P}}$ of ${\mathbb{R}}^{n}$ such
that ${\mathcal{X}}\subset{\mathcal{P}}$ and
$\Sigma\subset\operatorname{RecCone}({\mathcal{P}})$.

  2. (2)

Assume that $\Sigma$ is unimodular and pick a ${\mathcal{P}}$ as in the
previous item. Then there exists an integer $d$ and a refinement
${\mathcal{P}}^{\prime}$ of $d{\mathcal{P}}$ which is strongly unimodular and
such that $\Sigma\subset\operatorname{RecCone}({\mathcal{P}}^{\prime})$.

###### Proof.

(1) Since $\operatorname{RecCone}({\mathcal{X}})\subset\Sigma$, the cones
$C(\sigma)$, $\sigma\in{\mathcal{X}}$, together with the cones
$\rho\times\\{0\\}$, $\rho\in\Sigma$ form a pointed fan $\widetilde{\Sigma}$
in ${\mathbb{R}}^{n+1}$. By [Ewa96, Theorem III.2.8], any fan can be completed
to a complete fan. Let $\widetilde{\Sigma}^{\prime}$ be a completion of
$\widetilde{\Sigma}$. Then it is straightforward to check that
$\widetilde{\Sigma}^{\prime}\cap({\mathbb{R}}^{n}\times\\{1\\})$ is a complete
subdivision with the desired properties.

(2) This is very similar to [HK12]. We include a short proof streamlined for
our purposes. Let ${\mathcal{P}}$ be a subdivision from the previous item and
denote by $C({\mathcal{P}})$ the correponding fan in ${\mathbb{R}}^{n+1}$ with
support ${\mathbb{R}}^{n}\times{\mathbb{R}}_{\geq 0}$. By [Ful98, Section 2.6]
there exists a unimodular fan $\Sigma^{\prime}$ that refines
$C({\mathcal{P}})$ without subdividing any of the unimodular cones of
$C({\mathcal{P}})$, in particular, the cones $\rho\times\\{0\\}$,
$\rho\in\Sigma$. Moreover, let $d$ be a common multiple of the (non-zero) last
coordinates of primitive generators of all the rays in $\Sigma^{\prime}$. Then
${\mathcal{P}}^{\prime}=\Sigma^{\prime}\cap({\mathbb{R}}^{n}\times\\{d\\})$ is
a strongly unimodular subdivision that refines $d{\mathcal{P}}$ and such that
$\Sigma\subset\operatorname{RecCone}({\mathcal{P}}^{\prime})$. ∎

###### Proof of 1.2 based on 4.7.

Performing a base change $t\to t^{d}$ and applying the Lemma 4.8, we may
assume that there exists a strongly unimodular subdivision ${\mathcal{P}}$ of
${\mathbb{R}}^{n}$ such that
${\mathcal{X}}=\\{\sigma\in{\mathcal{P}}:\sigma\subset X\\}$ and
$\Sigma\subset\widetilde{\Sigma}=\operatorname{RecCone}({\mathcal{P}})$. By
4.7 there exists a homeomorphism
${\mathbb{R}}\widetilde{\Sigma}\approx\mathfrak{R}{\mathbb{T}}\widetilde{\Sigma}$
which sends ${\mathbb{R}}\overline{{\mathbf{X}}}_{t}$ to
$\mathfrak{R}(\overline{X},{\mathcal{E}})$ and respects torus orbits. The
latter property ensures that it restricts to homeomorphism
${\mathbb{R}}\Sigma\approx\mathfrak{R}{\mathbb{T}}\Sigma$ with the analogous
property. ∎

In the remaining subsections, we will prove 4.7. The general strategy will be
as follows. A strictly unimodular subdivision ${\mathcal{P}}$ yields a semi-
stable degeneration, both on the classical and tropical side. Due to the nice
properties of semi-stable degenerations, the topology of
$({\mathbb{R}}\Sigma,{\mathbb{R}}\overline{{\mathbf{X}}}_{t})$ and
$(\mathfrak{R}{\mathbb{T}}\Sigma,\mathfrak{R}(\overline{X},{\mathcal{E}}))$
can be described in terms of the classical and tropical special fibre,
respectively. Then the last step is to relate the classical and tropical
special fibres, which boils down to a combinatorial statement about an
isomorphism of posets.

###  4.4. Regular CW complexes and pairs

In this subsection, we present a convenient framework for the description of
the homeomorphisms that we are about to construct.

Let $X$ be a Hausdorff topological space. A subset $\sigma\subset X$ is called
a _(regular) $n$-cell_ if there exists an homeomorphism $\alpha\colon
D^{n}\to\sigma$. Here, $D^{n}=\\{x\in{\mathbb{R}}^{n}:|x|\leq 1\\}$ denotes
the closed unit $n$-ball. Note that $\sigma$ is compact and hence closed in
$X$. We define the _relative interior_ of $\sigma$ by
$\sigma^{\diamond}=\alpha(({\mathcal{D}}^{n})^{\circ})$ and the _relative
boundary_ by $d\sigma=\sigma\setminus\sigma^{\diamond}=\alpha(\partial
D^{n})$. These notions are well-defined, since if given a second homeomorphism
$\alpha^{\prime}\colon D^{n}\to\sigma$, the composition
$\alpha^{\prime}\circ\alpha^{-1}\colon D^{n}\to D^{n}$ is a homeomorphism and
hence preserves $\partial D^{n}=S^{n-1}$. Note also that $n$, called the
dimension of $\sigma$, is well-defined since $D^{n}$ and $D^{m}$ are
homeomorphic if and only if $n=m$.

######  Definition 4.9.

Let $X$ be a Hausdorff topological space. A _finite regular CW complex
${\mathcal{X}}$_ with support $X$ is a finite collection of subsets of $X$
such that

  1. (1)

each $\sigma\in{\mathcal{X}}$ is a $n_{\sigma}$-cell of $X$ for some
$n_{\sigma}\in{\mathbb{N}}$,

  2. (2)

the relative interiors $\sigma^{\diamond}$, $\sigma\in{\mathcal{X}}$, form a
disjoint union of $X$,

  3. (3)

for each $\sigma$, the relative boundary $d\sigma$ is a union of cells of
smaller dimension.

A few comments: Given ${\mathcal{X}}$ according to the definition, we can
choose a homeomorphism $\alpha_{\sigma}\colon D^{n_{\sigma}}\to\sigma\subset
X$ for every $\sigma\in{\mathcal{X}}$. By [Hat02, Proposition A.2] (note that
condition (iii) in _loc. cit._ is not needed/automatic since we restrict to
the finite case), the $\alpha_{\sigma}$ yield characteristic maps of a CW
complex supporting $X$. Since each $\alpha_{\sigma}$ is a homeomorphism, this
CW complex is regular according to the standard definition, see [LW69, Chapter
3, Definition 1.1]. Moreover, choosing different $\alpha^{\prime}_{\sigma}$,
we obtain a strictly equivalent CW complex, see [LW69, p. 7]. This justifies
our simplified definition and the naming. We also use
$|{\mathcal{X}}|=X=\bigcup_{\sigma\in{\mathcal{X}}}\sigma$ to denote the
support of ${\mathcal{X}}$. We consider ${\mathcal{X}}$ as a poset with the
partial ordering given by inclusion of cells. We will sometimes refer to this
a the _cell poset_ of ${\mathcal{X}}$ to emphasize that we forget all
geometric information and only consider the poset structure. The previous
discussion also implies that regular CW complexes in the sense of our
definition admit barycentric subdivisions. For convenience, we express this in
the following statement.

######  Proposition 4.10.

Let ${\mathcal{X}}$ be a regular CW complex. Then there exists a homeomorphism
$\beta\colon|O({\mathcal{X}})|\to|{\mathcal{X}}|$ from the simplicial complex
$O({\mathcal{X}})$ (the order complex of the poset ${\mathcal{X}}$) to
${\mathcal{X}}$ such that the image of the simplex in $O({\mathcal{X}})$
labelled by the chain $\sigma_{1}\subsetneq\dots\subsetneq\sigma_{k}$ lies in
$\sigma_{k}^{\diamond}$. Such a
$\beta\colon|O({\mathcal{X}})|\to|{\mathcal{X}}|$ is called a _barycentric
subdivision_ of ${\mathcal{X}}$.

###### Proof.

This is essentially [LW69, Theorem 1.7], but for later reference we give a
sketch. Fix characteristic maps $\alpha_{\sigma}\colon
D^{n_{\sigma}}\to\sigma$ for every $\sigma\in{\mathcal{X}}$. We proceed by
induction on the $k$-skeleton ${\mathcal{X}}^{(k)}$ of ${\mathcal{X}}$. The
initial case $k=0$ is trivial. Assume that a barycentric subdivision
$\beta^{(k)}\colon|O({\mathcal{X}}^{(k)})|\to|{\mathcal{X}}^{(k})|$ has
already been constructed. Given a $k+1$-cell $\sigma$, pulling back via
$\alpha_{\sigma}\colon S^{k}\to\partial\sigma\subset|{\mathcal{X}}^{(k)}|$ we
obtain a triangulation on $S^{k}$. Coning over $0\in D^{k+1}$ produces a
triangulation of $D^{k+1}$, and hence $\sigma$, labelled canonically by chains
of the form $\sigma_{1}\subsetneq\dots\subsetneq\sigma_{l}\subset\sigma$.
Repeating this for every $k+1$-cell $\sigma$, we obtain the desired
$\beta_{k+1}$. ∎

A topological pair $\tau\subset\sigma$ is called a standard pair of cells of
codimension $k$ if there exists an homeomorphism $\alpha\colon D^{n}\to\sigma$
such that $\alpha^{-1}(\tau)=\\{x\in{\mathcal{D}}^{n}:x_{1}=\dots=x_{k}=0\\}$.
In particular, $\sigma$ is an $n$-cell, $\tau$ is a $n-k$-cell, and
$d\tau=\tau\cap d\sigma$ as well as
$\tau^{\diamond}=\tau\cap\sigma^{\diamond}$.

######  Definition 4.11.

Let $X$ be a Hausdorff topological space. A _finite regular CW pair_ of
codimension $k$ is a tuple $({\mathcal{X}},Y)$ such that ${\mathcal{X}}$ is a
finite regular CW complex and $Y\subset|{\mathcal{X}}|$ is a subset such that
for each $\sigma\in{\mathcal{X}}$ either $\sigma\cap Y=\emptyset$ or
$\sigma\cap Y\subset\sigma$ is a standard pair of cells of codimension $k$.

It follows easily from the definition that in this case

${\mathcal{Y}}=\\{\sigma\cap Y:\sigma\in{\mathcal{X}},\sigma\cap
Y\neq\emptyset\\}$

is a regular CW complex, $\dim{\mathcal{Y}}=\dim{\mathcal{X}}-k$ and
$\partial(\sigma\cap Y)=\partial\sigma\cap Y$ as well as $(\sigma\cap
Y)^{\diamond}=\sigma^{\diamond}\cap Y$. Note that $Y=|{\mathcal{Y}}|$ is a
closed subset of $|{\mathcal{X}}|$. It will be convenient to use the notation

${\mathcal{X}}_{Y}=\\{\sigma\in{\mathcal{X}}:\sigma\cap Y\neq\emptyset\\}.$

By definition, we have a canonical bijection between ${\mathcal{X}}_{Y}$ and
${\mathcal{Y}}$. Since we only consider _finite_ regular CW complexes and
pairs, we will drop the finite from now on.

The notion of barycentric subdivisions extends to regular CW pairs in the
following sense.

######  Proposition 4.12.

Let $({\mathcal{X}},Y)$ be a regular CW pair with $X=|{\mathcal{X}}|$. Then
there exists a barycentric subdivision $\beta\colon|O({\mathcal{X}})|\to X$ of
${\mathcal{X}}$ such that the image of the simplicial subcomplex
$O({\mathcal{X}}_{Y})\subset O({\mathcal{X}})$ (given by the subposet
${\mathcal{X}}_{Y}\subset{\mathcal{X}}$) is equal to $Y$. Moreover, in this
case the restricted map $\beta_{Y}\colon|O({\mathcal{X}}_{Y})|\to Y$ is a
barycentric subdivision of ${\mathcal{Y}}$ using the identification
$O({\mathcal{X}}_{Y})=O({\mathcal{Y}})$.

###### Proof.

We just have to slightly adapt the argument given for Proposition 4.10: We
again choose characteristic maps $\alpha_{\sigma}\colon
D^{n_{\sigma}}\to\sigma$ for every $\sigma\in{\mathcal{X}}$, but additionally
assume that
$\alpha_{\sigma}^{-1}(Y)=\\{x\in{\mathcal{D}}^{n}:x_{1}=\dots=x_{k}=0\\}$
whenever $\sigma\cap Y\neq\emptyset$. Note that for any $m$ the pair
$({\mathcal{X}}^{(m)},Y\cap|{\mathcal{X}}^{(m)}|)$ is a regular CW pair as
well. Hence the same recursive coning procedure as in Proposition 4.10
produces the desired homeomorphism. Finally, given a chain
$\sigma_{1}\subsetneq\dots\subsetneq\sigma_{l}$ in ${\mathcal{X}}_{Y}$, the
image of the corresponding simplex lies in $\sigma_{l}^{\diamond}\cap
Y=(\sigma_{l}\cap Y)^{\diamond}$, which shows that the restricted map
$\beta_{Y}\colon|O({\mathcal{X}}_{Y})|\to Y$ is a barycentric subdivision of
${\mathcal{Y}}$. ∎

The following theorem states that, up to homeomorphisms, a regular CW complex
${\mathcal{X}}$ is determined by the poset structure on ${\mathcal{X}}$, and
similarly, a regular CW pair is determined by the pair of posets
${\mathcal{X}}_{Y}\subset{\mathcal{X}}$. This follows easily from the
existence of barycentric subdivisions.

######  Theorem 4.13.

Let $({\mathcal{X}},Y)$ and $({\mathcal{X}}^{\prime},Y^{\prime})$ be two
regular CW pairs. Assume that there exists poset isomorphism
$\varphi:{\mathcal{X}}\to{\mathcal{X}}^{\prime}$ that sends
${\mathcal{X}}_{Y}$ to ${\mathcal{X}}^{\prime}_{Y^{\prime}}$. Then there
exists a homeomorphism $\Phi\colon|{\mathcal{X}}|\to|{\mathcal{X}}^{\prime}|$
that sends a cell $\sigma$ to the cell $\varphi(\sigma)$ and sends $Y$ to
$Y^{\prime}$.

###### Proof.

The poset isomorphism $\varphi$ yields an induced simplicial homeomorphism
$O(\varphi)\colon|O({\mathcal{X}})|\to|O({\mathcal{X}}^{\prime})|$ that sends
the subcomplex $O({\mathcal{X}}_{Y})$ to the subcomplex
$O({\mathcal{X}}^{\prime}_{Y^{\prime}})$. Pre- and postcomposing $O(\varphi)$
with barycentric subdivisions as provided in Proposition 4.12 yields the
desired homeomorphism $\Phi$. ∎

###  4.5. The local cases

In this subsection we will deal with the local case of patchworking which, for
non-singular tropical subvarieties means the case of linear spaces: We fix a
real linear subvariety $\textbf{L}\subset({\mathbb{C}}^{*})^{n}$ of dimension
$d$. Its locus of real points is
${\mathbb{R}}\textbf{L}\subset({\mathbb{R}}^{*})^{n}$. We denote by
$L=\operatorname{Trop}(\textbf{L})$ the support of the associated matroid fan
and by ${\mathcal{E}}={\mathcal{E}}_{\textbf{L}}$ the associated real phase
structure on $L$. As usual,
$\mathfrak{R}L=\mathfrak{R}(L,{\mathcal{E}})\subset\mathfrak{R}{\mathbb{R}}^{n}\approx({\mathbb{R}}^{*})^{n}$
denotes the associated patchwork. It suffices to consider a single orthant,
and by symmetry, we focus on the positive orthant labelled by
$\varepsilon=(0,\dots,0)\in{\mathbb{Z}}_{2}^{n}$. We set
${\mathbb{R}}_{>}=(0,\infty)$ and

$\displaystyle{\mathbb{R}}_{>}\textbf{L}$
$\displaystyle:={\mathbb{R}}\textbf{L}\cap{\mathbb{R}}_{>}^{n}$
$\displaystyle\subset{\mathbb{R}}_{>}^{n},$ $\displaystyle\mathfrak{R}_{>}L$
$\displaystyle=\mathfrak{R}L\cap{\mathbb{R}}^{n}((0,\dots,0))$
$\displaystyle\subset{\mathbb{R}}^{n}.$

Note that ${\mathbb{R}}_{>}\textbf{L}$ is non-empty if and only if
$\mathfrak{R}_{>}L$ is non-empty by definition of ${\mathcal{E}}$.

Let $\Sigma$ be a complete unimodular fan. The closure of $\mathfrak{R}_{>}L$
in ${\mathbb{T}}\Sigma$ is denoted by $\mathfrak{R}_{\geq}L$. It is equal to
$\mathfrak{R}\overline{L}\cap{\mathbb{T}}\Sigma((0,\dots,0))$. The closure of
the positive orthant in ${\mathbb{R}}\Sigma$ is denoted by
${\mathbb{R}}_{\geq}\Sigma$ (note that ${\mathbb{R}}_{\geq}\Sigma$ can also be
described as the toric variety over the semigroup ${\mathbb{R}}_{\geq}$, so
the notation is consistent). The closure of ${\mathbb{R}}_{>}\textbf{L}$ in
${\mathbb{R}}_{\geq}\Sigma$ is denoted by ${\mathbb{R}}_{\geq}\textbf{L}$. It
is equal to ${\mathbb{R}}\overline{\textbf{L}}\cap{\mathbb{R}}_{\geq}\Sigma$.
The goal of this subsection is to prove that under suitable conditions the
pairs ${\mathbb{R}}_{\geq}\textbf{L}\subset{\mathbb{R}}_{\geq}\Sigma$ and
$\mathfrak{R}_{\geq}L\subset{\mathbb{T}}\Sigma$ are standard pairs of cells of
dimension $(d,n)$.

Note that $L$ is a union of a collection of cones in $\Sigma$ if and only if
$\Sigma_{L}=\\{\sigma\in\Sigma:\sigma\subset L\\}$ is a polyhedral subdivision
for $L$. We want to prove the following two analogous statements, which we
state separately for more convenient referencing.

######  Theorem 4.14.

Assume that ${\mathbb{R}}_{>}\textbf{L}$ is non-empty. Assume that $L$ is a
union of cones in $\Sigma$. Then
${\mathbb{R}}_{\geq}\textbf{L}\subset{\mathbb{R}}_{\geq}\Sigma$ is a standard
pair of cells of dimension $(d,n)$.

######  Theorem 4.15.

Assume that $\mathfrak{R}_{>}L$ is non-empty. Assume that $L$ is a union of
cones in $\Sigma$. Then $\mathfrak{R}_{\geq}L\subset{\mathbb{T}}\Sigma$ is a
standard pair of cells of dimension $(d,n)$.

We proceed in several steps.

######  Lemma 4.16.

Assume that ${\mathbb{R}}_{>}\textbf{L}$ (and hence $\mathfrak{R}_{>}L$) is
non-empty. Then the pairs
${\mathbb{R}}_{\geq}\textbf{L}\subset{\mathbb{R}}_{\geq}{\mathbb{P}}^{n}$ and
$\mathfrak{R}_{\geq}L\subset{\mathbb{T}}{\mathbb{P}}^{n}$ are standard pairs
of cells of dimension $(d,n)$.

We emphasize that in this lemma we do not require that $L$ is a union of cones
of $\Sigma$.

###### Proof.

It is clear that
${\mathbb{R}}_{\geq}\textbf{L}\subset{\mathbb{R}}_{\geq}{\mathbb{P}}^{n}$ is a
standard pair of cells: In projective coordinates,
${\mathbb{R}}_{\geq}{\mathbb{P}}^{n}$ can be described as the standard simplex
$x_{0},\dots,x_{n}\geq 0$, $x_{0}+\cdots+x_{n}=1$. Since
${\mathbb{R}}_{\geq}\textbf{L}$ is the intersection of this simplex with a
linear subspace which intersects its interior, the claim follows.

To finish, we construct a homeomorphism between
${\mathbb{R}}_{\geq}\textbf{L}\subset{\mathbb{R}}_{\geq}{\mathbb{P}}^{n}$ and
$\mathfrak{R}_{\geq}L\subset{\mathbb{T}}{\mathbb{P}}^{n}$. As explained above,
we regard
${\mathbb{R}}_{\geq}\textbf{L}\subset{\mathbb{R}}_{\geq}{\mathbb{P}}^{n}$ as
the intersection of the simplex with a linear space. We consider a barycentric
subdivision of ${\mathbb{R}}_{\geq}{\mathbb{P}}^{n}$ where the barycenter of a
face is chosen in ${\mathbb{R}}_{\geq}\textbf{L}$ whenever the intersection
with the relative interior of this face is non-empty. We obtain a pair of
simplicial complexes with the following characteristics:

  * •

vertices of ${\mathbb{R}}_{\geq}{\mathbb{P}}^{n}$: non-empty subsets of
$E=\\{0,\dots,n\\}$.

  * •

simplices of ${\mathbb{R}}_{\geq}{\mathbb{P}}^{n}$: chains under inclusion.

  * •

vertices of ${\mathbb{R}}_{\geq}\textbf{L}$: flats of $M_{\textbf{L}}$ that
touch $\varepsilon=(0,\dots,0)$, that is,
${\mathcal{F}}_{lv}^{{\mathcal{M}}_{\textbf{L}}}((0,\dots,0))$ from Example
3.3.

  * •

simplices of ${\mathbb{R}}_{\geq}\textbf{L}$: chains of
${\mathcal{F}}_{lv}^{{\mathcal{M}}_{\textbf{L}}}((0,\dots,0))$ under
inclusion.

As explained in Example 3.3,
$\mathfrak{R}_{\geq}L\subset{\mathbb{T}}{\mathbb{P}}^{n}$ is a pair of
simplicial complexes with exactly the same poset structure, which induces a
simplicial homeomorphism between the pairs. ∎

We now first deal with the tropical side, that is, with the pair
$\mathfrak{R}_{\geq}L\subset{\mathbb{T}}\Sigma$.

######  Lemma 4.17.

For any complete unimodular fan $\Sigma$, the space ${\mathbb{T}}\Sigma$ is a
ball. Moreover, given a subset $X\subset{\mathbb{R}}^{n}$ which is a union of
cones in $\Sigma$, the pair $\overline{X}\subset{\mathbb{T}}\Sigma$ is
homeomorphic to the pair of PL-stars at $0$ of $X\subset{\mathbb{R}}^{n}$.

We use PL-stars only for simplicity, since they are well-defined up to PL-
homeomorphism. We write $\operatorname{Pstar}(X)=\operatorname{Pstar}_{0}(X)$
for the PL-star at the origin $0\in{\mathbb{R}}^{n}$.

###### Proof.

Given a cone $\sigma\in\Sigma$ of dimension $k$, choosing primitive generators
$v_{1},\dots,v_{k}$ we obtain an identification with $[0,\infty)^{k}$. The
closure $\overline{\sigma}$ in ${\mathbb{T}}\Sigma$ is then canonically
homeomorphic to an infinite cube $[0,\infty]^{k}$. More generally, suppose $X$
is the support of a subset of cones of $\Sigma$ and denote this subset by
$\Sigma(X)\subset\Sigma$. Then the closure $\overline{X}$ in
${\mathbb{T}}\Sigma$ (or, equivalently, in ${\mathbb{T}}\Sigma(X)$) is
obtained by gluing these infinite cubes $\overline{\sigma}$,
$\sigma\in\Sigma(X)$, as a direct limit governed by the poset $\Sigma(X)$.

Fix a homeomorphism $[0,\infty]\to[0,1]$ that sends $0$ to $0$. This induces
an homeomorphism between $\overline{\sigma}$ and the (finite) cube
$C_{\sigma}$ in $\sigma$ whose vertices are the points $\sum_{i\in I}v_{i}$,
$I\subset\\{1,\dots,k\\}$. Moreover, these homeomorphisms glue to a
homeomorphism between the direct limit $\overline{X}$ and the set

$\operatorname{Pstar}(X)=\bigcup_{\sigma\in\Sigma(X)}C_{\sigma}.$

As the notation suggests, the set $\operatorname{Pstar}(X)$ is a PL-star for
$X$ in $0$. Finally, applying the same reasoning to ${\mathbb{R}}^{n}$ and
$\Sigma({\mathbb{R}}^{n})=\Sigma$, we obtain a homeomorphism of pairs between
$\overline{X}\subset\overline{{\mathbb{R}}^{n}}={\mathbb{T}}\Sigma$ and
$\operatorname{Pstar}(X)\subset\operatorname{Pstar}({\mathbb{R}}^{n})$. Of
course, $\operatorname{Pstar}({\mathbb{R}}^{n})$ is a ball. ∎

We can now finish the proof of 4.15.

###### Proof of 4.15.

By Lemma 4.16, we now that
$\mathfrak{R}_{\geq}{L}\subset{\mathbb{T}}{\mathbb{P}}^{n}$ is a standard pair
of cells. Since $0\in{\mathbb{R}}^{n}$ corresponds to a point in the relative
interior of this pair, it follows that
$\operatorname{Pstar}(\mathfrak{R}_{>}L)\subset\operatorname{Pstar}({\mathbb{R}}^{n})$
is a standard pair of cells. By Lemma 4.17, this implies that
$\mathfrak{R}_{\geq}L\subset{\mathbb{T}}\Sigma$ is a standard pair of cells. ∎

We now turn our attention to the pair
${\mathbb{R}}_{\geq}\textbf{L}\subset{\mathbb{R}}_{\geq}\Sigma$.

###### Proof of 4.14.

As we mentioned in Remark 2.10, ${\mathbb{R}}_{\geq}\Sigma$ is homeomorphic to
${\mathbb{T}}\Sigma$ via the extended logarithm map $\operatorname{Log}$.
Hence by Lemma 4.17, ${\mathbb{R}}_{\geq}\Sigma$ is a ball. We now proceed by
induction on $\dim(\textbf{L})=d$. For $d=0$, the statement is trivial. We now
prove the induction step $d-1\to d$.

By our first remark, the subsets
${\mathbb{R}}_{\geq}\operatorname{star}_{\rho}\Sigma$, $\rho\in\Sigma$, form a
regular CW complex (with support ${\mathbb{R}}_{\geq}\Sigma$). By the
induction assumption, we know that for $\rho\subset L$ of positive dimension,
the pair
${\mathbb{R}}_{\geq}\textbf{L}\cap{\mathbb{R}}_{\geq}\operatorname{star}_{\rho}\Sigma\subset{\mathbb{R}}_{\geq}\operatorname{star}_{\rho}\Sigma$
is a standard pair of cells, while for $\rho\not\subset L$ the intersection
${\mathbb{R}}_{\geq}\textbf{L}\cap{\mathbb{R}}_{\geq}\operatorname{star}_{\sigma}\Sigma$
is empty. It follows that
$\partial{\mathbb{R}}_{\geq}\textbf{L}\subset\partial{\mathbb{R}}_{\geq}\Sigma$
is a regular CW pair.

The analogous argument shows the corresponding tropical pair
$\partial\mathfrak{R}_{\geq}L\subset\partial{\mathbb{T}}\Sigma$ is a regular
CW pair with the same poset structure (since
${\mathbb{R}}_{\geq}\textbf{L}\cap{\mathbb{R}}_{\geq}\operatorname{star}_{\rho}\Sigma\neq\emptyset$
if and only if $0\in{\mathcal{E}}(\rho)$ if and only if
$\mathfrak{R}_{\geq}L\cap{\mathbb{T}}\operatorname{star}_{\rho}\Sigma\neq\emptyset$).
Hence by 4.13 the two pairs are homeomorphic. It follows that both pairs are
standard pairs of spheres, since by 4.15
$\mathfrak{R}_{\geq}L\subset{\mathbb{T}}\Sigma$ is a standard pair of cells.

Since $L$ is a union of cones in $\Sigma$, $\overline{\textbf{L}}$ is
transversal with respect to the toric boundary of ${\mathbb{C}}\Sigma$ in the
following sense: For any $p\in\overline{\textbf{L}}$, the torus orbit
containing $p$ and $\overline{\textbf{L}}$ intersect transversally in $p$.
This follows from Proposition 4.18. Transversality with the toric boundary
implies that there exists a neighbourhood
$\partial{\mathbb{R}}_{\geq}\Sigma\subset U\subset{\mathbb{R}}_{\geq}\Sigma$
which is homeomorphic to $\partial{\mathbb{R}}_{\geq}\Sigma\times[0,1]$ and
such that this homeomorphism identifies ${\mathbb{R}}_{\geq}\textbf{L}\cap U$
with $\partial{\mathbb{R}}_{\geq}\textbf{L}\times[0,1]$, see [Rau23, Theorem
6.1]. If follows that we may replace $\Sigma$ by any complete unimodular fan
supporting $L$, since by the previous argument
$\partial{\mathbb{R}}_{\geq}\textbf{L}\subset\partial{\mathbb{R}}_{\geq}\Sigma$
is a standard pair of spheres for any such choice of $\Sigma$. Hence we assume
from now on that $\Sigma=\Pi^{n}$ is the permutahedral fan in
${\mathbb{R}}^{n}$.

The permutahedral variety
${\mathbb{R}}\Pi^{n}\to{{\mathbb{R}}{\mathbb{P}}}^{n}$ is obtained from
${{\mathbb{R}}{\mathbb{P}}}^{n}$ from consecutively blowing up (strict
transforms) of coordinate subspaces of ${{\mathbb{R}}{\mathbb{P}}}^{n}$
starting with points, lines, etc. We know from Lemma 4.16 that
${\mathbb{R}}_{\geq}\textbf{L}\subset{\mathbb{R}}_{\geq}{\mathbb{P}}^{n}$ is a
standard pair of cells, so it remains to prove that the topology does not
change under the aforementioned blow-ups. For clarity, we denote the closures
in ${\mathbb{R}}_{\geq}{\mathbb{P}}^{n}$ and ${\mathbb{R}}_{\geq}\Pi^{n}$ by
${\mathbb{R}}_{\geq}^{\mathbb{P}}\textbf{L}$ and
${\mathbb{R}}_{\geq}^{\Pi}\textbf{L}$, respectively. The local model for all
these blow-ups is blowing up ${\mathbb{R}}_{\geq}^{k}\times{\mathbb{R}}^{n-k}$
along $\\{0\\}^{k}\times{\mathbb{R}}^{n-k}$ (the image of L after this change
of coordinates is still a linear subvariety in $({\mathbb{C}}^{*})^{n}$ which
we call L again by abuse of notation). We may assume that
$0\in{\mathbb{R}}_{\geq}^{\mathbb{P}}\textbf{L}$. In order to compare
${\mathbb{R}}_{\geq}^{\mathbb{P}}\textbf{L}$ and
${\mathbb{R}}_{\geq}^{\Pi}\textbf{L}$ it is sufficient to consider one slice
${\mathbb{R}}_{\geq}^{k}\times\\{0\\}^{n-k}$, that is, we may further reduce
to the situation of the blow-up $\operatorname{Bl}_{0}{\mathbb{R}}_{\geq}^{n}$
of ${\mathbb{R}}_{\geq}^{n}$ in the origin $0$. By abuse of notation, we now
denote by ${\mathbb{R}}_{\geq}^{\mathbb{P}}\textbf{L}$ and
${\mathbb{R}}_{\geq}^{\Pi}\textbf{L}$ their restriction to
${\mathbb{R}}_{\geq}^{n}$ and $\operatorname{Bl}_{0}{\mathbb{R}}_{\geq}^{n}$.

The blow-up can be constructed explicitly as

${\mathbb{R}}_{\triangle}^{n}=\\{x\in{\mathbb{R}}_{\geq}^{n}:\sum_{i}x_{i}\geq
1\\}$

with blow-down map given by $\pi\colon x\mapsto(1-\frac{1}{\sum_{i}x_{i}})x$.
Under this identification,
${\mathbb{R}}_{\geq}^{\Pi}\textbf{L}\subset\operatorname{Bl}_{0}{\mathbb{R}}_{\geq}^{n}$
is mapped to
${\mathbb{R}}_{\geq}^{\triangle}\textbf{L}:={\mathbb{R}}_{\geq}^{\mathbb{P}}\textbf{L}\cap{\mathbb{R}}_{\triangle}^{n}\subset{\mathbb{R}}_{\triangle}^{n}$.
We now construct a homeomorphism
$\varphi\colon{\mathbb{R}}_{\triangle}^{n}\to{\mathbb{R}}_{\geq}^{n}$ such
that
$\varphi({\mathbb{R}}_{\geq}^{\triangle}\textbf{L})={\mathbb{R}}_{\geq}^{\mathbb{P}}\textbf{L}$.
We fix a point $a\in{\mathbb{R}}_{>}\textbf{L}$ such that $\sum_{i}a_{i}=1$.
For any $I\subsetneq\\{1,\dots,n\\}$, we consider the simplices

$\displaystyle S_{I}=\operatorname{Conv}(a,2a,e_{i},i\in I),$ $\displaystyle
T_{I}=\operatorname{Conv}(0,2a,e_{i},i\in I).$

Then $\varphi$ is the piecewise linear map that maps each $S_{I}$ to $T_{I}$
by sending $a\mapsto 0$ and being identity on all other vertices as well as
outside of the union of the $S_{I}$. Clearly, $\varphi$ yields a homeomorphism
with
$\varphi({\mathbb{R}}_{\geq}^{\triangle}\textbf{L})={\mathbb{R}}_{\geq}^{\mathbb{P}}\textbf{L}$.
∎

###  4.6. Semi-stable degenerations — classical side

In this section we continue with the notation and set up of 4.7. We first
recall the main elements and explain its connection to [Rau23].

Again let ${\mathcal{P}}$ be a strongly unimodular subdivision of
${\mathbb{R}}^{n}$. As mentioned, the set of recession cones
$\Sigma=\operatorname{RecCone}({\mathcal{P}})$ is a pointed unimodular
complete fan. We set $\widetilde{{\mathcal{P}}}=C({\mathcal{P}})$.

Note that ${\mathbb{C}}\widetilde{{\mathcal{P}}}$ is a $(n+1)$-dimensional
non-singular toric variety that comes together with a toric morphism
$f\colon{\mathbb{C}}\widetilde{{\mathcal{P}}}\to{\mathbb{C}}$. The generic
fibre $f^{-1}(t)$, $t\neq 0$, is the toric variety ${\mathbb{C}}\Sigma$. The
orbits of the torus action on the special fibre
${\mathbb{C}}\widetilde{{\mathcal{P}}}_{0}=f^{-1}(0)$ are in canonical
bijection with the faces $\sigma$ of ${\mathcal{P}}$. Their closures are the
toric varieties ${\mathbb{C}}\operatorname{star}_{\sigma}{\mathcal{P}}$.

We set

${\mathbb{C}}{\mathcal{P}}:=f^{-1}({\mathcal{D}}).$

By abuse of notation, we will use the same letter $f$ for the restriction
$f\colon{\mathbb{C}}{\mathcal{P}}\to{\mathcal{D}}$, and also write
${\mathbb{C}}{\mathcal{P}}_{0}={\mathbb{C}}\widetilde{{\mathcal{P}}}_{0}=f^{-1}(0)$.

Now let ${\mathbf{X}}$ be an analytic family over ${\mathcal{D}}^{*}$. Note
that
$({\mathbb{C}}^{*})^{n}\times{\mathcal{D}}^{*}\subset{\mathbb{C}}{\mathcal{P}}=f^{-1}({\mathcal{D}})$.
We denote by $\overline{{\mathbf{X}}}$ the closure of ${\mathbf{X}}$ in
${\mathbb{C}}{\mathcal{P}}$, and set
${\mathbf{X}}_{0}=\overline{{\mathbf{X}}}\cap{\mathbb{C}}{\mathcal{P}}_{0}$.

######  Proposition 4.18.

Under the assumptions of 4.7, the following holds true.

  1. (1)

For any $\sigma\in{\mathcal{P}}$, the scheme-theoretic intersection
$\overline{{\mathbf{X}}}\cap{\mathbb{C}}{\mathcal{O}}_{C(\sigma)}$ in
${\mathbb{C}}{\mathcal{P}}$ is equal to
${\mathbf{X}}_{\sigma}=\operatorname{in}_{\sigma}{\mathbf{X}}/{\mathbf{T}}_{\sigma}$
(note that
${\mathbb{C}}{\mathcal{O}}_{C(\sigma)}\cong{\mathbb{C}}{\mathcal{O}}_{\sigma}$).

  2. (2)

The scheme $\overline{{\mathbf{X}}}$ is non-singular and for any
$\sigma\in{\mathcal{P}}$ the intersection
$\overline{{\mathbf{X}}}\cap{\mathbb{C}}{\mathcal{O}}_{C(\sigma)}$ in
${\mathbb{C}}{\mathcal{P}}$ is transversal.

###### Proof.

By Proposition 4.2, the subdivision
${\mathcal{X}}=\\{\sigma\in{\mathcal{P}}:\sigma\subset X\\}$ is tropical.
Then, item (1) is for example contained in [Spe05, Proposition 2.4.4]. Item
(2) follows since the intersection is non-singular and of expected dimension
by (1). ∎

As before, given a polyhedron $\sigma$ in ${\mathbb{R}}^{n}$, we use the
shorthand

${\mathbb{Z}}_{2}(\sigma)={\mathbb{Z}}_{2}^{n}/T_{{\mathbb{Z}}_{2}}(\sigma).$

Given the data of 4.7, we define the following poset:

$Q({\mathcal{P}})=\\{(\sigma,\varepsilon):\sigma\in{\mathcal{P}},\varepsilon\in{\mathbb{Z}}_{2}(\operatorname{RecCone}(\sigma))\\},$
(4.2)

cf. $Q(\Sigma)$ from Section 2.4. The partial order is given by

$(\sigma,\varepsilon)\leq(\tau,\delta):\Longleftrightarrow\tau\subset\sigma,\varepsilon=\pi(\delta),$

where
$\pi\colon{\mathbb{Z}}_{2}(\operatorname{RecCone}(\tau))\to{\mathbb{Z}}_{2}(\operatorname{RecCone}(\sigma))$
denotes the canonical projection. We also consider the subposet

$Q({\mathcal{X}},{\mathcal{E}})=\\{(\sigma,\varepsilon)\in
Q({\mathcal{P}}):\sigma\in{\mathcal{X}},\varepsilon\in\pi({\mathcal{E}}(\sigma))\\}\subset
Q({\mathcal{P}}).$ (4.3)

Here,
$\pi\colon{\mathbb{Z}}_{2}^{n}\to{\mathbb{Z}}_{2}(\operatorname{RecCone}(\sigma))$
denotes the canonical projection.

######  Theorem 4.19.

Under the assumptions of 4.7 and for sufficiently small and positive
$t\in{\mathcal{D}}^{*}\cap{\mathbb{R}}$, we can equip ${\mathbb{R}}\Sigma$
with the structure of a regular CW complex such that

  1. (1)

the poset $Q({\mathcal{P}})$ is isomorphic to the face poset of this CW
complex via an assignment $(\sigma,\varepsilon)\mapsto C(\sigma,\varepsilon)$,

  2. (2)

the cell $C(\sigma,\varepsilon)$ is contained in
${\mathbb{R}}\operatorname{star}_{\rho}\Sigma\subset{\mathbb{R}}\Sigma$ if and
only if $\rho\subset\operatorname{RecCone}(\sigma)$,

  3. (3)

the pair ${\mathbb{R}}\overline{{\mathbf{X}}}_{t}\subset{\mathbb{R}}\Sigma$ is
a regular CW pair with respect to this CW complex and the set of cells
intersecting ${\mathbb{R}}\overline{{\mathbf{X}}}_{t}$ corresponds to
$Q({\mathcal{X}},{\mathcal{E}})$,

$Q({\mathcal{X}},{\mathcal{E}})=\\{(\sigma,\varepsilon)\in
Q({\mathcal{P}}):C(\sigma,\varepsilon)\cap{\mathbb{R}}\overline{{\mathbf{X}}}_{t}\neq\emptyset\\}.$

###### Proof of 4.19.

The map $f\colon{\mathbb{C}}{\mathcal{P}}\to{\mathcal{D}}$ is a totally real
semi-stable degeneration, cf. [Rau23, Lemma 9.1]. It then follows by [Rau23,
Theorem 1.1] that the generic real fibre ${\mathbb{R}}\Sigma$ is homeomorphic
to a certain real-oriented blow-up
$\operatorname{Bl}_{+}({\mathbb{R}}{\mathcal{P}}_{0})$ of the special real
fibre ${\mathbb{R}}{\mathcal{P}}_{0}$. Instead of defining
$\operatorname{Bl}_{+}({\mathbb{R}}{\mathcal{P}}_{0})$ carefully, we apply
[Rau23, Corollary 9.3] which states that
$\operatorname{Bl}_{+}({\mathbb{R}}{\mathcal{P}}_{0})$ carries a
stratification into closed subsets $C(\tau,\varepsilon)$ labelled by the
elements $(\tau,\varepsilon)\in Q({\mathcal{P}})$ whose inclusion relations
are given by the partial order on ${\mathcal{P}}$. More precisely,
$C(\sigma,\varepsilon)$ is actually a copy of
${\mathbb{R}}\operatorname{star}_{\sigma}{\mathcal{P}}(\pi(\varepsilon))$,
with
$\pi\colon{\mathbb{Z}}_{2}(\operatorname{RecCone}(\sigma))\to{\mathbb{Z}}_{2}(\sigma)$
the canonical projection. It follows that $C(\sigma,\varepsilon)$ is an
$(n-\dim\sigma)$-cell by 4.14. The fact that these cells form a regular CW
complex satisfying the properties stated in items (1) and (2) is part of
[Rau23, Corollary 9.3].

Let us now proceed to item (3). It follows from Proposition 4.18 that that
${\mathbf{X}}$ together with the non-special toric boundary divisors in
${\mathbb{C}}{\mathcal{P}}$ (labelled by the rays of $\Sigma$) form a
transversal collection of submanifolds in the sense of [Rau23, Section 6].
Hence by [Rau23, Theorem 6.1], for sufficiently small positive $t$, the CW
structure from the previous paragraph can be chosen such that the pair
${\mathbb{R}}\overline{{\mathbf{X}}}_{t}\cap C(\sigma,\varepsilon)\subset
C(\sigma,\varepsilon)$ is homeomorphic to
$\overline{{\mathbf{X}}}_{0}\cap{\mathbb{R}}\operatorname{star}_{\sigma}{\mathcal{P}}(\varepsilon){\mathcal{P}}\subset{\mathbb{R}}\operatorname{star}_{\sigma}{\mathcal{P}}(\varepsilon)$.
By Proposition 4.18 we have
$\overline{{\mathbf{X}}}_{0}\cap{\mathbb{C}}\operatorname{star}_{\sigma}{\mathcal{P}}=\overline{{\mathbf{X}}}_{\sigma}$.
Hence the pair in the last item is equal to
${\mathbb{R}}\overline{{\mathbf{X}}}_{\sigma}(\varepsilon)\subset{\mathbb{R}}\operatorname{star}_{\sigma}{\mathcal{P}}(\varepsilon)$.

By definition, ${\mathbb{R}}{\mathbf{X}}_{\sigma}(\varepsilon)$ is non-empty
if and only if $(\sigma,\varepsilon)\in Q({\mathcal{X}},{\mathcal{E}})$. If
non-empty, ${\mathbf{X}}_{\sigma}$ is linear up to a coordinate change in
${\text{GL}}(n,{\mathbb{Z}})$ and
$\operatorname{Trop}({\mathbf{X}}_{\sigma})=|\operatorname{star}_{\sigma}{\mathcal{X}}|$
is a union of cones in $\operatorname{star}_{\sigma}{\mathcal{P}}$. Hence, by
4.14 the pair
${\mathbb{R}}\overline{{\mathbf{X}}}_{\sigma}(\varepsilon)\subset{\mathbb{R}}\operatorname{star}_{\sigma}{\mathcal{P}}(\varepsilon)$
is a standard pair of cells. This proves the statement. ∎

We note that the homeomorphisms from [Rau23] used above are relative and
stratified versions of similar results in the literature: In [Par98, Section 3
and 4], similar results are proven in the context retracting a family to the
special fibre. More general statements in the setting of log geometry and the
Kato-Nakayama space appear in [NO10, Theorem 5.1] and [Arg21, Proposition
6.4].

###  4.7. Semi-stable degenerations — tropical side

In this subsection, we consider the pair of spaces
$(\mathfrak{R}{\mathbb{T}}\Sigma,\mathfrak{R}(\overline{X},{\mathcal{E}}))$
arising from patchworking $\overline{X}$ with a real phase structure
${\mathcal{E}}$ from Section 2.4. Even though this pair comes with natural CW
structures from the beginning, it will be convenient to study it via the
tropical analogue of the the semi-stable degeneration
${\mathbb{C}}{\mathcal{P}}$ used in Section 4.6. We will use this approach to
construct a tropical _special fibre_
$({\mathbb{T}}{\mathcal{P}}_{\infty},\overline{X}_{\infty})$. We will
demonstrate a homeomorphism of the pairs
$({\mathbb{T}}{\mathcal{P}}_{\infty},\overline{X}_{\infty})$ and
$({\mathbb{T}}\Sigma,\overline{X})$, see Proposition 4.30. This produces a
homeomorphism of the pairs
$(\mathfrak{R}{\mathbb{T}}\Sigma,\mathfrak{R}(\overline{X},{\mathcal{E}}))$
and
$(\mathfrak{R}{\mathbb{T}}{\mathcal{P}}^{+}_{\infty},\mathfrak{R}(\overline{X},{\mathcal{E}})^{+}_{\infty})$.
We then finish the proof of 4.7 by showing that
$(\mathfrak{R}{\mathbb{T}}{\mathcal{P}}^{+}_{\infty},\mathfrak{R}(\overline{X},{\mathcal{E}})^{+}_{\infty})$,
and hence
$(\mathfrak{R}{\mathbb{T}}\Sigma,\mathfrak{R}(\overline{X},{\mathcal{E}}))$,
is a regular CW pair whose poset structure is the same as the one described in
4.19.

We fix a strongly unimodular subdivision ${\mathcal{P}}$ of ${\mathbb{R}}^{n}$
and denote by $\Sigma=\operatorname{RecCone}({\mathcal{P}})$ its pointed
unimodular recession fan. We denote by
$\widetilde{{\mathcal{P}}}=C({\mathcal{P}})$ the pointed unimodular fan with
support ${\mathbb{R}}^{n}\times{\mathbb{R}}_{\geq}$ described in Section 4.6.
The associated $(n+1)$-dimensional tropical toric variety
${\mathbb{T}}{{\mathcal{P}}}:={\mathbb{T}}\widetilde{{\mathcal{P}}}$ comes
with a canonical projection to ${\mathbb{T}}$. The generic fibre
${\mathbb{T}}{\mathcal{P}}_{t}$, $t\in{\mathbb{R}}$, is a copy of the toric
variety ${\mathbb{T}}\Sigma$. The special fibre over $-\infty$, denoted by
${\mathbb{T}}{{\mathcal{P}}}_{\infty}$, is covered by the tropical toric
varieties associated to the stars of all faces in ${\mathcal{P}}$,

${\mathbb{T}}{{\mathcal{P}}}_{\infty}=\bigcup_{\tau\in{\mathcal{P}}}{\mathbb{T}}\operatorname{star}_{\tau}{\mathcal{P}}.$
(4.4)

Note that by Lemma 4.17 this decomposition is a regular CW complex whose face
poset, via the given identification, is isomorphic to
${\mathcal{P}}^{\text{op}}$ (${\mathcal{P}}$ with reversed partial order).

Given a subcomplex ${\mathcal{X}}\subset{\mathcal{P}}$, we denote by
$\overline{{\mathcal{X}}}$ the closure of ${\mathcal{X}}$ as polyhedral
complex in ${\mathbb{T}}\Sigma$ described in Equation 2.1. The underlying sets
are denoted by $X$ and $\overline{X}$. Similarly, we denote by
$C({\mathcal{X}})=\\{C(\sigma):\sigma\in{\mathcal{X}}\\}$ the associated
subcomplex of $\widetilde{{\mathcal{P}}}$, by $C(X)$ its support and by
$\overline{C({\mathcal{X}})}$ the closure of $C({\mathcal{X}})$ as a
polyhedral complex in ${\mathbb{T}}{\tilde{{\mathcal{P}}}}$.

######  Definition 4.20.

The support of the polyhedral complex

$\overline{{\mathcal{X}}}_{\infty}=\overline{C({\mathcal{X}})}\cap{\mathbb{T}}{{\mathcal{P}}}_{\infty}.$

is the _tropical special fibre_ of $X$ associated to ${\mathcal{X}}$. We
denote it by $\overline{X}_{\infty}$.

######  Example 4.21.

Let us consider the small example of the subdivision ${\mathcal{P}}$ of
${\mathbb{R}}$ with vertices $\\{0,\pm 1\\}$ and $1$-cells
$(-\infty,-1]\cup[-1,0]\cup[0,1]\cup[1,+\infty)$. In Figure 3 we draw the fan
$\tilde{P}=C({\mathcal{P}})$ on the left hand side. If we set
$X=\left\\{-1,0,1\right\\}$, then $\overline{C(X)}$ consists of the three rays
of $\tilde{{\mathcal{P}}}$.

On the right hand side of Figure 3 is the tropical toric variety
${\mathbb{T}}{\tilde{{\mathcal{P}}}}$. The special fibre
${\mathbb{T}}{{\mathcal{P}}}_{\infty}$ is depicted in red. Moreover, the
special fiber $\overline{X}_{\infty}$ is the contained in the closure of
$C(X)\in{\mathbb{T}}{\tilde{{\mathcal{P}}}}$ and consists of the three points
in blue.

$\bullet$$(0,0)$$\bullet$$(-1,1)$$\bullet$$(1,1)$${\mathcal{P}}$$\bullet$$\bullet$$\bullet$$\bullet$
Figure 3. On the left hand side is the fan $C({\mathcal{P}})$ from Example
4.21. The polyhedral subdivision ${\mathcal{P}}$ is drawn at height one of
${\mathbb{R}}^{2}$ and the subcomplex $X$ in ${\mathcal{P}}$ consists of the
three points. On the right is the tropical toric variety
${\mathbb{T}}{\tilde{{\mathcal{P}}}}$ with the special fiber
${\mathbb{T}}{{\mathcal{P}}}_{\infty}$ drawn in red. The special fiber
$\overline{X}_{\infty}$ is contained in ${\mathbb{T}}{{\mathcal{P}}}_{\infty}$
and consists of the three points drawn in blue.

######  Proposition 4.22.

The polyhedral complexes $\overline{{\mathcal{X}}}$ and
$\overline{{\mathcal{X}}}_{\infty}$ are regular CW complexes with supports
$\overline{X}$ and $\overline{X}_{\infty}$, respectively.

This is an immediate consequence of the following lemma.

######  Lemma 4.23.

Let $\sigma\subset{\mathbb{R}}^{n}$ be a strictly unimodular polyhedron
containing a vertex. Set $\rho=\operatorname{RecCone}(\sigma)$. Then
$\overline{\sigma}\subset{\mathbb{T}}U_{\rho}$ is a $\dim(\sigma)$-cell whose
relative boundary $d\overline{\sigma}$ is equal to the union of proper faces
of $\overline{\sigma}$.

###### Proof.

Since $\sigma$ is a strictly unimodular polyhedron containing at least one
vertex, we have $\sigma=\kappa+\rho$, where $\kappa$ is a bounded simplex such
that $T(\kappa)$ and $T(\rho)$ have trivial intersection $\\{0\\}$. Moreover,
the closure of $\sigma$ satisfies $\overline{\sigma}=\kappa+\overline{\rho}$.
Since $\rho$ is homeomorphic to $[0,\infty)^{k}$, and $\overline{\rho}$ is
homeomorphic to $[0,\infty]^{k}$, we have that $\overline{\sigma}$ is
homeomorphic to $\kappa\times[0,\infty]^{k}$, which proves the claim. ∎

Next, we give a description of the combinatorial structure of
$\overline{{\mathcal{X}}}_{\infty}$ in terms of ${\mathcal{X}}$. Given any
poset $P$, we denote by $\operatorname{Int}(P)$ the poset of (closed)
intervals in $P$. Here, an interval is a subset of $P$ of the form
$[a,b]=\\{c\in P:a\leq c\leq b\\}$ for $a\leq b\in P$. The partial order on
$\operatorname{Int}(P)$ is given by inclusion, meaning

$[a,b]\leq[c,d]\Longleftrightarrow c\leq a,b\leq d.$

######  Lemma 4.24.

The cell poset $\overline{{\mathcal{X}}}_{\infty}$ is isomorphic to
$\operatorname{Int}({\mathcal{X}})$.

###### Proof.

By definition, the faces of $\overline{{\mathcal{X}}}_{\infty}$ are the non-
empty sets among

$\overline{C(\sigma)}\cap{\mathbb{T}}\operatorname{star}_{\tau}{\mathcal{P}}$

for $\sigma\in\overline{{\mathcal{X}}}_{\infty}$, $\tau\in{\mathcal{P}}$. But
$\overline{C(\sigma)}\cap{\mathbb{T}}\operatorname{star}_{\tau}{\mathcal{P}}$
is non-empty if and only if $C(\sigma)\cap\operatorname{relint}(C(\tau))$ is
non-empty ([MR, Proposition 3.2.4 (d)]) if and only if $\tau\subset\sigma$.
Hence the claim follows. ∎

######  Remark 4.25.

Applying the construction to ${\mathcal{X}}={\mathcal{P}}$, we obtain a
polyhedral complex $\overline{{\mathcal{X}}}_{\infty}$ whose support is
${\mathbb{T}}{\mathcal{P}}_{\infty}$. By Lemma 4.23 and Lemma 4.24
$\overline{{\mathcal{X}}}_{\infty}$ is a regular CW complex isomorphic as a
poset to $\operatorname{Int}({\mathcal{P}})$. On the other hand, we already
equipped ${\mathbb{T}}{\mathcal{P}}_{\infty}$ with a regular CW structures
given by Equation 4.4 whose cell poset is isomorphic to
${\mathcal{P}}^{\text{op}}$. In fact, it is easy to see that
$\overline{{\mathcal{X}}}_{\infty}$ is a cubical subdivision of
${\mathbb{T}}{\mathcal{P}}_{\infty}$ which, on the level of cell posets,
corresponds to passing from ${\mathcal{P}}^{\text{op}}$ to
$\operatorname{Int}({\mathcal{P}}^{\text{op}})$. Notice that the poset
$\operatorname{Int}({\mathcal{P}}^{\text{op}})$ is in fact equal to
$\operatorname{Int}({\mathcal{P}})$. The following example illustrates this
process. For clarity, we write ${\mathbb{T}}{\mathcal{P}}_{\infty}$ for the CW
complex from Equation 4.4 and
${\mathbb{T}}{\mathcal{P}}_{\infty}^{\text{cub}}$ for the CW complex
$\overline{{\mathcal{X}}}_{\infty}$ when ${\mathcal{X}}={\mathcal{P}}$.

Figure 4. On the left is a strongly unimodular subdivision ${\mathbb{R}}^{2}$.
In the middle the corresponding tropical special fibre, and on the right its
cubical subdivision (the new cells of dimension less than $1$ are colored in
red).

######  Example 4.26.

In Figure 4, we illustrate Remark 4.25 for a strongly unimodular subdivision
${\mathcal{P}}$ of ${\mathbb{R}}^{2}$ (left-hand side of Figure 4). The
tropical special fibre ${\mathbb{T}}{\mathcal{P}}_{\infty}$ from Equation 4.4
is depicted on the middle of the figure. On the right hand-side is depicted
${\mathcal{P}}_{\infty}$, called also
${\mathbb{T}}{\mathcal{P}}_{\infty}^{cub}$. As a cell complex, it is
isomorphic to $\operatorname{Int}({\mathcal{P}})$.

To see that $({\mathbb{T}}{\mathcal{P}}_{\infty},\overline{X}_{\infty})$ and
$({\mathbb{T}}\Sigma,\overline{X})$ are homeomorphic pairs, we will interpret
$\overline{{\mathcal{X}}}_{\infty}$ as a certain type of cubical subdivision
of $\overline{{\mathcal{X}}}$, which is explained next.

A polyhedron $\kappa\in{\mathbb{R}}^{n}$ is _bounded_ if
$\operatorname{RecCone}(\kappa)=\\{0\\}$. The subcomplex of bounded faces of
${\mathcal{X}}$ is denoted by

$\operatorname{Bounded}({\mathcal{X}})=\\{\kappa\in{\mathcal{X}}:\operatorname{RecCone}(\kappa)=\\{0\\}\\}.$

Notice that $\operatorname{Bounded}({\mathcal{X}})$ can be viewed as a
subcomplex of both ${\mathcal{X}}$ and $\overline{{\mathcal{X}}}$. We will now
construct polyhedral complexes refining ${\mathcal{X}}$ and
$\overline{{\mathcal{X}}}$ by first taking a cubical subdivision of
$\operatorname{Bounded}({\mathcal{P}})$ and then extending this subdivision in
a suitable way to infinity. We consider the subposet of
$\operatorname{Int}(\operatorname{Bounded}({\mathcal{X}}))\times\Sigma$ given
by

$P({\mathcal{X}}):=\\{([\lambda,\kappa],\rho)\in\operatorname{Int}(\operatorname{Bounded}({\mathcal{X}}))\times\Sigma:\kappa+\rho\in{\mathcal{X}}\\}.$
(4.5)

######  Definition 4.27.

For any $\gamma\in\operatorname{Bounded}({\mathcal{P}})$, choose a point
$p_{\gamma}$ in the relative interior of $\gamma$. The _bounded-cubical
subdivision of ${\mathcal{X}}$_ is the collection of (not necessarily
rational) polyhedra in ${\mathbb{R}}^{n}$ given by

$\displaystyle\operatorname{BC}({\mathcal{X}})$
$\displaystyle:=\\{F([\lambda,\kappa],\rho):([\lambda,\kappa],\rho)\in
P({\mathcal{X}})\\},$ $\displaystyle F([\lambda,\kappa],\rho)$
$\displaystyle:=\operatorname{Conv}\\{p_{\gamma}:\lambda\subset\gamma\subset\kappa\\}+\rho$
$\displaystyle\subset{\mathbb{R}}^{n}.$

Here, $\operatorname{Conv}$ denotes the convex hull. The _bounded-cubical
subdivision of $\overline{{\mathcal{X}}}$_ is the closure of
$\operatorname{BC}({\mathcal{X}})$ in ${\mathbb{T}}\Sigma$,

$\operatorname{BC}(\overline{{\mathcal{X}}})=\overline{\operatorname{BC}({\mathcal{X}})}.$

We note that since we are only interested in topological properties, the non-
rationality of $\operatorname{BC}({\mathcal{X}})$ does not present a problem.
Alternatively, under the assumption that ${\mathcal{X}}$ is
${\mathbb{Q}}$-rational, which is satisfied in all our applications, the
points $p_{\sigma}$ can be chosen in ${\mathbb{Q}}^{n}$ which makes
$\operatorname{BC}({\mathcal{X}})$ rational.

######  Example 4.28.

In Figure 5, we consider on the left the same subdivision as in Example 4.26.
On the rest of the figure, we illustrate Definition 4.27. In the middle part,
we choose a point (in red) for each bounded cell. On the right hand side is
depicted $\overline{\operatorname{BC}({\mathcal{P}})}$. The right picture is
the same as the one in Figure 5: this is an illustration of Proposition 4.30.

Figure 5. On the left the same subdivision as in Figure 4, on the middle in
red one point for each bounded cell, and on the right the bounded cubical
subdivision.

######  Lemma 4.29.

The collection of polyhedra $\operatorname{BC}({\mathcal{X}})$ is a polyhedral
complex in ${\mathbb{R}}^{n}$ refining ${\mathcal{X}}$ and such that
$\operatorname{RecCone}(\operatorname{BC}({\mathcal{X}}))=\operatorname{RecCone}({\mathcal{X}})$.
The collection of polyhedra $\operatorname{BC}(\overline{{\mathcal{X}}})$ is a
polyhedral complex refining $\overline{{\mathcal{X}}}$. Moreover,
$\operatorname{BC}(\overline{{\mathcal{X}}})$ is a regular CW complex.

###### Proof.

The second phrase follows from the first. In the first statement, the claim
about recession fans and the fact that the cones of
$\operatorname{BC}({\mathcal{X}})$ cover $X$ is obvious by construction. We
focus on showing that $\operatorname{BC}({\mathcal{X}})$ is a polyhedral
complex.

For $\lambda\subset\kappa\in\operatorname{Bounded}({\mathcal{X}})$, we set
$F(\lambda,\kappa):=F([\lambda,\kappa],0)$. By definition, these faces form a
cubical subdivision of $\operatorname{Bounded}({\mathcal{X}})$. The remaining
faces are of the form $F(\lambda,\kappa)+\rho$ for
$\kappa+\rho\in{\mathcal{X}}$. Since $\kappa+\rho$ is strictly unimodular, it
follows that any face of $F(\lambda,\kappa)+\rho$ is of the form
$F(\lambda^{\prime},\kappa^{\prime})+\rho^{\prime}$ where
$\lambda\subset\lambda^{\prime}\subset\kappa^{\prime}\subset\kappa$ and
$\rho^{\prime}\subset\rho$. Again since $\kappa+\rho$ is strictly unimodular,
it is clear that $\kappa^{\prime}+\rho^{\prime}\in{\mathcal{X}}$, hence
$\operatorname{BC}({\mathcal{X}})$ is closed under taking faces.

Let now $F([\lambda,\kappa],\rho)$ and
$F([\lambda^{\prime},\kappa^{\prime}],\rho^{\prime})$ be two faces of
$\operatorname{BC}({\mathcal{X}})$ with non-empty intersection. Since
$F([\lambda,\kappa],\rho)\subset\kappa+\rho$ and
$F([\lambda^{\prime},\kappa^{\prime}],\rho^{\prime})\subset\kappa^{\prime}+\rho^{\prime}$,
the intersection $(\kappa+\rho)\cap(\kappa^{\prime}+\rho^{\prime})$ is a non-
empty face of ${\mathcal{X}}$. Since ${\mathcal{X}}$ is a strictly unimodular,
$(\kappa+\rho)\cap(\kappa^{\prime}+\rho^{\prime})=(\kappa\cap\kappa^{\prime})+(\rho\cap\rho^{\prime})=\kappa^{\prime\prime}+\rho^{\prime\prime}$,
where $\kappa^{\prime\prime}=\kappa\cap\kappa^{\prime}$ and
$\rho^{\prime\prime}=\rho\cap\rho^{\prime}$. Since $F([\lambda,\kappa],\rho)$
and $F([\lambda^{\prime},\kappa^{\prime}],\rho^{\prime})$ intersect with
$\kappa^{\prime\prime}+\rho^{\prime\prime}$, we conclude
$\lambda,\lambda^{\prime}\subset\kappa^{\prime\prime}$ and

$\displaystyle F([\lambda,\kappa],\rho)\cap
F([\lambda^{\prime},\kappa^{\prime}],\rho^{\prime})$
$\displaystyle=F([\lambda,\kappa^{\prime\prime}],\rho)\cap
F([\lambda^{\prime},\kappa^{\prime\prime}],\rho^{\prime})$
$\displaystyle=F([\lambda,\kappa^{\prime\prime}],\rho^{\prime\prime})\cap
F([\lambda^{\prime},\kappa^{\prime\prime}],\rho^{\prime\prime})$
$\displaystyle=F([\lambda^{\prime\prime},\kappa^{\prime\prime}],\rho^{\prime\prime}),$

where $\lambda^{\prime\prime}$ is the smallest face of $\kappa^{\prime\prime}$
containing $\lambda$ and $\lambda^{\prime}$.

For the third statement, we use a simple variation of Lemma 4.23 and conclude
that each of the polyhedra in $\operatorname{BC}({\mathcal{X}})$ are
homeomorphic to balls. ∎

The relationship between $\operatorname{BC}(\overline{{\mathcal{P}}})$ and
${\mathbb{T}}{{\mathcal{P}}}_{\infty}^{\text{cub}}$ is as follows.

######  Proposition 4.30.

There exists a canonical isomorphism of regular CW complexes from
$\operatorname{BC}(\overline{{\mathcal{P}}})$ to
${\mathbb{T}}{{\mathcal{P}}}_{\infty}^{\text{cub}}$ which respects toric
strata and such that for any ${\mathcal{X}}\subset{\mathcal{P}}$ the
subcomplex $\operatorname{BC}(\overline{{\mathcal{X}}})$ is mapped to the
subcomplex $\overline{{\mathcal{X}}}_{\infty}$. In particular, the pairs
$({\mathbb{T}}\Sigma,\overline{X})$ and
$({\mathbb{T}}{{\mathcal{P}}}_{\infty},\overline{X}_{\infty})$ are
homeomorphic.

###### Proof.

We have already established in Proposition 4.22 and Lemma 4.29 that
$\operatorname{BC}(\overline{{\mathcal{P}}})$ and
${\mathbb{T}}{{\mathcal{P}}}_{\infty}^{\text{cub}}$ are regular CW complexes.
In order to constuct a CW homeomorphism, by 4.13 it is sufficient to construct
an isomorphism between their cell posets. We use the short-hand
$B=\operatorname{Bounded}({\mathcal{P}})$.

First we note that since ${\mathcal{X}}$ is strictly unimodular, the poset
${\mathcal{X}}$ is isomorphic to the subposet of $B\times\Sigma$ consisting of
elements $(\kappa,\rho)$ such that $\kappa+\rho\in{\mathcal{X}}$. For the same
reasons, the poset $\operatorname{Int}({\mathcal{X}})$ is isomorphic to the
subposet $P^{\prime}({\mathcal{X}})$ of
$\operatorname{Int}(B)\times\operatorname{Int}(\Sigma)$ consisting of elements
$([\lambda,\kappa],[\eta,\rho])$ such that $\kappa+\rho\in{\mathcal{X}}$, via
the map $([\lambda,\kappa],[\eta,\rho])\mapsto(\lambda+\eta,\kappa+\rho)$.

By Definition 4.27, the face poset of $\operatorname{BC}({\mathcal{X}})$ is
isomorphic to $P({\mathcal{X}})\subset\operatorname{Int}(B)\times\Sigma$ from
Equation 4.5. By Lemma 4.29 the recession fan of
$\operatorname{BC}({\mathcal{X}})$ is $\Sigma$. Hence by Equation 2.2 the face
poset of $\operatorname{BC}(\overline{{\mathcal{X}}})$ is isomorphic to
$P^{\prime}({\mathcal{X}})=\operatorname{Int}(B)\times\operatorname{Int}(\Sigma)$.
By the previous remark $P^{\prime}({\mathcal{X}})$ is isomorphic to
$\operatorname{Int}({\mathcal{X}})$ which by Lemma 4.24 is isomorphic to the
cell poset of $\overline{{\mathcal{X}}}_{\infty}$.

Taking the closure of the bounded cubical subdivision for
${\mathcal{X}}={\mathcal{P}}$ yields a CW homeomorphism between
$\operatorname{BC}(\overline{{\mathcal{P}}})$ and
${\mathbb{T}}{{\mathcal{P}}}_{\infty}^{\text{cub}}$ which clearly has the
desired properties. ∎

We now return to the patchwork setup. Recall that the patchwork of the
tropical toric variety ${\mathbb{T}}{\mathcal{P}}$ is defined by

$\mathfrak{R}{\mathbb{T}}{\mathcal{P}}:=\bigsqcup_{\varepsilon\in{\mathbb{Z}}_{2}^{n+1}}{\mathbb{T}}{\mathcal{P}}({\varepsilon})/\sim.$

We have a canonical map (toric morphism)
$\mathfrak{R}{\mathbb{T}}{\mathcal{P}}\to\mathfrak{R}{\mathbb{T}}$ where the
latter space is just a union of two copies of ${\mathbb{T}}$ glued along
$-\infty$. We denote the special fibre by
$\mathfrak{R}{\mathbb{T}}{\mathcal{P}}_{\infty}$. Given
$(\sigma,\varepsilon)$,
$\sigma\in{\mathcal{P}},\varepsilon\in{\mathbb{Z}}_{2}(\sigma)$, the
assignment
$(\sigma,\varepsilon)\mapsto{\mathbb{T}}\operatorname{star}_{\sigma}{\mathcal{P}}(\varepsilon)\subset\mathfrak{R}{\mathbb{T}}{\mathcal{P}}_{\infty}$
gives $\mathfrak{R}{\mathbb{T}}{\mathcal{P}}_{\infty}$ the structure of a
regular CW complex whose cell poset is isomorphic to

$\\{(\sigma,\varepsilon):\sigma\in{\mathcal{P}},\varepsilon\in{\mathbb{Z}}_{2}(\sigma)\\}.$

This is not quite the space we need: In analogy to the real-oriented blow-up
used in the classical case, we define the _tropical positive special fibre_
$\mathfrak{R}{\mathbb{T}}{\mathcal{P}}_{\infty}^{+}$ as follows. Recall the
poset

$Q({\mathcal{P}})=\\{(\sigma,\varepsilon):\sigma\in{\mathcal{P}},\varepsilon\in{\mathbb{Z}}_{2}(\operatorname{RecCone}(\sigma))\\}$

from Equation 4.2. Note that in comparison to the face poset of
$\mathfrak{R}{\mathbb{T}}{\mathcal{P}}_{\infty}$, in the definition of
$Q({\mathcal{P}})$ we use $\operatorname{RecCone}(\sigma)$ instead of
$\sigma$. The assignment
$(\sigma,\varepsilon)\mapsto{\mathbb{T}}\operatorname{star}_{\sigma}{\mathcal{P}}$
yields a directed system of topological spaces and we define
$\mathfrak{R}{\mathbb{T}}{\mathcal{P}}_{\infty}^{+}$ as the limit. By slight
abuse of notation, we denote the copy of
${\mathbb{T}}\operatorname{star}_{\sigma}{\mathcal{P}}$ labelled by
$(\sigma,\varepsilon)$ as
$\mathfrak{R}{\mathbb{T}}\operatorname{star}_{\sigma}{\mathcal{P}}(\varepsilon)$.
We have a map
$\mathfrak{R}{\mathbb{T}}{\mathcal{P}}_{\infty}^{+}\to\mathfrak{R}{\mathbb{T}}{\mathcal{P}}_{\infty}$
which identifies strata labelled by $(\sigma,\varepsilon)$ and
$(\sigma,\varepsilon^{\prime})$ if $\varepsilon=\varepsilon^{\prime}$ modulo
$T_{{\mathbb{Z}}_{2}}(\sigma)$.

Now let ${\mathcal{X}}\subset{\mathcal{P}}$ be a subcomplex and
${\mathcal{E}}$ be a real phase structure for ${\mathcal{X}}$. Each star
$\operatorname{star}_{\tau}{\mathcal{X}}$ carries an induced real phase
structure $\operatorname{star}_{\tau}{\mathcal{E}}$ given by quotienting each
${\mathcal{E}}(\sigma)$ by ${\mathbb{T}}_{{\mathbb{Z}}_{2}}(\tau)$. We define
the _positive special fibre_
$\mathfrak{R}(\overline{{\mathcal{X}}},{\mathcal{E}})^{+}_{\infty}$ of
$({\mathcal{X}},{\mathcal{E}})$ as the polyhedral complex in
$\mathfrak{R}{\mathbb{T}}{\mathcal{P}}_{\infty}^{+}$ whose intersection with
the stratum
${\mathbb{T}}\operatorname{star}_{\sigma}{\mathcal{P}}(\varepsilon)$ labelled
by $(\sigma,\varepsilon)\in Q({\mathcal{P}})$ is given by