# A study of compatible deformations in non-Archimedean geometry

John Welliaveetil

###### Abstract.

In 2010, Hrushovski–Loeser showed that the Berkovich analytification of a
quasi-projective variety over a non-Archimedean valued field admits a
deformation retraction onto a finite simplicial complex. In this article, we
adapt the tools and methods developed by Hrushovski–Loeser to study if such
deformation retractions can be obtained to be compatible with respect to a
given morphism. Amongst other results, we show that compatible deformation
retractions exist over a constructible partition of the base and prove the
general statement in the case of a morphism of relative dimension 1 where the
target is a smooth connected curve.

###### Contents

  1. 1 Introduction
    1. 1.1 Hrushovski-Loeser spaces
    2. 1.2 Compatible deformations exist generically
      1. 1.2.1 Sketch of the proof of Theorem 1.1
    3. 1.3 When the base is a curve
      1. 1.3.1 When $\phi$ is of relative dimension $1$.
    4. 1.4 Locally trivial morphisms
    5. 1.5 Related work
  2. 2 Hrushovski-Loeser spaces
    1. 2.1 The theory ACVF
    2. 2.2 Definable sets
    3. 2.3 The language $\mathcal{L}_{\mathcal{G}}$.
    4. 2.4 Stably dominated types
      1. 2.4.1 Types
      2. 2.4.2 Definable types
      3. 2.4.3 Stably dominated types
    5. 2.5 The Hrushovski-Loeser space
      1. 2.5.1 Canonical Extensions
      2. 2.5.2 Simple points
  3. 3 The Berkovich space $B_{\mathbf{F}}(X)$
  4. 4 Required tools
    1. 4.1 Preliminary simplifications
    2. 4.2 The inflation homotopy for families
  5. 5 Generic deformations
    1. 5.1 The initial set-up
    2. 5.2 The relative curve homotopy for families
    3. 5.3 Proof of Theorem 1.1
    4. 5.4 The relative tropical homotopy
      1. 5.4.1 Preliminaries
    5. 5.5 Completing the proof of Theorem 1.1
  6. 6 When the base is a curve
    1. 6.1 Initial reductions
    2. 6.2 The inflation homotopy
    3. 6.3 Relative curves homotopy
    4. 6.4 Theorem 6.1 and consequences
      1. 6.4.1 Deformation retractions for curves
  7. 7 Locally trivial morphisms
    1. 7.1 Locally trivial morphisms
      1. 7.1.1 Compatible deformations for $\mathbb{P}^{1}$-bundles

##  1\. Introduction

Over the course of the twentieth century there have been several approaches,
each with its merits, towards developing a theory of geometry over non-
Archimedean valued fields similar to the theory over the complex numbers.
However it was only in the late 1980’s that Berkovich proposed a theory that
provided for the first time non-Archimedean analytic spaces with good
topological properties similar to those that one takes for granted in complex
geometry.

Berkovich proved several important results that shed light on the topological
nature of such analytic spaces. In [1], he showed that Berkovich analytic
spaces are locally compact and locally path connected. Furthermore, there is
an analytification functor that takes varieties over a non-Archimedean
complete field $K$ to Berkovich $K$-analytic spaces and several GAGA type
results that serve to connect the topology of the analytification of the
variety to scheme theoretic properties of the variety. In [2], he showed that
every smooth $K$-analytic space is locally contractible. More generally, if
$X$ is locally isomorphic to a strictly $K$-analytic domain of a smooth
$K$-analytic space then $X$ is locally contractible. The question of local
contractibility and more general questions regarding the nature of the
homotopy type of a general $K$-analytic space remained open until 2010, when
Hrushovski and Loeser used techniques from Model theory to study the homotopy
types of the Berkovich analytification of quasi-projective varieties over a
valued field. In [7], they showed that the homotopy types of such
analytifications were determined by finite simplicial complexes embedded in
the analytifications. Furthermore, they proved that these spaces are locally
contractible and there can be only finitely many distinct homotopy types of
the analytifications of quasi-projective varieties that vary in a family. It
is important to note that they did not require any assumption of smoothness on
the underlying varieties.

The work of Hrushovski and Loeser in [7] offers a theory of non-Archimedean
geometry which enables one to use powerful techniques and tools from Model
theory such as the notion of definability and model theoretic compactness. A
brief overview of [7] can be found in the excellent survey article [3].
Inspired by the results and ideas in [7], we study the extent to which the
deformations of the analytifications of quasi-projective varieties onto finite
simplicial complexes is functorial. More precisely, we study the validity of
Statement 1.

By a variety over a field $F$, we mean a reduced and separated scheme of
finite type over $F$. We fix a field $K$ that is non-Archimedean non-trivially
real valued, algebraically closed and complete.

$\mathbf{Statement}$ $\mathbf{1:}$ _Let $\phi\colon V^{\prime}\to V$ be a flat
surjective morphism between quasi-projective $K$-varieties of finite type.
There exist deformation retractions $H\colon I\times V^{\mathrm{an}}\to
V^{\mathrm{an}}$ and $H^{\prime}\colon I\times V^{\prime\mathrm{an}}\to
V^{\prime\mathrm{an}}$ which are compatible with the morphism
$\phi^{\mathrm{an}}$ i.e. the following diagram commutes._

${I\times V^{\prime\mathrm{an}}}$${V^{\prime\mathrm{an}}}$${I\times
V^{\mathrm{an}}}$${V^{\mathrm{an}}}$$\scriptstyle{H^{\prime}}$$\scriptstyle{id\times\phi^{\mathrm{an}}}$$\scriptstyle{\phi^{\mathrm{an}}}$$\scriptstyle{H}$

_Furthermore, if $e$ denotes the end point of the interval $I$ then the images
of the deformations $\Upsilon:=H(e,V^{\mathrm{an}})$ and
$\Upsilon^{\prime}=H^{\prime}(e,V^{\prime\mathrm{an}})$ are homeomorphic to
finite simplicial complexes._

###  1.1. Hrushovski-Loeser spaces

As mentioned above, in order to study the homotopy types of certain non-
Archimedean analytic spaces, Hrushovski and Loeser introduced a model
theoretic analogue of the Berkovich space. The first order properties of non-
Archimedean valued fields can be described within the theory ACVF of
algebraically closed valued fields. Any model of ACVF is an algebraically
closed valued field $L$ whose value group we denote $\Gamma(L)$ and residue
field we denote $k(L)$. Let $\mathrm{val}\colon L\to\Gamma(L)$ denote the
valuation. Note that we write the group structure on $\Gamma(L)$ additively.

Let $F$ be a valued field. Analogous to the analytification functor, to any
variety $V$ over $F$, Hrushovski–Loeser associate a functor $\widehat{V}$ that
takes a model $L$ of ACVF extending $F$ to the set $\widehat{V}(L)$ of
$L$-definable stably dominated types that concentrate on $V$. The set
$\widehat{V}(L)$ can be endowed with a topology whose pre-basic open sets are
of the form $\\{p\in\widehat{U}|\mathrm{val}(f)_{*}(p)\in O\\}$ where $O$ is
an open subspace of the value group $\Gamma_{\infty}(L)$,
111$\Gamma_{\infty}(L):=\Gamma(L)\cup\\{\infty\\}$. $U$ is a Zariski open
subset of $V\times_{F}L$, $f$ is regular on $U$, $\mathrm{val}(f)\colon
U\to\Gamma_{\infty}$ is the function $u\mapsto\mathrm{val}(f(u))$ and
$\mathrm{val}(f)_{*}$ which is induced by $\mathrm{val}(f)$ maps types on $U$
to types on $\Gamma_{\infty}$. A map $f\colon V^{\prime}\to V$ of
$F$-varieties induces a continuous map
$\widehat{f}\colon\widehat{V^{\prime}}\to\widehat{V}$. Furthermore, we have an
injection $V\hookrightarrow\widehat{V}$ such that the topology induced on $V$
is the valuative topology. More generally, this formalism allows us to define
$\widehat{S}$ when $S$ is any definable subset of $V$ and endow it with the
induced topology.

The space $\widehat{V}$ is closely related to the associated Berkovich space
and homotopies constructed on $\widehat{V}$ induce homotopies on
$V^{\mathrm{an}}$ (cf. [7, §14]). Furthermore, if $L$ is an extension of $F$
that is maximally complete and $\Gamma(L)=\mathbb{R}$ then
$\widehat{V}(L)=V_{L}^{\mathrm{an}}$. Lastly, $\widehat{V}$ is almost always
not definable, it is however strictly pro-$F$-definable i.e. a pro-object in
the category of $F$-definable sets.

As stated previously, one of the goals of [7] is to study the homotopy type of
the space $\widehat{V}$. Theorem 11.1.1 in loc.cit. implies in particular that
given a quasi-projective $F$-variety $V$ there exists a deformation retraction
$H\colon I\times\widehat{V}\to\widehat{V}$ such that the image of $H$ is an
iso-definable subset $\Upsilon\subset\widehat{V}$ that admits an $F$-definable
homeomorphism with a definable subset of $\Gamma_{\infty}^{w}$ where $w$ is a
finite $F$-definable set. The interval $I$ is a generalized interval which
means that it is obtained by glueing copies of $[0,\infty]$ end to end.

###  1.2. Compatible deformations exist generically

In §5, we prove a version of Statement 1 for Hrushovski–Loeser spaces that
holds generically over the base. We show that if $\phi\colon V^{\prime}\to V$
is a morphism between quasi-projective varieties then there exists an open
dense subspace $U\subset V$ such that we have deformations $H^{\prime}$ and
$H$ of $\widehat{\phi^{-1}(U)}$ and $\widehat{U}$ respectively which are
compatible with the morphism $\widehat{\phi}$. Since our result holds only
generically, we do not require that the morphism $\phi\colon V^{\prime}\to V$
be flat. The precise statement is as follows.

######  Theorem 1.1.

Let $\phi\colon V^{\prime}\to V$ be a morphism between quasi-projective
$K$-varieties whose image is dense. Let $G$ be a finite algebraic group acting
on $V^{\prime}$ which restricts to a well defined action along the fibres of
the morphism $\phi$ and $\xi_{i}\colon V^{\prime}\to\Gamma_{\infty}$ be a
finite collection of $K$-definable functions. There exists an open dense
subset $U\subseteq V$ such that if $V^{\prime}_{U}:=\phi^{-1}(U)$ then there
exists a generalized interval $I$, deformation retractions $H\colon
I\times\widehat{U}\to\widehat{U}$ and $H^{\prime}\colon
I\times\widehat{V^{\prime}_{U}}\to\widehat{V^{\prime}_{U}}$ which satisfy the
following properties.

  1. (1)

The images of $H^{\prime}$ and $H$ are iso-definable $\Gamma$-internal subsets
of $\widehat{V^{\prime}_{U}}$ and $\widehat{U}$ respectively. Let
$\Upsilon^{\prime}$ and $\Upsilon$ denote the images of $H^{\prime}$ and $H$
respectively.

  2. (2)

The homotopy $H^{\prime}$ is invariant for the action of the group $G$,
respects the levels of the functions $\xi_{i}$ and is compatible with the
homotopy $H$.

  3. (3)

The homotopy $H^{\prime}$ is Zariski generalizing.

  4. (4)

If the fibres of the morphism $\phi$ are pure over an open dense subset of $V$
and of dimension $n$ then for every $z\in\Upsilon$, $\Upsilon^{\prime}_{z}$ is
pure of dimension $n$ where the notion of dimension for iso-definable
$\Gamma$-internal sets is as introduced in [7, §8.3]. If $Y\subset V$ is an
irreducible component then $\Upsilon\cap Y$ is pure of dimension
$\mathrm{dim}(Y)$.

  5. (5)

If $Z\subset V$ is a Zariski closed subset such that for every irreducible
component $Y$ of $V$, $Z\cap Y$ is strictly contained in $Y$ then
$\widehat{Z}\cap\Upsilon=\emptyset$.

  6. (6)

If the variety $V$ is integral, projective and normal then the deformation
retraction $H$ extends to a deformation retraction
$I\times\widehat{V}\to\widehat{V}$ which preserves the closed subset
$\widehat{V\smallsetminus U}$. Furthermore, if $d_{bord}\colon
V\to\Gamma_{\infty}$ denotes the schematic distance [7, §3.12] to
$V\smallsetminus U$ then $H$ respects $d_{bord}$. (In the event that $V$
satisfies assertions (1)-(5), we assume that (6) is vacuously true.)

Theorem 1.1 implies the following theorem concerning the analytifications of
morphisms between quasi-projective varieties. It can be easily deduced using
[7, Corollary 14.1.6].

######  Corollary 1.2.

Let $\phi\colon V^{\prime}\to V$ be a surjective morphism between quasi-
projective $K$-varieties. There exists a finite partition $\mathcal{V}$ of $V$
into locally closed sub-varieties such that for every $W\in\mathcal{V}$, there
exists a generalized real interval $I_{W}$ and a pair of deformation
retractions

$H^{\prime}_{W}\colon
I_{W}\times{V^{\prime}_{W}}^{\mathrm{an}}\to{V^{\prime}_{W}}^{\mathrm{an}}$

and

$H_{W}\colon I_{W}\times W^{\mathrm{an}}\to W^{\mathrm{an}}$

which are compatible with respect to the morphism
$(\phi_{|V^{\prime}_{W}})^{\mathrm{an}}$ and whose images are homeomorphic to
finite simplicial complexes.

####  1.2.1. Sketch of the proof of Theorem 1.1

If $V=\mathrm{Spec}(K)$ then Theorem 1.1 is essentially [7, Theorem 11.1.1].
Recall that in this case, the construction of the homotopy $H^{\prime}$ is the
composition of inflation and curve homotopies built using the following
commutative diagram

${V^{\prime}_{1}}$${E}$${F}$${V^{\prime}}$${\mathbb{P}^{m}}$${\mathrm{Spec}(K)}$$\scriptstyle{g^{\prime}}$$\scriptstyle{b^{\prime}}$$\scriptstyle{p}$$\scriptstyle{b}$$\scriptstyle{g}$$\scriptstyle{\phi}$

where the square is cartesian, $b$ is the blow up of $\mathbb{P}^{m}$ at a
point, $F:=\mathbb{P}^{m-1}\subset\mathbb{P}^{m}$ and $p$ is the projection.

To prove Theorem 1.1, we make use of the tools developed in [7]. As a first
step, we use Lemma 4.2 to reduce to the case where $V$ is integral, normal and
projective, $V^{\prime}$ is projective and there exists a Zariski open dense
subset $U\subseteq V$ such that the fibres of the morphism $\phi$ are pure and
equidimensional over $U$. In this situation, after shrinking $U$, Lemma 4.3
implies the following diagram

${V^{\prime}_{1U}}$${E\times U}$${F\times
U}$${V^{\prime}_{U}}$${\mathbb{P}^{m}\times
U}$${U}$$\scriptstyle{g^{\prime}_{U}}$$\scriptstyle{b^{\prime}_{U}}$$\scriptstyle{p_{U}}$$\scriptstyle{b_{U}}$$\scriptstyle{g_{U}}$$\scriptstyle{\phi_{U}}$

where the square is cartesian, $b_{U}$ is of the form $b\times\mathrm{id}_{U}$
where $b\colon E\to\mathbb{P}^{m}$ is the blow up at a point and
$p_{U}:=p\times\mathrm{id}_{U}$ where $p\colon E\to F$ is the projection map.

The deformation retraction $H^{\prime}$ as required by Theorem 1.1 is the
composition of homotopies $H^{\prime}_{\Gamma f}\circ H^{\prime}_{bf}\circ
H^{\prime}_{curvesf}\circ H^{\prime}_{inff}\circ H^{\prime}_{inff-primary}$.
The homotopy $H^{\prime}_{curvesf}$ is constructed on the Hrushovski–Loeser
space associated to a suitable constructible subset of
$\widehat{V^{\prime}_{1U}}$ while making use of the fact that
$p^{\prime}_{U}:=p_{U}\circ g^{\prime}_{U}$ is a fibration of curves. We can
extend to a homotopy on the entire space $\widehat{V^{\prime}_{1U}}$ by
composing with an inflation homotopy that makes use of the finite morphism
$g^{\prime}_{U}$ and by choosing an appropriate horizontal divisor over $U$
using Lemma 4.6 that guarantees that outside of this divisor, for every $u\in
U$, the morphism $\widehat{V^{\prime}_{1u}}\to\widehat{E_{u}}$ is a
homeomorphism locally around the simple points. We hence get a homotopy on
$\widehat{V^{\prime}_{1U}}$ such that the fibres of the morphism
$\widehat{p^{\prime}_{U}}$ restricted to this image are $\Gamma$-internal. We
then show as in [7, Theorem 6.4.4] that we can reduce to proving Theorem 1.1
for a certain pseudo-Galois cover of $F\times U$. Note that it is not
automatic that a homotopy of $V^{\prime}_{1U}$ that restricts to a well
defined homotopy on the exceptional divisor descends to a homotopy on
$V^{\prime}_{U}$. To rectify this, we begin with an inflation homotopy
$H^{\prime}_{inff-primary}$ on $\widehat{V^{\prime}_{U}}$ that enables us to
escape the exceptional divisor. The composition of the sequence of inflation
and curve homotopies does not fix their image. We construct as in [7, §11.5] a
tropical homotopy $H^{\prime}_{\Gamma f}$ on an iso-definable
$\Gamma$-internal subset of $\widehat{V^{\prime}_{1U}}$ such that the
composition $H^{\prime}$ is indeed a deformation retraction. At every stage,
if necessary, we shrink the open set $U$.

An important point to note is that the homotopies ($H^{\prime}_{inff-
primary}$), $H^{\prime}_{inff}$, $H^{\prime}_{curvesf}$ and
$H^{\prime}_{\Gamma f}$ move along the fibres of the morphism
$(\widehat{\phi_{U}})$, $\widehat{\phi_{U}\circ b_{U}}$. Hence, we see that
$H^{\prime}$ firsts seeks to deform $\widehat{V^{\prime}_{U}}$ into a subset
whose fibres are $\Gamma$-internal over $U$ followed by a deformation induced
by an appropriate deformation retraction of the base. This is the content of
Proposition 5.2.

###  1.3. When the base is a curve

In §6, we study Statement 1 when the base is a smooth connected curve. While
Theorem 1.1 shows the existence of deformations generically over the base,
Theorem 6.1 proves that if the base is a smooth connected curve then we can
get compatible homotopies over Zariski open neighbourhoods of an arbitrary
point. The precise statement is Theorem 6.1. As before, [7, Corollary 14.1.6]
implies the following analogous statement for Berkovich spaces.

######  Corollary 1.3.

Let $S$ be a smooth connected $K$-curve and $X$ be a quasi-projective
$K$-variety. Let $\phi\colon X\to S$ be a surjective morphism such that every
irreducible component of $X$ dominates $S$. Let $s\in S(K)$. There exists a
Zariski open subset $U\subset S$ containing $s$ and a pair of deformation
retractions

$H^{\prime}\colon I\times X_{U}^{\mathrm{an}}\to X_{U}^{\mathrm{an}}$

and

$H\colon I\times U^{\mathrm{an}}\to U^{\mathrm{an}}$

which are compatible with respect to the morphism $\phi^{\mathrm{an}}$ and
whose images are homeomorphic to finite simplicial complexes. The interval $I$
is a generalized real interval.

One observes from the proof of Theorem 1.1 that the existence of the curve and
inflation homotopies relies on the existence of divisors which are flat over
the base. The reason we can prove the stronger result when the base is of
dimension $1$ is a consequence of the following fact. Let $\phi\colon X\to S$
be as in the statement of Theorem 6.1 and assume in addition that the fibres
of $\phi$ are pure. Let $D\subset X$ be Zariski closed and generically of
codimension 1 and pure in the fibres of $\phi$ i.e. there exists an open set
$U\subset X$ such that if $s\in U$ then $D\cap X_{s}$ is of codimension $1$ in
$X_{s}$. We suppose in addition that if $\eta\in S$ is the generic point of
$S$ then $\overline{D_{\eta}}=D$ where $\overline{D_{\eta}}$ denotes the
Zariski closure of $D_{\eta}$ in $X$. It then follows that $D$ is of
codimension 1 and pure in every fibre of $\phi$. This follows from
dimensionality concerns.

####  1.3.1. When $\phi$ is of relative dimension $1$.

One of the advantages of working within the framework of Hrushovski–Loeser is
the flexibility it allows us when constructing our homotopies. For instance,
the construction of the inflation homotopy in [7, Lemma 10.3.2] makes use of a
suitable _cut-off_ to extend a homotopy to the entire space. This is the
motivation behind Lemma 6.6 which shows how we may extend homotopies that move
along the fibres of a projective morphism in such a way that the extensions
coincide with the original homotopy over a reasonably large subspace of the
base. This result is the key ingredient in our proof of Theorem 6.9 which
verifies Statement 1 when $S$ is a smooth connected curve, the morphism $\phi$
is flat and the fibres of $\phi$ are of dimension $1$. Since the fibres are of
dimension $1$, we go through the steps in the proof of Theorem 6.1 to find
that for every $s\in S$, there exists a neighbourhood $U$ of $s$ and a
homotopy $h_{U}\colon[0,\infty]\times X_{U}\to\widehat{X_{U}/U}$ whose image
is relatively $\Gamma$-internal. We then extend suitable cut-offs of the
homotopies $h_{U}$ to the whole of $X$ using Lemma 6.6. As before, we have the
following analogous result for Berkovich spaces.

######  Corollary 1.4.

Let $S$ be a smooth connected $K$-curve and $X$ be a quasi-projective
$K$-variety. Let $\phi\colon X\to S$ be a surjective morphism such that every
irreducible component of $X$ dominates $S$. We assume in addition that the
fibres of $\phi$ are of dimension $1$. There exists a pair of deformation
retractions

$H^{\prime}\colon I\times X^{\mathrm{an}}\to X^{\mathrm{an}}$

and

$H\colon I\times S^{\mathrm{an}}\to S^{\mathrm{an}}$

which are compatible with respect to the morphism $\phi^{\mathrm{an}}$ and
whose images are homeomorphic to finite simplicial complexes. The interval $I$
is a generalized real interval.

###  1.4. Locally trivial morphisms

In §7, we study the validity of Statement 1 in the context of _locally trivial
morphisms_ of relative dimension $1$. A morphism $\phi\colon X\to S$ is said
to be locally trivial if for every point $s\in S(K)$, there exists a Zariski
open neighbourhood $U$ of $s$ and a $U$-isomorphism $f_{U}\colon X_{U}\to
V\times U$ where $V$ is a quasi-projective $K$-variety that is independent of
$s$ and $U$. In the event that $\widehat{V}$ admits a homotopy whose image is
iso-definable $\Gamma$-internal and the generalized interval over which the
homotopy runs is $[0,\infty]$, we can employ Lemma 6.6 to verify the existence
of compatible homotopies on $\widehat{X}$ and $\widehat{S}$ whose images are
iso-definable and $\Gamma$-internal. This is the content of Corollary 7.2.

In the case when $V=\mathbb{P}^{1}_{K}$, we refer to the morphism as a
$\mathbb{P}^{1}$-bundle. The class of $\mathbb{P}^{1}$-bundles allows us
examples of objects for which relative homotopies can be constructed
explicitly. We choose a system of coordinates on $\mathbb{P}^{1}$. Recall from
[7, §7.5] that if we are given a divisor $D$ on $\mathbb{P}^{1}$ then we have
a canonical deformation retraction of $\widehat{\mathbb{P}^{1}}$ onto the
convex hull of $D$. In the case of a trivial family i.e. if $\phi\colon X\to
S$ is isomorphic over $S$ to $S\times\mathbb{P}^{1}$ and $D\subset X$ is a
divisor that is finite and surjective over $S$, Hrushovski and Loeser
construct deformations of $\widehat{X}$ that move along the fibres of the
morphism $\widehat{\phi}$ and whose image over a simple point $s\in S$ is the
convex hull of the divisor $D_{s}$ in $\widehat{\mathbb{P}^{1}}\times\\{s\\}$.
In Theorem 7.7, we show that the existence of a suitable horizontal divisor
$D$ on $X$ enables us to glue the standard homotopies on each trivializing
chart with respect to the restriction of the divisor $D$ to that chart.

###  1.5. Related work

An interesting variation of Statement 1 can be found in the tropical geometry
literature. In [11], Ulirsch constructs a functorial tropicalization morphism
for fine and saturated log schemes. More precisely, let $k$ be a trivally
valued field. Given a fine and saturated log scheme $X$ which is locally of
finite type over $k$, we can define a generalized cone complex $\Sigma_{X}$
and construct a continuous tropicalization morphism $\mathrm{trop}_{X}\colon
X^{\beth}\to\overline{\Sigma}_{X}$ where $\overline{\Sigma}_{X}$ is the
canonical extension of $\Sigma_{X}$. Theorem 1.1 of loc.cit. proves an
analogue of Statement 1 in this context by showing that if $f\colon
X^{\prime}\to X$ is a morphism of fine and saturated log schemes locally of
finite type over $k$ then we have an induced morphism of generalized cone
complexes $\Sigma(f)\colon\Sigma_{X^{\prime}}\to\Sigma_{X}$ such that
$\Sigma(f)\circ\mathrm{trop}_{X^{\prime}}=\mathrm{trop}_{X}\circ f^{\beth}$.

Acknowledgements: The work presented here is inspired and motivated by the
theory of non-Archimedean geometry introduced in [7]. We are greatly indebted
to the tools and techniques developed in this paper. We are grateful to
François Loeser for his patient explanations of several difficult parts of
loc.cit. and his advice regarding this project. We must also thank Yimu yin
and Andreas Gross for the many discussions. We are grateful as well to Kavli,
IPMU for the support during the writing of the article.

##  2\. Hrushovski-Loeser spaces

Model theory seeks to understand mathematical structures, the first order
sentences that hold true in these structures and those sets which are defined
by first order formulae. Thus, model theory gives us tools applicable in a
variety of settings alongside providing us a framework which permits us to
relate different mathematical structures. The Hrushovski-Loeser space allows
us to make use of these powerful tools to study the topology of objects that
occur in non-Archimedean geometry. This approach which was first introduced in
2010 opens up a completely new perspective when thinking of certain Berkovich
spaces. In what follows, we provide a brief introduction of the Hrushovski-
Loeser space. The details can be found in [7]. We assume that the reader is
familiar with the fundamental notions of model theory such as _languages,
structures, formulae, theories_ and _types._ Chapters 1 - 4 of [8] explain
these concepts perfectly. The following two sections are more or less
reproductions of parts of the paper [13].

###  2.1. The theory ACVF

Hrushovski-Loeser spaces can be thought of as a model theoretic analogue of
Berkovich spaces, developed within the framework of ACVF - the theory of
_algebraically closed non-trivially valued fields_. Note that working within
ACVF does not prevent us from studying objects which defined over valued
fields which are not algebraically closed. Let us now introduce the language
$\mathcal{L}_{k,\Gamma}$ to describe ACVF and then explain why we extend this
language to $\mathcal{L}_{\mathcal{G}}$.

######  Definition 2.1.

The multi-sorted language $\mathfrak{L}_{k,\Gamma}$ _is given by specifying
the following set of data._

  1. (1)

_A set $\mathfrak{S}$ of sorts - consisting of a valued field sort
$\mathrm{VF}$, a residue field sort $k$ and a value group sort $\Gamma$._

  2. (2)

_A set of function symbols
$\mathfrak{F}=\\{+_{\mathrm{VF}},\times_{\mathrm{VF}},-_{\mathrm{VF}},+_{k},\times_{k},-_{k},+_{\Gamma},\mathrm{val},\mathrm{Res}\\}$
which are defined on appropriate sorts. For instance,
$+_{\mathrm{VF}}\colon\mathrm{VF}\times\mathrm{VF}\to\mathrm{VF}$,
$\mathrm{val}\colon\mathrm{VF}^{*}\to\Gamma$,
$\mathrm{Res}\colon\mathrm{VF}^{2}\to k$ and so on._

  3. (3)

_A set of relation symbols $\mathfrak{R}=\\{<\\}$ defined on specific sorts.
For instance, $<\subset\Gamma\times\Gamma$. _

  4. (4)

_A set of constant symbols
$\mathfrak{C}=\\{0_{\mathrm{VF}},1_{\mathrm{VF}},0_{k},1_{k},0_{\Gamma}\\}$
which are sort specific._

ACVF is that $\mathcal{L}_{k,\Gamma}$-theory whose models are algebraically
closed non-trivially valued fields. Formally, we provide the following
definition.

######  Definition 2.2.

_The $\mathcal{L}_{k,\Gamma}$-theory_ $\mathrm{ACVF}$ _consists of the set of
sentences such that if $M$ is a model of $\mathrm{ACVF}$ then $\mathrm{VF}(M)$
is an algebraically closed valued field with valuation $\mathrm{val}$, value
group $\Gamma(M)$ and residue field $k(M)$. It follows that the value group
$\Gamma(M)$ is a non-trivial dense, linear ordered abelian group, the residue
field $k(M)$ is an algebraically closed field, the map
$\mathrm{val}\colon\mathrm{VF}(M)^{*}\to\Gamma(M)$ is a surjective
homomorphism that satisfies the strong triangle inequality i.e. if
$x,y\in\mathrm{VF}(M)^{*}$ then_

$\mathrm{val}(x+y)\geq\mathrm{min}\\{\mathrm{val}(x),\mathrm{val}(y)\\}.$

_Lastly, the function $\mathrm{Res}$ maps $(x,y)\in\mathrm{VF}^{2}$ to the
residue in $k(M)$ of $xy^{-1}$ if $\mathrm{val}(xy^{-1})\geq 0$ and $0$
otherwise, such that $\mathrm{Res}$ is surjective, $\mathrm{Res}(\\_,1)$ is a
homomorphism etc._

_Notation_ : In this section, let $K$ be a model of ACVF. We often abuse
notation and use $K$ itself to denote the valued field $\mathrm{VF}(K)$ and
$k$ to denote the residue field $k(K)$.

######  Remark 2.3.

  1. (1)

_A classical result of A. Robinson states that the completions of ACVF are the
theories $\mathrm{ACVF}_{p,q}$ where the residue field has characteristic $q$,
the valued field has characteristic $p$ and either $p=0$ or $p\neq 0$ and
$q=p$. Complete theories are defined in [8, Definition 2.2.1]._

  2. (2)

_An important point to note is that ACVF admits elimination of quantifiers in
the language $\mathcal{L}_{k,\Gamma}$. Quantifier elimination is defined in
[8, Definition 3.1.1]._

Although the language $\mathcal{L}_{k,\Gamma}$ describes ACVF, it does not
eliminate imaginaries. To rectify this problem we expand the language
$\mathcal{L}_{k,\Gamma}$ by adding certain sorts which we refer to as
_geometric_ sorts. We provide more details below.

###  2.2. Definable sets

We introduce the notion of a definable set as in [7, Section 2.1] which is of
a more geometric flavour than that presented in [8]. Let $\mathcal{L}$ be a
multi-sorted language and $T$ be a complete $\mathcal{L}$-theory which admits
quantifier elimination. We fix a large saturated model $\mathbb{U}$ of $T$ and
assume that any model of $T$ which is of any interest to us will be contained
in $\mathbb{U}$. Furthermore, we can assume that if $A\subset\mathbb{U}$ is of
cardinality strictly less than the cardinality of $\mathbb{U}$ then any type
consisting of formulae defined over $A$ has a realization in $\mathbb{U}$.
This technical condition will be of use to us when we discuss the notion of
definable types in §2.4.2. By a set of parameters, we will mean a _small_
subset of $\mathbb{U}$. Recall that a subset is said to be small if it is of
cardinality strictly smaller than the degree of saturation of $\mathbb{U}$
which we assume to be $\mathrm{card}(\mathbb{U})$.

Let $C\subset\mathbb{U}$ be a parameter set. We extend the language
$\mathcal{L}$ to $\mathcal{L}_{C}$ by adding constant symbols corresponding to
the elements of $C$. We can expand the theory $T$ to $T_{C}$ whose models are
those models of $T$ that contain the set $C$. An $\mathcal{L}_{C}$-formula
$\phi$ can be used to define a functor $Z_{\phi}$ from the category whose
objects are models of $T_{C}$ and morphisms are elementary embeddings to the
category of sets. Suppose that the formula $\phi$ involves the variables
$x_{1},\ldots,x_{m}$ where for every $i\in\\{1,\ldots,m\\}$, $x_{i}$ is
specific to a sort $S_{i}$ respectively. Then given an
$\mathfrak{L}_{C}$-model ${M}$ of T i.e. a model of T that contains $C$, we
set

$\displaystyle Z_{\phi}({M})=\\{\bar{a}\in S_{1}(M)\times\ldots\times
S_{m}(M)|{M}\models\phi(\bar{a})\\}.$

Clearly, $Z_{\phi}$ is well defined.

######  Definition 2.4.

  * •

_A_ $C$-definable set $Z$ _is a functor from the category whose objects are
models of $T_{C}$ and morphisms are elementary embeddings to the category of
sets such that there exists an $\mathfrak{L}_{C}$-formula $\phi$ and
$Z=Z_{\phi}$. _

  * •

_Let $X$ and $Y$ be $C$-definable sets. A morphism $f\colon X\to Y$ is said to
be _$C$ -definable_ if its graph is a $C$-definable set. _

The definable set $Z_{\phi}$ is fully determined by evaluating it on
$\mathbb{U}$ i.e. by the set $Z_{\phi}(\mathbb{U})$.

######  Example 2.5.

_Consider the following examples of three different classes of definable sets
in ACVF using the language $\mathcal{L}_{k,\Gamma}$. Observe that these
objects appear naturally in the study of Berkovich spaces, tropical geometry
and algebraic geometry._

  1. (1)

Let $K$ be a model of ACVF and $A\subset K$ be a set of parameters. Let
$n\in\mathbb{N}$. A semi-algebraic subset of $K^{n}$ is a finite boolean
combination of sets of the form

$\\{x\in K^{n}|\mathrm{val}(f(x))\geq\mathrm{val}(g(x))\\}$

where $f,g$ are polynomials with coefficients in $A$. One verifies that semi-
algebraic sets extend naturally to define $A$-definable sets in ACVF.

  2. (2)

Let $G=\Gamma(K)$ where $K$ is as above. A $G$-rational polyhedron in $G^{n}$
is a finite Boolean combination of subsets of the form

$\\{(a_{1},\ldots,a_{n})\in G^{n}|\sum_{i}z_{i}a_{i}\leq c\\}$

where $z_{i}\in\mathbb{Z}$ and $c\in G$. Such objects extend naturally to
define a $K$-definable subset of $\Gamma^{n}$ where $\Gamma$ is the value
group sort.

  3. (3)

Any constructible subset of $k^{n}$ gives a definable set in a natural way.

######  Remark 2.6.

_Let $F$ be a non-trivially valued field and let $V$ be an $F$-variety. The
variety $V$ defines in a natural way a functor $Z_{V}$ as follows. Let $K$ be
a model of ACVF that extends $F$. We then set_

$Z_{V}(K):=V\times_{F}K(K).$

_We add appropriate sorts to ACVF so that any such functor associated to a
variety is a definable set._

###  2.3. The language $\mathcal{L}_{\mathcal{G}}$.

As mentioned earlier, the theory ACVF does not eliminate imaginaries in the
language $\mathcal{L}_{k,\Gamma}$. In this section, we briefly introduce an
extension $\mathcal{L}_{\mathcal{G}}$ of $\mathcal{L}_{k,\Gamma}$ within which
ACVF eliminates imaginaries. In fact, ACVF also admits quantifier elimination
in the language $\mathcal{L}_{\mathcal{G}}$. The Hrushovski-Loeser spaces are
defined in the language $\mathcal{L}_{\mathcal{G}}$ and require elimination of
imaginaries for some of their fundamental properties, for instance pro-
definability (cf. [7, Lemma 2.5.1, Theorem 3.1.1]).

######  Definition 2.7.

_A theory $T$ is said to_ eliminate imaginaries _if for any model $M\models
T$, any collection of sorts $S_{1},\ldots,S_{m}$ and any $\emptyset$-definable
equivalence relation $E$ on $S_{1}(M)\times\ldots\times S_{m}(M)$ there exists
a definable function $f$ on $S_{1}(M)\times\ldots\times S_{m}(M)$ whose
codomain is a product of sorts and is such that $aEb$ if and only if
$f(a)=f(b)$._

Suppose a complete theory $T$ in a language $\mathcal{L}$ does not eliminate
imaginaries. We can then extend $T$ to a complete theory $T^{eq}$ over a
language $\mathcal{L}^{eq}$ so that $T^{eq}$ eliminates imaginaries. Indeed,
for every $\emptyset$-definable equivalence relation $E$ on a product of sorts
$S_{1}\times\ldots\times S_{m}$, we add a sort to $\mathcal{L}$ corresponding
to the quotient $(S_{1}\times\ldots\times S_{m})/E$ and a function symbol
representing the map $\overline{a}\mapsto\overline{a}/E$. This gives the
language $\mathcal{L}^{eq}$. Every model $M$ of $T$ extends canonically to a
model $M^{eq}$ of $T^{eq}$. The new sorts that were added to $\mathcal{L}$ are
referred to as the imaginary sorts and their elements are called imaginaries.

Given a definable set $X$ in $T$, we can associate to it an element in
$\mathbb{U}^{eq}$ called its _code_ as follows. Suppose

$X(\mathbb{U})=\\{{x}\in\mathbb{U}^{n}|\mathbb{U}\models\phi({x},a)\\}$

where $x$ and $a$ in the definition are tuples. We define an equivalence
relation by setting $y_{1}Ey_{2}$ if $\forall
x(\phi(x,y_{1})\leftrightarrow\phi(x,y_{2}))$. The element $a/E$ belongs to
$\mathbb{U}^{eq}$ and we refer to it as the code of $X$.

The language $\mathcal{L}_{\mathcal{G}}$ is obtained from
$\mathcal{L}_{k,\Gamma}$ by adjoining to it the _geometric sorts_ $S_{n}$ and
$T_{n}$ for $n\geq 1$. The sort $S_{n}$ is the collection of codes for all
free rank $R$-submodules of $K^{n}$ where $R$ is the valuation ring given by
$\\{x\in K|\mathrm{val}(x)\geq 0\\}$. Given $s\in S_{n}$ for some $n$, let
$\Lambda(s)$ denote the corresponding free rank $n$ $R$-submodule of $K^{n}$.
If $\mathcal{M}$ denotes the maximal ideal of $R$ then let $T_{n}$ be the set
of codes for the elements in $\bigcup_{s\in
S_{n}}\Lambda(s)/\mathcal{M}\Lambda(s)$.

###  2.4. Stably dominated types

Let $F$ be a valued field and let $V$ be a $F$-variety. The Hrushovski-Loeser
space $\widehat{V}$ associated to $V$ is the space of stably dominated types
that concentrate on $V$. We begin with a discussion of the notion of a type
followed by that of a definable type which is central to this story. We
closely follow the treatment in §2.3 of [7].

####  2.4.1. Types

Let $\mathcal{L}$ be a language and $T$ be a complete $\mathcal{L}$-theory. If
$z$ is a set of variables, we use $\mathcal{F}_{z}$ to denote the set of
$\mathcal{L}$-formulae with variables in $z$ up to equivalence in the theory
$T$.

######  Definition 2.8.

An $n$-type $p=p(x_{1},\ldots,x_{n})$ _is a subset of
$\mathcal{F}_{\\{x_{1},\ldots,x_{n}\\}}$ such that $p(x_{1},\ldots,x_{n})$ is
satisfiable in $T$ i.e. there exists a model $M$ of $T$ and
$(a_{1},\ldots,a_{n})\in M^{n}$ such that for every $f\in p$, $M\models
f(a_{1},\ldots,a_{n})$. Furthermore, we say that_ the type $p$ is complete _if
for every $\phi\in\mathcal{F}_{\\{x_{1},\ldots,x_{n}\\}}$ either $\phi\in p$
or $\neg\phi\in p$._

######  Remark 2.9.

  1. (1)

_Let $p$ be a complete $n$-type. Let $M$ be a model of $T$ such that there
exists $a:=(a_{1},\ldots,a_{n})$ that satisfies $p$. Since $p$ is complete,
observe that_

$p=\\{f\in\mathcal{F}_{\\{x_{1},\ldots,x_{n}\\}}|M\models f(a)\\}.$

_In this case, we say that $a$ _realizes_ the type $p$. In general, given an
$n$-tuple $\alpha\in\mathbb{U}^{n}$, we write $\mathrm{tp}(\alpha)$ to denote
the type generated by $\alpha$ i.e._

$\mathrm{tp}(\alpha):=\\{f\in\mathcal{F}_{\\{x_{1},\ldots,x_{n}\\}}|\mathbb{U}\models
f(\alpha)\\}.$

  2. (2)

_The definition above concerns types consisting of formulae without
parameters. However, it is natural and necessary to work with types defined
over a set of parameters. Suppose $A\subset\mathbb{U}$ is a small set of
parameters then a complete $n$-type defined over $A$ will be a complete
$n$-type in the language $\mathcal{L}_{A}$. Using this formalism and the
notion of realizations, one sees that types provide us a tool with which to
probe the first order properties of elements in elementary extensions._

######  Example 2.10.

_Let $\mathcal{L}_{r}:=\\{+,-,.,0,1\\}$ denote the language of rings where
$+,-,.$ are the binary function symbols and $0,1$ are the constant symbols.
Let $q$ be a prime number or zero. Let $\mathrm{ACF}_{q}$ denote the theory of
algebraically closed fields of characteristic $q$ i.e. the
$\mathcal{L}_{r}$-theory whose models are algebraically closed fields of
characteristic $q$. In this context, the complete $n$-types in
$\mathrm{ACF}_{q}$ take on a recognizable form from algebraic geometry. _

_Let $L$ be a model of $\mathrm{ACF}_{q}$. Let $L^{\prime}\subset L$ be a
subfield. Let $p=p(x_{1},\ldots,x_{n})$ be a type defined over $L^{\prime}$.
We can associate to $p$ an ideal $I_{p}\subset L^{\prime}[x_{1},\ldots,x_{n}]$
by setting_

$I_{p}:=\\{f\in L^{\prime}[x_{1},\ldots,x_{n}]|f(x_{1},\ldots,x_{n})=0\in
p\\}.$

_It can be shown that the association $p\mapsto I_{p}$ defines a bijection
between the set of complete $n$-types defined over $F$ and the prime ideals of
$L^{\prime}[x_{1},\ldots,x_{n}]$ (cf. [8, Example 4.1.14]). Given an ideal
$I\subset L^{\prime}[x_{1},\ldots,x_{n}]$, let
$Z(I)\subset\mathrm{Spec}(L^{\prime}[x_{1},\ldots,x_{n}])$ denote its
vanishing locus. We say that $p$ is the_ generic type _of the irreducible
closed subvariety $Z(I_{p})$._

####  2.4.2. Definable types

As in the previous section, we work in the complete $\mathcal{L}$-theory $T$.
Let $A\subset\mathbb{U}$ be a small set of parameters. Let $z$ be a set of
variables and $\mathcal{F}^{A}_{z}$ denote the set of $\mathcal{L}_{A}$
formulae with free variables in $z$ upto equivalence in $T_{A}$. Let $p$ be a
complete $n$-type defined over $A$. Since for every
$\phi\in\mathcal{F}^{A}_{z}$, either $\phi\in p$ or $\phi\notin p$, one may
think of $p$ as a Boolean retraction from $\mathcal{F}^{A}_{z}$ to the two
element Boolean algebra. Equivalently, if $U$ is an $A$-definable set whose
formula belongs to $p$, then one can see $p$ as a uniform decision to include
or exclude $A$-definable subsets $V$ of $U$ according to whether $a\in
V(\mathbb{U})$ or $a\notin V(\mathbb{U})$ where $a$ is a realization of $p$.
Note however that these decisions are restricted to only those subsets of $U$
that are $A$-definable. To broaden the scope of this definition to express
decisions on subsets which are not necessarily $A$-definable, we introduce the
notion of a definable type. Before doing so, we discuss an example we hope
will be illuminating.

######  Example 2.11.

_As in Example 2.10, let $q$ be a prime number or zero. Let $\mathcal{L}_{r}$
be the language of rings and $\mathrm{ACF}_{q}$ denote the theory of
algebraically closed fields of characteristic $q$. Let $L$ be a field of
characteristic $q$. In Example 2.10, we showed that there exists a bijection
between the complete $n$-types defined over $L$ and the irreducible sub-
varieties of $\mathbb{A}^{n}_{L}$. Let $p$ denote the complete $1$-type
defined over $L$ that corresponds to $\mathbb{A}^{1}_{L}$. By this we mean
that $Z(I_{p})=\mathbb{A}^{1}_{L}$ where $I_{p}\subset L[x]$ is the ideal
associated to $p$ and defined in Example 2.10._

_Let $L^{\prime}$ be an extension of $L$ and $p^{\prime}$ denote the complete
$1$-type corresponding to $\mathbb{A}^{1}_{L^{\prime}}$. One sees that if one
were to restrict $p^{\prime}$ to $L$ i.e. consider only those formulae in
$p^{\prime}$ with parameters in $L$ then we get $p$. In other words,_

$p^{\prime}_{|L}=p.$

_The geometric object $\mathbb{A}^{1}$ can be defined over any field and a
type cannot fully express this flexibility. We see that the correct notion is
that of a definable type which provides us with a compatible family of types,
each element of which is associated to a model of ACF. _

######  Definition 2.12.

_Let $x=\\{x_{1},\ldots,x_{m}\\}$. An_ $\emptyset$-definable type _is a
function_

$d_{p}x\colon\mathcal{F}_{x,y_{1},\ldots,}\to\mathcal{F}_{y_{1},\ldots,}$

_such that for any $y=\\{y_{1},\ldots,y_{n}\\}$, $d_{p}x$ restricts to a
Boolean retraction $\mathcal{F}_{x,y}\to\mathcal{F}_{y}$. If
$A\subset\mathbb{U}$ is a set of parameters then an $A$-definable type is a
$\emptyset$-definable type in the theory $T_{A}$._

######  Remark 2.13.

_Let $A\subset\mathbb{U}$ be a small subset and let $p$ be an $A$-definable
type._

  1. (1)

_The definable type $p$ provides us with a compatible family of types. Indeed,
if $M$ is a model of $T$ that contains $A$, then $p_{|M}$ is the type defined
over $M$ consisting of those formulae $\phi(x,b_{1},\ldots,b_{r})$ such that
$M\models d_{p}x(\phi)(b_{1},\ldots,b_{r})$._

  2. (2)

_Let $X$ be a definable set. We say that $p$ _concentrates_ on $X$ if all the
realizations of the type $p_{|\mathbb{U}}$ belong to $X$._

  3. (3)

_Let $X$ and $Y$ be $A$-definable sets and $f\colon X\to Y$ be an
$A$-definable map. Let $S^{A}_{def,X}$ denote the set of $A$-definable types
on $X$. The map $f$ induces a map $f_{*}\colon S^{A}_{def,X}\to S^{A}_{def,Y}$
such that $d_{f_{*}(p)}y(\phi(y,z))=d_{p}x(\phi(f(x),z))$._

####  2.4.3. Stably dominated types

The definition of a stably dominated type for a general theory $T$ is slightly
involved and not necessary for our purposes. We may hence restrict our
attention to working within the theory ACVF in the language
$\mathcal{L}_{\mathcal{G}}$.

######  Definition 2.14.

_Let $A\subset\mathbb{U}$ be a set of parameters. A type $p$ over $A$ is said
to be_ almost orthogonal to $\Gamma$ _to $\Gamma$ if for any realization $a$
of $p$, $\Gamma(A(a))=\Gamma(A)$ where $A(a)$ denotes the definable closure of
the set $A\cup\\{a\\}$ and $\Gamma(A):=\Gamma(\mathbb{U})\cap dcl(A)$. An
$A$-definable type $p$ is said to be orthogonal to $\Gamma$ if for every
structure $B$ that contains $A$, the type $p_{|B}$ is almost orthogonal to
$\Gamma$._

A stably dominated type is a definable type which in some sense does not
enlarge the value group sort. More precisely, we mean the following.

######  Proposition 2.15.

[7, Proposition 2.9.1] In $\mathrm{ACVF}$, an $A$-definable type $p$ is
_stably dominated_ if and only if it is orthogonal to $\Gamma$.

######  Remark 2.16.

_Let $A\subset\mathbb{U}$ be a small set of parameters and let $X$ and $Y$ be
$A$-definable sets. Let $g\colon X\to Y$ be an $A$-definable morphism. Recall
that we have a map $g_{*}\colon S^{A}_{def,X}\to S^{A}_{def,Y}$. One checks
that the map $g_{*}$ restricts to a well defined function from the set of
$A$-definable stably dominated types that concentrate on $X$ to the set of
$A$-definable stably dominated types that concentrate on $Y$._

_Let $f\colon X\to\Gamma$ be an $A$-definable function. Suppose $p$ is a
stably dominated type that concentrates on $X$ and is defined over $A$. One
verifies from the definition that $f_{*}(p)$ concentrates at a point i.e. if
$M$ is a model of ACVF then $(f_{*}(p))_{|M}$ contains the formula $x=a$ for
some $a\in\Gamma(A)$._

Stably dominated types in ACVF are controlled by the residue field which is
the _stable_ part of the theory. A precise formulation of this can be found in
[5, §2.2]. For an introduction to stability theory see the chapter by M.
Ziegler in [9].

######  Example 2.17.

_Let $a\in\mathrm{VF}(\mathbb{U})$ and $\alpha\in\Gamma(\mathbb{U})$. Let
$B(a,\alpha)$ denote the definable set such that if $M$ is a model of ACVF
that contains $\\{a,\alpha\\}$ then_

$B(a,\alpha)(M):=\\{x\in M|\mathrm{val}(x-a)\geq\alpha\\}.$

_In other words, $B(a,\alpha)$ is the closed ball around $a$ of radius
$\alpha$._

_We can associate to $B(a,\alpha)$ a stably dominated type called its generic
type which we denote $p_{B(a,\alpha)}$. The definable type $p_{B(a,\alpha)}$
is determined by a definable type concentrated on a geometric object over the
residue field sort in the following sense. Let us suppose $a=0$ and $\alpha=0$
which corresponds to the closed unit ball. Let $\mathrm{red}\colon B(0,0)\to
k$ be the reduction map i.e. $\mathrm{red}(x):=\mathrm{Res}(x,1)$. Let $L$ be
a model of ACVF. We then have that $b$ is a realization of
$(p_{B(a,\alpha)})_{|L}$ if and only if $\mathrm{red}(b)$ is a realization of
the generic type of $\mathbb{A}^{1}_{k(L)}$._

###  2.5. The Hrushovski-Loeser space

We define the Hrushovski-Loeser space and provide a fundamental example of
such an object.

######  Definition 2.18.

(The space $\widehat{X}$) _Let $C\subset\mathbb{U}$ be a small set of
parameters. Let $X$ be a $C$-definable set. We define $\widehat{X}$ to be the
functor from the category whose objects are models of $\mathrm{ACVF}$ that
contain $C$ and morphisms are elementary embeddings to the category of sets
such that if $M$ is a model that contains $C$ then $\widehat{X}(M)$ is the set
of $M$-definable stably dominated types that concentrate on $X$._

Let $F$ be a valued field. Given a quasi-projective $F$-variety $V$, we can
endow the space $\widehat{V}$ with a topology in a natural fashion which is
analogous to the topology of the Berkovich space $V^{\mathrm{an}}$. We first
define the topology on the space $\widehat{V}(\mathbb{U})$ and then restrict
this definition to $\widehat{V}(M)$ where $M$ is any model of $\widehat{V}$.

######  Definition 2.19.

  1. (1)

_A pre-basic open set of $\widehat{V}(\mathbb{U})$ is of the form_

$\\{p\in\widehat{O}|\mathrm{val}(f)_{*}(p)\in W\\}$

_where $O\subset V(\mathbb{U})$ is a Zariski open subset of $V$ with
parameters in $\mathbb{U}$, $f$ a regular function on $O$ defined over
$\mathbb{U}$ and $W$ an open subset of $\Gamma_{\infty}(\mathbb{U})$. Here
$\mathrm{val}(f)$ denotes the definable map given by
$O(\mathbb{U})\to\mathrm{VF}(\mathbb{U})\to\Gamma_{\infty}(\mathbb{U})$. (As
$\mathrm{val}(f)_{*}(p)$ is a stably dominated type on $\Gamma_{\infty}$, it
is constant.)_

  2. (2)

_The set $\widehat{V}(\mathbb{U})\times\Gamma_{\infty}^{l}(\mathbb{U})$ is
given the product topology and if
$X(\mathbb{U})\subset\widehat{V}(\mathbb{U})\times\Gamma_{\infty}^{l}(\mathbb{U})$
then we let $X(\mathbb{U})$ have the subspace topology._

######  Remark 2.20.

_Let $C\subset\mathbb{U}$ be a small set of parameters. Let $f\colon X\to Y$
be a $C$-definable map of $C$-definable sets. The map $f_{*}\colon
S^{C}_{def,X}\to S^{C}_{def,Y}$ restricts to a well defined map
$\widehat{f}\colon\widehat{X}\to\widehat{Y}$. Furthermore, if
$X(\mathbb{U})\subset{V}(\mathbb{U})$ for some variety $V$,
$Y(\mathbb{U})\subset{V^{\prime}}(\mathbb{U})$ for some variety $V^{\prime}$
and $r\in\mathbb{N}$ and $f$ is the restriction of a regular map then the map
$\widehat{f}$ is continuous._

Let $X$ be an $F$-definable subset contained in
$\mathrm{VF}^{n}\times\Gamma_{\infty}^{l}$ for some $n,l\in\mathbb{N}$. If $M$
is a structure that contains $F$ then we endow $\widehat{X}(M)$ with the
topology whose open sets are the $M$-definable open sets of $\widehat{X}$.
Note that it does not make sense to provide $\widehat{X}(M)$ with the subspace
topology via the inclusion $\widehat{X}(M)\subset\widehat{X}(\mathbb{U})$
since the resulting space would be discrete.

Note that if $V$ is an algebraic variety defined over $F$ then the space
$\widehat{V}(M)$ is Hausdorff for every model $M$ of ACVF that extends $F$.

######  Example 2.21.

_Observe that the affine line defines an $\emptyset$-definable set which we
denote $\mathbb{A}^{1}$ (cf. Remark 2.6). As a set
$\widehat{\mathbb{A}^{1}}(\mathbb{U})$ consists of the generic types of balls
$p_{B(a,\alpha)}$ (cf. Example 2.17) where $a\in\mathbb{U}$ and
$\alpha\in\Gamma(\mathbb{U})$ as well as the _simple_ points
$b\in\mathbb{U}$._

_Let $M$ be a model of ACVF. The topology of the space
$\widehat{\mathbb{A}^{1}}(M)$ can be described as follows. Let $m\in M$. By an
open ball around $m$ of radius $\alpha$, we mean the set
$O(m,\alpha):=\\{x\in\mathbb{U}|\mathrm{val}(x-m)>\alpha\\}$. A fundamental
system of open neighbourhoods in $\widehat{\mathbb{A}^{1}}(M)$ of $m$ is given
by the family $\\{\widehat{O(m,\alpha)}(M)\\}_{\alpha\in\Gamma(M)}$. Let
$B:=B(a,\alpha)$ be a closed ball with $\alpha\in\Gamma(M)$. A fundamental
system of open neighbourhoods of $p_{B}$ is given by sets of the form
$\widehat{O}$ where $O$ is a $M$-definable open ball from which finitely many
$M$-definable closed balls are removed and $\widehat{O}$ contains the point
$p_{B}$. This description can be deduced from Holly’s theorem [6] which
effectively says that a definable subset of $\mathrm{VF}$ admits a swiss
cheese decomposition i.e. it is the disjoint union of balls from which
finitely many sub-balls are removed. By a ball in $\mathrm{VF}$, we mean a set
of the form $\\{x\in\mathrm{VF}|\mathrm{val}(x-a)\mbox{ }\kappa\mbox{
}\alpha\\}$ where $\kappa\in\\{>,\geq\\}$ and $\alpha\in\Gamma_{\infty}$. _

_After fixing coordinates, we have the following equality of sets_

$\widehat{\mathbb{P}^{1}}(\mathbb{U})=\widehat{\mathbb{A}^{1}}(\mathbb{U})\cup\\{\infty\\}$

_where $\infty$ is an $\emptyset$-definable point. The space
$\widehat{\mathbb{A}^{1}}(M)$ is an open subspace of
$\widehat{\mathbb{P}^{1}}(M)$. The open neighbourhoods of $\infty$ in
$\widehat{\mathbb{P}^{1}}(M)$ consist of the complements in
$\widehat{\mathbb{P}^{1}}(M)$ of the spaces $\widehat{B(0,\alpha)}(M)$ where
$\alpha\in\Gamma(M)$. _

######  Remark 2.22.

_Let $C\subset\mathbb{U}$ be a set of parameters. Let $\mathrm{Def}_{C}$ be
the category whose objects are those sets which are definable with parameters
in $C$ and morphisms are $C$-definable maps. By a $C$-pro-definable set, we
mean a pro-object in $\mathrm{Def}_{C}$ indexed by a small partially ordered
set. Let $X$ be a $C$-definable set. By [7, Theorem 3.1.1], there exists a
$C$-pro-definable set $E$ such that for every model $M$ of ACVF that contains
$C$, we have a canonical identification_

$\widehat{X}(M)=E(M).$

The main theorem - [7, Theorem 11.1.1] proved by Hrushovski-Loeser in [7]
implies in particular that the homotopy type of the Hrushovski-Loeser space
associated to a quasi-projective variety over a valued field is determined by
a relatively simple object - a definable subset of $\Gamma_{\infty}^{n}$ for
some $n\in\mathbb{N}$. The precise statement is as in Theorem 2.24.

######  Remark 2.23.

_Note that when using the machinery of Hrushovski-Loeser to construct _pro-
definable_ deformation retractions $H\colon I\times\widehat{V}\to\widehat{V}$
onto a $\Gamma$-internal set, where $V$ is a quasi-projective variety of
dimension greater than 1, we can no longer suppose that the interval $I$ is
$[0,\infty]$. It is usually a_ generalized interval _which consists of glueing
end to end copies of $[0,\infty]$. A more in depth, explanation can be found
in [7, §3.9]._

######  Theorem 2.24.

Let $V$ be a quasi-projective variety over a valued field $F$ and let $X$ be a
definable subset of $V\times\Gamma^{l}_{\infty}$ over some base set
$A\subset\mathrm{VF}\cup\Gamma$, with $F=\mathrm{VF}(A)$. Then there exists an
$A$-definable deformation retraction

$H\colon I\times\widehat{X}\to\widehat{X}$

with image an iso-definable subset $\Upsilon$ which is definably homeomorphic
to a definable subset of $\Gamma_{\infty}^{w}$, for some finite $A$-definable
set $w$. One can furthermore require the following additional properties for
$H$ to hold simultaneously.

  1. (1)

Given finitely many $A$-definable functions $\xi_{i}\colon
X\to\Gamma_{\infty}$ with canonical extension
$\widehat{\xi_{i}}\colon\widehat{X}\to\Gamma_{\infty}$, one can choose $H$ to
respect the $\widehat{\xi_{i}}$, i.e. to satisfy
$\widehat{\xi_{i}}(H(t,x))=\widehat{\xi_{i}}(x)$ for all $(t,x)\in
I\times\widehat{X}$. In particular, finitely many definable subsets $U$ of $X$
can be preserved, in the sense that $H$ restricts to a homotopy of
$\widehat{U}$.

  2. (2)

Assume given, in addition, a finite algebraic group $G$ acting on $V$ and
leaving $X$ globally invariant. Then the retraction $H$ can be chosen to be
equivariant with respect to the $G$-action.

  3. (3)

Assume $l=0$. The homotopy $H$ is Zariski generalizing, i.e. for any Zariski
open subset $U$ of $V$, $\widehat{U}\cap X$ is invariant under $H$.

  4. (4)

The homotopy $H$ is such that for every $x\in\widehat{X}$,
$H(e_{I},H(t,x))=H(e_{I},x)$ for every $t$ and $x$.

  5. (5)

One has $H(e_{I},X)=\Upsilon$, i.e. $\Upsilon$ is the image of the simple
points.

  6. (6)

Assume $l=0$ and $X=V$. Given a finite number of closed irreducible
subvarieties $W_{i}$ of V , one can demand that $\Upsilon\cap\widehat{W_{i}}$
has pure dimension $\mathrm{dim}(W_{i})$.

Observe that the theorem not only describes precisely the homotopy type of the
space $\widehat{X}$ but also allows us to construct pro-definable deformation
retractions with considerable flexibility. It is this fact that we will
exploit in §5. Note that all deformation retractions considered and
constructed in this paper satisfy assertions (3), (4) and (5) above.

####  2.5.1. Canonical Extensions

Let $C\subset\mathbb{U}$ be a small set of parameters and let $X$ be a
$C$-definable set. Let $Y$ be a $C$-definable subset and $f\colon
X\to\widehat{Y}$ be a $C$-definable map. In this situation, we can extend the
map $f$ to a well defined map $\widehat{f}\colon\widehat{X}\to\widehat{Y}$
which we call the _canonical extension of $f$_. We do so as follows.

Let $p\in\widehat{X}(M)$ where $M$ is a model of ACVF that contains $C$. Let
$c\models p_{|M}$ and $d\models f(c)_{|M(c)}$ (cf. Remark 2.9). By [7,
Proposition 2.6.5], the type $\mathrm{tp}(cd|M)$ is stably dominated. It
follows that $\mathrm{tp}(d|M)$ is stably dominated as well. We set
$\widehat{f}(p):=\mathrm{tp}(d|M)$. The map $\widehat{f}$ is well defined and
can be shown to be pro-$C$-definable.

When $X\subset\mathbb{P}^{m}\times\Gamma_{\infty}^{n}$, it is natural to ask
what hypothesis must be placed on the definable map $f$ to ensure that the
induced map $\widehat{f}$ is continuous. To this end, we introduce the $v$ and
$g$-topologies on $X$. The $v$-topology is the topology induced by the
valuation on the ground field and is hence a well defined topology. The
$g$-topology on the other hand is a Grothendieck topology and if $U\subset X$
is both $v$ and $g$ open then $\widehat{U}$ is open in $\widehat{X}$.

######  Definition 2.25.

($v$ and $g$ \- open sets) _Let $V$ be an algebraic variety defined over a
valued field $F$. A definable set $U\subset V$ is v-open if it is open for the
valuation topology on $V$. A definable set $G$ is g-open if it is a positive
Boolean combination of Zariski closed and open subsets and sets of the form
$\\{u:\mathrm{val}(f)(u)<\mathrm{val}(g)(u)\\}$ where $f,g$ are regular
functions on some Zariski open set. More generally, if $U$ is a definable
subset of the variety $V$ then a set $W\subset U$ is v-open (g-open) if it is
of the form $U\cap O$ where $O$ is v-open (g-open). If $X\subset
V\times\Gamma_{\infty}^{n}$ is a definable set then $X$ is v-open (g-open) if
its pullback via $\mathrm{id}\times\mathrm{val}$ to $V\times\mathbb{A}^{n}$ is
v-open (g-open). _

######  Remark 2.26.

_Observe that the $v$-topology on $\Gamma$ is discrete while the neighborhoods
of $\infty$ are the same as those defined by the order topology. The
$g$-topology of $\Gamma_{\infty}$ when restricted to $\Gamma$ coincides with
the order topology while the point $\infty$ is isolated. It follows that the
$v+g$ topology which is the topology generated by the class of sets which are
both $v$ and $g$-open, induces the order topology on $\Gamma_{\infty}$. In
general the $v,g$ and $v+g$-open sets are definable. Observe that in the case
of a variety $V$ over a valued field $F$, the collection of $v$-open sets
definable over $F$ generate the valuation topology on $V$. The $g$-topology
however does not necessarily generate a topology._

######  Definition 2.27.

(v-continuity and g-continuity) _Let $V$ be an algebraic variety over a valued
field $F$ or a definable subset of such a variety. A definable function
$h\colon V\to\Gamma_{\infty}$ is called v-continuous (resp. g-continuous) if
the pullback of any v-open (resp. g-open) set is v-open (resp. g-open). A
function $h\colon V\to W$ with $W$ an affine $F$-variety is called
v-continuous (resp. g-continuous) if, for any regular function $f\colon
W\to\mathbb{A}^{1}$ , $\mathrm{val}\circ f\circ h$ is v-continuous (resp.
g-continuous). _

Let $V$ be an algebraic variety and
$W\subset\mathbb{P}^{m}\times\Gamma_{\infty}^{n}$ be a definable set. We endow
$\widehat{W}$ with the subspace topology from
$\widehat{\mathbb{P}^{m}}\times\Gamma_{\infty}^{n}$. Let $f\colon
V\to\widehat{W}$ be a well defined pro-definable function such that for every
open subset $O\subset\widehat{W}$, $f^{-1}(O)$ is $v+g$-open. In this case, we
say that the map $f$ is $v+g$-continuous. By [7, Lemma 3.8.2], $f$ induces a
continuous pro-definable map $\widehat{f}\colon\widehat{V}\to\widehat{W}$.

####  2.5.2. Simple points

Let $V$ be an $A$-definable set. For $x\in V$, the definable type
$\mathrm{tp}(x/\mathbb{U})$ which concentrates on the point $x$ is stably
dominated. It follows that $\mathrm{tp}(x/\mathbb{U})$ is an element of
$\widehat{V}(\mathbb{U})$. We can thus view $V$ as a subset of $\widehat{V}$.
This subset of points in $\widehat{V}$ is called the set of _simple points_.

######  Lemma 2.28.

([7], Lemma 3.6.1) Let $X$ be a definable subset of $\mathrm{VF}^{n}$.

  1. (1)

The set of simple points of $\widehat{X}$ (which we identify with $X$) is an
iso-definable [7, Definition 2.2.2] and relatively definable [7, §2.2] dense
subset of $\widehat{X}$. If $M$ is a model of $\mathrm{ACVF}$ then $X(M)$ is
dense in $\widehat{X}(M)$.

  2. (2)

The induced topology on $X$ agrees with the valuation topology on $X$.

######  Remark 2.29.

_Let $V$ be an algebraic variety and
$W\subset\mathbb{P}^{m}\times\Gamma_{\infty}^{n}$. We endow $\widehat{W}$ with
the subspace topology from
$\widehat{\mathbb{P}^{m}}\times\Gamma_{\infty}^{n}$. Let $f\colon
V\to\widehat{W}$ be a $v+g$-continuous map. Since the simple points are dense
in $V$, we see that there exists exactly one morphism
$\widehat{V}\to\widehat{W}$ that extends $f$._

##  3\. The Berkovich space $B_{\mathbf{F}}(X)$

We provide a model theoretic reinterpretation of the Berkovich space, one for
which a connection with the space of stably dominated types can be easily
made. Our presentation follows [7, Chapter 14].

Let $F$ be a real valued field and let
$\mathbb{R}_{\infty}:=\mathbb{R}\cup\\{\infty\\}$. Let $\mathbf{F}$ denote the
structure defined by the pair $(F,\mathbb{R}_{\infty})$. Let $V$ be a quasi-
projective $F$-variety. As a set, the Berkovich space $B_{\mathbf{F}}(V)$ is
defined as follows.

######  Definition 3.1.

_Let $X$ be an $\mathbf{F}$-definable subset of $V\times\Gamma_{\infty}^{l}$
for some $l\in\mathbb{N}$. Let_ $B_{\mathbf{F}}(X)$ _be the set of almost
orthogonal to $\Gamma$, $\mathbf{F}$-types which concentrate on $X$. The
notion of an almost orthogonal to $\Gamma$ type was introduced in Definition
2.14._

Let $f:X\to\Gamma_{\infty}$ be an $\mathbf{F}$-definable function. Observe
that if $p\in B_{\mathbf{F}}(X)$ is such that $a\models p$ then
$f(a)\in\mathbb{R}_{\infty}$ depends only on the type $p$ i.e. if
$a_{1}\models p$ and $a_{2}\models p$ then $f(a_{1})=f(a_{2})$. We set
$f(p):=f(a)$. Thus we have a well defined function $f\colon
B_{\mathbf{F}}(X)\to\mathbb{R}_{\infty}$.

######  Definition 3.2.

(Topology on $B_{\mathbf{F}}(X)$) _Let $X$ be an $\mathbf{F}$-definable subset
of the $F$-variety $V$. The set $B_{\mathbf{F}}(X)$ is endowed with the
topology generated by pre-basic open sets of the form $\\{q\in
B_{\mathbf{F}}(X\cap U)|\mathrm{val}(f)_{*}(q)\in W\\}$ where $U\subset V$ is
an open affine subspace, $f$ is a regular function on $U$ and
$W\subset\mathbb{R}_{\infty}$ is an open interval._

######  Lemma 3.3.

The spaces $V^{\mathrm{an}}$ and $B_{\mathbf{F}}(V)$ are canonically
homeomorphic.

###### Proof.

This is proved in [7, 14.1]. ∎

We relate the space $B_{\mathbf{F}}(V)$ to the space $\widehat{V}$. Let $L$ be
an algebraically closed, spherically complete valued field which contains
$\mathbf{F}$ as a substructure and whose residue field is the algebraic
closure of the residue field $k(F)$ of $F$ and value group $\Gamma(L)$ is
$\mathbb{R}$. Such a field is unique up to isomorphism over the structure
$\mathbf{F}$. We fix one such copy and call it $F^{max}$.

######  Lemma 3.4.

There exists a surjective continuous function

$\pi_{F,V}\colon\widehat{V}(F^{max})\to B_{\mathbf{F}}(V)$

such that if $X$ is an $\mathbf{F}$-definable subset of $V$ then
$\pi_{F,V}^{-1}(B_{\mathbf{F}}(X))=\widehat{X}(F^{max})$.

###### Proof.

The map is constructed in [7, §14.1]. The surjectivity of $\pi_{F,V}$ and the
assertion regarding $\pi_{F,V}^{-1}(B_{\mathbf{F}}(X))$ is proved in [7, Lemma
14.1.1] and [7, Proposition 14.1.2]. Nonetheless, we repeat the construction
here since the map $\pi_{F,V}$ figures prominently in §6.

Let $p$ be a stably dominated type defined over $F^{max}$ that concentrates on
$V$. Then $p_{|F^{max}}$ is an $F^{max}$-type. Let $\pi_{F,V}(p)$ denote the
$\mathbf{F}$-type defined by those formulae with parameters in $\mathbf{F}$
that are contained in $p_{|F^{max}}$. Let $a\in V$ be such that
$\pi_{F,V}(p)=\mathrm{tp}(a/\mathbf{F})$. We must have that
$\Gamma(\mathbf{F}(a))\subseteq\Gamma(F^{max}(a))=\Gamma(F^{max})=\Gamma(\mathbf{F})$.
It follows that $\pi_{F,V}$ is a well defined function. It can be checked that
it is continuous as well.

We show that $\pi_{F,V}$ is surjective. Let $p\in B_{\mathbf{F}}(V)$ and $a$
be a realization of $p$. By [5, Theorem 12.18 (ii)], the type
$\mathrm{tp}(a|F^{max})$ extends to an $F^{max}$-stably dominated type which
concentrates on $V$, thus defining an element of $\widehat{V}(F^{max})$. ∎

######  Remark 3.5.

_The purpose of this section was to emphasize the extent to which the
Berkovich space and the Hrushovski-Loeser space are closely related. In fact,
when working over certain models of ACVF, the two spaces coincide. Indeed,
using the notation from Lemma 3.4, we see that if $F=F^{max}$ then by [7,
Lemma 14.1.1] $\pi$ is a homeomorphism._

##  4\. Required tools

Our goal in this section is to develop the tools required to prove the
principal results of Sections §5 and §6.

###  4.1. Preliminary simplifications

We use the following lemmas to show that in most situations under
consideration, we may suppose that we have a morphism between projective
varieties whose fibres are pure over some open dense subset of the base.

######  Lemma 4.1.

Let $V$ be a quasi-projective $K$-variety. Let $\phi\colon V^{\prime}\to V$ be
a morphism between quasi-projective $K$-varieties whose image is dense. Let
$G$ be a finite algebraic group acting on $V^{\prime}$ that restricts to a
well defined action along the fibres of the morphism $\phi$ and $\xi_{i}\colon
V^{\prime}\to\Gamma_{\infty}$ be a finite collection of definable functions.
There exists projective $K$-varieties $\overline{V}$, $\overline{V^{\prime}}$
and a finite type surjective morphism
$\overline{\phi}\colon\overline{V^{\prime}}\to\overline{V}$ with the following
properties.

  1. (1)

The varieties $\overline{V^{\prime}}$ and $\overline{V}$ are pure of dimension
$\mathrm{dim}(V^{\prime})$ and $\mathrm{dim}(V)$ respectively.

     * •

In the event that $V$ is a smooth connected $K$-curve, we can suppose
$\overline{V}$ is the unique smooth projective $K$-curve that contains $V$ as
a Zariski open dense subset.

     * •

If $V$ is integral and normal then we can take $\overline{V}$ to be integral
and normal as well.

  2. (2)

There exists embeddings $i\colon V\hookrightarrow\overline{V}$ and
$i^{\prime}\colon V^{\prime}\hookrightarrow\overline{V^{\prime}}$ whose images
are locally closed subspaces.

  3. (3)

The morphism $\overline{\phi}$ extends the induced morphism $\phi\colon
i^{\prime}(V^{\prime})\to i(V)$.

  4. (4)

The fibres of the morphism $\overline{\phi}$ are pure and projective over some
open dense subset of $\overline{V}$. When $V$ is a smooth connected curve and
$\phi$ is flat, the morphism $\overline{\phi}$ is flat over $\overline{V}$ and
all fibres of $\overline{\phi}$ are pure.

  5. (5)

The variety $\overline{V^{\prime}}$ admits an action of the group $G$ that
extends the action of $G$ on $V^{\prime}$. The action of the group $G$
restricts to a well defined morphism on the fibres of $\overline{\phi}$.

  6. (6)

The functions $\xi_{i}$ extend to definable functions
$\xi^{\prime}_{i}\colon\overline{V^{\prime}}\to\Gamma_{\infty}$.

###### Proof.

As the varieties $V$ and $V^{\prime}$ are quasi-projective, there exists
$n,m\in\mathbb{N}$ such that we can identify $V$ and $V^{\prime}$ with locally
closed subspaces of $\mathbb{P}^{n}_{K}$ and $\mathbb{P}^{m}_{K}$
respectively. Let $a\colon V\hookrightarrow\mathbb{P}^{n}_{K}$ and
$a^{\prime}\colon V^{\prime}\hookrightarrow\mathbb{P}^{m}_{K}$ denote the
respective immersions. Let $\overline{V}$ denote a projective variety
contained in $\mathbb{P}^{n}_{K}$ that is pure, of dimension $\mathrm{dim}(V)$
and contains $V$ (cf. [7, §11.2]). In the event that $V$ is a smooth connected
$K$-curve, let $\overline{V}$ be the unique smooth projective $K$-curve that
contains $V$ as a Zariski open dense subset. If $V$ is integral and normal,
then we can replace $\overline{V}$ with its normalization. Note that this
normalization is a projective variety and contains $V$ as an open dense
subset. After increasing $n$ if necesssary, we abuse notation and assume that
we have a closed immersion $\overline{V}\hookrightarrow\mathbb{P}^{n}_{K}$.

Let $A^{\prime}:=(\mathbb{P}_{K}^{m})^{G}$. The group $G$ acts on the
projective variety $A^{\prime}$ in the following fashion. Let
$\mathbf{x}:=(x_{h})_{h\in G}\in A^{\prime}$ and $g\in G$ then we set
$g(\mathbf{x}):=(x_{hg})_{h\in G}$. Let $b^{\prime}\colon
V^{\prime}\hookrightarrow A^{\prime}$ denote the immersion defined by
$v\mapsto(a^{\prime}(hv))_{h\in G}$. Observe that $b^{\prime}$ is a
$G$-equivariant embedding.

We have an embedding $c\colon V^{\prime}\to A^{\prime}\times\overline{V}$
given by $c(v):=(b^{\prime}(v),a(\phi(v)))$. Let $p_{2}\colon
A^{\prime}\times\overline{V}\to\overline{V}$ denote the projection morphism
onto the second coordinate. Observe that $p_{2}$ is a $G$-equivariant morphism
where $\overline{V}$ is endowed with the trivial action. We identify
$V^{\prime}$ with its image in $A^{\prime}\times\overline{V}$ via the
embedding $c$. Likewise, we identify the morphism $\phi$ with the restriction
of $p_{2}$ to $c(V^{\prime})$.

Let $S\subset\overline{V}$ denote the set of images of the generic points of
$V^{\prime}$ for the morphism $\phi$ and the generic points of $\overline{V}$.
Given a point $x\in\overline{V}$, let $A^{\prime}_{x}$ denote the fibre over
$x$ for the morphism $p_{2}$. Recall from [4, Exercise II.3.10] that
$A^{\prime}_{x}$ is homeomorphic to the subspace $p_{2}^{-1}(x)\subset
A^{\prime}\times\overline{V}$. For every point $\eta\in S$, we choose a
$G$-equivariant pure projective variety $X(\eta)\subset A^{\prime}_{\eta}$
that contains $V^{\prime}\cap A^{\prime}_{\eta}$ and whose closure in
$A^{\prime}\times\overline{V}$ is of dimension $\mathrm{dim}(V^{\prime})$ (cf.
[7, §11.2]). It follows that $\bigcup_{\eta\in S}X(\eta)$ is a $G$-invariant
subset of $A^{\prime}\times\overline{V}$. Let $\mathrm{cl}({X(\eta)})$ denote
the Zariski closure of $X(\eta)$ in $A^{\prime}\times\overline{V}$ and we
define $\overline{V^{\prime}}:=\bigcup_{\eta\in S}\mathrm{cl}({X(\eta)})$
endowed with the reduced induced closed subscheme structure. Since $G$
stabilizes $\bigcup_{\eta\in S}\mathrm{cl}({X(\eta)})$, we see that $G$ must
stabilize $\overline{V^{\prime}}$. Hence, $\overline{V^{\prime}}$ is a
projective variety which is pure of $\mathrm{dim}(V^{\prime})$ that contains
$V^{\prime}$ and extends the action of the group $G$.

Observe that if $V$ is a smooth connected curve and $\phi$ is flat, then $S$
as defined above consists of exactly the generic point of $V$. In this
situation, our construction implies in addition that every irreducible
component of $\overline{V^{\prime}}$ dominates $\overline{V}$. It follows that
the morphism $\overline{V^{\prime}}\to\overline{V}$ is flat. The purity of the
fibres follows from [4, Corollary III 9.6].

Let us now return to the general situation. Using [4, Corollary III.9.6] and
[10, Tag 052B], we deduce that the fibres of the restriction
${p_{2}}_{|\overline{V^{\prime}}}\colon\overline{V^{\prime}}\to\overline{V}$
are pure and equidimensional over an open dense subset of $\overline{V}$.
Observe that the morphism ${p_{2}}_{|\overline{V^{\prime}}}$ is
$G$-equivariant. We define
$\overline{\phi}:={p_{2}}_{|\overline{V^{\prime}}}$. The functions
$\xi_{i}\colon V^{\prime}\to\Gamma_{\infty}$ extend to definable functions on
$\overline{V^{\prime}}$ by setting $\xi^{\prime}_{i}(v):=\xi_{i}(v)$ for $v\in
V^{\prime}$ and $\infty$ otherwise. ∎

######  Lemma 4.2.

Suppose that Theorem 1.1 and Proposition 5.2 are true in the following
situation.

  1. (1)

Let $G$ be a finite algebraic group acting on projective $K$-varieties
$V^{\prime}$ and $V$ where the action on $V$ is trivial.

  2. (2)

The variety $V$ is integral and normal.

  3. (3)

We are given a $G$-equivariant morphism $\phi\colon V^{\prime}\to V$ whose
fibres are pure over some dense open subset $U\subset V$.

  4. (4)

We are given a finite family $\\{\xi_{i}\colon
V^{\prime}\to\Gamma_{\infty}\\}_{i\in I}$ of $K$-definable functions.

Then Theorem 1.1 and Proposition 5.2 are true in general.

###### Proof.

We first consider the case of Theorem 1.1. Let the data be as given in Theorem
1.1. That is to say, let $V$ be a pure quasi-projective $K$-variety and let
$\phi\colon V^{\prime}\to V$ be a morphism between quasi-projective varieties
with dense image. Let $G$ be a finite algebraic group acting on $V^{\prime}$
which restricts to a well defined action along the fibres of the morphism
$\phi$ and $\xi_{i}\colon V^{\prime}\to\Gamma_{\infty}$ be a finite collection
of $K$-definable functions.

We deduce without difficulty that we can assume $V$ is integral and normal. By
Lemma 4.1, there exists projective $K$-varieties $\overline{V}$,
$\overline{V^{\prime}}$ and a finite type surjective morphism
$\overline{\phi}\colon\overline{V^{\prime}}\to\overline{V}$ satisfying
conditions (1)-(6) of 4.1. Observe from our construction in the proof of Lemma
4.1 that the variety $V^{\prime}$ is not necessarily dense in
$\overline{V^{\prime}}$. Let $V^{\prime}_{1}$ denote the closure of
$V^{\prime}$ in $\overline{V^{\prime}}$. In our situation, since $V^{\prime}$
is locally closed, it is open in $V^{\prime}_{1}$. To the family of definable
functions
$\\{\xi^{\prime}_{i}\colon\overline{V^{\prime}}\to\Gamma_{\infty}\\}_{i}$
which extend the functions $\xi_{i}$, we add the valuations of the
characteristic functions of the closed subvariety
$V^{\prime}_{1}\subset\overline{V^{\prime}}$ which we denote $\epsilon_{j}$
i.e. the family $\\{\epsilon_{j}\\}_{j}$ is such that
$\bigcap_{j}\epsilon_{j}^{-1}(\infty)=V^{\prime}_{1}$. The hypothesis of the
lemma implies the existence of a Zariski dense open subset
$W\subset\overline{V}$ such that if $W^{\prime}:=\phi^{-1}(W)$ then there
exists compatible deformation pairs $(H^{\prime},\Upsilon^{\prime})$ on
$\widehat{W^{\prime}}$ and $(H,\Upsilon)$ on $\widehat{W}$ satisfying
assertions (1)-(6) of Theorem 1.1. This implies in particular that the
deformation $H^{\prime}$ restricts to a deformation of
$\widehat{W^{\prime}\cap V^{\prime}_{1}}$. The set $W^{\prime}\cap V^{\prime}$
is open in $V^{\prime}_{1}$. Theorem 1.1 asserts that the deformation
$H^{\prime}$ is Zariski generalizing. Every Zariski open subset of
$W^{\prime}\cap V^{\prime}_{1}$ is of the form $O\cap W^{\prime}\cap
V^{\prime}_{1}$ where $O$ is a Zariski open subset of $W^{\prime}$. It follows
that the restriction of the deformation $H^{\prime}$ to
$\widehat{W^{\prime}\cap V^{\prime}_{1}}$ is Zariski generalizing as well.
This implies that $H^{\prime}$ restricts to a well defined deformation of
$\widehat{W^{\prime}\cap V^{\prime}}$. Recall that the map $V^{\prime}\to V$
is a well defined morphism of quasi-projective varieties whose image is dense.
We can hence shrink $W$ so that $W^{\prime}\cap V^{\prime}\to W\cap V$ is
surjective. It follows that $H$ restricts to a well defined deformation of
$\widehat{W\cap V}$. The restrictions of the deformations $H^{\prime}$ and $H$
to $W^{\prime}\cap V^{\prime}$ and $W\cap V$ must be compatible since
$H^{\prime}$ and $H$ are compatible. The proof in the case of Proposition 5.2
is an easy adaptation of the arguments above. ∎

######  Lemma 4.3.

Let $f\colon V^{\prime}\to V$ be a projective morphism of $K$-varieties such
that the fibres of $f$ are pure of dimension $m$. Let $G$ be a finite
algebraic group acting on $V^{\prime}$ such that the morphism $f$ is
$G$-equivariant when $V$ is endowed with the trivial action. For every $v\in
V(K)$, there exists a Zariski open neighbourhood $U\subset V$ of $v$ such that
the morphism $f\colon f^{-1}(U)\to U$ factors through a finite surjective
$G$-equivariant morphism $p\colon f^{-1}(U)\to\mathbb{P}^{m}\times U$ over $U$
where the $G$-action on $\mathbb{P}^{m}\times U$ is taken to be trivial.

Suppose we are given a horizontal divisor 222We say that $D$ is a horizontal
divisor if it does not contain any irreducible component of any fibre of the
morphism $f$. $D\subset V^{\prime}_{U}$. There then exists a $K$-point
$z\in\mathbb{P}^{m}$ such that after shrinking $U$ if necessary but
maintaining that it contains $v$, we have the following commutative diagram

${V^{\prime}_{1U}}$${E\times U}$${\mathbb{P}^{m-1}\times
U}$${V^{\prime}_{U}}$${\mathbb{P}^{m}\times
U}$${U}$$\scriptstyle{p^{\prime}}$$\scriptstyle{b^{\prime}}$$\scriptstyle{b\times
id}$$\scriptstyle{q}$$\scriptstyle{f}$$\scriptstyle{p}$

where $b\colon E\to\mathbb{P}^{m}$ is the blow up at the point $z$, the
restriction of $(q\circ p^{\prime})$ to $D$ is finite surjective onto
$\mathbb{P}^{m-1}\times U$ and the square in the diagram is cartesian.

###### Proof.

This is a relative version of [7, Lemma 11.2.1]. We may replace $V^{\prime}$
with $V^{\prime}/G$ and assume that the action of $G$ is trivial. Let $v\in
V$. Let $n\in\mathbb{N}$ be the smallest natural number greater than or equal
to $m$ such that there exists a Zariski open neighbourhood $U$ of $v$ and the
morphism $f$ factors through a finite morphism $i\colon
V^{\prime}_{U}\rightarrow\mathbb{P}^{n}\times U$. The fact that there exists
such an $n$ is because $f$ is projective. Let $V^{\prime}_{v}$ denote the
fibre over $v$. If $m=n$ then we have nothing to prove. Suppose $n>m$. Let
$(z,v)\in(\mathbb{P}^{n}\times U)(K)$ be a point that is not contained in
$i(V^{\prime}_{v})$. Let $C:=i(V^{\prime}_{U})\cap(\\{z\\}\times
U)\subset\mathbb{P}^{n}\times U$. Observe that $C$ is a closed subset of
$\mathbb{P}^{n}\times U$ and hence $p_{2}(C)$ is a closed subset of $U$ where
$p_{2}$ is the projection $\mathbb{P}^{n}\times U\to U$. Furthermore, our
choice of $z\in\mathbb{P}^{n}$ implies that $v\notin p_{2}(C)$. We abuse
notation and call the complement of $C$ in $U$, $U$ as well. By construction,
for every $u\in U$, $(z,u)\notin i(V^{\prime}_{u})$. Let
$p\colon\mathbb{P}^{n}\smallsetminus\\{z\\}\to\mathbb{P}^{n-1}$ denote the
projection through the point $z$. It follows that the map
$p\times\mathrm{id}\colon(\mathbb{P}^{n}\smallsetminus\\{z\\})\times
U\to\mathbb{P}^{n-1}\times U$ restricts to a finite morphism $i_{1}\colon
V^{\prime}_{U}\to\mathbb{P}^{n-1}\times U$. This contradicts our assumption
that $n$ was minimally chosen.

We now prove the second part of the Lemma. Let $D\subset V^{\prime}_{U}$ be a
horizontal divisor. Let $z\in\mathbb{P}^{m}(K)\times\\{v\\}$ be a point which
is not contained in the image $p(D)$. We now argue as before. Let
$C^{\prime}:=p(D)\cap(\\{z\\}~\times U)\subset\mathbb{P}^{m}\times U$. We have
that $C^{\prime}$ is closed and hence $p_{2}(C^{\prime})\subset U$ is a closed
subspace that does not contain $v$. We shrink $U$ so that it does not
intersect $p_{2}(C^{\prime})$. Let $b\colon E\to\mathbb{P}^{m}$ be the blow up
at the point $z$. Let
$V^{\prime}_{1U}:=V^{\prime}_{U}\times_{(\mathbb{P}^{m}\times U)}(E\times U)$.
We thus have the diagram above and the remaining assertions can be checked
from the construction. ∎

######  Remark 4.4.

_Observe in Lemma 4.3 that $V^{\prime}_{1U}$ comes equipped with a natural
action by the group $G$ and the morphism $b^{\prime}\colon V^{\prime}_{1U}\to
V^{\prime}_{U}$ is $G$-equivariant._

######  Lemma 4.5.

Let $F$ be a valued field. Let $f\colon V^{\prime}\to V$ be a finite
surjective morphism between pure $K$-varieties where $V$ is assumed to be
normal. Assume there exists a proper closed subset $D\subset V$ which
satisfies the following property.

  * •

If $n$ denotes the supremum of the values $\mathbf{card}\\{f^{-1}(v)\\}$ as
$v$ varies along $V$ then for every $u\in U:=V\smallsetminus D$,
$n=\mathbf{card}\\{f^{-1}(u)\\}$.

We then have that for every $v\in U$ and every point $v^{\prime}\in
f^{-1}(v)$, there exists definable sets $W\subset{V}$ containing $v$ and
$W^{\prime}\subset V^{\prime}$ containing $v^{\prime}$ such that $\widehat{W}$
and $\widehat{W^{\prime}}$ are open in $\widehat{V}$ and
$\widehat{V^{\prime}}$ respectively and $\widehat{f}$ restricts to a
homeomorphism from $\widehat{W^{\prime}}$ onto $\widehat{W}$.

###### Proof.

Let $F$ be a model of ACVF such that $v\in U(F)$. Let $U^{\prime}:=f^{-1}(U)$.
For every $v^{\prime}\in U^{\prime}$ such that $v^{\prime}\mapsto v$ we choose
a suitably small $F$-definable $v+g$-open subset
$W^{\prime\prime}_{v^{\prime}}\subset{U^{\prime}}$ such that if
$v^{\prime}_{1}\neq v^{\prime}_{2}$ are distinct preimages of $v$ then
$W^{\prime\prime}_{v^{\prime}_{1}}\cap
W^{\prime\prime}_{v^{\prime}_{2}}=\emptyset$. Let $W$ be the $F$-definable set
$\bigcap_{v^{\prime}\in f^{-1}(v)}{f}(W^{\prime\prime}_{v^{\prime}})$. By [7,
Corollary 9.7.4], the morphism $\widehat{f}$ is open. Furthermore, since $f$
is proper, [7, Lemma 4.2.26] implies that $\widehat{f}$ is a definably closed
map. It follows that $\widehat{W}$ is an pro-$F$-definable open neighbourhood
of $v$. For any $v^{\prime}\mapsto v$, let
$W^{\prime}_{v^{\prime}}:={f}^{-1}(W)\cap W^{\prime\prime}_{v^{\prime}}$.

We claim that ${f}^{-1}(W)=\bigcup_{v^{\prime}\in
f^{-1}(v)}W^{\prime}_{v^{\prime}}$. By definition of
$W^{\prime}_{v^{\prime}}$, we see that $\bigcup_{v^{\prime}\in
f^{-1}(v)}W^{\prime}_{v^{\prime}}\subseteq{f}^{-1}(W)$. Hence, we are left to
show that if $w^{\prime}\in{U^{\prime}}$ is such that ${f}(w^{\prime})\in W$
then for some $v^{\prime}\in f^{-1}(v)$, $w^{\prime}\in
W^{\prime}_{v^{\prime}}$. Let $w:={f}(w^{\prime})$. Since $w\in U$,
$\mathbf{card}({f}^{-1}(w))=n$ and by construction for every $v^{\prime}\in
f^{-1}(v)$, there exists $w^{\prime}_{v^{\prime}}\in W^{\prime}_{v^{\prime}}$
such that ${f}(w^{\prime}_{v^{\prime}})=w$. Since the
$W^{\prime}_{v^{\prime}}$ are mutually disjoint, the points
$w^{\prime}_{v^{\prime}}$ must account for all the preimages of $w$. This
implies that for some $v^{\prime}$, $w^{\prime}_{v^{\prime}}=w$. We have thus
verified the claim.

Observe that for every $v^{\prime}\in f^{-1}(v)$, $f$ restricts to a bijection
from $W^{\prime}_{v^{\prime}}$ to $W$. By [7, Lemma 4.2.6], this implies that
$\widehat{f}$ restricts to a bijection
$\widehat{W^{\prime}_{v^{\prime}}}\to\widehat{W}$. The morphism $\widehat{f}$
is clopen when restricted to $\bigcup_{v^{\prime}\in
f^{-1}(v^{\prime})}\widehat{W^{\prime}_{v^{\prime}}}$. By construction, for
every $v^{\prime}$, $\widehat{W^{\prime}_{v^{\prime}}}$ is open. It follows
that $\widehat{f}$ restricts to a homeomorphism from each
$\widehat{W^{\prime}_{v^{\prime}}}$ onto $\widehat{W}$. ∎

In the following lemma, when we write $v\in V$ where $V$ is a quasi-projective
variety defined over a field $F$, we mean $v\in V(\mathbb{U}^{\prime})$ where
$\mathbb{U}^{\prime}$ where $\mathbb{U}^{\prime}$ is chosen to be a suitable
universal domain for the theory ACF.

######  Lemma 4.6.

Let $F$ be a field and let $\phi\colon V^{\prime}\to V$ be a morphism of
quasi-projective $F$-varieties whose fibres are pure of dimension $m$ for some
$m\in\mathbb{N}$. We assume that the morphism $\phi$ factors through a finite
surjective morphism $f\colon V^{\prime}\to P\times V$ via the projection
$p\colon P\times V\to V$ where $P$ is an irreducible $F$-variety. There then
exists a Zariski open dense subset $U\subseteq V$ and a Zariski closed subset
$T\subset P\times U$ satisfying the following properties.

  1. (1)

The restriction $p_{|T}\colon T\to U$ is flat.

  2. (2)

Given $u\in U$, let $n(u)$ denote the supremum of the set
$\\{\mathbf{card}\\{f^{-1}(x,u)\\}|x\in P\\}$. We then have that

$\\{(x,u)\in P\times U|\mathbf{card}\\{f^{-1}(x,u)\\}<n(u)\\}\subseteq T.$

  3. (3)

When $V$ is a smooth curve and the morphism $\phi$ is flat, we can take $U=V$.

###### Proof.

We can assume at the outset that $V$ is integral. We begin by defining an ACF-
definable set $E\subset P\times V$ as follows.

Let $M$ denote the supremum of the set
$\\{\mathbf{card}\\{f^{-1}(x,u)\\}|(x,u)\in P\times V\\}$. There exists an
ACF-definable partition $S_{1},\ldots,S_{M}$ of $V$ such that if $s\in S_{i}$
then

$\mathrm{sup}\\{\mathbf{card}\\{f^{-1}(x,s)\\}|x\in P\\}=i.$

Let $E^{\prime}=\bigcup_{1\leq i\leq M}E^{\prime}_{i}$ where $E^{\prime}_{i}$
is the ACF-definable subset of $P\times S_{i}$ consisting of the set of pairs
$(x,s)$ such that

$\mathbf{card}\\{f^{-1}(x,s)\\}=i.$

Observe that $E^{\prime}$ is $F$-definable. Let $E$ be the complement of
$E^{\prime}$ in $P\times V$. Hence $E$ is a constructible subset of $P\times
V$ and for every $v\in V$, $E_{v}$ is of dimension strictly smaller than $m$.
Hence $\mathrm{dim}(E)<\mathrm{dim}(V)+m$. Let $\overline{E}$ denote the
Zariski closure of $E$ in $P\times V$.

We now show how to choose $U$ as required by the lemma. Let $Z\subset V$ be
the set of points $v\in V$ such that $\overline{E}$ contains $P\times\\{v\\}$.
The set $Z$ is ACF-definable. Since the dimension of $E$ is strictly smaller
than $\mathrm{dim}(V)+m$, there exists a Zariski open subset $U$ of $V$ that
is disjoint from $Z$. Let $T:=\overline{E}\cap(P\times U)$. We now shrink $U$
further if necessary so that the restriction ${p}_{|T}\colon T\to U$ is flat.
This verifies assertions (1) and (2) of the lemma.

Suppose that $V$ is a smooth curve and the morphism $\phi$ is flat. Since
$\mathrm{dim}(E)\leq m$, we see that $\overline{E}$ cannot contain any fibre
of the form $P\times\\{v\\}$ for some $v\in V$. Indeed, if $\overline{E}$
contains $P\times\\{v\\}$ for some $v\in V$ then $P\times\\{v\\}$ must be an
irreducible component of $\overline{E}$ and $E\cap P\times\\{v\\}$ must be
dense in $P\times\\{v\\}$. This is not possible because
$\mathrm{dim}(E_{v})<m$.

If $Y\subset\overline{E}$ is an irreducible component whose image via $p$ is a
closed point $v\in V(F)$ then we can identify $Y$ with a Zariski closed subset
of $P_{v}$ and write $T(Y):=Y\times V$. If on the other hand $Y$ is an
irreducible component of $\overline{E}$ which surjects onto $V$ via $p$ then
we set $T(Y):=Y$. If $Y_{1},\ldots,Y_{m}$ are the irreducible components of
$\overline{E}$ then let $T:=\bigcup_{i}T(Y_{i})$. By construction $T$
satisfies assertions (1) and (2) when $U=V$.

∎

###  4.2. The inflation homotopy for families

######  Lemma 4.7.

Let $V$ be a quasi-projective variety over a valued field $F$. Let $m\colon
V\times V\to\Gamma_{\infty}$ be a definable metric as constructed in the proof
of [7, Lemma 3.10.1]. Let $D\subset V$ be a $v$-closed subset and $x\in
V\smallsetminus D$. Then the set $\\{m(x,d)|d\in D\\}\subset\Gamma$ has a
supremum in $\Gamma$.

###### Proof.

Observe that the definable metric as constructed in the proof of [7, Lemma
3.10.1] is such that for any $z\in V$ and $\gamma\in\Gamma$, if
$B(z,\gamma):=\\{y\in V|m(y,z)\geq\gamma\\}$ then the family
$\\{B(z,\gamma)\\}_{\gamma}$ is a fundamental system of $v$-open
neighbourhoods of $y$ in $V$. Hence, if $\\{m(x,d)|d\in D\\}$ does not have a
supremum in $\Gamma$ then we see that $x$ must be a limit point of $D$ for the
$v$-topology. However, this is not possible since $D$ is $v$-closed. ∎

######  Lemma 4.8.

Let $F$ be a valued field and let $\phi\colon V^{\prime}\to V$ be a morphism
of quasi-projective $F$-varieties which satisfies the following properties.

  1. (1)

The fibres of the morphism $\phi$ are pure of dimension $m$ for some
$m\in\mathbb{N}$.

  2. (2)

There exists a proper closed subset $D^{\prime}\subset V^{\prime}$ such that
we have a map $f\colon V^{\prime}\smallsetminus
D^{\prime}\to\mathbb{A}^{m}\times V$. Furthermore, for every $v\in V$ and
$x\in\mathbb{A}^{m}\times\\{v\\}$, if $x^{\prime}\in f^{-1}(x)$ then there
exists definable sets $W\subseteq{\mathbb{A}^{m}\times\\{v\\}}$ containing $x$
and $W^{\prime}\subseteq V^{\prime}_{v}$ containing $x^{\prime}$ such that
$\widehat{W}$ and $\widehat{W^{\prime}}$ are open in
$\widehat{\mathbb{A}^{m}\times\\{v\\}}$ and $\widehat{V^{\prime}_{v}}$
respectively and $\widehat{f}_{v}$ restricts to a homeomorphism from
$\widehat{W^{\prime}}$ onto $\widehat{W}$.

  3. (3)

Let $\\{\xi_{i}\colon V^{\prime}\smallsetminus
D^{\prime}\to\Gamma_{\infty}\\}_{i}$ be a finite family of $v$-continuous
functions which are definable over $F$. Furthermore, for every $i$ and every
$v\in V$, $\xi_{i}^{-1}(\infty)\cap V^{\prime}_{v}$ is either empty or the
union of irreducible components of $V^{\prime}_{v}$.

  4. (4)

Let $G$ be a finite group acting on $V^{\prime}$ such that $D^{\prime}$ is
$G$-invariant and the morphisms $\phi$ and $f$ are $G$-equivariant when we
endow $V$ and $\mathbb{A}^{m}\times V$ with the trivial action.

Then there exists a $G$-equivariant homotopy
$H^{\prime}_{inff}\colon[0,\infty]\times\widehat{V^{\prime}}\to\widehat{V^{\prime}}$
with the following properties.

  1. (a)

The homotopy $H^{\prime}_{inff}$ restricts to well defined homotopies along
the fibres of the morphism $\widehat{\phi}$.

  2. (b)

Let $v\in V$ and $X\subset{\phi}^{-1}(v)$ be a Zariski closed subset of
dimension strictly smaller than $\mathrm{dim}({\phi}^{-1}(v))$. Then
$\widehat{X}\cap
H^{\prime}_{inff}(0,\widehat{V^{\prime}})\subset\widehat{D^{\prime}}$.

  3. (c)

The homotopy $H^{\prime}_{inff}$ can be taken to be $G$-invariant and respects
the levels of the functions $\xi_{i}$.

Our notation _inff_ is a concatenation of the short forms $inf$ for inflation
and $f$ for families.

###### Proof.

We adapt the proof of [7, Lemma 10.3.2] to prove the lemma. Let
$h_{0}\colon[0,\infty]\times\mathbb{A}^{m}\to\widehat{\mathbb{A}^{m}}$ be the
standard homotopy which sends $(t,x)$ to the generic type of the closed
polydisk around $x$ of valuative radius $(t,\ldots,t)$. We abuse notation and
use $h_{0}$ to denote the homotopy $[0,\infty]\times(\mathbb{A}^{m}\times
V)\to\widehat{(\mathbb{A}^{m}\times V)/V}$ defined by
$(t,(a,v))\mapsto(h_{0}(t,a)\otimes v)$. The fact that this is a well defined
_homotopy_ follows from [7, Lemma 9.8.3].

By condition (3) of the lemma and [7, Lemma 7.3.4], for each
$u:=(x,v)\in\mathbb{A}^{m}\times V$, there exists a $\gamma_{0}(u)\in\Gamma$
such that for any $v^{\prime}\in f^{-1}(v)$, the path $t\mapsto h_{0}(t,u)$
for $t\in[\gamma_{0}(u),\infty]$ lifts uniquely to a definable path
$\widehat{V^{\prime}_{v}}$ starting from $v^{\prime}$. Furthermore, observe
that for every $t\in[0,\infty]$, $\widehat{p}(h_{0}(t,u))=v$. It follows that
if $v^{\prime}\in V^{\prime}$ is such that $f(v^{\prime})=u$ then any lift of
the path $t\mapsto h_{0}(t,u)$ starting from $v^{\prime}$ must belong to
$\widehat{V^{\prime}_{v}}$. Thus, for any $v^{\prime}\in f^{-1}(v)$, the path
$t\mapsto h_{0}(t,u)$ for $t\in[\gamma_{0}(u),\infty]$ lifts uniquely to a
path in $\widehat{V^{\prime}}$ starting from $v^{\prime}$. The remainder of
the proof can be carried out using more or less the same arguments as in the
proof of [7, Lemma 10.3.2]. Note that $V^{\prime}$ is not a projective
variety. Hence instead of using [7, Lemma 4.2.29] as in the proof of [7, Lemma
10.3.2], we use Lemma 4.7. ∎

##  5\. Generic deformations

###  5.1. The initial set-up

The goal of this section is to summarize the constructions of Lemmas 4.3 and
4.6 and introduce notation that will be used in the proofs of Proposition 5.2
and Theorem 1.1.

Let $\phi\colon V^{\prime}\to V$ be a projective morphism of $K$-varieties
such that the fibres of $\phi$ are pure of dimension $m$. We suppose that $V$
is integral. As before, let $G$ be a finite algebraic group acting on
$V^{\prime}$ such that the morphism $\phi$ is $G$-equivariant when $V$ is
endowed with the trivial action. Let $\\{\xi_{i}\colon
V^{\prime}\to\Gamma_{\infty}\\}_{i}$ be a finite family of $v+g$-continuous
$K$-definable functions.

By Lemma 4.3, there exists a Zariski open neighbourhood $U\subset V$ such that
the morphism $\phi\colon\phi^{-1}(U)\to U$ factors through a finite surjective
$G$-equivariant morphism $g_{U}\colon\phi^{-1}(U)\to\mathbb{P}^{m}\times U$
over $U$ where the $G$-action on $\mathbb{P}^{m}\times U$ is taken to be
trivial. Let $V^{\prime}_{U}:=\phi^{-1}(U)$. We may shrink $U$ further and
assume that all generic points of $V^{\prime}_{U}$ belong to the fibre over
the generic point of $U$. We have the following commutative diagram.

${V^{\prime}_{U}}$${\mathbb{P}^{m}\times
U}$${U}$$\scriptstyle{\phi_{U}}$$\scriptstyle{g_{U}}$

We apply the following steps to choose a horizontal divisor in
$V^{\prime}_{U}$.

  1. (1)

We apply Lemma 4.6 to the diagram above to obtain a divisor
$D_{U}\subset\mathbb{P}^{m}\times U$ which satisfies the conditions of 4.6 and
$(\mathbb{P}^{m}\times U)\smallsetminus
D^{\prime}_{U}\subseteq\mathbb{A}^{m}\times U$ for some copy of
$\mathbb{A}^{m}$ in $\mathbb{P}^{m}$. Let $D^{\prime}_{U}:=g_{U}^{-1}(D_{U})$.

  2. (2)

For every $i$, let $Y_{i}:=\xi_{i}^{-1}(\infty)$. Since $\xi_{i}$ is
$v+g$-continuous, $Y_{i}$ is a Zariski closed subset of $V^{\prime}_{U}$. We
shrink $U$ further if necessary and suppose that for every $i$, the generic
points of $Y_{i}$ lie on the fibre over the generic point of $U$. Furthermore,
for every $i$, and every irreducible component $Y_{ij}$ of $Y_{i}$, we can
shrink $U$ so that the map $Y_{ij}\to U$ is flat. It follows from [4,
Corollary 9.6] that the fibres of $\phi_{|Y_{ij}}$ are pure and equi-
dimensional.

For every $i$, let $\mathcal{J}_{i}$ denote those indices $j$ such that the
irreducible component $Y_{ij}$ of $Y_{i}$ is of dimension strictly less than
$\mathrm{dim}(V^{\prime}_{U})$. We enlarge $D^{\prime}_{U}$ so that for every
$i$ and $j\in\mathcal{J}_{i}$, $Y_{ij}\subset D^{\prime}_{U}$.

  3. (3)

We shrink $U$ further and enlarge $D^{\prime}_{U}$ so that $D^{\prime}_{U}$
remains a horizontal divisor and $V^{\prime}_{U}\smallsetminus D^{\prime}_{U}$
is the disjoint union of irreducible varieties whose fibres over $U$ are pure
and equidimensional. This is possible by first choosing a closed subset $A$ of
the generic fibre $V^{\prime}_{\eta}$ such that
$V^{\prime}_{\eta}\smallsetminus A$ is the disjoint union of irreducible
$k(\eta)$-varieties. Note that
$\mathrm{dim}(A)<\mathrm{dim}(V^{\prime}_{\eta})$. We can shrink $U$ so that
the Zariski closure $A^{\prime}$ of $A$ in $V^{\prime}_{U}$ will satisfy the
required property. We then replace $D^{\prime}_{U}$ with the union
$D^{\prime}_{U}\cup A^{\prime}$.

  4. (4)

We have $D^{\prime}_{U}=g_{U}^{-1}(g_{U}(D^{\prime}_{U}))$. Hence,
$D^{\prime}_{U}$ is $G$-invariant.

By Lemma 4.3, there exists a $K$-point $z\in\mathbb{P}^{m}$ such that after
shrinking $U$ if necessary we can extend the diagram above to get the
following commutative diagram.

${V^{\prime}_{1U}}$${E\times U}$${\mathbb{P}^{m-1}\times
U}$${V^{\prime}_{U}}$${\mathbb{P}^{m}\times
U}$${U}$$\scriptstyle{g^{\prime}_{U}}$$\scriptstyle{b_{U}^{\prime}}$$\scriptstyle{b\times
id}$$\scriptstyle{p_{U}}$$\scriptstyle{\phi_{U}}$$\scriptstyle{g_{U}}$

where $b\colon E\to\mathbb{P}^{m}$ is the blow up at the point $z$, the
restriction of $(p_{U}\circ g^{\prime}_{U})$ to $D^{\prime}_{U}$ is finite
surjective onto $\mathbb{P}^{m-1}\times U$ and the square in the diagram is
cartesian.

Let $Z^{\prime}_{U}\subset V^{\prime}_{U}$ be the preimage for the morphism
$g_{U}$ of the closed subset $\\{z\\}\times U$. Let
$\phi^{\prime}_{U}:=\phi_{U}\circ b^{\prime}_{U}$,
$Z^{\prime}_{1U}:={b^{\prime}_{U}}^{-1}(Z^{\prime}_{U})$,
$D^{\prime}_{1U}:={b^{\prime}_{U}}^{-1}(D^{\prime}_{U})$ and
$D^{\prime}_{11U}:=D^{\prime}_{1U}\cup Z^{\prime}_{1U}$. For every $i$, let
$\xi_{1i}:=\xi_{i}\circ b^{\prime}_{U}$.

In §5.2, we adapt the construction in [7, §11.3] of the relative curve
homotopy to the fibration $p^{\prime}_{U}\colon V^{\prime}_{1U}\to F\times U$
where $F:=\mathbb{P}^{m-1}$ and $p^{\prime}_{U}:=p_{U}\circ g^{\prime}_{U}$.

###  5.2. The relative curve homotopy for families

Let the notation be as in §5.1. There exists an open subset $W\subset F\times
U$ such that $p_{U}^{-1}(W)\subset E\times U$ is isomorphic to
$\mathbb{P}^{1}\times W$. Indeed, if $p\colon E\to F$ denotes the projection
map then there exists a Zariski open subset $F_{0}\subset F$ such that
$p^{-1}(F_{0})=F_{0}\times\mathbb{P}^{1}$. By definition,
$p_{U}=p\times\mathrm{id}\colon E\times U\to F\times U$. Let $W:=F_{0}\times
U$. By construction, $p^{\prime}_{U}$ restricts to a finite morphism from
$D^{\prime}_{11U}$ onto $F\times U$.

Let $A:={p^{\prime}_{U}}^{-1}(W)\subset V^{\prime}_{1U}$ and
$B:=p_{U}^{-1}(W)\subset E\times U$. Furthermore, we can shrink $W$ if
necessary so that the map $g^{\prime}_{U}\colon A\to B$ factors through $A\to
A^{\prime}\to B$ where $A\to A^{\prime}$ is radicial and for every $w\in W$,
$A^{\prime}_{w}\to B_{w}=\mathbb{P}^{1}$ is generically étale. This is
possible because the above property is true over the generic point of $W$.
Using that the morphism $F\times U\to U$ is flat and hence open, we can shrink
$U$ so that the projection $W\to U$ is surjective.

_The homotopy $H^{\prime}_{curvesf}$._

We fix three points - $0,1$ and $\infty$ on $\mathbb{P}^{1}$. This is to make
sure that the notions of standard homotopy and closed ball are well defined.
Given a divisor $X$ on $B$, [7, §10.2] implies that there exists a well
defined definable map $\psi_{X}\colon[0,\infty]\times B\to\widehat{B/W}$ which
fixes $\widehat{X}$ and if $w\in W$ then $\psi_{X}$ restricts to a well
defined homotopy on $\widehat{B_{w}}$. Furthermore, if $X\cap B_{w}$ is
finite, then the image of the homotopy restricted to the fibre $B_{w}$ is a
$\Gamma$-internal subset of $\widehat{B_{w}}$. We emphasize that this map is a
priori not continuous unless for instance we add additional hypothesis on $X$.
By definition, for $w\in W$, the fibres of the map $\widehat{B/W}\to W$ are
copies of $\widehat{\mathbb{P}^{1}}$. The definable map $\psi_{X}$ is
constructed using the _standard homotopy_ (cf. [7, §7.5, p.105]) on
$\mathbb{P}^{1}$ and then defining cut offs of this homotopy via the divisor
$X$.

######  Lemma 5.1.

After shrinking the open set $U$ if necessary, there exists a constructible
set $C\subset V^{\prime}_{1U}$ with the following properties.

  1. (1)

The set $C$ maps surjectively onto $F\times U$ via $p^{\prime}_{U}$.

  2. (2)

There exists an open subset $W^{\prime}\subset F\times U$ such that
$V^{\prime}_{1U}$ contains ${p^{\prime}_{U}}^{-1}(W^{\prime})$.

  3. (3)

Let $H^{\prime}_{inff}$ be the homotopy obtained from Lemma 4.8 associated to
the divisor $D^{\prime}_{11U}$ and the morphism $g^{\prime}_{U}\colon
V^{\prime}_{1U}\smallsetminus D^{\prime}_{11U}\to E\times U$. (By
construction, the image of $g^{\prime}_{U}$ is contained in a copy of
$\mathbb{A}^{m}\times U$). The image
$H^{\prime}_{inff}(0,\widehat{V^{\prime}_{1U}})$ is contained in
$\widehat{C}$.

  4. (4)

There exists a deformation retraction
$h^{\prime}_{curvesf}\colon[0,\infty]\times C\to\widehat{C/F\times U}$. The
image
$\Upsilon^{\prime}_{2}:=h^{\prime}_{curvesf}(0,C)\subset\widehat{V^{\prime}_{1U}/F\times
U}$ is iso-definable and relatively $\Gamma$-internal over $F\times U$.
Furthermore, $h^{\prime}_{curvesf}$ respects the levels of the functions
$\xi_{1i}$ for every $i$ and is $G$-equivariant.

###### Proof.

By [7, Lemma 11.3.2], there exists a divisor $X$ on $B$ such that for any
divisor $X^{\prime}$ containing $X$, $\psi_{X^{\prime}}\colon[0,\infty]\times
B\to\widehat{B/W}$ lifts uniquely to a definable map $h\colon[0,\infty]\times
A\to\widehat{A/W}$ which is fibrewise a homotopy. We enlarge $X$ so that it
contains the image $g^{\prime}_{U}(D^{\prime}_{11U}\cap A)$ and the divisor
$\infty\times W$. We use

$h^{\prime}_{curvesf}\colon[0,\infty]\times A\to\widehat{A/W}$

to denote the unique lift of $\psi_{X}$. By Lemma 10.2.2 in loc.cit, we can
enlarge $X$ so that the lift $h^{\prime}_{curvesf}$ preserves the levels of
the restrictions of the functions $\xi_{1i}$. As the lift is unique, it is
$G$-invariant.

Note from the construction in [7, Lemma 11.3.2] that
$\mathrm{dim}(X)=\mathrm{dim}(W)$. Observe that $W$ is an irreducible
$K$-variety. It follows that the Zariski closure of the set $\\{w\in
W|\mathbb{P}^{1}\times\\{w\\}\subset X\\}$ is of dimension strictly smaller
than $\mathrm{dim}(W)$. Let $W^{\prime}\subset W$ be a Zariski open subset
over which $X$ is finite. We replace $A$ with
${p^{\prime}_{U}}^{-1}(W^{\prime})$ and $B$ with $p_{U}^{-1}(W^{\prime})$. It
then follows that $\psi_{X}\colon[0,\infty]\times B\to\widehat{B/W^{\prime}}$
is a well defined _homotopy_ and its lift is a well defined homotopy
$h^{\prime}_{curvesf}\colon[0,\infty]\times A\to\widehat{A/W^{\prime}}$ which
is $G$-invariant and in addition preserves the levels of the functions
$\xi_{1i}$. Using that the morphism $F\times U\to U$ is flat and hence open,
we can shrink $U$ so that the projection $W^{\prime}\to U$ is surjective.

We extend $h^{\prime}_{curvesf}$ to a definable map $[0,\infty]\times A\cup
D^{\prime}_{11U}\to\widehat{A\cup D^{\prime}_{11U}}$ by setting
$h^{\prime}_{curvesf}(t,x)=x$ for every $x\in D^{\prime}_{11U}$. By Lemma
11.3.3 in loc.cit., the map $h^{\prime}_{curvesf}$ is a well defined homotopy
with canonical extension
$H^{\prime}_{curvesf}\colon[0,\infty]\times\widehat{A\cup
D^{\prime}_{11U}}\to\widehat{A\cup D^{\prime}_{11U}}$. _We set $C:=A\cup
D^{\prime}_{11U}$_.

It remains to verify the inequality
$H^{\prime}_{inff}(e,\widehat{V^{\prime}_{1U}})\subset\widehat{C}$. Observe
that by construction the projection $W^{\prime}\to U$ is surjective and hence
the complement $S$ of $W^{\prime}$ in $F\times U$ cannot contain any subset of
the form $F\times u$ where $u\in U$. As a result for any $u\in U$,
${p^{\prime}}_{U}^{-1}(S)\cap{\phi^{\prime}}_{U}^{-1}(u)$ is a closed subset
whose dimension is strictly smaller than that of
${\phi^{\prime}}_{U}^{-1}(u)$. The inflation property (cf. Lemma 4.8 (2)) of
the homotopy $H^{\prime}_{inff}$ implies the result. ∎

We now prove the following proposition which is related to [7, Proposition
11.7.1] in that we treat a family of quasi-projective varieties parametrized
by a quasi-projective $K$-variety. While loc.cit. shows that there exists a
family of deformation retractions uniform over the base, we show that there
exists a global homotopy that restricts to a homotopy on each of the fibres
but compromise by treating only a suitable open subset of the family.

######  Proposition 5.2.

Let $V$ be a pure quasi-projective $K$-variety. Let $\phi\colon V^{\prime}\to
V$ be a morphism between quasi-projective varieties whose image is dense. Let
$G$ be a finite algebraic group acting on $V^{\prime}$ which restricts to a
well defined action along the fibres of the morphism $\phi$ and
$\\{\xi_{i}\colon V^{\prime}\to\Gamma_{\infty}\\}_{i}$ be a finite collection
of $K$-definable functions. There exists an open dense subset $U\subset V$
such that if $V^{\prime}_{U}:=\phi^{-1}(U)$ then there exists a generalized
interval $I$ and a homotopy $h^{\prime}_{rel}\colon I\times
V^{\prime}_{U}\to\widehat{V^{\prime}_{U}/U}$ which satisfies the following
properties.

  1. (1)

The image of $h^{\prime}_{rel}$ is a relatively $\Gamma$-internal subset of
$\widehat{V^{\prime}_{U}/U}$.

  2. (2)

The homotopy $h^{\prime}_{rel}$ is invariant for the action of the group $G$
and respects the levels of the functions $\xi_{i}$.

  3. (3)

The homotopy $h^{\prime}_{rel}$ is Zariski generalizing.

###### Proof.

Lemma 4.2 allows us to reduce to the case of a morphism $\phi\colon
V^{\prime}\to V$ satisfying assertions (1)-(4) of 4.2. We make use of the
notation and constructions introduced in §5.1. The simplifications in §5.1
show that it suffices to prove the theorem for the morphism
$\phi^{\prime}_{U}\colon V^{\prime}_{1U}\to U$. Indeed, any relative homotopy
on $V^{\prime}_{1U}$ that restricts to a well defined homotopy on
$Z^{\prime}_{1U}$ must descend to a homotopy on $V^{\prime}_{U}$. This is
because for every $v\in U$, the image of $Z^{\prime}_{1Uv}$ for the morphism
$b^{\prime}_{U}$ is $Z^{\prime}_{Uv}$ which is the disjoint union of Zariski
closed points and a relative homotopy that restricts to a well defined
homotopy on $Z^{\prime}_{1U}$ will also restrict to a well defined homotopy on
each connected component of $Z^{\prime}_{1Uv}$. By [7, Lemma 3.9.4], for every
$v$ the restriction of the homotopy to $V^{\prime}_{1Uv}$ descends to a
homotopy on $V^{\prime}_{Uv}$, and likewise the homotopy on $V^{\prime}_{1U}$
descends to a homotopy on $V^{\prime}_{U}$.

We proceed to prove the proposition by induction on the dimension of the
fibres of the morphism $\phi^{\prime}_{U}$. Note that when the dimesion of the
generic fibre is $0$, there is nothing to prove. Let $H^{\prime}_{inff}$ be
the homotopy obtained from Lemma 4.8 associated to the divisor
$D^{\prime}_{11U}$ and the morphism $g^{\prime}_{U}\colon
V^{\prime}_{1U}\smallsetminus D^{\prime}_{11U}\to E\times U$. By 5.1, there
exists homotopies $H^{\prime}_{inff}\colon
I_{1}\times\widehat{V^{\prime}_{1U}}\to\widehat{V^{\prime}_{1U}}$ and
$h^{\prime}_{curvesf}\colon I_{2}\times\widehat{C/F\times
U}\to\widehat{C/F\times U}$ where $C$ is a constructible subset of
$V^{\prime}_{1U}$ where $I_{1}=I_{2}=[0,\infty]$. Recall from the construction
of the homotopy $h^{\prime}_{curvesf}$ that there exists a Zariski open subset
$W^{\prime}\subset F\times U$ such that the image of $h^{\prime}_{curvesf}$
when restricted to those fibres over points in $W^{\prime}$ is of dimension
$1$. Let $H^{\prime}_{curvesf}\colon I_{2}\times\widehat{C}\to\widehat{C}$
denote the canonical extension of the homotopy $h^{\prime}_{curvesf}$. By
construction, the composition $H^{\prime}_{curvesf}\circ
H^{\prime}_{inff}\colon(I_{2}+I_{1})\times\widehat{V^{\prime}_{1U}}\to\widehat{V^{\prime}_{1U}}$
is a well defined homotopy. Let $\Upsilon^{\prime}_{2}$ denote the image
$h^{\prime}_{curvesf}(0,C)\subset\widehat{V^{\prime}_{1U}/F\times U}$. By
construction, $\Upsilon^{\prime}_{2}$ is iso-definable and relatively
$\Gamma$-internal over $F\times U$. By Lemma 6.4.1 in [7],
$\widehat{\Upsilon^{\prime}_{2}}$ can be identifed with a pro-definable subset
of $\widehat{V^{\prime}_{1U}}$ such that over a point $u\in F\times U$,
$\widehat{\Upsilon^{\prime}_{2}}_{u}=\Upsilon^{\prime}_{2u}$. The image of
$H^{\prime}_{curvesf}\circ H^{\prime}_{inff}$ is contained in
$\widehat{\Upsilon^{\prime}_{2}}$. By construction, $H^{\prime}_{curvesf}\circ
H^{\prime}_{inff}$ is compatible with the homotopy on $\widehat{U}$ that fixes
every point.

######  Lemma 5.3.

There exists a pseudo-Galois cover (cf. [7, §2.12]) $f\colon F^{\prime}\to
F\times U$ and a morphism $\kappa\colon\Upsilon^{\prime}_{2}\times_{(F\times
U)}F^{\prime}\to F^{\prime}\times\Gamma_{\infty}^{M}$ for some
$M\in\mathbb{N}$ such that the restriction
$\widehat{\kappa}_{|\widehat{\Upsilon^{\prime}_{2F^{\prime}}}}\colon\widehat{\Upsilon^{\prime}_{2F^{\prime}}}\to\widehat{F^{\prime}}\times\Gamma^{M}_{\infty}$
is a homeomorphism onto its image where
$\Upsilon^{\prime}_{2F^{\prime}}:=\Upsilon^{\prime}_{2}\times_{(F\times
U)}F^{\prime}$.

###### Proof.

Recall that $V$ is a projective variety. Let $d_{v}\colon V\to[0,\infty]$
denote the schematic distance to $V_{bord}$ where $V_{bord}:=V\smallsetminus
U$. By construction, the morphism $g^{\prime}_{U}\colon V^{\prime}_{1U}\to
E\times U$ is finite and hence projective. It follows that the morphism
$V^{\prime}_{1U}\xrightarrow{g^{\prime}_{U}}E\times U$ factors through an
embedding $j^{\prime}\colon V^{\prime}_{1U}\to\mathbb{P}_{K}^{N}\times E\times
U$ for some $N\in\mathbb{N}$. Let $V^{\prime}_{1}$ be the closure in
$\mathbb{P}_{K}^{N}\times E\times V$ of $V^{\prime}_{1U}$. Let
$p^{\prime}\colon V^{\prime}_{1}\to V$ be the composition of the morphisms
$V^{\prime}_{1}\to E\times V$ which is the restriction of a projection and
$E\times V\to V$. Let $d^{\prime}_{v}:=d_{v}\circ p^{\prime}$.

Similarly, let $\rho\colon V^{\prime}_{1}\to[0,\infty]$ denote the schematic
distance from the closure of $D^{\prime}_{11U}$ in $V^{\prime}_{1}$ and
$\eta\colon V^{\prime}_{1}\to[0,\infty]$ denote the schematic distance to the
preimage of $(F\times V)\smallsetminus W^{\prime}$ where $W^{\prime}\subset
F\times U$ is as in the statement of Lemma 5.1.

For $\gamma\in[0,\infty)$, let
$V_{\gamma}:=\\{x\in\overline{V}|d_{v}(x)\leq\gamma\\}$ and let
$V^{\prime}_{1V_{\gamma}}$ denote its preimage in $V^{\prime}_{1}$ i.e. the
set $\\{x\in\overline{V^{\prime}_{1}}|d^{\prime}_{v}(x)\leq\gamma\\}$. Observe
that $\widehat{V_{\gamma}}$ is definably compact since it is a closed subset
of the definably compact space $\widehat{V}$. Furthermore, $V_{\gamma}\subset
U$. For similar reasons, $\widehat{V^{\prime}_{1V_{\gamma}}}$ is definably
compact and also $V^{\prime}_{1V_{\gamma}}\subset V^{\prime}_{1U}$. Observe
that $\widehat{\Upsilon^{\prime}_{2\gamma}}$ is $\sigma$-compact with respect
to the restrictions of $\rho$ and $\eta$.

For ease of notation, let $S:=F\times U$ and for $\gamma\in\Gamma$,
$S_{\gamma}:=F\times V_{\gamma}$. By [7, Lemma 6.4.2], there exists a pseudo-
Galois cover $f\colon F^{\prime}\to S$ and a morphism
$\kappa\colon\Upsilon^{\prime}_{2}\times_{S}F^{\prime}\to
F^{\prime}\times\Gamma_{\infty}^{M}$ over $F^{\prime}$ for some
$M\in\mathbb{N}$ such that
$\widehat{\kappa}_{|\widehat{\Upsilon^{\prime}_{2F^{\prime}}}}\colon\widehat{\Upsilon^{\prime}_{2F^{\prime}}}\to\widehat{F^{\prime}}\times\Gamma^{M}_{\infty}$
is a continuous injection where
$\Upsilon^{\prime}_{2F^{\prime}}:=\Upsilon^{\prime}_{2}\times_{S}F^{\prime}$.

We fix $\gamma\in\Gamma$. We claim that $\widehat{\kappa}$ restricts to a
homeomorphism on the closed subspace
$\widehat{\Upsilon^{\prime}_{2F^{\prime}_{\gamma}}}$ where
$F^{\prime}_{\gamma}$ is the preimage of $S_{\gamma}$ via $f$ and
$\Upsilon^{\prime}_{2F^{\prime}_{\gamma}}:=\Upsilon^{\prime}_{2}\times_{S_{\gamma}}F^{\prime}_{\gamma}$.
We apply [7, Lemma 6.4.3] to
$\Upsilon^{\prime}_{2}\subset\widehat{V^{\prime}_{1V_{\gamma}}/S_{\gamma}}$ to
verify the claim. Note that strictly speaking, loc.cit. requires that
$S_{\gamma}$ be a quasi-projective variety. However, one checks that the steps
of the proof can be carried out in our situation.

Let $V_{o\gamma}:=\\{x\in{V}|d_{v}(x)<\gamma\\}$, $S_{o\gamma}:=F\times
V_{o\gamma}$ and $F^{\prime}_{o\gamma}:=f^{-1}(S_{o\gamma})$. We have shown
that $\widehat{\kappa}$ restricts to a homeomorphism on
$\widehat{\Upsilon^{\prime}_{2F^{\prime}_{o\gamma}}}$. To conclude that
$\widehat{\kappa}$ is a homeomorphism on
$\widehat{\Upsilon^{\prime}_{2F^{\prime}}}$, it suffices to verify that the
image of $\widehat{\Upsilon^{\prime}_{2F^{\prime}_{o\gamma}}}$ via
$\widehat{\kappa}$ is open in
$\widehat{\kappa}(\widehat{\Upsilon^{\prime}_{2F^{\prime}}})$. This follows
from the fact that by construction,

$\widehat{\kappa}(\widehat{\Upsilon^{\prime}_{2F^{\prime}_{o\gamma}}})=\widehat{\kappa}(\widehat{\Upsilon^{\prime}_{2F^{\prime}}})\cap(\widehat{F^{\prime}_{o\gamma}}\times\Gamma^{M}_{\infty})$

since $\kappa$ is over $F^{\prime}$ and hence also over $S$ and $U$. Note that
$\widehat{F^{\prime}_{o\gamma}}\times\Gamma^{M}_{\infty}$ is open in
$\widehat{F^{\prime}}\times\Gamma^{M}_{\infty}$.

∎

By Lemma 5.3, there exists a pseudo-Galois cover $f\colon F^{\prime}\to
F\times U$ and a morphism $\kappa\colon\Upsilon^{\prime}_{2}\times_{(F\times
U)}F^{\prime}\to F^{\prime}\times\Gamma_{\infty}^{M}$ for some
$M\in\mathbb{N}$ such that the restriction
$\widehat{\kappa}_{|\widehat{\Upsilon^{\prime}_{2F^{\prime}}}}\colon\widehat{\Upsilon^{\prime}_{2F^{\prime}}}\to\widehat{F^{\prime}}\times\Gamma^{M}_{\infty}$
is a homeomorphism onto its image where
$\Upsilon^{\prime}_{2F^{\prime}}:=\Upsilon^{\prime}_{2}\times_{(F\times
U)}F^{\prime}$.

We shrink $U$ if necessary and assume that it is normal. Using the arguments
in [7, 6.4.4], there exists a finite collection of $K$-definable functions
$\mu_{j}\colon F^{\prime}\to\Gamma_{\infty}$ such that, if $O\subset U$ is a
Zariski open set, for $I$ a generalised interval, a homotopy
$a^{\prime}_{f}\colon I\times F\times O\to\widehat{F\times O}$ which lifts to
a homotopy $a^{\prime}\colon I\times f^{-1}(F\times
O)\to\widehat{f^{-1}(F\times O)}$ that preserves the levels of the functions
$\mu_{j}$ also induces a homotopy $a^{\prime\prime}\colon
I\times\widehat{\Upsilon^{\prime}_{2}}\cap\widehat{{\phi^{\prime}_{U}}^{-1}(O)}\to\widehat{\Upsilon^{\prime}_{2}}\cap\widehat{{\phi^{\prime}_{U}}^{-1}(O)}$
that is $G$-invariant and respects the levels of the functions $\xi_{i}$.

Observe that the fibres of the morphism $p_{2}\circ f$ are pure over $U$. Let
$G^{\prime}:=\mathrm{Aut}(F^{\prime}/F\times U)$. Let $p_{2}\colon F\times
U\to U$ denote the projection map onto the second coordinate. Observe that the
group $G^{\prime}$ acts on $F^{\prime}$ along the fibres of the composition
$p_{2}\circ f$. We apply the induction hypothesis to the morphism $p_{2}\circ
f\colon F^{\prime}\to U$ along with the definable functions $\mu_{j}$ and the
group $G^{\prime}$. Note that the morphism $p_{2}\circ f$ is projective. It
follows that after shrinking $U$ if necessary, we have a well defined homotopy

$H^{\prime}_{bf}\colon
I_{3}\times\widehat{\Upsilon^{\prime}_{2}}\cap\widehat{{\phi^{\prime}_{U}}^{-1}(U)}\to\widehat{\Upsilon^{\prime}_{2}}\cap\widehat{{\phi^{\prime}_{U}}^{-1}(U)}.$

Let
$\Upsilon^{\prime}_{2U}:=\Upsilon^{\prime}_{2}\cap{\phi^{\prime}_{U}}^{-1}(U)$.
Observe that by construction, $H^{\prime}_{bf}$ restricts to a well defined
homotopy
$I\times{\Upsilon^{\prime}_{2U}}\to\widehat{\Upsilon^{\prime}_{2U}/U}$ whose
image is relatively $\Gamma$-internal over $U$. Furthermore, the composition,
$H^{\prime}_{bf}\circ H^{\prime}_{curvesf}\circ H^{\prime}_{inff}$ restricts
to a well defined homotopy $(I_{3}+I_{2}+I_{1})\times
V^{\prime}_{1U}\to\widehat{V^{\prime}_{1U}/U}$ which fulfils the assertions of
the proposition. ∎

###  5.3. Proof of Theorem 1.1

We begin by verifying Theorem 1.1 when the dimension of the generic fibre is
zero.

######  Lemma 5.4.

Let $\phi\colon V^{\prime}\to V$ be a morphism satisfying assertions (1)-(4)
of 4.2. We assume in addition that the dimension of the generic fibre is $0$.
Then, the conclusion of Theorem 1.1 is true.

###### Proof.

There exists a Zariski open subset $U\subseteq V$ such that $\phi_{U}\colon
V^{\prime}_{U}\to U$ is finite. It follows that $\widehat{V^{\prime}_{U}/U}$
is relatively $\Gamma$-internal and iso-definable. By our assumption in 4.2,
$V$ and hence $U$ is normal. Let $d\colon V\to\Gamma_{\infty}$ be the
schematic distance to the closed subset $V\smallsetminus U$. Observe that
$\widehat{V^{\prime}_{U}}$ is $\sigma$-compact with respect to the function
$\widehat{d_{|U}}\circ\widehat{\phi_{U}}$. By [7, Theorem 6.4.4], there exists
a finite pseudo-Galois cover $\pi\colon U^{\prime}\to U$ and finitely many
definable functions $\\{\xi^{\prime}_{j}\\}_{j}$ such that any deformation
retraction of $\widehat{U}$ that lifts to a deformation retraction on
$\widehat{U^{\prime}}$ that respects the definable functions
$\xi^{\prime}_{j}$ will lift to a deformation retraction of
$\widehat{V^{\prime}_{U}}$ that is equivariant for the action of the group $G$
and respects the definable functions $\xi_{i}$. We may hence assume
$V^{\prime}_{U}\to U$ is a pseudo-Galois cover and
$G=\mathrm{Gal}(K(V^{\prime}_{U})/K(U))$ where $K(U)$ is the function field of
$U$.

We shrink $U$ and assume $V^{\prime}_{U}$ is normal. We replace $V^{\prime}$
with the normalization of $V$ in $K(V^{\prime}_{U})$. We can extend the
functions $\xi^{\prime}_{j}$ to definable functions on $V^{\prime}$. Let
$D^{\prime}$ denote the preimage of $V\smallsetminus U$ for the morphism
$V^{\prime}\to V$. Recall that in the proof of [7, Theorem 11.1.1] applied to
the variety $V^{\prime}$, the group $G$ and the functions $\xi^{\prime}_{j}$,
we choose a divisor of $V^{\prime}$ with suitable properties and the first
homotopy on $\widehat{V^{\prime}}$ that we construct is an inflation homotopy
with respect to this divisor. We shrink $U$ if necessary and hence enlarge
$D^{\prime}$ so that we can take it to be the divisor in the proof of loc.cit.
for $V^{\prime}$. We add the schematic distance to $D^{\prime}$ to the
definable functions $\xi^{\prime}_{j}$.

We apply loc.cit. to the variety $V^{\prime}$, the group $G$ and functions
$\xi^{\prime}_{j}$ to get a deformation retraction $H^{\prime}\colon
I\times\widehat{V^{\prime}}\to\widehat{V^{\prime}}$ on $\widehat{V^{\prime}}$
which is $G$-equivariant. We suppose in addition that the image of
$H^{\prime}$ is pure of dimension $\mathrm{dim}(V^{\prime})$. The deformation
retraction $H^{\prime}$ descends to a deformation retraction on $\widehat{V}$
and the restrictions of $H^{\prime}$ and $H$ to $\widehat{V^{\prime}_{U}}$ and
$\widehat{U}$ respectively, clearly satisfy (1) - (4) and (6) of Theorem 1.1.
To verify property (5) for $H$, it suffices to verify that if $Z\subset
V^{\prime}$ is a proper Zariski closed subset then $\widehat{Z}\cap
H^{\prime}(e,\widehat{V^{\prime}_{U}})$ is empty where $e$ is the end point of
the generalized interval $I$. Note by construction, $H^{\prime}$ is the
composition of homotopies $H^{\prime}_{\Gamma}\circ H^{\prime}_{b}\circ
H^{\prime}_{curves}\circ H^{\prime}_{inf}$ where $H^{\prime}_{inf}$ is an
inflation homotopy with respect to $D^{\prime}$. Furthermore, the image of
$H^{\prime}$ is contained in the image of $H^{\prime}_{inf}$. It follows that
if $\widehat{Z}$ intersects the image of $H^{\prime}$ non-trivially then it
must intersect the image of $H^{\prime}_{inf}$ non-trivially. By construction
of $H^{\prime}_{inf}$, this implies that $\widehat{Z}\cap
H^{\prime}(e,\widehat{V^{\prime}})\subseteq\widehat{D^{\prime}}$. It follows
that $\widehat{Z}\cap H^{\prime}(e,\widehat{V^{\prime}_{U}})$ is empty.

∎

###### Proof.

(of Theorem 1.1) Let the data be as given in Theorem 1.1. It suffices to treat
the case of a morphism $\phi\colon V^{\prime}\to V$ satisfying assertions
(1)-(4) of 4.2. We proceed by induction on the dimension of the generic fibre.
When the generic fibre has dimension $0$, Theorem 1.1 is true by Lemma 5.4.

We proceed to the general case. Suppose the dimension of the generic fibre is
greater than $0$. We make use of the notation and construction introduced in
§5.1. By Lemma 5.1, there exists a homotopy
$h^{\prime}_{curvesf}\colon[0,\infty]\times\widehat{C/F\times
U}\to\widehat{C/F\times U}$ where $C$ is a certain constructible subset of
$V^{\prime}_{1U}$. Let
$H^{\prime}_{curvesf}\colon[0,\infty]\times\widehat{C}\to\widehat{C}$ denote
the canonical extension of the homotopy $h^{\prime}_{curvesf}$. Let
$\Upsilon^{\prime}_{2}$ denote the image
$h^{\prime}_{curvesf}(0,C)\subset\widehat{V^{\prime}_{1U}/F\times U}$. By
construction, $\Upsilon^{\prime}_{2}$ is iso-definable and relatively
$\Gamma$-internal over $F\times U$. By Lemma 6.4.1 in [7],
$\widehat{\Upsilon^{\prime}_{2}}$ can be identifed with a pro-definable subset
of $\widehat{V^{\prime}_{1U}}$ such that over a point $u\in F\times U$,
$\widehat{\Upsilon^{\prime}_{2}}_{u}=\Upsilon^{\prime}_{2u}$. By construction,
$H^{\prime}_{curvesf}$ is compatible with the homotopy that acts trivially on
$\widehat{U}$.

By Lemma 5.3, there exists a pseudo-Galois cover $f\colon F^{\prime}\to
F\times U$ and a morphism $\kappa\colon\Upsilon^{\prime}_{2}\times_{(F\times
U)}F^{\prime}\to F^{\prime}\times\Gamma_{\infty}^{M}$ for some
$M\in\mathbb{N}$ such that the restriction
$\widehat{\kappa}_{|\widehat{\Upsilon^{\prime}_{2F^{\prime}}}}\colon\widehat{\Upsilon^{\prime}_{2F^{\prime}}}\to\widehat{F^{\prime}}\times\Gamma^{M}_{\infty}$
is a homeomorphism onto its image where
$\Upsilon^{\prime}_{2F^{\prime}}:=\Upsilon^{\prime}_{2}\times_{(F\times
U)}F^{\prime}$.

Using the arguments in [7, 6.4.4], there exists a finite collection of
$K$-definable functions $\mu_{j}\colon F^{\prime}\to\Gamma_{\infty}$ such
that, if $O\subset U$ is a Zariski open set, for $I$ a generalised interval, a
homotopy $a^{\prime}_{f}\colon I\times F\times O\to\widehat{F\times O}$ which
lifts to a homotopy $a^{\prime}\colon I\times f^{-1}(F\times
O)\to\widehat{f^{-1}(F\times O)}$ that preserves the levels of the functions
$\mu_{j}$ also induces a homotopy $a^{\prime\prime}\colon
I\times\widehat{\Upsilon^{\prime}_{2}}\cap\widehat{{\phi^{\prime}_{U}}^{-1}(O)}\to\widehat{\Upsilon^{\prime}_{2}}\cap\widehat{{\phi^{\prime}_{U}}^{-1}(O)}$
that is $G$-invariant and respects the levels of the functions $\xi_{i}$.

Let $p_{2}\colon F\times U\to U$ denote the projection map onto the second
coordinate. Observe that the fibres of the morphism $p_{2}\circ f$ are pure
over $U$. Let $G^{\prime}:=\mathrm{Aut}(F^{\prime}/F\times U)$. Observe that
the group $G^{\prime}$ acts on $F^{\prime}$ along the fibres of the
composition $p_{2}\circ f$. We apply the induction hypothesis to the morphism
$p_{2}\circ f\colon F^{\prime}\to U$ along with the definable functions
$\mu_{j}$ and the group $G^{\prime}$. It follows that we can shrink $U$ so
that if
$\Upsilon^{\prime}_{2,U}:=\Upsilon^{\prime}_{2}\cap{\phi^{\prime}_{U}}^{-1}(U)$
then we have a well defined homotopy

$H^{\prime}_{bf}\colon
I_{bf}\times\widehat{\Upsilon^{\prime}_{2,U}}\to\widehat{\Upsilon^{\prime}_{2,U}}$

whose image is an iso-definable $\Gamma$-internal subset of
$\widehat{V^{\prime}_{1U}}$. By construction, the composition
$H^{\prime}_{bf}\circ H^{\prime}_{curvesf}$ is a well defined homotopy on
$\widehat{C}$.

######  Remark 5.5.

_Observe that in the proof of Proposition 5.2, the homotopy we constructed on
$\widehat{V^{\prime}_{1U}/U}$ automatically descends to a homotopy on
$\widehat{V^{\prime}_{U}/U}$. As explained before, this is because for every
$u\in U$, $V^{\prime}_{1u}\to V^{\prime}_{u}$ is isomorphic outside the finite
set of closed points $Z^{\prime}_{u}\subset V^{\prime}_{u}$. Hence, as long as
the induced homotopy on $\widehat{V^{\prime}_{1u}}$ preserves
$\widehat{Z^{\prime}_{1u}}$ where $Z^{\prime}_{1u}$ is the preimage of
$Z^{\prime}_{u}$ for the morphism $V^{\prime}_{1u}\to V^{\prime}_{u}$._

_Note that such an argument does not hold in the case of Theorem 1.1 since we
must construct a homotopy on the base $U$ as well. To resolve this issue, we
construct an inflation homotopy $H^{\prime}_{inff-primary}$ on
$V^{\prime}_{U}$ whose image does not contain $Z^{\prime}_{U}$ and is hence
contained in $\widehat{V^{\prime}_{1U}}$. We then proceed to construct
suitable homotopies on $\widehat{V^{\prime}_{1U}}$._

###  5.4. The relative tropical homotopy

Let $\Upsilon^{\prime}_{bcf}$ denote the image of the composition of
homotopies

$H^{\prime}_{bf}\circ H^{\prime}_{curvesf}.$

Our goal in this section runs parallel to [7, §11.5]. We construct a homotopy
$H^{\prime}_{\Gamma f}$ on a subset of $\Upsilon^{\prime}_{bcf}$ and
homotopies $H^{\prime}_{inff}$ and $H^{\prime}_{inff-primary}$ by applying
Lemma 4.8 such that the composition

$H^{\prime}_{\Gamma f}\circ H^{\prime}_{bf}\circ H^{\prime}_{curvesf}\circ
H^{\prime}_{inff}\circ H^{\prime}_{inff-primary}$

is a well defined _deformation retraction_ i.e. it fixes its image.

An important fact to note is that we will no longer shrink the base $U$ to
adapt the proofs in [7, §11.5] to our relative setting. We construct the
homotopy $H^{\prime}_{\Gamma f}$ such that its image outside of
$\widehat{D^{\prime}_{1U}}$ will be controlled completely by definable
functions in $\Gamma$. This enables us to choose the inflation homotopies
$H^{\prime}_{inff}$ and $H^{\prime}_{inff-primary}$ so that they fix the image
of $H^{\prime}_{\Gamma f}$ as well. Furthermore, $H^{\prime}_{\Gamma f}$
restricts to well defined homotopies along the fibres of
$\widehat{\phi_{U}}_{|\widehat{\Upsilon^{\prime}_{bcf}}}$. It is hence
compatible with $\widehat{\phi_{U}}_{|\widehat{\Upsilon^{\prime}_{bcf}}}$.

####  5.4.1. Preliminaries

By our induction hypothesis, $H^{\prime}_{bf}$ descends to a deformation
retraction $H_{b}\colon I_{bf}\times\widehat{U}\to\widehat{U}$ whose image is
a $\Gamma$-internal iso-definable set which we shall denote $\Upsilon_{b}$.
Furthermore, $H_{b}$ satisfies properties (5) and (6) of Theorem 1.1. For ease
of notation, until the end of the proof of Lemma 5.7, we write
$\Upsilon^{\prime}$ in place of $\Upsilon^{\prime}_{bcf}$ and $\Upsilon$ in
place of $\Upsilon_{b}$.

We now choose continuous injective maps on $\Upsilon^{\prime}$ and $\Upsilon$
into the value group sort and construct a homotopy
$H^{\prime\mathrm{trop}}_{\Gamma f}$ on the image of $\Upsilon^{\prime}$ for
this map. However, since we are making use of two inflation homotopies (cf.
Remark 5.5), where $H^{\prime}_{inff-primary}$ acts on $V^{\prime}_{U}$ and
not $V^{\prime}_{1U}$, we must take this into consideration when choosing
coordinates in the value group sort for $\Upsilon^{\prime}$.

Recall the closed subsets $Z^{\prime}_{1U}\subset V^{\prime}_{1U}$,
$Z^{\prime}_{U}\subset V^{\prime}_{U}$ such that for every fibre over $u\in
U$, $Z^{\prime}_{1u}$ is the exceptional divisor of the map
$V^{\prime}_{1u}\to V^{\prime}_{u}$ and $V^{\prime}_{1U}\smallsetminus
Z^{\prime}_{1U}$ is isomorphic to $V^{\prime}_{U}\smallsetminus
Z^{\prime}_{U}$. We identify $V^{\prime}_{U}\smallsetminus Z^{\prime}_{U}$
with the subset $V^{\prime}_{1U}\smallsetminus Z^{\prime}_{1U}\subset
V^{\prime}_{1U}$. If
$\Upsilon^{\prime}_{1}:=\Upsilon^{\prime}\smallsetminus\widehat{Z^{\prime}_{1U}}$
and $\Upsilon^{\prime}_{2}:=\Upsilon^{\prime}\cap\widehat{Z^{\prime}_{1U}}$
then we have a decomposition,
$\Upsilon^{\prime}=\Upsilon^{\prime}_{1}\sqcup\Upsilon^{\prime}_{2}$. Observe
that $\Upsilon^{\prime}_{1}\subset\widehat{V^{\prime}_{U}}$. By [7, Theorem
6.2.8], there exists a $K$-definable map $\alpha_{0}\colon
V^{\prime}_{U}\to\Gamma_{\infty}^{N_{1}}$ such that $\widehat{\alpha_{0}}$ is
continuous and its restriction to $\Upsilon^{\prime}_{1}$ is injective.
Similarly, there exists a $K$-definable map $\alpha_{1}\colon
V^{\prime}_{1U}\to\Gamma_{\infty}^{N_{2}}$ such that $\widehat{\alpha_{1}}$ is
continuous and its restriction to $\Upsilon^{\prime}_{2}$ is injective. We
abuse notation and write $\alpha_{0}$ and $\alpha_{1}$ in place of
$\widehat{\alpha_{0}}$ and $\widehat{\alpha_{1}}$ respectively. Let
$\alpha^{\prime}_{0}$ denote the composition
$\widehat{V^{\prime}_{1U}}\xrightarrow{\widehat{b_{U}}}\widehat{V^{\prime}_{U}}\xrightarrow{{\alpha_{0}}}\Gamma_{\infty}^{N_{1}}$.

Once again, by [7, Theorem 6.2.8], there exists a $K$-definable map
$\alpha\colon U\to\Gamma_{\infty}^{M}$ such that the induced map
$\widehat{\alpha}\colon\widehat{U}\to\Gamma_{\infty}^{M}$ is continuous and
restricts to an injective map on $\Upsilon$. We abuse notation and use
$\alpha$ itself in place of $\widehat{\alpha}$. The proof of [7, Theorem
6.2.8] shows that if $x\colon\Gamma_{\infty}^{M}\to\Gamma_{\infty}$ is a
coordinate then
$(x\circ\alpha)^{-1}(\\{z\in\Gamma_{\infty}^{M}|x(z)=\infty\\})$ is of the
form $\widehat{Z}$ where $Z$ is some Zariski closed subset of $U$. Since $U$
is irreducible, we can assume that for every coordinate $x$, $x\circ\alpha$ is
not identically $\infty$ on $\widehat{U}$. Hence the locus of points $u\in U$
such that $x\circ\alpha(u)<\infty$ for every coordinate $x$ on
$\Gamma_{\infty}^{M}$ is a Zariski open dense subset of $U$. By property (5),
we deduce that we have a continuous injective definable map
$\alpha\colon\Upsilon\to\Gamma^{M}$.

As in [7, §11.5], we may assume that $G$ acts on the coordinates of
$\Gamma_{\infty}^{N_{1}}$ and on the coordinates of $\Gamma_{\infty}^{N_{2}}$
such that the map $x\mapsto(\alpha^{\prime}_{0}(x),\alpha_{1}(x))$ is
$G$-equivariant. We simplify notation and write
$f:=\widehat{\phi^{\prime}_{U}}_{|\Upsilon^{\prime}}$ and $N:=N_{1}+N_{2}$. We
define
$\alpha^{\prime}\colon\Upsilon^{\prime}\hookrightarrow\Gamma_{\infty}^{N+M}$
by $\alpha^{\prime}(x):=(\alpha^{\prime}_{0}(x),\alpha_{1}(x),\alpha(f(x)))$.
Observe that $\alpha^{\prime}$ is a well defined, injective, continuous
$K$-definable map such that

$f^{\prime}\circ\alpha^{\prime}=\alpha\circ f$

where $f^{\prime}$ is the projection
$\Gamma_{\infty}^{N+M}\to\Gamma_{\infty}^{M}$. We abuse notation and write
$\xi_{i}$ for the functions $\xi_{i}\circ\alpha^{\prime-1}$.

Let $W^{\prime}$ and $W$ denote the images of $\Upsilon^{\prime}$ and
$\Upsilon$ via the maps $\alpha^{\prime}$ and $\alpha$ respectively. Note that
the action of $G$ on $W^{\prime}$ restricts to a well defined action on the
fibres of $f^{\prime}_{|W^{\prime}}$. Let $\rho_{0}\colon
V^{\prime}\to[0,\infty]$ denote the schematic distance from the closure of
$D^{\prime}_{U}$ in $V^{\prime}$. Let $\eta$ and $\rho$ be as in the proof of
Lemma 5.3.

Let $d_{bord}\colon V\to\Gamma_{\infty}$ denote the schematic distance to the
closed subset $V\smallsetminus U$. After modifying the injection
$\alpha^{\prime}$, we can assume that there exists coordinates
$x_{h}$,$x_{hh}$,$x_{v}$ on $\Gamma_{\infty}^{N}$ and $x_{c}$ on
$\Gamma_{\infty}^{M}$ such that
$x_{hh}\circ\alpha^{\prime}$,$x_{h}\circ\alpha^{\prime}$,
$x_{v}\circ\alpha^{\prime}$ and $x_{c}\circ\alpha$ correspond to $\rho$,
$\rho_{0}\circ b^{\prime}_{U}$, $\eta$ and $d_{bord}$ respectively where
$b^{\prime}_{U}$ is the map $V^{\prime}_{1U}\to V_{U}$.

######  Lemma 5.6.

The space $W^{\prime}\cap\Gamma^{N+M}$ is closed in $\Gamma^{N+M}$.

###### Proof.

Let $x\in\Gamma^{N+M}$ be a limit point of the set