# Non-arithmetic uniformization of metric spaces attached to unitary Shimura
varieties

Olivier de Gaay Fortman111 _Date:_ . _Mathematics Subject Classification:_
20F65, 22E40, 53C35.

###### Abstract

Abstract: We develop a new method of constructing non-arithmetic lattices in
the projective orthogonal group $\text{PO}(n,1)$ for every integer $n$ larger
than one. The technique is to consider anti-holomorphic involutions on a
complex arithmetic ball quotient, glue their fixed loci along geodesic
subspaces, and show that the resulting metric space carries canonically the
structure of a complete real hyperbolic orbifold. The volume of various of
these non-arithmetic orbifolds can be explicitly calculated.

##  1 Introduction

To study locally symmetric spaces $M=\Gamma\backslash G/K$ of finite volume,
one would like to classify lattices in semi-simple Lie groups. This difficult
problem can be simplified somewhat by considering the notion of
_arithmeticity_. On the one hand, arithmetic lattices can be classified in a
way which is close to the classification of semi-simple algebraic groups over
number fields, something which is well-understood. On the other hand, if a
semi-simple Lie group $G$ is non-compact and not isogenous to
${\textnormal{PO}}(n,1)$ or ${\textnormal{PU}}(n,1)$, then every irreducible
lattice $\Gamma\subset G$ is arithmetic [Mar75, Cor92, GS92]. Thus, to
describe all non-arithmetic lattices, one is left with the problem of
classifying non-arithmetic lattices in ${\textnormal{PO}}(n,1)$ and in
${\textnormal{PU}}(n,1)$, a complicated task which remains completely open in
general.

In various cases, when studying variations of Hodge structure attached to the
cohomology of algebraic varieties, lattices in ${\textnormal{PU}}(n,1)$ arise
in a natural way. Indeed, there are moduli spaces of complex varieties whose
period map identifies it with a quotient of complex hyperbolic space. The ball
quotients that arise in this way are often arithmetic [Pic83, Shi64, Kon00,
ACT02a], but may happen to be non-arithmetic as well [DM86]. Similar
constructions can be carried out to uniformize moduli of real varieties as
quotients of real hyperbolic space by lattices
$\Gamma\subset{\textnormal{PO}}(n,1)$, see e.g. [AY98, ACT06, ACT07, ACT10,
Chu11, HR18].

A striking result in the latter direction was given by Allcock, Carlson and
Toledo, who identified the moduli space of stable real cubic surfaces with a
four-dimensional non-arithmetic real ball quotient [ACT10]. This ball quotient
is assembled from various pieces, each of which is a real arithmetic ball
quotient that, over an open subset, parametrizes moduli of smooth real cubic
surfaces of one topological type.

The starting point of this paper is the question whether one can glue the
fixed loci of anti-holomorphic involutions on an arbitrary unitary Shimura
variety, in a way that does not depend on real moduli theory, and such that
the resulting space is naturally a real ball quotient. Our first goal is to
prove that such glueing can be done, leading to a construction which
generalizes [ACT06, ACT07, ACT10] and seems an orbifold analogue of the
construction of Gromov–Piatetski-Shapiro [GPS87]. For indeed, our second goal
is to show that many of the lattices that arise in this way, are in fact non-
arithmetic.

To explain this, let $n\in\mathbb{Z}_{\geq 1}$. Consider a CM field $K$ and a
hermitian $\mathcal{O}_{K}$-lattice $\Lambda_{n}$ of signature $(n,1)$ for one
infinite place of $K$ and definite for others. To the lattice $\Lambda_{n}$,
one can associate a complex ball quotient
$\Gamma\setminus\mathbb{B}^{n}(\mathbb{C})$. Moreover, each anti-unitary
involution $\alpha\colon\Lambda_{n}\to\Lambda_{n}$ defines a real ball
quotient
$\Gamma_{\alpha}\setminus\mathbb{B}^{n}_{\alpha}(\mathbb{R})\coloneqq{\textnormal{Stab}}_{\Gamma}(\mathbb{B}^{n}(\mathbb{C})^{\alpha})\backslash\mathbb{B}^{n}(\mathbb{C})^{\alpha}$.
We glue these real ball quotients together along a certain orthogonal
hyperplane arrangement ${\mathscr{H}}\subset\mathbb{B}^{n}(\mathbb{C})$. The
resulting space $X(\Lambda_{n})$ is naturally a path metric space, see
Proposition 4.7. Our first main result says that, in fact, $X(\Lambda_{n})$
carries a complete hyperbolic orbifold structure:

######  Theorem 1.1 (c.f. Theorem 4.1).

For each component $X(\Lambda_{n})^{+}\subset X(\Lambda_{n})$ there exists a
lattice $\Gamma_{n}^{+}\subset{\textnormal{PO}}(n,1)$ and a canonical isometry
$X(\Lambda_{n})^{+}\cong\Gamma_{n}^{+}\setminus\mathbb{B}^{n}(\mathbb{R})$.

Applying Theorem 1.1 to the case where $K=\mathbb{Q}(\zeta_{3})$ and

$\Lambda_{n}=\mathbb{Z}[\zeta_{3}]^{n,1}=\left(\mathbb{Z}[\zeta_{3}]^{n+1},{\textnormal{diag}}(-1,1,\dotsc,1)\right),$
(1)

one obtains a sequence of real hyperbolic orbifolds $X(\Lambda_{n})$ with
interesting properties. Indeed, $X(\Lambda_{2})$ is connected, its orbifold
fundamental group $\Gamma_{2}^{+}$ is non-arithmetic, and $X(\Lambda_{2})$
immerses totally geodesically into a connected component $X(\Lambda_{n})^{+}$
of $X(\Lambda_{n})$ for each $n\in\mathbb{Z}_{\geq 2}$ (c.f. Theorem 8.5). By
[BC05, Proposition 15.2.2], this implies that
$\Gamma_{n}^{+}\subset{\mathrm{PO}}(n,1)$ is non-arithmetic. Thus, we obtain:

######  Theorem 1.2 (c.f. Theorem 8.7).

Let $n\in\mathbb{Z}_{\geq 2}$ and let
$\Lambda_{n}=\mathbb{Z}[\zeta_{3}]^{n,1}$. There exists a connected component
$X(\Lambda_{n})^{+}\subset X(\Lambda_{n})$ such that the lattice
$\Gamma_{n}^{+}\subset{\textnormal{PO}}(n,1)$ underlying the complete
hyperbolic orbifold $X(\Lambda_{n})^{+}$ is non-arithmetic.

Theorem 1.1 also has applications to the theory of moduli spaces of real
algebraic varieties. For some moduli stacks of GIT stable hypersurfaces
${\mathcal{M}}_{s}$, one can consider the hermitian lattice $\Lambda_{n}$ that
arises as the cohomology of the finite cover of projective space ramified
along a member of the moduli space, and define an isomorphism
${\mathcal{M}}_{s}(\mathbb{R})\cong X(\Lambda_{n})$ for
$n=\dim({\mathcal{M}}_{s})$. In such cases, an application of Theorem 1.1
gives a uniformization of ${\mathcal{M}}_{s}(\mathbb{R})$ by real hyperbolic
space. For instance, for cubic surfaces and binary sextics, one retrieves the
main results of [ACT06, ACT07, ACT10] in this way.

Applying Theorem 1.1 to $K=\mathbb{Q}(\zeta_{5})$ and
$\Lambda_{n}=(\mathbb{Z}[\zeta_{5}]^{n+1},{\textnormal{diag}}(-\lambda,1,\dotsc,1))$,
where $\lambda=\frac{\sqrt{5}-1}{2}\in\mathbb{R}$, yields further examples of
non-arithmetic lattices. In a follow-up paper, see [GF23], we will pursue this
direction for $n=2$, by using the theory developed here to investigate the
structure of the moduli space of stable real binary quintics. An analysis of
the period map and an application of Theorem 1.1 will show that, also in that
case, $X(\Lambda_{2})$ is connected and underlies a non-arithmetic ball
quotient. As before, this has the following consequence.

######  Theorem 1.3.

Let $K=\mathbb{Q}(\zeta_{5})$. Let $\Phi=\left\\{\sigma_{1},\sigma_{2}\colon
K\to\mathbb{C}\right\\}$ be the CM type with
$\sigma_{i}(\zeta_{5})=\zeta_{5}^{i}$ for $i=1,2$. Define a sequence of
hermitian $\mathcal{O}_{K}$-lattices $\Lambda_{n}$ as follows:

$\Lambda_{n}=\left(\mathbb{Z}[\zeta_{5}]^{n+1},{\textnormal{diag}}(-\lambda,1,\dotsc,1)\right)\quad{\textnormal{\emph{with}}}\quad\lambda=\zeta_{5}+\zeta_{5}^{-1}=(\sqrt{5}-1)/2.$

For each $n\in\mathbb{Z}_{\geq 2}$, there is a connected component
$X(\Lambda_{n})^{+}\subset X(\Lambda_{n})$ such that the lattice
$\Gamma_{n}^{+}\subset{\textnormal{PO}}(n,1)$ attached to $X(\Lambda_{n})^{+}$
via Theorem 1.1 is non-arithmetic.

The results of Sections 7 and 8 will reduce the proof of Theorem 1.3 to the
case $n=2$. The remaining part of the proof (i.e. the non-arithmeticity of
$\Gamma_{2}^{+}$) requires a new technique, namely the use of equivariant
Hodge theory to identify $X(\Lambda_{2})$ with the moduli space of stable real
binary quintics. This will be done in [GF23].

###  1.1 Outline of the glueing construction

Let $(K,\Psi)$ be a CM type, where $K$ is a CM field of degree $2g$ over
$\mathbb{Q}$ with ring of integers $\mathcal{O}_{K}$, and let $\Lambda$ be a
hermitian $\mathcal{O}_{K}$-lattice. Thus, $\Lambda$ is a finite free
$\mathcal{O}_{K}$-module equipped with a non-degenerate hermitian form
$h\colon\Lambda\times\Lambda\to\mathcal{O}_{K}$. Let $n\in\mathbb{Z}_{\geq 1}$
and assume that $h$ has signature $(n,1)$ with respect to one embedding
$(\tau:K\to\mathbb{C})\in\Psi$ and is positive definite for the other elements
in $\Psi$. Let $\mathbb{C}H^{n}$ be the space of negative lines in
$\Lambda\otimes_{\mathcal{O}_{K},\tau}\mathbb{C}$, define
$\Gamma=\textnormal{Aut}(\Lambda)$ and let
$P\Gamma=\textnormal{Aut}(\Lambda)/\mu_{K}$, where
$\mu_{K}\subset\mathcal{O}_{K}^{\ast}$ is the group of finite units. Finally,
let $P{\mathscr{A}}$ be the quotient of the set of anti-unitary involutions
$\alpha:\Lambda\to\Lambda$ by $\mu_{K}$.

Attached to the hermitian lattice $\Lambda$ there is a Shimura variety
${\textnormal{Sh}}_{K}(G,X)$, see [Ach20, Section 5.3], with complex
uniformization
${\textnormal{Sh}}_{K}(G,X)(\mathbb{C})=\Gamma\setminus\mathbb{C}H^{n}$, see
Corollary 5.9 or [Shi63, Theorem 2]. Consider the hyperplane arrangement

${\mathscr{H}}=\bigcup_{\begin{subarray}{c}r\in\Lambda\colon\\\
h(r,r)=1\end{subarray}}\langle
r_{\mathbb{C}}\rangle^{\perp}\subset\mathbb{C}H^{n},$

and assume that ${\mathscr{H}}$ is an _orthogonal arrangement_ , i.e. that the
following holds:

######  Condition 1.4 (c.f. [ACT02]).

Any two different hyperplanes $\langle r_{\mathbb{C}}\rangle^{\perp},\langle
t_{\mathbb{C}}\rangle^{\perp}$ in ${\mathscr{H}}$ intersect orthogonally or
not at all, i.e. either $h(r,t)=0$ or $\langle
r_{\mathbb{C}}\rangle^{\perp}\cap\langle
t_{\mathbb{C}}\rangle^{\perp}=\emptyset$.

In many cases, Condition 1.4 is automatically satisfied:

######  Theorem 1.5 (c.f. Theorem 6.2).

If the different ideal ${\mathfrak{D}}_{K}\subset\mathcal{O}_{K}$ is generated
by a purely imaginary element $\eta\in{\mathfrak{D}}_{K}$ such that
$\Im(\varphi(\eta))>0$ for every $\varphi\in\Psi$, then Condition 1.4 holds.

The condition that ${\mathfrak{D}}_{K}\subset\mathcal{O}_{K}$ is generated by
a purely imaginary element $\eta\in\mathcal{O}_{K}$ holds whenever $K$ is
cyclotomic or imaginary quadratic, see Proposition 6.4.

The glueing construction is performed by assembling the different copies

$\mathbb{R}H^{n}_{\alpha}\coloneqq\left(\mathbb{C}H^{n}\right)^{\alpha}\subset\mathbb{C}H^{n},\quad\alpha\in
P{\mathscr{A}}$

of real hyperbolic space $\mathbb{R}H^{n}$ along the hyperplane arrangement
${\mathscr{H}}$. See Definition 3.9 and Remark 3.10 for the precise
formulation of the equivalence relation. This gives a topological space
$Y(\Lambda)$, acted upon by $P\Gamma$. Define $P\Gamma_{\alpha}\subset
P\Gamma$ to be the stabilizer of $\mathbb{R}H^{n}_{\alpha}$. Theorem 1.1 can
then be reformulated as follows.

######  Theorem 1.6 (c.f. Theorem 4.1).

The space $Y(\Lambda)$ admits a path metric such that the natural map
$Y(\Lambda)\to\mathbb{C}H^{n}$ is a local isometric embedding. This metric on
$Y(\Lambda)$ induces a complete path metric on $X(\Lambda)=P\Gamma\setminus
Y(\Lambda)$, which extends canonically to a real hyperbolic orbifold structure
of finite volume. The subspace

$\coprod_{\alpha\in P\Gamma\setminus
P{\mathscr{A}}}P\Gamma_{\alpha}\setminus\left(\mathbb{R}H^{n}_{\alpha}-{\mathscr{H}}\right)\subset
X(\Lambda)$

is an open suborbifold, and for each connected component
$X(\Lambda)^{+}\subset X(\Lambda)$ there exists a lattice
$\Gamma^{+}\subset\textnormal{PO}(n,1)$ and an isomorphism of hyperbolic
orbifolds

$X(\Lambda)^{+}\cong\Gamma^{+}\setminus\mathbb{R}H^{n}.$

######  Remarks 1.7.

  1. 1.

The volume of $X(\Lambda)$ equals the volume of $\coprod_{\alpha\in
P\Gamma\setminus
P{\mathscr{A}}}P\Gamma_{\alpha}\setminus\mathbb{R}H^{n}_{\alpha}$, and hence
the sum of the volumes of $P\Gamma_{\alpha}\setminus\mathbb{R}H^{n}_{\alpha}$
for $\alpha\in P\Gamma\setminus P{\mathscr{A}}$.

  2. 2.

Crucial in the proof of the finiteness of the volume of $X(\Lambda)$ are: (i)
the fact that
$P\Gamma_{\alpha}\subset{\textnormal{Isom}}(\mathbb{R}H^{n}_{\alpha})$ is an
arithmetic lattice for $\alpha\in P{\mathscr{A}}$; (ii) the fact that the set
$P\Gamma\backslash P{\mathscr{A}}$ is finite. We prove these facts in Theorem
2.7 and Proposition 2.9.

  3. 3.

Our construction relies on Condition 1.4, saying that the hyperplane
arrangement ${\mathscr{H}}\subset\mathbb{C}H^{n}$ is an orthogonal arrangement
in the sense of [ACT02]. In fact, there is always an orthogonal arrangement
${\mathcal{H}}\subset\mathbb{C}H^{n}$ attached to $h$ in such a way that
${\mathcal{H}}={\mathscr{H}}$ when ${\mathscr{H}}$ is orthogonal (c.f. Remark
6.5), and one can glue the different copies ${\mathbb{R}}H^{n}_{\alpha}$ of
real hyperbolic space along the arrangement ${\mathcal{H}}$ to obtain a
complete hyperbolic orbifold as in Theorem 1.6. We will not prove this.

  4. 4.

The fact that ${\mathscr{H}}\subset\mathbb{C}H^{n}$ is an orthogonal
arrangement implies that for $n>1$, the orbifold fundamental
$\pi_{1}^{\textnormal{orb}}\left(P\Gamma\setminus\left(\mathbb{C}H^{n}-{\mathscr{H}}\right)\right)$
is not a lattice in any Lie group with finitely many connected components, see
[ACT02, Theorem 1.2].

###  1.2 Further remarks and questions

######  Remark 1.8.

As mentioned above, our construction can be seen as an orbifold analogue of
the construction of non-arithmetic lattices provided by Gromov and
Piatetski–Shapiro [GPS87]. Apart from the analogy, their construction is
technically quite different from ours. Gromov and Piatetski–Shapiro glue
hyperbolic manifolds with boundary whose fundamental groups are Zariski dense
in ${\mathrm{PO}}(n,1)$ and lie in non-commensurable arithmetic lattices. In
this paper, we glue real hyperbolic orbifolds with boundary and corners whose
orbifold fundamental groups are generally not Zariski dense in
${\mathrm{PO}}(n,1)$ (but do lie in arithmetic lattices that are generally not
all commensurable). In light of these differences, it makes sense to ask
whether the lattices we construct are essentially different, in the sense that
they (or some of them) are not commensurable with any of the non-arithmetic
lattices constructed via the method of [GPS87]. More generally, one could ask
if our lattices are commensurable with any other non-arithmetic lattice
currently known to exist.

######  Remark 1.9.

Let $X(\Lambda)$ be a hyperbolic orbifold as constructed in Theorem 1.6. As
indicated in Remark 1.7.1, we have ${\mathrm{Vol}}(X(\Lambda))=\sum_{\alpha\in
P\Gamma\setminus
P{\mathscr{A}}}{\mathrm{Vol}}(P\Gamma_{\alpha}\setminus\mathbb{R}H^{n}_{\alpha})$.
Moreover, if ${\mathscr{A}}\neq\emptyset$ then $\textnormal{Gal}(K/F)$ acts on
the group $P\Gamma$ and there is a bijection between $P\Gamma\setminus
P{\mathscr{A}}$ and the first non-abelian Galois cohomology group
${\mathrm{H}}^{1}(G,P\Gamma)$, see Proposition 2.9. In various cases, this
gives a way to explicitly calculate the volume of the hyperbolic orbifold
$X(\Lambda)$, or of one of its connected components $X(\Lambda)^{+}$.

######  Remark 1.10.

Let us elaborate on Remark 1.9 by analyzing the case $K=\mathbb{Q}(\zeta_{3})$
and $\Lambda_{n}=\mathbb{Z}[\zeta_{3}]^{n,1}$, see (1). For $n\geq 2$, the
volume of the connected non-arithmetic hyperbolic orbifold
$X(\Lambda_{n})^{+}=X(\mathbb{Z}[\zeta_{3}]^{n,1})^{+}$ (see Theorem 1.2) can
be bounded from below, and in some cases precisely determined, in the
following way. For each $i\in\left\\{0,\dotsc,n\right\\}$, define an anti-
unitary involution
$\alpha_{i}\colon\mathbb{Z}[\zeta_{3}]^{n,1}\to\mathbb{Z}[\zeta_{3}]^{n,1}$ as
follows:

$\displaystyle\alpha_{i}\colon\mathbb{Z}[\zeta_{3}]^{n,1}\to\mathbb{Z}[\zeta_{3}]^{n,1},\quad\alpha_{i}(x_{0},\dotsc,x_{n})=\left(\bar{x}_{0},-\bar{x}_{1},\dotsc,-\bar{x}_{i},\bar{x}_{i+1},\dotsc,\bar{x}_{n}\right).$

Let $G=\textnormal{Gal}(\mathbb{Q}(\zeta_{3})/\mathbb{Q})$ and
$P\Gamma={\mathrm{PU}}(\mathbb{Z}[\zeta_{3}]^{n,1})$. One can show that the
$n+1$ classes $[\alpha_{0}],\dotsc,[\alpha_{n}]\in P\Gamma\setminus
P{\mathscr{A}}={\mathrm{H}}^{1}(G,P\Gamma)$ are pairwise distinct, see Lemma
7.2. For $i=0,\dotsc,n$, consider then the integral quadratic form

$\Psi_{i}(x_{0},\dotsc,x_{n})=-x_{0}^{2}+\sqrt{3}x_{1}^{2}+\cdots+\sqrt{3}x_{i}^{2}+x_{i+1}^{2}+\cdots+x_{n}^{2}.$

One has
$P\Gamma_{\alpha_{i}}\setminus\mathbb{R}H^{n}_{\alpha_{i}}\cong{\mathrm{PO}}(\Psi_{i},\mathbb{Z})\setminus\mathbb{R}H^{n}$
and hence (see Theorem 1.6), we have

$\coprod_{i=0}^{n}{\mathrm{PO}}(\Psi_{i},\mathbb{Z})\setminus\left(\mathbb{R}H^{n}-{\mathscr{H}}\right)\subset
X(\Lambda_{n})^{+}=X(\mathbb{Z}[\zeta_{3}]^{n,1})^{+}.$

In particular,

$\displaystyle\sum_{i=0}^{n}{\mathrm{Vol}}\left({\mathrm{PO}}(\Psi_{i},\mathbb{Z})\setminus\mathbb{R}H^{n}\right)\leq{\mathrm{Vol}}\left(X(\Lambda_{n})^{+}\right).$
(2)

Moreover, for each $n\geq 2$ such that the cardinality of
${\mathrm{H}}^{1}(G,P\Gamma)$ equals $n+1$, one has
$X(\Lambda_{n})=X(\Lambda_{n})^{+}$ and the inequality in (2) is actually an
equality. This happens for instance when $n=2$ or $n=4$. There are other
values of $n\geq 3$ (such as $n=3$, in which case
$\\#{\mathrm{H}}^{1}(G,P\Gamma)=n+2$), for which one has equality in (2) even
though $X(\Lambda_{n})^{+}\subsetneq X(\Lambda_{n})$, i.e. $X(\Lambda_{n})$ is
not connected.

Finally, there is a closed formula for the volume
${\mathrm{Vol}}\left({\mathrm{PO}}(\Psi_{i},\mathbb{Z})\setminus\mathbb{R}H^{n}\right)$
of the arithmetic orbifold
${\mathrm{PO}}(\Psi_{i},\mathbb{Z})\setminus\mathbb{R}H^{n}$, c.f. [Pra89],
[Bel04, Section 3] and [BE12].

###  1.3 Overview of this paper

This paper is structured as follows. In Section 2 we introduce some notation
and prove several preliminary results, which will be needed in Sections 3 and
4. In Section 3, we construct the glued space $X(\Lambda)=P\Gamma\setminus
Y(\Lambda)$, starting from a hermitian $\mathcal{O}_{K}$-lattice $\Lambda$ of
hyperbolic signature. In Section 4, we provide the glued space $X(\Lambda)$
with a real hyperbolic orbifold structure. The goal of Section 5 will be to
provide the ball quotient $P\Gamma\setminus\mathbb{C}H^{n}$ with a modular
interpretation, which will be used in Section 6 to prove Theorem 1.5. In
Sections 7 and 8, we prove Theorem 1.2.

###  1.4 Acknowledgements

I emphasize the influence of the work of Allcock, Carlson and Toledo [ACT10]
on the development of the content of this article. I thank Emiliano Ambrosi,
Olivier Benoist, Nicolas Bergeron, Samuel Bronstein, Frans Oort, Pierre Py,
Emre Sertöz, Nicolas Tholozan and Domenico Valloni for stimulating
conversations and comments on earlier versions of this paper.

This research has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No754362  and from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme under grant
agreement No948066 (ERC-StG RationAlgic).

##  2 Preliminaries

The goal of Sections 2.1 \- 2.3 is to introduce the definitions needed for the
glueing construction, which will be carried out in Section 3. We also
establish some preliminary results. In Section 2.4, we will provide examples
of hermitian $\mathcal{O}_{K}$-lattices $\Lambda$ satisfying the signature
condition, together with anti-unitary involutions
$\alpha\colon\Lambda\to\Lambda$.

###  2.1 Arithmetic ball quotients and anti-unitary involutions

Let $n$ be a positive integer. Let $K$ be a CM field over $\mathbb{Q}$, with
totally real subfield $F\subset K$ and non-trivial element
$\sigma\in\textnormal{Gal}(K/F)$. Let $\mathcal{O}_{K}$ (resp.
$\mathcal{O}_{F}$) be the ring of integers of $K$ (resp. $F$), and assume the
following:

######  Condition 2.1.

The different ideal ${\mathfrak{D}}_{K}\subset\mathcal{O}_{K}$ is generated by
an element $\eta\in{\mathfrak{D}}_{K}$ that satisfies $\sigma(\eta)=-\eta$.

For an element $x\in K$, we will sometimes use the notation
$\overline{x}=\sigma(x)$. Fix an element $\eta\in\mathcal{O}_{K}$ as in
Condition 2.1, and let $\Psi$ be a set of embeddings

$\displaystyle\Psi\subset\textnormal{Hom}(K,\mathbb{C})\quad\quad{\textnormal{
such that}}\quad\quad\Psi\cup\Psi\sigma=\textnormal{Hom}(K,\mathbb{C})$

and such that

$\displaystyle\Im\left(\psi(\eta)\right)>0\quad\quad{\textnormal{ for each
}}\quad\quad\psi\in\Psi.$

Fix an embedding

$\tau\colon K\to\mathbb{C}$

with $\tau\in\Psi$. Let $\Lambda$ be a free $\mathcal{O}_{K}$-module of rank
$n+1$ equipped with a non-degenerate hermitian form

$h\colon\Lambda\times\Lambda\to\mathcal{O}_{K}$

of the following signature $(r_{\psi},s_{\psi})$ with respect to $\Psi$: we
have

$\displaystyle\begin{split}(r_{\psi},s_{\psi})\quad=\quad\begin{cases}(n,1)\quad&{\textnormal{
for }}\psi=\tau\in\Psi,\\\ (n+1,0)\quad&{\textnormal{ for
}}\psi\in\Psi{\textnormal{ with }}\psi\neq\tau.\end{cases}\end{split}$ (3)

In other words: $h$ is an $\mathcal{O}_{F}$-bilinear form,
$\mathcal{O}_{K}$-linear in its first argument, satisfies
$h(y,x)=\sigma(h(x,y))$ for $x,y\in\Lambda$, and the complex vector space
$\Lambda\otimes_{\mathcal{O}_{K},\psi}\mathbb{C}$ admits a basis $\\{e_{i}\\}$
such that $(h^{\psi}(e_{i},e_{j}))_{ij}$ is a diagonal matrix with $r_{\psi}$
diagonal entries equal to $1$ and $s_{\psi}$ to $-1$. Here,
$h^{\psi}\colon\Lambda\otimes_{\mathcal{O}_{K},\psi}\mathbb{C}\times\Lambda\otimes_{\mathcal{O}_{K},\psi}\mathbb{C}\to\mathbb{C}$
is the hermitian form attached to $h$ via the embedding $\psi$. Define

$V=\Lambda\otimes_{\mathcal{O}_{K},\tau}\mathbb{C}.$

Let $m$ be the largest positive integer for which the $m$-th cyclotomic field
$\mathbb{Q}(\zeta_{m})$ can be embedded in $K$, where $\zeta_{m}=e^{2\pi
i/m}\in\mathbb{C}$. Let $\zeta\in K$ be a primitive $m$-th root of unity in
$K$, and define

$\mu_{K}=\langle\zeta\rangle\subset{\mathcal{O}}_{K}^{\ast}\subset\mathcal{O}_{K}.$

Moreover, define $\Gamma$ to be the unitary group of $\Lambda$, and $P\Gamma$
as its quotient by $\mu_{K}$:

$\Gamma=U(\Lambda)(\mathcal{O}_{K})=\textnormal{Aut}_{\mathcal{O}_{K}}(\Lambda,h)\quad{\textnormal{
and }}\quad P\Gamma=\Gamma/\mu_{K}.$

A norm one vector $r\in\Lambda$ is called a short root. Let
${\mathscr{R}}\subset\Lambda$ be the set of short roots. For
$r\in{\mathscr{R}}$, define isometries $\phi_{r}^{i}\colon V\to V$ as follows:

$\phi_{r}(x)=x-(1-\zeta)h(x,r)\cdot
r,\quad\phi_{r}^{i}(x)=x-(1-\zeta^{i})h(x,r)\cdot r,\quad
i\in(\mathbb{Z}/m)^{\ast}.$

Note that $\phi^{i}_{r}\in\Gamma$ for $r\in{\mathscr{R}}$, and that
$\phi_{r}^{i}=\phi_{r}\circ\cdots\circ\phi_{r}$ ($i$ times). In particular,
$\phi_{r}^{m}=\textnormal{id}$. Let $\mathbb{P}(V)$ be the projective space of
lines in $V$, and let

$\mathbb{C}H^{n}=\left\\{\ell=[v]\in\mathbb{P}(V)\mid
h(v,v)<0\right\\}\subset\mathbb{P}(V)$

be the space of negative lines in $V$. Define

$H_{r}=\\{x\in\mathbb{C}H^{n}:h(x,r)=0\\}\;{\textnormal{ for
}}\;r\in{\mathscr{R}},\quad{\textnormal{ and
}}\quad{\mathscr{H}}=\bigcup_{r\in{\mathscr{R}}}H_{r}\;\;\;\subset\;\;\;\mathbb{C}H^{n}.$

###  2.2 Orthogonality of the hyperplane arrangement

It turns out that Condition 2.1 implies that the following condition is
satisfied.

######  Condition 2.2.

For $r,t\in{\mathscr{R}}$ with $H_{r}\neq H_{t}$ and $H_{r}\cap
H_{t}\neq\emptyset$, one has $h(r,t)=0$.

The fact that Condition 2.2 follows from Condition 2.1 will be proved in
Sections 5 and 6, see Theorem 6.2.

######  Remarks 2.3.

  1. 1.

Condition 2.1 is satisfied by quadratic and cyclotomic CM fields $K$, see
Proposition 6.4 in Section 6.

  2. 2.

Our glueing construction depends heavily on Condition 2.2; the only reason
that we assume Condition 2.1 is that it implies Condition 2.2 (c.f. Theorem
6.2). We do not know whether the converse implication holds true.

  3. 3.

There is a way to ensure that Condition 2.2 is automatic, see Remark 6.5, but
the definitions become more complicated. We do not pursue this direction here.

###  2.3 Anti-unitary involutions

Define an ${\mathcal{O}}_{F}$-linear map $\alpha\colon\Lambda\to\Lambda$ to be
anti-unitary if for all $x,y\in\Lambda$ and $b\in\mathcal{O}_{K}$, one has
$\alpha(b\cdot x)=\sigma(b)\cdot\alpha(x)$ and
$h(\alpha(x),\alpha(y))=\sigma(h(x,y))\in\mathcal{O}_{K}.$ Define
$\Gamma^{\prime}$ to be the group of unitary and anti-unitary
$\mathcal{O}_{F}$-linear bijections $\Lambda\xrightarrow{\sim}\Lambda$. Let
${\mathscr{A}}\subset\Gamma^{\prime}$ be the set of anti-unitary involutions
$\alpha\colon\Lambda\to\Lambda$. Then

$\displaystyle\mu_{K}\subset\Gamma\subset\Gamma^{\prime}\quad-\quad{\textnormal{
define }}\quad P\Gamma^{\prime}=\Gamma^{\prime}/\mu_{K}.$ (4)

Let $x\in K^{\ast}$. Observe that

$\displaystyle\left(x\in\mathcal{O}_{K}^{\ast}{\textnormal{ and
}}x\cdot\sigma(x)=1\right)\quad\iff\quad\left(x\in\mu_{K}\right).$ (5)

Indeed, for an embedding $\varphi\colon K\to\mathbb{C}$, one has
$\left|\varphi(x)\right|^{2}=\varphi(x\cdot\sigma(x))$; if
$x\in\mathcal{O}_{K}$, then $\left|\varphi(x)\right|=1$ for each $\varphi$ if
and only if $x$ is a root of $1$ [Mil08, Corollary 5.6].

######  Lemma 2.4.

Let $\textnormal{Isom}({\mathbb{C}}H^{n})$ be the group of isometries
$f\colon\mathbb{C}H^{n}\xrightarrow{\sim}\mathbb{C}H^{n}$. The natural
homomorphism $P\Gamma^{\prime}\to\textnormal{Isom}({\mathbb{C}}H^{n})$ is
injective.

###### Proof.

This follows readily from (5). ∎

The group $\mu_{K}$ acts on ${\mathscr{A}}$ by multiplication; define

$P{\mathscr{A}}=\mu_{K}\setminus{\mathscr{A}},\quad{\textnormal{ and }}\quad
C{\mathscr{A}}=P\Gamma\setminus P{\mathscr{A}},$

where $P\Gamma$ acts on $P{\mathscr{A}}$ by conjugation. Any $\alpha\in
P{\mathscr{A}}$ defines an anti-holomorphic involution

$\alpha:\mathbb{C}H^{n}\to\mathbb{C}H^{n};\quad{\textnormal{ define
}}\quad\mathbb{R}H^{n}_{\alpha}=(\mathbb{C}H^{n})^{\alpha}\subset\mathbb{C}H^{n}.$

For any element $\alpha\in{\mathscr{A}}$, the quadratic form $h|_{V^{\alpha}}$
on the real vector space

$\displaystyle
V^{\alpha}=\Lambda^{\alpha}\otimes_{\mathcal{O}_{F},\tau|_{F}}\mathbb{R}$ (6)

has hyperbolic signature. The following lemma is readily proved:

######  Lemma 2.5.

For $\alpha\in{\mathscr{A}}$, let $\mathbb{P}(V^{\alpha})$ be the real
projective space of lines in $V^{\alpha}$, and let
$\mathbb{R}H(V^{\alpha})\subset\mathbb{P}(V^{\alpha})$ be the space of
negative lines in $V^{\alpha}$. The canonical isomorphism
$\mathbb{P}(V^{\alpha})\cong\mathbb{P}(V)^{\alpha}$ restricts to an
isomorphism ${\mathbb{R}}H(V^{\alpha})\cong{\mathbb{R}}H^{n}_{\alpha}$. ∎

We conclude that $\mathbb{R}H^{n}_{\alpha}\subset\mathbb{C}H^{n}$ is isometric
to the real hyperbolic space of dimension $n$. We define

$P\Gamma_{\alpha}=\textnormal{Stab}_{P\Gamma}(\mathbb{R}H^{n}_{\alpha})\subset
P\Gamma\quad\quad{\textnormal{(the stabilizer of $\mathbb{R}H^{n}_{\alpha}$ in
$P\Gamma$).}}$

######  Lemma 2.6.

We have $P\Gamma_{\alpha}=N_{P\Gamma}(\alpha)\coloneqq\left\\{\gamma\in
P\Gamma\mid\gamma\circ\alpha=\alpha\circ\gamma\right\\}\subset P\Gamma.$

###### Proof.

See the proof of [HR18, Proposition 4.9]. ∎

Let $\alpha\in P{\mathscr{A}}$, consider the lattice $\Lambda^{\alpha}\subset
V^{\alpha}$ (see (6)) and the space of negative lines
$\mathbb{R}H(V^{\alpha})\subset\mathbb{P}(V^{\alpha})$. We have
${\textnormal{Isom}}(\mathbb{R}H(V^{\alpha}))\cong{\textnormal{Isom}}(\mathbb{R}H^{n})={\textnormal{PO}}(n,1)$
by Lemma 2.4. Let ${\textnormal{O}}(\Lambda^{\alpha})(\mathcal{O}_{F})$ be the
group of $\mathcal{O}_{F}$-linear isometries
$\gamma\colon\Lambda^{\alpha}\xrightarrow{\sim}\Lambda^{\alpha}$, and define
${\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})={\textnormal{O}}(\Lambda^{\alpha})(\mathcal{O}_{F})/\left\\{\pm
1\right\\}$. There are canonical embeddings

$P\Gamma_{\alpha}\hookrightarrow{\textnormal{Isom}}(\mathbb{R}H(V^{\alpha}))={\textnormal{PO}}(V^{\alpha})\quad{\textnormal{
and
}}\quad{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})\hookrightarrow{\textnormal{PO}}(V^{\alpha}).$

######  Theorem 2.7.

The groups $P\Gamma_{\alpha}$ and
${\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})$ are commensurable.

###### Proof.

We follow the ideas of the proof of [HR18, Theorem 4.10]. Let $g\in\Gamma$
such that $[g]\in P\Gamma_{\alpha}$. Recall that $\zeta\in K$ is a primitive
$m$-th root of unity in $K$ that generates
$\mu_{K}\subset\mathcal{O}_{K}^{\ast}$, see Section 2.1, and write
$m=2^{a}\cdot k$ with $k\in\mathbb{Z}_{\geq 1}$ odd. Let
$\alpha\in{\mathscr{A}}$ be a representative of $\alpha\in P{\mathscr{A}}$,
recall that $P\Gamma_{\alpha}=N_{P\Gamma}(\alpha)$ by Lemma 2.6, and define a
subgroup $P\Gamma_{\alpha}^{I}\subset P\Gamma_{\alpha}$ as follows:

$P\Gamma_{\alpha}^{I}=\left\\{[g]\in P\Gamma_{\alpha}\mid g\alpha
g^{-1}=\zeta^{i}\cdot\alpha\;{\textnormal{ with }}\;i\in 2\mathbb{Z}_{\geq
0}\;{\textnormal{ for every }}\;g\in[g]\right\\}.$

Note that for each $[g]\in P\Gamma_{\alpha}^{I}$ there is an element
$h\in[g]$, unique up to sign, such that $h\alpha h^{-1}=\alpha$. Consequently,
there is a natural embedding

$\displaystyle
P\Gamma_{\alpha}^{I}\hookrightarrow{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F}).$
(7)

In fact, one has
$P\Gamma_{\alpha}\cap{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})=P\Gamma_{\alpha}^{I}\subset{\textnormal{PO}}(V^{\alpha}),$
and this intersection has index at most two in $P\Gamma_{\alpha}$. Indeed,
either $P\Gamma_{\alpha}^{I}=P\Gamma_{\alpha}$ or there exists an element
$[g]\in P\Gamma_{\alpha}$ such that for every $g\in[g]$ one has $g\alpha
g^{-1}=\zeta^{i}\alpha$ for some odd $i\in\mathbb{Z}$, and for any two such
$g_{i}\in[g_{i}]\in P\Gamma_{\alpha}$, $i=1,2$, one has $[g_{1}g_{2}^{-1}]\in
P\Gamma_{\alpha}^{I}\subset P\Gamma_{\alpha}$.

It remains to show that $P\Gamma_{\alpha}^{I}$ has finite index in
${\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})$. To prove this, let
$i\colon\Lambda^{\alpha}\hookrightarrow\Lambda$ denote the inclusion. The
induced map
$i\colon\Lambda^{\alpha}\otimes_{\mathcal{O}_{F}}\mathcal{O}_{K}\hookrightarrow\Lambda$
has finite cokernel; let $d$ be its order. There is a map
$\varphi\colon\Lambda\to\Lambda^{\alpha}\otimes_{\mathcal{O}_{F}}\mathcal{O}_{K}$
such that $i\circ\varphi=d$ and $\varphi\circ i=d$, and such that the induced
map
$\varphi_{K}\colon\Lambda\otimes_{\mathcal{O}_{K}}K\to\Lambda^{\alpha}\otimes_{\mathcal{O}_{F}}K$
is $d$ times the inverse $i^{-1}$ of
$i\colon\Lambda^{\alpha}\otimes_{\mathcal{O}_{F}}K\xrightarrow{\sim}\Lambda\otimes_{\mathcal{O}_{K}}K$.
Define

${\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})^{I}=\left\\{[M]\in{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})\mid
i\circ M_{\mathcal{O}_{K}}\circ\varphi=d\cdot\overline{M}{\textnormal{ for
some }}\overline{M}\in\Gamma\right\\}.$

Let $[g]\in P\Gamma_{\alpha}^{I}$ and let $g\in[g]$ such that $g\alpha=\alpha
g$. Let $M_{g}=g|_{\Lambda^{\alpha}}$ be the induced map
$\Lambda^{\alpha}\to\Lambda^{\alpha}$; we have
$[M_{g}]\in{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})$. Furthermore,
as $i\circ(M_{g})_{\mathcal{O}_{K}}\circ\varphi=d\cdot
g\colon\Lambda\to\Lambda$, the embedding
$P\Gamma_{\alpha}^{I}\hookrightarrow{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})$
defined in (7) factors as

$P\Gamma_{\alpha}^{I}\hookrightarrow{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})^{I}\hookrightarrow{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F}).$

We claim that the first inclusion is an equality. Namely, let
$[M]\in{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})^{I}$. There exists
an element $\overline{M}\in\Gamma$ such that
$\overline{M}(\Lambda^{\alpha})=\Lambda^{\alpha}$ and
$\overline{M}|_{\Lambda^{\alpha}}=M$. Since $[\overline{M}]\in P\Gamma$
stabilizes $\mathbb{R}H^{n}_{\alpha}$, we have $[\overline{M}]\in
P\Gamma_{\alpha}$ by Lemma’s 2.5 and 2.6. Thus
$\overline{M}\alpha(\overline{M})^{-1}=\zeta^{i}\cdot\alpha$ for some
$i\in\mathbb{Z}_{\geq 0}$. The element $\overline{M}\in[\overline{M}]$
satisfies $\overline{M}(\Lambda^{\alpha})=\Lambda^{\alpha}$, which implies
that $i\equiv 0\bmod m$. Therefore, $[\overline{M}]\in P\Gamma_{\alpha}^{I}$,
proving the claim.

We conclude that

$\displaystyle
P\Gamma_{\alpha}^{I}={\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})^{I}\subset{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F}).$
(8)

Pick a basis for $\Lambda^{\alpha}$ over $\mathcal{O}_{F}$. This yields an
isomorphism
$(\Lambda^{\alpha},h|_{\Lambda^{\alpha}})\cong(\mathcal{O}_{F}^{n+1},Q_{\alpha})$
for some quadratic form $Q_{\alpha}$ on $\mathcal{O}_{F}^{n+1}$. If
$[M]\in{\textnormal{PO}}(Q_{\alpha})(\mathcal{O}_{F})$ satisfies
$M\equiv\textnormal{Id}\bmod d$, we can write $M=\textnormal{Id}+d\cdot N$ for
some element $N$ in the matrix algebra $M_{n+1}(\mathcal{O}_{F})$. This gives
$i\circ M_{\mathcal{O}_{K}}\circ\varphi=d\cdot\left(\textnormal{Id}+i\circ
N_{\mathcal{O}_{K}}\circ\varphi\right)\in d\cdot M_{n+1}(\mathcal{O}_{K})$,
which implies that
$[M]\in{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})^{I}$. Therefore,

$\left\\{[M]\in{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})\mid
M\equiv\textnormal{Id}\bmod
d\right\\}\subset{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})^{I}=P\Gamma_{\alpha}^{I}\subset{\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F}).$

In particular, $P\Gamma_{\alpha}^{I}$ has finite index in
${\textnormal{PO}}(\Lambda^{\alpha})(\mathcal{O}_{F})$, and we are done. ∎

######  Corollary 2.8.

The subgroup
$P\Gamma_{\alpha}\subset{\textnormal{PO}}(V^{\alpha})\cong{\textnormal{PO}}(n,1)$
is an arithmetic lattice in ${\textnormal{PO}}(V^{\alpha})$. ∎

We will also need the following finiteness result.

######  Proposition 2.9.

Suppose that ${\mathscr{A}}\neq\emptyset$. Let $\alpha_{0}\in{\mathscr{A}}$,
and define $G=\textnormal{Gal}(K/F)$.

  1. 1.

The group $G$ acts on $\Gamma$, and there is a bijection
$\Gamma\backslash{\mathscr{A}}\cong{\mathrm{H}}^{1}(G,\Gamma)$.

  2. 2.

The sets $\Gamma\backslash{\mathscr{A}}$ and $C{\mathscr{A}}=P\Gamma\backslash
P{\mathscr{A}}$ are finite.

###### Proof.

  1. 1.

We can let $G$ act on $\Gamma$ via the involution
$\sigma\colon\gamma\mapsto\alpha_{0}\circ\gamma\circ\alpha_{0}$. Consider

$\displaystyle{\mathrm{H}}^{1}(G,\Gamma)$
$\displaystyle={\mathrm{Z}}^{1}(G,\Gamma)/_{\sim}=\left\\{{\textnormal{$1$-cocycles
}}G\to\Gamma,f\colon s\mapsto a_{s}\right\\}/_{\sim}$

where $f\sim f^{\prime}$ if there exists $b\in\Gamma$ such that
$f^{\prime}(s)=b^{-1}f(s)^{s}b\in\Gamma$ for all $s\in G$. See [Ser73,
Chapitre I, Section 5.1]. If $g$ denotes the generator of $G$, the map
$f\mapsto f(g)$ defines a bijection between the set of $1$-cocycles
${\mathrm{Z}}^{1}(G,\Gamma)$ and the set of elements $\gamma\in\Gamma$ that
satisfy $\gamma\cdot\sigma(\gamma)=\textnormal{id}$. This yields a bijection

${\mathscr{A}}\xrightarrow{\sim}{\mathrm{Z}}^{1}(G,\Gamma),\quad\alpha\mapsto\alpha\circ\alpha_{0},$

with inverse given by $\gamma\mapsto\gamma\circ\alpha_{0}$. Let
$\alpha,\beta\in{\mathscr{A}}$ and $g\in\Gamma$. Then $g^{-1}\beta g=\alpha$
if and only if
$g^{-1}\left(\beta\circ\alpha_{0}\right)\sigma(g)=\alpha\circ\alpha_{0}$,
hence item 1 is proved.

  2. 2.

Since the canonical map $\Gamma\backslash{\mathscr{A}}\to C{\mathscr{A}}$ is
surjective, it suffices to prove that $\Gamma\backslash{\mathscr{A}}$ is
finite. By item 1, we need to show that ${\mathrm{H}}^{1}(G,\Gamma)$ is
finite. Note that
$\Gamma=\textnormal{Aut}_{\mathcal{O}_{K}}(\Lambda)=U(\Lambda)(\mathcal{O}_{K})$
arises as the group of $\mathcal{O}_{K}$-rational points of a linear algebraic
group
$U(\Lambda)_{K}\subset\textnormal{GL}(\Lambda\otimes_{\mathcal{O}_{K}}K)$ over
$K$. Moreover, the involution $\sigma\colon\Gamma\to\Gamma$ defined in item 1
extends to an involution $\sigma\colon U(\Lambda)_{K}\to U(\Lambda)_{K}$,
which implies that $\Gamma$ is an _arithmetic $G$-group_ in the sense of
[BS64, Section 3.4]. In particular, the set ${\mathrm{H}}^{1}(G,\Gamma)$ is
finite by [BS64, Proposition 3.8].

The proof is finished. ∎

###  2.4 Example

The goal of Section 2.4 is to work out an example. More precisely, we will
show that for any CM field $K$ over $\mathbb{Q}$, any CM type
$\Psi\subset\textnormal{Hom}(K,\mathbb{C})$ and any $n\in\mathbb{Z}_{\geq 1}$,
there exist a non-degenerate hermitian form $h$ on $\mathcal{O}_{K}^{n+1}$
that satisfies the signature condition (3) with respect to $\Psi$. We also
prove that in these examples, the set ${\mathscr{A}}$ of anti-unitary
involutions $\alpha$ on $(\mathcal{O}_{K}^{n+1},h)$ is non-empty.

Let $K$ be a CM field over $\mathbb{Q}$, and let $F\subset K$ be the maximal
totally real subfield of $K$.

######  Lemma 2.10.

There exists an element $\lambda\in\mathcal{O}_{F}$ with $\tau(\lambda)>0$ for
some embedding $\tau\colon F\to\mathbb{R}$ and $\psi(\lambda)<0$ for all other
$\psi\neq\tau\in\textnormal{Hom}(F,\mathbb{R})$.

###### Proof.

Let $\tilde{\lambda}\in\mathcal{O}_{F}-\mathbb{Z}$. Note that we can write
$\textnormal{Hom}(F,\mathbb{R})=\left\\{\tau,\psi_{1},\dotsc,\psi_{k}\right\\}$
with
$\tau(\tilde{\lambda})<\psi_{1}(\tilde{\lambda})<\cdots<\psi_{k}(\tilde{\lambda})$.
Let $r\in\mathbb{Q}$ be a rational number such that
$\tau(\tilde{\lambda})<r<\psi_{1}(\tilde{\lambda})$. Possibly after replacing
$\tilde{\lambda}$ by $n\cdot\tilde{\lambda}$ for some $n\in\mathbb{Z}_{\geq
1}$, we may assume that $r\in\mathbb{Z}$. The element
$\lambda=r-\tilde{\lambda}\in\mathcal{O}_{F}$ satisfies the requirements. ∎

Let $\lambda\in F$ be as in Lemma 2.10. Let
$\Lambda=(\mathcal{O}_{K}^{n+1},h)$, where $h$ is the hermitian form on
$\mathcal{O}_{K}^{n+1}$ given by ${\textnormal{diag}}(-\lambda,1,\dotsc,1)$.
Let $\Psi\subset\textnormal{Hom}(K,\mathbb{C})$ be any CM type and let
$\tau\in\Psi$ be a lift of $\tau\colon F\to\mathbb{R}$. By construction of
$\lambda$, the signature of $h^{\psi}$ is $(n,1)$ for $\psi=\tau$ and
$(n+1,0)$ for $\psi\neq\tau\in\Psi$. Thus, $h$ satisfies condition (3).

To obtain examples of anti-unitary involutions
$\alpha\colon\Lambda\to\Lambda$, consider the set $\left\\{\pm
1\right\\}^{n+1}$. Define, for each
$\epsilon=(\epsilon_{0},\dotsc,\epsilon_{n})\in\left\\{\pm 1\right\\}^{n+1}$,
an involution $\alpha_{\epsilon}$ as follows:

$\alpha_{\epsilon}\colon\Lambda\to\Lambda,\quad\alpha_{\epsilon}\left(x_{0},\dotsc,x_{n}\right)=\left(\epsilon_{0}\bar{x}_{0},\dotsc,\epsilon_{n}\bar{x}_{n}\right).$

For $x,y\in\Lambda,b\in\mathcal{O}_{K}$, one has $\alpha_{\epsilon}(b\cdot
x)=\bar{b}\cdot\alpha_{\epsilon}(x)$ and
$h(\alpha_{\epsilon}(x),\alpha_{\epsilon}(y))=\sigma(h(x,y))$ because
$\sigma(\lambda)=\lambda$. Hence $\alpha_{\epsilon}$ is an anti-unitary
involution for each $\epsilon\in\left\\{\pm 1\right\\}^{n+1}$.

##  3 Glueing real hyperbolic orbifolds

In this section, we construct a topological space by glueing together
arithmetic hyperbolic pieces. Afterwards, in Section 4, we provide this
topological space first with a path metric and then with a complete hyperbolic
orbifold structure.

###  3.1 Orthogonal hyperplanes and complex reflections

We continue with the notation and assumptions of Section 2.1. Note that
Condition 2.2 implies that if $H_{r_{1}},\dotsc,H_{r_{k}}$ for
$r_{i}\in{\mathscr{R}}$ are mutually distinct, and if their common
intersection is non-empty, then
$\cap_{i=1}^{k}H_{r_{i}}\subset\mathbb{C}H^{n}$ is a totally geodesic subspace
of codimension $k$. Note also that for any $r\in{\mathscr{R}}$, the element
$\phi_{r}\in\Gamma$ generates a finite subgroup
$\langle\phi_{r}\rangle\subset\Gamma$ of order $m$, and that the restriction
of the quotient map $\Gamma\to P\Gamma$ to this subgroup
$\langle\phi_{r}\rangle\subset\Gamma$ is injective. We will abuse notation, by
letting $\phi_{r}\in P\Gamma$ denote the image of $\phi_{r}\in\Gamma$ in
$P\Gamma$.

######  Definition 3.1.

Let ${\mathcal{H}}=\\{H_{r}\mid r\in{\mathscr{R}}\\}$. For
$x\in\mathbb{C}H^{n}$, define

${\mathcal{H}}(x)=\left\\{H\in{\mathcal{H}}\mid x\in H\right\\},\quad
G(x)=\langle\phi_{r}^{i}\in P\Gamma{\textnormal{ with
}}r\in{\mathscr{R}},i\in\mathbb{Z}/m\mid x\in H_{r}\rangle.$

The hyperplanes $H\in{\mathcal{H}}(x)$ are called the _nodes_ of $x$. We say
that _$x$ has $k$ nodes_ if the cardinality of ${\mathcal{H}}(x)$ is $k$.

######  Lemma 3.2.

Let $x\in{\mathbb{C}}H^{n}$ be an element with $k$ nodes. Then
$G(x)\cong(\mathbb{Z}/m)^{k}$.

###### Proof.

Let $r,t\in{\mathscr{R}}$. Then, for $z\in\Lambda$, one has

$\displaystyle\begin{split}\phi_{r}^{i}(\phi_{t}^{j}(z))=&\;z-(1-\zeta^{j})h(z,t)\cdot
t-(1-\zeta^{i})h(z,r)\cdot r\\\ &+(1-\zeta^{i})(1-\zeta^{j})h(z,t)h(t,r)\cdot
r.\end{split}$ (9)

Suppose that $H_{r},H_{t}\in{\mathcal{H}}(x)$, with $H_{r}\neq H_{t}$. By
Condition 2.2, we have $h(r,t)=0$; by equation (9), this implies that
$\phi_{r}^{i}\circ\phi_{t}^{j}=\phi_{t}^{j}\circ\phi_{r}^{i}$ for each
$i,j\in\mathbb{Z}/m$. We conclude that the group $G(x)$ is abelian. Suppose
that $H_{r}=H_{t}\in{\mathcal{H}}(x)$. By Lemma 3.3 below, we have that
$\phi_{t}=b\cdot\phi_{r}^{i}$ for some $i\in\mathbb{Z}/m$ and $b\in\mu_{K}$. ∎

######  Lemma 3.3.

Let $r\in {\mathscr{R}}$. Let $\phi:\mathbb{C}H^{n}\to\mathbb{C}H^{n}$ be an
isometry of order $m$ that restricts to the identity on
$H_{r}\subset\mathbb{C}H^{n}$. Then $\phi=\phi_{r}^{i}$ for some
$i\in\mathbb{Z}/m$.

###### Proof.

Let $\mathbb{H}^{n}_{\mathbb{C}}$ be the hyperbolic space attached to the
standard hermitian space $\mathbb{C}^{n,1}$ of dimension $n+1$. It is
classical that

$\displaystyle\text{Stab}_{U(n,1)}(\mathbb{H}^{n-1}_{\mathbb{C}})=U(n-1,1)\times
U(1).$

Thus, any $\phi\in U(n,1)$ that fixes $\mathbb{H}^{n-1}_{\mathbb{C}}$
pointwise lies in $\mathbb{C}^{\ast}\times U(1)$, where
$U(1)=\\{z\in\mathbb{C}^{\ast}:\left|z\right|^{2}=1\\}$. If
$\phi^{m}\in\mathbb{C}^{\ast}\times\left\\{1\right\\}$, then
$\phi\in\mathbb{C}^{\ast}\times\langle\zeta\rangle\subset U(n-1,1)\times
U(1)$. ∎

Lemma 2.4 allows us to view $P{\mathscr{A}}$ as a subset of
$\textnormal{Isom}({\mathbb{C}}H^{n})$, and also to view the groups
$G(x)\subset P\Gamma\subset P\Gamma^{\prime}$ (for any $x\in\mathbb{C}H^{n}$)
as subgroups of $\textnormal{Isom}(\mathbb{C}H^{n})$. Define

$\widetilde{Y}=\coprod_{\alpha\in P{\mathscr{A}}}\mathbb{R}H^{n}_{\alpha}.$

We will glue the different hyperbolic spaces $\mathbb{R}H^{n}_{\alpha}$, by
defining an equivalence relation $\sim$ on $\widetilde{Y}$. Before we define
it, we state and prove a couple of easy results.

######  Lemma 3.4.

Let $\alpha\in{\mathscr{A}}$ and $r\in{\mathscr{R}}$. Then
$\alpha\circ\phi_{r}^{i}=\phi_{\alpha(r)}^{-i}\circ\alpha$.

###### Proof.

Indeed, $\alpha(\phi_{r}^{i}(x))=\alpha(x-(1-\zeta^{i})h(x,r)\cdot
r)=\phi_{\alpha(r)}^{-i}(\alpha(x))$ for $x\in\Lambda$. ∎

######  Lemma 3.5.

Let $x\in{\mathbb{R}}H^{n}_{\alpha}$ and write
${\mathcal{H}}(x)=\left\\{H_{r_{1}},\dotsc,H_{r_{k}}\right\\}$, $r_{i}\in
{\mathscr{R}}$. For each $i\in\\{1,\dotsc,k\\}$ there is a unique
$j\in\\{1,\dotsc,k\\}$ with $\alpha(H_{r_{i}})=H_{\alpha(r_{i})}=H_{r_{j}}.$

###### Proof.

Let $\beta\in{\mathscr{A}}$ and $r\in{\mathscr{R}}$. Then
$\beta(H_{r})=H_{\beta(r)}.$ Since $x\in H_{r_{i}}$, we have
$x=\alpha(x)\in\alpha(H_{r_{i}})=H_{\alpha(r_{i})}$ for every $i$. In
particular, we have $H_{\alpha(r_{i})}\in{\mathcal{H}}(x)$ (see Definition
3.1), so that $H_{\alpha(r_{i})}=H_{r_{j}}$ for some $j$. ∎

######  Proposition 3.6.

Let $r,t\in{\mathscr{R}}$. The following are equivalent:

  1. 1.

One has $\phi_{r}=\phi_{t}\in\Gamma$.

  2. 2.

There exist $i,j\in\mathbb{Z}/m-\\{0\\}$ such that
$\phi_{r}^{i}=\phi_{t}^{j}\in\Gamma$.

  3. 3.

There exist $a,b\in\mathcal{O}_{K}-\\{0\\}$ with
$\left|a\right|^{2}=\left|b\right|^{2}$ such that $a\cdot r=b\cdot
t\in\mathcal{O}_{K}$.

###### Proof.

This follows readily from the definitions. ∎

######  Definition 3.7.

Let $\alpha\in P{\mathscr{A}}$ and $x\in{\mathbb{R}}H^{n}_{\alpha}$. Write
${\mathcal{H}}(x)=\left\\{H_{r_{1}},\dotsc,H_{r_{k}}\right\\}$, see Definition
3.1. By Lemma 3.5, the involution $\alpha$ induces an involution on the set
${\mathcal{H}}(x)$. Define $\alpha\colon I\to I$ as the resulting involution
on the set $I=\left\\{1,\dotsc,k\right\\}$.

######  Proposition 3.8.

Let $\alpha\in P{\mathscr{A}}$ and $x\in{\mathbb{R}}H^{n}_{\alpha}$. Write
${\mathcal{H}}(x)=\left\\{H_{r_{1}},\dotsc,H_{r_{k}}\right\\}$, and let
$g=\phi_{r_{1}}^{i_{1}}\circ\cdots\circ\phi_{r_{k}}^{i_{k}}\in G(x)$ for some
$i_{\nu}\in\mathbb{Z}/m$. The following are equivalent:

  1. 1.

We have $g\circ\alpha\in P{\mathscr{A}}$. Equivalently: $g\circ\alpha$ is an
involution.

  2. 2.

For each $\nu\in I$, we have $i_{\nu}\equiv i_{\alpha(\nu)}\bmod m$.

###### Proof.

This follows from Lemma’s 3.3, 3.4 and 3.5, and Proposition 3.6. ∎

###  3.2 The definition of the glued space

We are now in position to formulate for which $\alpha,\beta\in P{\mathscr{A}}$
we glue the space $\mathbb{R}H^{n}_{\alpha}$ to the space
$\mathbb{R}H^{n}_{\beta}$, and how we glue these spaces together.

######  Definition 3.9.

Define a relation $R\subset\widetilde{Y}\times\widetilde{Y}$ as follows. An
element

$(x_{\alpha},y_{\beta})\in\mathbb{R}H^{n}_{\alpha}\times\mathbb{R}H^{n}_{\beta}\subset\widetilde{Y}\times\widetilde{Y}$

is an element of $R$ if the following conditions are satisfied:

  1. 1.

The images of $x_{\alpha}$ and $y_{\beta}$ in $\mathbb{C}H^{n}$ under the
natural map $\widetilde{Y}\to\mathbb{C}H^{n}$ agree.

  2. 2.

If $\alpha\neq\beta$, then $x_{\alpha}=y_{\beta}$ lies in ${\mathscr{H}}$ and
$\beta=g\circ\alpha\in P{\mathscr{A}}$ for some element $g\in
G(x_{\alpha})=G(y_{\beta})$ (c.f. Lemma 2.4).

######  Remark 3.10.

Conditions 1 and 2 in Definition 3.9 say that we are identifying points of
$\mathbb{R}H^{n}_{\alpha}\cap{\mathscr{H}}$ and
$\mathbb{R}H^{n}_{\beta}\cap{\mathscr{H}}$ that have the same image in
$\mathbb{C}H^{n}$. But we do not glue all such points: the real structures
$\alpha$ and $\beta$ are required to differ by complex reflections in the
hyperplanes that pass through $x$. In fact, we will see below (see Lemma 4.2)
that the glueing rules can be rephrased as follows: we glue
$\mathbb{R}H^{n}_{\alpha}$ and $\mathbb{R}H^{n}_{\beta}$ along their
intersection, provided that for some (equivalently, any)
$x\in\mathbb{R}H^{n}_{\alpha}\cap\mathbb{R}H^{n}_{\beta}$, the real structures
$\alpha$ and $\beta$ differ by reflections in hyperplanes
$H_{r}\subset{\mathscr{H}}$ that pass through $x$.

######  Lemma 3.11.

The relation $R$ is an equivalence relation.

###### Proof.

Consider three elements $x_{\alpha},y_{\beta},z_{\gamma}\in\widetilde{Y}$. The
fact that $x_{\alpha}\sim x_{\alpha}$ is clear.

Suppose that $x_{\alpha}\sim y_{\beta}$. If $\alpha=\beta$ then
$x_{\alpha}=y_{\beta}\in\widetilde{Y}$ hence $y_{\beta}\sim x_{\alpha}$. If
$\alpha\neq\beta$ then
$x_{\alpha}=y_{\beta}\in{\mathscr{H}}\subset\mathbb{C}H^{n}$, and
$\beta=g\circ\alpha$ for $g\in G(x_{\alpha})=G(y_{\beta})$ as in Definition
3.9. Since $\alpha=g^{-1}\circ\beta$ with $g^{-1}\in G(x_{\alpha})$, this
shows that $y_{\beta}\sim x_{\alpha}$.

Suppose that $x_{\alpha}\sim y_{\beta}$ and $y_{\beta}\sim z_{\gamma}$; we
claim that $x_{\alpha}\sim z_{\gamma}$. We may and do assume that
$\alpha,\beta$ and $\gamma$ are different, which implies that
$x_{\alpha}=y_{\beta}=z_{\gamma}\in{\mathscr{H}}$, that $\gamma=h\circ\beta$
for some $h\in G(y_{\beta})$, and that $\beta=g\circ\alpha$ for some $g\in
G(x_{\alpha})$. We obtain $\gamma=h\circ\beta=h\circ g\circ\alpha$ for $h\circ
g\in G(x_{\alpha})=G(y_{\beta})=G(z_{\gamma})$. ∎

######  Lemma 3.12.

The action of $P\Gamma$ on $\mathbb{C}H^{n}$ induces an action of $P\Gamma$ on
$\widetilde{Y}$ which is compatible with the equivalence relation $R$.
Therefore, $P\Gamma$ acts naturally on $Y$. Moreover,
$P\Gamma\setminus\widetilde{Y}=\coprod_{\alpha\in
C{\mathscr{A}}}P\Gamma_{\alpha}\setminus\mathbb{R}H^{n}_{\alpha}$.

###### Proof.

If $\phi\in P\Gamma$, then
$\phi\left(\mathbb{R}H^{n}_{\alpha}\right)=\mathbb{R}H^{n}_{\phi\alpha\phi^{-1}}$
hence $P\Gamma$ acts on $\widetilde{Y}=\coprod_{\alpha\in
P{\mathscr{A}}}\mathbb{R}H^{n}_{\alpha}$, and

$P\Gamma\setminus\widetilde{Y}=P\Gamma\setminus\coprod_{\alpha\in
P{\mathscr{A}}}\mathbb{R}H^{n}_{\alpha}=\coprod_{\alpha\in
C{\mathscr{A}}}P\Gamma_{\alpha}\setminus\mathbb{R}H^{n}_{\alpha}.$

Now suppose that $x_{\alpha}\sim y_{\beta}\in\widetilde{Y}$ and $f\in
P\Gamma$. Then $f(x_{\alpha})\in\mathbb{R}H^{n}_{f\alpha f^{-1}}$ and
$f(y_{\beta})\in{\mathbb{R}}H^{n}_{f\beta f^{-1}}$. We claim that
$f(x_{\alpha})_{f\alpha f^{-1}}\sim f(y_{\beta})_{f\beta f^{-1}}.$ For this,
we may and do assume that $x_{\alpha}\neq y_{\beta}$, hence
$x_{\alpha}=y_{\beta}\in{\mathscr{H}}$ and $\beta=g\circ\alpha$ for some $g\in
G(x_{\alpha})$ as in Definition 3.9. In particular,
$f(x_{\alpha})=f(y_{\beta})$. Since $f\circ\phi_{r}^{i}\circ
f^{-1}=\phi_{f(r)}^{i}$ for each $r\in{\mathscr{R}}$ and $i\in\mathbb{Z}/m$,
and $h(x,r)=0$ if and only if $h(f(x),f(r))=0$, we have $fG(x)f^{-1}=G(f(x))$
for each $x\in{\mathbb{C}}H^{n}$. This proves that $f\beta
f^{-1}=f(g\circ\alpha)f^{-1}=fgf^{-1}\circ(f\alpha f^{-1})$ with $fgf^{-1}\in
G(f(x_{\alpha}))$, and therefore $f(x_{\alpha})\sim
f(y_{\beta})\in\widetilde{Y}$. ∎

######  Definition 3.13.

Define $Y(\Lambda)$ as the quotient of $\widetilde{Y}$ by the equivalence
relation $R$ introduced in Definition 3.9, and equip it with the quotient
topology. By Lemma 3.12, the group $P\Gamma$ acts on $Y(\Lambda)$. We define

$X(\Lambda)=P\Gamma\setminus Y(\Lambda),$

and call $X(\Lambda)$ the glued space attached to the hermitian
$\mathcal{O}_{K}$-lattice $\Lambda$.

##  4 The hyperbolic orbifold structure of the glued space

Section 4 is devoted to the proof of the following theorem.

######  Theorem 4.1.

  1. 1.

The space $Y(\Lambda)$ admits a path metric such that the natural map
$Y(\Lambda)\to\mathbb{C}H^{n}$ is a local isometric embedding. This metric on
$Y(\Lambda)$ induces a complete path metric on the glued space
$X(\Lambda)=P\Gamma\setminus Y(\Lambda)$.

  2. 2.

Each point $x\in X(\Lambda)$ admits an open neighborhood $U\subset X(\Lambda)$
which is isometric to the quotient of an open subset $V\subset\mathbb{R}H^{n}$
by a finite group of isometries.

  3. 3.

The glued space $X(\Lambda)$ has a canonical realization of a real hyperbolic
orbifold of finite volume.

  4. 4.

One has $\coprod_{\alpha\in
C{\mathscr{A}}}P\Gamma_{\alpha}\setminus\left(\mathbb{R}H^{n}_{\alpha}-{\mathscr{H}}\right)\subset
X(\Lambda)$ as an open suborbifold.

  5. 5.

The connected components of the real hyperbolic orbifold $X(\Lambda)$ are
uniformized by $\mathbb{R}H^{n}$: for each component $C\subset X(\Lambda)$
there exists a lattice $P\Gamma_{C}\subset\textnormal{PO}(n,1)$ and an
isomorphism of real hyperbolic orbifolds $C\cong
P\Gamma_{C}\setminus\mathbb{R}H^{n}$. Consequently,

$X(\Lambda)\cong\coprod_{C\in\pi_{0}(P\Gamma\setminus
K)}P\Gamma_{C}\setminus\mathbb{R}H^{n}.$

In several cases, the glued space $X(\Lambda)$ is connected: this happens when
$K=\mathbb{Q}(\zeta_{3})$ and $\Lambda=\mathbb{Z}[\zeta_{3}]^{2,1}$ or
$\Lambda=\mathbb{Z}[\zeta_{3}]^{4,1}$, see [ACT06, ACT10]. See [GF23] for an
example with $K=\mathbb{Q}(\zeta_{5})$. If
$\Lambda=\mathbb{Z}[\zeta_{3}]^{3,1}$, then $X(\Lambda)$ has two components,
see [ACT07, Remark 6].

###  4.1 The path metric of the glued space

We start with a lemma. We will need it in the proof of Lemma 4.4 below, which
will be used to define a path metric on $X(\Lambda)$ making it locally
isometric to quotients of $\mathbb{R}H^{n}$ by finite groups of isometries. It
also serves as a sanity check: if there exists one element
$x\in\mathbb{R}H^{n}_{\alpha}\cap\mathbb{R}H^{n}_{\beta}$ such that
$x_{\alpha}\sim x_{\beta}$, then one glues the entire space
$\mathbb{R}H^{n}_{\alpha}$ to the space $\mathbb{R}H^{n}_{\beta}$ along their
intersection in $\mathbb{C}H^{n}$.

######  Lemma 4.2.

  1. 1.

Let $g=\prod_{\nu=1}^{k}\phi_{r_{\nu}}^{i_{\nu}}\in\Gamma$ for some set
$\\{r_{\nu}\\}$ of mutually orthogonal short roots $r_{\nu}$, where
$i_{\nu}\not\equiv 0\bmod m$ for each $\nu$. Then
$\left(\mathbb{C}H^{n}\right)^{g}=\cap_{\nu=1}^{k}H_{r_{\nu}}$.

  2. 2.

Let $\alpha,\beta\in P{\mathscr{A}}$ and
$x\in\mathbb{R}H^{n}_{\alpha}\cap\mathbb{R}H^{n}_{\beta}$ such that
$x_{\alpha}\sim x_{\beta}$. Then $y_{\alpha}\sim y_{\beta}$ for every
$y\in\mathbb{R}H^{n}_{\alpha}\cap\mathbb{R}H^{n}_{\beta}$.

  3. 3.

The map $\widetilde{Y}\to\mathbb{C}H^{n}$ descends to a $P\Gamma$-equivariant
map ${\mathscr{P}}\colon Y(\Lambda)\to\mathbb{C}H^{n}$.

###### Proof.

1\. Let $y\in V$ be representing an element in
$\left(\mathbb{C}H^{n}\right)^{\phi}$. Since the $r_{i}$ are orthogonal, and
$g(y)=\lambda$ for some $\lambda\in\mathbb{C}^{\ast}$, we have

$\displaystyle
g(y)=\prod_{\nu=1}^{k}\phi_{r_{\nu}}^{i_{\nu}}(y)=y-\sum_{\nu=1}^{k}\left(1-\zeta^{i_{\nu}}\right)h(y,r_{\nu})r_{\nu}=\lambda
y,$ (10)

hence
$(1-\lambda)y=\sum_{i=1}^{\ell}\left(1-\zeta^{i_{\nu}}\right)h(y,r_{\nu})r_{\nu}\in
V$. But $y$ spans a negative definite subspace of $V$ while the $r_{\nu}$ span
a positive definite subspace, so that we must have
$1-\lambda=0=\sum_{\nu=1}^{k}\left(1-\zeta^{i_{\nu}}\right)h(y,r_{\nu})r_{\nu}$.
Since the $r_{\nu}$ are mutually orthogonal, they are linearly independent;
since $\zeta^{i_{\nu}}\neq 1$ we find $h(y,r_{\nu})=0$ for each $\nu$.
Conversely, if $x\in\cap H_{r_{\nu}}$, then $\phi_{r_{\nu}}^{i_{\nu}}(x)=x$
for each $\nu$.

2\. Since $x_{\alpha}\sim x_{\beta}$, there exists $g\in G(x)$ such that
$\beta=g\circ\alpha$. Write
${\mathcal{H}}(x)=\left\\{H_{r_{1}},\dotsc,H_{r_{k}}\right\\}$. Let
$y\in\mathbb{R}H^{n}_{\alpha}\cap\mathbb{R}H^{n}_{\beta}$. Then
$\alpha(y)=\beta(y)=y$ implies that $g(y)=y$. In particular,
$y\in\cap_{\nu}H_{r_{\nu}}$ by Part 1, which implies that
${\mathcal{H}}(x)\subset{\mathcal{H}}(y)$, which in turn implies that
$G(x)\subset G(y)$. We conclude that $g\in G(y)$. Hence $y_{\alpha}\sim
y_{\beta}$.

3\. If $x_{\alpha}\sim y_{\beta}$, then $x=y\in\mathbb{C}H^{n}$. ∎

By Lemma 4.2, we obtain continuous maps

${\mathscr{P}}\colon Y(\Lambda)\to\mathbb{C}H^{n}\quad{\textnormal{ and
}}\quad\overline{{\mathscr{P}}}:X(\Lambda)=P\Gamma\setminus Y(\Lambda)\to
P\Gamma\setminus\mathbb{C}H^{n}.$

Our next goal is to prove that each point $x\in Y(\Lambda)$ has a
neighbourhood $V\subset Y(\Lambda)$ that maps homeomorphically onto a finite
union $\cup_{i=1}^{\ell}\mathbb{R}H^{n}_{\alpha_{i}}\subset\mathbb{C}H^{n}$.
Hence $x$ has an open neighourhood $x\in U\subset V$ that identifies with an
open set in a union of copies of $\mathbb{R}H^{n}$ in $\mathbb{C}H^{n}$ under
the map ${\mathscr{P}}$. This allows us to define a metric on $Y(\Lambda)$ by
pulling back the metric on $\mathbb{C}H^{n}$.

######  Lemma 4.3.

For each compact set $Z\subset\mathbb{C}H^{n}$ there are only finitely
$\alpha\in P{\mathscr{A}}$ with $Z\cap\mathbb{R}H^{n}_{\alpha}\neq\emptyset$.

###### Proof.

The subgroup $P\Gamma^{\prime}\subset\textnormal{Isom}(\mathbb{C}H^{n})$ (see
(4) and Lemma 2.4) acts properly discontinuously on $\mathbb{C}H^{n}$. So if
$S$ is the set of $\alpha\in P{\mathscr{A}}$ such that $\alpha Z\cap
Z\neq\emptyset$, then $S$ is finite. In particular, $Z$ meets only finitely
many sets $\mathbb{R}H^{n}_{\alpha}$. ∎

Fix a point $f\in Y(\Lambda)$ and a point $x_{\alpha}\in\widetilde{Y}$ lying
above $f$. Let $\alpha_{1},\dotsc,\alpha_{\ell}$ be the elements in
$P{\mathscr{A}}$ such that $x_{\alpha_{i}}\sim x_{\alpha}$ for each $i\in
I\coloneqq\left\\{1,\dotsc,\ell\right\\}$ (since the group $G(x)$ is finite by
Lemma 3.2, these are finite in number).

Let $p:\widetilde{Y}\to Y(\Lambda)$ be the quotient map, and define

$Y_{f}=p\left(\coprod_{i=1}^{\ell}\mathbb{R}H^{n}_{\alpha_{i}}\right)\subset
Y(\Lambda).$ (11)

We prove that $Y(\Lambda)$ is locally isometric to opens in unions of real
hyperbolic subspaces of $\mathbb{C}H^{n}$. Indeed, we have the following:

######  Lemma 4.4.

  1. 1.

The set $Y_{f}$ is closed in $Y(\Lambda)$.

  2. 2.

We have
${\mathscr{P}}\left(Y_{f}\right)=\cup_{i=1}^{\ell}\mathbb{R}H^{n}_{\alpha_{i}}\subset\mathbb{C}H^{n}$,
and the map

${\mathscr{P}}_{f}:Y_{f}\to\bigcup_{i=1}^{\ell}\mathbb{R}H^{n}_{\alpha_{i}}$

induced by ${\mathscr{P}}$ is a homeomorphism.

  3. 3.

The set $Y_{f}\subset Y(\Lambda)$ contains an open neighborhood $U_{f}$ of $f$
in $Y(\Lambda)$.

###### Proof.

1\. One has

$p^{-1}\left(Y_{f}\right)=p^{-1}\left(p\left(\coprod_{i=1}^{\ell}\mathbb{R}H^{n}_{\alpha_{i}}\right)\right)=\bigcup_{i=1}^{\ell}p^{-1}\left(p\left(\mathbb{R}H^{n}_{\alpha_{i}}\right)\right)\subset\widetilde{Y}.$

Therefore, it suffices to show that
$p^{-1}\left(p\left(\mathbb{R}H^{n}_{\alpha_{i}}\right)\right)$ is closed in
$\widetilde{Y}$. But notice that $x_{\beta}\in
p^{-1}\left(p\left(\mathbb{R}H^{n}_{\alpha}\right)\right)$ if and only if
$x\in\mathbb{R}H^{n}_{\alpha}$ and $x_{\alpha}\sim x_{\beta}$, which implies
(Lemma 4.2) that $\mathbb{R}H^{n}_{\alpha}\cap\mathbb{R}H^{n}_{\beta}\subset
p^{-1}\left(p\left(\mathbb{R}H^{n}_{\alpha}\right)\right)$. Hence for any
$\alpha\in P{\mathscr{A}}$, one has

$p^{-1}\left(p\left(\mathbb{R}H^{n}_{\alpha}\right)\right)=\coprod_{\beta\sim\alpha}\mathbb{R}H^{n}_{\alpha}\cap\mathbb{R}H^{n}_{\beta},$

where $\beta\sim\alpha$ if and only if there exists
$x\in\mathbb{R}H^{n}_{\alpha}\cap\mathbb{R}H^{n}_{\beta}$ such that
$x_{\alpha}\sim x_{\beta}$. It follows that
$p^{-1}\left(p\left(\mathbb{R}H^{n}_{\alpha}\right)\right)\cap\mathbb{R}H^{n}_{\beta}$
is closed in $\mathbb{R}H^{n}_{\beta}$ for every $\beta\in P{\mathscr{A}}$.
But the $\mathbb{R}H^{n}_{\beta}$ are open in $\widetilde{Y}$ and cover
$\widetilde{Y}$, so that
$p^{-1}\left(p\left(\mathbb{R}H^{n}_{\alpha}\right)\right)$ is closed in
$\widetilde{Y}$.

2\. We have

${\mathscr{P}}_{f}(Y_{f})={\mathscr{P}}\left(p\left(\coprod_{i=1}^{\ell}\mathbb{R}H^{n}_{\alpha_{i}}\right)\right)=\widetilde{{\mathscr{P}}}\left(\coprod_{i=1}^{\ell}\mathbb{R}H^{n}_{\alpha_{i}}\right)=\bigcup_{i=1}^{\ell}\mathbb{R}H^{n}_{\alpha_{i}}\subset\mathbb{C}H^{n}.$

To prove injectivity, let $x_{\alpha_{i}},y_{\alpha_{j}}\in\widetilde{Y}$ and
suppose that $x=y\in\mathbb{C}H^{n}$. Then indeed, $x_{\alpha_{i}}\sim
y_{\alpha_{j}}$ because $\sim$ is an equivalence relation by Lemma 3.11.

Let $Z\subset\mathbb{C}H^{n}$ be a compact set. Let
$\widetilde{{\mathscr{P}}}:\widetilde{Y}\to\mathbb{C}H^{n}$ be the canonical
map. Remark that $Z$ meets only finitely many of the
$\mathbb{R}H^{n}_{\alpha}$ for $\alpha\in P{\mathscr{A}}$, see Lemma 4.3. Each
$Z\cap\mathbb{R}H^{n}_{\alpha}$ is closed in $Z$ since
$\mathbb{R}H^{n}_{\alpha}$ is closed in $\mathbb{C}H^{n}$, so each
$Z\cap\mathbb{R}H^{n}_{\alpha}$ is compact. We conclude that
${\widetilde{{\mathscr{P}}}}^{-1}(Z)=\coprod Z\cap\mathbb{R}H^{n}_{\alpha}$ is
compact. In particular, $\widetilde{{\mathscr{P}}}$ is closed.

Finally, we prove that ${\mathscr{P}}_{f}$ is closed. Let $Z\subset Y_{f}$ be
a closed set. Then $Z$ is closed in $Y$ by Part 1, hence $p^{-1}(Z)$ is closed
in $\widetilde{Y}$, hence $\widetilde{{\mathscr{P}}}\left(p^{-1}(Z)\right))$
is closed in $\mathbb{C}H^{n}$, so that

${\mathscr{P}}_{f}(Z)={\mathscr{P}}(Z)=\widetilde{{\mathscr{P}}}\left(p^{-1}\left(Z\right)\right)=\left(\widetilde{{\mathscr{P}}}\left(p^{-1}\left(Z\right)\right)\right)\cap\left(\cup_{i=1}^{\ell}\mathbb{R}H^{n}_{\alpha_{i}}\right)$

is closed in $\cup_{i=1}^{\ell}\mathbb{R}H^{n}_{\alpha_{i}}$.

3\. Let $x={\mathscr{P}}(f)\in\mathbb{C}H^{n}$. Since $\mathbb{C}H^{n}$ is
locally compact, there exists a compact set $Z\subset\mathbb{C}H^{n}$ and an
open set $U\subset\mathbb{C}H^{n}$ with $x\in U\subset Z$. Since $Z$ is
compact, it meets only finitely many of the
$\mathbb{R}H^{n}_{\beta}\subset\mathbb{C}H^{n}$ (Lemma 4.3). Consequently, the
same holds for $U$; define $V={\mathscr{P}}^{-1}(U)\subset Y(\Lambda)$. Define

${\mathscr{B}}=\\{\beta\in
P{\mathscr{A}}:U\cap\mathbb{R}H^{n}_{\beta}\neq\emptyset\\}.$

Also define, for $\beta\in P{\mathscr{A}}$,
$Z_{\beta}=p\left(\mathbb{R}H^{n}_{\beta}\right)\subset Y(\Lambda)$. Then

$f\in
V\subset\bigcup_{\beta\in{\mathscr{B}}}Z_{\beta}=\bigcup_{\begin{subarray}{c}\beta\in{\mathscr{B}}\\\
\beta(x)=x\end{subarray}}Z_{\beta}\bigcup_{\begin{subarray}{c}\beta\in{\mathscr{B}}\\\
\beta(x)\neq x\end{subarray}}Z_{\beta}.$

Since each $Z_{\beta}$ is closed in $Y(\Lambda)$ by the proof of part 1, there
is an open $V^{\prime}\subset V$ with

$f\in V^{\prime}\subset\bigcup_{\begin{subarray}{c}\beta\in{\mathscr{B}}\\\
\beta(x)=x\end{subarray}}Z_{\beta}=\bigcup_{\begin{subarray}{c}\beta\in{\mathscr{B}}\\\
\beta(x)=x\\\ x_{\beta}\sim
x_{\alpha}\end{subarray}}Z_{\beta}\bigcup_{\begin{subarray}{c}\beta\in{\mathscr{B}}\\\
\beta(x)=x\\\ x_{\beta}\not\sim x_{\alpha}\end{subarray}}Z_{\beta}$

Hence again there exists an open subset $V^{\prime\prime}\subset V^{\prime}$
with

$f\in
V^{\prime\prime}\subset\bigcup_{\begin{subarray}{c}\beta\in{\mathscr{B}}\\\
\beta(x)=x\\\ x_{\beta}\sim
x_{\alpha}\end{subarray}}Z_{\beta}\subset\bigcup_{\begin{subarray}{c}\beta\in
P{\mathscr{A}}\\\ \beta(x)=x\\\ x_{\beta}\sim
x_{\alpha}\end{subarray}}Z_{\beta}=Y_{f}.$

Therefore, $U_{f}\coloneqq V^{\prime\prime}\subset Y$ satisfies the
requirements. ∎

######  Lemma 4.5.

The topological space $Y(\Lambda)$ is Hausdorff.

###### Proof.

Let $f,f^{\prime}\in Y(\Lambda)$ be elements such that $f\neq f^{\prime}$.
First suppose that $f\not\in Y_{f^{\prime}}$. Since $Y_{f^{\prime}}$ is closed
in $Y(\Lambda)$ by Lemma 4.4, there is an open neighbourhood $U$ of $f$ such
that $U\cap U_{f^{\prime}}\subset U\cap Y_{f^{\prime}}=\emptyset$.

Next, suppose that $f\in Y_{f^{\prime}}$. Lift $f$ and $f^{\prime}$ to
elements $x_{\alpha},y_{\beta}\in\widetilde{Y}$. Assume first that $x=y$. This
means that ${\mathscr{P}}(f)={\mathscr{P}}(f^{\prime})$. Since
${\mathscr{P}}:Y_{f^{\prime}}\to\mathbb{C}H^{n}$ is injective, this implies
that $f=f^{\prime}$, contradiction. So we have $x\neq y\in\mathbb{C}H^{n}$.
But $\mathbb{C}H^{n}$ is Hausdorff, so there are open subsets
$\left(U\subset\mathbb{C}H^{n},V\subset\mathbb{C}H^{n}\right)$ such that $x\in
U$, $y\in V$ and $U\cap V=\emptyset$. Then
${\mathscr{P}}^{-1}(U)\cap{\mathscr{P}}^{-1}(V)=\emptyset$. ∎

######  Proposition 4.6.

$Y(\Lambda)$ is a path metric space, piecewise isometric to $\mathbb{R}H^{n}$.

###### Proof.

For each $f\in Y(\Lambda)$ there exists an open neighborhood $f\in
U_{f}\subset Y(\Lambda)$ such that ${\mathscr{P}}$ induces a homeomorphism
$Y\supset U_{f}\xrightarrow{\sim}{\mathscr{P}}(U_{f})\subset\mathbb{C}H^{n}$.
Indeed, this follows from Lemma 4.4. Pull back the metric on
${\mathscr{P}}(U_{f})$ to obtain a metric on $U_{f}$. Then define a metric on
$Y(\Lambda)$ as the largest metric which is compatible with the metric on each
open set $U_{f}$ and which preserves the lengths of paths. ∎

######  Proposition 4.7.

The path metric on $Y(\Lambda)$ descends to a path metric on $X(\Lambda)$.

###### Proof.

The metric on $Y(\Lambda)$ descends in any case to a pseudo-metric on
$X(\Lambda)$, and by [Gro07, Chapter 1], this is a metric if $P\Gamma$ acts by
isometries on $Y(\Lambda)$ with closed orbits. This is true: the fact that
$P\Gamma$ acts isometrically on $Y(\Lambda)$ comes from the
$P\Gamma$-equivariance of ${\mathscr{P}}:Y\to\mathbb{C}H^{n}$ (Lemma 4.2)
together with the construction of the metric on $Y(\Lambda)$ (Proposition
4.6). To check that the $P\Gamma$-orbits are closed in $Y(\Lambda)$, let $f\in
Y(\Lambda)$ with representative $x_{\alpha}\in\widetilde{Y}$. By equivariance
of $p:\widetilde{Y}\to Y(\Lambda)$, we have $p^{-1}\left(P\Gamma\cdot
f\right)=P\Gamma\cdot\left(p^{-1}f\right)$, so since $p$ is a quotient map, it
suffices to show that

$P\Gamma\cdot\left(p^{-1}f\right)=P\Gamma\cdot\bigcup_{x_{\beta}\sim
x_{\alpha}}x_{\beta}=\bigcup_{x_{\beta}\sim x_{\alpha}}P\Gamma\cdot x_{\beta}$

is closed in $\widetilde{Y}$, thus that each orbit $P\Gamma\cdot x_{\beta}$ is
closed in $\widetilde{Y}$. Since $P\Gamma$ is discrete, it suffices to show
that $P\Gamma$ acts properly on $\widetilde{Y}$. So let
$Z\subset\widetilde{Y}$ be any compact set: we claim that $\\{g\in
P\Gamma:gZ\cap Z\neq\emptyset\\}$ is a finite set. Indeed, for each $g\in
P\Gamma$, one has $\widetilde{{\mathscr{P}}}\left(gZ\cap Z\right)\subset
g\widetilde{{\mathscr{P}}}(Z)\cap\widetilde{{\mathscr{P}}}(Z)$, and the latter
is non-empty for only finitely many $g\in P\Gamma$, by properness of the
action of $P\Gamma$ on $\mathbb{C}H^{n}$.

Since $Y(\Lambda)$ is a path metric space, the same holds for $X(\Lambda)$,
see [Gro07]. ∎

###  4.2 The orbifold structure of the glued space

The next step is to prove that the glued space $X(\Lambda)=P\Gamma\setminus
Y(\Lambda)$ (see Definition 3.13) is locally isometric to quotients of open
sets in $\mathbb{R}H^{n}$ by finite groups of isometries.

######  Definition 4.8.

Let $f\in Y(\Lambda)$ with representative $x_{\alpha}\in\widetilde{Y}$. Thus,
$x$ is an element in ${\mathbb{C}}H^{n}$, and $\alpha\in P{\mathscr{A}}$ is
the class of an anti-unitary involution such that $\alpha(x)=x$.

  1. 1.

The _nodes_ of $f$ are by definition the nodes of $x_{\alpha}$ (see Definition
3.1). Thus, these are the hyperplanes $H\in{\mathcal{H}}(x)$, i.e. the
hyperplanes $H_{r}\in{\mathcal{H}}$ defined by short roots $r\in{\mathscr{R}}$
such that $x\in H_{r}$ (equivalently, such that $h(x,r)=0$).

  2. 2.

The number of nodes of $f$ is the cardinality of ${\mathcal{H}}(x)$.

  3. 3.

The anti-unitary involution $\alpha\in P{\mathscr{A}}$ induces an involution
on the set ${\mathcal{H}}(x)$ by Lemma 3.5. Let $H\in{\mathcal{H}}(x)$ be a
node. We call $H$ a _real node_ of $f$ if $\alpha(H)=H$. We call
$(H,\alpha(H))$ a _pair of complex conjugate nodes_ of $f$ if $\alpha(H)\neq
H$.

  4. 4.

If $k$ is the number of nodes of $f$, we generally write $k=2a+b$, with $a$
the number of pairs of complex conjugate nodes of $f$, and $b$ the number of
real nodes of $f$.

Fix again a point $f\in Y(\Lambda)$ and a point $x_{\alpha}\in\widetilde{Y}$
lying above $f$. Let $k=2a+b$ be the number of nodes of $f$. Thus
$x\in{\mathbb{R}}H^{n}_{\alpha}$, and there exist
$r_{1},\dotsc,r_{k}\in{\mathscr{R}}$ such that

${\mathcal{H}}(x)=\left\\{H_{r_{1}},\dotsc,H_{r_{k}}\right\\},\quad
G(x)=\langle\phi_{r_{1}},\dotsc,\phi_{r_{\ell}}\rangle\cong(\mathbb{Z}/m)^{k}.$

For $\beta\in P{\mathscr{A}}$, observe that $x_{\beta}\sim x_{\alpha}$ if and
only if $\alpha\circ\beta\in G(x)$. We relabel the $r_{i}$ so that they
satisfy the following condition:

$\displaystyle\begin{split}\alpha(H_{r_{i}})=\begin{cases}H_{r_{i+1}}{\textnormal{
for }}i{\textnormal{ odd and }}i\leq 2a,\\\ H_{r_{i-1}}{\textnormal{ for
}}i{\textnormal{ even and }}i\leq 2a,{\textnormal{ and }}\\\
H_{r_{i}}{\textnormal{ for
}}i\in\left\\{2a+1,\dotsc,k\right\\}.\end{cases}\end{split}$ (12)

In other words, $H_{r_{i}}$ is a real root if and only if $i>2a$, and
$\left(H_{r_{i}},H_{r_{i+1}}\right)$ is a pair of complex conjugate roots if
and only if $i<2a$ is odd.

######  Lemma 4.9.

Continue with the notation from above.

  1. 1.

Let $\beta\in P{\mathscr{A}}$ be such that $x_{\beta}\sim x_{\alpha}$. Then

$\beta=\prod_{i=1}^{a}\left(\phi_{r_{2i-1}}\circ\phi_{r_{2i}}\right)^{j_{i}}\circ\prod_{i=2a+1}^{k}\phi_{r_{i}}^{j_{i}}\circ\alpha$

for some $j_{1},\dotsc,j_{a},j_{2a+1},\dotsc,j_{k}\in\mathbb{Z}/m$. In
particular, there are $m^{a+b}$ such $\beta$.

  2. 2.

There is an isometry
$\mathbb{C}H^{n}\xrightarrow{\sim}{{\mathbb{B}}}^{n}(\mathbb{C})$ identifying
$x$ with the origin, $\phi_{r_{i}}$ with the map

${{\mathbb{B}}}^{n}(\mathbb{C})\to{{\mathbb{B}}}^{n}(\mathbb{C}),\;\;\;(t_{1},\dotsc,t_{i},\dotsc,t_{n})\mapsto(t_{1},\dotsc,\zeta
t_{i},\dotsc,t_{n}),$

and $\alpha$ with the map defined by

$t_{i}\mapsto\begin{cases}\bar{t}_{i+1}\;\;\;\textnormal{for $i$ odd and
$i\leq 2a$}\\\ \bar{t}_{i-1}\;\;\;\textnormal{for $i$ even and $i\leq 2a$}\\\
\bar{t}_{i}\;\;\;\;\;\;\textnormal{for $i>2a$}.\end{cases}$ (13)

###### Proof.

1\. This follows readily from Proposition 3.8.

2\. Since the $H_{r_{i}}$ are orthogonal by Condition 2.2, and their
intersection contains $x$, we can find coordinates $t_{1},\dotsc,t_{n+1}$ on
$V$ that induce an identification
$\left(V,h\right)\cong\mathbb{C}^{n,1}\coloneqq\left(\mathbb{C}^{n+1},H\right)$
with
$H(x,x)=\left|x_{1}\right|^{2}+\cdots+\left|x_{n}\right|^{2}-\left|x_{n+1}\right|^{2}$,
in such a way that $H_{r_{i}}\subset V$ is identified with the hyperplane
$\\{t_{i}=0\\}\subset\mathbb{C}^{n+1}$ and $x\in\cap_{i}H_{r_{i}}$ with the
point $(0,0,\dotsc,0,1)$. We will do this in the following way. Define

$T=\langle x\rangle\oplus\langle r_{1}\rangle\oplus\cdots\oplus\langle
r_{k}\rangle\subset V,\quad W=T^{\perp}=\\{w\in V\mid h(w,t)=0\;\;\forall t\in
T\\}.$

For each $i\in I=\\{1,\dotsc,k\\}$, we have $\alpha(r_{i})=\lambda_{i}\cdot
r_{\alpha(i)}$ for some $\lambda_{i}\in K$ (see Lemma 3.5, Definition 3.7, and
Lemmas 3.3 and 3.2). Observe that $\alpha(W)=W$. Since $W\subset\langle
x\rangle^{\perp}$, the hermitian space $(W,h|_{W})$ is positive definite. Let
$\\{w_{1},\dotsc,w_{n-k}\\}\subset W$ be an orthonormal basis such that
$\alpha(w_{i})=w_{i}$. Let $\\{e_{i}\\}_{i=1}^{n+1}$ be the standard basis of
$\mathbb{C}^{n+1}$, and define a $\mathbb{C}$-linear isomorphism

$\displaystyle\Phi\colon
V\xrightarrow{\sim}\mathbb{C}^{n+1},\quad\left(\frac{x}{h(x,x)}\mapsto
e_{n+1},\quad r_{i}\mapsto e_{i},\quad w_{i}\mapsto e_{i}\right).$ (14)

By (12), we have that $\alpha(r_{i})=\lambda_{i}\cdot r_{i+1}$ for $i$ odd and
$i\leq 2a$, that $\alpha(r_{i})=\lambda_{i}\cdot r_{i-1}$ for $i$ even and
$i\leq 2a$, and that $\alpha(r_{i})=\lambda_{i}\cdot r_{i}$ for $i>2a$. We
conclude that the anti-linear involution on $\mathbb{C}^{n+1}$ induced by
$\alpha$ and (14) corresponds to the matrix

$\alpha={\textnormal{diag}}(A_{12},A_{34},\dotsc,A_{2a-1,2a},\alpha_{2a+1},\dotsc,\alpha_{n}),\quad
A_{i,i+1}=\begin{pmatrix}0&\alpha_{i}\\\ \alpha_{i+1}&0\end{pmatrix},$

where each $\alpha_{i}$ is an anti-linear involution
$\mathbb{C}\to\mathbb{C}$, and $\alpha_{i}=\alpha_{i+1}$ for $i<2a$ odd. If
$\alpha_{i}(1)=\mu_{i}\in\mathbb{C}^{\ast}$, then
$\mu_{i}^{-1}\cdot\alpha_{i}$ is the complex conjugation map
$\mathbb{C}\to\mathbb{C}$. Since $\left|\mu_{i}\right|=1$, there exists
$\rho_{i}\in\mathbb{C}$ such that $\mu_{i}=\overline{\rho_{i}}/\rho_{i}$ and
$\left|\rho_{i}\right|=1$. This gives
$\mu_{i}^{-1}\cdot\alpha_{i}=\rho_{i}\cdot\alpha_{i}\cdot\rho_{i}^{-1}={\textnormal{conj}}\colon\mathbb{C}\to\mathbb{C}$.
The composition of $\Phi\colon V\to\mathbb{C}^{n+1}$ with the diagonal linear
transformation
${\textnormal{diag}}(\rho_{1},\dotsc,\rho_{n+1})\colon\mathbb{C}^{n+1}\to\mathbb{C}^{n+1}$
induces an isomorphism ${\mathbb{C}}H^{n}\cong{\mathbb{B}}^{n}(\mathbb{C})$
with the required properties. ∎

######  Definition 4.10.

  1. 1.

Define $A_{f}=\textnormal{Stab}_{P\Gamma}(f)$ to be the subgroup of $P\Gamma$
fixing $f\in Y(\Lambda)$. This contains the group
$G(x)\cong(\mathbb{Z}/m)^{k}$.

  2. 2.

Define $B_{f}$ as the subgroup of $G(x)$ generated by the order $m$ complex
reflections associated to the real nodes of $f$. Hence
$B_{f}=\langle\phi_{r_{i}}\rangle_{i>2a}\cong(\mathbb{Z}/m)^{b}$.

Consider the quotient map $p\colon\widetilde{Y}\to Y(\Lambda)$, and the subset
$Y_{f}\subset X(\Lambda)$, see (11).

######  Lemma 4.11.

The stabilizer $A_{f}$ of $f\in Y(\Lambda)$ preserves the subset $Y_{f}\subset
Y(\Lambda)$.

###### Proof.

Let $\psi\in A_{f}$, with $f=p(x_{\alpha})\in Y(\Lambda)$,
$x\in\cap_{i}H_{r_{i}}$. Then $\psi(x)_{\psi\alpha\psi^{-1}}\sim x_{\alpha}$.
Now let $p(y_{\beta})\in Y_{f}$. Then $\beta(x)=x$ and $x_{\alpha}\sim
x_{\beta}$. Hence $x_{\alpha}\sim\psi\cdot x_{\alpha}\sim\psi\cdot
x_{\beta}=\psi(x)_{\psi\beta\psi^{-1}}$. This implies that
$\psi\beta\psi^{-1}\circ\alpha\in G(x)$, so that
$p\left(\psi(y)_{\psi\beta\psi^{-1}}\right)\in Y_{f}$. ∎

We also need the following lemma. Write $m=2^{a}k$ with $k\neq 0\bmod 2$.

######  Lemma 4.12.

Let $T=\\{t\in\mathbb{C}:t^{m}\in\mathbb{R}\\}$. Then
$G=\langle\zeta_{m}\rangle$ acts on $T$ by multiplication. Moreover, each
element in $T/G$ has a unique representative of the form
$\zeta_{2^{a+1}}^{\epsilon}\cdot r\in T$ for $r\geq 0$ and
$\epsilon\in\\{0,1\\}$.

###### Proof.

Observe that $a\geq 1$. Moreover, for $t\in\mathbb{C}$, one has that
$t=r\zeta_{2m}^{j}$ for some $j\in\mathbb{Z}$ and $r\in\mathbb{R}$ if and only
if $t^{m}\in\mathbb{R}$. Since $\gcd(2,k)=1$, we have
$\zeta_{2^{a+1}}\cdot\zeta_{2^{a}k}=(\zeta_{2^{a+1}k})^{k+2}$. Raising both
sides to the power $b=(k+2)^{-1}\in(\mathbb{Z}/m)^{\ast}$ gives
$\zeta_{2m}=\zeta_{2^{a+1}}^{b}\cdot\zeta_{m}^{b}$. Consequently,
$t^{m}\in\mathbb{R}$ if and only if
$t=r\cdot\zeta_{2^{a+1}}^{bj}\cdot\zeta_{m}^{bj}$ for some $r\in\mathbb{R}$.
Finally, $\zeta_{2^{a+1}}^{u}\cdot\zeta_{2^{a}}^{v}=\zeta_{2^{a+1}}^{u+2v}$
hence
$\langle\zeta_{2^{a+1}}\rangle/\langle\zeta_{2^{a}}\rangle\cong\mathbb{Z}/2$.
∎

We obtain the key to Theorem 4.1.

######  Proposition 4.13.

With the above notation, the following assertions are true.

  1. 1.

If $f$ has no nodes, then $G(x)=B_{f}$ is trivial, and
$Y_{f}=\mathbb{R}H^{n}_{\alpha}\cong{{\mathbb{B}}}^{n}(\mathbb{R})$.

  2. 2.

If $f$ has only real nodes, then $B_{f}\setminus Y_{f}$ is isometric to
${{\mathbb{B}}}^{n}(\mathbb{R})$.

  3. 3.

If $f$ has $a$ pairs of complex conjugate nodes ($k=2a$), and no other nodes,
then $B_{f}\setminus Y_{f}=Y_{f}$ is the union of $m^{a}$ copies of
${{\mathbb{B}}}^{n}(\mathbb{R})$, any two of which meet along a
${{\mathbb{B}}}^{2c}(\mathbb{R})$ for some integer $c$ with $0\leq c\leq a$.

  4. 4.

If $f$ has $2a$ complex conjugate nodes and $b$ real nodes, then there is an
isometry between $B_{f}\setminus Y_{f}$ and the union of $m^{a}$ copies of
${{\mathbb{B}}}^{n}(\mathbb{R})$ identified along common
${{\mathbb{B}}}^{2c}(\mathbb{R})^{\prime}s$, that is, the set $Y_{f}$ of case
3 above.

  5. 5.

In each case, $A_{f}$ acts transitively on the indicated copies of
${{\mathbb{B}}}^{n}(\mathbb{R})$. If ${{\mathbb{B}}}^{n}(\mathbb{R})$ is any
one of them, and $\Gamma_{f}=(A_{f}/B_{f})_{{{\mathbb{B}}}^{n}(\mathbb{R})}$
its stabilizer, then the natural map

$\Gamma_{f}\setminus{{\mathbb{B}}}^{n}(\mathbb{R})\to\left(A_{f}/B_{f}\right)\setminus\left(B_{f}\setminus
Y_{f}\right)=A_{f}\setminus Y_{f}$

is an isometry of path metrics.

###### Proof.

1\. This is clear.

2\. Suppose then that $f$ has $k$ real nodes. Then in the local coordinates
$t_{i}$ of Lemma 4.9.2, we have that
$\alpha:{{\mathbb{B}}}^{n}(\mathbb{C})\to{{\mathbb{B}}}^{n}(\mathbb{C})$ is
defined by $\alpha(t_{i})=\bar{t}_{i}$. Part 1 of the same lemma shows that
any $\beta\in P{\mathscr{A}}$ fixing $x$ such that $x_{\alpha}\sim x_{\beta}$
is of the form

${{\mathbb{B}}}^{n}(\mathbb{C})\to{{\mathbb{B}}}^{n}(\mathbb{C}),\;\;\;(t_{1},\dotsc,t_{i},\dotsc,t_{n})\mapsto(\bar{t}_{1}\zeta^{j_{1}},\dotsc,\bar{t}_{k}\zeta^{j_{k}},\bar{t}_{k+1},\dotsc,\bar{t}_{n}).$

Since $f$ has $k$ real nodes and no complex conjugate nodes, we have (writing
$j=(j_{1},\dotsc,j_{k})$ and
$\alpha_{j}=\prod_{i=1}^{k}\phi_{r_{i}}^{j_{i}}\circ\alpha$):

$Y_{f}\cong\bigcup_{j_{1},\dotsc,j_{k}=1}^{m}\mathbb{R}H^{n}_{\alpha_{j}}\cong\left\\{(t_{1},\dotsc,t_{n})\in{{\mathbb{B}}}^{n}(\mathbb{C}):t_{1}^{m},\dotsc,t_{k}^{m},t_{k+1},\dotsc,t_{n}\in\mathbb{R}\right\\}.$

Each of the $2^{k}$ subsets

$\displaystyle
K_{f,\epsilon_{1},\dotsc,\epsilon_{k}}\coloneqq\left\\{(t_{1},\dotsc,t_{n})\in{{\mathbb{B}}}^{n}(\mathbb{C}):\zeta_{2^{a+1}}^{-\epsilon_{1}}t_{1},\dotsc,\zeta_{2^{a+1}}^{-\epsilon_{k}}t_{k}\in\mathbb{R}_{\geq
0}\textnormal{ and }t_{k+1},\dotsc,t_{n}\in\mathbb{R}\right\\},$

indexed by $\epsilon_{1},\dotsc,\epsilon_{k}\in\\{0,1\\}$, is isometric to the
closed region in ${{\mathbb{B}}}^{n}(\mathbb{R})$ bounded by $k$ mutually
orthogonal hyperplanes. By Lemma 4.12, their union $U$ is a fundamental domain
for $B_{f}$, in the sense that it maps homeomorphically and piecewise-
isometrically onto $B_{f}\setminus Y_{f}$. Under its path metric, $U=\cup
K_{f,\epsilon_{1},\dotsc,\epsilon_{k}}$ is isometric to
${{\mathbb{B}}}^{n}(\mathbb{R})$ by the following map:

$U\to{{\mathbb{B}}}^{n}(\mathbb{R}),\;\;\;(t_{1},\dotsc,t_{k})\mapsto\left((-\zeta_{2^{a+1}})^{-\epsilon_{1}}t_{1},\dotsc,(-\zeta_{2^{a+1}})^{-\epsilon_{k}}t_{k},t_{k+1},\dotsc,t_{n}\right).$

This identifies $B_{f}\setminus Y_{f}$ with the standard
${{\mathbb{B}}}^{n}(\mathbb{R})\subset{{\mathbb{B}}}^{n}(\mathbb{C})$.

3\. Now suppose $f$ has $k=2a$ nodes $H_{r_{1}},\dotsc,H_{r_{2a}}$. There are
now $m^{a}$ anti-isometric involutions $\alpha_{j_{i}}$ fixing $x$ and such
that $x_{\alpha_{j_{i}}}\sim x_{\alpha}$: they are given in the coordinates
$t_{i}$ as follows, taking $j=(j_{1},\dotsc,j_{a})\in(\mathbb{Z}/m)^{a}$:

$\alpha_{j}:(t_{1},\dotsc,t_{n})\mapsto(\bar{t}_{2}\zeta^{j_{1}},\bar{t}_{1}\zeta^{j_{1}},\dotsc,\bar{t}_{2a}\zeta^{j_{a}},\bar{t}_{2a-1}\zeta^{j_{a}},\bar{t}_{2a+1},\dotsc,\bar{t}_{n}).$

So any fixed-point set $\mathbb{R}H^{n}_{\alpha_{j}}$ is identified with

$\displaystyle{{\mathbb{B}}}^{n}(\mathbb{R})_{\alpha_{j}}=\left\\{(t_{1},\dotsc,t_{n})\in{{\mathbb{B}}}^{n}(\mathbb{C})\colon
t_{i}=\bar{t}_{i-1}\zeta^{j_{i}}\textnormal{ for $i\leq 2a$ even,
}t_{i}\in\mathbb{R}\textnormal{ for $i>2a$}\right\\},$

and we have

$Y_{f}\cong\bigcup_{j}\mathbb{B}^{n}(\mathbb{R})_{\alpha_{j}}=\left\\{(t_{1},\dotsc,t_{n})\in{{\mathbb{B}}}^{n}(\mathbb{C})\colon
t_{i}^{m}=\bar{t}_{i-1}^{m}\textnormal{if $i\leq 2a$ even,
}t_{i}\in\mathbb{R}\textnormal{ if }i>2a\right\\}.$

These $m^{a}$ copies of ${{\mathbb{B}}}^{n}(\mathbb{R})$ meet at the origin of
${{\mathbb{B}}}^{n}(\mathbb{C})$; in fact, for $j\neq j^{\prime}$, the space
${{\mathbb{B}}}^{n}(\mathbb{R})_{\alpha_{j}}$ meets
${{\mathbb{B}}}^{n}(\mathbb{R})_{\alpha_{j^{\prime}}}$ in a
${{\mathbb{B}}}^{2c}(\mathbb{R})$ if $c$ is the number of pairs
$(j_{i},j^{\prime}_{i})$ with $j_{i}=j^{\prime}_{i}$.

4\. Now we treat the general case. In the local coordinates $t_{i}$, any anti-
unitary involutions fixing $x$ and equivalent to $\alpha$ is of the form

$\displaystyle(t_{1},\dotsc,t_{n})\mapsto(\bar{t}_{2}\zeta^{j_{1}},\bar{t}_{1}\zeta^{j_{1}},\dotsc,\bar{t}_{2a}\zeta^{j_{a}},\bar{t}_{2a-1}\zeta^{j_{a}},\bar{t}_{2a+1}\zeta^{j_{2a+1}},\dotsc,\bar{t}_{k}\zeta^{j_{k}},\bar{t}_{k+1},\dotsc,\bar{t}_{n})$

for some
$j=(j_{1},\dotsc,j_{a},j_{2a+1},\dotsc,j_{k})\in(\mathbb{Z}/m)^{a+b}$. We now
have $B_{f}\cong(\mathbb{Z}/m)^{b}$ acting by multiplying the $t_{i}$ for
$2a+1\leq i\leq k$ by powers of $\zeta$, and there are $m^{a+b}$ anti-unitary
involutions $\alpha_{j}$. We have

$\displaystyle
Y_{f}\cong\bigcup_{j_{1},\dotsc,j_{k}=1}^{m}\mathbb{R}H^{n}_{\alpha_{j}}\cong$
$\displaystyle\left\\{(t_{1},\dotsc,t_{n})\in{{\mathbb{B}}}^{n}(\mathbb{C})\mid
t_{2}^{m}=\bar{t}_{1}^{m},\dotsc,t_{2a}^{m}=\bar{t}_{2a-1}^{m},t_{2a+1}^{m},\dotsc,t_{k}^{m},t_{k+1},\dotsc,t_{n}\in\mathbb{R}\right\\}.$

We look at subsets $K_{f,\epsilon_{1},\dotsc,\epsilon_{k}}\subset Y_{f}$
again, this time defined as

$\displaystyle\left\\{t\in{{\mathbb{B}}}^{n}(\mathbb{C})\mid
t_{i}^{m}=\bar{t}_{i-1}^{m}\textnormal{ $i\leq 2a$ even,
}\zeta_{2^{a+1}}^{-\epsilon_{i}}t_{i}\in\mathbb{R}_{\geq 0}\textnormal{
$2a<i\leq k$, }t_{i}\in\mathbb{R},i>k\right\\}.$

As before, the natural map $U\coloneqq\bigcup_{\epsilon}K_{f,\epsilon}\to
B_{f}\setminus Y_{f}$ is an isometry. Define

$\widetilde{Y}_{f}=\left\\{(t_{1},\dotsc,t_{n})\in{{\mathbb{B}}}^{n}(\mathbb{C}):t_{i}^{m}=\bar{t}_{i-1}^{m}\textnormal{
for $i\leq 2a$ even, }t_{i}\in\mathbb{R},\textnormal{ for }i>2a\right\\}.$

Under its path metric,
$U=\cup_{\epsilon}K_{f,\epsilon_{1},\dotsc,\epsilon_{k}}$ is isometric to
$\widetilde{Y}_{f}$ by the following map:

$\displaystyle U\to\widetilde{Y}_{f},\;\;\;$
$\displaystyle(t_{1},\dotsc,t_{k})\mapsto$
$\displaystyle\left(t_{1},\dotsc,t_{2a},(-\zeta_{2^{a+1}})^{-\epsilon_{1}}t_{2a+1},\dotsc,(-\zeta_{2^{a+1}})^{-\epsilon_{k}}t_{k},t_{k+1},\dotsc,t_{n}\right).$

Hence $B_{f}\setminus Y_{f}\cong\widetilde{Y}_{f}$; but since
$\widetilde{Y}_{f}$ is what $Y_{f}$ was in case 3, we are done.

5\. The transitivity of $A_{f}$ on the copies of
${{\mathbb{B}}}^{n}(\mathbb{R})$ follows from the fact that $G(x)\subset
A_{f}$ contains transformations multiplying $t_{1},\dotsc,t_{2a}$ by powers of
$\zeta$, hence the map $(t_{i}\mapsto\zeta^{u}t_{i},t_{i-1}\mapsto t_{i-1})$
maps those $(t_{i-1},t_{i})$ with $t_{i}=\bar{t}_{i-1}\zeta^{j_{i}}$ to those
$(t_{i-1},t_{i})$ with $t_{i}=\bar{t}_{i-1}\zeta^{j_{1}+u}$. So if $B$ is any
one of the copies of ${{\mathbb{B}}}^{n}(\mathbb{R})$, and
$G=(A_{f}/B_{f})_{H}$ is its stabilizer, then it remains to prove that
$G\setminus B\to A_{f}\setminus Y_{f}$ is an isometry. Surjectivity follows
from the transitivity of $A_{f}$ on the
${{\mathbb{B}}}^{n}(\mathbb{R})^{{}^{\prime}}s$. It is a piecewise isometry so
we only need to prove injectivity. This will follow from the following
elementary lemma.

######  Lemma 4.14.

Let a group $G$ act on a set $X$, which is a union $X=\cup_{i\in I}Y_{i}$ of
subsets $Y_{i}\subset X$. Suppose that for all $y\in X$, the stabilizer of $y$
in $G$ acts transitively on the sets $Y_{i}$ containing $y$. Then for any
$i_{0}\in I$, if $H_{i_{0}}\subset G$ is the stabilizer of $Y_{i_{0}}$, then
the natural map $H_{i_{0}}\setminus Y_{i_{0}}\to G\setminus X$ is injective.

###### Proof.

Let $x,y\in Y_{i_{0}}$ and $g\in G$ such that $g\cdot x=y$. Then $y=gx\in
gY_{i_{0}}$. Since also $y\in Y_{i_{0}}$, there is an element
$h\in\text{Stab}_{G}(y)$ such that $hgY_{i_{0}}=Y_{i_{0}}$ and $hg(x)=h(y)=y$.
Let $f=hg$; then $f\in H_{i_{0}}$ and $f\cdot x=y$. ∎

Let us use the lemma. Suppose that $y\in B_{f}\setminus Y_{f}$. We need to
prove that $\text{Stab}_{A_{f}/B_{f}}(y)$ acts transtivitely on the copies of
${{\mathbb{B}}}^{n}(\mathbb{R})$ containing $y$. There exists

$j=(j_{1},\dotsc,j_{a},j_{2a+1},\dotsc,j_{k})\in(\mathbb{Z}/m)^{a+b}$

such that $y=(t_{1},\dotsc,t_{n})$ with $t_{i}=\bar{t}_{i-1}\zeta^{j_{i}}$ for
$i\leq 2a$ even, $t_{i}=\bar{t}_{i-1}\zeta^{j_{i}}$ for $2a<i\leq k$, and
$t_{i}\in\mathbb{R}$ for $i>k$. If all $t_{i}$ are non-zero, then
$y\in\cup_{j^{\prime}}\mathbb{R}H^{n}_{\alpha_{j^{\prime}}}$ is only contained
in $\mathbb{R}H^{n}_{\alpha_{j}}$, so there is nothing to prove. Let us
suppose that $t_{1}=t_{2}=0$ and the other $t_{i}$ are non-zero. Then $y$ is
contained in all the $\mathbb{R}H^{n}_{\alpha_{j^{\prime}}}$ with
$j_{i}^{\prime}=j_{i}$ for $i\geq 2$; there are $m$ of them. The stabilizer of
$y$ multiplies $t_{1}$ and $t_{2}$ by powers of $\zeta$ and leaves the other
$t_{i}$ invariant; it acts transitively on the
$\mathbb{R}H^{n}_{\alpha_{j^{\prime}}}$ containing $y$ for if
$t_{2}=\bar{t}_{1}\zeta^{j_{1}^{\prime}}$ then
$\zeta^{(j_{1}^{\prime\prime}-j_{1}^{\prime})}t_{2}=\bar{t}_{1}\zeta^{j_{1}^{\prime\prime}}$.
The general case is similar. ∎

We need one more lemma before we can prove Theorem 4.1:

######  Lemma 4.15.

The maps ${\mathscr{P}}\colon Y(\Lambda)\to\mathbb{C}H^{n}$ and
$\overline{{\mathscr{P}}}\colon X(\Lambda)\to P\Gamma\setminus\mathbb{C}H^{n}$
are proper with finite fibers.

###### Proof.

The map ${\mathscr{P}}\colon Y(\Lambda)\to\mathbb{C}H^{n}$ is proper with
finite fibers because any compact set in $\mathbb{C}H^{n}$ meets only finitely
many $(\mathbb{R}H^{n}_{\alpha})^{{}^{\prime}}s$ for $\alpha\in
P{\mathscr{A}}$ (see Lemma 4.3), and ${\mathscr{P}}$ carries each
$p\left(\mathbb{R}H^{n}_{\alpha}\right)\subset Y(\Lambda)$ homeomorphically
onto $\mathbb{R}H^{n}_{\alpha}\subset\mathbb{C}H^{n}$. Since ${\mathscr{P}}$
is $P\Gamma$-equivariant, the induced map $\overline{{\mathscr{P}}}\colon
X(\Lambda)\to P\Gamma\backslash\mathbb{C}H^{n}$ is proper with finite fibers
as well. ∎

###### Proof of Theorem 4.1.

1. The path metric on $Y(\Lambda)$ was provided in Proposition 4.6, and the path metric on $X(\Lambda)$ was given in Proposition 4.7. The map ${\mathscr{P}}\colon Y\to\mathbb{C}H^{n}$ is a local embedding by Lemma 4.4, which was used to define the metric on $Y(\Lambda)$ (c.f. Proposition 4.6). Thus, almost by definition, ${\mathscr{P}}$ is a local isometric embedding. The space $P\Gamma\setminus\mathbb{C}H^{n}$ is complete, and ${\mathscr{P}}$ and $\overline{{\mathscr{P}}}$ are proper by Lemma 4.15. Therefore, the metric sapce $X(\Lambda)$ is complete as well.

2. $[f]\in X(\Lambda)$ be the image of $f\in Y(\Lambda)$. Then $[f]$ has an open neighborhood isometric to the quotient of an open set $W$ in $\mathbb{R}H^{n}$ by a finite group of isometries $\Gamma_{f}$. Indeed, take $Y_{f}\subset Y(\Lambda)$ as in Equation (11), and $f\in U_{f}\subset Y_{f}$ as in Lemma 4.4.2. We let $A_{f}=P\Gamma_{f}$ be the stabilizer of $f$ in $P\Gamma$ as before, and take an $A_{f}$-equivariant open neighborhood $V_{f}\subset U_{f}$ such that $A_{f}\setminus V_{f}\subset P\Gamma\setminus Y(\Lambda)$. By Proposition 4.13.5, we know that $A_{f}\setminus Y_{f}$ is isometric to $\Gamma_{f}\setminus\mathbb{R}H^{n}$ for some finite group of isometries of $\mathbb{R}H^{n}$. Thus, $A_{f}\setminus V_{f}$ is isometric to some open set $W^{\prime}$ in $\Gamma_{f}\setminus\mathbb{R}H^{n}$. Take $W\subset\mathbb{R}H^{n}$ to be the preimage of $W^{\prime}$.

3. _Claim:_ For any path metric space $X$ locally isometric to quotients of $\mathbb{R}H^{n}$ by finite groups of isometries, there is a unique real-hyperbolic orbifold structure on $X$ whose path metric is the given one. _Proof of the Claim:_ If $U$ and $U^{\prime}$ are connected open subsets of $\mathbb{R}H^{n}$ and $\Gamma$ and $\Gamma^{\prime}$ finite groups of isometries of $\mathbb{R}H^{n}$ preserving $U$ and $U^{\prime}$ respectively, then any isometry $\bar{\phi}:\Gamma\setminus U\to\Gamma^{\prime}\setminus U^{\prime}$ extends to an isometry $\phi:\mathbb{R}H^{n}\to\mathbb{R}H^{n}$ such that $\phi(U)=U^{\prime}$ and $\phi\Gamma{\phi}^{-1}=\Gamma^{\prime}\subset\textnormal{Isom}(\mathbb{R}H^{n})$. We conclude that $P\Gamma\setminus Y(\Lambda)$ is naturally a real hyperbolic orbifold.

We show that the complete real hyperbolic orbifold $X(\Lambda)$ has finite
volume. For each $\alpha\in P{\mathscr{A}}$, the natural map
$P\Gamma_{\alpha}\backslash\mathbb{R}H^{n}_{\alpha}\to X(\Lambda)$ is a closed
immersion, and $X(\Lambda)$ is the union of the images of these maps, when
$\alpha$ ranges over $C{\mathscr{A}}=P\Gamma\backslash P{\mathscr{A}}$. The
hyperbolic orbifold $P\Gamma_{\alpha}\backslash\mathbb{R}H^{n}_{\alpha}$ has
finite volume by Corollary 2.8, and the set $C{\mathscr{A}}$ is finite by
Proposition 2.9. Thus, $X(\Lambda)$ has finite volume.

4. Let us show that

$O\coloneqq\coprod_{\alpha\in
C{\mathscr{A}}}[P\Gamma_{\alpha}\setminus\left(\mathbb{R}H^{n}_{\alpha}-{\mathscr{H}}\right)]\subset
X(\Lambda)=P\Gamma\setminus Y(\Lambda)$

as hyperbolic orbifolds. It suffices to show the following:

_Claim:_ For those $f=p(x_{\alpha})\in Y(\Lambda)$ that have no nodes, the
stabilizer $A_{f}=P\Gamma_{f}\subset P\Gamma$ of $f\in Y(\Lambda)$ and the
stabilizer $P\Gamma_{\alpha,x}\subset P\Gamma_{\alpha}$ of
$x\in\mathbb{R}H^{n}_{\alpha}$ agree as subgroups of $P\Gamma$. _Proof of the
Claim:_ To prove that $A_{f}=P\Gamma_{\alpha,x}$, we first observe that
$p:\widetilde{Y}\to Y(\Lambda)$ induces an isomorphism between
$P\Gamma_{x_{\alpha}}$, the stabilizer of $x_{\alpha}\in\widetilde{Y}$ and
$P\Gamma_{f}$, the stabilizer of $f=[x,\alpha]\in Y(\Lambda)$. So it suffices
to show that $P\Gamma_{x_{\alpha}}=P\Gamma_{\alpha,x}$. Indeed, the normalizer
$N_{P\Gamma}(\alpha)$ and the stabilizer $P\Gamma_{\alpha}\subset P\Gamma$ of
$\alpha$ in $P\Gamma$ are equal, which implies that
$P\Gamma_{\alpha,x}=P\Gamma_{x_{\alpha}}$ because

$\displaystyle\left\\{g\in P\Gamma_{\alpha}:gx=x\right\\}$
$\displaystyle=\left\\{g\in N_{P\Gamma}(\alpha):gx=x\right\\}$
$\displaystyle=\left\\{g\in P\Gamma:g\cdot x_{\alpha}=(g(x),g\alpha
g^{-1})=x_{\alpha}\right\\}.$

The claim is proved. Part 4 of the theorem can be deduced from it as follows.
Let $f=p(x_{\alpha})\in Y(\Lambda)$ have no nodes. We have
$Y_{f}=\mathbb{R}H^{n}_{\alpha}$, hence

$A_{f}\setminus\mathbb{R}H^{n}_{\alpha}=A_{f}\setminus
Y_{f}=\Gamma_{f}\setminus\mathbb{R}H^{n}\quad{\textnormal{with}}\quad\Gamma_{f}=A_{f}\setminus
B_{f}=A_{f}.$

By construction, an orbifold chart of the glued space $P\Gamma\setminus
Y(\Lambda)$ is given by

$W\to A_{f}\setminus W\subset
P\Gamma_{\alpha}\setminus\mathbb{R}H^{n}_{\alpha}\subset Y(\Lambda)$

for an invariant open subset $W$ of $\mathbb{R}H^{n}_{\alpha}$ containing $x$.
Because $A_{f}=P\Gamma_{\alpha,x}$ by the claim, this is also an orbifold
chart for $O$ at the point $x_{\alpha}$.

5. The real-hyperbolic orbifold $X(\Lambda)=P\Gamma\setminus Y(\Lambda)$ is complete by Part 1, so the uniformization of the connected components of $X(\Lambda)$ follows from the Ehresmann–Thurston uniformization theorem for $(G,X)$-orbifolds, see [Thu80, Proposition 13.3.2] or [MMA91]. This concludes the proof of Theorem 4.1.∎

##  5 Unitary Shimura varieties

The goal of this section is to prove Proposition 5.7, which describes the
complex ball quotient $P\Gamma\setminus\mathbb{C}H^{n}$ defined in Section 2.1
in terms of moduli of abelian varieties with $\mathcal{O}_{K}$-action of
hyperbolic signature, and Proposition 5.11, which interprets the divisor
$P\Gamma\setminus{\mathscr{H}}$ as the locus of abelian varieties $A$ that
admit an $\mathcal{O}_{K}$-linear homomorphism
$\mathbb{C}^{g}/\Psi(\mathcal{O}_{K})\to A$ of polarized
$\mathcal{O}_{K}$-linear abelian varieties. This has two applications:

1\. Consider a relative uniform cyclic cover (see e.g. [AV04])
${\mathscr{X}}\to P\to S$, where $P=\mathbb{P}^{1}_{S}$ (resp.
$\mathbb{P}^{3}_{S}$), the fibers of ${\mathscr{X}}\to S$ are curves (resp.
threefolds with ${\mathrm{H}}^{0,3}=0$) and the induced hermitian form on
middle cohomology has hyperbolic signature. Since the image
${\mathfrak{I}}=P(S(\mathbb{C}))\subset P\Gamma\setminus\mathbb{C}H^{n}$ of
the period map $P\colon S(\mathbb{C})\to P\Gamma\setminus\mathbb{C}H^{n}$ is
contained in the locus of principally polarized abelian varieties whose theta
divisor is irreducible, one has ${\mathfrak{I}}\subset
P\Gamma\setminus\left(\mathbb{C}H^{n}-{\mathscr{H}}\right).$

2\. Using the theory of this section, we will be able to show that Condition
2.1 implies Condition 2.2, saying that the hyperplanes in the arrangement
${\mathscr{H}}\subset\mathbb{C}H^{n}$ are orthogonal along their intersection.
See Theorem 6.2 in the next Section 6.

###  5.1 Alternating and hermitian forms

The goal of this subsection is to prove two lemmas. They will later be used to
show that the ball quotient $P\Gamma\setminus\mathbb{C}H^{n}$ of Section 3 is
a moduli space of abelian varieties, and then to interpret the divisor
$P\Gamma\setminus{\mathscr{H}}$ in terms of moduli of abelian varieties. This
will be the key to Theorem 6.2.

Let $M$ be a CM field, with ring of integers $\mathcal{O}_{M}$, and let
$\sigma\colon M\to M$ be the involution induced by complex conjugation on
$\mathbb{C}$. Let $\Lambda$ be a free $\mathcal{O}_{M}$-module of rank $n+1$
for some $n\in\mathbb{N}$. Let ${\mathfrak{D}}_{M}\subset\mathcal{O}_{M}$ be
the different ideal.

######  Lemma 5.1.

The assignment $T\mapsto\textnormal{Tr}_{M/\mathbb{Q}}\circ T$ defines a
bijection between:

  1. 1.

The set of skew-hermitian forms
$T:\Lambda_{\mathbb{Q}}\times\Lambda_{\mathbb{Q}}\to M$.

  2. 2.

The set of alternating forms
$E\colon\Lambda_{\mathbb{Q}}\times\Lambda_{\mathbb{Q}}\to\mathbb{Q}$ with
$E(a\cdot x,y)=E(x,\sigma(a)\cdot y)$.

Under this correspondence, $T(\Lambda,\Lambda)\subset{\mathfrak{D}}_{M}^{-1}$
if and only if $E(\Lambda,\Lambda)\subset\mathbb{Z}$.

###### Proof.

Let $T\colon\Lambda_{\mathbb{Q}}\times\Lambda_{\mathbb{Q}}\to M$ be as in 1.
Define $E_{T}=\text{Tr}_{M/\mathbb{Q}}\circ T$. Since $T$ is skew-hermitian,
we have, for each $x,y\in\Lambda_{\mathbb{Q}}$, that

$\text{Tr}_{M/\mathbb{Q}}T(x,y)=-\text{Tr}_{M/\mathbb{Q}}\overline{T(y,x)}.$

Since $M/\mathbb{Q}$ is separable, for $x\in M$, we have
$\text{Tr}_{M/\mathbb{Q}}(x)=\sum_{1\leq i\leq
g}\left(\tau_{i}(x)+\tau_{i}\sigma(x)\right)$, see [Ste08, (7-1)]. Therefore,
$\text{Tr}_{M/\mathbb{Q}}(\sigma(x))=\text{Tr}_{M/\mathbb{Q}}(x)$, so that
$E_{T}(x,y)=-E_{T}(y,x)$ for any $x,y\in\Lambda_{\mathbb{Q}}$. The property in
2 is easily checked.

Conversely, let
$E:\Lambda_{\mathbb{Q}}\times\Lambda_{\mathbb{Q}}\to\mathbb{Q}$ be as in 2.
Choose a basis $\\{b_{1},\dotsc,b_{n+1}\\}\subset\Lambda$ for $\Lambda$ over
$\mathcal{O}_{M}$. Define $Q$ to be the induced map $M^{n+1}\times
M^{n+1}\to\mathbb{Q}$ and consider the map $M\to\mathbb{Q}$, $a\mapsto
Q(a\cdot e_{i},e_{j})$. Since the trace pairing

$M\times M\to\mathbb{Q},\quad(x,y)\mapsto\text{Tr}_{M/\mathbb{Q}}(xy)$

is non-degenerate [Stacks, Tag 0BIE], there is a unique element $t_{ij}\in M$
such that $Q(a\cdot e_{i},e_{j})=\text{Tr}_{M/\mathbb{Q}}(a\cdot t_{ij})$ for
every $a\in M$. This gives a matrix $(t_{ij})_{ij}\in M_{n+1}(K)$ such that
$\sigma(t_{ij})=-t_{ji}$, and the basis $\\{b_{i}\\}$ induces a skew-hermitian
form $T_{E}:\Lambda_{\mathbb{Q}}\times\Lambda_{\mathbb{Q}}\to M$. This
provides the bijection; the other claim from the definition of
${\mathfrak{D}}_{M}^{-1}\subset M$: it is the trace dual of $\mathcal{O}_{M}$,
see [Ser79, Chapter III, §3]. ∎

######  Examples 5.2.

  1. 1.

Suppose that $M=\mathbb{Q}(\sqrt{\Delta})$ is imaginary quadratic over
$\mathbb{Q}$, with discriminant $\Delta$ and non-trivial Galois automorphism
$a\mapsto\sigma(a)$. Let $E\colon\Lambda\times\Lambda\to\mathbb{Z}$ be an
alternating form with $E(a\cdot x,y)=E(x,\sigma(a)\cdot y)$. The form
$T\colon\Lambda\times\Lambda\to{\mathfrak{D}}_{M}^{-1}=(\sqrt{\Delta})^{-1}$
is defined as

$T(x,y)=\frac{E(\sqrt{\Delta}\cdot x,y)+E(x,y)\sqrt{\Delta}}{2\sqrt{\Delta}}.$

  2. 2.

Suppose that $M=\mathbb{Q}(\zeta_{p})$ where $\zeta_{p}=e^{2\pi
i/p}\in\mathbb{C}$ for some prime number $p>2$. Let
$E:\Lambda\times\Lambda\to\mathbb{Z}$ be an alternating form with $E(a\cdot
x,y)=E(x,\sigma(a)\cdot y)$. Then
${\mathfrak{D}}_{M}=\left(p/(\zeta_{p}-\zeta_{p}^{-1})\right)$ and

$T:\Lambda\times\Lambda\to{\mathfrak{D}}_{M}^{-1}\quad{\textnormal{ is defined
as }}\quad T(x,y)=\frac{1}{p}\sum_{j=0}^{p-1}\zeta_{p}^{j}\cdot
E\left(x,\zeta_{p}^{j}\cdot y\right).$

Consider a pair
$\left(E\colon\Lambda_{\mathbb{Q}}\times\Lambda_{\mathbb{Q}}\to\mathbb{Q},T:\Lambda_{\mathbb{Q}}\times\Lambda_{\mathbb{Q}}\to
K\right)$ as in Lemma 5.1, and suppose that $E$ is non-degenerate. To any
embedding $\phi\colon M\to\mathbb{C}$ be an embedding, one can associate a
skew-hermitian form
$T^{\phi}\colon\Lambda\otimes_{\mathcal{O}_{M},\phi}\mathbb{C}\times\Lambda\otimes_{\mathcal{O}_{M},\phi}\mathbb{C}\to\mathbb{C}$
by putting

$\displaystyle\begin{split}T^{\phi}(x\otimes\lambda,y\otimes\mu)=\lambda\overline{\mu}\cdot\phi\left(T(x,y)\right)\end{split}$

and extending linearly. Remark that, on
$\Lambda_{\mathbb{C}}=\Lambda\otimes_{\mathbb{Z}}\mathbb{C}$, one has the
skew-hermitian form $A(x,y)=E_{\mathbb{C}}(x,\bar{y})$. The composition
$\left(\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\phi}\to\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\to\Lambda\otimes_{\mathcal{O}_{M},\phi}\mathbb{C}$
is an isomorphism; we define $A^{\phi}$ as the restriction of $A$ to the
subspace

$\left(\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\phi}=\Lambda\otimes_{\mathcal{O}_{M},\phi}\mathbb{C}\subset\Lambda_{\mathbb{C}}.$

Note that
$\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\cong\oplus_{\phi:M\to\mathbb{C}}\left(\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\phi}.$
For $x\in\Lambda\otimes_{\mathbb{Z}}\mathbb{C}$, let
$x_{\phi}\in\left(\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\phi}$ denote
the image of $x$ under the projection
$\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\to\left(\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\phi}$.

######  Lemma 5.3.

Let $\phi\colon M\to\mathbb{C}$ be an embedding. We have an equality
$T^{\phi}=A^{\phi}$ of skew-hermitian forms
$\left(\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\phi}\times\left(\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\phi}\to\mathbb{C}$.
More precisely, we have
$A(x,y)=\sum_{\phi:M\to\mathbb{C}}T^{\phi}(x_{\phi},y_{\phi})$ for every
$x,y\in\Lambda\otimes_{\mathbb{Z}}\mathbb{C}$.

###### Proof.

This is straightforward and we leave the proof to the reader. ∎

###  5.2 Moduli of abelian varieties with an action by a CM field

In the rest of Section 5, we fix the following notation.

######  Notation 5.4.

  1. 1.

Let $M$ be a CM field over $\mathbb{Q}$, and define $\sigma\colon M\to M$ as
the involution that corresponds to complex conjugation on $\mathbb{C}$.

  2. 2.

Let $\eta\in\mathcal{O}_{M}$ be a non-zero element such that
$\sigma(\eta)=-\eta$.

  3. 3.

Let $\Lambda$ be a free $\mathcal{O}_{M}$-module of rank $n+1$ for some
$n\in\mathbb{Z}_{\geq 0}$ equipped with a non-degenerate hermitian form
${\mathfrak{h}}:\Lambda\times\Lambda\to\eta\cdot{\mathfrak{D}}_{M}^{-1}\subset
K$.

  4. 4.

Let $\Phi\subset\textnormal{Hom}(M,\mathbb{C})$ be the CM type such that
$\Im\left(\phi(\eta)\right)>0$ for all $\phi\in\Phi$.

These data define a skew-hermitian form
$T\colon\Lambda\times\Lambda\to{\mathfrak{D}}_{M}^{-1}$ by putting
$T\coloneqq\eta^{-1}\cdot{\mathfrak{h}}.$ The form $T$ is in turn attached to
an alternating form (see Lemma 5.1):

$E\colon\Lambda\times\Lambda\to\mathbb{Z}\; {\textnormal{ such that
}}\;E(ax,y)=E(x,\sigma(a)y)\;{\textnormal{ for all
}}\;a\in\mathcal{O}_{M},\;x,y\in\Lambda.$

Write $V_{\phi}=\Lambda_{\mathbb{Q}}\otimes_{K,\phi}\mathbb{C}$ and define
${\mathfrak{h}}^{\phi}\colon V_{\phi}\times V_{\phi}\to\mathbb{C}$ to be the
hermitian form restricting to $\phi\circ{\mathfrak{h}}$ on $\Lambda$. Let
$(r_{\phi},s_{\phi})$ be the signature of the hermitian form
${\mathfrak{h}}^{\phi}$.

Let $A$ be a complex abelian variety, $\iota$ a homomorphism
${\mathcal{O}}_{M}\to\text{End}(A)$, and $\lambda$ a polarization $A\to
A^{\vee}$, satisfying the following (compare [KR14, Part I, §2.1]):

######  Conditions 5.5.

  1. 1.

We have $\iota(a)^{\dagger}=i(\sigma(a))$, where
$\dagger\colon\textnormal{End}(A)_{\mathbb{Q}}\to\textnormal{End}(A)_{\mathbb{Q}}$
denotes the Rosati involution.

  2. 2.

We have
$\textnormal{char}(t,\iota(a)|\textnormal{Lie}(A))=\prod_{\phi\in\Phi}(t-a^{\phi})^{r_{\phi}}\cdot(t-a^{\phi\sigma})^{s_{\phi}}\;\in\;\mathbb{C}[t].$

Here, $\textnormal{char}(t,\iota(a)|\textnormal{Lie}(A))\in\mathbb{C}[t]$
denotes the characteristic polynomial of $\iota(a)$. Any abelian variety $A$
as above satisfies $\dim A=g(n+1)$, where $g=[M\colon\mathbb{Q}]$.

Define
$E_{A}:{\mathrm{H}}_{1}(A,\mathbb{Z})\times{\mathrm{H}}_{1}(A,\mathbb{Z})\to\mathbb{Z}$
to be the alternating form corresponding to $\lambda$. The condition on the
Rosati involution implies that $E_{A}(\iota(a)x,y)=E_{A}(x,\iota(\sigma(a))y)$
for $x,y\in{\mathrm{H}}_{1}(A,\mathbb{Q})$. Define a hermitian form
${\mathfrak{h}}_{A}\colon{\mathrm{H}}_{1}(A,\mathbb{Z})\times{\mathrm{H}}_{1}(A,\mathbb{Z})\to\eta\cdot{\mathfrak{D}}_{M}^{-1}$
as ${\mathfrak{h}}_{A}=\eta\cdot T_{A}$. Here,
$T_{A}\colon{\mathrm{H}}_{1}(A,\mathbb{Z})\times{\mathrm{H}}_{1}(A,\mathbb{Z})\to{\mathfrak{D}}_{M}^{-1}$
is the skew-hermitian form attached to $E_{A}$ via Lemma 5.1.

######  Definition 5.6.

  1. 1.

Let $\widetilde{\textnormal{Sh}}_{M}(\Lambda)$ be the set of isomorphism
classes of four-tuples $(A,\iota,\lambda,j)$, where $(A,\iota,\lambda)$ is as
above and satisfies Conditions 5.5, and where
$j\colon{\mathrm{H}}_{1}(A,\mathbb{Z})\to\Lambda$ is an isomorphism of
$\mathcal{O}_{M}$-modules compatible with the alternating forms $E_{A}$ and
$E$ (equivalently, with the hermitian forms ${\mathfrak{h}}_{A}$ and
${\mathfrak{h}}$).

  2. 2.

Let ${\mathbb{D}}(V_{\phi})$ be the space of negative $s_{\phi}$-planes in the
hermitian space $(V_{\phi},{\mathfrak{h}}^{\phi})$.

We have the following proposition which is due to Shimura, see [Shi63,
Theorems 1 & 2] or [Shi64, §1]. We give a different proof since it will imply
Proposition 5.11 below, whereas we did not know how to deduce Proposition 5.11
from _loc. cit._ We remark that Shimura assumes $\Lambda$ to be an $R$-module,
for an order $R\subset\mathcal{O}_{M}$ which is not necessarily maximal; our
proof carries over, but we do not need this.

######  Proposition 5.7.

There is a canonical bijection
$\widetilde{\textnormal{Sh}}_{M}(\Lambda)\cong\prod_{\phi\in\Phi}{\mathbb{D}}(V_{\phi})$.

###### Proof.

Let $(A,\iota,\lambda,j)$ be a representative of an isomorphism class that
defines a point in $\widetilde{\textnormal{Sh}}_{M}(\Lambda)$. Consider the
Hodge decomposition
${\mathrm{H}}_{1}(A,\mathbb{C})={\mathrm{H}}^{-1,0}\oplus{\mathrm{H}}^{0,-1}$.
For each $\phi\in\Phi$, there is a decomposition

${\mathrm{H}}_{1}(A,\mathbb{C})_{\phi}={\mathrm{H}}^{-1,0}_{\phi}\oplus{\mathrm{H}}^{0,-1}_{\phi},$
(15)

with $\dim{\mathrm{H}}^{-1,0}_{\phi}=r_{\phi}$ and
$\dim{\mathrm{H}}^{0,-1}_{\phi}=s_{\phi}$. The latter holds because
$\overline{{\mathrm{H}}^{-1,0}_{\phi\sigma}}={\mathrm{H}}^{0,-1}_{\phi}$. By
Lemma 5.3, the forms $\phi(\eta)E_{A,\mathbb{C}}(x,\bar{y})$ and
${{\mathfrak{h}}}_{A}^{\phi}(x,y)$ agree as hermitian forms on the complex
vector space
${\mathrm{H}}_{1}(A,\mathbb{Z})\otimes_{\mathcal{O}_{M},\phi}\mathbb{C}$.
Since $\Im(\phi(\eta))>0$ for every $\phi\in\Phi$, the decomposition of
${\mathrm{H}}_{1}(A,\mathbb{C})_{\phi}$ in (15) is a decomposition into a
positive definite $r_{\phi}$-dimensional subspace and a negative definite
$s_{\phi}$-dimensional subspace. The isomorphism
$j\colon{\mathrm{H}}_{1}(A,\mathbb{Q})\to\Lambda_{\mathbb{Q}}$ induces an
isometry $j_{\phi}\colon{\mathrm{H}}_{1}(A,\mathbb{C})_{\phi}\to V_{\phi}$ for
every $\phi$, and so we obtain a negative $s_{\phi}$-plane
$j_{\phi}({\mathrm{H}}^{0,-1}_{\phi})$ in the hermitian space $V_{\phi}$ for
all $\phi$. Reversing the argument shows that given a negative
$s_{\phi}$-plane $X_{\phi}\subset V_{\phi}$ for every $\phi$, there is a
canonical polarized abelian variety $A={\mathrm{H}}^{-1,0}/\Lambda$, acted
upon by $\mathcal{O}_{M}$ and inducing the planes $X_{\phi}\subset V_{\phi}$.
∎

######  Definition 5.8.

  1. 1.

Let $\textnormal{Sh}_{M}(\Lambda)$ be the set of isomorphism classes of
polarized abelian varieties with $\mathcal{O}_{M}$-action $(A,\iota,\lambda)$,
satisfying Conditions 5.5, such that ${\mathrm{H}}_{1}(A,\mathbb{Z})$ is
isomorphic to $\Lambda$ as hermitian $\mathcal{O}_{M}$-modules.

  2. 2.

Let $\Gamma(\Lambda)=\textnormal{Aut}_{\mathcal{O}_{M}}(\Lambda)$; this is the
group of $\mathcal{O}_{M}$-linear automorphisms of $\Lambda$ preserving the
hermitian form
${\mathfrak{h}}:\Lambda\times\Lambda\to\eta\cdot{\mathfrak{D}}_{M}^{-1}$.

The bijection in Proposition 5.7 being $\Gamma(\Lambda)$-equivariant, we
obtain the following:

######  Corollary 5.9.

There is a canonical bijection
$\textnormal{Sh}_{M}(\Lambda)\cong\Gamma(\Lambda)\setminus\prod_{i=1}^{g}{\mathbb{D}}(V_{i})$.
∎

###  5.3 Moduli of abelian varieties in the hyperplane arrangement

The set of embeddings $\Phi$ introduced in Notation 5.4 defines a map
$\Phi:\mathcal{O}_{M}\to\mathbb{C}^{g}$, which yields a complex torus
$\mathbb{C}^{g}/\Phi(\mathcal{O}_{M})$. The map

$\displaystyle Q\colon M\times M\to\mathbb{Q},\quad
Q(x,y)=\text{Tr}_{M/\mathbb{Q}}(\eta^{-1}x\bar{y})$

is a non-degenerate $\mathbb{Q}$-bilinear form with
$Q(ax,y)={Q}(x,\sigma(a)y)$ for $a,x,y\in M$.

######  Lemma 5.10.

Suppose that the element $\eta\in\mathcal{O}_{M}$ of Notation 5.4.2 satisfies
$\eta^{-1}\in{\mathfrak{D}}_{M}^{-1}$. Then $Q$ defines a polarization on the
complex torus $\mathbb{C}^{g}/\Phi(\mathcal{O}_{M})$.

###### Proof.

Remark that $Q(\mathcal{O}_{M},\mathcal{O}_{M})\subset\mathbb{Z}$ because
$\eta^{-1}\in{\mathfrak{D}}_{M}^{-1}$. By [Mil20, Example 2.9 & Footnote 16],
the result follows. ∎

Suppose from now on that

$\displaystyle\begin{split}(r_{\phi},s_{\phi})\quad=\quad\begin{cases}(n,1)\quad&{\textnormal{
if }}\phi=\tau,\\\ (n+1,0)\quad&{\textnormal{ if
}}\phi\neq\tau.\end{cases}\end{split}$ (16)

The space $W\coloneqq\Lambda\otimes_{\mathcal{O}_{M},\tau}\mathbb{C}$ is
equipped with the hermitian form ${\mathfrak{h}}^{\tau}\colon W\times
W\to\mathbb{C}$. Define

$\displaystyle\mathbb{C}H(W)=\left\\{{\textnormal{lines }}\ell\subset
W\mid{\mathfrak{h}}^{\tau}(x,x)<0{\textnormal{ for }}0\neq x\in\ell\right\\},$

and let ${\mathscr{H}}=\cup_{{\mathfrak{h}}(r,r)=1}\langle
r_{\mathbb{C}}\rangle^{\perp}\subset\mathbb{C}H(W)$.

######  Proposition 5.11.

Suppose that $\eta^{-1}\in{\mathfrak{D}}_{M}^{-1}$, so that
$\left(\mathbb{C}^{g}/\Phi(\mathcal{O}_{M}),Q\right)$ is a polarized abelian
variety by Lemma 5.10. Under the bijection
$\widetilde{\textnormal{Sh}}_{M}(\Lambda)\cong\mathbb{C}H(W)$ of Proposition
5.7, the subset ${\mathscr{H}}\subset\mathbb{C}H(W)$ is identified with the
set of isomorphism classes of marked polarized $\mathcal{O}_{M}$-linear
abelian varieties $(A,\iota,\lambda,j)$ admitting an $\mathcal{O}_{M}$-linear
homomorphism $f\colon\mathbb{C}^{g}/\Phi(\mathcal{O}_{M})\to A$ of polarized
abelian varieties.

###### Proof.

Consider an isomorphism class
$[(A,\iota,\lambda,y)]\in\widetilde{\textnormal{Sh}}_{M}(\Lambda)$
corresponding to a point $[x]\in{\mathbb{C}}H^{n}$. We may assume that
$A={\mathrm{H}}^{-1,0}/\Lambda$ with
$\Lambda\otimes_{\mathbb{Z}}\mathbb{C}={\mathrm{H}}^{-1,0}\oplus{\mathrm{H}}^{0,-1}$,
and that $T_{A}=T$. Let $f\colon\mathbb{C}^{g}/\Phi(\mathcal{O}_{M})\to A$ be
a homomorphism as in the proposition. We obtain a homomorphism
$\mathcal{O}_{M}\to\Phi(\mathcal{O}_{M})\to{\mathrm{H}}_{1}(A,\mathbb{Z})=\Lambda$
which, for simplicity, we also denote by $f\colon\mathcal{O}_{M}\to\Lambda$.
Let $r\in\Lambda$ be the image of $1\in\mathcal{O}_{M}$. The fact that
$Q=f^{\ast}E_{A}$ implies that $T_{Q}=f^{\ast}T_{A}=f^{\ast}T$. Therefore, we
have $\eta^{-1}=T_{Q}(1,1)=T_{A}(f(1),f(1))=T(f(1),f(1))=T(r,r),$ so that
${\mathfrak{h}}(r,r)=\eta\cdot T(r,r)=1$. We claim that
${\mathfrak{h}}(x,r_{\tau})=0$, where the element
$r_{\tau}\in\left(\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\tau}$ is the
image of $r\in\Lambda$. To see this, write $\Phi(\mathcal{O}_{M})=L,$ and
$L\otimes\mathbb{C}=W^{-1,0}\oplus W^{0,-1}.$ Notice that
$\left(L\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\tau}=W^{-1,0}_{\tau}$.
Consequently, since the composition

$W^{-1,0}_{\tau}=\left(L\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\tau}\to\left(\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\tau}={\mathrm{H}}^{-1,0}_{\tau}\oplus{\mathrm{H}}^{0,-1}_{\tau}$

factors through the inclusion of ${\mathrm{H}}^{-1,0}_{\tau}$ into
$\left(L\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\tau}$, we see that
$r_{\tau}=r^{-1,0}_{\tau}\in{\mathrm{H}}^{-1,0}_{\tau}=({\mathrm{H}}^{0,-1}_{\tau})^{\perp}=\langle
x\rangle^{\perp},$ and the claim follows.

Conversely, let $[x]\in\langle
r_{\mathbb{C}}\rangle^{\perp}\subset{\mathscr{H}}$ with $r\in\Lambda$ such
that ${\mathfrak{h}}(r,r)=1$ and consider the marked abelian variety
$A={\mathrm{H}}^{-1,0}/\Lambda$ corresponding to $[x]$. Define a homomorphism
$f\colon\mathcal{O}_{M}\to\Lambda$ by $f(1)=r$. Then $f$ can be shown to be a
morphism of Hodge structures using the fact that its $\mathbb{C}$-linear
extension preserves the eigenspace decompositions. We obtain an
$\mathcal{O}_{M}$-linear homorphism
$f\colon\mathbb{C}^{g}/\Phi(\mathcal{O}_{M})\to A$. The fact that
${\mathfrak{h}}(r,r)=1$ implies that $f$ preserves the polarizations on both
sides. ∎

##  6 Orthogonal hyperplane arrangements

Let $M$ be a CM field of degree $2g$ over $\mathbb{Q}$, with involution
$\sigma\colon M\to M$ that corresponds to complex conjugation on $\mathbb{C}$.
Assume that $M$ satisfies the following:

######  Condition 6.1.

There exists an element $\eta\in\mathcal{O}_{M}$ satisfying the condition that

$\displaystyle{\mathfrak{D}}_{M}=(\eta)\quad{\textnormal{ and
}}\quad\sigma(\eta)=-\eta.$ (17)

Fix an element $\eta\in\mathcal{O}_{M}$ that satisfies (17). Let
$\Phi\subset\textnormal{Hom}(M,\mathbb{C})$ be the unique CM type such that
$\Im(\varphi(\eta))>0$ for all $\varphi\in\Phi$. Let $\Lambda$ be a free
$\mathcal{O}_{M}$-module of rank $n+1$ equipped with a hermitian form
${\mathfrak{h}}\colon\Lambda\times\Lambda\to\mathcal{O}_{M}$ of signature
$(n,1)$ with respect to one $\tau\in\Phi$ and of signature $(n+1,0)$ with
respect to all other $\phi\neq\tau\in\Phi$. Let
$W=\Lambda\otimes_{\mathcal{O}_{M},\tau}\mathbb{C}$ and define
$\mathbb{C}H(W)$ as the space of negative lines in the hermitian space $W$.
For an element $r\in\Lambda$, let $r_{\tau}$ be its image in
$\left(\Lambda\otimes_{\mathbb{Z}}\mathbb{C}\right)_{\tau}=\Lambda\otimes_{\mathcal{O}_{M},\tau}\mathbb{C}$.
If ${\mathfrak{h}}(r,r)=1$, we define $\langle
r_{\tau}\rangle^{\perp}\subset\mathbb{C}H(W)$ as the set of negative lines
$\ell\subset W$ with ${\mathfrak{h}}(r_{\tau},\ell)=0$.

######  Theorem 6.2.

Assume Condition 6.1. Let $r,t\in\Lambda$ such that
${\mathfrak{h}}(r,r)={\mathfrak{h}}(t,t)=1$. Suppose that $\langle
r_{\tau}\rangle^{\perp}\cap\langle t_{\tau}\rangle^{\perp}\neq\emptyset$ and
$\langle r_{\tau}\rangle^{\perp}\neq\langle
t_{\tau}\rangle^{\perp}\subset\mathbb{C}H(W)$. Then ${\mathfrak{h}}(r,t)=0$.

###### Proof.

Consider the complex torus $B\coloneqq\mathbb{C}^{g}/\Phi(\mathcal{O}_{M})$.
Since the different ideal ${\mathfrak{D}}_{M}$ is the principal ideal
$(\eta)\subset\mathcal{O}_{M}$, we have

$\displaystyle\\{x\in
M\colon\textnormal{Tr}_{M/\mathbb{Q}}\left(x\eta^{-1}\mathcal{O}_{M}\right)\subset\mathbb{Z}\\}=\\{x\in
M\colon
x\cdot\eta^{-1}\mathcal{O}_{M}\subset\eta^{-1}\mathcal{O}_{M}\\}=\mathcal{O}_{M}.$

Therefore, polarization on $B$ attached to the alternating form

$Q\colon\Phi(\mathcal{O}_{M})\times\Phi(\mathcal{O}_{M})\to\mathbb{Z},\quad\left(\Phi(x),\Phi(y)\right)\mapsto\textnormal{Tr}_{M/\mathbb{Q}}\left(\eta^{-1}\cdot
x\cdot y\right),$

see Lemma 5.10, is a _principal_ polarization on $B$.

Consider the moduli space ${\textnormal{Sh}}_{M}(\Lambda)$ attached to the
hermitian $\mathcal{O}_{M}$-lattice $\Lambda$ as in Definition (5.8). Let
$[x]\in\langle r_{\tau}\rangle^{\perp}\cap\langle
t_{\tau}\rangle^{\perp}\subset\mathbb{C}H(W)$, and let $[(A,\iota,\lambda)]$
be the moduli point in ${\mathrm{Sh}}_{M}(\Lambda)$ corresponding to the image
of $[x]$ in $P\Gamma\setminus\mathbb{C}H(W)$, see Corollary 5.9. By
Proposition 5.11, the roots $r$ and $t$ induce $\mathcal{O}_{M}$-linear
embeddings $f_{1}\colon B\hookrightarrow A$ and $f_{2}\colon B\hookrightarrow
A$ of polarized abelian varieties. By [BL04, Corollary 5.3.13], there exist
abelian subvarieties $C_{i}\subset A$, $i=1,2$ such that $A\cong B\times
C_{1}$ and $A\cong B\times C_{2}$ as polarized abelian varieties, where $B$
and $C_{i}$ are endowed with the polarizations $\lambda_{B}$ and
$\lambda_{C_{i}}$ induced by their embedding into $A$. Note that $B$ is a
simple abelian variety because
$\textnormal{End}(B)\otimes_{\mathbb{Z}}\mathbb{Q}=\mathcal{O}_{M}\otimes_{\mathbb{Z}}\mathbb{Q}=K$
is a field. In particular $(B,\lambda_{B})$ is non-decomposable as a
principally polarized principally polarized abelian variety.

By [Deb96], the decomposition of $(A,\lambda)$ into non-decomposable polarized
abelian subvarieties is unique, in the strong sense that if
$(A_{i},\lambda_{i})$, $i\in\\{1,\dotsc,r\\}$ and $(B_{j},\mu_{j})$,
$j\in\\{1,\dotsc,m\\}$ are polarized abelian subvarieties such that the
natural homomorphisms $\prod_{i}(A_{i},\lambda_{i})\to(A,\lambda)$ and
$\prod_{j}(B_{j},\lambda_{j})\to(A,\lambda)$ are isomorphisms, then $r=m$ and
there exists a permutation $\sigma$ on $\\{1,\dotsc,r\\}$ such that $B_{j}$
and $A_{\sigma(j)}$ are equal as polarized abelian subvarieties of
$(A,\lambda)$, for every $j\in\\{1,\dotsc,r\\}$. Thus, for the abelian
subvarieties $B_{i}=\phi_{i}(B)\subset A$, we have either that
$B_{1}=B_{2}\subset A$, or that $B_{1}\cap B_{2}=\\{0\\}\subset A$.

Suppose that $B_{1}=B_{2}$. Then $\mathcal{O}_{M}\cdot
r=\phi_{1}(\mathcal{O}_{M})=\phi_{2}(\mathcal{O}_{M})=\mathcal{O}_{M}\cdot
t\subset\Lambda,$ hence $r=\lambda t$ for some
$\lambda\in\mathcal{O}_{M}^{\ast}$; but then $\langle
r_{\tau}\rangle^{\perp}=\langle t_{\tau}\rangle^{\perp}\subset\mathbb{C}H(W)$
which is absurd. Therefore, we must have $A=B_{1}\times B_{2}\times C$ as
polarized abelian varieties, for some polarized abelian subvariety $C$ of $A$.
This implies that

$\displaystyle{\mathrm{H}}^{-1,0}=\text{Lie}(A)=\textnormal{Lie}(B_{1})\times\textnormal{Lie}(B_{2})\times\textnormal{Lie}(C),$
(18)

which is orthogonal for the positive definite hermitian form
$iE_{\mathbb{C}}(x,\bar{y})$ on ${\mathrm{H}}^{-1,0}$.

Observe that $r_{\tau}=r^{-1,0}_{\tau}\in{\mathrm{H}}^{-1,0}_{\tau}$ and
$t_{\tau}=t^{-1,0}_{\tau}\in{\mathrm{H}}^{-1,0}_{\tau}$: see the proof of
Proposition 5.11. By Lemma 5.3, we have

$\displaystyle{\mathfrak{h}}(r,t)={\mathfrak{h}}^{\tau}(r_{\tau},t_{\tau})=\tau(\eta)\cdot
T^{\tau}_{\mathbb{C}}(r_{\tau},t_{\tau})=\tau(\eta)\cdot
E_{\mathbb{C}}(r_{\tau},\bar{t_{\tau}})=\tau(\eta)\cdot
E_{\mathbb{C}}(r_{\tau}^{-1,0},\overline{t_{\tau}^{-1,0}}).$

Since $r_{\tau}^{-1,0}\in\textnormal{Lie}(B_{1})$ and
$t_{\tau}^{-1,0}\in\textnormal{Lie}(B_{2})$, we have
$iE_{\mathbb{C}}(r_{\tau}^{-1,0},\overline{t_{\tau}^{-1,0}})=0$ by the
orthogonality of (18). We conclude that ${\mathfrak{h}}(r,t)=0$ and the proof
is finished. ∎

######  Example 6.3.

Suppose that $K=\mathbb{Q}(\sqrt{d})\subset\mathbb{C}$ for some
$d\in\mathbb{Z}$ with $d<0$. Note that Condition 6.1 is satisfied. Moreover,
there is an alternative, more elementary proof of Theorem 6.2 in this case,
see [ACT02a, Lemma 7.29]. Namely, let
$r,t\in\Lambda\subset\Lambda\otimes_{\mathcal{O}_{K}}\mathbb{C}=W$ and suppose
that ${\mathfrak{h}}(r,r)={\mathfrak{h}}(t,t)=1$, that $\langle
r\rangle^{\perp}\neq\langle t\rangle^{\perp}$ and that $\langle
r\rangle^{\perp}\cap\langle
t\rangle^{\perp}\neq\emptyset\subset\mathbb{C}H(W)$. Let $[x]\in\langle
r\rangle^{\perp}\cap\langle t\rangle^{\perp}$ and consider the decomposition
$W=\langle x\rangle\oplus\langle x\rangle^{\perp}$. Since $r,t\in\langle
x\rangle^{\perp}$, and the signature of $W$ is $(n,1)$, they span a positive
definite subspace $\langle r,t\rangle\subset W$. Therefore, the matrix

$\begin{pmatrix}{\mathfrak{h}}(r,r)&{\mathfrak{h}}(r,t)\\\
{\mathfrak{h}}(t,r)&{\mathfrak{h}}(t,t)\end{pmatrix}=\begin{pmatrix}1&{\mathfrak{h}}(r,t)\\\
{\mathfrak{h}}(t,r)&1\end{pmatrix}$

is positive definite, hence $\left|{\mathfrak{h}}(r,t)\right|^{2}<1$. Since
$\left|{\mathfrak{h}}(r,t)\right|^{2}\in\mathbb{Z}$, we have
${\mathfrak{h}}(r,t)=0$.

We conclude Section 6 by showing that the condition on the different ideal
${\mathfrak{D}}_{M} \subset\mathcal{O}_{M}$ in Theorem 6.2 (see Condition 6.1)
is satisfied in several natural cases.

######  Proposition 6.4.

Suppose that $M/\mathbb{Q}$ is an imaginary quadratic extension, or that
$M=\mathbb{Q}(\zeta_{n})$ is a cyclotomic field for some integer $n\geq 3$.
Then ${\mathfrak{D}}_{M}=(\beta)\subset\mathcal{O}_{M}$ for some element
$\beta\in\mathcal{O}_{M}$ such that $\sigma(\beta)=-\beta$.

###### Proof.

If $M/\mathbb{Q}$ is imaginary quadratic with discriminant $\Delta$, then
${\mathfrak{D}}_{M}=(\sqrt{\Delta})$ and the assertion is immediate. Let
$n\geq 3$ be an integer, and consider the fields
$M=\mathbb{Q}(\zeta_{n})\supset F=\mathbb{Q}(\alpha)$, where
$\alpha=\zeta_{n}+\zeta_{n}^{-1}$. Since
$\mathcal{O}_{M}=\mathbb{Z}[\zeta_{n}]$ we have
$\mathcal{O}_{M}=\mathcal{O}_{F}[\zeta_{n}]$. Notice that $f(x)=x^{2}-\alpha
x+1\in\mathcal{O}_{F}[x]$ is the minimal polynomial of $\zeta_{n}$ over $F$.
We have $f^{\prime}(\zeta_{n})=2\zeta_{n}-\alpha=\zeta_{n}-\zeta_{n}^{-1}$
hence
${\mathfrak{D}}_{M/F}=\left(f^{\prime}(\zeta_{n})\right)=\left(\zeta_{n}-\zeta_{n}^{-1}\right)$,
see [Neu99]. By [Lia76], we know that $\mathcal{O}_{F}=\mathbb{Z}[\alpha]$.
Moreover, if $g(x)\in\mathbb{Z}[x]$ is the minimal polynomial of $\alpha$ over
$\mathbb{Q}$, then ${\mathfrak{D}}_{F/\mathbb{Q}}=(g^{\prime}(\alpha))$. Since
${\mathfrak{D}}_{M/\mathbb{Q}}={\mathfrak{D}}_{M/F}{\mathfrak{D}}_{F/\mathbb{Q}}$
(c.f. [Neu99, Chapter III, Proposition 2.2]), we obtain
${\mathfrak{D}}_{M/\mathbb{Q}}={\mathfrak{D}}_{M/F}{\mathfrak{D}}_{F/\mathbb{Q}}=\left((\zeta_{n}-\zeta_{n}^{-1})\cdot
g^{\prime}(\alpha)\right).$ ∎

######  Remark 6.5.

Consider triples $(M,\beta,\Lambda)$ of the following form: $M$ is a CM field
with totally real subfield $F$, $\beta$ is an element in
$\mathcal{O}_{M}-\mathcal{O}_{F}$ such that $\beta^{2}\in\mathcal{O}_{F}$, and
$\Lambda$ is a finite free $\mathcal{O}_{M}$-module equipped with a hermitian
form
${\mathfrak{h}}\colon\Lambda\times\Lambda\to\beta\cdot{\mathfrak{D}}_{M}^{-1}$
of signature $(n,1)$ with respect to one
$\tau\in\Phi=\left\\{\varphi\in\textnormal{Hom}(K,\mathbb{C})\mid\Im(\varphi(\beta))>0\right\\}$
and positive definite with respect to every $\varphi\neq\tau\in\Phi$. It turns
out that can associate an orthogonal hyperplane arrangement

${\mathcal{H}}(M,\beta,\Lambda)=\bigcup_{f}H_{f}\subset\mathbb{C}H^{n}=\left\\{{\textnormal{negative
lines in }}\Lambda\otimes_{\mathcal{O}_{M},\tau}\mathbb{C}\right\\}$

to _every_ such triple $(M,\beta,\Lambda)$, in such a way that
${\mathcal{H}}(M,\beta,\Lambda)={\mathscr{H}}=\cup_{{\mathfrak{h}}(r,r)=1}\langle
r_{\mathbb{C}}\rangle^{\perp}$ if ${\mathfrak{D}}_{M}=(\beta)$. Indeed, for
such a triple, let ${\mathscr{S}}$ be the set of fractional ideals
${\mathfrak{a}}\subset M$ for which there exist $b\in\mathcal{O}_{F}$ such
that ${\mathfrak{D}}_{M}{\mathfrak{a}}\overline{{\mathfrak{a}}}=(b\beta)$. By
[Wam99, Theorem 4], the set ${\mathscr{S}}$ is not empty. For each
${\mathfrak{a}}\in{\mathscr{S}}$, pick an element $b$ as above and define
$\eta({\mathfrak{a}})=b\beta\in\mathcal{O}_{M}$. The complex torus
$B=\mathbb{C}^{g}/\Phi({\mathfrak{a}})$ is polarized by the Riemann form
$Q:\Phi({\mathfrak{a}})\times\Phi({\mathfrak{a}})\to\mathbb{Z}$,
$(\Phi(x),\Phi(y))\mapsto\textnormal{Tr}_{M/\mathbb{Q}}(\eta({\mathfrak{a}})^{-1}\cdot
x\bar{y})$, and this is a principal polarization [Wam99, Theorem 3]. Let
${\mathscr{R}}$ be the set of embeddings $f\colon{\mathfrak{a}}\to\Lambda$,
${\mathfrak{a}}\in{\mathscr{S}}$, such that
${\mathfrak{h}}(f(x),f(y))=x\bar{y}$ for $x,y\in{\mathfrak{a}}$. For
$f\in{\mathscr{R}}$, one obtains a hyperplane

$H_{f}=\\{x\in\mathbb{C}H^{n}\colon{\mathfrak{h}}^{\tau}(x,f({\mathfrak{a}}))=0\\}\subset\mathbb{C}H^{n}.$

Then the hyperplane arrangement
${\mathcal{H}}(M,\beta,\Lambda)=\cup_{f\in{\mathscr{R}}}H_{f}$ is orthogonal
by arguments similar to those used to prove Proposition 5.11 and Theorem 6.2.

##  7 Hermitian lattices over the Eisenstein integers

The goal of this section is to study particular examples of the spaces
$X(\Lambda)$ obtained via Theorem 4.1, namely those obtained by considering
the field $\mathbb{Q}(\zeta_{3})$ and the hermitian lattice
$\mathbb{Z}[\zeta_{3}]^{n,1}$. The ultimate goal is to prove Theorem 1.2.

###  7.1 Eisenstein integers

In Section 7, we fix the following notation: we let $K=\mathbb{Q}(\zeta_{3})$,
and for each $n\in\mathbb{Z}_{\geq 2}$, we define a lattice
$\Lambda_{n}=\left(\mathbb{Z}[\zeta_{3}]^{n+1},h_{n}\right)$ with
$h_{n}={\textnormal{diag}}(-1,1,\dotsc,1)$. As in Sections 2.1 \- 2.3, we make
the following definitions, although this time we label each object by the
integer $n\in\mathbb{Z}_{\geq 2}$ that is naturally attached to it:

$\displaystyle\Gamma(n)$
$\displaystyle=\textnormal{Aut}(\Lambda_{n})=\textnormal{Aut}_{\mathbb{Z}[\zeta_{3}]}\left(\mathbb{Z}[\zeta_{3}]^{n+1},h_{n}\right),\quad\quad\quad\quad\;P\Gamma(n)=\mu_{K}\backslash\Gamma(n)$
$\displaystyle{\mathscr{A}}_{n}$ $\displaystyle=\left\\{{\textnormal{anti-
unitary involutions
}}\alpha\colon\Lambda_{n}\to\Lambda_{n}\right\\},\quad\quad
P{\mathscr{A}}_{n}=\mu_{K}\backslash{\mathscr{A}}_{n},$ $\displaystyle
C{\mathscr{A}}_{n}$ $\displaystyle=P\Gamma(n)\backslash P{\mathscr{A}}_{n}.$

By Theorem 6.2 and Proposition 6.4, the hyperplane arrangement
${\mathscr{H}}_{n}=\cup_{r\in{\mathscr{R}}_{n}}H_{r}$ is an orthogonal
arrangement, where ${\mathscr{R}}_{n}=\left\\{r\in\Lambda_{n}\mid
h_{n}(r,r)=1\right\\}$. We can thus perform the glueing construction of
Section 3 to obtain a sequence of metric spaces $X(\Lambda_{n})$, with
$n\in\mathbb{Z}_{\geq 2}$. For each $n\in\mathbb{Z}_{\geq 2}$, the metric on
$X(\Lambda_{n})$ extends to a complete real hyperbolic orbifold structure,
hence its connected components are quotients of $\mathbb{R}H^{n}$ by discrete
groups of isometries (see Theorem 4.1.5). The goal of Section 7 is to prove
Theorem 1.2: for every $n\geq 2$, there exists a connected component
$X(\Lambda_{n})^{+}$ of $X(\Lambda_{n})$ such that the lattice
$\Gamma_{n}^{+}\subset{\mathrm{PO}}(n,1)$ underlying $X(\Lambda_{n})^{+}$ is
non-arithmetic.

For $i=0,1,2$, define
$\epsilon_{i}=(\epsilon_{i,1},\epsilon_{1,2},\epsilon_{i,3})\in\left\\{\pm
1\right\\}^{3}$ by $\epsilon_{0}=(1,1,1),\epsilon_{1}=(1,-1,1)$ and
$\epsilon_{2}=(1,-1,-1)$ (compare Section 2.4). For each
$i\in\left\\{0,1,2\right\\}$, we define an anti-unitary involution
$\alpha_{i}$ on $\Lambda_{2}$ as follows:

$\alpha_{i}\colon\Lambda_{2}\to\Lambda_{2},\quad\alpha_{i}(x_{0},x_{1},x_{2})=(\epsilon_{0,i}\cdot\bar{x}_{0},\epsilon_{1,i}\cdot\bar{x}_{1},\epsilon_{2,i}\cdot\bar{x}_{2}).$
(19)

######  Lemma 7.1.

The involutions $\alpha_{0},\alpha_{1},\alpha_{2}$ are pairwise not
$\Gamma(2)$-conjugate, and each anti-unitary involution of $\Lambda_{2}$ is
$\Gamma(2)$-conjugate to exactly one of the $\pm\alpha_{i}$. Moreover, the
composition

$\coprod_{i=0}^{2}\mathbb{R}H^{2}_{\alpha_{i}}\to Y(\Lambda_{2})\to
P\Gamma(2)\setminus Y(\Lambda_{2})=X(\Lambda_{2})$

is surjective, and the orbifold $X(\Lambda_{2})$ is connected.

###### Proof.

Let $\theta=\zeta_{3}-\zeta_{3}^{-1}=\sqrt{-3}\in\mathbb{Z}[\zeta_{3}]$ and
consider the vector space $W_{2}=\Lambda_{2}/\theta\Lambda_{2}$. For
$\alpha\in{\mathscr{A}}$, let $\alpha\colon W_{2}\to W_{2}$ be the induced
involution, let $q_{2}\colon W_{2}\to\mathbb{F}_{3}$ be the quadratic form
$q_{2}(x)=h_{2}(x,x)\bmod\theta$, and define $D(\alpha)=\dim(V^{\alpha})$ and
$T(\alpha)=\det(q_{2}|_{V^{\alpha}})\in\mathbb{F}_{3}^{\ast}/(\mathbb{F}_{3}^{\ast})^{2}=\\{\pm
1\\}$. The restrictions of $q_{2}$ to the fixed spaces $V^{\alpha_{i}}\subset
V$ for $\alpha_{0},\alpha_{1}$ and $\alpha_{2}$ have pairwise distinct
conjugacy invariant $(D(\alpha_{i}),T(\alpha_{i}))$ (see Lemma 7.2 below),
which proves the first statement. Next, one observes that the subspace
$\coprod_{\alpha\in
C{\mathscr{A}}_{2}}P\Gamma(2)_{\alpha}\setminus\left(\mathbb{R}H^{2}_{\alpha}-{\mathscr{H}}_{2}\right)\subset
X(\Lambda_{2})$ (see Theorem 4.1.4) is naturally homeomorphic to the moduli
space of stable real binary sextics with one double root at $\infty$ and no
other double roots (c.f. [ACT06, Section 5]). As in the proof of Theorem 4 of
[ACT10], one deduces that each anti-unitary involution of $\Lambda_{2}$ must
be $\Gamma(2)$-conjugate to exactly one of the $\pm\alpha_{i}$. The statements
follow. ∎

For each $i\in\left\\{0,1,\dotsc,n\right\\}$, define an anti-unitary
involution $\beta_{i}$ as follows:

$\displaystyle\begin{split}\beta_{i}&\colon\Lambda_{n}\to\Lambda_{n},\quad\beta_{i}(x_{0},\dotsc,x_{n})=\left(\bar{x}_{0},-\bar{x}_{1},\dotsc,-\bar{x}_{i},\bar{x}_{i+1},\dotsc,\bar{x}_{n}\right).\end{split}$
(20)

######  Lemma 7.2.

The elements $\pm\beta_{i}\in{\mathscr{A}}$ for $i=0,\dotsc,n$ are pairwise
non $\Gamma(n)$-conjugate. In particular, the classes $\beta_{i}\in
P{\mathscr{A}}_{n}$ are pairwise non $P\Gamma(n)$-conjugate.

###### Proof.

Consider the element $\theta=\zeta_{3}-\zeta_{3}^{-1}$ and define
$W_{n}=\Lambda_{n}/\theta\Lambda_{n}$. Consider the quadratic form
$q_{n}\colon W_{n}\to\mathbb{F}_{3}$ defined as
$q_{n}(x)=h_{n}(x,x)\bmod\theta.$ Note that, for $i\geq 1$ and
$x=(x_{0},x_{i+1}\dotsc,x_{n})\in W_{n}^{\beta_{i}}$, one has
$q_{n}(x)=-x_{0}^{2}+x_{i+1}^{2}+\cdots+x_{n}^{2}$. Similarly, for
$x=(x_{1},\dotsc,x_{i})\in V^{-\beta_{i}}$, one has
$q_{n}(x)=x_{1}^{2}+\cdots+x_{i}^{2}$. For an anti-unitary involution
$\alpha\colon\Lambda_{n}\to\Lambda_{n}$, define $D(\alpha)=\dim(V^{\alpha})$
and $T(\alpha)=\det(q_{n}|_{V^{\alpha}})$, which are $\Gamma(n)$-conjugacy
invariants. The above shows that
$\left(D(\beta_{i}),T(\beta_{i})\right)=\left(n-i+1,-1\right)$ and
$\left(D(-\beta_{i}),T(-\beta_{i})\right)=\left(i,1\right)$ for
$i=0,1,\dotsc,n$. The lemma follows. ∎

###  7.2 Maps between glued spaces

Let $\Psi$ be the embedding of hermitian $\mathcal{O}_{K}$-lattices
$\Lambda_{2}\to\Lambda_{n}$ that sends $(x_{0},x_{1},x_{2})\in\Lambda_{2}$ to
$\left(x_{0},x_{1},x_{2},0,0,\dotsc,0\right)\in\Lambda_{n}$. View
$\Lambda_{2}$ as a sublattice of $\Lambda_{n}$ via $\Psi$, and write

$\displaystyle\Lambda_{n}=\Lambda_{2}\oplus\left(\Lambda_{2}\right)^{\perp}.$
(21)

Using the canonical basis of $\Lambda_{n}$, we may view
$\Gamma(n)=\textnormal{Aut}(\Lambda_{n})$ as a subgroup of
$\textnormal{GL}_{n+1}(\mathcal{O}_{K})$. This gives an embedding

$\displaystyle j\colon\Gamma(2)\to\Gamma(n),\quad
M\mapsto(M,\textnormal{Id}).$ (22)

The natural totally geodesic embedding
$\nu\colon\mathbb{C}H^{2}\hookrightarrow\mathbb{C}H^{n}$ induces a totally
geodesic embedding

$\displaystyle\nu\colon\quad\widetilde{Y}_{2}^{\\#}\coloneqq\coprod_{i=0}^{2}\mathbb{R}H^{2}_{\alpha_{i}}\hookrightarrow\coprod_{i=0}^{2}\mathbb{R}H^{n}_{\beta_{i}}\subset\coprod_{\alpha\in
P{\mathscr{A}}_{n}}\mathbb{R}H^{n}_{\alpha}\eqqcolon\widetilde{Y}_{n}.$ (23)

As the composition
${\mathrm{SO}}(2,1)\to{\mathrm{O}}(2,1)\to{\mathrm{PO}}(2,1)={\textnormal{Isom}}(\mathbb{R}H^{2})$
is an isomorphism, there are natural embeddings
${\mathrm{PO}}(2,1)\hookrightarrow{\mathrm{PO}}(n,1)$ and
${\textnormal{PO}}(\Lambda_{2}^{\alpha_{i}})(\mathbb{Z})\hookrightarrow{\textnormal{PO}}(\Lambda_{n}^{\beta_{i}})(\mathbb{Z})$
for $i=0,1,2$. Moreover, as

$P\Gamma(2)_{\alpha_{i}}=\textnormal{Stab}_{P\Gamma(2)}\left(\mathbb{R}H^{2}_{\alpha_{i}}\right)={\mathrm{PO}}(\Lambda_{2}^{\alpha_{i}})(\mathbb{Z})$

by [ACT06, (5.1)], one can use the equality (8) to observe that (22) induces,
for each $i\in\left\\{0,1,2\right\\}$, an embedding $j\colon
P\Gamma(2)_{\alpha_{i}}\hookrightarrow P\Gamma(n)_{\beta_{i}}$. The map
$\nu\colon\widetilde{Y}_{2}^{\\#}\to\widetilde{Y}_{n}$ defined in (23) is
equivariant with respect to these embeddings $j$.

By Lemma 7.2, the classes of $\beta_{0},\beta_{1}$ and $\beta_{2}$ in
$P{\mathscr{A}}_{n}$ are pairwise not $P\Gamma(n)$-conjugate, so that the
induced map of hyperbolic orbifolds

$\displaystyle
O_{2}\coloneqq\coprod_{i=0}^{2}P\Gamma(2)_{\alpha_{i}}\setminus\left(\mathbb{R}H^{2}_{\alpha_{i}}-{\mathscr{H}}_{2}\right)\to\coprod_{\alpha\in
C{\mathscr{A}}_{n}}P\Gamma(n)_{\alpha}\setminus\left(\mathbb{R}H^{n}_{\alpha}-{\mathscr{H}}_{n}\right)\eqqcolon
O_{n}$ (24)

induces an injective map on sets of connected components
$\pi_{0}(O_{2})\to\pi_{0}(O_{n})$.

######  Lemma 7.3.

For each $n\geq 2$, there exists a natural map $\iota\colon X(\Lambda_{2})\to
X(\Lambda_{n}).$

###### Proof.

This follows readily from the following assertions:

1\. The map $\nu\colon\widetilde{Y}_{2}^{\\#}\to\widetilde{Y}_{n}$ defined in
(23) is compatible with the equivalence relations on both sides (see
Definition 3.9).

2\. Let $Y(\Lambda_{2})^{\\#}\subset Y(\Lambda_{2})$ be the image of
$\widetilde{Y}_{2}^{\\#}$ in $Y(\Lambda_{2})$ under the canonical map
$\widetilde{Y}_{2}\to Y(\Lambda_{2})$. The resulting map of topological spaces

$\displaystyle\phi\colon Y(\Lambda_{2})^{\\#}\to Y(\Lambda_{n})$ (25)