1. Introduction
For interesting classes of complex varieties, there is a period map that identifies the moduli space with an open subset of an arithmetic quotient of a hermitian symmetric domain. Classical examples include abelian varieties, K3-surfaces, and configurations of points on the line. To study moduli of real algebraic varieties, several authors have analyzed the equivariance of the complex period map with respect to the action of complex conjugation on cohomology [Reference KharlamovKha76, Reference NikulinNik79, Reference KharlamovKha84, Reference Seppälä and SilholSS89, Reference YoshidaYos98, Reference Apéry and YoshidaAY98, Reference Degtyarev, Itenberg and KharlamovDIK00]. An important difference between the complex and the real case is that moduli spaces of smooth real varieties are often not connected. This implies that a real period map has to be defined on each connected component of the moduli space separately; in favorable cases, this defines an isomorphism between any such component and the quotient of a Riemannian manifold by a discrete group of isometries (see e.g. [Reference Gross and HarrisGH81, Reference Allcock, Carlson and ToledoACT10, Reference ChuChu11, Reference Heckman and RiekenHR18]).
To salvage the non-connectedness of the real moduli space, one can sometimes define a slightly larger moduli space by allowing mild singularities. The idea is that, in such a larger space, the smooth varieties of one topological type do now deform into smooth varieties of another topological type, making the moduli space connected. In their beautiful paper [Reference Allcock, Carlson and ToledoACT10], Allcock, Carlson and Toledo showed that, for cubic surfaces, the real period maps defined on the various connected components of the moduli space of smooth surfaces extend to a global period map, defined on the moduli space of stable real cubic surfaces. In this way, they identified the latter with a single non-arithmetic real hyperbolic quotient. They proved analogous results for moduli of stable binary sextics, and stable binary sextics with a double root at infinity [Reference Allcock, Carlson and ToledoACT06; Reference Allcock, Carlson and ToledoACT07].
It turns out that binary quintics provide a new example of this phenomenon. Let
$X \cong \mathbf{A}^6_{\mathbf{R}}$
be the real algebraic variety that parametrizes homogeneous polynomials
$F \in {\mathbf{R}}[x,y]$
of degree five. Let
$X_0 \, \subset X$
parametrize polynomials with distinct roots, and
$X_s \, \subset X$
polynomials with roots of multiplicity at most two (i.e. stable in the sense of geometric invariant theory). The principal goal of this paper is to study the geometry of the moduli space of stable real binary quintics
\begin{align*} \mathcal{M}_s({\mathbf{R}}) := {\rm GL}_2({\mathbf{R}}) \setminus X_s({\mathbf{R}}) \supset {\rm GL}_2({\mathbf{R}}) \setminus X_0({\mathbf{R}}) =: \mathcal{M}_0({\mathbf{R}}). \end{align*}
Let
$P_s \, \subset {\mathbf{P}}^1({\mathbf{C}})^5$
be the set of five-tuples
$(x_1, \dotsc , x_5)$
such that no three
$x_i \in {\mathbf{P}}^1({\mathbf{C}})$
coincide (cf. [Reference Mumford and SuominenMS72]), and let
$P_0 \, \subset P_s$
be the subset of five-tuples whose coordinates are distinct. These spaces are naturally acted upon by
${\mathfrak{S}}_5$
, the symmetric group on five letters. Moreover, complex conjugation
$\sigma \colon {\mathbf{P}}^1({\mathbf{C}})^5 \to {\mathbf{P}}^1({\mathbf{C}})^5$
induces an anti-holomorphic involution
$\sigma \colon P_s/{\mathfrak{S}}_5 \to P_s/{\mathfrak{S}}_5$
that preserves
$P_0/{\mathfrak{S}}_5$
. Let
$(P_0/{\mathfrak{S}}_5)({\mathbf{R}})$
and
$(P_s/{\mathfrak{S}}_5)({\mathbf{R}})$
denote the respective fixed loci. Then
\begin{align*} \mathcal{M}_0({\mathbf{R}}) \cong {\rm PGL}_2({\mathbf{R}}) \setminus (P_0/{\mathfrak{S}}_5)({\mathbf{R}}) \;\;\; \;\;\; \mathrm{and} \;\;\; \;\;\; \mathcal{M}_s({\mathbf{R}}) \cong {\rm PGL}_2({\mathbf{R}}) \setminus (P_s/{\mathfrak{S}}_5)({\mathbf{R}}). \end{align*}
In other words,
$\mathcal{M}_0({\mathbf{R}})$
is the space of subsets
$S \, \subset {\mathbf{P}}^1({\mathbf{C}})$
of cardinality
$|S| = 5$
that are stable under complex conjugation modulo real projective transformations; in
$\mathcal{M}_s({\mathbf{R}})$
one or two pairs of points are allowed to collapse.
By the Deligne–Mostow theory, the coarse moduli space
$\mathcal{M}_s({\mathbf{C}})= {\rm GL}_2({\mathbf{C}}) \setminus X_s({\mathbf{C}})$
of stable complex binary quintics has a complex hyperbolic orbifold structure. Indeed, for five distinct points
$u_1, \dotsc , u_5 \in \mathbf{A}^1({\mathbf{C}}) \, \subset {\mathbf{P}}^1({\mathbf{C}})$
, the projective model of the normalization of the affine curve
$z^5 = (x-u_1)^2 \cdots (x-u_5)^2$
is a smooth curve
$C$
of genus six; this curve
$C$
carries an automorphism of order five that induces an automorphism on the space
${\rm H}^0(C, \Omega^1_C)$
of holomorphic one-forms on
$C$
whose
$e^{2 \pi i/5}$
-eigenspace defines a line in the corresponding eigenspace in
${\rm H}^1(C({\mathbf{C}}),{\mathbf{C}})$
. This line is negative for a natural hermitian form, and hence one can associate to
$\left \{ u_1, \dotsc , u_5 \right \}$
a point in a certain two-dimensional complex ball quotient
$P\Gamma \setminus \mathbf{C} H^2$
. This construction was already known to Shimura; see [Reference ShimuraShi63; Reference ShimuraShi64]. By varying the subset
$\left \{ u_1, \dotsc , u_5 \right \}$
of points on
${\mathbf{P}}^1({\mathbf{C}})$
, or rather the associated complex binary quintic, one obtains a period map
$\mathcal{M}_0({\mathbf{C}}) = {\rm GL}_2({\mathbf{C}}) \setminus X_0({\mathbf{C}}) \to P\Gamma \setminus \mathbf{C} H^2$
(see Section 4.1 for details). The results of Deligne and Mostow [Reference Deligne and MostowDM86] imply that this period map extends to an isomorphism of complex analytic spaces
$ \mathcal{M}_s({\mathbf{C}}) = {\rm GL}_2({\mathbf{C}}) \setminus X_s({\mathbf{C}}) \xrightarrow {\sim } P\Gamma \setminus {\mathbf{C}} H^2$
; see Theorem 4.1. Since strictly stable quintics correspond to points in a hyperplane arrangement
${\mathscr{H}} \, \subset {\mathbf{C}} H^2$
(see Proposition 4.3), one thus obtains an isomorphism
\begin{align} \mathcal{M}_0({\mathbf{C}}) = {\rm GL}_2({\mathbf{C}}) \setminus X_0({\mathbf{C}}) \xrightarrow {\sim } P\Gamma \setminus \left ({\mathbf{C}} H^2 - {\mathscr{H}}\kern2pt \right). \end{align}
By investigating the equivariance of the period map with respect to suitable anti-holomorphic involutions
$\alpha _j \colon {\mathbf{C}} H^2 \to {\mathbf{C}} H^2$
, we obtain the following real analogue.
Theorem 1.1.
For
$j \in \{0,1,2\}$
, let
${\mathscr{M}}_{j}$
be the connected component of
$\mathcal{M}_0({\mathbf{R}})$
parametrizing
${\rm Gal}({\mathbf{C}}/{\mathbf{R}})$
-stable subsets
$S \, \subset {\mathbf{P}}^1({\mathbf{C}})$
with
$2j$
complex and
$5 - 2j$
real points. The period map induces an isomorphism of real analytic orbifolds
\begin{align} {\mathscr{M}}_j \xrightarrow {\sim } P \Gamma _j \setminus \left ({\mathbf{R}} H^2 - {\mathscr{H}}_j \kern1pt \right). \end{align}
Here
${\mathbf{R}} H^2$
is the real hyperbolic plane,
${\mathscr{H}}_j$
a union of geodesic subspaces in
$ {\mathbf{R}} H^2$
, and
$P\Gamma _j$
an arithmetic lattice in
${\rm PO}(2,1)$
. Moreover, the lattices
$P\Gamma _j$
are projective orthogonal groups attached to explicit quadratic forms over
${\mathbf{Z}}[\lambda ]$
,
$\lambda = \zeta _5 + \zeta _5^{-1} = (\sqrt 5 - 1)/2$
; see (41).
In particular, Theorem 1.1 endows each connected component
${\mathscr{M}}_j \, \subset \mathcal{M}_0({\mathbf{R}})$
with a hyperbolic metric. Since one can deform the topological type of a
${\rm Gal}({\mathbf{C}}/{\mathbf{R}})$
-stable five-element subset of
${\mathbf{P}}^1({\mathbf{C}})$
by allowing two points to collide, the compactification
$\mathcal{M}_s({\mathbf{R}}) \supset \mathcal{M}_0({\mathbf{R}})$
is connected. One may wonder whether the metrics on the components
${\mathscr{M}}_j$
extend to a metric on the whole of
$\mathcal{M}_s({\mathbf{R}})$
. If so, what does the resulting space look like at the boundary? Our main result answers these questions in the following way.
Theorem 1.2.
There exists a complete hyperbolic metric on
$\mathcal{M}_s({\mathbf{R}})$
that restricts to the metrics on
${\mathscr{M}}_j$
induced by (2). Let
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
denote the resulting metric space, and define
$\Gamma _{3,5,10}$
as the group
\begin{align} \Gamma _{3,5,10} = \langle a, b, c \mid a^2 = b^2 = c^2 = (ab)^3 = (ac)^5 = (bc)^{10} = 1 \rangle. \end{align}
There exist an open embedding
$\coprod _j P \Gamma _j \setminus \left ({\mathbf{R}} H^2 - {\mathscr{H}}_j\, \right ) \hookrightarrow \Gamma _{3,5,10}\setminus {\mathbf{R}} H^2$
and an isometry
\begin{align} \overline {{\mathscr{M}}}_{\mathbf{R}} \cong \Gamma _{3,5,10} \setminus {\mathbf{R}} H^2 \end{align}
that extend the real analytic orbifold isomorphisms (2) in Theorem 1.1. In particular,
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
is isometric to the hyperbolic triangle
$\Delta _{3,5,10}$
with angles
$\pi /3, \pi /5, \pi /10$
; see Figure 1.
Figure 1. The moduli space of stable real binary quintics as the hyperbolic triangle
$\Delta _{3,5,10} \, \subset {\mathbf{R}} H^2$
. Here
$\lambda = \zeta _5 + \zeta _5^{-1} = (\sqrt 5 - 1)/2$
and
$\omega = \zeta _3$
(where
$\zeta _n = e^{2 \pi i/n} \in {\mathbf{C}}$
for
$n \in {\mathbf{Z}}_{\geq 3}$
).
Note that the closure
\begin{align*}\overline {{\mathscr{M}}}_0 \, \subset \overline {{\mathscr{M}}}_{\mathbf{R}}\end{align*}
of
${\mathscr{M}}_0$
in
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
is the moduli space of stable configurations of five real points on
${\mathbf{P}}^1_{\mathbf{R}}$
. This moduli space was investigated by Apéry and Yoshida in [Reference Apéry and YoshidaAY98], who proved that
$\overline {{\mathscr{M}}}_0$
is the hyperbolic triangle with angles
$\pi /2, \pi /4$
and
$\pi /5$
. From this, together with the knowledge of the angles of
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
and the fact that the two hyperplanes in Figure 1 intersect orthogonally, one can deduce the remaining angles of the closures
$\overline {{\mathscr{M}}}_j \, \subset \overline {{\mathscr{M}}}_{\mathbf{R}}$
of the subsets
${\mathscr{M}}_j \, \subset \overline {{\mathscr{M}}}_{\mathbf{R}}$
(
$j \in \left \{ 0,1,2 \right \}$
).
Theorem 1.3.
Consider Figure 1. For
$j=0,1,2$
, let
$\overline {{\mathscr{M}}}_j \, \subset \overline {{\mathscr{M}}}_{\mathbf{R}}$
be the closure of
${\mathscr{M}}_j \, \subset \overline {{\mathscr{M}}}_{\mathbf{R}}$
.
-
(i) The angle of
$\overline {{\mathscr{M}}}_0$
at
$(0,-1,\infty ,\infty ,1)$
is
$\pi /2$
, and its angle at
$(0,-1,-1,\infty ,\infty )$
is
$\pi /4$
. -
(ii) The angle of
$\overline {{\mathscr{M}}}_1$
at
$(0,-1, \infty ,\infty ,1)$
is
$\pi /2$
, and its angle at
$(0,-i,\infty ,\infty ,i)$
is
$\pi /2$
. -
(iii) The angle of
$\overline {{\mathscr{M}}}_2$
at
$(0,-1,-1,\infty ,\infty )$
is
$\pi /4$
, and its angle at
$(0,-i,\infty ,\infty ,i)$
is
$\pi /2$
.
Remark 1.4.
-
(i) The lattice
$\Gamma _{3,5,10} \, \subset {\rm PO}(2,1)$
is non-arithmetic; see [Reference TakeuchiTak77]. -
(ii) The topological space
$\mathcal{M}_s({\mathbf{R}})$
underlies two topological orbifold structures: the natural orbifold structure on
$ {\rm GL}_2({\mathbf{R}}) \setminus X_s({\mathbf{R}})$
and the orbifold structure on
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
induced by (4). These orbifold structures only differ at one point of
$\mathcal{M}_s({\mathbf{R}})$
, which is
$(\infty , i, i, -i, -i)$
(see Figure 1). The stabilizer group of
$\mathcal{M}_s({\mathbf{R}})$
at
$(\infty , i, i, -i, -i)$
is isomorphic to
${\mathbf{Z}}/2$
, whereas the stabilizer group of
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
at
$(\infty , i, i, -i, -i)$
is isomorphic to the dihedral group of order
$20$
. -
(iii) Important ingredients in the proof of Theorem 1.2 are:
-
(a) the fact that under the complex period map (1), moduli of singular binary quintics correspond to points in the quotient of a certain hyperplane arrangement
${\mathscr{H}} \, \subset {\mathbf{C}} H^2$
; and -
(b) the fact that the hyperplane arrangement
${\mathscr{H}} \, \subset {\mathbf{C}} H^2$
is an orthogonal arrangement in the sense of [Reference Allcock, Carlson, Toledo and ClemensACT02a].
We prove (a) in Proposition 4.3, and (b) holds by [Reference de Gaay FortmanGF24, Theorem 2.5 and Example 2.12].
-
Remark 1.5. Let
$PX_0({\mathbf{C}})$
denote the space of
${\mathbf{C}}^\ast$
-equivalence classes of smooth complex binary quintics
$F \in {\mathbf{C}}[x,y]$
. The natural map
$P_0 \to PX_0({\mathbf{C}})$
induces a
${\rm PGL}_2({\mathbf{C}})$
-equivariant isomorphism
$P_0/{\mathfrak{S}}_5 \xrightarrow {\sim } PX_0({\mathbf{C}})$
, and the quotient
$\mathcal{M}_0({\mathbf{C}}) = {\rm PGL}_2({\mathbf{C}}) \setminus PX_0({\mathbf{C}})$
is the moduli space of smooth complex binary quintics. It turns out that neither
$\pi _1 \left ( PX_0({\mathbf{C}})\, \right )$
nor
$\pi _1^{{\rm orb}}\left ( \mathcal{M}_0({\mathbf{C}})\, \right )$
is a lattice in any Lie group with finitely many connected components. In view of [Reference Allcock, Carlson, Toledo and ClemensACT02a, Theorem 1.2], this follows from the isomorphism
$\mathcal{M}_0({\mathbf{C}}) \cong P\Gamma \setminus \left ({\mathbf{C}} H^2 - {\mathscr{H}}\, \right )$
(see (1) above) and the orthogonality of the hyperplane arrangement
${\mathscr{H}} \, \subset {\mathbf{C}} H^2$
(see Remark 1.4(iii)(b) above).
1.1 Overview of this paper
In Section 3, we recall known results on families of quintic covers of the complex projective line, branched along a binary quintic. We consider moduli of complex binary quintics in Section 4. In particular, we show that the Deligne–Mostow theory provides an isomorphism between the space of stable complex binary quintics and an arithmetic ball quotient. In Section 5, we prove that moduli of stable real binary quintics are in one-to-one correspondence with points in the real hyperbolic quotient space
$P\Gamma _{\mathbf{R}}\setminus {\mathbf{R}} H^2$
defined by a lattice
$P\Gamma _{\mathbf{R}} \, \subset {\rm PO}(2,1)$
. We calculate
$P\Gamma _{\mathbf{R}}$
in Section 6: it is conjugate to the lattice
$\Gamma _{3,5,10}$
defined in (3). In Section 7, we study monodromy groups of moduli spaces of smooth binary quintics over
$\mathbf{C}$
and over
$\mathbf{R}$
, and prove Theorems 1.2 and 1.3. In Section 8, we use [Reference de Gaay FortmanGF24, Theorem 1.8] and the main results of this paper to provide an explicit sequence
$\left \{ \Gamma _n \, \subset {\rm PO}(n,1) \right \}_{n \geq 2}$
of non-arithmetic lattices
$\Gamma _n$
, with
$\Gamma _2 = P\Gamma _{\mathbf{R}}$
.
2. Notation
Let
$K$
be the cyclotomic field
$ {\mathbf{Q}}(\zeta )$
, where
\begin{align*}\zeta := \zeta _5 = e^{2 \pi i /5} \in {\mathbf{C}}.\end{align*}
The ring of integers
${\mathcal{O}}_K$
of
$K$
is
${\mathbf{Z}}[\zeta ]$
(see e.g. [Reference NeukirchNeu99, Chapter I, Proposition 10.2]). Let
$\mu _K \, \subset {\mathcal{O}}_K^\ast$
be the torsion subgroup of the unit group
${\mathcal{O}}_K^\ast$
, and recall that
$\mu _K$
is cyclic of order ten, generated by
$-\zeta$
. Define an involution
$\rho \colon K \to K$
by
$\rho (\zeta ) = \zeta ^{-1}$
, and let
$F = K^\rho$
be the maximal totally real subfield of
$K$
. Recall that
$F$
is generated over
$\mathbf{Q}$
by the element
\begin{align*}\lambda := \zeta + \zeta ^{-1} = (\sqrt 5 - 1)/2.\end{align*}
Define
\begin{align*} \eta = \zeta^2 - \zeta ^{-2} \in {\mathcal{O}}_K, \end{align*}
and consider the different ideal
${\mathfrak{D}}_K \, \subset {\mathcal{O}}_K$
(see e.g. [Reference NeukirchNeu99, Chapter III, Section 2, Definition 2.1] or [Reference SerreSer79, Chapter III, Section 3]). We have
$ (\zeta - \zeta ^{-1}) \cdot (\zeta^2 - \zeta ^{-2}) \cdot (\zeta^3 - \zeta ^{-3}) \cdot (\zeta^4 - \zeta ^{-4}) = 5,$
and hence
$ (\eta )^4 = (5)$
as ideals of
${\mathcal{O}}_K$
. This implies (see e.g. [Reference NeukirchNeu99, Chapter III, Theorem 2.6]) that
\begin{align*} {\mathfrak{D}}_K = \left (5/ \eta\, \right ) = (\eta )^3. \end{align*}
3. Preliminaries on quintic covers of the projective line branched along five points
In this section, we recollect some known results on quintic covers
$C \to {\mathbf{P}}^1_{\mathbf{C}}$
ramified along five points with local monodromy
$\exp (4 \pi i / 5)$
around each point. Some of these results are well known but hard to find in the literature; we state and prove these for the convenience of the reader. Other results of this section are available in the literature, but we formulate them in a different manner.
Recall from the introduction that
$X \cong \mathbf{A}^6_{\mathbf{R}}$
is the real algebraic variety that parametrizes homogeneous polynomials of degree five,
$X_0$
the subvariety of polynomials with distinct roots, and
$X_s \, \subset X$
the subvariety of polynomials with roots of multiplicity at most two (i.e. non-zero polynomials whose class in the associated projective space is stable in the sense of geometric invariant theory [Reference Mumford, Fogarty and KirwanMFK94] for the action of
${\rm SL}_{2, {\mathbf{R}}}$
on it).
3.1 Refined Hodge numbers of a quintic cover of
${\mathbf{P}}^1$
branched along five points
Let
$F \in X_0({\mathbf{C}})$
be a smooth binary quintic; thus
$F = F(x,y) \in {\mathbf{C}}[x,y]$
is a homogeneous polynomial of degree five whose zeros in
${\mathbf{P}}^1({\mathbf{C}})$
all have multiplicity one. Let
$t_1, \dotsc , t_5 \in {\mathbf{P}}^1({\mathbf{C}})$
be the zeros of
$F$
, with
$t_j = [u_j \colon v_j]$
in homogeneous coordinates of
${\mathbf{P}}^1({\mathbf{C}})$
, for
$j=1, \dotsc , 5$
. Let
\begin{align} C_F \to {\mathbf{P}}^1_{\mathbf{C}} \end{align}
be the cyclic quintic cover with branch points
$t_1, \dotsc , t_5 \in {\mathbf{P}}^1({\mathbf{C}})$
and local monodromy
\begin{align} \exp (2 \pi i \cdot 2/5) \in \mu _5 \end{align}
around
$t_j$
for each
$j \in \left \{ 1, \dotsc , 5 \right \}$
. If the points
$t_j = [u_j \colon 1]$
are all in
$\mathbf{A}^1({\mathbf{C}})$
, the cover (5) is, in affine coordinates, given by the normalization of the curve defined by the equation
\begin{align*} z^5 = (x - u_1)^2 \cdot (x - u_2)^2 \cdots (x-u_5)^2, \end{align*}
with
$\zeta \in \mu _5 \, \subset {\mathbf{C}}$
acting by
$(x,z) \mapsto (x, \zeta \cdot z)$
. We have
$g=6$
for the genus
$g = g(C_F)$
of the curve
$C_F$
. Let
$JC_F$
be the Jacobian of the curve
$C_F$
, so that
\begin{align*} JC_F({\mathbf{C}})= {\rm H}^1(C_F({\mathbf{C}}), {\mathcal{O}}_{C_F})/{\rm H}^1(C_F({\mathbf{C}}), {\mathbf{Z}})(1), \end{align*}
with weight
$-1$
Hodge decomposition
\begin{align} {\rm H}_1(JC_F({\mathbf{C}}), {\mathbf{C}}) = {\rm H}^1(C_F({\mathbf{C}}), {\mathbf{Z}})(1) \otimes {\mathbf{C}} = {\rm H}^{-1,0}(JC_F) \oplus {\rm H}^{0,-1}(JC_F). \end{align}
Note that
${\rm H}^{0,-1}(JC_F)$
is naturally isomorphic to the space
${\rm H}^0(C_F, \Omega^1)$
of global holomorphic differentials on the curve
$C_F$
. The order five automorphism
\begin{align*}\zeta \colon C_F \xrightarrow {\sim }C_F\end{align*}
defined above induces an embedding of rings
\begin{align*} \varphi \colon {\mathbf{Z}}[\zeta ] \to {{\rm End}}(JC_F), \quad \varphi (\zeta ) = (\zeta ^{-1})^\ast , \end{align*}
which is compatible with the Hodge decomposition (7). For
$k \in \left \{ 1,2,3,4 \right \}$
, define
\begin{align*} {\rm H}^{0,-1}(JC_F)_{\zeta ^k} = \left \{ x \in {\rm H}^{0,-1}(JC_F) \mid \varphi (\zeta ) = \zeta ^k \right \} \, \subset {\rm H}^{0,-1}(JC_F) = {\rm H}^{1,0}(C_F) = {\rm H}^0(C_F, \Omega^1), \end{align*}
and define
${\rm H}^{-1,0}(JC_F)_{\zeta ^k} \, \subset {\rm H}^{-1,0}(JC_F)$
in a similar way.
The way to calculate the refined Hodge numbers
$h^{0,-1}(JC_F)_{\zeta ^k} = \dim {\rm H}^{0,-1}(JC_F)_{\zeta ^k}$
and
$h^{-1,0}(JC_F)_{\zeta ^k}= \dim {\rm H}^{-1,0}(JC_F)_{\zeta ^k}$
for
$k=1,2,3,4$
is well known; the result is as follows.
Lemma 3.1.
Let
$F \in X_0({\mathbf{C}})$
be a smooth binary quintic, and let
$JC_F$
be the Jacobian of the cyclic cover
$C_F \to {\mathbf{P}}^1({\mathbf{C}})$
associated to
$F$
as in (5). One has the following refined Hodge numbers:
\begin{align*} h^{0,-1}(JC_F)_{\zeta } & = 1, \;\;\; h^{0,-1}(JC_F)_{\zeta^2} = 3, \;\;\; h^{0,-1}(JC_F)_{\zeta^3} = 0, \;\;\; h^{0,-1}(JC_F)_{\zeta ^{4}} = 2, \\ h^{-1,0}(JC_F)_{\zeta } & = 2, \;\;\; h^{-1,0}(JC_F)_{\zeta^2} = 0, \;\;\; h^{-1,0}(JC_F)_{\zeta^3} = 3, \;\;\; h^{-1,0}(JC_F)_{\zeta ^{4}} = 1. \end{align*}
Proof. This follows from the Hurwitz–Chevalley–Weil formula (see [Reference Moonen and OortMO13, Proposition 5.9]). Alternatively, see [Reference LooijengaLoo07, Lemma 4.2].
3.2 The hermitian lattice
We fix a smooth binary quintic
$F_0 \in X_0({\mathbf{C}})$
. Let
$C \to {\mathbf{P}}^1_{\mathbf{C}}$
be the cyclic cover of
${\mathbf{P}}^1_{\mathbf{C}}$
associated to
$F_0$
as in (5), and consider the Jacobian variety with
${\mathcal{O}}_K$
-action
\begin{align*} \left (JC ,\quad \varphi \colon {\mathcal{O}}_K = {\mathbf{Z}}[\zeta ] \to {{\rm End}}(A\right). \end{align*}
See Section 3.1 above. Define
$\Lambda$
as the free
${\mathcal{O}}_K$
-module
\begin{align*} \Lambda ={\rm H}^1(C({\mathbf{C}}), {\mathbf{Z}}). \end{align*}
The canonical principal polarization of the Jacobian
$JC$
is given by a symplectic pairing
\begin{align*} E \colon \Lambda \times \Lambda = {\rm H}^1(C({\mathbf{C}}), {\mathbf{Z}}) \times {\rm H}^1(C({\mathbf{C}}), {\mathbf{Z}}) \to {\rm H}^2(C({\mathbf{C}}),{\mathbf{Z}}) = {\mathbf{Z}} \end{align*}
which satisfies
$E(\varphi (a)x,y) = E(x, \varphi (\rho (a))y)$
for each
$a \in {\mathcal{O}}_K$
and
$x,y \in \Lambda$
, where
$\rho$
is the automorphism
$K \xrightarrow {\sim } K, \zeta \mapsto \zeta ^{-1}$
(see Section 2). Consider the different ideal
${\mathfrak{D}}_K \, \subset {\mathcal{O}}_K$
, and define a skew-hermitian form
$T$
on
$\Lambda$
as follows:
\begin{align*} T \colon \Lambda \times \Lambda \to {\mathfrak{D}}_K^{-1}, \quad T(x,y) = \frac {1}{5}\sum _{j=0}^{4}\zeta ^jE\left ( x, \varphi (\zeta )^j y\right)\!. \end{align*}
By [Reference de Gaay FortmanGF24, Example 11.2.2], this is the skew-hermitian form corresponding to
$E$
, via [Reference de Gaay FortmanGF24, Lemma 11.1]. Let
$\eta \in {\mathcal{O}}_K$
be the purely imaginary element
\begin{align*} \eta = \zeta^2 - \zeta ^{-2} \in {\mathcal{O}}_K, \end{align*}
and note that the different ideal
${\mathfrak{D}}_K \, \subset {\mathcal{O}}_K$
is generated by
$5/\eta$
(see Section 2). We obtain a hermitian form on the free
${\mathcal{O}}_K$
-module
$\Lambda$
as follows:
\begin{equation} {\mathfrak{h}} \colon \Lambda \times \Lambda \to {\mathcal{O}}_K, \;\;\; {\mathfrak{h}}(x,y) = \frac {5}{\eta } \cdot T(x,y) = \frac {1}{\zeta^2 - \zeta ^{-2}} \cdot \sum _{j=0}^{4}\zeta ^jE\left ( x, \varphi (\zeta )^j y\, \right)\!. \end{equation}
By [Reference de Gaay FortmanGF24, Lemma 11.1], we have that
$(\Lambda , {\mathfrak{h}})$
is unimodular as a hermitian lattice over
${\mathcal{O}}_K$
, because
$(\Lambda , E)$
is unimodular as an alternating lattice over
$\mathbf{Z}$
.
3.3 The signature
Recall that a CM type on a CM field
$L$
of degree
$2g$
218 over
$\mathbf{Q}$
is a set
$\Phi \subset {{\rm Hom}}(L, {C})$
of
$g$
embeddings of
$L$
into
$\mathbf{C}$
, such that for each embedding
$\varphi \colon L \to \mathbf{C}$224$
, either
$\varphi$
lies in
$\Phi$
or the complex conjugate of
$\varphi$
does.
Define a CM type
$\Phi \, \subset {{\rm Hom}}(K,{\mathbf{C}})$
as
\begin{align} \Phi = \left \{ \sigma _1, \sigma _2 \colon K \to {\mathbf{C}} \right \}, \quad \sigma _k(\zeta ) = \zeta ^k \quad {\rm for}\quad k=1,2. \end{align}
Observe that, for
$k=1,2$
, we have
\begin{align*} \Lambda \otimes _{{\mathcal{O}}_K, \sigma _k}{\mathbf{C}}= \left (\Lambda \otimes _{\mathbf{Z}} {\mathbf{C}}\right )_{\zeta ^k} = {\rm H}^1(C({\mathbf{C}}),{\mathbf{C}})_{\zeta ^k}. \end{align*}
Lemma 3.2.
For
$k \in \left \{ 1,2 \right \}$
, consider the hermitian form
\begin{align*} {\mathfrak{h}}^{\sigma _k} \colon {\rm H}^1(C({\mathbf{C}}),{\mathbf{C}})_{\zeta ^k} \times {\rm H}^1(C({\mathbf{C}}),{\mathbf{C}})_{\zeta ^k} \to {\mathbf{C}}. \end{align*}
Then
${{\rm sign}}({\mathfrak{h}}^{\sigma _1}) = (2,1)$
and
${{\rm sign}}({\mathfrak{h}}^{\sigma _2}) = (3,0)$
, where
${{\rm sign}}({\mathfrak{h}}^{\sigma _k})$
is the signature of
${\mathfrak{h}}^{\sigma _k}$
.
Proof. Write
$\Lambda _{\mathbf{C}} = \Lambda \otimes _{\mathbf{Z}} {\mathbf{C}} = {\rm H}^1(C({\mathbf{C}}),{\mathbf{C}})$
. For each embedding
$\phi \colon K \to {\mathbf{C}}$
, the restriction of the hermitian form
\begin{align*}\phi \left (5/\eta\, \right ) \cdot E_{\mathbf{C}}(x, \bar{y}) \colon \Lambda _{\mathbf{C}} \times \Lambda _{\mathbf{C}} \to {\mathbf{C}}\end{align*}
to
$(\Lambda _{\mathbf{C}})_{\phi } \, \subset \Lambda _{\mathbf{C}}$
coincides with
$ {\mathfrak{h}}^{\phi }$
by [Reference de Gaay FortmanGF24, Lemma 11.3]. Moreover, the hermitian form
\begin{align*} i \cdot E_{\mathbf{C}}(x, \bar{y}) \colon {\rm H}^1(C({\mathbf{C}}),{\mathbf{C}}) \times {\rm H}^1(C({\mathbf{C}}),{\mathbf{C}}) \to {\mathbf{C}} \end{align*}
is positive definite on
$ {\rm H}^0(C, \Omega^1) = {\rm H}^{1,0}(C) = {\rm H}^{0,-1}(JC)$
and negative definite on
${\rm H}^{-1,0}(JC) = {\rm H}^{0,1}(C)$
(see [Reference VoisinVoi02, Théorème 6.32]). As
$\Im (\sigma _1(\eta )) \gt 0$
and
$\Im (\sigma _2(\eta )) \lt 0$
, we have
$\Im (\sigma _1(5/\eta )) \lt 0$
and
$\Im (\sigma _2(5/\eta )) \gt 0$
. Consequently, the hermitian form
\begin{align*} \sigma _1(5/\eta )\cdot E_{\mathbf{C}}(x, \bar{y}) = {\mathfrak{h}}^{\sigma _1}(x,y) \colon {\rm H}^1(C({\mathbf{C}}),{\mathbf{C}})_\zeta \times {\rm H}^1(C({\mathbf{C}}),{\mathbf{C}})_\zeta \to {\mathbf{C}} \end{align*}
is negative definite on
${\rm H}^{0,-1}(JC)_\zeta$
and positive definite on
${\rm H}^{-1,0}(JC)_{\zeta }$
, so that
$ {{\rm sign}}({\mathfrak{h}}^{\sigma _1}) = (2,1)$
by Lemma 3.1. Similarly, the hermitian form
\begin{align*} \sigma _2(5/\eta )\cdot E_{\mathbf{C}}(x, \bar{y}) = {\mathfrak{h}}^{\sigma _2}(x,y) \colon {\rm H}^1(C({\mathbf{C}}),{\mathbf{C}})_{\zeta^2} \times {\rm H}^1(C({\mathbf{C}}),{\mathbf{C}})_{\zeta^2} \to {\mathbf{C}} \end{align*}
is positive definite on
${\rm H}^{0,-1}(JC)_{\zeta^2}$
and negative definite on
${\rm H}^{-1,0}(JC)_{\zeta^2}$
. Hence, using Lemma 3.1 again, we conclude that
${{\rm sign}}({\mathfrak{h}}^{\sigma _2}) = (3,0)$
.
3.4 The monodromy representation
Consider the real algebraic variety
$X_0$
introduced in Section 1. Let
$D \, \subset {\rm GL}_2({\mathbf{C}})$
be the subgroup
$D = \left \{ \zeta ^j\cdot {{\rm Id}} \right \} \, \subset {\rm GL}_2({\mathbf{C}})$
of scalar matrices of the form
$\zeta ^j \cdot {{\rm Id}}$
with
$j \in \left \{ 0,1,2,3,4 \right \}$
and
${{\rm Id}} \in {\rm GL}_2({\mathbf{C}})$
the identity two-by-two matrix. Define
\begin{align} G({\mathbf{C}}) = {\rm GL}_2({\mathbf{C}})/D. \end{align}
The group
$G({\mathbf{C}})$
acts from the left on
$X_0({\mathbf{C}})$
in the following way: if
\begin{align*}F = F(x,y) \in {\mathbf{C}}[x,y]\end{align*}
is a binary quintic, we may view
$F$
as a function
${\mathbf{C}}^2 \to {\mathbf{C}}$
, and define
$g \cdot F=F(g^{-1})$
for
$g \in G({\mathbf{C}})$
. This gives a canonical isomorphism of complex analytic orbifolds
\begin{align*} \mathcal{M}_0({\mathbf{C}}) = G({\mathbf{C}}) \setminus X_0({\mathbf{C}}), \end{align*}
where
$\mathcal{M}_0$
is the moduli stack of smooth binary quintics. Define
\begin{align} g \colon {\mathscr{C}} \to X_0 \end{align}
as the universal family of cyclic quintic covers
$C \to {\mathbf{P}}^1$
ramified along a smooth binary quintic
$\{F=0\} \, \subset {\mathbf{P}}^1$
with local monodromy given by (6), and let
\begin{align*} J=J({\mathscr{C}}) \to X_0\end{align*}
be the relative Jacobian of
${\mathscr{C}}/X_0$
. By Lemma 3.2,
$J$
is a polarized abelian scheme of relative dimension six over
$X_0$
, equipped with
${\mathcal{O}}_K$
-action of signature
$\{(2,1), (3,0)\}$
with respect to
$\Phi = \{\sigma _1, \sigma _2\}$
. Let
$\mathbf{V} = R^1g_\ast {\mathbf{Z}}$
be the local system of hermitian
${\mathcal{O}}_K$
-modules underlying the abelian scheme
$J/X_0$
. Attached to
$\mathbf{V}$
and the base point
$F_0 \in X_0({\mathbf{C}})$
, we have a representation
\begin{align} \rho _\Gamma \colon \pi _1(X_0({\mathbf{C}}), F_0) \to \Gamma := {{\rm Aut}}_{{\mathcal{O}}_K}(\Lambda , {\mathfrak{h}}) \end{align}
whose composition with the quotient map
$\Gamma \to P\Gamma = \Gamma / \mu _K$
defines a homomorphism
\begin{align} P(\rho _\Gamma ) \colon \pi _1(X_0({\mathbf{C}}), F_0) \to P\Gamma. \end{align}
3.5 Marked binary quintics and points on the projective line
Let
$F \in X_0({\mathbf{C}})$
and consider the hypersurface
$ Z_F := \{F=0\} \, \subset {\mathbf{P}}^1_{\mathbf{C}}$
. A marking of
$F$
is an ordering
$m \colon Z_F({\mathbf{C}}) \xrightarrow {\sim } \left \{ 1, 2, 3,4,5 \right \}$
of the five-element set
$Z_F({\mathbf{C}}) = \left \{ x \in {\mathbf{P}}^1({\mathbf{C}}) \mid F(x)=0 \right \}$
.
Remark 3.3. Let
$F \in X_0({\mathbf{C}})$
. To give a marking of
$F$
is to give an isomorphism of rings
${\rm H}^0(Z_F({\mathbf{C}}), {\mathbf{Z}}) \xrightarrow {\sim } {\mathbf{Z}}^5$
. Moreover, sending a ring automorphism of
${\mathbf{Z}}^5$
to the induced permutation of the canonical basis of
$\left \{ e_1, \dotsc , e_5 \right \} \, \subset {\mathbf{Z}}^5$
defines a group isomorphism
${{\rm Aut}}_{{\rm ring}}({\mathbf{Z}}^5) \cong {\mathfrak{S}}_5$
.
Define
$\mathcal{N}_0$
as the set of marked smooth complex binary quintics
$(F, m)$
, and consider the map that forgets the marking:
\begin{align} \mathcal{N}_0 \to X_0({\mathbf{C}}), \quad \quad (F,m) \mapsto F. \end{align}
To provide
$\mathcal{N}_0$
with a complex manifold structure such that (14) is a finite covering map, we define
$\mathcal{N}_0$
in a slightly different but equivalent way. Define an algebraic subgroup
\begin{align*} T:= \left \{ (\lambda _1, \dotsc , \lambda _5) \in ({\mathbf{C}}^\ast )^5 \mid \lambda _1 \cdots \lambda _5=1 \right \} \, \subset ({\mathbf{C}}^\ast )^5. \end{align*}
The group
$T$
acts freely on
$({\mathbf{C}}^2 - \left \{ 0 \right \})^5$
in the usual way, i.e. by letting
$(\lambda _1, \dotsc , \lambda _5) \in T$
act as
\begin{align*} \left ( (a_1, b_1), \dotsc , (a_5, b_5)\, \right ) \mapsto \left ( \lambda _1 \cdot (a_1, b_1), \dotsc , \lambda _5 \cdot (a_5, b_5) \right). \end{align*}
As
$\dim (T)=4$
, the quotient variety
$ \mathcal{N} := ({\mathbf{C}}^2 - \left \{ 0 \right \})^5 / T$
is a complex manifold of dimension six, equipped with a holomorphic principal
${\mathbf{C}}^\ast$
-bundle map
\begin{align} f \colon \mathcal{N} \to {\mathbf{P}}^1({\mathbf{C}})^5, \quad \quad \left (v_1, \dotsc , v_5\, \right ) \mapsto \left ([v_1], \dotsc , [v_5]\right)\!, \end{align}
where
${\mathbf{C}}^\ast$
acts on
$\mathcal{N}$
by
$\lambda \cdot (v_1, v_2, \dotsc , v_5) = (\lambda v_1, v_2, \dotsc , v_5) = (v_1, \lambda v_2, \dotsc , v_5) = \cdots = (v_1, v_2, \dotsc , \lambda v_5) \in \mathcal{N}$
for
$\lambda \in {\mathbf{C}}^\ast$
and
$(v_1, v_2, \dotsc , v_5) \in \mathcal{N}$
. For
$i,j \in \left \{ 1, \dotsc , 5 \right \}$
with
$i\lt j$
, define
$\overline {\Delta }_{ij} = \left \{ (x_1, \dotsc , x_5) \in {\mathbf{P}}^1({\mathbf{C}})^5 \mid x_i = x_j \right \}$
, and let
\begin{align*}\Delta _{ij} := f^{-1}(\overline {\Delta }_{ij}), \end{align*}
where
$f$
is the principal bundle map (15). Since
$\overline {\Delta }_{ij} \, \subset {\mathbf{P}}^1({\mathbf{C}})^5$
is Zariski closed in
${\mathbf{P}}^1({\mathbf{C}})^5$
, we see that
$\Delta _{ij}$
is Zariski closed in
$\mathcal{N}$
. Define a Zariski open subset
$\mathcal{N}_0 \, \subset \mathcal{N}$
as follows:
\begin{align*} \mathcal{N}_0 := \mathcal{N} - \bigcup _{i \lt j} \Delta _{ij}. \end{align*}
Lemma 3.4.
The complex manifold
$\mathcal{N}_0$
is connected and there is a canonical bijection between
$\mathcal{N}_0$
and the set of marked smooth complex binary quintics
$(F,m)$
. Under this bijection, the map
\begin{align} \mathcal{N}_0 \to X_0({\mathbf{C}}), \quad \left ((a_1, b_1), (a_2, b_2), \dotsc , (a_5, b_5)\, \right ) \mapsto \prod _{j=1}^5 \left (b_j\cdot x - a_j \cdot y\, \right ) \end{align}
corresponds to the map (14) that forgets the marking, and (16) is a Galois covering with Galois group
${\mathfrak{S}}_5$
.
Proof. Note that
$\mathcal {N}_0$
is connected because
$(\mathbf{C}^2 - \{0\})^2$
is connected. For
\begin{align}v = ((a_1, b_2&), \dotsc, &\\(a_5, b_5)) \in \mathcal N_0\end{align}
, define
$F_v = \prod_{j=1}^5 (b_j \cdot x - a_j \cdot y)$
. The polynomial
$F_v$
is a smooth binary quintic, equipped with the marking
$m_v$
defined by the ordering
$\{[a_1 \colon b_1], \dotsc, [a_5 \colon b_5])\}$
of the roots
$[a_j \colon b_j] \in \mathbf{P}^1(\mathbf{C})$
of
$F_v$
. The association
$v \mapsto (F_v, m_v)$
defines the bijection alluded to in the first statement of the lemma. The rest is clear.
In a similar way, we define, for
$i,j,k \in \left \{ 1, \dotsc , 5 \right \}$
with
$i\lt j\lt k$
, closed subsets
\begin{align*} \overline \Delta _{ijk} := \left \{ (x_1, \dotsc , x_5) \in {\mathbf{P}}^1({\mathbf{C}})^5 \mid x_i = x_j = x_k \right \}, \quad \quad \Delta _{ijk} := f^{-1}(\overline \Delta _{ijk}) \, \subset \mathcal{N}, \end{align*}
and put
\begin{align*} \mathcal{N}_s := \mathcal{N} - \bigcup _{i\lt j\lt k} \Delta _{ijk}. \end{align*}
The space
$\mathcal{N}_s$
is equipped with a finite ramified covering map
\begin{align*} \mathcal{N}_s \to X_s({\mathbf{C}}), \quad \quad \left ((a_1, b_1), (a_2, b_2), \dotsc , (a_5, b_5)\, \right )\mapsto \prod _{j=1}^5 (b_j \cdot x- a_j \cdot y) \in {\mathbf{C}}[x,y], \end{align*}
that commutes with (16) and the natural open embedding
$\mathcal{N}_0 \, \subset \mathcal{N}_s$
. Observe that
$\mathcal{N}_s \to X_s({\mathbf{C}})$
is the Fox completion (cf. [Reference FoxFox57] or [Reference Deligne and MostowDM86, Section 8.1]) of the spread
$\mathcal{N}_0 \to X_0({\mathbf{C}}) \hookrightarrow X_s({\mathbf{C}})$
.
3.6 Monodromy
Choose a marking
$m_0$
lying over our base point
$F_0 \in X_0({\mathbf{C}})$
. By Lemma 3.4, we have a surjective homomorphism
\begin{equation} \rho _5 \colon \pi _1(X_0({\mathbf{C}}), F_0) \twoheadrightarrow {\mathfrak{S}}_5\end{equation}
whose kernel is given by the image of the natural embedding
$\pi _1(\mathcal{N}_0, m_0) \hookrightarrow \pi _1(X_0({\mathbf{C}}), F_0)$
. Composing the latter with the maps
$\rho _\Gamma$
and
$P(\rho _\Gamma )$
of (12) and (13) yields homomorphisms
\begin{equation} \mu \colon \pi _1(\mathcal{N}_0, m_0) \to \Gamma \quad \quad {\rm and} \quad \quad P(\mu ) \colon \pi _1(\mathcal{N}_0, m_0) \to P\Gamma. \end{equation}
Consider the three-dimensional
$\mathbf{F}_5$
vector space
$\Lambda /(1 - \zeta )\Lambda$
, as well as the quadratic space
\begin{align*} W := \left (\Lambda /(1-\zeta )\Lambda , {\mathfrak{q}}\, \right ) \end{align*}
over
$\mathbf{F}_5$
. Here,
$\mathfrak{q}$
is the quadratic form obtained by reducing
$\mathfrak{h}$
modulo
$(1-\zeta )\Lambda$
. Define two groups
$\Gamma _\theta$
and
$P\Gamma _\theta$
as follows:
\begin{align*} \Gamma _\theta = {{\rm Ker}}\left ( \Gamma \to {{\rm Aut}}(W)\, \right ), \quad P\Gamma _\theta = {{\rm Ker}}\left ( P\Gamma \to P{{\rm Aut}}(W)\, \right). \end{align*}
The following proposition seems to be due to Terada [Reference TeradaTer83] and Yamazaki and Yoshida [Reference Yamazaki and YoshidaYY84].
Proposition 3.5.
The image of the map
$P(\mu )$
defined in (18) is the group
$P\Gamma _\theta$
. Moreover, the map
$P\Gamma \to P{{\rm Aut}}(W)$
is surjective, and the induced homomorphism
\begin{align*}\overline {\rho }_\Gamma \colon {\mathfrak{S}}_5 = \pi _1(X_0({\mathbf{C}}), F_0)/ \pi _1 \left ( \mathcal{N}_0, m_0\, \right )\to P\Gamma /P\Gamma _\theta = P{{\rm Aut}}(W)\end{align*}
is an isomorphism. As a consequence, we obtain the following commutative diagram with exact rows.
Proof. The equality
$P(\mu )\left (\pi _1(\mathcal{N}_0,m_0)\right ) = P\Gamma _\theta$
follows from
$\mu \left (\pi _1(\mathcal{N}_0,m_0)\right ) = \Gamma _\theta$
. For the latter, see [Reference Yamazaki and YoshidaYY84, Propositions 4.2 and 4.3]. The group
$P{{\rm Aut}}(W) \cong {\rm PO}_3(\mathbf{F}_5)$
is isomorphic to
${\mathfrak{S}}_5$
, and
$\overline {\rho }_\Gamma \colon {\mathfrak{S}}_5 \to P\Gamma /\Gamma _\theta$
is an isomorphism (cf. [Reference Yamazaki and YoshidaYY84, Propositions 4.2 and 4.3] and [Reference Apéry and YoshidaAY98, p. 10]).
Corollary 3.6.
The monodromy representation
$P(\rho _\Gamma ) \colon \pi _1(X_0({\mathbf{C}}), F_0) \to P\Gamma$
is surjective.
3.7 Framed binary quintics, nodal binary quintics and local monodromy
By a framing of a point
$F \in X_0({\mathbf{C}})$
we mean a projective equivalence class
$[f]$
, where
\begin{align*} f \colon {\mathbf{V}}_F = {\rm H}^1(C_F({\mathbf{C}}), {\mathbf{Z}}) \xrightarrow {\sim } \Lambda \end{align*}
is an
${\mathcal{O}}_K$
-linear isometry: two such isometries are in the same class if and only if they differ by an element in
$\mu _K = ({\mathcal{O}}_K^\ast )_{{\rm tors}}$
. Let
$\mathcal{F}_0$
be the collection of all framings of all points
$F \in X_0({\mathbf{C}})$
. The set
$\mathcal{F}_0$
can naturally be given the structure of a complex manifold, in a way similar to the procedure described in [Reference Allcock, Carlson and ToledoACT02b, (3.9)]. In what follows, we consider
$\mathcal{F}_0$
as a complex manifold.
Lemma 3.7.
-
(i) The complex manifold
$\mathcal{F}_0$
is connected and the map
(20)that forgets the framing is a Galois covering map, with Galois group
\begin{equation} \mathcal{F}_0 \to X_0({\mathbf{C}}), \quad \quad (F, [f]) \mapsto F \end{equation}
$P\Gamma$
.
-
(ii) The spaces
$P\Gamma _\theta \setminus \mathcal{F}_0$
and
$ \mathcal{N}_0$
are isomorphic as covering spaces of
$X_0({\mathbf{C}})$
. In particular, there is a covering map
$\mathcal{F}_0 \to \mathcal{N}_0$
with Galois group
$P\Gamma _\theta$
.
Proof. Indeed,
$\mathcal{F}_0$
is connected by Corollary 3.6. The isomorphism
$P\Gamma _\theta \setminus \mathcal{F}_0 \cong \mathcal{N}_0$
of covering spaces of
$X_0({\mathbf{C}})$
follows from the isomorphism
$P\Gamma /P\Gamma _\theta \cong {\mathfrak{S}}_5$
as quotients of
$\pi _1(X_0({\mathbf{C}}), F_0)$
, which was shown in Proposition 3.5. The lemma follows.
Lemma 3.8.
The subvariety
$\Delta := X_s - X_0$
is an irreducible normal crossings divisor in
$X_s$
.
Proof. The irreducibility of
$\Delta$
is well known; see e.g. [Reference VoisinVoi02, Section 14.1.1]. Let
$(p,F)$
be a point on the incidence variety
$\mathcal{I} = \left \{ (p, F) \in {\mathbf{P}}^1 \times X_s \mid p \in {{\rm Sing}}(F) \right \}$
, choose a hyperplane away from
$p$
, and view
$p \in {\mathbf{C}}$
and
$F$
as a polynomial
$f(x)$
. Since
$f^{\prime\prime}(p)$
is non-zero, the tangent map
\begin{align*}Tq \colon T_{(p,f)} (\mathcal{I}) \to T_f(X_s) = X({\mathbf{C}})\end{align*}
of the projection
$q \colon \mathcal{I} \to X_s$
is injective, and its image consists of the linear subspace
$X_p \, \subset X$
of binary quintics that contain
$p$
as a root. If
$f$
has
$k$
double points
$p_j$
, where
$k \in \left \{ 1,2 \right \}$
, then
$\Delta$
is locally isomorphic to the union
$\cup _{j=1}^k X_{p_j}$
. These
$X_{p_j}$
intersect transversally, and we are done.
Definition 3.9. A node
$p \in Z_F({\mathbf{C}}) \, \subset {\mathbf{P}}^1({\mathbf{C}})$
of a binary quintic
$F$
is a double point, i.e. a zero with multiplicity two. For
$k=1, 2$
, let
$\Delta _k \, \subset \Delta = X_s - X_0$
be the locus of stable binary quintics with exactly
$k$
nodes.
The following result is due to Deligne and Mostow [Reference Deligne and MostowDM86].
Lemma 3.10.
The local monodromy transformations of
$\mathcal{F}_0 \to X_0({\mathbf{C}})$
around every
$F \in \Delta$
are of finite order. More precisely, if
$F \in \Delta _k$
, then the local monodromy group around
$F$
is isomorphic to
$({\mathbf{Z}}/10)^k$
.
Proof. Let
$F_1 \in \Delta _1$
be a binary quintic with one node. Consider the universal family
${\mathscr{C}} \to X_s$
of quintic covers of
${\mathbf{P}}^1$
ramified along a stable binary quintic, whose restriction to
$X_0$
is (11). Let
$D \, \subset X_s({\mathbf{C}})$
be an open disc transverse to
$\Delta _1$
at
$F_1$
. For
$F \in D^\ast$
, one obtains a monodromy transformation
$T \colon {\rm H}^1(C_F({\mathbf{C}}), {\mathbf{Z}}) \to {\rm H}^1(C_F({\mathbf{C}}), {\mathbf{Z}})$
induced by a vanishing cycle, and
$T$
has order ten by [Reference Deligne and MostowDM86, Proposition 9.2]. Similarly, if
$F_2 \in \Delta _2$
has two nodes, we may choose an embedding
$D^2 \, \subset X_s({\mathbf{C}})$
of the polydisc
$D^2 = D \times D$
transversal to
$\Delta _2$
at
$F_2$
. Since distinct nodes have orthogonal vanishing cycles, the local monodromy transformations commute.
In the following corollary, we let
$D = \left \{ z \in {\mathbf{C}} \colon \left | z \right | \lt 1 \right \}$
denote the open unit disc, and
$D^\ast = D - \{0\}$
the punctured open unit disc.
Corollary 3.11. There is an essentially unique branched cover
\begin{align*} \pi \colon \mathcal{F}_s \to X_s({\mathbf{C}}), \end{align*}
with
$\mathcal{F}_s$
a complex manifold, such that for any
$x \in \Delta$
, any open
$x \in U \, \subset X_s({\mathbf{C}})$
with
\begin{align*} U \cong D^k \times D^{6-k} \quad {\rm and} \quad U \cap X_0({\mathbf{C}}) \cong (D^\ast )^k \times D^{6-k},\end{align*}
and any component
$U^{\prime}$
of
$\pi ^{-1}(U) \, \subset \mathcal{F}_s$
, there is an isomorphism
$U' \cong D^k \times D^{6-k}$
such that the composition
\begin{align*} D^k \times D^{6-k} \cong U' \to U \cong D^6 \; \mathrm{is\ the\ map}\ (z_1, \dotsc , z_6) \mapsto (z_1^{10}, \dotsc , z_k^{10}, z_{k+1}, \dotsc , z_6). \end{align*}
Proof. In view of [Reference BeauvilleBea09, Lemma 7.2] (see also [Reference FoxFox57] and [Reference Deligne and MostowDM86, Section 8.1]), this follows from Lemma 3.10.
The group
$G({\mathbf{C}}) = {\rm GL}_2({\mathbf{C}})/D$
(see (10)) acts on
$\mathcal{F}_0$
over its action on
$X_0$
. Explicitly, if
$g \in G({\mathbf{C}})$
and if
$([\phi ], \phi \colon {\mathbf{V}}_{F} \xrightarrow {\sim } \Lambda )$
is a framing of
$F \in X_0({\mathbf{C}})$
, then
\begin{align*} \big([\phi \circ g^\ast ], \phi \circ g^\ast \colon {\mathbf{V}}_{g\cdot F} \xrightarrow {\sim } \Lambda \big) \end{align*}
is a framing of
$g\cdot F \in X_0({\mathbf{C}})$
. This is a left action. The group
$P\Gamma$
also acts on
$\mathcal{F}_0$
from the left, and the actions of
$P\Gamma$
and
$G({\mathbf{C}})$
on
$\mathcal{F}_0$
commute. By functoriality of the Fox completion, the action of
$G({\mathbf{C}})$
on
$\mathcal{F}_0$
extends to an action of
$G({\mathbf{C}})$
on
$\mathcal{F}_s$
.
Lemma 3.12.
The group
$G({\mathbf{C}}) = {\rm GL}_2({\mathbf{C}})/D$
acts freely on
$\mathcal{F}_s$
.
Proof. Consider the natural action of
$G({\mathbf{C}})$
on
$\mathcal{N}_s$
, and the action of
$G({\mathbf{C}})$
on
$P\Gamma _\theta \setminus \mathcal{F}_s$
induced by the action of
$G({\mathbf{C}})$
on
$\mathcal{F}_s$
. With respect to these actions, the isomorphism of ramified covering spaces
$P\Gamma _\theta \setminus \mathcal{F}_s \cong \mathcal{N}_s$
of
$X_s({\mathbf{C}})$
that results from Lemma 3.7(ii) is
$G({\mathbf{C}})$
-equivariant. In particular, the natural ramified covering map
$\mathcal{F}_s \to \mathcal{N}_s$
is
$G({\mathbf{C}})$
-equivariant, and so it suffices to show that
$G({\mathbf{C}})$
acts freely on
$\mathcal{N}_s$
.
To this end, note that
$\mathcal{N}_s$
admits a natural
${\mathbf{C}}^\ast$
-quotient map
\begin{align} \mathcal{N}_s \to P_s, \end{align}
where
$P_s \, \subset {\mathbf{P}}^1({\mathbf{C}})^5$
is the space of stable ordered five-tuples in
${\mathbf{P}}^1({\mathbf{C}})$
introduced in Section 1, and where
${\mathbf{C}}^\ast$
acts on
$\mathcal{N}_s$
by
$\lambda \cdot (v_1, v_2, \dotsc , v_5) = (\lambda v_1, v_2, \dotsc , v_5) = (v_1, \lambda v_2, \dotsc , v_5) = \cdots = (v_1, v_2, \dotsc , \lambda v_5) \in \mathcal{N}_s$
for
$\lambda \in {\mathbf{C}}^\ast$
and
$(v_1, v_2, \dotsc , v_5) \in \mathcal{N}_s$
(see the description of
$\mathcal{N}_s$
in Section 3.5), and (21) is equivariant for the natural homomorphism
$G({\mathbf{C}}) \to {\rm PGL}_2({\mathbf{C}})$
. Let
$g \in {\rm GL}_2({\mathbf{C}})$
and
$x \in \mathcal{N}_s$
such that
$gx = x$
. Since any element of
$ {\rm PGL}_2({\mathbf{C}})$
that fixes three distinct points on
${\mathbf{P}}^1({\mathbf{C}})$
is the identity, we have that
${\rm PGL}_2({\mathbf{C}})$
acts freely on
$P_s$
. Therefore,
$g \in {\mathbf{C}}^\ast \, \subset {\rm GL}_2({\mathbf{C}})$
. Let
$F \in X_s({\mathbf{C}})$
be the image of
$x \in \mathcal{N}_s$
; then
\begin{align*}F(x,y) = gF(x,y) = F(g^{-1}(x,y)) = F(g^{-1}x, g^{-1}y) = g^{-5}F(x,y).\end{align*}
Thus, we have
$g^{5} = 1 \in {\mathbf{C}}^\ast$
, and we conclude that
$g \in D \, \subset {\rm GL}_2({\mathbf{C}})$
.
4. Deligne–Mostow uniformization of the moduli space of complex binary quintics
In this section, we show that the results of Deligne and Mostow [Reference Deligne and MostowDM86] yield an isomorphism of complex analytic spaces
$\mathcal{M}_s({\mathbf{C}}) \cong P\Gamma \setminus {\mathbf{C}} H^2$
. This map is induced by the Riemann extension
$\mathcal{P}_s \colon \mathcal{F}_s \to \mathbf{C} H^2$
of a holomorphic map
$\mathcal{P} \colon \mathcal{F}_0 \to \mathbf{C} H^2$
whose definition follows rather directly from the set-up and results of the previous section. We also show that this isomorphism induces an isomorphism between the divisor
$G({\mathbf{C}}) \setminus \Delta ({\mathbf{C}}) = \mathcal{M}_s({\mathbf{C}}) - \mathcal{M}_0({\mathbf{C}})$
and the divisor
$P\Gamma \setminus {\mathscr{H}} \, \subset P\Gamma \setminus {\mathbf{C}} H^2$
defined by a certain hyperplane arrangement
${\mathscr{H}} \, \subset \mathbf{C} H^2$
, and prove that
$\mathcal{P}_s$
identifies binary quintics with
$k$
nodes (
$k=1,2$
) with points in
$\mathscr{H}$
where exactly
$k$
hyperplanes meet.
4.1 The period map
Define a complex hermitian vector space
$V$
as
\begin{align*} V = \left ( \Lambda \otimes _{{\mathcal{O}}_K, \sigma _1} {\mathbf{C}}, {\mathfrak{h}}^{\sigma _1}\right ), \end{align*}
where
${\mathfrak{h}}^{\sigma _1}$
is the hermitian form defined in Section 3.3. Let
${\mathbf{C}} H^2$
be the space of negative lines in
$V$
. Using [Reference de Gaay FortmanGF24, Proposition 11.7] and Lemma 3.2, we see that the abelian scheme
$J \to X_0$
induces a holomorphic map, the period map:
\begin{equation} \mathcal{P} \colon \mathcal{F}_0 \to {\mathbf{C}} H^2.\end{equation}
Explicitly, if
$(F, [f]) \in \mathcal{F}_0$
is the framing
$[f\colon {\rm H}^1(C_F({\mathbf{C}}), {\mathbf{Z}}) \xrightarrow {\sim }\Lambda ]$
of the binary quintic
$F \in X_0({\mathbf{C}})$
, and if
$JC_F$
is the Jacobian of the curve
$C_F$
, then
\begin{align*}f \left ( {\rm H}^{0,-1}(JC_F)_{\zeta }\, \right ) = f \left ({\rm H}^{1,0}(C_F)_{\zeta }\, \right ) \, \subset {\rm H}^1(C({\mathbf{C}}), {\mathbf{C}})_{\zeta } = \Lambda \otimes _{{\mathcal{O}}_K, \sigma _1} {\mathbf{C}} = V\end{align*}
is a negative line in
$V$
, and we have
$\mathcal{P}(F , [f] ) = f \left ({\rm H}^{1,0}(C_F)_{\zeta }\, \right ) \in {\mathbf{C}} H^2$
. The map
$\mathcal{P}$
is holomorphic, and descends to a morphism of complex analytic spaces
\begin{align*} \mathcal{M}_0({\mathbf{C}}) =G({\mathbf{C}}) \setminus X_0({\mathbf{C}}) \to P\Gamma \setminus {\mathbf{C}} H^2. \end{align*}
Moreover, by Riemann extension, (22) extends to a
$G({\mathbf{C}})$
-equivariant holomorphic map
\begin{align} \mathcal{P}_s\colon \mathcal{F}_s \to {\mathbf{C}} H^2. \end{align}
Theorem 4.1 (Deligne–Mostow). The period map (23) induces an isomorphism of complex manifolds
\begin{align} G({\mathbf{C}}) \setminus \mathcal{F}_s \cong {\mathbf{C}} H^2. \end{align}
Taking
$P\Gamma$
-quotients gives an isomorphism of complex analytic spaces
\begin{equation} \mathcal{M}_s({\mathbf{C}}) = G({\mathbf{C}}) \setminus X_s({\mathbf{C}}) \cong P\Gamma \setminus {\mathbf{C}} H^2.\end{equation}
Proof. Recall that
$P_0 \, \subset {\mathbf{P}}^1({\mathbf{C}})^5$
is the set of
$(x_1, \dotsc , x_5) \in {\mathbf{P}}^1({\mathbf{C}})^5$
such that all
$x_i$
are distinct. In accordance with [Reference Deligne and MostowDM86], define
$Q = G({\mathbf{C}}) \setminus \mathcal{N}_0 = {\rm PGL}_2({\mathbf{C}}) \setminus P_0$
and
$Q_{{\rm st}} := G({\mathbf{C}}) \setminus \mathcal{N}_s = {\rm PGL}_2({\mathbf{C}}) \setminus P_s$
. Fix a base point
$0 \in Q$
whose image in
${\mathfrak{S}}_5 \setminus Q$
coincides with the image of
$F_0 \in X_0({\mathbf{C}})$
under the canonical map
$X_0({\mathbf{C}}) \to \mathcal{M}_0({\mathbf{C}}) = {\mathfrak{S}}_5 \setminus Q$
.
By Lemma 3.7, we have that
$\mathcal{F}_0$
is a covering space of
$\mathcal{N}_0$
, with Galois group
$P\Gamma _\theta$
. In [Reference Deligne and MostowDM86], Deligne and Mostow define
$\widetilde Q \to Q$
to be the covering space corresponding to the monodromy representation
$\pi _1(Q,0) \to P\Gamma$
; since the image of this homomorphism is
$P\Gamma _\theta$
(see Proposition 3.5), it follows that
$G({\mathbf{C}}) \setminus \mathcal{F}_0 \cong \widetilde Q$
as covering spaces of
$Q$
. Consequently, if
$\widetilde Q_{{\rm st}} \to Q_{{\rm st}}$
denotes the Fox completion (cf. [Reference FoxFox57], [Reference Deligne and MostowDM86, Section 8.1]) of the spread
\begin{align*}\widetilde Q \to Q \hookrightarrow Q_{{\rm st}},\end{align*}
then there is an isomorphism
$G({\mathbf{C}}) \setminus \mathcal{F}_s \cong \widetilde Q_{{\rm st}}$
of branched covering spaces of
$ Q_{{\rm st}}$
. We obtain the following commutative diagram, in which the horizontal arrows on the left are isomorphisms.
The map
$\widetilde {Q}_{{\rm st}} \to {\mathbf{C}} H^2$
is an isomorphism by [Reference Deligne and MostowDM86, (3.11)]. It follows that
$Q_{{\rm st}} \to P\Gamma _\theta \setminus {\mathbf{C}} H^2$
and
${\mathfrak{S}}_5 \setminus Q_{{\rm st}} \to P\Gamma \setminus {\mathbf{C}} H^2$
are isomorphisms as well. Therefore, we are done if we can show that the composition
$\mathcal{F}_0 \to \widetilde Q \to {\mathbf{C}} H^2$
agrees with the period map
$\mathcal{P}$
of equation (22). This follows from [Reference Deligne and MostowDM86, (2.23) and (12.9)].
4.2 Nodal binary quintics and orthogonal hyperplanes
Consider the CM type
$\Phi = \left \{ \sigma _1, \sigma _2 \right \} \, \subset {{\rm Hom}}(K,{\mathbf{C}})$
defined in (9), the hermitian
${\mathcal{O}}_K$
-lattice
$(\Lambda , {\mathfrak{h}})$
defined in (8), and the following sets (cf. [Reference de Gaay FortmanGF24, Sections 2.2 and 2.3]):
\begin{align} \mathcal{H} = \left \{ H_r \, \subset \mathbf{C} H^2 \mid r \in {\mathscr{R}} \right \} \quad {{\rm and}} \quad {\mathscr{H}}\ = \bigcup _{H\in \mathcal{H}}H \, \subset \mathbf{C} H^2. \end{align}
Here,
${\mathscr{R}}\, \subset \Lambda$
is the set of short roots, i.e. the set of
$r \in \Lambda$
with
${\mathfrak{h}}(r,r)=1$
, and for each
$r \in {\mathscr{R}}$
,
$H_r \, \subset {\mathbf{C}} H^2$
is the hyperplane of elements
$x \in {\mathbf{C}} H^2$
that are orthogonal to
$r$
.
Proposition 4.2.
The hyperplane arrangement
${\mathscr{H}} \, \subset \mathbf{C} H^2$
is an orthogonal arrangement in the sense of [Reference Allcock, Carlson, Toledo and ClemensACT02a]. In other words, any two different hyperplanes
$H_1,H_2\in \mathcal{H}$
either meet orthogonally, or not at all.
Proof. By Lemma 3.2, we have that
$\mathfrak{h}$
has signature
$(2,1)$
with respect to the embedding
$\sigma _1 \colon K \hookrightarrow {\mathbf{C}}$
, and signature
$(3,0)$
with respect to the embedding
$\sigma _2 \colon K \hookrightarrow {\mathbf{C}}$
, where
$\sigma _1$
and
$\sigma _2$
are defined in (9). Therefore, the result follows from [Reference de Gaay FortmanGF24, Theorem 2.5] and [Reference de Gaay FortmanGF24, Example 2.12].
Proposition 4.3. The map (25) induces an isomorphism of complex analytic spaces
\begin{align*} \mathcal{M}_0({\mathbf{C}}) = G({\mathbf{C}}) \setminus X_0({\mathbf{C}}) \cong P\Gamma \setminus \left ({\mathbf{C}} H^2 - {\mathscr{H}}\, \right). \end{align*}
Proof. We have
$\mathcal{P}_s(\mathcal{F}_0) \, \subset {\mathbf{C}} H^2 - {\mathscr{H}}$
by [Reference de Gaay FortmanGF24, Proposition 11.12], because the Jacobian of a smooth curve cannot contain a non-trivial abelian subvariety whose induced polarization is principal. Therefore, we have
$\mathcal{P}_s^{-1}({\mathscr{H}}\ ) \, \subset \mathcal{F}_s - \mathcal{F}_0$
. Since
$\mathcal{F}_s$
is irreducible (it is smooth by Corollary 3.11 and connected by Lemma 3.7(i)), the analytic space
$\mathcal{P}_s^{-1}({\mathscr{H}}\ )$
is a divisor. Since
$\mathcal{F}_s - \mathcal{F}_0$
is also a divisor by Corollary 3.11, we have
$\mathcal{P}_s^{-1}({\mathscr{H}}\ ) = \mathcal{F}_s - \mathcal{F}_0$
.
Define
$\widetilde \Delta = \mathcal{F}_s - \mathcal{F}_0$
, and for
$k \in \left \{ 1,2 \right \}$
, let
$\widetilde \Delta _k \, \subset \widetilde \Delta$
be the inverse image of
$\Delta _k$
in
$\widetilde \Delta$
under the map
$\widetilde \Delta \to \Delta$
. Here,
$\Delta _k \, \subset \Delta$
is the subvariety defined in Definition 3.9. Moreover, for
$k=1,2$
, define
${\mathscr{H}}_k \, \subset {\mathscr{H}}$
as the set
\begin{align*}{\mathscr{H}}_k := \left \{ x \in \mathbf{C} H^2 \colon \left | \mathcal{H}(x) \right | = k \right\}\!.\end{align*}
Thus, this is the locus of points in
$\mathscr{H}$
where exactly
$k$
hyperplanes meet. For
$r \in {\mathscr{R}}$
, define an isometry
\begin{align*}\phi _r \colon \Lambda \to \Lambda , \quad \quad \phi _r(x) = x - (1-\zeta ) {\mathfrak{h}}(x,r) \cdot r.\end{align*}
Let
$[\phi _r] \in P\Gamma$
be the image of
$\phi _r \in \Gamma$
in the group
$P\Gamma = \Gamma /\mu _K$
, and for
$x \in \mathbf{C} H^2$
, define
\begin{align} G(x) := \langle [\phi _r] \mid r\in {\mathscr{R}} \mid {\mathfrak{h}}(x,r)=0 \rangle \, \subset P\Gamma. \end{align}
Lemma 4.4.
-
(i) The period map
$\mathcal{P}_s$
of (23) satisfies
$\mathcal{P}_s(\widetilde \Delta _k) \, \subset {\mathscr{H}}_k$
. -
(ii) Let
$\tilde F \in \widetilde \Delta _k \, \subset \mathcal{F}_s$
and
$x = \mathcal{P}_s(\tilde F) \in {\mathscr{H}}_k \, \subset \mathbf{C} H^2$
. Let
$P\Gamma _{\tilde F} \, \subset P\Gamma$
be the stabilizer of
$\tilde F$
in
$P\Gamma$
. Then
$P\Gamma _{\tilde F}= G(x)$
, where
$G(x) \cong ({\mathbf{Z}}/10)^k$
is as in (27).
Proof.
-
(i) We know that
$\mathcal{P}_s$
induces an isomorphism
$G({\mathbf{C}}) \setminus \widetilde \Delta \xrightarrow {\sim } {\mathscr{H}}\ $
by Theorem 4.1 and Proposition 4.3. This map must identify the smooth (respectively singular) locus of one analytic variety with the smooth (respectively singular) locus of the other, from which the result follows. -
(ii) This follows from Lemma 3.10 and Corollary 3.11 (compare [Reference Allcock, Carlson and ToledoACT02b, (3.10)] and [Reference Allcock, Carlson and ToledoACT10, Lemma 10.3]).
5. The moduli space of real binary quintics
With the period map for complex binary quintics in place, we turn to the construction of the period map for real binary quintics. Define
$\kappa$
as the anti-holomorphic involution
\begin{align*} \kappa \colon X_0({\mathbf{C}}) \to X_0({\mathbf{C}}), \quad F(x,y) = \sum _{i+j=5}a_{ij}x^i y^j \;\; \mapsto \;\; \overline {F(x,y)}= \sum _{i+j=5}\overline {a_{ij}} x^i y^j. \end{align*}
Definition 5.1. (Compare [Reference de Gaay FortmanGF24, Sections 3.1 and 3.2].)
-
(i) An
${\mathcal{O}}_F$
-linear bijection
$\phi \colon \Lambda \xrightarrow {\sim } \Lambda$
is called anti-unitary if
$\phi (\mu \cdot x) = \overline \mu \cdot \phi (x)$
and
${\mathfrak{h}}(\phi (x), \phi (y)) = \overline {{\mathfrak{h}}(x,y)}$
for
$x,y \in \Lambda$
,
$\mu \in {\mathcal{O}}_K$
. -
(ii) Let
$\Gamma^{\prime}$
be the group of unitary and anti-unitary
${\mathcal{O}}_F$
-linear bijections of
$\Lambda$
. Let
$P\Gamma^{\prime} = \Gamma^{\prime}/\mu _K$
. -
(iii) Let
$\mathscr{A}$
be the set of anti-unitary involutions
$\alpha \colon \Lambda \to \Lambda$
, and define
$P{\mathscr{A}} = \mu _K \setminus {\mathscr{A}}$
. -
(iv) For
$\alpha \in P{\mathscr{A}}$
, define
$\mathbf{R} H^2_\alpha \, \subset \mathbf{C} H^2$
as the fixed space of
$\alpha$
, i.e.
${\mathbf{R}} H^2_\alpha = ({\mathbf{C}} H^2)^\alpha$
. -
(v) For
$\alpha \in P{\mathscr{A}}$
, let
$P\Gamma _\alpha \, \subset P\Gamma$
be the stabilizer of the subspace
$\mathbf{R} H^2_\alpha = (\mathbf{C} H^2)^\alpha \, \subset \mathbf{C} H^2$
.
For
$\alpha \in P{\mathscr{A}}$
, the notation
$\mathbf{R} H^2_\alpha$
reflects the fact that
$\mathbf{R} H^2_\alpha$
is isometric to the real hyperbolic plane
$\mathbf{R} H^2$
; see [Reference de Gaay FortmanGF24, Lemma 3.4].
For each
$\alpha \in P{\mathscr{A}}$
, there is a natural anti-holomorphic involution
$\alpha \colon \mathcal{F}_0 \to \mathcal{F}_0$
lying over the anti-holomorphic involution
$\kappa \colon X_0({\mathbf{C}}) \to X_0({\mathbf{C}})$
. To define
$\alpha$
, consider a framed binary quintic
$(F, [f]) \in \mathcal{F}_0$
, where
$f\colon {\mathbf{V}}_F \to \Lambda$
is an
${\mathcal{O}}_K$
-linear isometry. Let
$C_F \to {\mathbf{P}}^1_{\mathbf{C}}$
be the induced quintic cover of
${\mathbf{P}}^1_{\mathbf{C}}$
. Complex conjugation
${\mathbf{P}}^1({\mathbf{C}}) \to {\mathbf{P}}^1({\mathbf{C}})$
induces a bijection
$Z_F({\mathbf{C}}) \xrightarrow {\sim } Z_{\kappa (F)}({\mathbf{C}})$
, and extends to an anti-holomorphic diffeomorphism
$ \sigma _F \colon C_F({\mathbf{C}}) \to C_{\kappa (F)}({\mathbf{C}})$
with pull-back
$\sigma _F^\ast \colon \mathbf{V}_{\kappa (F)} \to \mathbf{V}_F$
. The composition
$\alpha \circ f \circ \sigma _F^{\ast } \colon \mathbf{V}_{\kappa (F)} \to \Lambda$
induces a framing of
$\kappa (F) \in X_0({\mathbf{C}})$
, and we define
\begin{align*} \alpha (F, [f]) := \left (\kappa (F), [\alpha \circ f \circ \sigma _F^\ast ]\, \right ) \in \mathcal{F}_0. \end{align*}
Although we have chosen a representative
$\alpha \in {\mathscr{A}}$
of the class
$\alpha \in P{\mathscr{A}}$
, the element
$\alpha (F, [f]) \in \mathcal{F}_0$
does not depend on this choice.
Consider the covering map
$\mathcal{F}_0 \to X_0({\mathbf{C}})$
defined in (20) in Lemma 3.7, and define
$\mathcal{F}_0({\mathbf{R}})$
as the preimage of
$X_0({\mathbf{R}})$
in the space
$\mathcal{F}_0$
. Then
\begin{align} \mathcal{F}_0({\mathbf{R}}) = \coprod _{\alpha \in P{\mathscr{A}}} \mathcal{F}_0^\alpha \, \subset \mathcal{F}_0. \end{align}
To see why the union in (28) is disjoint, let
$\alpha \in P{\mathscr{A}}$
; then
\begin{align*} \mathcal{F}_0^\alpha = \left \{ (F, [f]) \in \mathcal{F}_0 \colon \kappa (F) = F \mathrm{and} [f \circ \sigma ^\ast _{F} \circ f^{-1}] = \alpha \right\}\!.\end{align*}
Thus, for
$\alpha , \beta \in P{\mathscr{A}}$
and
$(F, [f]) \in \mathcal{F}_0^\alpha \cap \mathcal{F}_0^\beta$
, we have
$\alpha = [f \circ \sigma _F^\ast \circ f^{-1}] = \beta$
.
Lemma 5.2.
Let
$\alpha \in P{\mathscr{A}}$
. The anti-holomorphic involution
$\alpha \colon \mathcal{F}_0 \to \mathcal{F}_0$
commutes with the period map
$\mathcal{P} \colon \mathcal{F}_0 \to {\mathbf{C}} H^2$
and the anti-holomorphic involution
$\alpha \colon \mathbf{C} H^2 \to \mathbf{C} H^2$
.
Proof. If
${{\rm conj}} \colon {\mathbf{C}} \to {\mathbf{C}}$
is complex conjugation, then for any
$F \in X_0({\mathbf{C}})$
, the induced map
$\sigma ^\ast _F \otimes {{\rm conj}} \colon \mathbf{V}_{\kappa (F)}\otimes _{\mathbf{Z}} {\mathbf{C}} \to \mathbf{V}_F \otimes _{\mathbf{Z}} {\mathbf{C}}$
is anti-linear and preserves the Hodge decompositions [Reference SilholSil89, Chapter I, Lemma 2.4] as well as the eigenspace decompositions.
By Lemma 5.2, we obtain a real period map
Let
$\sigma \colon {\rm GL}_2({\mathbf{C}}) \to {\rm GL}_2({\mathbf{C}})$
be the anti-holomorphic involution that sends a matrix to its complex conjugate, and note that
$\sigma$
descends to an anti-holomorphic involution
$\sigma \colon G({\mathbf{C}}) \to G({\mathbf{C}})$
; define
\begin{align*}G({\mathbf{R}}) := G({\mathbf{C}})^\sigma. \end{align*}
Lemma 5.3.
The natural map
${\rm GL}_2({\mathbf{R}}) \to G({\mathbf{R}})$
is an isomorphism.
Proof. Indeed, if
$M \in {\rm GL}_2({\mathbf{C}})$
is a matrix such that
$\sigma (M) = \zeta ^j \cdot M$
for some
$j \in \left \{ 0, \dotsc , 4 \right \}$
, then we can write
$\zeta ^j = \zeta ^{2k}$
for some
$k \in \left \{ 0, \dotsc , 4 \right \}$
, and hence
$\sigma (\zeta ^k \cdot M) = \zeta ^{-k}\zeta ^j \cdot M = \zeta ^k \cdot M$
, so that
$\zeta ^k \cdot M \in {\rm GL}_2({\mathbf{R}})$
. This proves that
${\rm GL}_2({\mathbf{R}}) \to G({\mathbf{R}})$
is surjective. Injectivity follows from the fact that the kernel of
${\rm GL}_2({\mathbf{R}}) \to G({\mathbf{R}})$
is spanned by the elements of
$D = \left \{ 1, \zeta , \dotsc , \zeta^4 \right \}$
that are invariant under complex conjugation; the only such element is
$1 \in D$
.
The map (29) is constant on
$G({\mathbf{R}})$
-orbits, as the same is true for
$\mathcal{P} \colon \mathcal{F}_0 \to {\mathbf{C}} H^2$
. By abuse of notation, we write
$\mathbf{R} H^2_\alpha - {\mathscr{H}}\ = \mathbf{R} H^2_\alpha - \left ({\mathscr{H}}\, \cap \mathbf{R} H^2_\alpha\, \right )$
for
$\alpha \in P{\mathscr{A}}$
.
Proposition 5.4.
The period map (29) descends to a
$P\Gamma$
-equivariant diffeomorphism
\begin{align} \mathcal{P}^{\mathbf{R}} \colon \mathcal{M}_0({\mathbf{R}})^f := G({\mathbf{R}}) \setminus \mathcal{F}_0({\mathbf{R}}) \cong \coprod _{\alpha \in P{\mathscr{A}}} \left ({\mathbf{R}} H^2_\alpha - {\mathscr{H}}\ \right). \end{align}
Let
$C{\mathscr{A}} \, \subset P{\mathscr{A}}$
be a set of representatives for the action of
$P\Gamma$
on
$P{\mathscr{A}}$
. By
$P\Gamma$
-equivariance, the map (30) induces an isomorphism of real-analytic orbifolds
\begin{equation} \mathcal{P}^{\mathbf{R}} \colon \mathcal{M}_0({\mathbf{R}}) = G({\mathbf{R}}) \setminus X_0({\mathbf{R}}) \cong \coprod _{\alpha \in C{\mathscr{A}}}P\Gamma _\alpha \setminus \left ( {\mathbf{R}} H^2_\alpha - {\mathscr{H}}\ \right). \end{equation}
Proof. We follow the proof of [Reference Allcock, Carlson and ToledoACT10, Theorem 3.3]. The first thing to observe is that the map
\begin{align*}\mathcal{P}^{\mathbf{R}} \colon G({\mathbf{R}}) \setminus \mathcal{F}_0({\mathbf{R}}) \to \coprod _{\alpha \in P{\mathscr{A}}} \left ({\mathbf{R}} H^2_\alpha - {\mathscr{H}}\ \right)\end{align*}
is a local diffeomorphism, as the same holds for
$\mathcal{P} \colon G({\mathbf{C}}) \setminus \mathcal{F}_0 \to {\mathbf{C}} H^2 - {\mathscr{H}}\ $
by Theorem 4.1. To prove the surjectivity and injectivity of
$\mathcal{P}^{\mathbf{R}}$
, notice that the arguments used in [Reference Allcock, Carlson and ToledoACT10] to prove the analogous claims for cubic surfaces readily carry over to our situation.
Our next goal is to prove the real analogue of Theorem 4.1. By the naturality of the Fox completion, for every
$\alpha \in P{\mathscr{A}}$
the involution
$\alpha \colon \mathcal{F}_0 \to \mathcal{F}_0$
extends to an involution
$\alpha$
on
$\mathcal{F}_s$
.
Lemma 5.5.
The restriction of the map
$\mathcal{P}_s \colon \mathcal{F}_s \to {\mathbf{C}} H^2$
to
$\mathcal{F}_s^\alpha$
induces a diffeomorphism
\begin{align*}\mathcal{P}_s^{\alpha } \colon G({\mathbf{R}}) \setminus \mathcal{F}_s^\alpha \cong {\mathbf{R}} H^2_\alpha. \end{align*}
Proof. The map
$\mathcal{P}_s^{\alpha } \colon G({\mathbf{R}}) \setminus \mathcal{F}_s^\alpha \to {\mathbf{R}} H^2_\alpha$
is a local diffeomorphism because its differential at any point is an isomorphism by Theorem 4.1. Let us prove that
$\mathcal{P}_s^\alpha$
is injective. Apply [Reference Allcock, Carlson and ToledoACT10, Lemma 3.5] with
$X = \mathcal{F}_0$
,
$G = G({\mathbf{C}})$
and
$\phi = \alpha$
; then
$X^\phi = \mathcal{F}_s^\alpha$
and
$Z = G({\mathbf{R}})$
. Note that we may apply this lemma because
$G({\mathbf{C}})$
acts freely on
$\mathcal{F}_s$
(see Lemma 3.12). The conclusion is that the map
\begin{align*}G({\mathbf{R}}) \setminus \mathcal{F}_0^\alpha \to G({\mathbf{C}}) \setminus \mathcal{F}_0 = {\mathbf{C}} H^2 - {\mathscr{H}}\end{align*}
is injective. To prove the surjectivity of
$\mathcal{P}_s^{\alpha }$
, one uses [Reference Allcock, Carlson and ToledoACT10, Lemma 11.2] to see that the map
$G({\mathbf{R}}) \setminus \mathcal{F}_s^{\alpha } \to G({\mathbf{C}}) \setminus {\mathbf{C}} H^2$
is proper. By Proposition 5.4, its image contains the dense open subset
$\mathcal{P}_s(\mathcal{F}_0^\alpha ) = {\mathbf{R}} H^2_\alpha - {\mathscr{H}}$
, so
$\mathcal{P}_s^\alpha$
is surjective.
Definition 5.6. (See [Reference de Gaay FortmanGF24, Definition 1.1].) Define an equivalence relation
$\sim$
on the disjoint union
$\coprod _{\alpha \in P{\mathscr{A}}}{\mathbf{R}} H^2_\alpha$
in the following way. Consider two points
$(x,\beta ) \in \mathbf{R} H^2_\beta \, \subset \coprod _{\alpha \in P{\mathscr{A}}}{\mathbf{R}} H^2_\alpha$
and
$(y,\gamma ) \in \mathbf{R} H^2_\gamma \, \subset \coprod _{\alpha \in P{\mathscr{A}}}{\mathbf{R}} H^2_\alpha$
. Then
$(x,\beta ) \sim (y,\gamma )$
if
$x = y \in \mathbf{C} H^2$
and
$\gamma \circ \beta \in G(x)$
.
By [Reference de Gaay FortmanGF24, Lemma 4.4], the action of
$P\Gamma$
on
$\coprod _{\alpha \in P{\mathscr{A}}}{\mathbf{R}} H^2_\alpha$
is compatible with this equivalence relation; define
\begin{align*}Y := \left (\coprod _{\alpha \in P{\mathscr{A}}}{\mathbf{R}} H^2_\alpha\, \right ) / \sim , \quad \quad M := P\Gamma \setminus Y.\end{align*}
Theorem 5.7.
Let
$C{\mathscr{A}} \, \subset P{\mathscr{A}}$
be a set of representatives for the action of
$P\Gamma$
on
$P{\mathscr{A}}$
. There exists a canonical real hyperbolic orbifold structure on the topological space
$M = P\Gamma \setminus Y$
together with a natural open immersion of hyperbolic orbifolds
\begin{align*} \coprod _{\alpha \in C{\mathscr{A}}} P\Gamma _\alpha \setminus \left ( \mathbf{R} H^2_\alpha - {\mathscr{H}}\, \right ) \hookrightarrow M. \end{align*}
Moreover, for each connected component
$M_i \, \subset M$
there exists a lattice
$P\Gamma _{\mathbf{R}}^i \, \subset {\rm PO}(2,1)$
and an isomorphism of real hyperbolic orbifolds
$M_i \cong P\Gamma _{\mathbf{R}}^i \setminus \mathbf{R} H^2$
.
Proof. By Lemma 3.2 and [Reference de Gaay FortmanGF24, Example 2.12], this is a special case of [Reference de Gaay FortmanGF24, Theorem 1.2].
Let
$p \colon \coprod _{\alpha \in P{\mathscr{A}}}{\mathbf{R}} H^2_\alpha \to Y$
be the quotient map, consider the map
$\pi \colon \mathcal{F}_s \to X_s({\mathbf{C}})$
(see Corollary 3.11), and define a union of embedded real submanifolds of
$\mathcal{F}_s$
as follows:
\begin{align*} \mathcal{F}_s({\mathbf{R}}) = \bigcup _{\alpha \in P{\mathscr{A}}} \mathcal{F}_s^\alpha = \pi ^{-1}\left (X_s({\mathbf{R}}\right). \end{align*}
We arrive at the main theorem of Section 5.
Theorem 5.8. There is a smooth map
\begin{align} \mathcal{P}_s^{{\mathbf{R}}} \colon \coprod _{\alpha \in P{\mathscr{A}}} \mathcal{F}_s^\alpha \to \coprod _{\alpha \in P{\mathscr{A}}}{\mathbf{R}} H^2_\alpha \end{align}
that extends the real period map (29). The map (32) induces the following commutative diagram of topological spaces, in which
${\mathscr{P}}_s^{\mathbf{R}}$
and
${\mathscr{T}}_s^{\ \mathbf{R}}$
are homeomorphisms.
Proof. The existence of
$\mathcal{P}_s^{{\mathbf{R}}}$
follows from the compatibility between
$\mathcal{P}_s$
and the involutions
$\alpha \in P{\mathscr{A}}$
. We claim that the composition
$p \circ \mathcal{P}_s^{{\mathbf{R}}}$
factors through
$\mathcal{F}_s({\mathbf{R}})$
. To prove this, let
$(f,\alpha )$
and
$(g, \beta )$
be elements of the disjoint union
$\coprod _{\gamma \in P{\mathscr{A}}} \mathcal{F}_s^\gamma$
, with
$f \in \mathbf{R} H^2_\alpha$
and
$g \in \mathbf{R} H^2_\beta$
. Then
$(f, \alpha )$
and
$(g,\beta )$
have the same image in
$\mathcal{F}_s({\mathbf{R}})$
if and only if
$f = g \in \mathcal{F}_s^\alpha \cap \mathcal{F}_s^\beta$
, in which case
$ \mathcal{P}_s(f) = \mathcal{P}_s(g) =: x \in {\mathbf{R}} H^2_\alpha \cap \mathbf{R} H^2_\beta$
. Let
$(x, \alpha )$
and
$(y,\beta )$
be elements of the disjoint union
$\widetilde Y = \coprod _{\gamma \in P{\mathscr{A}}}\mathbf{R} H^2_\gamma$
, with
$x \in \mathbf{R} H^2_\alpha$
and
$y =x \in \mathbf{R} H^2_\beta$
. We need to prove that
$(x,\alpha ) \sim (x,\beta ) \in \widetilde Y$
, for the equivalence relation
$\sim$
on
$\widetilde Y$
defined in Definition 5.6. Note that
$\alpha \beta \in P\Gamma _f$
, and that
$\mathcal{P}_s$
induces an isomorphism
$ P\Gamma _f \cong G(x)$
(see Lemma 4.4). Hence
$\alpha \beta \in G(x)$
so that
$(x,\alpha ) \sim (x,\beta )$
, proving what we want. We conclude that the composition
$p \circ \mathcal{P}_s^{{\mathbf{R}}}$
factors through a map
$\mathcal{P}_s^{\mathbf{R}} \colon \mathcal{F}_s({\mathbf{R}}) \to Y\!$
.
Next, we prove the
$G({\mathbf{R}})$
-equivariance of
$\mathcal{P}_s^{{\mathbf{R}}}$
. Suppose that
$ f \in \mathcal{F}_s^\alpha , g \in \mathcal{F}_s^\beta$
such that
$ a \cdot f=g \in \mathcal{F}_s({\mathbf{R}})$
for some
$ a \in G({\mathbf{R}})$
. Then
$x:= \mathcal{P}_s(f) = \mathcal{P}_s(g) \in {\mathbf{C}} H^2$
, so we need to show that
$\alpha \beta \in G(x)$
. The actions of
$G({\mathbf{C}})$
and
$P\Gamma$
on
${\mathbf{C}} H^2$
commute, and the same holds for the actions of
$G({\mathbf{R}})$
and
$P\Gamma^{\prime}$
on
$\mathcal{F}_s^{\mathbf{R}}$
, where
$P\Gamma^{\prime}$
is the group defined in Definition 5.1. It follows that
$ \alpha (g) = \alpha (a \cdot f) = a \cdot \alpha (f) = a \cdot f=g,$
and hence
$g \in \mathcal{F}_s^\alpha \cap \mathcal{F}_s^\beta$
. This implies in turn that
$(\alpha \circ \beta ) (g) = g$
, and hence
$\alpha \beta \in P\Gamma _g \cong G(x)$
(see Lemma 4.4). Therefore
$(x,\alpha ) \sim (x,\beta )$
, so that
$\alpha \beta \in G(x)$
as desired.
Let us prove that
${\mathscr{P}}_s^{\mathbf{R}}$
is injective. To do so, let
$ f \in \mathcal{F}_s^\alpha$
and
$g \in \mathcal{F}_s^\beta$
and suppose that these elements have the same image in
$Y$
. Thus,
$ x:= \mathcal{P}_s(f) = \mathcal{P}_s(g) \in {\mathbf{R}} H^2_\alpha \cap \mathbf{R} H^2_\beta $
, and
$\beta = \phi \circ \alpha$
for some
$\phi \in G(x)$
. We have
$\phi \in G(x) \cong P\Gamma _f$
(Lemma 4.4), and hence
$ \beta (f) = \phi \left (\alpha (f)\right ) = \phi (f) = f$
. Therefore
$f,g \in \mathcal{F}_s^\beta$
; since
$\mathcal{P}_s(f) = \mathcal{P}_s(g)$
, it follows from Lemma 5.5 that there exists
$a \in G({\mathbf{R}})$
such that
$a \cdot f=g$
. This proves the injectivity of
${\mathscr{P}}_s^{\mathbf{R}}$
. The surjectivity of
${\mathscr{P}}_s^{\mathbf{R}}\colon G({\mathbf{R}}) \setminus \mathcal{F}_s({\mathbf{R}}) \to Y$
is straightforward: it follows from the surjectivity of
$\mathcal{P}_s^{{\mathbf{R}}}$
(see Lemma 5.5). Finally, we claim that
${\mathscr{P}}_s^{\mathbf{R}}$
is open. Let
$U \, \subset G({\mathbf{R}}) \setminus \mathcal{F}_s^{\mathbf{R}}$
be open. Let
$V$
be the preimage of
$U$
in
$\coprod _{\alpha \in P{\mathscr{A}}}\mathcal{F}_s^\alpha$
. Then
$ V = (\mathcal{P}_s^{{\mathbf{R}}})^{-1}\left ( p^{-1}\left ({\mathscr{P}}_s^{\mathbf{R}}(U)\right )\right )$
, hence
$ \mathcal{P}_s^{{\mathbf{R}}}\left ( V\, \right ) = p^{-1}\left ({\mathscr{P}}_s^{\mathbf{R}}(U)\right )$
, so that it suffices to show that
$\mathcal{P}_s^{\mathbf{R}}(V)$
is an open subset of
$\coprod _{\alpha \in P{\mathscr{A}}} {\mathbf{R}} H^2_\alpha$
. This follows, because
$\mathcal{P}_s^{{\mathbf{R}}}$
is open, being the coproduct of the maps
$\mathcal{F}_s^\alpha \to {\mathbf{R}} H^2_\alpha$
, which are open since they have surjective differential at each point.
Corollary 5.9.
Let
$C{\mathscr{A}} \, \subset P{\mathscr{A}}$
be a set of representatives for the action of
$P\Gamma$
on
$P{\mathscr{A}}$
. Then there is a lattice
$P\Gamma _{\mathbf{R}} \, \subset {\rm PO}(2,1)$
, an open immersion of hyperbolic orbifolds
\begin{align} \coprod _{\alpha \in C{\mathscr{A}}}P\Gamma _\alpha \setminus \left ( {\mathbf{R}} H^2_\alpha - {\mathscr{H}}\, \right ) \hookrightarrow P\Gamma _{\mathbf{R}} \setminus {\mathbf{R}} H^2, \end{align}
and a homeomorphism
\begin{equation} \mathcal{P}_s^{{\mathbf{R}}} \colon \mathcal{M}_s({\mathbf{R}}) = G({\mathbf{R}}) \setminus X_s({\mathbf{R}}) \cong P\Gamma _{\mathbf{R}} \setminus {\mathbf{R}} H^2, \end{equation}
such that the restriction of (34) to
$\mathcal{M}_0({\mathbf{R}}) \, \subset \mathcal{M}_s({\mathbf{R}})$
coincides with the isomorphism (31).
Remark 5.10. The proof of Corollary 5.9 also shows that
$\mathcal{M}_s({\mathbf{R}})$
is homeomorphic to a complete hyperbolic orbifold in the cases where
$\mathcal{M}_s$
is the stack of cubic surfaces or of binary sextics over
$\mathbf{R}$
. This strategy to uniformize the real moduli space differs from the one used in [Reference Allcock, Carlson and ToledoACT06, Reference Allcock, Carlson and ToledoACT07, Reference Allcock, Carlson and ToledoACT10], since we first glue the real ball quotients together (by using the general construction of [Reference de Gaay FortmanGF24]) and then prove that the real moduli space is homeomorphic to the resulting glued space.
6. The moduli space of real binary quintics as a hyperbolic triangle
Consider the moduli space
$\mathcal{M}_s({\mathbf{R}}) = {\rm GL}_2({\mathbf{R}}) \setminus X_s({\mathbf{R}})$
of stable real binary quintics. Let
$\left | \mathcal{M}_s({\mathbf{R}}) \right |$
be the underlying topological space of
$\mathcal{M}_s({\mathbf{R}})$
.
Definition 6.1. Let
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
be the orbifold with
$\left | \mathcal{M}_s({\mathbf{R}}) \right |$
as underlying space whose orbifold structure is induced by the homeomorphism (34) and the natural orbifold structure of
$P\Gamma _{\mathbf{R}} \setminus \mathbf{R} H^2$
.
The goal of Section 6 is to prove the following result.
Theorem 6.2.
Consider the lattice
$P\Gamma _{\mathbf{R}} \, \subset {\rm PO}(2,1)$
(see Corollary 5.9) and the hyperbolic orbifold
$\overline {{\mathscr{M}}}_{\mathbf{R}} \cong P\Gamma _{\mathbf{R}} \setminus \mathbf{R} H^2$
(see Definition 6.1). Then
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
is isometric to the hyperbolic triangle with angles
$\pi /3, \pi /5, \pi /10$
. In particular,
$P\Gamma _{\mathbf{R}}$
is conjugate to the lattice
$\Gamma _{3,5,10}$
defined in (3).
To prove Theorem 6.2, we need to understand the orbifold structure of
$\mathcal{M}_s({\mathbf{R}})$
and how this structure differs from the orbifold structure of the quotient space
$P\Gamma _{\mathbf{R}} \setminus {\mathbf{R}} H^2$
(see Corollary 5.9). To this end, we will first analyze the orbifold structure of
$\mathcal{M}_s({\mathbf{R}})$
by listing its stabilizer groups.
6.1 Automorphism groups of stable real binary quintics
Recall that there is a canonical orbifold isomorphism
\begin{align*} \mathcal{M}_s({\mathbf{R}}) = G({\mathbf{R}}) \setminus X_s({\mathbf{R}}) = {\rm PGL}_2({\mathbf{R}}) \setminus (P_s/{\mathfrak{S}}_5)({\mathbf{R}}). \end{align*}
Thus, to list those groups that occur as the automorphism groups of a binary quintic is to classify the stabilizer groups
${\rm PGL}_2({\mathbf{R}})_x$
of points
$x = (x_1, \dotsc , x_5) \in (P_s/{\mathfrak{S}}_5)({\mathbf{R}})$
.
Proposition 6.3.
Consider the stabilizer group
${\rm PGL}_2({\mathbf{C}})_x$
of a point
$x \in P_0/{\mathfrak{S}}_5$
. If
${\rm PGL}_2({\mathbf{C}})_x$
is non-trivial, then
${\rm PGL}_2({\mathbf{C}})_x$
is isomorphic to one of
${\mathbf{Z}}/2, {\mathbf{Z}}/4, D_3$
or
$D_5$
. Moreover, the conjugacy class of each such subgroup of
${\rm PGL}_2({\mathbf{C}})$
is unique. If
$H$
equals any of the subgroups
${\mathbf{Z}}/4$
,
$D_3$
or
$D_5$
of
$ {\rm PGL}_2({\mathbf{C}})$
, then there is a unique
${\rm PGL}_2({\mathbf{C}})$
-orbit in
$P_0/{\mathfrak{S}}_5$
with stabilizer group
$H$
.
Proof. By [Reference BeauvilleBea10, Theorem 4.2], any finite subgroup of
${\rm PGL}_2({\mathbf{C}})$
is isomorphic to
${\mathbf{Z}}/n$
,
$D_n$
(the dihedral group of order
$2n$
),
${\mathfrak{A}}_4$
,
${\mathfrak{S}}_4$
or
${\mathfrak{A}}_5$
, and there is only one conjugacy class for each of these groups. Let
$H$
be any of these groups, considered as a subgroup of
${\rm PGL}_2({\mathbf{C}})$
. Assume that, with respect to the action of
$H$
on the finite subsets of
${\mathbf{P}}^1({\mathbf{C}})$
, one has
\begin{align*} H \cdot \left \{ z_1, \dotsc , z_5 \right \} = \left \{ z_1, \dotsc , z_5 \right \} \, \subset {\mathbf{P}}^1({\mathbf{C}}) \quad {\rm for} \quad z_1, \dotsc , z_5 \in {\mathbf{P}}^1({\mathbf{C}}) \quad {\rm distinct.} \end{align*}
This gives a homomorphism
$\rho \colon H \to {\mathfrak{S}}_5$
as follows: for an element
$j \in \left \{ 1,2,3,4,5 \right \}$
, we let
$\rho (h)(j) \in \left \{ 1,2,3,4,5 \right \}$
be the element with
$z_{\rho (h)(j)} = h \cdot z_j$
.
Note that
$\rho$
is injective, as
$h \cdot z_i = z_i$
for each
$i$
implies
$h = {{\rm id}}$
. Therefore,
\begin{align*} H \in \left \{ {\mathbf{Z}}/2, {\mathbf{Z}}/3, {\mathbf{Z}}/4, {\mathbf{Z}}/5, {\mathbf{Z}}/6, D_3, D_4, D_5, {\mathfrak{A}}_4, {\mathfrak{S}}_4, {\mathfrak{A}}_5 \right\}\!. \end{align*}
Next, assume that
$H = {\rm Stab}_{{\rm PGL}_2({\mathbf{C}})}(x)$
for the five-element subset
$x = \left \{ z_1, \dotsc , z_5 \right \} \, \subset {\mathbf{P}}^1({\mathbf{C}})$
. Suppose that
$\phi \in H$
is an element of order three. Note that there must be three distinct elements
$z_i \in x$
with
$\phi (z_i) \neq z_i$
. We may assume that these are
$z_1, z_2$
and
$z_3$
. Moreover, we may assume that
$\phi (z_1) = z_2$
,
$\phi (z_2) = z_3$
and
$\phi (z_3) = z_1$
. By replacing
$\phi$
by
$g \phi g^{-1}$
for some
$g \in {\rm PGL}_2({\mathbf{C}})$
, we may assume that
$z_1=1, z_2 = \zeta _3$
and
$z_3 = \zeta _3^2$
, and that
$\phi (z) = \zeta _3 \cdot z$
for
$z \in {\mathbf{P}}^1({\mathbf{C}})$
. This gives
$x = \left \{ 1, \zeta _3, \zeta _3^2, z_4, z_5 \right \} \, \subset {\mathbf{P}}^1({\mathbf{C}})$
. As
$\phi (z_4) \neq z_5$
, we have
$\phi (z_4) = z_4$
and
$\phi (z_5) = z_5$
, so that
\begin{align} x = \left \{ 1, \zeta _3, \zeta _3^2, 0, \infty \right \} \, \subset {\mathbf{P}}^1({\mathbf{C}}). \end{align}
Let
$\nu \in {\rm PGL}_2({\mathbf{C}})$
be the element with
$\nu (z)=1/z$
for
$z \in {\mathbf{P}}^1({\mathbf{C}})$
. Then (35) implies that
$\nu$
is contained in
$H$
, and one readily observes that
$H = D_3$
. We conclude that
\begin{align*} H = {\rm Stab}_{{\rm PGL}_2({\mathbf{C}})}(x) \in \left \{ {\mathbf{Z}}/2, {\mathbf{Z}}/4, {\mathbf{Z}}/5, D_3, D_4, D_5 \right\}\!. \end{align*}
It remains to exclude
${\mathbf{Z}}/5$
and
$D_4$
. Suppose that
$H$
contains an element
$\phi$
of order five. As above, one can readily show that one may assume that
\begin{align*} x = \left \{ 1, \zeta _5, \zeta _5^2, \zeta _5^3, \zeta _5^4 \right \} \, \subset {\mathbf{P}}^1({\mathbf{C}}), \quad \phi (z) = \zeta _5 \cdot z \ {\rm for} \ z \in {\mathbf{P}}^1({\mathbf{C}}). \end{align*}
This implies that the element
$\nu \in {\rm PGL}_2({\mathbf{C}})$
as defined above is contained in
$H$
, and
$H = D_5$
. Finally, assume
$H$
contains an element
$\phi$
of order four. We may assume that
\begin{align*} x = \left \{ 1, i, -1, -i, 0 \right \}, \quad \phi (z) = i \cdot z \ {\rm for} \ z \in {\mathbf{P}}^1({\mathbf{C}}). \end{align*}
As a consequence, we have
$H = {\mathbf{Z}}/4$
, and the proof is finished.
We proceed to prove the real analogue of Proposition 6.3.
Proposition 6.4.
Let
$x \in (P_s/ {\mathfrak{S}}_5)({\mathbf{R}})$
such that the stabilizer group
${\rm PGL}_2({\mathbf{R}})_x \, \subset {\rm PGL}_2({\mathbf{R}})$
of
$x$
is non-trivial. Then its stabilizer group
${\rm PGL}_2({\mathbf{R}})_x$
is isomorphic to
${\mathbf{Z}}/2$
,
$D_3$
or
$D_5$
. For each
$n \in \{3,5\}$
, there is a unique
${\rm PGL}_2({\mathbf{R}})$
-orbit of points
$x$
in
$(P_s/ {\mathfrak{S}}_5)({\mathbf{R}})$
with stabilizer
$D_n$
.
Proof. We have an injection
$ (P_s/{\mathfrak{S}}_5)({\mathbf{R}}) \hookrightarrow P_s/{\mathfrak{S}}_5$
which is equivariant for the embedding
${\rm PGL}_2({\mathbf{R}}) \hookrightarrow {\rm PGL}_2({\mathbf{C}})$
. In particular,
${\rm PGL}_2({\mathbf{R}})_x \, \subset {\rm PGL}_2({\mathbf{C}})_x$
for every
$x \in (P_s/{\mathfrak{S}}_5)({\mathbf{R}})$
. Note that none of the groups appearing in Proposition 6.3 have subgroups isomorphic to
$D_2 = {\mathbf{Z}}/2 \rtimes {\mathbf{Z}}/2$
or
$D_4 = {\mathbf{Z}}/2 \rtimes {\mathbf{Z}}/4$
. Consider the involution
$ \nu = (z \mapsto 1/z) \in {\rm PGL}_2({\mathbf{R}})$
. We will prove the proposition by using the following steps.
Step 1: Let
$\tau \in {\rm PGL}_2({\mathbf{R}})$
. Consider a subset
$S = \{x,y,z\} \, \subset {\mathbf{P}}^1({\mathbf{C}})$
stabilized by complex conjugation, such that
$\tau (x) = x$
,
$\tau (y) = z$
and
$\tau (z) = y$
. There is a transformation
$g \in {\rm PGL}_2({\mathbf{R}})$
that maps
$S$
to either
$\{-1, 0, \infty \}$
or
$\{-1, i, -i\}$
, and that satisfies
$g \tau g^{-1} = \nu = (z \mapsto 1/z) \in {\rm PGL}_2({\mathbf{R}})$
. In particular,
$\tau^2 = {{\rm id}}$
.
Proof of Step 1. Indeed, two transformations
$g,h \in {\rm PGL}_2({\mathbf{C}})$
that satisfy
$g(x_i) = h(x_i)$
for three different points
$x_1, x_2, x_3 \in {\mathbf{P}}^1({\mathbf{C}})$
are necessarily equal.
Step 2: There is no
$\phi \in {\rm PGL}_2({\mathbf{R}})$
of order four that fixes a point
$x \in (P_s/{\mathfrak{S}}_5)({\mathbf{R}})$
.
Proof of Step 2. By [Reference BeauvilleBea10, Theorem 4.2], all subgroups
$G \, \subset {\rm PGL}_2({\mathbf{R}})$
that are isomorphic to
${\mathbf{Z}}/4$
are conjugate to each other. Since the transformation
$I\colon z \mapsto (z-1)/(z+1)$
is of order four, it gives a representative
$G_I = \langle I \rangle$
of this conjugacy class. It is easily shown that
$I$
cannot fix any point
$x \in (P_s/{\mathfrak{S}}_5)({\mathbf{R}})$
.
Step 3: Define
$\rho \in {\rm PGL}_2({\mathbf{R}})$
by
$\rho (z) = \frac {-1}{z+1}$
. Let
$x = (x_1, \dotsc , x_5) \in (P_s/{\mathfrak{S}}_5)({\mathbf{R}})$
. Let
$\phi \in {\rm PGL}_2({\mathbf{R}})$
be of order three, with
$\phi (x) = x$
. There is a transformation
$g \in {\rm PGL}_2({\mathbf{R}})$
mapping
$x$
to
$(-1, \infty , 0, \omega , \omega^2)$
with
$\omega$
a primitive third root of unity. The stabilizer of
$x$
is the subgroup of
${\rm PGL}_2({\mathbf{R}})$
generated by
$\rho$
and
$\nu$
. In particular, we have
${\rm PGL}_2({\mathbf{R}})_x \cong D_3$
.
Proof of Step 3. By Step 1, there are elements
$x_1, x_2, x_3$
which form an orbit under
$\phi$
. Since complex conjugation preserves this orbit, one element in it is real; since
$g$
is defined over
$\mathbf{R}$
, they are all real. Let
$g \in {\rm PGL}_2({\mathbf{R}})$
such that
$g(x_1)=-1$
,
$g(x_2) = \infty$
and
$g(x_3)=0$
. Define
$\kappa = g \phi g^{-1}$
. Then
$\kappa^3 = {{\rm id}}$
, and
$\kappa$
preserves
$\{-1, \infty , 0\}$
and sends
$-1$
to
$\infty$
and
$\infty$
to
$0$
. Consequently,
$\kappa (0)=-1$
, and it follows that
$\kappa = \rho$
. Hence
$x$
is equivalent to an element of the form
$(-1, \infty , 0, \alpha , \beta )$
with
$\beta = \bar{\alpha}$
and
$\alpha^2 + \alpha+1=0$
.
Recall that
$ \lambda = \zeta _5 + \zeta _5^{-1} \in {\mathbf{R}}$
. Define
$ \gamma \in {\rm PGL}_2({\mathbf{R}})$
by
$\gamma (z) = \frac {(\lambda+1)z - 1}{z+1}$
for
$ z \in {\mathbf{P}}^1({\mathbf{C}}).$
Step 4: Let
$x = (x_1, \dotsc , x_5) \in (P_s/{\mathfrak{S}}_5)({\mathbf{R}})$
. Suppose
$x$
is stabilized by a subgroup of
${\rm PGL}_2({\mathbf{R}})$
of order five. There is a transformation
$g \in {\rm PGL}_2({\mathbf{R}})$
mapping
$x$
to
$z = (0, -1, \infty , \lambda+1, \lambda )$
and identifying the stabilizer of
$x$
with the subgroup of
${\rm PGL}_2({\mathbf{R}})$
generated by
$\gamma$
and
$\nu$
. In particular, the stabilizer
${\rm PGL}_2({\mathbf{R}})_x$
of
$x$
is isomorphic to
$D_5$
.
Proof of Step 4.
Let
$\phi \in {\rm PGL}_2({\mathbf{R}})_x$
be an element of order five. Using Step 1, one shows that the
$x_i$
are pairwise distinct, and we may assume that
$x_i = \phi ^{i-1}(x_1)$
for
$i=2, \dotsc , 5$
. Since there is one real
$x_i$
and
$\phi$
is defined over
$\mathbf{R}$
, all
$x_i$
are real.
Let
$z = \left \{ 0, -1, \infty , \lambda+1, \lambda \right \}$
. Note that
$z$
is the orbit of
$0$
under
$\gamma \colon z \mapsto ((\lambda+1)z - 1)/$
$(z+1)$
. The reflection
$\nu \colon z \mapsto 1/z$
preserves
$z$
as well: we have
$\lambda+1 = - (\zeta _5^2 + \zeta _5^{-2}) = - \lambda^2+2$
, so that
$\lambda (\lambda+1)=1$
. We conclude that
${\rm PGL}_2({\mathbf{R}})_z \cong D_5$
. Thus, by Proposition 6.3, there exists
$g \in {\rm PGL}_2({\mathbf{C}})$
such that
$g(x_1)=0$
,
$g(x_2)=-1$
,
$g(x_3) = \infty$
,
$g(x_4) = \lambda+1$
and
$g(x_5) = \lambda$
, and such that
$g{\rm PGL}_2({\mathbf{R}})_xg^{-1} = {\rm PGL}_2({\mathbf{R}})_z$
. Since all
$x_i$
and
$z_i \in z$
are real, we have
$\bar{g}(x_i) = z_i$
for each
$i$
. Hence,
$g$
and
$\bar{g}$
coincide on three points, which implies that
$g = \bar{g}$
, i.e.
$g \in {\rm PGL}_2({\mathbf{R}})$
.
By Steps 1–4 above, together with Proposition 6.3, we are done.
6.2 Comparing the orbifold structures
There are two orbifold structures on the space
$\left | \mathcal{M}_s({\mathbf{R}}) \right |$
. On the one hand, one has the natural orbifold structure on
$\mathcal{M}_s({\mathbf{R}})$
by considering it as the real locus of a smooth separated Deligne–Mumford stack over
$\mathbf{R}$
(see [Reference de Gaay FortmanGF22, Proposition 2.12]); this is the orbifold structure of the quotient
$ G({\mathbf{R}}) \setminus X_s({\mathbf{R}})$
. On the other hand, one has the orbifold structure
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
introduced in Definition 6.1. The goal of Section 6.2 is to calculate the difference between these orbifold structures.
We first show that there are no cone points in the orbifold
${\rm PGL}_2({\mathbf{R}}) \setminus (P_s/ {\mathfrak{S}}_5)({\mathbf{R}})$
. These are orbifold points whose stabilizer group is
${\mathbf{Z}}/n$
(
$n \geq 2$
) acting on the orbifold chart by rotations. By Proposition 6.4, the fact that
${\rm PGL}_2({\mathbf{R}}) \setminus (P_s/ {\mathfrak{S}}_5)({\mathbf{R}})$
has no cone points follows from the following lemma.
Lemma 6.5.
Let
$x=(x_1, \dotsc , x_5) \in (P_s/{\mathfrak{S}}_5)({\mathbf{R}})$
such that
${\rm PGL}_2({\mathbf{R}})_x = \langle \tau \rangle$
has order two. There is a
${\rm PGL}_2({\mathbf{R}})_x$
-stable open neighborhood
$U \, \subset (P_s/{\mathfrak{S}}_5)({\mathbf{R}})$
of
$x$
such that
${\rm PGL}_2({\mathbf{R}})_x \setminus U \to \mathcal{M}_s({\mathbf{R}})$
is injective, and a homeomorphism
$\phi \colon (U,x) \to (B,0)$
for
$0 \in B \, \subset {\mathbf{R}}^2$
an open ball, such that
$\phi$
identifies
${\rm PGL}_2({\mathbf{R}})_x$
with
${\mathbf{Z}}/2$
acting on
$B$
by reflections in a line through
$0$
.
Proof.
Using Step 1 in the proof of Proposition 6.4, one checks that the only possibilities for the element
$x=(x_1, \dotsc , x_5) \in (P_s/{\mathfrak{S}}_5)({\mathbf{R}})$
are
$(-1, 0, \infty , \beta , \beta ^{-1})$
,
$(-1, i, -i, \beta , \beta ^{-1})$
,
$ (-1, -1, \beta , 0, \infty )$
,
$ (-1, -1, \beta , i, -i)$
,
$(0,0, \infty , \infty , -1)$
and
$(-1, i, i, - i, -i)$
for some
$\beta \in {\mathbf{P}}^1({\mathbf{R}})$
.
To analyze the difference between the two orbifolds
$\mathcal{M}_s({\mathbf{R}})$
and
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
, we also need the following general lemma.
Lemma 6.6.
Let
$X$
be a set and let
$\Gamma$
and
$G$
be groups with commuting actions on
$X$
. Let
$x \in X$
with images
$\bar{x} \in \Gamma \setminus X$
and
$[x] \in G \setminus X$
. Let
$\Gamma _{[x]}$
be the stabilizer of
$[x] \in G \setminus X$
in
$\Gamma$
, and let
$G_{\bar{x}}$
be the stabilizer of
$\bar{x}$
in
$G$
. Then for each
$\gamma \in \Gamma _{[x]}$
there exists an element
$\phi (\gamma ) \in G_{\bar{x}}$
, unique up to multiplication by an element of
$G_x$
, such that
$\gamma \cdot x = \phi (\gamma ) \cdot x$
; moreover, the map
\begin{align} \Gamma _{[x]}/\Gamma _x \to G_{\bar{x}}/G_x, \quad \quad \gamma \mapsto \phi (\gamma ) \end{align}
is an isomorphism.
Proof.
The map (36) is well-defined because if
$g, g' \in G$
are such that
$\gamma \cdot x=g \cdot x=g' \cdot x$
then
$(g^{\prime})^{-1}g \in G_x$
. Since the construction is symmetric in
$\Gamma$
and
$G$
, the analogous map
$G_{\bar{x}}/G_x \to \Gamma _{[x]}/\Gamma _x$
is also well-defined. The latter is a left and right inverse of (36), and we are done.
Recall (see [Reference ThurstonThu80, Proposition 13.3.1]), that the singular locus of a two-dimensional orbifold has the following types of local models: (i)
${\mathbf{R}}^2/({\mathbf{Z}}/2)$
, where
${\mathbf{Z}}/2$
acts on
${\mathbf{R}}^2$
by reflection in the
$y$
-axis (mirror points); (ii)
${\mathbf{R}}^2/({\mathbf{Z}}/n)$
, with
${\mathbf{Z}}/n$
acting by rotations (cone points of order
$n$
); and (iii)
${\mathbf{R}}^2/D_n$
, with
$D_n$
the dihedral group of order
$2n$
(corner reflectors of order
$n$
).
Proposition 6.7.
Consider the orbifold structures
$\mathcal{M}_s({\mathbf{R}})$
and
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
on the space
$\left | \mathcal{M}_s({\mathbf{R}}) \right |$
.
-
(i) For
$x_0 \in \mathcal{M}_s({\mathbf{R}})$
, the isomorphism class of stabilizer groups of
$\mathcal{M}_s({\mathbf{R}})$
and
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
at
$x_0$
differ if and only if
$x_0 \in \mathcal{M}_s({\mathbf{R}})$
is the moduli point attached to the five-tuple
$(\infty , i, i, -i, -i)$
. -
(ii) The stabilizer group of
$\mathcal{M}_s({\mathbf{R}})$
at the point
$x_0$
is isomorphic to
${\mathbf{Z}}/2$
, whereas the stabilizer group of
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
at
$x_0$
is isomorphic to the dihedral group
$D_{10}$
of order
$20$
. -
(iii) The orbifold
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
has no cone points and three corner reflectors, whose angles are
$\pi /3, \pi /5$
and
$\pi /10$
.
Proof.
The statements can be deduced from Proposition 6.4, Lemmas 6.5 and 6.6, and [Reference de Gaay FortmanGF24, Proposition 5.19]. To show how this works, let us introduce some notation. Let
$\tilde F \in \mathcal{F}_s({\mathbf{R}})$
with image
$F \in X_s({\mathbf{R}})$
. Let
$f \in Y$
be the image of
$[\tilde F] \in G({\mathbf{R}}) \setminus \mathcal{F}_s({\mathbf{R}})$
under the isomorphism
$G({\mathbf{R}}) \setminus \mathcal{F}_s({\mathbf{R}}) \xrightarrow {\sim } Y$
of Theorem 5.8. Let
$P\Gamma _f \, \subset P\Gamma$
be the stabilizer of
$f \in Y$
in the group
$P\Gamma$
. Let
$k \in \left \{ 0,1,2 \right \}$
be the number of nodes of
$F$
, and write
$k=2a + b$
, where
$a$
is the number of pairs of complex conjugate nodes and
$b$
the number of real nodes. Then the image
$x \in {\mathbf{C}} H^2$
of
$\tilde F$
under (23) lies on exactly
$k$
distinct mutually orthogonal hyperplanes
$H \in \mathcal{H}$
, with
$\mathcal{H}$
the set defined in (26). Since
$F \in X_s({\mathbf{R}})$
, we have that
$\tilde F \in \mathcal{F}_s^\alpha$
for some
$\alpha \in P{\mathscr{A}}$
. We get
$x \in \mathbf{R} H^n_\alpha$
.
Let
$\mathcal{H}(x) \, \subset \mathcal{H}$
be the set of hyperplanes
$H \in \mathcal{H}$
such that
$x \in H$
. Then
$a$
equals the number of pairs of hyperplanes
$H_1, H_2 \in \mathcal{H}(x)$
with
$\alpha (H_1) = H_2$
, and
$b$
equals the number of hyperplanes
$H \in \mathcal{H}(x)$
with
$\alpha (H) = H$
. Define
$B_f \, \subset P\Gamma _f$
as the group generated by reflections in all
$H \in \mathcal{H}(x)$
such that
$\alpha (H) = H$
. Consider the quotient map
$p \colon \sqcup _{\alpha \in P{\mathscr{A}}} {\mathbf{R}} H^2_\alpha \to Y\!$
, let
$\alpha _1, \dotsc , \alpha _\ell \in P{\mathscr{A}}\ $
be the elements such that
$(x,\alpha _i) \sim (x,\alpha )$
, and define
$Y_f = \cup _{i=1}^\ell p({\mathbf{R}} H^2_{\alpha _i}) \, \subset Y$
. The subgroup
$P\Gamma _f \, \subset P\Gamma$
preserves the subset
$Y_f \, \subset Y$
by [Reference de Gaay FortmanGF24, Lemma 5.9]. Moreover, by [Reference de Gaay FortmanGF24, Proposition 5.19.4], there is an isometry between
$B_f \setminus Y_f$
and the union of
$10^a$
copies of
$\mathbf{B}^2({\mathbf{R}})$
. Let
$\mathbf{B}$
be any one of these copies of
${\mathbf{B}}^2({\mathbf{R}})$
, and define
\begin{align*}S_f := {\rm Stab}_{P\Gamma _f/B_f}(\mathbf{B}),\end{align*}
the stabilizer of
$\mathbf{B}$
in the group
$P\Gamma _f / B_f$
. By construction of the orbifold
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
(see [Reference de Gaay FortmanGF24, Propositions 5.1 and 5.19]), the group
$S_f$
is a representative of the isomorphism class of stabilizer groups of the orbifold
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
at the moduli point
$[F] \in \mathcal{M}_s({\mathbf{R}})$
induced by
$F$
. Clearly, the stabilizer
$G({\mathbf{R}})_F \, \subset G({\mathbf{R}})$
of
$F \in X_s({\mathbf{R}})$
yields the isomorphism class of stabilizer groups of the orbifold
$\mathcal{M}_s({\mathbf{R}})$
at the moduli point
$[F] \in \mathcal{M}_s({\mathbf{R}})$
. In particular, we need to compare the isomorphism classes of the groups
$S_f$
and
$G({\mathbf{R}})_F$
. To do so, we claim that there is a canonical isomorphism
\begin{align} G({\mathbf{R}})_F \xrightarrow {\sim } P\Gamma _f / G(x). \end{align}
Indeed, the actions of the groups
$P\Gamma$
and
$G({\mathbf{R}})$
on
$\mathcal{F}_s({\mathbf{R}})$
commute, so we can apply Lemma 6.6. Recall that
$\tilde F \in \mathcal{F}_s({\mathbf{R}})$
has images
$F \in X_s({\mathbf{R}})$
and
$f \in Y$
, and that the map
$\mathcal{F}_s({\mathbf{R}}) \to Y$
factors through an isomorphism
$G({\mathbf{R}}) \setminus \mathcal{F}_s({\mathbf{R}}) \xrightarrow {\sim } Y$
. Moreover, if
$P\Gamma _{\tilde F} \, \subset P\Gamma$
denotes the stabilizer of
$\tilde F$
in
$P\Gamma$
, then
$P\Gamma _{\tilde F} = G(x)$
by Lemma 4.4. As the group
$G({\mathbf{C}})$
acts freely on
$\mathcal{F}_s$
by Lemma 3.12, the group
$G({\mathbf{R}})$
acts freely on
$\mathcal{F}_s({\mathbf{R}})$
. Thus, (37) follows from Lemma 6.6.
If
$F$
has no nodes (
$k=0$
), then
$G(x)$
is trivial by [Reference de Gaay FortmanGF24, Proposition 5.19.1], and
$P\Gamma _f = S_f$
. Thus,
$G({\mathbf{R}})_F \cong S_f$
in view of (37).
If
$F$
has only real nodes, then
$B_f = G(x)$
and
$P\Gamma _f/G(x) =S_f$
. Thus,
$G({\mathbf{R}})_F \cong S_f$
by (37).
Finally, suppose that
$F$
has a pair of complex nodes (
$a=1$
and
$b=0$
). The zero set of
$F$
defines a five-tuple
$\underline {z}= (z_1, \dotsc , z_5) \in {\mathbf{P}}^1({\mathbf{C}})^5$
, well defined up to the
${\rm PGL}_2({\mathbf{R}}) \times {\mathfrak{S}}_5$
action on
${\mathbf{P}}^1({\mathbf{C}})$
, where
$z_1 \in {\mathbf{P}}^1({\mathbf{R}})$
and
$z_3 = \bar{z}_2 = z_5 = \bar{z}_4 \in {\mathbf{P}}^1({\mathbf{C}}) \setminus {\mathbf{P}}^1({\mathbf{R}})$
. Write
$\underline {z} = (w, z, \bar{z}, z, \bar{z})$
with
$w \in {\mathbf{P}}^1({\mathbf{R}})$
and
$z \in {\mathbf{P}}^1({\mathbf{C}}) \setminus {\mathbf{P}}^1({\mathbf{R}})$
. There is a unique
$T \in {\rm PGL}_2({\mathbf{R}})$
such that
$T(w) = \infty$
and
$T(z) = i$
. This gives
$T(\underline z) = (\infty , i , -i, i, -i)$
. In particular,
$F$
is unique up to isomorphism.
We have
$G(x) \cong ({\mathbf{Z}}/10)^2$
, and as there are no real nodes,
$B_f$
is trivial. With respect to the isometry
${\mathbf{C}} H^2({\mathbf{C}}) \xrightarrow {\sim } {\mathbf{B}}^2({\mathbf{C}})$
of [Reference de Gaay FortmanGF24, Lemma 5.16], the anti-holomorphic involutions
$\alpha _j\colon {\mathbf{B}}^2({\mathbf{C}}) \to {\mathbf{B}}^2({\mathbf{C}})$
induced by the
$\alpha _j \in P{\mathscr{A}}$
are
$(t_1, t_2) \mapsto (\bar{t}_2 \zeta ^j , \bar{t}_1 \zeta ^j)$
, for
$j \in {\mathbf{Z}}/10$
. The fixed point sets are given by
${\mathbf{B}}^2({\mathbf{R}})_{\alpha _j} = \{t_2 = \bar{t}_1 \zeta ^j\}\, \subset {\mathbf{B}}^2({\mathbf{C}})$
. The subgroup
$E_f \, \subset G(x)$
that stabilizes
${\mathbf{B}}^2({\mathbf{R}})_{\alpha _j}$
is the group
$E_f \cong {\mathbf{Z}}/10$
generated by the transformations
$(t_1, t_2) \mapsto (\zeta t_1, \zeta ^{-1} t_2)$
. There is only one non-trivial
$T \in {\rm PGL}_2({\mathbf{R}})$
fixing
$\infty$
and preserving
$\{i, -i\} \, \subset {\mathbf{P}}^1({\mathbf{C}})$
, and
$T$
has order two, so
$G({\mathbf{R}})_F = {\mathbf{Z}}/2$
. Therefore, by (37), we have an exact sequence of groups
$ 0 \to G(x) \cong ({\mathbf{Z}}/10)^2 \to P\Gamma _f \to {\mathbf{Z}}/2 \to 0$
inducing an exact sequence
$0 \to E_f \cong {\mathbf{Z}}/10 \to S_f \to {\mathbf{Z}}/2 \to 0$
. By Proposition 6.4 and Lemma 6.5, the proposition follows.
6.3 The real moduli space as a hyperbolic triangle
The goal of Section 6.3 is to prove Theorem 6.2.
The results in the above Sections 6.1 and 6.2 give the orbifold singularities of
$\overline {{\mathscr{M}}}_{{\mathbf{R}}}$
together with their stabilizer groups. In order to determine the hyperbolic orbifold structure of
$\overline {{\mathscr{M}}}_{{\mathbf{R}}}$
, we also need to know the underlying topological space
$\left | \mathcal{M}_s({\mathbf{R}}) \right |$
of
$\overline {{\mathscr{M}}}_{{\mathbf{R}}}$
. The first observation is that
$\mathcal{M}_s({\mathbf{R}})$
is compact. Indeed, it is classical that the topological space
$\mathcal{M}_s({\mathbf{C}}) = G({\mathbf{C}}) \setminus X_s({\mathbf{C}})$
, parametrizing complex stable binary quintics, is compact. This follows from the proper surjective map
$\overline {M}_{0,5}({\mathbf{C}})/ {\mathfrak{S}}_5 \to \mathcal{M}_s({\mathbf{C}})$
and the properness of the stack of stable five-pointed curves
$\overline {M}_{0,5}$
[Reference KnudsenKnu83], or from the fact that
$\mathcal{M}_s({\mathbf{C}})$
is homeomorphic to a compact ball quotient [Reference ShimuraShi64]. Moreover, the map
$\mathcal{M}_s({\mathbf{R}}) \to \mathcal{M}_s({\mathbf{C}})$
is proper, which proves the compactness of
$\mathcal{M}_s({\mathbf{R}})$
. The second observation is that
$\mathcal{M}_s({\mathbf{R}})$
is connected, since
$X_s({\mathbf{R}})$
is obtained from the euclidean space
$X({\mathbf{R}}) = \{F \in {\mathbf{R}}[x,y]\colon F\ \mathrm{homogeneous\ and} \deg (F)=5\}$
by removing a subspace of codimension two. In the following lemma we generalize both of these observations.
Lemma 6.8.
The moduli space
$\mathcal{M}_s({\mathbf{R}})$
is homeomorphic to a closed disc
$\overline D \, \subset {\mathbf{R}}^2$
.
Proof. The idea is to show that the following holds.
-
(i) For each
$j \in \{0,1,2\}$
, the embedding
${\mathscr{M}}_j \hookrightarrow \overline {{\mathscr{M}}}_j \, \subset \mathcal{M}_s({{\mathbf{R}}})$
of the connected component
${\mathscr{M}}_j$
of
$\mathcal{M}_0({\mathbf{R}})$
into its closure in
$\mathcal{M}_s({{\mathbf{R}}})$
is homeomorphic to the embedding
$D \hookrightarrow \overline {D}$
of the open unit disc into the closed unit disc in
${\mathbf{R}}^2$
. -
(ii) We have
$\mathcal{M}_s({\mathbf{R}}) = \overline {{\mathscr{M}}}_0 \cup \overline {{\mathscr{M}}}_1 \cup \overline {{\mathscr{M}}}_2$
, and this gluing corresponds, up to homeomorphism, to the gluing of three closed discs
$\overline {D}_i \, \subset {\mathbf{R}}^2$
as in Figure 1.
To prove this, one considers the moduli spaces of real smooth (respectively stable) genus zero curves with five real marked points, as well as twists of this space. Define two anti-holomorphic involutions
$\sigma _i\colon {\mathbf{P}}^1({\mathbf{C}})^5 \to {\mathbf{P}}^1({\mathbf{C}})^5$
by
$ \sigma _1(x_1, x_2, x_3, x_4, x_5) = (\bar{x}_1, \bar{x}_2, \bar{x}_3, \bar{x}_5, \bar{x}_4),$
and
$ \sigma (x_1, x_2, x_3, x_4, x_5) = (\bar{x}_1, \bar{x}_3, \bar{x}_2, \bar{x}_5, \bar{x}_4).$
Then define
\begin{align*} P_0^1({\mathbf{R}}) = P_0^{\sigma _1}, \;\;\; P_s^1({\mathbf{R}}) = P_1({\mathbf{C}})^{\sigma _1}, \;\;\; P_0^2({\mathbf{R}}) = P_0^{\sigma _2}, \;\;\; P_s^2({\mathbf{R}}) = P_1({\mathbf{C}})^{\sigma _2}. \end{align*}
It is clear that
$ {\mathscr{M}}_0 = {\rm PGL}_2({\mathbf{R}}) \setminus P_0({\mathbf{R}}) / {\mathfrak{S}}_5$
. Similarly, we have
\begin{align*} {\mathscr{M}}_1 = {\rm PGL}_2({\mathbf{R}}) \setminus P_0^1({\mathbf{R}}) / {\mathfrak{S}}_3 \times {\mathfrak{S}}_2 \quad \mathrm{and} \quad {\mathscr{M}}_2 = {\rm PGL}_2({\mathbf{R}}) \setminus P_0^2({\mathbf{R}}) / {\mathfrak{S}}_2 \times {\mathfrak{S}}_2. \end{align*}
Moreover, we have
$\overline {{\mathscr{M}}}_0 = {\rm PGL}_2({\mathbf{R}}) \setminus P_s({\mathbf{R}}) / {\mathfrak{S}}_5$
. We define
\begin{align*} \overline {{\mathscr{M}}}_1 = {\rm PGL}_2({\mathbf{R}}) \setminus P_s^1({\mathbf{R}}) / {\mathfrak{S}}_3 \times {\mathfrak{S}}_2, \quad \mathrm{and} \quad \overline {{\mathscr{M}}}_2 = {\rm PGL}_2({\mathbf{R}}) \setminus P_s^2({\mathbf{R}}) / {\mathfrak{S}}_2 \times {\mathfrak{S}}_2. \end{align*}
Each
$\overline {{\mathscr{M}}}_j$
is homeomorphic to a closed disc in
${\mathbf{R}}^2$
. Moreover, the natural maps
$\overline {{\mathscr{M}}}_j \to \mathcal{M}_s({\mathbf{R}})$
are closed embeddings of topological spaces, and one can check that the images glue to form
$\mathcal{M}_s({\mathbf{R}})$
in the prescribed way. We leave the details to the reader.
Proof of Theorem
6.2. To any closed two-dimensional orbifold
$O$
one can associate a set of natural numbers
$S_O = \{n_1, \dotsc , n_k; m_1, \dotsc , m_l\}$
by letting
$k$
be the number of cone points of
$X_O$
,
$l$
the number of corner reflectors,
$n_i$
the order of the
$i$
th cone point and
$m_j$
the order of the
$j$
th corner reflector (see [Reference ThurstonThu80, Proposition 13.3.1]). A closed two-dimensional orbifold
$O$
is determined, up to orbifold-structure preserving homeomorphism, by its underlying space
$X_O$
and the set
$S_O$
[Reference ThurstonThu80]. By Lemma 6.8,
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
is homeomorphic to a closed disc in
${\mathbf{R}}^2$
. By Proposition 6.7,
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
has no cone points and three corner reflectors whose angles are
$\pi /3, \pi /5$
and
$\pi /10$
. This implies that
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
and
$\Delta _{3,5,10}$
are isomorphic as topological orbifolds. Consequently, the orbifold fundamental group of
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
is abstractly isomorphic to the group
$\Gamma _{3,5,10}$
defined in (3).
Let
$\phi \colon \Gamma _{3,5,10} \hookrightarrow {\rm PSL}_2({\mathbf{R}})$
be any embedding such that
$X:=\phi \left (\Gamma _{3,5,10}\right ) \setminus {\mathbf{R}} H^2$
is a hyperbolic orbifold; we claim that there is a fundamental domain
$\Delta$
for
$X$
isometric to
$\Delta _{3,5,10}$
. To see this, consider the generator
$a \in \Gamma _{3,5,10}$
. Since
$\phi (a)^2=1$
, there exists a geodesic
$L_1 \, \subset {\mathbf{R}} H^2$
such that
$\phi (a) \in {\rm PSL}_2({\mathbf{R}}) = {\rm Isom}({\mathbf{R}} H^2)$
is the reflection across
$L_1$
. Next, consider the generator
$b \in \Gamma _{3,5,10}$
. There exists a geodesic
$L_2 \, \subset {\mathbf{R}} H^2$
such that
$\phi (b)$
is the reflection across
$L_2$
, and we have
$L_2 \cap L_1 \neq \emptyset$
. Let
$x \in L_1 \cap L_2$
. Then
$\phi (a)\phi (b)$
is an element of order three that fixes
$x$
, and hence is a rotation around
$x$
. Therefore, one of the angles between
$L_1$
and
$L_2$
must be
$\pi /3$
. Finally, we know that
$\phi (c)$
is an element of order two in
${\rm PSL}_2({\mathbf{R}})$
, and hence a reflection across a line
$L_3$
. As
$L_3 \cap L_2 \neq \emptyset$
and
$L_3 \cap L_1 \neq \emptyset$
, the three geodesics
$L_i \, \subset {\mathbf{R}} H^2$
enclose a hyperbolic triangle. As the orders of the three elements
$\phi (a)\phi (b)$
,
$\phi (a)\phi (c)$
and
$\phi (b)\phi (c)$
are respectively three, five and ten, the three interior angles of the triangle are
$\pi /3$
,
$\pi /5$
and
$\pi /10$
. Thus,
$X$
is isometric to
$\Delta _{3,5,10}$
.
Consequently,
$P\Gamma _{\mathbf{R}} \setminus \mathbf{R} H^2$
is isometric to
$\Gamma _{3,5,10} \setminus \mathbf{R} H^2$
. It follows that the lattices
$P\Gamma _{\mathbf{R}}$
and
$\Gamma _{3,5,10}$
are conjugate in
${\rm PO}(2,1)$
(see e.g. [Reference RatcliffeRat99, Lemma 1]).
7. The monodromy groups
In this section, we describe the monodromy group
$P\Gamma$
attached to the moduli space
$X_0({\mathbf{C}})$
, as well as the groups
$P\Gamma _\alpha$
appearing in Proposition 5.4. As for the lattice
$(\Lambda , {\mathfrak{h}})$
(see (8)), we have the following.
Theorem 7.1 (Shimura). There is an isomorphism of hermitian
${\mathcal{O}}_K$
-lattices
\begin{align*}\left ( \Lambda , {\mathfrak{h}}\, \right ) \cong \left ( {\mathcal{O}}_K^3, {\rm diag}\left (- \lambda , 1,1\right )\, \right ), \quad \lambda = \zeta _5 + \zeta _5^{-1} = \frac {\sqrt 5 - 1}{2}.\end{align*}
Proof. See [Reference ShimuraShi64, Section 6] as well as item (5) in the table on page 1 of that paper.
Write
$\Lambda = {\mathcal{O}}_K^3$
and
${\mathfrak{h}} = {\rm diag}(- \lambda , 1, 1)$
. Consider the
$\mathbf{F}_5$
-vector space
$W = \Lambda / (1 - \zeta _5)\Lambda$
equipped with the quadratic form
$q = {\mathfrak{h}} \bmod \theta$
. Define three anti-isometric involutions as follows:
\begin{align} \alpha _0\colon & (x_0,x_1,x_2) \mapsto (\bar{x}_0, \;\;\;\bar{x}_1,\;\;\bar{x}_2), \nonumber\\ \alpha _1 \colon & (x_0,x_1,x_2) \mapsto (\bar{x}_0, - \bar{x}_1, \;\;\bar{x}_2), \nonumber\\ \alpha _2 \colon & (x_0,x_1,x_2) \mapsto (\bar{x}_0, -\bar{x}_1, -\bar{x}_2). \end{align}
Lemma 7.2.
An anti-unitary involution of
$\Lambda$
is
$\Gamma$
-conjugate to exactly one of the
$\pm \alpha _j$
. In particular,
$\left | P\Gamma \setminus P{\mathscr{A}}\ \right | = 3$
and the
$\alpha _i$
defined in (
38
) form a set of representatives for
$P\Gamma \setminus P{\mathscr{A}}$
.
Proof.
For isometries
$\alpha \colon W \to W$
, the dimension and determinant of the fixed space
$(W^\alpha , q|_{W^\alpha })$
are conjugacy-invariant. Using this, one can show that the elements
$\pm \alpha _i$
are pairwise non
$\Gamma$
-conjugate. Moreover,
$\left | P\Gamma \setminus P{\mathscr{A}}\ \right | = \left | \pi _0(\mathcal{M}_0({\mathbf{R}})) \right | = 3$
by Proposition 5.4 and Theorem 7.1.
Define
\begin{align*}\theta = \zeta _5 - \zeta _5^{-1} \in {\mathcal{O}}_K.\end{align*}
Note that
$|\theta |^2 = \frac {\sqrt {5} + 5}{2}$
. The fixed lattices of the anti-unitary involutions
$\alpha _i$
defined in (38) are
\begin{align} \Lambda ^{\alpha _0} & = {\mathbf{Z}}[\lambda ] \oplus {\mathbf{Z}}[\lambda ] \oplus {\mathbf{Z}}[\lambda ], \nonumber\\ \Lambda ^{\alpha _1} & = {\mathbf{Z}}[\lambda ] \oplus \theta {\mathbf{Z}}[\lambda ] \oplus {\mathbf{Z}}[\lambda ],\nonumber\\ \Lambda ^{\alpha _2} & = {\mathbf{Z}}[\lambda ] \oplus \theta {\mathbf{Z}}[\lambda ] \oplus \theta {\mathbf{Z}}[\lambda ], \end{align}
where
\begin{align} \lambda = \zeta _5 + \zeta _5^{-1} = \frac {\sqrt 5 - 1}{2}. \end{align}
Restricting
$\mathfrak{h}$
to the
$\Lambda ^{\alpha _j}$
yields quadratic forms
$q_0$
,
$q_1$
and
$q_2$
on
${\mathbf{Z}}[\lambda ]^3$
defined as follows:
\begin{align} q_0(x_0, x_1, x_2) & = - \lambda x_0^2 + x_1^2 + x_2^2, \nonumber\\ q_1(x_0, x_1, x_2) & = - \lambda x_0^2 + \left (\frac {\sqrt {5} + 5}{2}\right ) \cdot x_1^2 + x_2^2, \nonumber\\ q_2(x_0, x_1, x_2) & = - \lambda x_0^2 + \left (\frac {\sqrt {5} + 5}{2}\right ) \cdot x_1^2 + \left (\frac {\sqrt {5} + 5}{2}\right ) \cdot x_2^2, \end{align}
We consider
${\mathbf{Z}}[\lambda ]$
(with
$\lambda$
as in (40)) as a subring of
$\mathbf{R}$
via the embedding that sends
$\lambda$
to a positive element.
Theorem 7.3.
Consider the quadratic forms
$q_j$
defined in (
41
), with
$\lambda$
as in (
40
). There is a union of geodesic subspaces
${\mathscr{H}}_j \, \subset {\mathbf{R}} H^2$
$\left ( j \in \{0,1,2\}\, \right )$
and an isomorphism of hyperbolic orbifolds
\begin{equation} \mathcal{M}_0({\mathbf{R}}) \cong \coprod _{j=0}^2 {\rm PO}(q_j,{\mathbf{Z}}[\lambda ]) \setminus \left ({\mathbf{R}} H^2 - {\mathscr{H}}_j \kern1pt\right)\!. \end{equation}
Proof. By Proposition 5.4 and Lemma 7.2, we obtain an isomorphism
\begin{align*}\mathcal{M}_0({\mathbf{R}}) \cong \coprod _{j=0}^2 P\Gamma _{\alpha _j} \setminus ({\mathbf{R}} H^2_{\alpha _j} - {\mathscr{H}}\kern3pt).\end{align*}
Note that
$P\Gamma _{\alpha _j}=N_{P\Gamma }(\alpha _j)$
for the normalizer
$N_{P\Gamma }(\alpha _j)$
of
$\alpha _j$
in
$P\Gamma$
. If
$h_j$
denotes the restriction of
$\mathfrak{h}$
to
$\Lambda ^{\alpha _j}$
, there is a natural embedding
\begin{align*} \iota _j \colon N_{P\Gamma }(\alpha _j) \hookrightarrow {\rm PO}(\Lambda ^{\alpha _j},h_j, {\mathbf{Z}}[\lambda ]).\end{align*}
We claim that
$\iota _j$
is an isomorphism. This holds because the natural homomorphism
$ \pi _j \colon N_\Gamma (\alpha _j) \to {\rm O}(\Lambda ^{\alpha _j}, h_j)$
is surjective, where
$N_\Gamma (\alpha _j) = \{g \in \Gamma \colon g \circ \alpha _j = \alpha _j \circ g \}$
is the normalizer of
$\alpha _j$
in
$\Gamma$
. The surjectivity of
$\pi _j$
follows in turn from the equality
\begin{align} \Lambda = \mathcal{O}_K \cdot \Lambda ^{\alpha _j} + \mathcal{O}_K \cdot \theta \left (\Lambda ^{\alpha _j}\right )^\vee \, \subset K^3, \end{align}
and (43) follows from (39). Since
${\rm PO}(\Lambda ^{\alpha _j},h_j, {\mathbf{Z}}[\lambda ]) = {\rm PO}(q_j,{\mathbf{Z}}[\lambda ])$
, we are done.
Proof of Theorem
1.3. In [Reference Apéry and YoshidaAY98], Apéry and Yoshida proved that
$\overline {{\mathscr{M}}}_0$
is the hyperbolic triangle with angles
$\pi /2, \pi /4$
and
$\pi /5$
. As the two hyperplanes in Figure 1 intersect orthogonally, this implies that the bottom angle of the triangle
$\overline {{\mathscr{M}}}_0$
in Figure 1 (i.e. its angle at
$(0,-1,\infty ,\infty ,1)$
) equals
$\pi /2$
, and that the angle of
$\overline {{\mathscr{M}}}_0$
at
$(0,-1,-1,\infty ,\infty )$
equals
$\pi /4$
. One deduces that the left angle of
$\overline {{\mathscr{M}}}_1$
is
$\pi /2$
, and that the angle of
$\overline {{\mathscr{M}}}_2$
at
$(0,-1,-1,\infty ,\infty )$
equals
$\pi /4$
.
For a hyperbolic triangle with angles
$\alpha , \beta , \gamma$
and sides
$a,b,c$
such that
$a$
is the side opposite to
$\alpha$
,
$b$
the side opposite to
$\beta$
and
$\gamma$
the side opposite to
$c$
, one has the hyperbolic law of cosines
\begin{align} \cosh (c) = \frac {\cos (\alpha )\cos (\beta )+\cos (\gamma )}{\sin (\alpha )\sin (\beta )}. \end{align}
Applying (44) to the triangles
$\overline {{\mathscr{M}}}_0$
and
$\overline {{\mathscr{M}}}_{\mathbf{R}}$
, one can calculate the length of the side of
$\overline {{\mathscr{M}}}_2$
that connects
$(0,-1,-1,\infty ,\infty )$
and
$(\infty , i, i, -i, -i)$
. Applying (44) again, it follows that the angle of
$\overline {{\mathscr{M}}}_2$
at the point
$(0,-i,\infty ,\infty ,i)$
is
$\pi /2$
. Thus, the angle of
$\overline {{\mathscr{M}}}_1$
at
$(0,-i,\infty ,\infty ,i)$
is also
$\pi /2$
.
8. Non-arithmetic lattices in the projective orthogonal group
In a previous paper we proved a result, see [Reference de Gaay FortmanGF24, Theorem 1.8], that has the following consequence. For
$n \geq 2$
, define
\begin{align*} {\mathscr{L}}_{\zeta _5}^{\, n}(\lambda ) := \left ({\mathbf{Q}}(\zeta _5), {\mathbf{Z}}[\zeta _5]^{n,1}_\lambda\right)\!, \quad \quad \lambda = \zeta _5 + \zeta _5^{-1} = (\sqrt 5 - 1)/2. \end{align*}
Here,
$ {\mathbf{Z}}[\zeta _5]^{n,1}_\lambda$
is the free
${\mathbf{Z}}[\zeta _5]$
-module of rank
$n+1$
equipped with the hermitian form
$h$
defined as
$h(x,y) = -\lambda \cdot x_0 \bar{y}_0 + \cdots + x_n \bar{y}_n$
. Then
${\mathscr{L}}_{\zeta _5}^n(\lambda )$
is a hermitian lattice of rank
$n+1$
in the sense of [Reference de Gaay FortmanGF24, Definition 2.2] (indeed, this follows from [Reference de Gaay FortmanGF24, Example 2.12]). For each
$n \geq 2$
, perform the gluing construction of [Reference de Gaay FortmanGF24, Definition 1.1] to associate to the hermitian lattice
${\mathscr{L}}_{\zeta _5}^n(\lambda )$
a topological space
$ M( {\mathscr{L}}_{\zeta _5}^n(\lambda ))$
. By [Reference de Gaay FortmanGF24, Theorem 1.2], there exists a canonical real hyperbolic orbifold structure on
$M( {\mathscr{L}}_{\zeta _5}^n(\lambda ) )$
such that each connected component of
$M( {\mathscr{L}}_{\zeta _5}^n(\lambda ) )$
is isomorphic to the quotient of real hyperbolic
$n$
-space
$\mathbf{R} H^n$
by a lattice in
${\rm PO}(n,1)$
. Define an anti-unitary involution
$\alpha _0 \colon {\mathbf{Z}}[\zeta _5]^{n,1}_\lambda \to {\mathbf{Z}}[\zeta _5]^{n,1}_\lambda$
by
$\alpha _0(x) = \bar{x}$
, let
\begin{align*}M\left ( {\mathscr{L}}_{\zeta _5}^n(\lambda ) , \alpha _0\, \right ) \, \subset M\left ( {\mathscr{L}}_{\zeta _5}^n(\lambda )\, \right )\end{align*}
be the connected component that contains the image of the natural map
$\mathbf{R} H^n_{\alpha _0} \to M( {\mathscr{L}}_{\zeta _5}^n(\lambda ) )$
, and let
\begin{align*} \Gamma _{\zeta _5}^n(\lambda ) \, \subset {\rm PO}(n,1) \end{align*}
be a lattice such that
$M( {\mathscr{L}}_{\zeta _5}^n(\lambda ) , \alpha _0) \cong \Gamma _{\zeta _5}^n(\lambda ) \setminus \mathbf{R} H^n$
(compare [Reference de Gaay FortmanGF24, p. 7]).
By combining [Reference de Gaay FortmanGF24, Theorem 1.8] with the main results of this paper, one can prove the following result.
Theorem 8.1.
For each
$n \geq 2$
, the lattice
$\Gamma _{\zeta _5}^n(\lambda ) \, \subset {\rm PO}(n,1)$
is non-arithmetic.
Proof.
Write
${\mathbf{Z}}[\zeta _5]^{2,1}_\lambda = \Lambda$
; this abuse of notation is harmless in view of Theorem 7.1. Let
$\Gamma = {{\rm Aut}}(\Lambda )$
. By Lemma 7.2, an anti-unitary involution of
$\Lambda$
is
$\Gamma$
-conjugate to exactly one of the involutions
$\pm \alpha _j$
defined in (38). We can therefore apply [Reference de Gaay FortmanGF24, Theorem 1.8], which implies that
$\Gamma _{\zeta _5}^n(\lambda ) \, \subset {\rm PO}(n,1)$
is non-arithmetic for each
$n \geq 2$
provided that
$\Gamma _{\zeta _5}^2(\lambda ) \, \subset {\rm PO}(2,1)$
is non-arithmetic. In other words, we are reduced to the case
$n=2$
. By Theorem 7.1, the lattice
$\Gamma _{\zeta _5}^2(\lambda ) \, \subset {\rm PO}(2,1)$
is conjugate to the lattice
$P\Gamma _{\mathbf{R}} \, \subset {\rm PO}(2,1)$
defined in Corollary 5.9. Moreover, by Theorem 6.2, the lattice
$P\Gamma _{\mathbf{R}}$
is conjugate to the lattice
$\Gamma _{3,5,10} \, \subset {\rm PO}(2,1)$
defined in (3). Finally, by Takeuchi’s classification of arithmetic triangle groups (see [Reference TakeuchiTak77]), the subgroup
$\Gamma _{3,5,10} \, \subset {\rm PO}(2,1)$
is non-arithmetic. Thus,
$\Gamma _{\zeta _5}^2(\lambda )$
is non-arithmetic, and the theorem follows.
Acknowledgements
This project was carried out partly at the ENS in Paris and partly at the Leibniz University in Hannover. I thank my former PhD advisor Olivier Benoist for his guidance and support. I thank Romain Branchereau, Samuel Bronstein, Nicolas Tholozan and Frans Oort for useful discussions. I thank Nicolas Bergeron for pointing me to Takeuchi’s paper on hyperbolic triangle groups.
I would like to thank the referee for his or her valuable comments on this paper.
Funding Statement
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754362 
and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 948066 (ERC-StG RationAlgic).
Conflicts of Interest
None.
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