This follows from the Hurwitz-Chevalley-Weil formula, see [MO13, Proposition
5.9]. Alternatively, see [CT99, Section 5]. ∎

Fix a point $F_{0}\in X_{0}(\mathbb{C})$ and let

$C=\\{z^{5}=F_{0}(x,y)\\}\subset\mathbb{P}^{2}_{\mathbb{C}}$ (6)

be the corresponding cyclic cover of $\mathbb{P}^{1}_{\mathbb{C}}$. Let

$\left(A=J(C)=\text{Pic}^{0}(C),\quad\lambda\colon
A\to{\widehat{A}},\quad\iota\colon\mathcal{O}_{K}=\mathbb{Z}[\zeta]\to\textnormal{End}(A)\right)$

be the Jacobian of $C$, viewed as a principally polarized abelian variety of
dimension six equipped with an $\mathcal{O}_{K}$-action compatible with the
polarization, see (4.43).

Write $\Lambda={\mathrm{H}}_{1}(A(\mathbb{C}),\mathbb{Z})$. We have
$\Lambda\otimes_{\mathbb{Z}}\mathbb{C}={\mathrm{H}}^{-1,0}\oplus{\mathrm{H}}^{0,-1}$,
the Hodge decomposition of $\Lambda\otimes_{\mathbb{Z}}\mathbb{C}$. Define a
CM-type $\Psi\subset\textnormal{Hom}(K,\mathbb{C})$ as follows:

$\displaystyle\tau_{i}:K\to\mathbb{C},\quad\tau_{1}(\zeta)=\zeta^{3},\quad\tau_{2}(\zeta)=\zeta^{4};\quad\quad\Psi=\left\\{\tau_{1},\tau_{2}\right\\}.$
(7)

Since
${\mathrm{H}}^{-1,0}=\textnormal{Lie}(A)={\mathrm{H}}^{1}(C,\mathcal{O}_{C})={\mathrm{H}}^{0,1}(C)$,
Lemma 5.6 implies that

$\dim_{\mathbb{C}}{\mathrm{H}}^{-1,0}_{\tau_{1}}=2,\;\;\;\dim_{\mathbb{C}}{\mathrm{H}}^{-1,0}_{\tau_{1}\sigma}=1,\;\;\;\dim_{\mathbb{C}}{\mathrm{H}}^{-1,0}_{\tau_{2}}=3,\;\;\;\dim_{\mathbb{C}}{\mathrm{H}}^{-1,0}_{\tau_{2}\sigma}=0.$
(8)

Define $\eta=5/(\zeta-\zeta^{-1})$. Then ${\mathfrak{D}}_{K}=(\eta)$ (see
Lemma 4.52). Let

$E:\Lambda\times\Lambda\to\mathbb{Z}$

be the alternating form corresponding to the polarization $\lambda$ of the
abelian variety $A$. For $a\in\mathcal{O}_{K}$ and $x,y\in\Lambda$, we have
$E(\iota(a)x,y)=E(x,\iota(a^{\sigma})y)$. Define

$T\colon\Lambda\times\Lambda\to{\mathfrak{D}}_{K}^{-1},\quad
T(x,y)=\frac{1}{5}\sum_{j=0}^{4}\zeta^{j}E\left(x,\iota(\zeta)^{j}y\right).$

By Example 4.40.2, this is the skew-hermitian form corresponding to $E$ via
Lemma 4.39. We obtain a hermitian form on the free $\mathcal{O}_{K}$-module
$\Lambda$ as follows:

${\mathfrak{h}}:\Lambda\times\Lambda\to\mathcal{O}_{K},\;\;\;{\mathfrak{h}}(x,y)=\eta
T(x,y)=(\zeta-\zeta^{-1})^{-1}\sum_{j=0}^{4}\zeta^{j}E\left(x,\iota(\zeta)^{j}y\right).$
(9)

By Lemma 4.39, the hermitian lattice $(\Lambda,{\mathfrak{h}})$ is unimodular,
because $(\Lambda,E)$ is unimodular. For each embedding
$\varphi:K\to\mathbb{C}$, the restriction of the hermitian form
$\varphi(\eta)\cdot E_{\mathbb{C}}(x,\bar{y})$ on $\Lambda_{\mathbb{C}}$ to
$(\Lambda_{\mathbb{C}})_{\varphi}\subset\Lambda_{\mathbb{C}}$ coincides with
${\mathfrak{h}}^{\varphi}$ by Lemma 4.41. Since
$\Im(\tau_{i}(\zeta-\zeta^{-1}))<0$ for $i=1,2$, the signature of
${\mathfrak{h}}^{\tau_{i}}$ on
$V_{i}=\Lambda\otimes_{\mathcal{O}_{K},\tau_{i}}\mathbb{C}$ is

$\displaystyle{\textnormal{sign}}({\mathfrak{h}}^{\tau_{i}})=\begin{cases}({\mathrm{h}}^{-1,0}_{\tau_{1}},h^{0,-1}_{\tau_{1}})=(2,1)&\textnormal{
for }i=1,\quad\textnormal{ and }\\\
({\mathrm{h}}^{-1,0}_{\tau_{2}},h^{0,-1}_{\tau_{2}})=(3,0)&\textnormal{ for
}i=2.\end{cases}$ (10)

#####  2 The monodromy representation

Consider the real algebraic variety $X_{0}$ introduced in Section 1. Let
$D\subset\textnormal{GL}_{2}(\mathbb{C})$ be the subgroup
$D=\left\\{\zeta^{i}\cdot
I_{2}\right\\}\subset\textnormal{GL}_{2}(\mathbb{C})$ of scalar matrices
$\zeta^{i}\cdot I_{2}$, where $I_{2}\in\textnormal{GL}_{2}(\mathbb{C})$ is the
identity matrix of rank two, and define

$\displaystyle{\mathbb{G}}(\mathbb{C})=\textnormal{GL}_{2}(\mathbb{C})/D.$
(11)

The group $\mathbb{G}(\mathbb{C})$ acts from the left on $X_{0}(\mathbb{C})$
in the following way: if $F(x,y)\in\mathbb{C}[x,y]$ is a binary quintic, we
may view $F$ as a function $\mathbb{C}^{2}\to\mathbb{C}$, and define $g\cdot
F=F(g^{-1})$ for $g\in\mathbb{G}(\mathbb{C})$. This gives a canonical
isomorphism of complex analytic orbifolds

${\mathcal{M}}_{0}(\mathbb{C})=\mathbb{G}(\mathbb{C})\setminus
X_{0}(\mathbb{C}),$

where ${\mathcal{M}}_{0}$ is the moduli stack of smooth binary quintics.

Consider two families

$\pi:{\mathscr{C}}\to X_{0}\quad{\textnormal{ and }}\quad\phi:J\to X_{0},$

defined as follows. We define $\pi$ as the universal family of cyclic covers
$C\to\mathbb{P}^{1}$ ramified along a smooth binary quintic
$\\{F=0\\}\subset\mathbb{P}^{1}$. We let $\phi$ be the relative Jacobian of
$\pi$. By Section 1, $\phi$ is an abelian scheme of relative dimension six
over $X_{0}$, with $\mathcal{O}_{K}$-action of signature $\\{(2,1),(3,0)\\}$
with respect to $\Psi=\\{\tau_{1},\tau_{2}\\}$.

Let ${\mathbb{V}}=R^{1}\pi_{\ast}\mathbb{Z}$ be the local system of hermitian
$\mathcal{O}_{K}$-modules underlying the abelian scheme $J/X_{0}$. Attached to
${\mathbb{V}}$, we have a representation

$\rho^{\prime}:\pi_{1}(X_{0}(\mathbb{C}),F_{0})\to\Gamma,\quad\Gamma=\textnormal{Aut}_{\mathcal{O}_{K}}(\Lambda,{\mathfrak{h}}),$

whose composition with the quotient map $\Gamma\to P\Gamma=\Gamma/\mu_{K}$
defines a homomorphism

$\rho:\pi_{1}(X_{0}(\mathbb{C}),F_{0})\to P\Gamma.$ (12)

We shall see that $\rho$ is surjective, see Corollary 5.8 below.

#####  3 Marked binary quintics

For $F\in X_{0}(\mathbb{C})$, define $Z_{F}$ as the hypersurface

$Z_{F}=\\{F=0\\}\subset\mathbb{P}^{1}_{\mathbb{C}}.$

A marking of $F$ is a ring isomorphism
$m:{\mathrm{H}}^{0}(Z_{F}(\mathbb{C}),\mathbb{Z})\xrightarrow{\sim}\mathbb{Z}^{5}$.
To give a marking is to give a labelling of the points $p\in
Z_{F}(\mathbb{C})$. Let ${\mathcal{N}}_{0}$ be the space of marked binary
quintics $(F,m)$; this is a manifold, equipped with a holomorphic map

$\displaystyle{\mathcal{N}}_{0}\to X_{0}(\mathbb{C}).$ (13)

Let

$\psi:{\mathcal{Z}}\to X_{0}(\mathbb{C})$

be the universal complex binary quintic, and consider the local system
$H=\psi_{\ast}\mathbb{Z}$ of stalk
$H_{F}={\mathrm{H}}^{0}(Z_{F}(\mathbb{C}),\mathbb{Z})$ for $F\in
X_{0}(\mathbb{C})$. Then $H$ corresponds to a monodromy representation

$\tau:\pi_{1}(X_{0}(\mathbb{C}),F_{0})\to{\mathfrak{S}}_{5}.$ (14)

It can be shown that $\tau$ is surjective using the results of [Bea86a]. This
implies that (13) is covering space, i.e. that ${\mathcal{N}}_{0}$ is
connected.

If we choose a marking
$m_{0}:{\mathrm{H}}^{0}(Z_{F_{0}}(\mathbb{C}),\mathbb{Z})\cong\mathbb{Z}^{5}$
lying over our base point $F_{0}\in X_{0}(\mathbb{C})$, we obtain an embedding
$\pi_{1}\left({\mathcal{N}}_{0},m_{0}\right)\hookrightarrow\pi_{1}(X_{0}(\mathbb{C}),F_{0})$,
whose composition with $\rho$ in (12) defines a homomorphism

$\mu:\pi_{1}({\mathcal{N}}_{0},m_{0})\to P\Gamma.$ (15)

Define $\theta=\zeta-\zeta^{-1}$ and consider the $3$-dimensional
${\mathbb{F}}_{5}$ vector space $\Lambda/\theta\Lambda$ and the quadratic
space

$W\coloneqq\left(\Lambda/\theta\Lambda,q\right),$

where $q$ is the quadratic form obtained by reducing ${\mathfrak{h}}$ modulo
$\theta\Lambda$. Define two groups $\Gamma_{\theta}$ and $P\Gamma_{\theta}$ as
follows:

$\Gamma_{\theta}=\textnormal{Ker}\left(\Gamma\to\textnormal{Aut}(W)\right),\quad
P\Gamma_{\theta}=\textnormal{Ker}\left(P\Gamma\to
P\textnormal{Aut}(W)\right)\subset\text{PU}(2,1).$

Remark that the composition ${\mathcal{N}}_{0}\to X_{0}(\mathbb{C})\to
X_{s}(\mathbb{C})$ admits an essentially unique completion
${\mathcal{N}}_{s}\to X_{s}(\mathbb{C})$, see [Fox57] or [DM86, §8]. Here
${\mathcal{N}}_{s}$ a manifold and ${\mathcal{M}}_{s}\to X_{s}(\mathbb{C})$ is
a ramified covering space.

######  Proposition 5.7.

The image of $\mu$ in (15) is the group $P\Gamma_{\theta}$, and the induced
homomorphism
$\pi_{1}(X_{0}(\mathbb{C}),F_{0})/\pi_{1}\left({\mathcal{N}}_{0},m_{0}\right)={\mathfrak{S}}_{5}\to
P\Gamma/P\Gamma_{\theta}$ is an isomorphism. In other words, we obtain the
following commutative diagram with exact rows:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}({\mathcal{N}}_{0},m_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{\pi_{1}(X_{0}(\mathbb{C}),F_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\;\;\;\;\;\;\tau}$$\textstyle{{\mathfrak{S}}_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\sim}$$\scriptstyle{\gamma}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\Gamma_{\theta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\textnormal{Aut}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
(16)

###### Proof.

Consider the quotient

$Q=\mathbb{G}(\mathbb{C})\setminus{\mathcal{N}}_{0}=\textnormal{PGL}_{2}(\mathbb{C})\setminus
P_{0}(\mathbb{C}),$

where $P_{0}~\subset(\mathbb{P}^{1}_{\mathbb{R}})^{5}$ is the subvariety of
distinct five-tupes, see Section 5. Let $0\in Q$ be the image of
$m_{0}\in{\mathcal{N}}_{0}$. In [DM86], Deligne and Mostow define a hermitian
space bundle $B_{Q}\to Q$ over $Q$ whose fiber over $0\in Q$ is
$\mathbb{C}H^{2}$. Consequently, writing
$V_{1}=\Lambda\otimes_{\mathcal{O}_{K},\tau_{1}}\mathbb{C}$, this gives a
monodromy representation

$\pi_{1}(Q,0)\to\textnormal{PU}(V_{1},{\mathfrak{h}}^{\tau_{1}})\cong\textnormal{PU}(2,1)$

whose image we denote by $\Gamma_{\text{DM}}$. Kondō has shown that in fact,
$\Gamma_{\text{DM}}=P\Gamma_{\theta}$ [Kon07, Theorem 7.1]. Since
${\mathcal{N}}_{0}\to Q$ is a covering space (the action of
$\mathbb{G}(\mathbb{C})$ on ${\mathcal{N}}_{0}$ being free) we have an
embedding $\pi_{1}({\mathcal{N}}_{0},m_{0})\hookrightarrow\pi_{1}(Q,0)$ whose
composition with $\pi_{1}(Q,0)\to\textnormal{PU}(2,1)$ is the map
$\mu:\pi_{1}({\mathcal{N}}_{0},m_{0})\to P\Gamma\subset\textnormal{PU}(2,1)$.

To prove that the image of $\mu$ is $P\Gamma_{\theta}$, it suffices to give a
section of the map ${\mathcal{N}}_{0}\to Q$. Indeed, such a section induces a
retraction of $\pi_{1}({\mathcal{N}}_{0},m_{0})\hookrightarrow\pi_{1}(Q,0)$,
so that the images of these two groups in $\textnormal{PU}(2,1)$ are the same.

To define such a section, observe that if
$\Delta\subset\mathbb{P}^{1}(\mathbb{C})^{5}$ is the union of all hyperplanes
$\\{x_{i}=x_{j}\\}\subset\mathbb{P}^{1}(\mathbb{C})^{5}$ for $i\neq j$, then

$\displaystyle Q=\textnormal{PGL}_{2}(\mathbb{C})\setminus P_{0}(\mathbb{C})$
$\displaystyle=\textnormal{PGL}_{2}(\mathbb{C})\setminus\left(\mathbb{P}^{1}(\mathbb{C})^{5}-\Delta\right)$
$\displaystyle\cong\\{(x_{4},x_{5})\in\mathbb{C}^{2}:x_{i}\neq 0,1\textnormal{
and }x_{1}\neq x_{2}\\}.$

The section $Q\to{\mathcal{N}}_{0}$ may then be defined by sending
$(x_{4},x_{5})$ to the binary quintic

$F(x,y)=x(x-y)y(x-x_{4}\cdot y)(x-x_{5}\cdot y)\in X_{0}(\mathbb{C}),$

marked by the labelling of its roots
$\\{[0:1],[1:1],[1:0],[x_{4}:1],[x_{5}:1]\\}$.

It remains to prove that the homomorphism $\gamma:{\mathfrak{S}}_{5}\to
P\Gamma/P\Gamma_{\theta}$ appearing on the right in (16) is an isomorphism. We
use Theorem 5.34, proven by Shimura in [Shi64], which says that

$(\Lambda,{\mathfrak{h}})\cong\left(\mathcal{O}_{K}^{3},\textnormal{diag}\left(1,1,\frac{1-\sqrt{5}}{2}\right)\right).$

It follows that

$P\Gamma/P\Gamma_{\theta}=P\textnormal{Aut}(W)\cong\textnormal{PO}_{3}({\mathbb{F}}_{5})\cong{\mathfrak{S}}_{5}.$

Next, consider the manifold ${\mathcal{N}}_{s}$. Remark that
${\mathfrak{S}}_{5}$ embeds into
$\textnormal{Aut}(\mathbb{G}(\mathbb{C})\setminus{\mathcal{N}}_{s})$.
Moreover, there is a natural isomorpism

$p\colon\mathbb{G}(\mathbb{C})\setminus{\mathcal{N}}_{s}\cong
P\Gamma_{\theta}\setminus\mathbb{C}H^{2},\quad\quad{\textnormal{see~\cite[cite]{[\@@bibref{}{DeligneMostow,
kondo5points}{}{}]}.}}$

(See also (18).) The two compositions

${\mathfrak{S}}_{5}\subset\textnormal{Aut}(\mathbb{G}(\mathbb{C})\setminus{\mathcal{N}}_{s})\cong\textnormal{Aut}(P\Gamma_{\theta}\setminus\mathbb{C}H^{2})\;{\textnormal{
and }}\;{\mathfrak{S}}_{5}\to
P\Gamma/P\Gamma_{\theta}\subset\textnormal{Aut}(P\Gamma_{\theta}\setminus\mathbb{C}H^{2})$

agree, because of the equivariance of $p$ with respect to $\gamma$. Thus,
$\gamma$ is injective. ∎

######  Corollary 5.8.

The monodromy representation $\rho$ in (12) is surjective. $\hfill\qed$

#####  4 Framed binary quintics

By a framing of a point $F\in X_{0}(\mathbb{C})$ we mean a projective
equivalence class $[f]$, where

$f\colon\mathbb{V}_{F}={\mathrm{H}}^{1}(C_{F}(\mathbb{C}),\mathbb{Z})\to\Lambda$

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 $\mu_{K}$. Let
${\mathcal{F}}_{0}$ be the collection of all framings of all points $x\in
X_{0}(\mathbb{C})$. The set ${\mathcal{F}}_{0}$ is naturally a complex
manifold, by arguments similar to those in [ACT02a]. Note that Corollary 5.8
implies that ${\mathcal{F}}_{0}$ is connected, hence

$\displaystyle{\mathcal{F}}_{0}\to X_{0}(\mathbb{C})$ (17)

is a covering, with Galois group $P\Gamma$.

######  Lemma 5.9.

The spaces $P\Gamma_{\theta}\setminus{\mathcal{F}}_{0}$ and
${\mathcal{N}}_{0}$ are isomorphic as covering spaces of $X_{0}(\mathbb{C})$.
In particular, there is a covering map ${\mathcal{F}}_{0}\to{\mathcal{N}}_{0}$
with Galois group $P\Gamma_{\theta}$.

###### Proof.

We have $P\Gamma/P\Gamma_{\theta}\cong{\mathfrak{S}}_{5}$ as quotients of
$P\Gamma$, see Proposition 5.7. ∎

######  Lemma 5.10.

$\Delta\coloneqq X_{s}(\mathbb{C})-X_{0}(\mathbb{C})$ is an irreducible normal
crossings divisor of $X_{s}(\mathbb{C})$.

###### Proof.

The proof is similar to the proof of Proposition 6.7 in [Bea09]. ∎

######  Lemma 5.11.

The local monodromy transformations of ${\mathcal{F}}_{0}\to
X_{0}(\mathbb{C})$ around every $x\in\Delta$ are of finite order. More
precisely, if $x\in\Delta$ lies on the intersection of $k$ local components of
$\Delta$, then the local monodromy group around $x$ is isomorphic to
$(\mathbb{Z}/10)^{k}$.

###### Proof.

See [DM86, Proposition 9.2] or [CT99, Proposition 6.1] for the generic case,
i.e. when a quintic $Z=\\{F=0\\}\subset\mathbb{P}^{1}_{\mathbb{C}}$ aquires
only one node. In this case, the local equation of the singularity is
$x^{2}=0$, hence the curve $C_{F}$ acquires a singularity of the form
$y^{5}+x^{2}=0$. If the quintic acquires two nodes, then $C_{F}$ acquires two
such singularities; the vanishing cohomology splits as an orthogonal direct
sum, hence the local monodromy transformations commute. ∎

In the following corollary, we let
$D=\left\\{z\in\mathbb{C}\mid\left|z\right|<1\right\\}$ denote the open unit
disc, and $D^{\ast}=D-\\{0\\}$ the punctured open unit disc.

######  Corollary 5.12.

There is an essentially unique branched cover $\pi:{\mathcal{F}}_{s}\to
X_{s}(\mathbb{C})$, with ${\mathcal{F}}_{s}$ a complex manifold, such that for
any $x\in\Delta$, any open $x\in U\subset X_{s}(\mathbb{C})$ with $U\cong
D^{k}\times D^{6-k}$ and $U\cap X_{0}(\mathbb{C})\cong(D^{\ast})^{k}\times
D^{6-k}$, and any connected component $U^{\prime}$ of
$\pi^{-1}(U)\subset{\mathcal{F}}_{s}$, there is an isomorphism
$U^{\prime}\cong D^{k}\times D^{6-k}$ such that the composition

$D^{k}\times D^{6-k}\cong U^{\prime}\to U\cong D^{6}\;\text{ is the map
}\;(z_{1},\dotsc,z_{6})\mapsto(z_{1}^{r_{1}},\dotsc,z_{k}^{r_{k}},z_{k+1},\dotsc,z_{6}).$

###### Proof.

See [Bea09, Lemma 7.2]. See also [Fox57] and [DM86, Section 8]. ∎

The group $\mathbb{G}(\mathbb{C})=\textnormal{GL}_{2}(\mathbb{C})/D$ (see
(11)) acts on ${\mathcal{F}}_{0}$ over its action on $X_{0}$. Explicitly, if
$g\in\mathbb{G}(\mathbb{C})$ and if $([\phi],\phi:\mathbb{V}_{F}\cong\Lambda)$
is a framing of $F\in X_{0}(\mathbb{C})$, then

$\left([\phi\circ g^{\ast}],\phi\circ g^{\ast}:\mathbb{V}_{g\cdot
F}\to\Lambda\right)$

is a framing of $g\cdot F\in X_{0}(\mathbb{C})$. 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 $\mathbb{G}(\mathbb{C})$ on ${\mathcal{F}}_{0}$
commute. By functoriality of the Fox completion, the action of
$\mathbb{G}(\mathbb{C})$ on ${\mathcal{F}}_{0}$ extends to an action of
$\mathbb{G}(\mathbb{C})$ on ${\mathcal{F}}_{s}$.

######  Lemma 5.13.

The group $\mathbb{G}(\mathbb{C})=\textnormal{GL}_{2}(\mathbb{C})/D$ acts
freely on ${\mathcal{F}}_{s}$.

###### Proof.

The functoriality of the Fox completion gives an action of
$\mathbb{G}(\mathbb{C})$ on ${\mathcal{N}}_{s}$ such that, by Lemma 5.9, there
is a $\mathbb{G}(\mathbb{C})$-equivariant commutative diagram

---
$\textstyle{P\Gamma_{\theta}\setminus{\mathcal{F}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{{\mathcal{N}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{s}(\mathbb{C}).}$

In particular, it suffices to show that $\mathbb{G}(\mathbb{C})$ acts freely
on ${\mathcal{N}}_{s}$. Note that ${\mathcal{N}}_{0}$ admits a natural
$\mathbb{G}_{m}$-covering map ${\mathcal{N}}_{0}\to P_{0}(\mathbb{C})$ where
$P_{0}(\mathbb{C})\subset\mathbb{P}^{1}(\mathbb{C})^{5}$ is the space of
distinct ordered five-tuples in $\mathbb{P}^{1}(\mathbb{C})$ introduced in
Section 1. Consequently, there is a $\mathbb{G}_{m}$-quotient map
${\mathcal{N}}_{s}\to P_{s}(\mathbb{C})$, where $P_{s}(\mathbb{C})$ is the
space of stable ordered five-tuples, and this map is equivariant for the
homomorphism
$\textnormal{GL}_{2}(\mathbb{C})\to\textnormal{PGL}_{2}(\mathbb{C})$.

Let $g\in\textnormal{GL}_{2}(\mathbb{C})$ and $x\in{\mathcal{N}}_{s}$ such
that $gx=x$. It is clear that $\textnormal{PGL}_{2}(\mathbb{C})$ acts freely
on $P_{s}(\mathbb{C})$. Therefore, $g=\lambda\in\mathbb{C}^{\ast}$. Let $F\in
X_{s}(\mathbb{C})$ be the image of $x\in{\mathcal{N}}_{s}$; then

$gF(x,y)=F(g^{-1}(x,y))=F(\lambda^{-1}x,\lambda^{-1}y)=\lambda^{-5}F(x,y).$

The equality $gF=F$ implies that $\lambda^{5}=1\in\mathbb{C}$, from which we
conclude that $\lambda\in\langle\zeta\rangle$. Therefore,
$[g]=[\textnormal{id}]\in\mathbb{G}(\mathbb{C})=\textnormal{GL}_{2}(\mathbb{C})/D$.
∎

#####  5 Complex uniformization

Consider the hermitian space
$V_{1}=\Lambda\otimes_{\mathcal{O}_{K},\tau_{1}}\mathbb{C}$; define
$\mathbb{C}H^{2}$ to be the space of negative lines in $V_{1}$. Using
Proposition 4.45 we see that the abelian scheme $J\to X_{0}$ induces a
$\mathbb{G}(\mathbb{C})$-equivariant morphism
${\mathcal{P}}:{\mathcal{F}}_{0}\to\mathbb{C}H^{2}$. Explicitly, if
$(F,[f])\in{\mathcal{F}}_{0}$ is the framing
$[f:{\mathrm{H}}^{1}(C_{F}(\mathbb{C}),\mathbb{Z})\xrightarrow{\sim}\Lambda]$
of the binary quintic $F\in X_{0}(\mathbb{C})$, and $A_{F}$ is the Jacobian of
the curve $C_{F}$, then

${\mathcal{P}}:{\mathcal{F}}_{0}\to\mathbb{C}H^{2},\quad{\mathcal{P}}\left(F,[f]\right)=f\left(H^{0,-1}(A_{F})_{\tau_{1}}\right)=f\left(H^{1,0}(C_{F})_{\zeta^{3}}\right)\in\mathbb{C}H^{2}.$
(18)

The map ${\mathcal{P}}$ is holomorphic, and descends to a morphism of complex
analytic spaces

${\mathcal{M}}_{0}(\mathbb{C})=\mathbb{G}(\mathbb{C})\setminus
X_{0}(\mathbb{C})\to P\Gamma\setminus\mathbb{C}H^{2}.$ (19)

By Riemann extension, (18) extends to a $\mathbb{G}(\mathbb{C})$-equivariant
holomorphic map

$\displaystyle\overline{{\mathcal{P}}}:{\mathcal{F}}_{s}\to\mathbb{C}H^{2}.$
(20)

######  Theorem 5.14 (Deligne–Mostow).

The period map (20) induces an isomorphism of complex manifolds

$\displaystyle{\mathcal{M}}_{s}^{f}(\mathbb{C})\coloneqq\mathbb{G}(\mathbb{C})\setminus{\mathcal{F}}_{s}\cong\mathbb{C}H^{2}.$
(21)

Taking $P\Gamma$-quotients gives an isomorphism of complex analytic spaces

${\mathcal{M}}_{s}(\mathbb{C})=\mathbb{G}(\mathbb{C})\setminus
X_{s}(\mathbb{C})\cong P\Gamma\setminus\mathbb{C}H^{2}.$ (22)

###### Proof.

In [DM86], Deligne and Mostow define $\widetilde{Q}\to Q$ to be the covering
space corresponding to the monodromy representation
$\pi_{1}(Q,0)\to\textnormal{PU}(2,1)$; since the image of this homomomorphism
is $P\Gamma_{\theta}$ (see the proof of Proposition 5.7), it follows that
$\mathbb{G}(\mathbb{C})\setminus{\mathcal{F}}_{0}\cong\widetilde{Q}$ as
covering spaces of $Q$. Consequently, if $\widetilde{Q}_{\textnormal{st}}$ is
the Fox completion of the spread

$\widetilde{Q}\to Q\to
Q_{\textnormal{st}}:=\mathbb{G}(\mathbb{C})\setminus{\mathcal{N}}_{s}=\textnormal{PGL}_{2}(\mathbb{C})\setminus
P_{s}(\mathbb{C}),$

then there is an isomorphism
$\mathbb{G}(\mathbb{C})\setminus{\mathcal{F}}_{s}\cong\widetilde{Q}_{\textnormal{st}}$
of branched covering spaces of $Q_{\textnormal{st}}$. We obtain commutative
diagrams, where the lower right morphism uses (16):

$\textstyle{\mathbb{G}(\mathbb{C})\setminus{\mathcal{F}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim\;\;\;\;}$$\textstyle{\widetilde{Q}_{\textnormal{st}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}H^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{G}(\mathbb{C})\setminus{\mathcal{N}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim\;\;\;\;}$$\textstyle{Q_{\textnormal{st}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\Gamma_{\theta}\setminus\mathbb{C}H^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{G}(\mathbb{C})\setminus
X_{s}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim\;\;\;\;}$$\textstyle{Q_{\textnormal{st}}/{\mathfrak{S}}_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\Gamma\setminus\mathbb{C}H^{2}.}$

The map $\widetilde{Q}_{\textnormal{st}}\to\mathbb{C}H^{2}$ is an isomorphism
by [DM86, (3.11)]. Therefore, we are done if the composition
$\mathbb{G}(\mathbb{C})\setminus{\mathcal{F}}_{0}\to\widetilde{Q}\to\mathbb{C}H^{2}$
agrees with the period map ${\mathcal{P}}$ of equation (18). This follows from
[DM86, (2.23) and (12.9)]. ∎

######  Proposition 5.15.

The isomorphism (22) induces an isomorphism of complex analytic spaces

${\mathcal{M}}_{0}(\mathbb{C})=\mathbb{G}(\mathbb{C})\setminus
X_{0}(\mathbb{C})\cong
P\Gamma\setminus\left(\mathbb{C}H^{2}-{\mathscr{H}}\right).$ (23)

###### Proof.

We have
$\overline{{\mathcal{P}}}({\mathcal{F}}_{0})\subset\mathbb{C}H^{2}-{\mathscr{H}}$
by Proposition 4.48, because the Jacobian of a smooth curve cannot contain an
abelian subvariety whose induced polarization is principal. Therefore, we have
$\overline{{\mathcal{P}}}^{-1}({\mathscr{H}})\subset{\mathcal{F}}_{s}-{\mathcal{F}}_{0}$.
Since ${\mathcal{F}}_{s}$ is irreducible (it is smooth by Corollary 5.12 and
connected by Corollary 5.8), the analytic space
$\overline{{\mathcal{P}}}^{-1}({\mathscr{H}})$ is a divisor. Since
${\mathcal{F}}_{s}-{\mathcal{F}}_{0}$ is also a divisor by Corollary 5.12, we
have

$\overline{{\mathcal{P}}}^{-1}({\mathscr{H}})={\mathcal{F}}_{s}-{\mathcal{F}}_{0}$

and we are done.

Alternatively, let $H_{0,5}$ be the moduli space of degree $5$ covers of
$\mathbb{P}^{1}$ ramified along five distinct marked points [HM98, §2.G]. The
period map

$H_{0,5}(\mathbb{C})\to P\Gamma\setminus\mathbb{C}H^{2},$

that sends the moduli point of a curve $C\to\mathbb{P}^{1}$ to the moduli
point of the $\mathbb{Z}[\zeta]$-linear Jacobian $J(C)$, extends to the stable
compactification $\overline{H}_{0,5}(\mathbb{C})\supset H_{0,5}(\mathbb{C})$
because the curves in the limit are of compact type. Since the divisor
${\mathscr{H}}\subset\mathbb{C}H^{2}$ parametrizes abelian varieties that are
products of lower dimensional ones by Proposition 4.48, the image of the
boundary is exactly $P\Gamma\setminus{\mathscr{H}}$. ∎

####  3 Moduli of real binary quintics

Having understood the period map for complex binary quintics, we turn to the
period map of real binary quintics in this Section 3.

#####  1 The period map for stable real binary quintics

Define $\kappa$ as the anti-holomorphic involution

$\kappa\colon X_{0}(\mathbb{C})\to X_{0}(\mathbb{C}),\quad
F(x,y)=\sum_{i+j=5}a_{ij}x^{i}y^{j}\mapsto\overline{F(x,y)}=\sum_{i+j=5}\overline{a_{ij}}x^{i}y^{j}.$

Let ${\mathscr{A}}$ be the set of anti-unitary involutions
$\alpha:\Lambda\to\Lambda$, and
$P{\mathscr{A}}=\mu_{K}\setminus{\mathscr{A}}$, see Section 4. For each
$\alpha\in P{\mathscr{A}}$, there is a natural anti-holomorphic involution
$\alpha~\colon{\mathcal{F}}_{0}\to{\mathcal{F}}_{0}$ such that the following
diagram commutes:

$\textstyle{{\mathcal{F}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{{\mathcal{F}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{X_{0}(\mathbb{C}).}$

It is defined as follows. Consider a framed binary quintic
$(F,[f])\in{\mathcal{F}}_{0}$, where $f:\mathbb{V}_{F}\to\Lambda$ is an
$\mathcal{O}_{K}$-linear isometry. Let $C_{F}\to\mathbb{P}^{1}_{\mathbb{C}}$
be the quintic cover defined by a smooth binary quintic $F\in
X_{0}(\mathbb{C})$. Complex conjugation
${\textnormal{conj}}\colon\mathbb{P}^{2}(\mathbb{C})\to\mathbb{P}^{2}(\mathbb{C})$
induces an anti-holomorphic map

$\sigma_{F}:C_{F}(\mathbb{C})\to C_{\kappa(F)}(\mathbb{C}).$

Consider the pull-back
$\sigma_{F}^{\ast}:{\mathbb{V}}_{\kappa(F)}\to{\mathbb{V}}_{F}$ of $\sigma$.
The composition

$\mathbb{V}_{\kappa(F)}\xrightarrow{\sigma_{F}^{\ast}}\mathbb{V}_{F}\xrightarrow{f}\Lambda\xrightarrow{\alpha}\Lambda$

induces a framing of $\kappa(F)\in X_{0}(\mathbb{C})$, and we define

$\alpha(F,[f])\coloneqq\left(\kappa(F),[\alpha\circ
f\circ\sigma_{F}^{\ast}]\right)\in{\mathcal{F}}_{0}.$

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 ${\mathcal{F}}_{0}\to X_{0}(\mathbb{C})$ introduced
in (3), and define

$\displaystyle{\mathcal{F}}_{0}(\mathbb{R})=\bigsqcup_{\alpha\in
P{\mathscr{A}}}{\mathcal{F}}_{0}^{\alpha}\subset{\mathcal{F}}_{0}$ (24)

as the preimage of $X_{0}(\mathbb{R})$ in the space ${\mathcal{F}}_{0}$. To
see why the union on the left hand side of (24) is disjoint, observe that

${\mathcal{F}}_{0}^{\alpha}=\left\\{(F,[f])\in{\mathcal{F}}_{0}:\kappa(F)=F\textnormal{
and }[f\circ\sigma^{\ast}_{F}\circ f^{-1}]=\alpha\right\\}.$

Thus, for $\alpha,\beta\in P{\mathscr{A}}$ and
$(F,[f])\in{\mathcal{F}}_{0}^{\alpha}\cap{\mathcal{F}}_{0}^{\beta}$, we have
$\alpha=[f\circ\sigma\circ f^{-1}]=\beta$.

######  Lemma 5.16.

The anti-holomorphic involution
$\alpha\colon{\mathcal{F}}_{0}\to{\mathcal{F}}_{0}$ defined by $\alpha\in
P{\mathscr{A}}$ makes the period map
${\mathcal{P}}\colon{\mathcal{F}}_{0}\to\mathbb{C}H^{2}$ equivariant for the
$\mathbb{G}(\mathbb{C})$-actions on both sides.

###### Proof.

Indeed, if ${\textnormal{conj}}\colon\mathbb{C}\to\mathbb{C}$ is complex
conjugation, then for any $F\in X_{0}(\mathbb{C})$, the induced map

$\sigma^{\ast}_{F}\otimes{\textnormal{conj}}\colon{\mathbb{V}}_{\kappa(F)}\otimes_{\mathbb{Z}}\mathbb{C}\to{\mathbb{V}}_{F}\otimes_{\mathbb{Z}}\mathbb{C}$

is anti-linear, preserves the Hodge decompositions [Sil89, Chapter I, Lemma
(2.4)] as well as the eigenspace decompositions. ∎

We obtain a real period map

(27)

Define $\mathbb{G}(\mathbb{R})=\textnormal{GL}_{2}(\mathbb{R})$. The map (27)
is constant on $\mathbb{G}(\mathbb{R})$-orbits, since the same is true for the
complex period map ${\mathcal{P}}\colon{\mathcal{F}}_{0}\to\mathbb{C}H^{2}$.

By abuse of notation, we for $\alpha\in P{\mathscr{A}}$, we write
${\mathbb{R}}H^{2}_{\alpha}-{\mathscr{H}}={\mathbb{R}}H^{2}_{\alpha}-\left({\mathscr{H}}\cap{\mathbb{R}}H^{2}_{\alpha}\right)$.

######  Proposition 5.17.

The real period map (27) descends to a $P\Gamma$-equivariant diffeomorphism

$\displaystyle{\mathcal{M}}_{0}(\mathbb{R})^{f}\coloneqq\mathbb{G}(\mathbb{R})\setminus{\mathcal{F}}_{0}(\mathbb{R})\cong\coprod_{\alpha\in
P{\mathscr{A}}}\mathbb{R}H^{2}_{\alpha}-{\mathscr{H}}.$ (28)

By $P\Gamma$-equivariance, the map (28) induces an isomorphism of real-
analytic orbifolds

${\mathcal{P}}_{\mathbb{R}}\colon{\mathcal{M}}_{0}(\mathbb{R})=\mathbb{G}(\mathbb{R})\setminus
X_{0}(\mathbb{R})\cong\coprod_{\alpha\in
C{\mathscr{A}}}P\Gamma_{\alpha}\setminus\left(\mathbb{R}H^{2}_{\alpha}-{\mathscr{H}}\right).$
(29)

###### Proof.

This follows from [ACT10, proof of Theorem 3.3]. It is crucial that the
actions of $G$ and $P\Gamma$ on ${\mathcal{F}}_{0}$ commute and are free,
which is the case, see Corollary 5.13. ∎

#####  2 The period map for smooth real binary quintics

Our next goal will be to prove the real analogue of the isomorphisms (21) and
(21) in Theorem 5.14. We need a lemma, a definition, and then two more lemmas.

Consider the CM-type $\Psi=\left\\{\tau_{1},\tau_{2}\right\\}$ defined in (7),
the hermitian $\mathcal{O}_{K}$-lattice $(\Lambda,{\mathfrak{h}})$ defined in
(9), and the sets (c.f. Definition 4.11):

${\mathcal{H}}=\left\\{H_{r}\subset{\mathbb{C}}H^{2}\mid
r\in{\mathscr{R}}\right\\},~\quad{\textnormal{
and~}}\quad{\mathscr{H}}=\cup_{H\in{\mathcal{H}}}H\subset{\mathbb{C}}H^{2}.$

Here, ${\mathscr{R}}\subset\Lambda$ is the set of short roots (see Section 1).

######  Lemma 5.18.

The hyperplane arrangement ${\mathscr{H}}~\subset{\mathbb{C}}H^{2}$ satisfies
Condition 4.9, that is: any two distinct $H_{1},H_{2}\in{\mathcal{H}}$ either
meet orthogonally, or not at all.

###### Proof.

Condition 4.49.1 holds because $K$ does not contain proper CM-subfields. By
Lemma 4.52, we have that Condition 4.49.2 is satisfied. By equation (10),
Condition 4.49.3 holds. By Theorem 4.50, we obtain the desired result. ∎

######  Definition 5.19.

  1. 1.

For $k=1,2$, define
$\Delta_{k}\subset\Delta=X_{s}(\mathbb{C})-X_{0}(\mathbb{C})$ to be the locus
of stable binary quintics with exactly $k$ nodes. Define
$\widetilde{\Delta}={\mathcal{F}}_{s}-{\mathcal{F}}_{0}$, and let
$\widetilde{\Delta}_{k}\subset\widetilde{\Delta}$ be the inverse image of
$\Delta_{k}$ in $\widetilde{\Delta}$ under the map
$\widetilde{\Delta}\to\Delta$.

  2. 2.

For $k=1,2$, define ${\mathscr{H}}_{k}\subset{\mathscr{H}}$ as the set
${\mathscr{H}}_{k}=\left\\{x\in{\mathbb{C}}H^{2}\mid\left|{\mathcal{H}}(x)\right|=k\right\\}$.
Thus, this is the locus of points in ${\mathscr{H}}$ where exactly $k$
hyperplanes meet.

######  Lemma 5.20.

  1. 1.

The period map $\overline{{\mathcal{P}}}$ of (20) satisfies
$\overline{{\mathcal{P}}}(\widetilde{\Delta}_{k})\subset{\mathscr{H}}_{k}$.

  2. 2.

If $f\in\widetilde{\Delta}_{k}$,
$x=\overline{{\mathcal{P}}(f)}\in~{\mathbb{C}}H^{2}$, and
${\mathcal{H}}(x)=\left\\{H_{r_{1}},\dotsc,H_{r_{k}}\right\\}$ for
$r_{i}\in{\mathscr{R}}$, then $\overline{{\mathcal{P}}}$ induces a group
isomorphism $P\Gamma_{f}\cong G(x)$. ∎

The naturality of the Fox completion implies that for $\alpha\in
P{\mathscr{A}}$, the anti-holomorphic involution
$\alpha:{\mathcal{F}}_{0}\to{\mathcal{F}}_{0}$ extends to an anti-holomorphic
involution $\alpha:{\mathcal{F}}_{s}\to{\mathcal{F}}_{s}$.

######  Lemma 5.21.

For every $\alpha\in P{\mathscr{A}}$, the restriction of
$\overline{{\mathcal{P}}}:{\mathcal{F}}_{s}\to\mathbb{C}H^{2}$ to
${\mathcal{F}}_{s}^{\alpha}$ defines a diffeomorphism
$\mathbb{G}(\mathbb{R})\setminus{\mathcal{F}}_{s}^{\alpha}\cong\mathbb{R}H^{2}_{\alpha}$.

###### Proof.

See [ACT10, Lemma 11.3]. It is essential that $G$ acts freely on
${\mathcal{F}}_{s}$, which holds by Corollary 5.13. ∎

We arrive at the main theorem of Section 3. Define

${\mathcal{F}}_{s}(\mathbb{R})=\bigcup_{\alpha\in
P{\mathscr{A}}}{\mathcal{F}}_{s}^{\alpha}=\pi^{-1}\left(X_{s}(\mathbb{R})\right).$

This is not a manifold because of the ramification of
$\pi:{\mathcal{F}}_{s}\to X_{s}(\mathbb{C})$, but a union of embedded
submanifolds.

######  Theorem 5.22.

There is a smooth map

$\displaystyle\overline{{\mathcal{P}}}_{\mathbb{R}}:\coprod_{\alpha\in
P{\mathscr{A}}}{\mathcal{F}}_{s}^{\alpha}\to\coprod_{\alpha\in
P{\mathscr{A}}}\mathbb{R}H^{2}_{\alpha}=\widetilde{Y}$ (30)

that extends the real period map (27). The map (30) induces the following
commutative diagram of topological spaces, in which
${\mathscr{P}}_{\mathbb{R}}$ and ${\mathfrak{P}}_{\mathbb{R}}$ are
homeomorphisms:

$\textstyle{\coprod_{\alpha\in
P{\mathscr{A}}}{\mathcal{F}}_{s}^{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{{\mathcal{P}}}_{\mathbb{R}}}$$\textstyle{\widetilde{Y}=\coprod_{\alpha\in
P{\mathscr{A}}}\mathbb{R}H^{2}_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}_{s}(\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{{\mathcal{P}}}_{\mathbb{R}}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{M}}_{s}(\mathbb{R})^{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{G}(\mathbb{R})\setminus{\mathcal{F}}_{s}(\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathscr{P}}_{\mathbb{R}}}$$\scriptstyle{\sim}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{M}}_{s}(\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{G}(\mathbb{R})\setminus
X_{s}(\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{{\mathfrak{P}}}_{\mathbb{R}}}$$\scriptstyle{\sim}$$\textstyle{P\Gamma\setminus
Y.}$

###### Proof.

The existence of $\overline{{\mathcal{P}}}_{\mathbb{R}}$ follows from the
compatibility with the involutions $\alpha\in P{\mathscr{A}}$. We first show
that the composition

$\coprod_{\alpha\in
P{\mathscr{A}}}{\mathcal{F}}_{s}^{\alpha}\xrightarrow{\overline{{\mathcal{P}}}_{\mathbb{R}}}\widetilde{Y}\xrightarrow{p}Y$

factors through ${\mathcal{F}}_{s}(\mathbb{R})$. Now $f_{\alpha}$ and
$g_{\beta}\in\coprod_{\alpha\in P{\mathscr{A}}}{\mathcal{F}}_{s}^{\alpha}$
have the same image in ${\mathcal{F}}_{s}(\mathbb{R})$ if and only if
$f=g\in{\mathcal{F}}_{s}^{\alpha}\cap{\mathcal{F}}_{s}^{\beta}$, in which case

$x\coloneqq\overline{{\mathcal{P}}}(f)=\overline{{\mathcal{P}}}(g)\in\mathbb{R}H^{2}_{\alpha}\cap{\mathbb{R}}H^{2}_{\beta},$

so we need to show is that $x_{\alpha}\sim x_{\beta}\in\widetilde{Y}$. For
this, note that $\alpha\beta\in P\Gamma_{f}\cong(\mathbb{Z}/10)^{k}$, and
$\overline{{\mathcal{P}}}$ induces an isomorphism $P\Gamma_{f}\cong G(x)$ by
Lemma 5.20. Hence $\alpha\beta\in G(x)$ so that indeed, $x_{\alpha}\sim
x_{\beta}$.

Let us prove the $\mathbb{G}(\mathbb{R})$-equivariance of
$\overline{{\mathcal{P}}}_{\mathbb{R}}$. Suppose that

$f\in{\mathcal{F}}_{s}^{\alpha},g\in{\mathcal{F}}_{s}^{\beta}\quad\mid\quad
a\cdot f=g\in{\mathcal{F}}_{s}(\mathbb{R})~\quad{\textnormal{ for some
}}~\quad a\in\mathbb{G}(\mathbb{R}).$

Then
$x\coloneqq\overline{{\mathcal{P}}}(f)=\overline{{\mathcal{P}}}(g)\in\mathbb{C}H^{2}$,
so we need to show that $\alpha\beta\in G(x)$. The actions of
$\mathbb{G}(\mathbb{C})$ and $P\Gamma$ on $\mathbb{C}H^{2}$ commute, and the
same holds for the actions of $\mathbb{G}(\mathbb{R})$ and $P\Gamma^{\prime}$
on ${\mathcal{F}}_{s}^{\mathbb{R}}$. It follows that

$\alpha(g)=\alpha(a\cdot f)=a\cdot\alpha(f)=a\cdot f=g,$

hence $g\in{\mathcal{F}}_{s}^{\alpha}\cap{\mathcal{F}}_{s}^{\beta}$. This
implies in turn that $\alpha\beta(g)=g$, hence $\alpha\beta\in
P\Gamma_{g}\cong G(x)$, so that indeed, $x_{\alpha}\sim x_{\beta}$.

To prove that ${\mathscr{P}}_{\mathbb{R}}$ is injective, let again
$f_{\alpha},g_{\beta}\in\coprod_{\alpha\in
P{\mathscr{A}}}{\mathcal{F}}_{s}^{\alpha}$ and suppose that they have the same
image in $Y$. This implies that

$x\coloneqq\overline{{\mathcal{P}}}(f)=\overline{{\mathcal{P}}}(g)\in\mathbb{R}H^{2}_{\alpha}\cap{\mathbb{R}}H^{2}_{\beta},$

and that $\beta=\phi\circ\alpha$ for some $\phi\in G(x)$. We have $\phi\in
G(x)\cong P\Gamma_{f}$ (by Lemma 5.20) hence

$\beta(f)=\phi\left(\alpha(f)\right)=\phi(f)=f.$ (31)

Therefore $f,g\in{\mathcal{F}}_{s}^{\beta}$; since
$\overline{{\mathcal{P}}}(f)=\overline{{\mathcal{P}}}(g)$, it follows from
Lemma 5.21 that there exists $a\in\mathbb{G}(\mathbb{R})$ such that $a\cdot
f=g$. This proves injectivity of ${\mathscr{P}}_{\mathbb{R}}$, as desired.

The surjectivity of
${\mathscr{P}}_{\mathbb{R}}:\mathbb{G}(\mathbb{R})\setminus{\mathcal{F}}_{s}(\mathbb{R})\to
Y$ is straightforward, using the surjectivity of
$\overline{{\mathcal{P}}}_{\mathbb{R}}$, which follows from Lemma 5.21.

Finally, we claim that ${\mathscr{P}}_{\mathbb{R}}$ is open. Let
$U\subset\mathbb{G}(\mathbb{R})\setminus{\mathcal{F}}_{s}^{\mathbb{R}}$, and
write
$U={\mathscr{P}}_{\mathbb{R}}^{-1}{\mathscr{P}}_{\mathbb{R}}\left(U\right)$.
Let $V$ be the preimage of $U$ in $\coprod_{\alpha\in
P{\mathscr{A}}}{\mathcal{F}}_{s}^{\alpha}$. Then

$V=\overline{{\mathcal{P}}}_{\mathbb{R}}^{-1}\left(p^{-1}\left({\mathscr{P}}_{\mathbb{R}}(U)\right)\right)$

and hence

$\overline{{\mathcal{P}}}_{\mathbb{R}}\left(V\right)=p^{-1}\left({\mathscr{P}}_{\mathbb{R}}(U)\right).$

The map $\overline{{\mathcal{P}}}_{\mathbb{R}}$ is open, being the coproduct
of the maps ${\mathcal{F}}_{s}^{\alpha}\to\mathbb{R}H^{2}_{\alpha}$, which are
open since they have surjective differential at each point. Thus
${\mathscr{P}}_{\mathbb{R}}(U)$ is open in $Y$. ∎

######  Corollary 5.23.

There is a lattice $P\Gamma_{\mathbb{R}}\subset\textnormal{PO}(2,1)$, an
inclusion of orbifolds

$\displaystyle\coprod_{\alpha\in
C{\mathscr{A}}}P\Gamma_{\alpha}\setminus\left(\mathbb{R}H^{2}_{\alpha}-{\mathscr{H}}\right)\hookrightarrow
P\Gamma_{\mathbb{R}}\setminus\mathbb{R}H^{2},$ (32)

and a homeomorphism

${\mathcal{M}}_{s}(\mathbb{R})=\mathbb{G}(\mathbb{R})\setminus
X_{s}(\mathbb{R})\cong P\Gamma_{\mathbb{R}}\setminus\mathbb{R}H^{2}$ (33)

such that (33) restricts to (29) with respect to (32).

###### Proof.

This follows directly from Theorems 4.24 and 5.22. ∎

######  Remark 5.24.

The proof of Theorem 5.22 also shows that ${\mathcal{M}}_{s}(\mathbb{R})$ is
homeomorphic to the glued space $P\Gamma\setminus Y$ (see Definition 4.22) if
${\mathcal{M}}_{s}$ is the stack of cubic surfaces or of binary sextics. This
strategy to uniformize the real moduli space differs from the one used in
[ACT10, ACT06, ACT07], since we first glue together the real ball quotients,
and then prove that our real moduli space is homeomorphic to the result.

#####  3 Automorphism groups of stable real binary quintics

Before we can finish the proof of Theorem 5.2, we need to understand the
orbifold structure of ${\mathcal{M}}_{s}(\mathbb{R})$, and how this structure
differs from the orbifold structure of the glued space $P\Gamma\setminus Y$.
In the current Section 3 we start by analyzing the orbifold structure of
${\mathcal{M}}_{s}(\mathbb{R})$, by listing its stabilizer groups. There is a
canonical orbifold isomorphism
${\mathcal{M}}_{s}(\mathbb{R})=\mathbb{G}(\mathbb{R})\setminus
X_{s}(\mathbb{R})=(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R}).$ Therefore, to list
the automorphism groups of binary quintics is to list the elements
$x=[\alpha_{1},\dotsc,\alpha_{5}]\in(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})$
whose stabilizer $\textnormal{PGL}_{2}(\mathbb{R})_{x}$ is non-trivial, and
calculate $\textnormal{PGL}_{2}(\mathbb{R})_{x}$ in these cases. This will be
our next goal.

######  Proposition 5.25.

All stabilizer groups
$\textnormal{PGL}_{2}(\mathbb{R})_{x}\subset\textnormal{PGL}_{2}(\mathbb{R})$
for points $x\in(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})$ are among
$\mathbb{Z}/2,D_{3},D_{5}$. For $n\in\\{3,5\\}$, there is a unique
$\textnormal{PGL}_{2}(\mathbb{R})$-orbit in
$(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})$ of points $x$ with stabilizer
$D_{n}$.

###### Proof.

We have an injection $(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})\hookrightarrow
P_{s}/{\mathfrak{S}}_{5}$ which is equivariant for the embedding
$\textnormal{PGL}_{2}(\mathbb{R})\hookrightarrow\textnormal{PGL}_{2}(\mathbb{C})$.
In particular,
$\textnormal{PGL}_{2}(\mathbb{R})_{x}\subset\textnormal{PGL}_{2}(\mathbb{C})_{x}$
for every $x\in(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})$. The groups
$\textnormal{PGL}_{2}(\mathbb{C})_{x}$ for equivalence classes of distinct
points $x\in P_{0}/{\mathfrak{S}}_{5}$ are calculated in [WX17, Theorem 22],
and such a group is isomorphic to $\mathbb{Z}/2,D_{3},\mathbb{Z}/4$ or
$D_{5}$. None of these have subgroups isomorphic to
$D_{2}=\mathbb{Z}/2\rtimes\mathbb{Z}/2$ or
$D_{4}=\mathbb{Z}/2\rtimes\mathbb{Z}/4$. Define an involution

$\nu\coloneqq(z\mapsto 1/z)\in\textnormal{PGL}_{2}(\mathbb{R}).$

The proof of Proposition 5.25 will follow from the following Lemmas 5.26,
5.27, 5.28 and 5.29.

######  Lemma 5.26.

Let $\tau\in\textnormal{PGL}_{2}(\mathbb{R})$. Consider a subset
$S=\\{x,y,z\\}~\subset\mathbb{P}^{1}(\mathbb{C})$ stabilized by complex
conjugation, such that $\tau(x)=x$, $\tau(y)=z$ and $\tau(z)=y$. There is a
transformation $g\in\textnormal{PGL}_{2}(\mathbb{R})$ that maps $S$ to either
$\\{-1,0,\infty\\}$ or $\\{-1,i,-i\\}$, and that satisfies $g\tau
g^{-1}=\nu=(z\mapsto 1/z)\in\textnormal{PGL}_{2}(\mathbb{R})$. In particular,
$\tau^{2}=\textnormal{id}$.

###### Proof.

Indeed, two transformations $g,h\in\textnormal{PGL}_{2}(\mathbb{C})$ that
satisfy $g(x_{i})=h(x_{i})$ for three different points
$x_{1},x_{2},x_{3}\in\mathbb{P}^{1}(\mathbb{C})$ are necessarily equal. ∎

######  Lemma 5.27.

There is no $x\in(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})$ stabilized by some
$\phi\in\textnormal{PGL}_{2}(\mathbb{R})$ of order $4$.

###### Proof.

By [Bea10, Theorem 4.2], all subgroups
$G\subset\textnormal{PGL}_{2}(\mathbb{R})$ that are isomorphic to
$\mathbb{Z}/4$ are conjugate to each other. Since the transformation
$I:z\mapsto(z-1)/(z+1)$ is of order $4$, it gives a representative
$G_{I}=\langle I\rangle$ of this conjugacy class. Hence, assuming there exists
$x$ and $\phi$ as in the lemma, possibly after replacing $x$ by $gx$ for some
$g\in\textnormal{PGL}_{2}(\mathbb{R})$, we may and do assume that $\phi=I$. On
the other hand, it is easily shown that $I$ cannot fix any
$x\in(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})$. ∎

Define

$\rho\in\textnormal{PGL}_{2}(\mathbb{R}),\;\;\;\rho(z)=\frac{-1}{z+1}.$

######  Lemma 5.28.

Let $x=(x_{1},\dotsc,x_{5})\in(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})$. Suppose
$\phi(x)=x$ for an element $\phi\in\textnormal{PGL}_{2}(\mathbb{R})$ of order
$3$. There is a transformation $g\in\textnormal{PGL}_{2}(\mathbb{R})$ mapping
$x$ to $z=(-1,\infty,0,\omega,\omega^{2})$ with $\omega$ a primitive third
root of unity, and the stabilizer of $x$ to the subgroup of
$\textnormal{PGL}_{2}(\mathbb{R})$ generated by $\rho$ and $\nu$. In
particular, $\textnormal{PGL}_{2}(\mathbb{R})_{x}$ is isomorphic to $D_{3}$.

###### Proof.

It follows from Lemma 5.26 that there must be three 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 $\mathbb{R}$, they are all real. Let
$g\in\textnormal{PGL}_{2}(\mathbb{R})$ such that $g(x_{1})=-1$,
$g(x_{2})=\infty$ and $g(x_{3})=0$. Define $\kappa=g\phi g^{-1}$. Then
$\kappa^{3}=\textnormal{id}$, 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 $z=(-1,\infty,0,\alpha,\beta)$. Moreover, $\beta=\bar{\alpha}$ and
$\alpha^{2}+\alpha+1=0$. ∎

Recall that $\zeta_{5}=e^{2i\pi/5}\in\mathbb{P}^{1}(\mathbb{C})$ and define

$\lambda=\zeta_{5}+\zeta_{5}^{-1}\in\mathbb{R},\;\;\;\;\;\;\gamma(z)=\frac{(\lambda+1)z-1}{z+1}\in\textnormal{PGL}_{2}(\mathbb{R}).$

######  Lemma 5.29.

Let $x=(x_{1},\dotsc,x_{5})\in(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})$. Suppose
$x$ is stabilized by a subgroup of $\textnormal{PGL}_{2}(\mathbb{R})$ of order
$5$. There is a transformation $g\in\textnormal{PGL}_{2}(\mathbb{R})$ mapping
$x$ to $z=(0,-1,\infty,\lambda+1,\lambda)$ and identifying the stabilizer of
$x$ with the subgroup of $\textnormal{PGL}_{2}(\mathbb{R})$ generated by
$\gamma$ and $\nu$. In particular, the stabilizer
$\textnormal{PGL}_{2}(\mathbb{R})_{x}$ of $x$ is isomorphic to $D_{5}$.

###### Proof.

Let $\phi\in\textnormal{PGL}_{2}(\mathbb{R})_{x}$ be an element of order $5$.
Using Lemma 5.26 one shows that $x$ must be smooth, i.e. all $x_{i}$ are
distinct, and $x_{i}=\phi^{i-1}(x_{1})$. Since there is one real $x_{i}$ and
$\phi$ is defined over $\mathbb{R}$, all $x_{i}$ are real. Now note that
$z=(0,-1,\infty,\lambda+1,\lambda)$ is the orbit of $0$ under
$\gamma:z\mapsto((\lambda+1)z-1)/(z+1)$. The reflection $\nu:z\mapsto 1/z$
preserves $z$ as well: if $\zeta=\zeta_{5}$ then $\lambda=\zeta+\zeta^{-1}$
hence $\lambda+1=-(\zeta^{2}+\zeta^{-2})=-\lambda^{2}+2$, so that
$\lambda(\lambda+1)=1$. So we have $\textnormal{PGL}_{2}(\mathbb{R})_{z}\cong
D_{5}$. By [WX17, Theorem 22], the point $z$ with its stabilizer
$\textnormal{PGL}_{2}(\mathbb{R})_{z}$ must be equivalent under
$\textnormal{PGL}_{2}(\mathbb{C})$ to the point
$(1,\zeta,\zeta^{2},\zeta^{3},\zeta^{4})$ with its stabilizer $\langle
x\mapsto\zeta x,x\mapsto 1/x\rangle$. Thus, there exists
$g\in\textnormal{PGL}_{2}(\mathbb{C})$ 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
$g\textnormal{PGL}_{2}(\mathbb{R})_{x}g^{-1}=\textnormal{PGL}_{2}(\mathbb{R})_{z}$.
Since all $x_{i}$ and $z_{i}\in z$ are real, we see that
$\bar{g}(x_{i})=z_{i}$ for every $i$, hence $g$ and $\bar{g}$ coincide on more
than $2$ points, hence $g=\bar{g}\in\textnormal{PGL}_{2}(\mathbb{R})$. ∎

Proposition 5.25 follows. ∎

#####  4 Binary quintics with automorphism group of order two

The goal of Section 4 is to prove that there are no cone points in the
orbifold
$\textnormal{PGL}_{2}(\mathbb{R})\setminus(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})$,
i.e. orbifold points whose stabilizer group is $\mathbb{Z}/n$ for some $n$
acting on the orbifold chart by rotations. By Proposition 5.25, this fact will
follow from the following:

######  Proposition 5.30.

Let $x=(x_{1},\dotsc,x_{5})\in(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})$ such
that $\textnormal{PGL}_{2}(\mathbb{R})_{x}=\langle\tau\rangle$ has order two.
There is a $\textnormal{PGL}_{2}(\mathbb{R})_{x}$-stable open neighborhood
$U\subset(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})$ of $x$ such that
$\textnormal{PGL}_{2}(\mathbb{R})_{x}\setminus
U\to{\mathcal{M}}_{s}(\mathbb{R})$ is injective, and a homeomorphism
$\phi:(U,x)\to(B,0)$ for $0\in B\subset\mathbb{R}^{2}$ an open ball, such that
$\phi$ identifies $\textnormal{PGL}_{2}(\mathbb{R})_{x}$ with $\mathbb{Z}/2$
acting on $B$ by reflections in a line through $0$.

###### Proof.

Using Lemma 5.26, one checks that the only possibilities for the element
$x=(x_{1},\dotsc,x_{5})\in(P_{s}/{\mathfrak{S}}_{5})(\mathbb{R})$ 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 $(-1,i,i,-i,-i)$. ∎

#####  5 Comparing the orbifold structures

Consider the moduli space ${\mathcal{M}}_{s}(\mathbb{R})$ of real stable
binary quintics.

######  Definition 5.31.

Let $\overline{{\mathscr{M}}}_{\mathbb{R}}$ be the hyperbolic orbifold with
$\left|{\mathcal{M}}_{s}(\mathbb{R})\right|$ as underlying space, whose
orbifold structure is induced by the homeomorphism of Corollary 5.23 and the
natural orbifold structure of
$P\Gamma_{\mathbb{R}}\setminus{\mathbb{R}}H^{2}$.

There are two orbifold structures on the space
$\left|{\mathcal{M}}_{s}(\mathbb{R})\right|$: the natural orbifold structure
of ${\mathcal{M}}_{s}(\mathbb{R})$, see Proposition 2.12 (i.e. the orbifold
structure of the quotient $\mathbb{G}(\mathbb{R})\setminus
X_{s}(\mathbb{R})$), and the orbifold structure
$\overline{{\mathscr{M}}}_{\mathbb{R}}$ introduced in Definition 5.31.

######  Proposition 5.32.

  1. 1.

The orbifold structures of ${\mathcal{M}}_{s}(\mathbb{R})$ and
$\overline{{\mathscr{M}}}_{\mathbb{R}}$ differ only at the moduli point
attached to the five-tuple $(\infty,i,i,-i,-i)$. The stabilizer group of
${\mathcal{M}}_{s}(\mathbb{R})$ at that moduli point is $\mathbb{Z}/2$,
whereas the stabilizer group of $\overline{{\mathscr{M}}}_{\mathbb{R}}$ at
that point is the dihedral group $D_{10}$ of order twenty.

  2. 2.

The orbifold $\overline{{\mathscr{M}}}_{\mathbb{R}}$ has no cone points and
three corner reflectors, whose orders are $\pi/3,\pi/5$ and $\pi/10$.

###### Proof.

The statements can be deduced from Proposition 4.37. The notation of that
proposition was as follows: for $f\in
Y\cong\mathbb{G}(\mathbb{R})\setminus{\mathcal{F}}_{s}(\mathbb{R})$ (see
Theorem 5.22) the group $A_{f}\subset P\Gamma$ is the stabilizer of $f\in K$.
Moreover, if $\tilde{f}\in{\mathcal{F}}_{s}(\mathbb{R})$ represents $f$ and if
$F=[\tilde{f}]\in X_{s}(\mathbb{R})$ has $k=2a+b$ nodes, then the image
$x\in\mathbb{C}H^{2}$ has $k=2a+b$ nodes in the sense of Definition 4.11. If
$F$ has no nodes ($k=0$), then $G(x)$ is trivial by Proposition 4.37.1 and
$G_{F}=A_{f}=\Gamma_{f}$. If $F$ has only real nodes, then $B_{f}=G(x)$ hence
$G_{F}=A_{f}/G(x)=A_{f}/B_{f}=\Gamma_{f}$. Now suppose that $a=1$ and $b=0$:
the equation $F$ defines a pair of complex conjugate nodes. In other words,
the zero set of $F$ defines a $5$-tuple
$\underline{\alpha}=(\alpha_{1},\dotsc,\alpha_{5})\in\mathbb{P}^{1}(\mathbb{C})$,
well-defined up to the
$\textnormal{PGL}_{2}(\mathbb{R})\times{\mathfrak{S}}_{5}$ action on
$\mathbb{P}^{1}$, where $\alpha_{1}\in\mathbb{P}^{1}(\mathbb{R})$ and
$\alpha_{3}=\bar{\alpha}_{2}=\alpha_{5}=\bar{\alpha}_{4}\in\mathbb{P}^{1}(\mathbb{C})\setminus\mathbb{P}^{1}(\mathbb{R})$.
So we may write
$\underline{\alpha}=(\rho,\alpha,\bar{\alpha},\alpha,\bar{\alpha})$ with
$\rho\in\mathbb{P}^{1}(\mathbb{R})$ and
$\alpha\in\mathbb{P}^{1}(\mathbb{C})\setminus\mathbb{P}^{1}(\mathbb{R})$. Then
there is a unique $T\in\textnormal{PGL}_{2}(\mathbb{R})$ such that
$T(\rho)=\infty$ and $T(\alpha)=i$. But this gives $T(x)=(\infty,i,-i,i,-i)$
hence $F$ is unique up to isomorphism. As for the stabilizer
$G_{F}=A_{f}/G(x)$, we have $G(x)\cong(\mathbb{Z}/10)^{2}$. Since there are no
real nodes, $B_{f}$ is trivial. By Proposition 4.37.3, $K_{f}$ is the union of
ten copies of ${\mathbb{B}}^{2}(\mathbb{R})$ meeting along a common point
$\mathbb{B}^{0}(\mathbb{R})$. In fact, in the local coordinates
$(t_{1},t_{2})$ around $f$, the
$\alpha_{j}:\mathbb{B}^{2}(\mathbb{C})\to\mathbb{B}^{2}(\mathbb{C})$ are
defined by $(t_{1},t_{2})\mapsto(\bar{t}_{2}\zeta^{j},\bar{t}_{1}\zeta^{j})$,
for $j\in\mathbb{Z}/10$, and so the fixed points sets are given by the
equations
$\mathbb{R}H^{2}_{j}=\\{t_{2}=\bar{t}_{1}\zeta^{j}\\}\subset\mathbb{B}^{2}(\mathbb{C})$,
$j\in\mathbb{Z}/10$. Notice that the subgroup $E\subset G(x)$ that stabilizes
$\mathbb{R}H^{2}_{j}$ is the cyclic group of order $10$ generated by the
transformations $(t_{1},t_{2})\mapsto(\zeta t_{1},\zeta^{-1}t_{2})$. There is
only one non-trivial transformation $T\in\textnormal{PGL}_{2}(\mathbb{R})$
that fixes $\infty$ and sends the subset
$\\{i,-i\\}\subset\mathbb{P}^{1}(\mathbb{C})$ to itself, and $T$ is of order
$2$. Hence $G_{F}=\mathbb{Z}/2$ so that we have an exact sequence
$0\to\mathbb{Z}/10\to\Gamma_{f}\to\mathbb{Z}/2\to 0$ and this splits since
$G_{F}$ is a subgroup of $\Gamma_{f}$. We are done by Propositions 5.25 and
5.30. ∎

#####  6 The real moduli space as a hyperbolic triangle

The goal of Section 6 is to show that $\overline{{\mathscr{M}}}_{\mathbb{R}}$
(see Definition 5.31) is isomorphic, as hyperbolic orbifolds, to the triangle
$\Delta_{3,5,10}$ in the real hyperbolic plane ${\mathbb{R}}H^{2}$ with angles
$\pi/3,\pi/5$ and $\pi/10$. The results in the above Sections 3, 4 and 5 give
the orbifold singularities of $\overline{{\mathscr{M}}}_{\mathbb{R}}$ together
with their stabilizer groups. In order to complete determine the hyperbolic
orbifold structure of $\overline{{\mathscr{M}}}_{\mathbb{R}}$, however, we
also need to know the underlying topological space
$\left|{\mathcal{M}}_{s}(\mathbb{R})\right|$ of
$\overline{{\mathscr{M}}}_{\mathbb{R}}$. The first observation is that
$\left|{\mathcal{M}}_{s}(\mathbb{R})\right|$ is compact. Indeed, it is
classical that the topological space
${\mathcal{M}}_{s}(\mathbb{C})=\mathbb{G}(\mathbb{C})\setminus
X_{s}(\mathbb{C})$, parametrizing complex stable binary quintics, is compact.
This follows from the fact that it is homeomorphic to
$\overline{M}_{0,5}(\mathbb{C})/{\mathfrak{S}}_{5}$, and the stack of stable
five-pointed curves $\overline{M}_{0,5}$ is proper [Knu83], or from the fact
that it is homeomorphic to a compact ball quotient [Shi64]. Moreover, the map
${\mathcal{M}}_{s}(\mathbb{R})\to{\mathcal{M}}_{s}(\mathbb{C})$ is proper,
which proves the compactness of ${\mathcal{M}}_{s}(\mathbb{R})$.

The second observation is that ${\mathcal{M}}_{s}(\mathbb{R})$ is connected,
since $X_{s}(\mathbb{R})$ is obtained from the euclidean space
$\\{F\in\mathbb{R}[x,y]:F\text{ homogeneous }\mid\deg(F)=5\\}$ by removing a
subspace of codimension at least two. We can prove more:

######  Lemma 5.33.

The moduli space ${\mathcal{M}}_{s}(\mathbb{R})$ of real stable binary
quintics is simply connected.

###### Proof.

The idea is to show that the following holds:

  1. 1.

For each $i\in\\{0,1,2\\}$, the embedding
${\mathscr{M}}_{i}\hookrightarrow\overline{{\mathscr{M}}}_{i}\subset{\mathcal{M}}_{s}({\mathbb{R}})$
of the connected component ${\mathscr{M}}_{i}$ of
${\mathcal{M}}_{0}(\mathbb{R})$ into its closure in
${\mathcal{M}}_{s}({\mathbb{R}})$ is homeomorphic to the embedding
$D\hookrightarrow\overline{D}$ of the open unit disc into the closed unit disc
in $\mathbb{R}^{2}$.

  2. 2.

We have
${\mathcal{M}}_{s}(\mathbb{R})=\overline{{\mathscr{M}}}_{0}\cup\overline{{\mathscr{M}}}_{1}\cup\overline{{\mathscr{M}}}_{2}$,
and this glueing corresponds up to homeomorphism to the glueing of three
closed discs $\overline{D}_{i}\subset\mathbb{R}^{2}$ as in Figure 1.

To do this, one considers the moduli spaces of real smooth (resp. stable)
genus zero curves with five real marked points [Knu83], as well as twists of
this space. Define two anti-holomorphic involutions
$\sigma_{i}:\mathbb{P}^{1}(\mathbb{C})^{5}\to\mathbb{P}^{1}(\mathbb{C})^{5}$
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

$P_{0}^{1}(\mathbb{R})=P_{0}(\mathbb{C})^{\sigma_{1}},\;\;\;P_{s}^{1}(\mathbb{R})=P_{1}(\mathbb{C})^{\sigma_{1}},\;\;\;P_{0}^{2}(\mathbb{R})=P_{0}(\mathbb{C})^{\sigma_{2}},\;\;\;P_{s}^{2}(\mathbb{R})=P_{1}(\mathbb{C})^{\sigma_{2}}.$

It is clear that ${\mathscr{M}}_{0}=\textnormal{PGL}_{2}(\mathbb{R})\setminus
P_{0}(\mathbb{R})/{\mathfrak{S}}_{5}$. Similarly, we have:

$\displaystyle{\mathscr{M}}_{1}=\textnormal{PGL}_{2}(\mathbb{R})\setminus
P_{0}^{1}(\mathbb{R})/{\mathfrak{S}}_{3}\times{\mathfrak{S}}_{2}\quad{\textnormal{
and }}\quad{\mathscr{M}}_{2}=\textnormal{PGL}_{2}(\mathbb{R})\setminus
P_{0}^{2}(\mathbb{R})/{\mathfrak{S}}_{2}\times{\mathfrak{S}}_{2}.$

Moreover, we have
$\overline{{\mathscr{M}}}_{0}=\textnormal{PGL}_{2}(\mathbb{R})\setminus
P_{s}(\mathbb{R})/{\mathfrak{S}}_{5}$. We define

$\displaystyle\overline{{\mathscr{M}}}_{1}=\textnormal{PGL}_{2}(\mathbb{R})\setminus
P_{s}^{1}(\mathbb{R})/{\mathfrak{S}}_{3}\times{\mathfrak{S}}_{2},\quad{\textnormal{
and
}}\quad\overline{{\mathscr{M}}}_{2}=\textnormal{PGL}_{2}(\mathbb{R})\setminus
P_{s}^{2}(\mathbb{R})/{\mathfrak{S}}_{2}\times{\mathfrak{S}}_{2}.$

Each $\overline{{\mathscr{M}}}_{i}$ is simply connected. Moreover, the natural
maps $\overline{{\mathscr{M}}}_{i}\to{\mathcal{M}}_{s}(\mathbb{R})$ are closed
embeddings of topological spaces, and one can check that the images glue to
form ${\mathcal{M}}_{s}(\mathbb{R})$ in the prescribed way. We leave the
details to the reader. ∎

###### Proof of Theorem 5.2.

To any closed $2$-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 $2m_{j}$ the order of the
$j$-th corner reflector. A closed $2$-dimensional orbifold $O$ is then
determined, up to orbifold-structure preserving homeomorphism, by its
underlying space $X_{O}$ and the set $S_{O}$ [Thu80]. By Lemma 5.33,
$\overline{{\mathscr{M}}}_{\mathbb{R}}$ is simply connected. By Proposition
5.32, $\overline{{\mathscr{M}}}_{\mathbb{R}}$ has no cone points and three
corner reflectors whose orders are $\pi/3,\pi/5$ and $\pi/10$. This implies
$\overline{{\mathscr{M}}}_{\mathbb{R}}$ and $\Delta_{3,5,10}$ are isomorphic
as topological orbifolds. Consequently, the orbifold fundamental group of
$\overline{{\mathscr{M}}}_{\mathbb{R}}$ is abstractly isomorphic to the group
$P\Gamma_{3,5,10}$ defined in (2).

Let $\phi:P\Gamma_{3,5,10}\hookrightarrow\text{PSL}_{2}(\mathbb{R})$ be any
embedding such that
$X:=\phi\left(P\Gamma_{3,5,10}\right)\setminus\mathbb{R}H^{2}$ is a hyperbolic
orbifold; we claim that there is a fundamental domain $\Delta$ of $X$
isometric to $\Delta_{3,5,10}$. Consider the generator $a\in
P\Gamma_{3,5,10}$. Since $\phi(a)^{2}=1$, there exists a geodesic
$L_{1}\subset\mathbb{R}H^{2}$ such that
$\phi(a)\in\text{PSL}_{2}(\mathbb{R})=\text{Isom}(\mathbb{R}H^{2})$ is the
reflection across $L_{1}$. Next, consider the generator $b\in
P\Gamma_{3,5,10}$. There exists a geodesic $L_{2}\subset\mathbb{R}H^{2}$ such
that $\phi(b)$ is the reflection across $L_{2}$. One easily shows that
$L_{2}\cap L_{1}\neq\emptyset$. Let $x\in L_{1}\cap L_{2}$. Then
$\phi(a)\phi(b)$ is an element of order three that fixes $x$, hence
$\phi(a)\phi(b)$ 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 $2$ in $\textnormal{PSL}_{2}(\mathbb{R})$, hence a
reflection across a line $L_{3}$. By the previous arguments, $L_{3}\cap
L_{2}\neq\emptyset$ and $L_{3}\cap L_{1}\neq\emptyset$. It also follows that
$x\in L_{3}\cap L_{2}\cap L_{1}=\emptyset$. Consequently, the three geodesics
$L_{i}\subset\mathbb{R}H^{2}$ enclose a hyperbolic triangle; the orders of
$\phi(a)\phi(b)$, $\phi(a)\phi(c)$ and $\phi(b)\phi(c)$ imply that the three
interior angles of the triangle are $\pi/3$, $\pi/5$ and $\pi/10$. ∎

####  4 The monodromy groups

In this section, we describe the monodromy group $P\Gamma$, as well as the
groups $P\Gamma_{\alpha}$ appearing in Proposition 5.17. As for the lattice
$(\Lambda,{\mathfrak{h}})$ (see (9)), we have:

######  Theorem 5.34 (Shimura).

There is an isomorphism of hermitian $\mathcal{O}_{K}$-lattices

$\left(\Lambda,{\mathfrak{h}}\right)\cong\left(\mathcal{O}_{K}^{3},\textnormal{diag}\left(1,1,\frac{1-\sqrt{5}}{2}\right)\right).$

###### Proof.

See [Shi64, Section 6] as well as item (5) in the table on page 1. ∎

Let us write $\Lambda=\mathcal{O}_{K}^{3}$ and
${\mathfrak{h}}=\textnormal{diag}(1,1,\frac{1-\sqrt{5}}{2})$ in the remaining
part of Section 4. Write
$\alpha=\zeta_{5}+\zeta_{5}^{-1}=\frac{\sqrt{5}-1}{2}$. Recall that
$\theta=\zeta_{5}-\zeta_{5}^{-1}$ and observe that
$|\theta|^{2}=\frac{\sqrt{5}+5}{2}$. Define three quadratic forms $q_{0}$,
$q_{1}$ and $q_{2}$ on $\mathbb{Z}[\alpha]^{3}$ as follows:

$\begin{split}q_{0}(x_{0},x_{1},x_{2})&=x_{0}^{2}+x_{1}^{2}-\alpha
x_{2}^{2},\\\ q_{1}(x_{0},x_{1},x_{2})&=|\theta|^{2}x_{0}^{2}+x_{1}^{2}-\alpha
x_{2}^{2},\\\
q_{2}(x_{0},x_{1},x_{2})&=|\theta|^{2}x_{0}^{2}+|\theta|^{2}x_{1}^{2}-\alpha
x_{2}^{2}.\end{split}$ (34)

We consider $\mathbb{Z}[\alpha]$ as a subring of $\mathbb{R}$ via the standard
embedding.

######  Theorem 5.35.

Consider the quadratic forms $q_{j}$ defined in (34). There is a union of
geodesic subspaces ${\mathscr{H}}_{j}\subset\mathbb{R}H^{2}$
($j\in\\{0,1,2\\}$) and an isomorphism of hyperbolic orbifolds

${\mathcal{M}}_{0}(\mathbb{R})\cong\coprod_{j=0}^{2}\textnormal{PO}(q_{j},\mathbb{Z}[\alpha])\setminus\left(\mathbb{R}H^{2}-{\mathscr{H}}_{j}\right).$
(35)

###### Proof.

Recall that $\theta=\zeta_{5}-\zeta_{5}^{-1}$; we consider the
${\mathbb{F}}_{5}$-vector space $W$ equipped with the quadratic form
$q={\mathfrak{h}}\mod\theta$. Define three anti-isometric involutions as
follows:

$\begin{split}\alpha_{0}:&(x_{0},x_{1},x_{2})\mapsto(\;\;\;\bar{x}_{0},\;\;\;\bar{x}_{1},\bar{x}_{2})\\\
\alpha_{1}:&(x_{0},x_{1},x_{2})\mapsto(-\bar{x}_{0},\;\;\;\bar{x}_{1},\bar{x}_{2})\\\
\alpha_{2}:&(x_{0},x_{1},x_{2})\mapsto(-\bar{x}_{0},-\bar{x}_{1},\bar{x}_{2}).\end{split}$
(36)

For isometries $\alpha:W\to W$, the dimension and determinant of the fixed
space $(W^{\alpha},q|_{W^{\alpha}})$ are conjugacy-invariant. Using this, one
easily shows that an anti-unitary involution of $\Lambda$ is
$\Gamma$-conjugate to exactly one of the $\pm\alpha_{j}$, hence
$C{\mathscr{A}}$ has cardinality $3$ and is represented by
$\alpha_{0},\alpha_{1},\alpha_{2}$ of (36). By Proposition 5.17, we obtain
${\mathcal{M}}_{0}(\mathbb{R})\cong\coprod_{j=0}^{2}P\Gamma_{\alpha_{j}}\setminus(\mathbb{R}H^{2}_{\alpha_{j}}-{\mathscr{H}})$
where each hyperbolic quotient
$P\Gamma_{\alpha_{j}}\setminus(\mathbb{R}H^{2}_{\alpha_{j}}-{\mathscr{H}})$ is
connected. Next, consider the fixed lattices

$\begin{split}\Lambda_{0}:=\Lambda^{\alpha_{0}}=\mathbb{Z}[\alpha]\oplus\mathbb{Z}[\alpha]\oplus\mathbb{Z}[\alpha]\\\
\Lambda_{1}:=\Lambda^{\alpha_{1}}=\theta\mathbb{Z}[\alpha]\oplus\mathbb{Z}[\alpha]\oplus\mathbb{Z}[\alpha]\\\
\Lambda_{2}:=\Lambda^{\alpha_{2}}=\theta\mathbb{Z}[\alpha]\oplus\theta\mathbb{Z}[\alpha]\oplus\mathbb{Z}[\alpha].\end{split}$
(37)

One easily shows that $P\Gamma_{\alpha_{j}}=N_{P\Gamma}(\alpha_{j})$ for the
normalizer $N_{P\Gamma}(\alpha_{j})$ of $\alpha_{j}$ in $P\Gamma$. Moreover,
if $h_{j}$ denotes the restriction of ${\mathfrak{h}}$ to
$\Lambda^{\alpha_{j}}$, then there is a natural embedding

$\iota:N_{P\Gamma}(\alpha_{j})\hookrightarrow\textnormal{PO}(\Lambda_{j},h_{j},\mathbb{Z}[\alpha]).$
(38)

We claim that $\iota$ is actually an isomorphism. Indeed, this follows from
the fact that the natural homomorphism $\pi:N_{\Gamma}(\alpha_{j})\to
O(\Lambda_{j},h_{j})$ is surjective, where
$N_{\Gamma}(\alpha_{j})=\\{g\in\Gamma:g\circ\alpha_{j}=\alpha_{j}\circ g\\}$
is the normalizer of $\alpha_{j}$ in $\Gamma$. The surjectivity of $\pi$
follows in turn from the equality

$\Lambda={\mathcal{O}}_{K}\cdot\Lambda_{j}+{\mathcal{O}}_{K}\cdot\theta\Lambda_{j}^{\vee}\subset
K^{3}$

which follows from (37). Since
$\textnormal{PO}(\Lambda_{j},h_{j},\mathbb{Z}[\alpha])=\textnormal{PO}(q_{j},\mathbb{Z}[\alpha])$,
we are done. ∎

##  Part 2 Real algebraic cycles

###  Chapter 6 Integral Fourier transforms

This chapter is based on joint work with Thorsten Beckmann.

####  1 Introduction

In the second part of this thesis, we focus on algebraic cycles on complex and
real abelian varieties. A central role in this study – which takes up Chapters
6, 7 and 8 – is played by a certain correspondence. It has since long been
known that for an abelian variety $A$ over a field $k$, with dual abelian
variety ${\widehat{A}}$, the Fourier transform

$\displaystyle{\mathcal{F}}_{A}\colon\textnormal{CH}(A)_{\mathbb{Q}}\xrightarrow{\sim}\textnormal{CH}({\widehat{A}})_{\mathbb{Q}}$
(1)

provides a powerful tool to study the $\mathbb{Q}$-linear algebraic cycles of
$A$. It is used to study the rational Chow ring of $A$, as well as the cycle
class map to rational cohomology. Recently, Moonen and Polishchuk [MP10]
initiated the study of the integrality aspects of the Fourier transform (1).
Indeed, it is natural to ask:

######  Question 6.1.

How does ${\mathcal{F}}_{A}$ interact with integral algebraic cycles?

The goal of the current Chapter 6 is to work on Question 6.1, building on the
results of Moonen–Polishchuk. The applications of _loc.cit._ primarily concern
the structure of the integral Chow rings themselves. We continue with their
study, but we also address the compatibility of Fourier transforms with
integral cycle class maps. Since Question 6.1 was phrased somewhat
imprecisely, let us explain in some detail the steps that we take during our
search for an answer:

  1. Chapter 6:

We further develop the theory of _integral Fourier transforms_ , on Chow rings
as well as on cohomology. The main result of this chapter (Theorem 6.9) will
provide necessary and sufficient conditions for the Fourier transform (1) to
lift to a motivic homomorphism between integral Chow groups.

  2. Chapter 7:

We apply the theory of Chapter 6 to complex abelian varieties. The main
outcome of this project is Theorem 7.1, which says that on a principally
polarized complex abelian variety $A$ whose minimal cohomology class is
algebraic, all integral Hodge classes of degree $2\dim(A)-2$ are algebraic.

  3. Chapter 8:

We apply the theory of Chapter 6 (which is developed for abelian varieties
over an arbitrary field $k$) to the case of abelian varieties over
$k=\mathbb{R}$. The principal outcome is that modulo torsion, every real
abelian threefold satisfies the real integral Hodge conjecture (Theorem 8.3).
Other applications of integral Fourier transforms include a detailed analysis
of the Hochschild-Serre filtration on equivariant cohomology of a real abelian
variety (Theorem 8.8).

Having lifted a veil of what to do with integral Fourier transforms, let us
now make Question 6.1 more precise. Let $g$ be a positive integer and let $A$
be an abelian variety of dimension $g$ over a field $k$. _Fourier transforms_
are correspondences between the derived categories, rational Chow groups and
cohomology of $A$ and ${\widehat{A}}$ attached to the Poincaré bundle
${\mathcal{P}}_{A}$ on $A\times{\widehat{A}}$ [Muk81, Bea82, Huy06]. On the
level of cohomology, the Fourier transform preserves integral $\ell$-adic
étale cohomology when $k=k_{s}$ (separable closure), and integral Betti
cohomology when $k=\mathbb{C}$. It is thus natural to ask whether the Fourier
transform on rational Chow groups preserves integral cycles modulo torsion or,
more generally, lifts to a homomorphism between integral Chow groups. This
question was raised by Moonen–Polishchuk [MP10] and Totaro [Tot21]. More
precisely, Moonen and Polishchuk gave a counterexample for abelian varieties
over non-closed fields and asked about the case of algebraically closed
fields.

The goal of Chapter 6 is to investigate this question further.

####  2 Integral Fourier transforms

The main result of Section 6 gives necessary and sufficient conditions for the
Fourier transform on rational Chow groups or cohomology to lift to a motivic
homomorphism between integral Chow groups. To get there, we need a precise
definition of "integral Fourier transform", which we introduce in this Section
2.

#####  1 Notation and conventions

We let $k$ be a field with separable closure $k_{s}$ and $\ell$ a prime number
different from the characteristic of $k$. For a smooth projective variety $X$
over $k$, we let $\textnormal{CH}(X)$ be the Chow group of $X$ and define
$\textnormal{CH}(X)_{\mathbb{Q}}=\textnormal{CH}(X)\otimes\mathbb{Q}$,
$\textnormal{CH}(X)_{\mathbb{Q}_{\ell}}=\textnormal{CH}(X)\otimes{\mathbb{Q}_{\ell}}$
and
$\textnormal{CH}(X)_{\mathbb{Z}_{\ell}}=\textnormal{CH}(X)\otimes{\mathbb{Z}_{\ell}}$.
We let
${\mathrm{H}}_{\textnormal{\'{e}t}}^{i}(X_{k_{s}},\mathbb{Z}_{\ell}(a))$ be
the $i$-th degree étale cohomology group with coeffients in
$\mathbb{Z}_{\ell}(a)$, $a\in\mathbb{Z}$.

Often, $A$ will denote an abelian variety of dimension $g$ over $k$, with dual
abelian variety ${\widehat{A}}$ and (normalized) Poincaré bundle on
${\mathcal{P}}_{A}$. The abelian group $\textnormal{CH}(A)$ will in that case
be equipped with two ring structures: the usual intersection product $\cdotp$
as well as the Pontryagin product $\star$. Recall that the latter is defined
as follows:

$\star\colon\textnormal{CH}(A)\times\textnormal{CH}(A)\to\textnormal{CH}(A),\quad
x\star y=m_{\ast}(\pi_{1}^{\ast}(x)\cdot\pi_{2}^{\ast}(y)).$

Here, as well as in the rest of the paper, $\pi_{i}$ denotes the projection
onto the $i$-th factor, $\Delta\colon A\to A\times A$ the diagonal morphism,
and $m\colon A\times A\to A$ the group law morphism of $A$.

For any abelian group $M$ and any element $x\in M$, we denote by
$x_{\mathbb{Q}}\in M\otimes_{\mathbb{Z}}\mathbb{Q}$ the image of $x$ in
$M\otimes_{\mathbb{Z}}\mathbb{Q}$ under the canonical homomorphism $M\to
M\otimes_{\mathbb{Z}}\mathbb{Q}$.

#####  2 Integral Fourier transforms and integral Hodge classes

For abelian varieties $A$ over $k=k_{s}$, it is unknown whether the Fourier
transform
${\mathcal{F}}_{A}\colon\textnormal{CH}(A)_{\mathbb{Q}}\to\textnormal{CH}({\widehat{A}})_{\mathbb{Q}}$
preserves the subgroups of integral cycles modulo torsion. A sufficient
condition for this to hold is that ${\mathcal{F}}_{A}$ lifts to a homomorphism
$\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$. In this section we
outline a second consequence of such a lift
$\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$ when $A$ is defined over
the complex numbers: the existence of an integral lift of ${\mathcal{F}}_{A}$
implies the integral Hodge conjecture for one-cycles on ${\widehat{A}}$.

Let $A$ be an abelian variety over $k$. The Fourier transform on the level of
Chow groups is the group homomorphism

${\mathcal{F}}_{A}\colon\textnormal{CH}(A)_{\mathbb{Q}}\to\textnormal{CH}({\widehat{A}})_{\mathbb{Q}}$

induced by the correspondence
$\textnormal{ch}({\mathcal{P}}_{A})\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$,
where $\textnormal{ch}({\mathcal{P}}_{A})$ is the Chern character of
${\mathcal{P}}_{A}$. Similarly, one defines the Fourier transform on the level
of étale cohomology:

${\mathscr{F}}_{A}\colon{\mathrm{H}}_{\textnormal{\'{e}t}}^{\bullet}(A_{k_{s}},\mathbb{Q}_{\ell}(\bullet))\to{\mathrm{H}}_{\textnormal{\'{e}t}}^{\bullet}({\widehat{A}}_{k_{s}},\mathbb{Q}_{\ell}(\bullet)).$

In fact, ${\mathscr{F}}_{A}$ preserves the integral cohomology classes and
induces, for each integer $j$ with $0\leq j\leq 2g$, an isomorphism [Bea82,
Proposition 1], [Tot21, page 18]:

${\mathscr{F}}_{A}\colon{\mathrm{H}}_{\textnormal{\'{e}t}}^{j}(A_{k_{s}},\mathbb{Z}_{\ell}(a))\to{\mathrm{H}}_{\textnormal{\'{e}t}}^{2g-j}({\widehat{A}}_{k_{s}},\mathbb{Z}_{\ell}(a+g-j)).$

Similarly, if $k=\mathbb{C}$, then $\textnormal{ch}({\mathcal{P}}_{A})$
induces, for each integer $i$ with $0\leq i\leq 2g$, an isomorphism of Hodge
structures

${\mathscr{F}}_{A}\colon{\mathrm{H}}^{i}(A,\mathbb{Z})\to{\mathrm{H}}^{2g-i}({\widehat{A}},\mathbb{Z})(g-i).$
(2)

In [MP10], Moonen and Polishchuk consider an isomorphism $\phi\colon
A\xrightarrow{\sim}{\widehat{A}}$, a positive integer $d$, and define the
notion of motivic integral Fourier transform of $(A,\phi)$ up to factor $d$.
The definition goes as follows. Let ${\mathcal{M}}(k)$ be the category of
effective Chow motives over $k$ with respect to ungraded correspondences, and
let $h(A)$ be the motive of $A$. Then a morphism

${\mathcal{F}}\colon h(A)\to h(A)$

in ${\mathcal{M}}(k)$ is a motivic integral Fourier transform of $(A,\phi)$ up
to factor $d$ if the following three conditions are satisfied: (i) the induced
morphism $h(A)_{\mathbb{Q}}\to h(A)_{\mathbb{Q}}$ is the composition of the
usual Fourier transform with the isomorphism $\phi^{\ast}\colon
h({\widehat{A}})_{\mathbb{Q}}\xrightarrow{\sim}h(A)_{\mathbb{Q}}$, (ii) one
has $d\cdot{\mathcal{F}}\circ{\mathcal{F}}=d\cdot(-1)^{g}\cdot[-1]_{\ast}$ as
morphisms from $h(A)$ to $h(A)$, and (iii) $d\cdot{\mathcal{F}}\circ
m_{\ast}=d\cdot\Delta^{\ast}\circ{\mathcal{F}}\otimes{\mathcal{F}}\colon
h(A)\otimes h(A)\to h(A)$.

For our purposes, we consider similar homomorphisms
$\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$. To make their existence
easier to verify (c.f. Theorem 6.9) we relax the above conditions:

######  Definition 6.2.

Let $A_{/k}$ be an abelian variety and let
${\mathcal{F}}\colon\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$ be a
group homomorphism. We call ${\mathcal{F}}$ a weak integral Fourier transform
if the following diagram commutes:

$\textstyle{\textnormal{CH}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}}$$\textstyle{\textnormal{CH}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textnormal{CH}(A)_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}_{A}}$$\textstyle{\textnormal{CH}({\widehat{A}})_{\mathbb{Q}}.}$
(3)

We call a weak integral Fourier transform ${\mathcal{F}}$ motivic if it is
induced by a cycle $\Gamma$ in $\textnormal{CH}(A\times{\widehat{A}})$ that
satisfies
$\Gamma_{\mathbb{Q}}=\textnormal{ch}({\mathcal{P}}_{A})\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$.
A group homomorphism
${\mathcal{F}}\colon\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$ is an
integral Fourier transform up to homology if the following diagram commutes:

$\textstyle{\textnormal{CH}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}}$$\textstyle{\textnormal{CH}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\oplus_{r\geq
0}{\mathrm{H}}_{\textnormal{\'{e}t}}^{2r}(A_{k_{s}},\mathbb{Z}_{\ell}(r))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathscr{F}}_{A}}$$\textstyle{\oplus_{r\geq
0}{\mathrm{H}}_{\textnormal{\'{e}t}}^{2r}({\widehat{A}}_{k_{s}},\mathbb{Z}_{\ell}(r)).}$
(4)

Similarly, a $\mathbb{Z}_{\ell}$-module homomorphism
${\mathcal{F}}_{\ell}\colon\textnormal{CH}(A)_{\mathbb{Z}_{\ell}}\to\textnormal{CH}({\widehat{A}})_{\mathbb{Z}_{\ell}}$
is called an $\ell$-adic integral Fourier transform up to homology if
${\mathcal{F}}_{\ell}$ is compatible with ${\mathscr{F}}_{A}$ and the
$\ell$-adic cycle class maps. Finally, an integral Fourier transform up to
homology ${\mathcal{F}}$ (resp. an $\ell$-adic integral Fourier transform up
to homology ${\mathcal{F}}_{\ell}$) is called motivic if it is induced by a
cycle $\Gamma\in\textnormal{CH}(A\times{\widehat{A}})$ (resp.
$\Gamma_{\ell}\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Z}_{\ell}})$
such that $cl(\Gamma)$ (resp. $cl(\Gamma_{\ell})$) equals
$\textnormal{ch}({\mathcal{P}}_{A})\in\oplus_{r\geq
0}{\mathrm{H}}_{\textnormal{\'{e}t}}^{2r}((A\times{\widehat{A}})_{k_{s}},\mathbb{Z}_{\ell}(r))$.

######  Remark 6.3.

If ${\mathcal{F}}\colon\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$ is
a weak integral Fourier transform, then ${\mathcal{F}}$ is an integral Fourier
transform up to homology, the $\mathbb{Z}_{\ell}$-module $\oplus_{r\geq
0}{\mathrm{H}}_{\textnormal{\'{e}t}}^{2r}({\widehat{A}}_{k_{s}},\mathbb{Z}_{\ell}(r))$
being torsion-free. If $k=\mathbb{C}$, then
${\mathcal{F}}\colon\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$ is an
integral Fourier transform up to homology if and only if ${\mathcal{F}}$ is
compatible with the Fourier transform
${\mathscr{F}}_{A}\colon{\mathrm{H}}^{\bullet}(A,\mathbb{Z})\to{\mathrm{H}}^{\bullet}({\widehat{A}},\mathbb{Z})$
on Betti cohomology.

The relation between integral Fourier transforms and Hodge classes is as
follows:

######  Lemma 6.4.

Let $A$ be a complex abelian variety and let
${\mathcal{F}}\colon\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$ be an
integral Fourier transform up to homology.

  1. 1.

For each $i\in\mathbb{Z}_{\geq 0}$, the integral Hodge conjecture for degree
$2i$ classes on $A$ implies the integral Hodge conjecture for degree $2g-2i$
classes on ${\widehat{A}}$.

  2. 2.

If ${\mathcal{F}}$ is motivic, then ${\mathscr{F}}_{A}$ induces a group
isomorphism
${\mathrm{Z}}^{2i}(A)\xrightarrow{\sim}{\mathrm{Z}}^{2g-2i}({\widehat{A}})$
and, therefore, the integral Hodge conjectures for degree $2i$ classes on $A$
and degree $2g-2i$ classes on ${\widehat{A}}$ are equivalent for all $i$.

###### Proof.

We can extend Diagram (4) to the following commutative diagram:

$\textstyle{\textnormal{CH}^{i}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl^{i}}$$\textstyle{\textnormal{CH}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}}$$\textstyle{\textnormal{CH}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textnormal{CH}_{i}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl_{i}}$$\textstyle{{\mathrm{H}}^{2i}(A,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{\bullet}(A,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathscr{F}}_{A}}$$\textstyle{{\mathrm{H}}^{\bullet}({\widehat{A}},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{2g-2i}({\widehat{A}},\mathbb{Z}).}$

The composition
${\mathrm{H}}^{2i}(A,\mathbb{Z})\to{\mathrm{H}}^{2g-2i}({\widehat{A}},\mathbb{Z})$
appearing on the bottom line agrees up to a suitable Tate twist with the map
${\mathscr{F}}_{A}$ of equation (2). Therefore, we obtain a commutative
diagram:

$\textstyle{\textnormal{CH}^{i}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl^{i}}$$\textstyle{\textnormal{CH}_{i}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl_{i}}$$\textstyle{\textnormal{Hdg}^{2i}(A,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\textnormal{Hdg}^{2g-2i}({\widehat{A}},\mathbb{Z}).}$
(5)

Thus the surjectivity of $cl^{i}$ implies the surjectivity of $cl_{i}$.
Moreover, if ${\mathcal{F}}$ is motivic, then replacing $A$ by ${\widehat{A}}$
and ${\widehat{A}}$ by ${\widehat{\widehat{A}\mkern 5.5mu}\mkern-5.5mu}{}$ in
the argument above shows that the images of $cl^{i}$ and $cl_{i}$ are
identified under the isomorphism
${\mathscr{F}}_{A}\colon\textnormal{Hdg}^{2i}(A,\mathbb{Z})\xrightarrow{\sim}\textnormal{Hdg}^{2g-2i}({\widehat{A}},\mathbb{Z})$
in diagram (5). ∎

####  3 Rational Fourier transforms

The above suggests that to prove Theorem 7.1, one would need to show that for
a complex abelian variety of dimension $g$ whose minimal Poincaré class
$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic, all classes of the form
$c_{1}({\mathcal{P}}_{A})^{i}/i!\in{\mathrm{H}}^{2i}(A\times{\widehat{A}},\mathbb{Z})$
are algebraic. With this goal in mind we shall study Fourier transforms on
rational Chow groups in Section 3, and investigate how these relate to
$\textnormal{ch}({\mathcal{P}}_{A})\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$.
In turns out that the cycles $c_{1}({\mathcal{P}}_{A})^{i}/i!$ in
$\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$ satisfy several relations
that are very similar to those proved by Beauville for the cycles
$\theta^{i}/i!\in\textnormal{CH}(A)_{\mathbb{Q}}$ in case $A$ is principally
polarized, see [Bea82]. Since we will need these results in any characteristic
in order to prove Theorem 7.6, we will work over our general field $k$, see
Section 1.

Let $A$ be an abelian variety over $k$. Define cycles

$\displaystyle\ell$
$\displaystyle=c_{1}({\mathcal{P}}_{A})\in\textnormal{CH}^{1}(A\times{\widehat{A}})_{\mathbb{Q}},$
$\displaystyle{\widehat{\ell}}$
$\displaystyle=c_{1}({\mathcal{P}}_{{\widehat{A}}})\in\textnormal{CH}^{1}({\widehat{A}}\times
A)_{\mathbb{Q}},$ $\displaystyle{\mathscr{R}}_{A}$
$\displaystyle=c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in\textnormal{CH}_{1}(A\times{\widehat{A}})_{\mathbb{Q}},\quad{\textnormal{
and }}$ $\displaystyle{\mathscr{R}}_{{\widehat{A}}}$
$\displaystyle=c_{1}({\mathcal{P}}_{{\widehat{A}}})^{2g-1}/(2g-1)!\in\textnormal{CH}_{1}({\widehat{A}}\times
A)_{\mathbb{Q}}.$

For $a\in\textnormal{CH}(A)_{\mathbb{Q}}$, define
$\mathrm{E}(a)\in\textnormal{CH}(A)_{\mathbb{Q}}$ as the $\star$-exponential
of $a$:

$\mathrm{E}(a)\coloneqq\sum_{n\geq 0}\frac{a^{\star
n}}{n!}\in\textnormal{CH}(A)_{\mathbb{Q}}.$

The key to Theorem 7.1 will be the following:

######  Lemma 6.5.

We have
$\textnormal{ch}({\mathcal{P}}_{A})=e^{\ell}=(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot{\mathscr{R}}_{A})\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$.

###### Proof.

The most important ingredient in the proof is the following:

Claim $(\ast)$: Consider the Fourier transform
${\mathcal{F}}_{A\times{\widehat{A}}}\colon\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}\to\textnormal{CH}({\widehat{A}}\times
A)_{\mathbb{Q}}$. One has

${\mathcal{F}}_{A\times{\widehat{A}}}(e^{\ell})=(-1)^{g}\cdot
e^{-{\widehat{\ell}}}\in\textnormal{CH}({\widehat{A}}\times A)_{\mathbb{Q}}.$

To prove Claim ($\ast$), we lift the desired equality in the rational Chow
group of ${\widehat{A}}\times A$ to an isomorphism in the derived category
${\mathrm{D}}^{b}({\widehat{A}}\times A)$ of ${\widehat{A}}\times A$. For
$X=A\times{\widehat{A}}$ the Poincaré line bundle $\mathcal{P}_{X}$ on
$X\times{\widehat{X}}\cong A\times{\widehat{A}}\times{\widehat{A}}\times A$ is
isomorphic to
$\pi_{13}^{\ast}\mathcal{P}_{A}\otimes\pi_{24}^{\ast}\mathcal{P}_{{\widehat{A}}}$.
Consider

$\Phi_{{{\mathcal{P}}}_{X}}({\mathcal{P}}_{A})\cong\pi_{34,\ast}\left(\pi_{13}^{\ast}{\mathcal{P}}_{A}\otimes\pi_{24}^{\ast}{\mathcal{P}}_{{\widehat{A}}}\otimes\pi_{12}^{\ast}{\mathcal{P}}_{A}\right)\in{\mathrm{D}}^{b}({\widehat{A}}\times
A)$ (6)

whose Chern character is exactly $\mathcal{F}_{X}(e^{\ell})$. Applying the
pushforward along the permutation map

$(123)\colon A\times{\widehat{A}}\times{\widehat{A}}\times
A\cong{\widehat{A}}\times A\times{\widehat{A}}\times A$

the object (6) becomes
$\pi_{14,\ast}\left(\pi_{12}^{\ast}\mathcal{P}_{{\widehat{A}}}\otimes\pi_{23}^{\ast}\mathcal{P}_{A}\otimes\pi_{34}^{\ast}\mathcal{P}_{{\widehat{A}}}\right)$
which is isomorphic to the Fourier–Mukai kernel of the composition

$\Phi_{\mathcal{P}_{{\widehat{A}}}}\circ\Phi_{\mathcal{P}_{A}}\circ\Phi_{\mathcal{P}_{{\widehat{A}}}}.$

Since $\Phi_{\mathcal{P}_{A}}\circ\Phi_{\mathcal{P}_{{\widehat{A}}}}$ is
isomorphic to $[-1_{{\widehat{A}}}]^{\ast}\circ[-g]$ by [Muk81, Theorem 2.2],
we have

$\Phi_{\mathcal{P}_{{\widehat{A}}}}\circ\Phi_{\mathcal{P}_{A}}\circ\Phi_{\mathcal{P}_{{\widehat{A}}}}\cong\Phi_{\mathcal{P}_{{\widehat{A}}}}\circ[-1_{{\widehat{A}}}]^{\ast}\circ[-g].$

This is the Fourier–Mukai transform with kernel
${\mathcal{E}}=\mathcal{P}_{{\widehat{A}}}^{\vee}[-g]\in{\mathrm{D}}^{b}({\widehat{A}}\times
A)$. By uniqueness of the Fourier–Mukai kernel of an equivalence [Orl97,
Theorem 2.2] and the fact that the Chern character of ${\mathcal{E}}$ equals
$(-1)^{g}\cdot e^{-{\widehat{\ell}}}\in\textnormal{CH}({\widehat{A}}\times
A)_{\mathbb{Q}}$, this finishes the proof of Claim ($\ast$).

Next, we claim that $(-1)^{g}\cdot{\mathcal{F}}_{{\widehat{A}}\times
A}(-{\widehat{\ell}})={\mathscr{R}}_{A}$. To see this, recall that for each
integer $i$ with $0\leq i\leq g$, there is a canonical Beauville decomposition

$\textnormal{CH}^{i}(A)_{\mathbb{Q}}=\oplus_{j=i-g}^{i}\textnormal{CH}^{i,j}(A)_{\mathbb{Q}},\quad\quad\quad{\textnormal{see
\cite[cite]{[\@@bibref{}{beauvilledecomposition}{}{}]}}}.$

Since the Poincaré bundle ${\mathcal{P}}_{A}$ is symmetric, we have
$\ell\in\textnormal{CH}^{1,0}(A\times{\widehat{A}})_{\mathbb{Q}}$ and hence
$\ell^{i}\in\textnormal{CH}^{i,0}(A\times{\widehat{A}})_{\mathbb{Q}}$. In
particular, we have
${\mathscr{R}}_{A}\in\textnormal{CH}^{2g-1,0}(A\times{\widehat{A}})_{\mathbb{Q}}$.
The fact that ${\mathcal{P}}_{A}$ is symmetric also implies - via Claim
($\ast$) - that we have ${\mathcal{F}}_{{\widehat{A}}\times A}((-1)^{g}\cdot
e^{-{\widehat{\ell}}})=e^{\ell}$. Indeed,

${\mathcal{F}}_{{\widehat{A}}\times
A}\circ{\mathcal{F}}_{A\times{\widehat{A}}}=[-1]^{\ast}\cdot(-1)^{2g}=[-1]^{\ast},$

see [DM91, Corollary 2.22]. Since ${\mathcal{F}}_{{\widehat{A}}\times A}$
identifies the group $\textnormal{CH}^{i,0}({\widehat{A}}\times
A)_{\mathbb{Q}}$ with the group
$\textnormal{CH}^{g-i,0}(A\times{\widehat{A}})$ (see [DM91, Lemma 2.18]), we
must indeed have

$(-1)^{g}\cdot{\mathcal{F}}_{{\widehat{A}}\times
A}(-{\widehat{\ell}})={\mathcal{F}}_{{\widehat{A}}\times
A}((-1)^{g+1}\cdot{\widehat{\ell}})=\frac{{\ell}^{2g-1}}{(2g-1)!}={\mathscr{R}}_{A}.$
(7)

For a $g$-dimensional abelian variety $X$ and any
$x,y\in\textnormal{CH}(X)_{\mathbb{Q}}$, one has ${\mathcal{F}}_{X}(x\cdot
y)=(-1)^{g}\cdot{\mathcal{F}}_{X}(x)\star{\mathcal{F}}_{X}(y)\in\textnormal{CH}({\widehat{X}})_{\mathbb{Q}}$,
see [Bea82, Proposition 3]. This implies (see also [MP10, §3.7]) that if $a$
is a cycle on $X$ such that
${\mathcal{F}}_{X}(a)\in\textnormal{CH}_{>0}({\widehat{X}})_{\mathbb{Q}}$,
then
${\mathcal{F}}_{X}(e^{a})=(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot{\mathcal{F}}_{X}(a))$.
This allows us to conclude that

$\displaystyle e^{\ell}$ $\displaystyle={\mathcal{F}}_{{\widehat{A}}\times
A}((-1)^{g}\cdot
e^{-{\widehat{\ell}}})=(-1)^{g}\cdot{\mathcal{F}}_{{\widehat{A}}\times
A}(e^{-{\widehat{\ell}}})$
$\displaystyle=(-1)^{g}\cdot{\mathrm{E}}({\mathcal{F}}_{{\widehat{A}}\times
A}(-{\widehat{\ell}}))=(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot{\mathscr{R}}_{A}),$

which finishes the proof. ∎

Next, assume that $A$ is equipped with a principal polarization $\lambda\colon
A\xrightarrow{\sim}{\widehat{A}}$, define $\ell=c_{1}({\mathcal{P}}_{A})$, and
let

$\Theta=\frac{1}{2}\cdot(\textnormal{id},\lambda)^{\ast}\ell\in\textnormal{CH}^{1}(A)_{\mathbb{Q}}$
(8)

be the symmetric ample class corresponding to the polarization. Here
$(\textnormal{id},\lambda)$ is the morphism $(\textnormal{id},\lambda)\colon
A\to A\times{\widehat{A}}$. One can understand the relation between

$\Gamma_{\Theta}\coloneqq\Theta^{g-1}/(g-1)!\in\textnormal{CH}_{1}(A)_{\mathbb{Q}}$

and
${\mathscr{R}}_{A}={\ell}^{2g-1}/(2g-1)!\in\textnormal{CH}_{1}(A\times{\widehat{A}})_{\mathbb{Q}}$
in the following way. Define

$\displaystyle j_{1}$ $\displaystyle\colon A\to A\times{\widehat{A}},\quad
x\mapsto(x,0),\quad{\textnormal{ and }}$ $\displaystyle j_{2}$
$\displaystyle\colon{\widehat{A}}\to A\times{\widehat{A}},\quad
y\mapsto(0,y).$

Let ${\widehat{\Theta}}\in\textnormal{CH}^{1}({\widehat{A}})_{\mathbb{Q}}$ be
the dual of $\Theta$, and define a one-cycle $\tau$ on $A\times{\widehat{A}}$
as follows:

$\displaystyle\tau\coloneqq
j_{1,\ast}(\Gamma_{\Theta})+j_{2,\ast}(\Gamma_{{\widehat{\Theta}}})-(\textnormal{id},\lambda)_{\ast}(\Gamma_{\Theta})\in\textnormal{CH}_{1}(A\times{\widehat{A}})_{\mathbb{Q}}.$

######  Lemma 6.6.

One has
$\tau=(-1)^{g+1}\cdot{\mathscr{R}}_{A}\in\textnormal{CH}_{1}(A\times{\widehat{A}})_{\mathbb{Q}}$.

###### Proof.

Identify $A$ and ${\widehat{A}}$ via $\lambda$. This gives
$\ell=m^{\ast}(\Theta)-\pi_{1}^{\ast}(\Theta)-\pi_{2}^{\ast}(\Theta)$, and the
Fourier transform becomes an endomorphism
${\mathcal{F}}_{A}\colon\textnormal{CH}(A)_{\mathbb{Q}}\to\textnormal{CH}(A)_{\mathbb{Q}}$.

We claim that

$\tau=(-1)^{g}\cdot\left(\Delta_{\ast}{\mathcal{F}}_{A}(\Theta)-j_{1,\ast}{\mathcal{F}}_{A}(\Theta)-j_{2,\ast}{\mathcal{F}}_{A}(\Theta)\right).$

For this, it suffices to show that
${\mathcal{F}}_{A}(\Theta)=(-1)^{g-1}\cdot\Theta^{g-1}/(g-1)!\in\textnormal{CH}_{1}(A)_{\mathbb{Q}}$.
Now ${\mathcal{F}}_{A}(e^{\Theta})=e^{-\Theta}$ by Lemma 6.7 below. Moreover,
since $\Theta$ is symmetric, we have
$\Theta\in\textnormal{CH}^{1,0}(A)_{\mathbb{Q}}$, hence
$\Theta^{i}/i!\in\textnormal{CH}^{i,0}(A)_{\mathbb{Q}}$ for each $i\geq 0$.
Therefore,
${\mathcal{F}}_{A}\left(\Theta^{i}/i!\right)\in\textnormal{CH}^{g-i,0}(A)_{\mathbb{Q}}$
by [DM91, Lemma 2.18]. This implies that, in fact,

${\mathcal{F}}_{A}\left(\Theta^{i}/i!\right)=(-1)^{g-i}\cdot\Theta^{g-i}/(g-i)!\in\textnormal{CH}^{g-i,0}(A)_{\mathbb{Q}}$

for every $i$. In particular, the claim follows.

Next, recall that ${\mathcal{F}}_{A\times
A}(\ell)=(-1)^{g+1}\cdot{\mathscr{R}}_{A}$, see Claim ($\ast$). So at this
point, it suffices to prove the identity

${\mathcal{F}}_{A\times
A}(\ell)=(-1)^{g}\cdot\left(\Delta_{\ast}{\mathcal{F}}_{A}(\Theta)-j_{1,\ast}{\mathcal{F}}_{A}(\Theta)-j_{2,\ast}{\mathcal{F}}_{A}(\Theta)\right).$

To prove this, we use the following functoriality properties of the Fourier
transform on the level of rational Chow groups. Let $X$ and $Y$ be abelian
varieties and let $f\colon X\to Y$ be a homomorphism with dual homomorphism
${\widehat{f}}\colon{\widehat{Y}}\to{\widehat{X}}$. We then have the following
equalities [MP10, (3.7.1)]:

$({\widehat{f}})^{\ast}\circ{\mathcal{F}}_{X}={\mathcal{F}}_{Y}\circ
f_{\ast},\hskip 8.53581pt{\mathcal{F}}_{X}\circ f^{\ast}=(-1)^{\dim X-\dim
Y}\cdot({\widehat{f}})_{\ast}\circ{\mathcal{F}}_{Y}.$ (9)

Since $\ell=m^{\ast}\Theta-\pi_{1}^{\ast}\Theta-\pi_{2}^{\ast}\Theta$, it
follows from Equation (9) that

$\displaystyle{\mathcal{F}}_{A\times A}(\ell)$
$\displaystyle={\mathcal{F}}_{A\times
A}\left(m^{\ast}\Theta\right)-{\mathcal{F}}_{A\times
A}\left(\pi_{1}^{\ast}\Theta\right)-{\mathcal{F}}_{A\times
A}\left(\pi_{2}^{\ast}\Theta\right)$
$\displaystyle=(-1)^{g}\cdot\left(\Delta_{\ast}{\mathcal{F}}_{A}(\Theta)-j_{1,\ast}{\mathcal{F}}_{A}(\Theta)-j_{2,\ast}{\mathcal{F}}_{A}(\Theta)\right).$

∎

######  Lemma 6.7 (Beauville).

Let $A$ be an abelian variety over $k$, principally polarized by
$\lambda\colon A\xrightarrow{\sim}{\widehat{A}}$, and define
$\Theta=\frac{1}{2}\cdot(\textnormal{id},\lambda)^{\ast}c_{1}({\mathcal{P}}_{A})\in\textnormal{CH}^{1}(A)_{\mathbb{Q}}$.
Identify $A$ and ${\widehat{A}}$ via $\lambda$. With respect to the Fourier
transform

${\mathcal{F}}_{A}\colon\textnormal{CH}(A)_{\mathbb{Q}}\xrightarrow{\sim}\textnormal{CH}(A)_{\mathbb{Q}},\quad\text{
one has }\quad{\mathcal{F}}_{A}(e^{\Theta})=e^{-\Theta}.$

###### Proof.

Our proof follows the proof of [Bea82, Lemme 1], but has to be adapted, since
$\Theta$ does not necessarily come from a symmetric ample line bundle on $A$.
Since one still has
$\ell=m^{\ast}\Theta-\pi_{1}^{\ast}\Theta-\pi_{2}^{\ast}\Theta$, the argument
can be made to work: one has

$\displaystyle{\mathcal{F}}_{A}(e^{\Theta})$
$\displaystyle=\pi_{2,\ast}\left(e^{\ell}\cdot\pi_{1}^{\ast}e^{\Theta}\right)$
$\displaystyle=\pi_{2,\ast}\left(e^{m^{\ast}\Theta-\pi_{2}^{\ast}\Theta}\right)=e^{-\Theta}\pi_{2,\ast}(m^{\ast}e^{\Theta})\in\textnormal{CH}(A)_{\mathbb{Q}}.$

For codimension reasons, one has

$\pi_{2,\ast}(m^{\ast}e^{\Theta})=\pi_{2,\ast}m^{\ast}(\Theta^{g}/g!)=\deg(\Theta^{g}/g!)\in\textnormal{CH}^{0}(A)_{\mathbb{Q}}\cong\mathbb{Q}.$

Pull back $\Theta^{g}/g!$ along $A_{k_{s}}\to A$ to see that

$\deg(\Theta^{g}/g!)=1\in\textnormal{CH}^{0}(A)_{\mathbb{Q}}\cong\textnormal{CH}^{0}(A_{k_{s}})_{\mathbb{Q}},$

since over $k_{s}$ the cycle $\Theta$ becomes the cycle class attached to a
symmetric ample line bundle. ∎

####  4 Divided powers of algebraic cycles

It was asked by Bruno Kahn whether there exists a PD-structure on the Chow
ring of an abelian variety over any field with respect to its usual
(intersection) product. There are counterexamples over non-closed fields: see
[Esn04], where Esnault constructs an abelian surface $X$ and a line bundle
${\mathcal{L}}$ on $X$ such that $c_{1}({\mathcal{L}})\cdot
c_{1}({\mathcal{L}})$ is not divisible by $2$ in $\textnormal{CH}_{0}(X)$.
However, the case of algebraically closed fields remains open [MP10, Section
3.2]. What we do know, is the following:

######  Theorem 6.8 (Moonen–Polishchuk).

Let $A$ be an abelian variety over $k$. The ring
$\left(\textnormal{CH}(A),\star\right)$ admits a canonical PD-structure
$\gamma$ on the ideal $\textnormal{CH}_{>0}(A)\subset\textnormal{CH}(A)$. If
$k=\bar{k}$, then $\gamma$ extends to a PD-structure on the ideal generated by
$\textnormal{CH}_{>0}(A)$ and the zero cycles of degree zero.

In particular, for each element $x\in\textnormal{CH}_{>0}(A)$ and each
$n\in\mathbb{Z}_{\geq 1}$, there is a canonical element
$x^{[n]}\in\textnormal{CH}_{>0}(A)$ such that $n!x^{[n]}=x^{\star n}$, see
[Sta18, Tag 07GM]. For $x\in\textnormal{CH}_{>0}(A)$, we may then define

$\mathrm{E}(x)=\sum_{n\geq 0}x^{[n]}\in\textnormal{CH}(A)$

as the $\star$-exponential of $x$ in terms of its divided powers.

Together with the results of Section 3, Theorem 6.8 enables us to provide
criteria for the existence of a motivic weak integral Fourier transform.
Recall that for an abelian variety $A$ over $k$, principally polarized by
$\lambda\colon A\xrightarrow{\sim}{\widehat{A}}$, we defined
$\Theta\in\textnormal{CH}^{1}(A)_{\mathbb{Q}}$ as the symmetric ample class
attached to $\lambda$, see equation (8).

######  Theorem 6.9.

Let $A_{/k}$ be an abelian variety of dimension $g$. The following are
equivalent:

  1. 1.

The one-cycle
${\mathscr{R}}_{A}=c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$
lifts to $\textnormal{CH}_{1}(A\times{\widehat{A}})$.

  2. 2.

The abelian variety $A$ admits a motivic weak integral Fourier transform.

  3. 3.

The abelian variety $A\times{\widehat{A}}$ admits a motivic weak integral
Fourier transform.

Moreover, if $A$ carries a symmetric ample line bundle that induces a
principal polarization $\lambda\colon A\xrightarrow{\sim}{\widehat{A}}$, then
the above statements are equivalent to the following equivalent statements:

  1. 4.

The two-cycle
${\mathscr{S}}_{A}=c_{1}({\mathcal{P}}_{A})^{2g-2}/(2g-2)!\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$
lifts to $\textnormal{CH}_{2}(A\times{\widehat{A}})$.

  2. 5.

The one-cycle
$\Gamma_{\Theta}=\Theta^{g-1}/(g-1)!\in\textnormal{CH}(A)_{\mathbb{Q}}$ lifts
to a one-cycle in $\textnormal{CH}(A)$.

  3. 6.

The abelian variety $A$ admits a weak integral Fourier transform.

  4. 7.

The Fourier transform ${\mathcal{F}}_{A}$ satisfies
${\mathcal{F}}_{A}\left(\textnormal{CH}(A)/(\textnormal{torsion})\right)\subset\textnormal{CH}({\widehat{A}})/(\textnormal{torsion})$.

  5. 8.

There exists a PD-structure on the ideal
$\textnormal{CH}^{>0}(A)/(\textnormal{torsion})\subset\textnormal{CH}(A)/(\textnormal{torsion})$.

###### Proof.

Suppose that 1 holds, and let
$\Gamma\in\textnormal{CH}_{1}(A\times{\widehat{A}})$ be a cycle such that
$\Gamma_{\mathbb{Q}}={\mathscr{R}}_{A}$. Then consider the cycle
$(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot\Gamma)\in\textnormal{CH}(A\times{\widehat{A}})$.
By Lemma 6.5, we have

$\displaystyle(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot\Gamma)_{\mathbb{Q}}=(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot\Gamma_{\mathbb{Q}})$
$\displaystyle=(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot{\mathscr{R}}_{A})=\textnormal{ch}({\mathcal{P}}_{A})\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}.$

Thus 2 holds. We claim that 3 holds as well. Indeed, consider the line bundle
${\mathcal{P}}_{A\times{\widehat{A}}}$ on the abelian variety
$X=A\times{\widehat{A}}\times{\widehat{A}}\times A$; one has that
${\mathcal{P}}_{A\times{\widehat{A}}}\cong\pi_{13}^{\ast}{\mathcal{P}}_{A}\otimes\pi_{24}^{\ast}{\mathcal{P}}_{{\widehat{A}}}$,
which implies that we have the following equality in
$\textnormal{CH}_{1}(X)_{\mathbb{Q}}$:

$\begin{split}{\mathscr{R}}_{A\times{\widehat{A}}}&=\frac{\left(\pi_{13}^{\ast}c_{1}({\mathcal{P}}_{A})+\pi_{24}^{\ast}c_{1}({\mathcal{P}}_{{\widehat{A}}})\right)^{4g-1}}{(4g-1)!}\\\
&=\frac{\pi_{13}^{\ast}c_{1}({\mathcal{P}}_{A})^{2g-1}\cdot\pi_{24}^{\ast}c_{1}({\mathcal{P}}_{{\widehat{A}}})^{2g}+\pi_{13}^{\ast}c_{1}({\mathcal{P}}_{A})^{2g}\cdot\pi_{24}^{\ast}c_{1}({\mathcal{P}}_{{\widehat{A}}})^{2g-1}}{(2g)!(2g-1)!}\\\
&=\frac{\pi_{13}^{\ast}c_{1}({\mathcal{P}}_{A})^{2g-1}\cdot\pi_{24}^{\ast}((2g)!\cdot[0]_{A\times{\widehat{A}}})+\pi_{13}^{\ast}((2g)!\cdot[0]_{{\widehat{A}}\times
A})\cdot\pi_{24}^{\ast}c_{1}({\mathcal{P}}_{{\widehat{A}}})^{2g-1}}{(2g)!(2g-1)!}\\\
&=\pi_{13}^{\ast}(\frac{c_{1}({\mathcal{P}}_{A})^{2g-1}}{(2g-1)!})\cdot\pi_{24}^{\ast}([0]_{A\times{\widehat{A}}})+\pi_{13}^{\ast}([0]_{{\widehat{A}}\times
A})\cdot\pi_{24}^{\ast}(\frac{c_{1}({\mathcal{P}}_{{\widehat{A}}})^{2g-1}}{(2g-1)!}).\end{split}$
(10)

We conclude that ${\mathscr{R}}_{A\times{\widehat{A}}}$ lifts to
$\textnormal{CH}_{1}(X)$ which, by the implication
$[\ref{motivicone}\implies\ref{motivictwo}]$ (that has already been proved),
implies that $A\times{\widehat{A}}$ admits a motivic weak integral Fourier
transform. On the other hand, the implication
$[\ref{motivicthree}\implies\ref{motivicone}]$ follows from the fact that
$(-1)^{g}\cdot{\mathcal{F}}_{{\widehat{A}}\times
A}(-{\widehat{\ell}})={\mathscr{R}}_{A}$ (see Equation (7)) and the fact that
an abelian variety admits a motivic weak integral Fourier transform if and
only if its dual abelian variety does. Therefore, we have
$[\ref{motivicone}\iff\ref{motivictwo}\iff\ref{motivicthree}]$.

From now on, assume that $A$ is principally polarized by $\lambda\colon
A\xrightarrow{\sim}A$, where $\lambda$ is the polarization attached to a
symmetric ample line bundle ${\mathcal{L}}$ on $A$. Moreover, in what follows
we shall identify ${\widehat{A}}$ and $A$ via $\lambda$.

Suppose that 4 holds and let $S_{A}\in\textnormal{CH}_{2}(A\times
A)=\textnormal{CH}^{2g-2}(A\times A)$ be such that
$(S_{A})_{\mathbb{Q}}={\mathscr{S}}_{A}\in\textnormal{CH}_{2}(A\times
A)_{\mathbb{Q}}$. Define
$\textnormal{CH}^{1,0}(A)\coloneqq\textnormal{Pic}^{\textnormal{sym}}(A)$ to
be the group of isomorphism classes of symmetric line bundes on $A$. Then
$S_{A}$ induces a homomorphism
${\mathcal{F}}\colon\textnormal{CH}^{1,0}(A)\to\textnormal{CH}_{1}(A)$ defined
as the composition

$\displaystyle\textnormal{CH}^{1,0}(A)\xrightarrow{\pi_{1}^{\ast}}\textnormal{CH}^{1}(A\times
A)\xrightarrow{\cdot S_{A}}$ $\displaystyle\textnormal{CH}^{2g-1}(A\times A)$
$\displaystyle=\textnormal{CH}_{1}(A\times
A)\xrightarrow{\pi_{2,\ast}}\textnormal{CH}_{1}(A).$

Since
${\mathcal{F}}_{A}\left(\textnormal{CH}^{1,0}(A)_{\mathbb{Q}}\right)\subset\textnormal{CH}_{1}(A)_{\mathbb{Q}}$
(see [DM91, Lemma 2.18]) we see that

$\textstyle{\textnormal{CH}^{1,0}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}}$$\textstyle{\textnormal{CH}_{1}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textnormal{CH}^{1,0}(A)_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}_{A}}$$\textstyle{\textnormal{CH}_{1}(A)_{\mathbb{Q}}}$
(11)

commutes. Since the line bundle ${\mathcal{L}}$ is symmetric, we have

$\displaystyle\begin{split}\Theta&=\frac{1}{2}\cdot(\textnormal{id},\lambda)^{\ast}c_{1}({\mathcal{P}}_{A})=\frac{1}{2}\cdot
c_{1}\left((\textnormal{id},\lambda)^{\ast}{\mathcal{P}}_{A}\right)\\\
&=\frac{1}{2}\cdot
c_{1}({\mathcal{L}}\otimes{\mathcal{L}})=c_{1}({\mathcal{L}})\in\textnormal{CH}^{1}(A)_{\mathbb{Q}}.\end{split}$
(12)

The class $c_{1}({\mathcal{L}})\in\textnormal{CH}^{1,0}(A)$ of the line bundle
${\mathcal{L}}$ thus lies above
$\Theta\in\textnormal{CH}^{1}(A)_{\mathbb{Q}}$. Therefore,
${\mathcal{F}}(c_{1}({\mathcal{L}}))\in\textnormal{CH}_{1}(A)$ lies above
$\Gamma_{\Theta}=(-1)^{g-1}{\mathcal{F}}_{A}(\Theta)$ by the commutativity of
(11), and 5 holds.

Suppose that 5 holds. Then 1 follows readily from Lemma 6.6. Moreover, if 2
holds, then $\textnormal{ch}({\mathcal{P}}_{A})\in\textnormal{CH}(A\times
A)_{\mathbb{Q}}$ lifts to $\textnormal{CH}(A\times A)$, hence in particular 4
holds. Since we have already proved that 1 implies 2, we conclude that
$[\ref{motiviczero}\implies\ref{motivicfour}\implies\ref{motivicone}\implies\ref{motivictwo}\implies\ref{motiviczero}]$.

The implications
$[\ref{motivictwo}\implies\ref{motivicfive}\implies\ref{motivicsix}]$ are
trivial. Assume that 7 holds. By Equation (12),
$\Theta\in\textnormal{CH}^{1}(A)_{\mathbb{Q}}$ lifts to
$\textnormal{CH}^{1}(A)$, hence

${\mathcal{F}}_{A}(\Theta)=(-1)^{g-1}\cdot\Gamma_{\Theta}$

lifts to $\textnormal{CH}_{1}(A)$, i.e. 5 holds.

Assume that 7 holds. The fact that
${\mathcal{F}}_{A}\left(\textnormal{CH}(A)/(\textnormal{torsion})\right)\subset\textnormal{CH}(A)/(\textnormal{torsion})$
implies that

$\displaystyle\textnormal{CH}(A)/(\textnormal{torsion})$
$\displaystyle={\mathcal{F}}_{A}\left({\mathcal{F}}_{A}\left(\textnormal{CH}(A)/(\textnormal{torsion})\right)\right)$
$\displaystyle\subset{\mathcal{F}}_{A}\left(\textnormal{CH}(A)/(\textnormal{torsion})\right)\subset\textnormal{CH}(A)/(\textnormal{torsion}).$

Thus the restriction of the Fourier transform ${\mathcal{F}}_{A}$ to
$\textnormal{CH}(A)/(\textnormal{torsion})$ defines an isomorphism

${\mathcal{F}}_{A}\colon\textnormal{CH}(A)/(\textnormal{torsion})\xrightarrow{\sim}\textnormal{CH}(A)/(\textnormal{torsion}).$

If $R$ is a ring and $\gamma$ a PD-structure on an ideal $I\subset R$, then
$\gamma$ extends to a PD-structure on $I/(\textnormal{torsion})\subset
R/(\textnormal{torsion})$. Thus, the ideal
$\textnormal{CH}_{>0}(A)/(\textnormal{torsion})$ of
$\textnormal{CH}(A)/(\textnormal{torsion})$ admits a PD-structure for the
Pontryagin product $\star$ by Theorem 6.8. Since ${\mathcal{F}}_{A}$ exchanges
Pontryagin and intersection products up to sign [Bea82, Proposition 3(ii)], it
follows that 8 holds. Since 8 trivially implies 5, we are done. ∎

######  Question 6.10 (Moonen–Polishchuk [MP10], Totaro [Tot21]).

Let $A$ be any principally polarized abelian variety over $k=\bar{k}$. Are the
equivalent conditions in Theorem 6.9 satisfied for $A$?

######  Remark 6.11.

For the Jacobian $A=J(C)$ of a hyperelliptic curve $C$, the answer to Question
6.10 is "yes" [MP10].

Similarly, there is a relation between integral Fourier transforms up to
homology and the algebraicity of minimal cohomology classes induced by
Poincaré line bundles and theta divisors.

######  Proposition 6.12.

Let $A$ be an abelian variety of dimension $g$ over $k$.

The following are equivalent:

  1. 1.

The class

$\rho_{A}\coloneqq
c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}_{\textnormal{\'{e}t}}((A\times{\widehat{A}})_{k_{s}},\mathbb{Z}_{\ell}(2g-1))$

is the class of a cycle in $\textnormal{CH}_{1}(A\times{\widehat{A}})$.

  2. 2.

The abelian variety $A$ admits a motivic integral Fourier transform up to
homology.

  3. 3.

The abelian variety $A\times{\widehat{A}}$ admits a motivic integral Fourier
transform up to homology.

Moreover, if $A$ carries an ample line bundle that induces a principal
polarization $\lambda\colon A\xrightarrow{\sim}{\widehat{A}}$, then the above
statements are equivalent to the following equivalent statements:

  1. 4.

The class

$\sigma_{A}\coloneqq
c_{1}({\mathcal{P}}_{A})^{2g-2}/(2g-2)!\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{4g-4}((A\times{\widehat{A}})_{k_{s}},\mathbb{Z}_{\ell}(2g-2))$

is the class of a cycle in $\textnormal{CH}_{2}(A\times{\widehat{A}})$.

  2. 5.

The class
$\gamma_{\theta}=\theta^{g-1}/(g-1)!\in{\mathrm{H}}^{2g-2}_{\textnormal{\'{e}t}}(A_{k_{s}},\mathbb{Z}_{\ell}(g-1))$
lifts to a cycle in $\textnormal{CH}_{1}(A)$.

  3. 6.

The abelian variety $A$ admits an integral Fourier transform up to homology.

###### Proof.

The proof of Theorem 6.9 can easily be adapted to this situation. ∎

######  Proposition 6.13.

  1. 1.

If $k=\mathbb{C}$, then each of the statements $\ref{motivicone-
new}-\ref{motivicfive-new}$ in Proposition 6.12 is equivalent to the same
statement with étale cohomology replaced by Betti cohomology.

  2. 2.

Proposition 6.12 remains valid if one replaces integral Chow groups in
statements 1, 4 and 5 by their tensor product with $\mathbb{Z}_{\ell}$ and
‘integral Fourier transform up to homology’ by ‘$\ell$-adic integral Fourier
transform up to homology’ in statements 2, 3 and 6.

###### Proof.

  1. 1.

In this case $\mathbb{Z}_{\ell}(i)=\mathbb{Z}_{\ell}$ and the Artin comparison
isomorphism

${\mathrm{H}}^{2i}_{\textnormal{\'{e}t}}(A,\mathbb{Z}_{\ell})\xrightarrow{\sim}{\mathrm{H}}^{2i}(A(\mathbb{C}),\mathbb{Z}_{\ell})$

[AGV71, III, Exposé XI] is compatible with the cycle class map. Since the map
${\mathrm{H}}^{2i}(A(\mathbb{C}),\mathbb{Z})\to{\mathrm{H}}_{\textnormal{\'{e}t}}^{2i}(A,\mathbb{Z}_{\ell})$
is injective, a class $\beta\in{\mathrm{H}}^{2i}(A(\mathbb{C}),\mathbb{Z})$ is
in the image of
$cl\colon\textnormal{CH}^{i}(A)\to{\mathrm{H}}^{2i}(A(\mathbb{C}),\mathbb{Z})$
if and only if its image
$\beta_{\ell}\in{\mathrm{H}}^{2i}_{\textnormal{\'{e}t}}(A,\mathbb{Z}_{\ell})$
is in the image of
$cl\colon\textnormal{CH}^{i}(A)\to{\mathrm{H}}^{2i}_{\textnormal{\'{e}t}}(A,\mathbb{Z}_{\ell})$.

  2. 2.

Indeed, for an abelian variety $A$ over $k$, the PD-structure on
$\textnormal{CH}_{>0}(A)\subset(\textnormal{CH}(A),\star)$ induces a PD-
structure on
$\textnormal{CH}_{>0}(A)\otimes\mathbb{Z}_{\ell}\subset(\textnormal{CH}(A)_{\mathbb{Z}_{\ell}},\star)$
by [Sta18, Tag 07H1], because the ring map
$(\textnormal{CH}(A),\star)\to(\textnormal{CH}(A)_{\mathbb{Z}_{\ell}},\star)$
is flat. The latter follows from the flatness of
$\mathbb{Z}\to\mathbb{Z}_{\ell}$.

∎

###  Chapter 7 One-cycles on abelian varieties

This chapter is based on joint work with Thorsten Beckmann.

####  1 Introduction

In this chapter we provide applications of the results developed in the
previous Chapter 6. These applications concern the cycle class map for curves
on an abelian variety $A$. More precisely, we will consider the integral Hodge
conjecture for one-cycles when $A$ is defined over $\mathbb{C}$, and the
integral Tate conjecture for one-cycles when $A$ is defined over the separable
closure of a finitely generated field.

To state the most important results of Chapter 7, let us recall how the
complex cycle class map was defined (see also Section 3). Whenever
$\iota\colon C\hookrightarrow A$ is a smooth curve, the image of the
fundamental class under the pushforward map

$\iota_{\ast}\colon{\mathrm{H}}_{2}(C,\mathbb{Z})\to{\mathrm{H}}_{2}(A,\mathbb{Z})\cong{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$

defines a cohomology class $[C]\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$. This
construction extends to one-cycles and factors modulo rational equivalence.
The cycle class map for curves on $A$ is the canonical homomorphism defined in
this way:

$cl\colon\textnormal{CH}_{1}(A)\to\textnormal{Hdg}^{2g-2}(A,\mathbb{Z}).$

It extends to a natural graded ring homomorphism
$cl\colon\textnormal{CH}(A)\to{\mathrm{H}}^{\bullet}(A,\mathbb{Z})$.

The liftability of the Fourier transform that we studied in Chapter 6 turns
out to have important consequences for the image of the cycle class map. An
element $\alpha\in{\mathrm{H}}^{\bullet}(A,\mathbb{Z})$ is called algebraic if
it is in the image of $cl$, and that $A$ satisfies the integral Hodge
conjecture for $i$-cycles if all Hodge classes in
${\mathrm{H}}^{2g-2i}(A,\mathbb{Z})$ are algebraic. Although the integral
Hodge conjecture fails in general [AH62, tre92, Tot97], it is an open question
for abelian varieties. The main result of Chapter 7 is as follows.

######  Theorem 7.1 (with T. Beckmann).

Let $A$ be a complex abelian variety of dimension $g$ with Poincaré bundle
${\mathcal{P}}_{A}$. The following three statements are equivalent:

  1. 1.

The cohomology class
$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic.

  2. 2.

The Chern character
$\textnormal{ch}({\mathcal{P}}_{A})=\exp(c_{1}({\mathcal{P}}_{A}))\in{\mathrm{H}}^{\bullet}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic.

  3. 3.

The integral Hodge conjecture for one-cycles holds for $A\times{\widehat{A}}$.

Any of these statements implies that

  1. 4.

The integral Hodge conjecture for one-cycles holds for $A$ and
${\widehat{A}}$.

Suppose that $A$ is principally polarized by
$\theta\in\textnormal{Hdg}^{2}(A,\mathbb{Z})$ and consider the following
statements:

  1. 5.

The minimal cohomology class
$\gamma_{\theta}\coloneqq\theta^{g-1}/(g-1)!\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$
is algebraic.

  2. 6.

The cohomology class
$c_{1}({\mathcal{P}}_{A})^{2g-2}/(2g-2)!\in{\mathrm{H}}^{4g-4}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic.

Then statements
$\ref{introitem:minimalpoincare}-\ref{introitem:minimalpoincare2}$ are
equivalent. If they hold, then the class
$\theta^{i}/i!\in{\mathrm{H}}^{2i}(A,\mathbb{Z})$ is algebraic for every
positive integer $i$.

Remark that Condition 5 is stable under products, so a product of principally
polarized abelian varieties satisfies the integral Hodge conjecture for one-
cycles if and only if each of the factors does. More importantly, if $J(C)$ is
the Jacobian of a smooth projective curve $C$ of genus $g$, then every
integral Hodge class of degree $2g-2$ on $J(C)$ is a $\mathbb{Z}$-linear
combination of curves classes:

######  Theorem 7.2.

Let $C_{1},\dotsc,C_{n}$ be smooth projective curves over $\mathbb{C}$. Then
the integral Hodge conjecture for one-cycles holds for the product of
Jacobians $J(C_{1})\times\cdots\times J(C_{n})$.

See Remark 7.9.1 for another approach towards Theorem 7.2 in the case $n=1$. A
second consequence of Theorem 7.1 is that the integral Hodge conjecture for
one-cycles on principally polarized complex abelian varieties is stable under
specialization, see Corollary 7.10. An application of somewhat different
nature is the following density result, proven in Section 2:

######  Theorem 7.3.

Let $\delta=(\delta_{1},\dotsc,\delta_{g})$ be positive integers such that
$\delta_{i}|\delta_{i+1}$ and let ${\mathsf{A}}_{g,\delta}(\mathbb{C})$ be the
coarse moduli space of polarized abelian varieties over $\mathbb{C}$ with
polarization type $\delta$. There is a countable union
$X\subset{\mathsf{A}}_{g,\delta}(\mathbb{C})$ of closed algebraic subvarieties
of dimension at least $g$, that satisfies the following property: $X$ is dense
in the analytic topology, and the integral Hodge conjecture for one-cycles
holds for those polarized abelian varieties whose isomorphism class defines a
point in $X$.

######  Remark 7.4.

The lower bound that we obtain on the dimension of the components of $X$
actually depends on $\delta$ and is often greater than $g$. For instance, when
$\delta=1$ and $g\geq 2$, there is a set $X$ as in the theorem, whose elements
are prime-power isogenous to products of Jacobians of curves. Therefore, the
components of $X$ have dimension $3g-3$ in this case, c.f. Remark 7.15.

One could compare Theorem 7.1 with the following statement, proven by
Grabowski [Gra04]: if $g$ is a positive integer such that the minimal class
$\gamma_{\theta}=\theta^{g-1}/(g-1)!$ of every principally polarized abelian
variety of dimension $g$ is algebraic, then every abelian variety of dimension
$g$ satisfies the integral Hodge conjecture for one-cycles. In this way, he
proved the integral Hodge conjecture for abelian threefolds, a result which
has been extended to smooth projective threefolds $X$ with $K_{X}=0$ by Voisin
and Totaro [Voi06, Tot21]. For abelian varieties of dimension greater than
three, not many unconditional statements seem to have been known.

In Section 3, we consider an abelian variety $A_{/\mathbb{C}}$ of dimension
$g$ and ask: if $n\in\mathbb{Z}_{\geq 1}$ is such that $n\cdot
c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})_{\textnormal{alg}}$,
can we bound the order of ${\mathrm{Z}}^{2g-2}(A)$ in terms of $g$ and $n$?
For a smooth complex projective $d$-dimensional variety $X$,
${\mathrm{Z}}^{2d-2}(X)$ is called the degree $2d-2$ Voisin group of $X$
[Per22], is a stably birational invariant [Voi07, Lemma 15], and related to
the unramified cohomology groups by Colliot-Thélène–Voisin and Schreieder
[CTV12, Sch20]. We prove that if $n\cdot
c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!$ is algebraic, then
$\gcd(n^{2},(2g-2)!)\cdot{\mathrm{Z}}^{2g-2}(A)=(0)$. In particular,
$(2g-2)!\cdot{\mathrm{Z}}^{2g-2}(A)=(0)$ for any $g$-dimensional complex
abelian variety $A$. Moreover, if $A$ is principally polarized by
$\theta\in\textnormal{NS}(A)$ and if
$n\cdot\gamma_{\theta}\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$ is algebraic, then
$n\cdot c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!$ is algebraic. Since it is
well known that for Prym varieties, the Hodge class $2\cdot\gamma_{\theta}$ is
algebraic, these observations lead to the following result (see also Theorem
7.19).

######  Theorem 7.5.

Let $A$ be a $g$-dimensional Prym variety over $\mathbb{C}$. Then
$4\cdot{\mathrm{Z}}^{2g-2}(A)=(0)$.

Recall that in our study of the liftability of the Fourier transform, carried
out in the previous Chapter 6, we considered abelian varieties defined over
arbitrary fields. This generality allows us now to obtain the analogue of
Theorem 7.1 over the separable closure $k$ of a finitely generated field.

A smooth projective variety $X$ of dimension $d$ over $k$ satisfies the
integral Tate conjecture for one-cycles over $k$ if, for every prime number
$\ell$ different from $\textnormal{char}(k)$ and for some finitely generated
field of definition $k_{0}\subset k$ of $X$, the cycle class map

$cl\colon\textnormal{CH}_{1}(X)_{\mathbb{Z}_{\ell}}=\textnormal{CH}_{1}(X)\otimes_{\mathbb{Z}}{\mathbb{Z}_{\ell}}\to\bigcup_{U}{\mathrm{H}}_{\textnormal{\'{e}t}}^{2d-2}(X,\mathbb{Z}_{\ell}(d-1))^{U}$
(1)

is surjective, where $U$ ranges over the open subgroups of
$\textnormal{Gal}(k/k_{0})$.

######  Theorem 7.6.

Let $A$ be an abelian variety of dimension $g$ over the separable closure $k$
of a finitely generated field.

  * •

The abelian variety $A$ satisfies the integral Tate conjecture for one-cycles
over $k$ if the cohomology class

$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z}_{\ell}(2g-1))$

is the class of a one-cycle with $\mathbb{Z}_{\ell}$-coefficients for every
prime number $\ell<(2g-1)!$ unequal to $\textnormal{char}(k)$.

  * •

Suppose that $A$ is principally polarized and let
$\theta_{\ell}\in{\mathrm{H}}^{2}_{\textnormal{\'{e}t}}(A,\mathbb{Z}_{\ell}(1))$
be the class of the polarization. The map (1) is surjective if

$\gamma_{\theta_{\ell}}\coloneqq\theta_{\ell}^{g-1}/(g-1)!\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{2g-2}(A,\mathbb{Z}_{\ell}(g-1))$

is in its image. In particular, if $\ell>(g-1)!$ then this always holds. Thus
$A$ satisfies the integral Tate conjecture for one-cycles if and only if
$\gamma_{\theta_{\ell}}$ is in the image of (1) for every prime number
$\ell<(g-1)!$ unequal to $\textnormal{char}(k)$.

Theorem 7.6 implies in particular that products of Jacobians of smooth
projective curves over $k$ satisfy the integral Tate conjecture for one-cycles
over $k$. Moreover, for an abelian variety $A_{K}$ over a number field
$K\subset\mathbb{C}$, the integral Hodge conjecture for one-cycles on
$A_{\mathbb{C}}$ is equivalent to the integral Tate conjecture for one-cycles
on $A_{\bar{K}}$ (Corollary 7.21), which in turn implies the integral Tate
conjecture for one-cycles on the geometric special fiber
$A_{\overline{k}({\mathfrak{p}})}$ of the Néron model of $A_{K}$ over
$\mathcal{O}_{K}$ for any prime ${\mathfrak{p}}\subset\mathcal{O}_{K}$ at
which $A_{K}$ has good reduction (Corollary 7.22). Finally, we obtain the
analogue of Theorem 7.3 in positive characteristic as well. The definition for
a smooth projective variety over the algebraic closure $k$ of a finitely
generated field to satisfy the integral Tate conjecture for one-cycles over
$k$ is analogous to the definition above (see e.g. [CP15]).

######  Theorem 7.7.

Let $k$ be the algebraic closure of a finitely generated field of
characteristic $p>0$. Let ${\mathsf{A}}_{g}$ be the coarse moduli space over
$k$ of principally polarized abelian varieties of dimension $g$ over $k$. Let
$X\subset{\mathsf{A}}_{g}(k)$ be the subset of moduli points attached to
principally polarized abelian varieties over $k$ that satisfy the integral
Tate conjecture for one-cycles over $k$. Then $X$ is Zariski dense in
${\mathsf{A}}_{g}$.

####  2 The integral Hodge conjecture

In this section we use the theory developed in Chapter 6 to prove Theorem 7.1.
We also prove some applications of Theorem 7.1: the integral Hodge conjecture
for one-cycles on products of Jacobians (Theorem 7.2), the fact that the
integral Hodge conjecture for one-cycles on principally polarized complex
abelian varieties is stable under specialization (Corollary 7.10) and density
of polarized abelian varieties satisfying the integral Hodge conjecture for
one-cycles (Theorem 7.3).

#####  1 Proof of the main theorem

Let us prove Theorem 7.1.

###### Proof of Theorem 7.1.

Suppose that 1 holds. Then 2 holds by Propositions 6.12 and 6.13.1. Suppose
that 2 holds. Then 4 follows from Lemma 6.4. So we have
$[\ref{introitem:minimalpoincare}\iff\ref{introitem:integralpoincare}\implies\ref{introitem:IHC}]$.
If 1 holds, then
$\rho_{A}=c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic, which implies that
$\rho_{{\widehat{A}}}\in{\mathrm{H}}^{4g-2}({\widehat{A}}\times A,\mathbb{Z})$
is algebraic. Therefore,
$\rho_{A\times{\widehat{A}}}\in{\mathrm{H}}^{8g-2}(A\times{\widehat{A}}\times{\widehat{A}}\times
A,\mathbb{Z})$ is algebraic by Equation (10). We then apply the implication
$[\ref{introitem:minimalpoincare}\implies\ref{introitem:IHC}]$ to the abelian
variety $A\times{\widehat{A}}$, which shows that 3 holds. Since
$[\ref{introitem:integralhodgeforproduct}\implies\ref{introitem:minimalpoincare}]$
is trivial, we have proven
$[\ref{introitem:minimalpoincare}\iff\ref{introitem:integralpoincare}\iff\ref{introitem:integralhodgeforproduct}\implies\ref{introitem:IHC}]$.

Next, assume that $A$ is principally polarized by
$\theta\in\textnormal{NS}(A)\subset{\mathrm{H}}^{2}(A,\mathbb{Z})$. The
directions $[\ref{introitem:IHC}\implies\ref{introitem:minimalclass}]$ and
$[\ref{introitem:integralpoincare}\implies\ref{introitem:minimalpoincare2}]$
are trivial and
$[\ref{introitem:minimalclass}\implies\ref{introitem:minimalpoincare}]$
follows from Propositions 6.12 and 6.13.1. We claim that 6 implies 4. Define

$\sigma_{A}=c_{1}({\mathcal{P}}_{A})^{2g-2}/(2g-2)!\in{\mathrm{H}}^{4g-4}(A\times{\widehat{A}},\mathbb{Z})$

and let $S\in\textnormal{CH}_{2}(A\times{\widehat{A}})$ be such that
$cl(S)=\sigma_{A}$. The squares in the following diagram commute:

$\textstyle{\textnormal{CH}^{1}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\scriptstyle{\pi_{1}^{\ast}}$$\textstyle{\textnormal{CH}^{1}(A\times{\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\scriptstyle{\cdot
S}$$\textstyle{\textnormal{CH}^{2g-1}(A\times{\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\scriptstyle{\pi_{2,\ast}}$$\textstyle{\textnormal{CH}_{1}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\textstyle{{\mathrm{H}}^{2}(A,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}^{\ast}}$$\textstyle{{\mathrm{H}}^{2}(A\times{\widehat{A}},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot\sigma_{A}}$$\textstyle{{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2,\ast}}$$\textstyle{{\mathrm{H}}^{2g-2}({\widehat{A}},\mathbb{Z}).}$
(2)

Since
${\mathscr{F}}_{A}=\pi_{2,\ast}\left(\textnormal{ch}({\mathcal{P}}_{A})\cdot\pi_{1}^{\ast}(-)\right)$
restricts to an isomorphism

${\mathscr{F}}_{A}\colon{\mathrm{H}}^{2}(A,\mathbb{Z})\xrightarrow{\sim}{\mathrm{H}}^{2g-2}({\widehat{A}},\mathbb{Z})$

by [Bea82, Proposition 1], the composition
$\pi_{2,\ast}\circ(-\cdot\sigma_{A})\circ\pi_{1}^{\ast}$ on the bottom row of
(2) is an isomorphism. By Lefschetz $(1,1)$, the map
$cl\colon\textnormal{CH}_{1}({\widehat{A}})\to\textnormal{Hdg}^{2g-2}({\widehat{A}},\mathbb{Z})$
is therefore surjective.

It remains to prove the algebraicity of the classes
$\theta^{i}/i!\in{\mathrm{H}}^{2i}(A,\mathbb{Z})$. This follows from Theorem
6.8 and the following equality, see [Bea82, Corollaire 2]):

$\frac{\theta^{i}}{i!}=\frac{\gamma_{\theta}^{\star
j}}{j!},\quad\gamma_{\theta}=\frac{\theta^{g-1}}{(g-1)!}\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z}),\quad
i+j=g.$

Therefore, the proof is finished. ∎

######  Corollary 7.8.

Let $A$ and $B$ be complex abelian varieties of respective dimensions
$g_{A},g_{B}$.

  * •

The Hodge classes
$\rho_{A}\in{\mathrm{H}}^{4g_{A}-2}(A\times{\widehat{A}},\mathbb{Z})$ and
$\rho_{B}\in{\mathrm{H}}^{4g_{B}-2}(B\times{\widehat{B}},\mathbb{Z})$ are
algebraic if and only if $A\times{\widehat{A}}$, $B\times{\widehat{B}}$,
$A\times B$ and ${\widehat{A}}\times{\widehat{B}}$ satisfy the integral Hodge
conjecture for one-cycles.

  * •

If $A$ and $B$ are principally polarized, then the integral Hodge conjecture
for one-cycles holds for $A\times B$ if and only if it holds for $A$ and $B$.

###### Proof.

The first statement follows from Theorem 7.1 and Equation (10). The second
statement follows from the fact that the minimal cohomology class of the
product $A\times B$ is algebraic if and only if the minimal cohomology classes
of the factors $A$ and $B$ are both algebraic. ∎

###### Proof of Theorem 7.2.

By Corollary 7.8 we may assume $n=1$, so let $C$ be a smooth projective curve.
Let $p\in C$ and consider the morphism $\iota\colon C\to J(C)$ defined by
sending a point $q$ to the isomorphism class of the degree zero line bundle
$\mathcal{O}(p-q)$. Then
$cl(\iota(C))=\gamma_{\theta}\in{\mathrm{H}}^{2g-2}(J(C),\mathbb{Z})$ by
Poincaré’s formula [ACGH85], so $\gamma_{\theta}$ is algebraic and the result
follows from Theorem 7.1. ∎

######  Remarks 7.9.

  1. 1.

Let us give another proof of Theorem 7.2 in the case $n=1$, i.e. let $C$ be a
smooth projective curve of genus $g$ and let us prove the integral Hodge
conjecture for one-cycles on $J(C)$ in a way that does not use Fourier
transforms. It is classical that any Abel-Jacobi map $C^{(g)}\to J(C)$ is
birational. On the other hand, the integral Hodge conjecture for one-cycles is
a birational invariant, see [Voi07, Lemma 15]. Therefore, to prove it for
$J(C)$ it suffices to prove it for $C^{(g)}$. One then uses [Bn02, Corollary
5] which says that for each $n\in\mathbb{Z}_{\geq 1}$, there is a natural
polarization $\eta$ on the $n$-fold symmetric product $C^{(n)}$ such that for
any $i\in\mathbb{Z}_{\geq 0}$, the map
$\eta^{n-i}\cup(-)\colon{\mathrm{H}}^{i}(C^{(n)},\mathbb{Z})\to{\mathrm{H}}^{2n-i}(C^{(n)},\mathbb{Z})$
is an isomorphism. In particular, the variety $C^{(n)}$ satisfies the integral
Hodge conjecture for one-cycles for any positive integer $n$.

  2. 2.

Along these lines, observe that the integral Hodge conjecture for one-cycles
holds not only for symmetric products of smooth projective complex curves but
also for any product $C_{1}\times\cdots\times C_{n}$ of smooth projective
curves $C_{i}$ over $\mathbb{C}$. Indeed, this follows readily from the
Künneth formula.

  3. 3.

Let $C$ be a smooth projective complex curve of genus $g$. Our proof of
Theorem 7.1 provides an explicit description of
$\textnormal{Hdg}^{2g-2}(J(C),\mathbb{Z})$ depending on
$\textnormal{Hdg}^{2}(J(C),\mathbb{Z})$. More generally, let $(A,\theta)$ be a
principally polarized abelian variety of dimension $g$, identify $A$ and
${\widehat{A}}$ via the polarization, and let
$\ell=c_{1}({\mathcal{P}}_{A})\in{\mathrm{H}}^{2}(A\times{\widehat{A}},\mathbb{Z})$.
Then $\ell=m^{\ast}(\theta)-\pi_{1}^{\ast}(\theta)-\pi_{2}^{\ast}(\theta)$,
which implies that

$\displaystyle\sigma_{A}=\frac{\ell^{2g-2}}{(2g-2)!}$
$\displaystyle=\sum_{\begin{subarray}{c}i,j,k\geq 0\\\
i+j+k=2g-2\end{subarray}}^{2g-2}(-1)^{j+k}\cdot
m^{\ast}\left(\frac{\theta^{i}}{i!}\right)\cdot\pi_{1}^{\ast}\left(\frac{\theta^{j}}{j!}\right)\cdot\pi_{2}^{\ast}\left(\frac{\theta^{k}}{k!}\right).$

Any $\beta\in\textnormal{Hdg}^{2g-2}(A,\mathbb{Z})$ is of the form
$\pi_{2,\ast}\left(\sigma_{A}\cdot\pi_{1}^{\ast}[D]\right)$, where $[D]=cl(D)$
for a divisor $D$ on $A$, as follows from (2). Therefore, any
$\beta\in\textnormal{Hdg}^{2g-2}(A,\mathbb{Z})$ may be written as

$\beta=\sum_{\begin{subarray}{c}i,j,k\geq 0\\\
i+j+k=2g-2\end{subarray}}^{2g-2}(-1)^{j+k}\cdot\pi_{2,\ast}\left(m^{\ast}\left(\frac{\theta^{i}}{i!}\right)\cdot\pi_{1}^{\ast}\left(\frac{\theta^{j}}{j!}\right)\cdot\pi_{1}^{\ast}[D]\right)\cdot\frac{\theta^{k}}{k!}.$
(3)

Returning to the case of a Jacobian $J(C)$ of a smooth projective curve $C$ of
genus $g$, the classes $\theta^{i}/i!$ appearing in (3) are effective
algebraic cycle classes. Indeed, for $p\in C$ and $d\in\mathbb{Z}_{\geq 1}$,
the image of the morphism $C^{d}\to J(C)$,
$(x_{i})\mapsto{\mathcal{O}}(\sum_{i}x_{i}-d\cdot p)$ defines a subvariety
$W_{d}(C)\subset J(C)$ and by Poincaré’s formula [ACGH85, §I.5] one has
$cl(W_{d}(C))=\theta^{g-d}/(g-d)!\in{\mathrm{H}}^{2g-2d}(J(C),\mathbb{Z})$.

Apart from Theorem 7.2, we obtain the following corollary of Theorem 7.1:

######  Corollary 7.10.

Let $A\to S$ be a principally polarized abelian scheme over a proper, smooth
and connected variety $S$ over $\mathbb{C}$. Let $X\subset S(\mathbb{C})$ be
the set of $x\in S(\mathbb{C})$ such that the abelian variety $A_{x}$
satisfies the integral Hodge conjecture for one-cycles. Then
$X=\cup_{i}Z_{i}(\mathbb{C})$ for some countable union of closed algebraic
subvarieties $Z_{i}\subset S$. In particular, if the integral Hodge conjecture
for one-cycles holds on $U(\mathbb{C})$ for a non-empty open subscheme $U$ of
$S$, then it holds on all of $S(\mathbb{C})$.

###### Proof.

Write ${\mathcal{A}}=A(\mathbb{C})$ and $B=S(\mathbb{C})$ and let
$\pi\colon{\mathcal{A}}\to B$ be the induced family of complex abelian
varieties. Let $g\in\mathbb{Z}_{\geq 0}$ be the relative dimension of $\pi$
and define, for $t\in S(\mathbb{C})$,

$\theta_{t}\in\textnormal{NS}({\mathcal{A}}_{t})\subset{\mathrm{H}}^{2}({\mathcal{A}}_{t},\mathbb{Z})$

to be the polarization of ${\mathcal{A}}_{t}$. There is a global section
$\gamma_{\theta}\in{\mathrm{R}}^{2g-2}\pi_{\ast}\mathbb{Z}$ such that for each
$t\in B$,

$\gamma_{\theta_{t}}=\theta_{t}^{g-1}/(g-1)!\in{\mathrm{H}}^{2g-2}({\mathcal{A}}_{t},\mathbb{Z}).$

Note that $\gamma_{\theta}$ is Hodge everywhere on $B$. For those $t\in B$ for
which $\gamma_{\theta_{t}}$ is algebraic, write $\gamma_{\theta_{t}}$ as the
difference of effective algebraic cycle classes on ${\mathcal{A}}_{t}$. This
gives a countable disjoint union

$\phi\colon\sqcup_{ij}H_{i}\times_{S}H_{j}\to S$

of products of relative Hilbert schemes $H_{i}/S$. By Lemma 7.11 below,
$\gamma_{\theta_{t}}$ is algebraic precisely for closed points $t$ in the
image $Y\subset S$ of $\phi$. By Theorem 7.1, $X=Y$. ∎

######  Lemma 7.11.

Let $S$ be an integral variety over $\mathbb{C}$, let ${\mathcal{A}}\to S$ be
a principally polarized abelian scheme of relative dimension $g$ over $S$ and
let ${\mathcal{C}}_{i}\subset{\mathcal{A}}$ for $i=1,\dotsc,k$ be relative