# On the integral Hodge conjecture
for real abelian threefolds

Olivier de Gaay Fortman

###### Abstract

We prove the real integral Hodge conjecture for several classes of real
abelian threefolds. For instance, we prove the property for real abelian
threefolds with connected real locus, and for real abelian threefolds $A$
which are the product $A=B\times E$ of an abelian surface $B$ and an elliptic
curve $E$ with connected real locus $E(\mathbb{R})$. Moreover, we show that
every real abelian threefold satisfies the real integral Hodge conjecture
modulo torsion, and reduce the principally polarized case to the Jacobian
case.

##  1 Introduction

The Hodge structure on the singular cohomology ring of a smooth projective
variety $X$ over $\mathbb{C}$ is a powerful tool to study the algebraic cycles
on $X$. Where the Hodge conjecture predicts that the space of rational Hodge
classes in any degree $2k$ is spanned by algebraic cycles, the _integral Hodge
conjecture_ asks for something stronger, namely that this is already true with
$\mathbb{Z}$-coefficients. As such, the integral Hodge conjecture is a
property rather than a conjecture: it may hold or fail depending on $k$ and
the geometry of $X$ [AH62, BCC92, Tot97, Voi06, CTV12, Sch19, BO20, Tot21,
BGF23].

Recently, the analogue of the integral Hodge conjecture for real algebraic
varieties has been formulated [BW20, BW20a]. Let $X$ be a smooth projective
variety over $\mathbb{R}$, and let
$G=\textnormal{Gal}(\mathbb{C}/\mathbb{R})$. Building on work of Krasnov
[Kra91, Kra94] and Van Hamel [Ham97], Benoist and Wittenberg define a subgroup
$\textnormal{Hdg}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))_{0}$ of the
$G$-equivariant cohomology group
${\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$ in the sense of Borel,
and study the cycle class map

$\displaystyle\textnormal{CH}_{i}(X)\to{\mathrm{\textnormal{Hdg}}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))_{0},\quad\quad
i+k=\dim(X).$ (1)

The _real integral Hodge conjecture for $i$-cycles_ refers to the property
that (1) is surjective. As in the complex situation, this property holds for
every $X$ if $i\in\\{\dim(X),\dim(X)-1,0\\}$ [Kra91, MH98, Ham97, BW20], but
may fail for other values of $i\in\left\\{0,1,\dotsc,\dim(X)\right\\}$.

Complex uniruled threefolds, as well as threefolds $X$ over $\mathbb{C}$ with
$K_{X}=0$, satisfy the integral Hodge conjecture by work of Grabowski, Voisin
and Totaro [Gra04, Voi06, Tot21]. In [BW20, Question 2.16], Benoist and
Wittenberg ask whether the same is true over $\mathbb{R}$. In fact, in [BW20a]
they provide positive answers for various classes of uniruled threefolds. For
real Calabi-Yau varieties, however, nothing seems to be known. †† _Date:_ In
particular, one may ask:

######  Question 1.1.

Do real abelian threefolds satisfy the real integral Hodge conjecture?

The goal of this paper is to provide evidence towards a positive answer to
Question 1.1.

Let us explain our results. If $A$ is a real abelian variety, then for each
$k\in{\mathbb{Z}}_{\geq 0}$ the following sequence is exact (see Lemma 2.2):

$\displaystyle
0\to\textnormal{Hdg}^{2k}_{G}(A(\mathbb{C}),\mathbb{Z}(k))_{0}[2]\to\textnormal{Hdg}^{2k}_{G}(A(\mathbb{C}),\mathbb{Z}(k))_{0}\to\textnormal{Hdg}^{2k}(A(\mathbb{C}),\mathbb{Z}(k))^{G}\to
0.$

######  Definition 1.2.

A real abelian variety $A$ satisfies the _real integral Hodge conjecture for
$i$-cycles modulo torsion_ if the following map is surjective:

$\displaystyle\textnormal{CH}_{i}(A)\to\textnormal{Hdg}^{2k}(A(\mathbb{C}),\mathbb{Z}(k))^{G},\quad\quad
i+k=\dim(A).$ (2)

We say that $A$ satisfies the _real integral Hodge conjecture modulo torsion_
if $A$ satisfies the real integral Hodge conjecture for $i$-cycles modulo
torsion for every $i\in\left\\{0,1,\dotsc,\dim(A)\right\\}$.

Our first main result is as follows.

######  Theorem 1.3.

Every abelian threefold over $\mathbb{R}$ satisfies the real integral Hodge
conjecture modulo torsion.

By using the Hochschild-Serre spectral sequence (see Section 2.2), one can
calculate the torsion rank of the equivariant cohomology of a real abelian
threefold. We obtain:

######  Corollary 1.4.

Let $A$ be an abelian threefold over $\mathbb{R}$ such that $A(\mathbb{R})$ is
connected. Then $A$ satisfies the real integral Hodge conjecture.

Our proof of Theorem 1.3 is inspired by Grabowski’s proof of the integral
Hodge conjecture for complex abelian threefolds [Gra04]. It consists of two
steps:

  1. 1.

Reduce the real integral Hodge conjecture for one-cycles modulo torsion for
abelian varieties of dimension $g$ to the algebraicity of the minimal class

$\gamma_{\theta}=\theta^{g-1}/(g-1)!\in\textnormal{Hdg}^{2g-2}(A(\mathbb{C}),\mathbb{Z}(g-1))^{G}$

for every principally polarized abelian variety $(A,\theta)$ of dimension $g$
over $\mathbb{R}$.

  2. 2.

Reduce the algebraicity of $\gamma_{\theta}$ on a principally polarized real
abelian threefold $A$ to the case where $A=J(C)$ is the Jacobian of a real
algebraic curve $C$ of genus three with non-empty real locus, where this is
clear.

An essential ingredient for the first reduction step is the fact that any
polarized abelian variety over $\mathbb{R}$ is isogenous to a principally
polarized one. Although the analogue of this statement over any algebraically
closed field is classical [Mum08], it fails over general fields [How01]. We
will prove this fact in Theorem 4.1.

As for step 2, our strategy is to use Hecke orbits. For $g\geq 1$, let
${\mathcal{A}}_{g}$ be the moduli stack of principally polarized abelian
varieties of dimension $g$, and consider the real-analytic topology on its
real locus ${\mathcal{A}}_{g}(\mathbb{R})$ (c.f. [GF22a, Definition 7.5 &
Theorem 8.1] or [GF22, Definition 2.7 & Theorem 2.15]). For integers $a$ and
$b$, define the _$(a,b)$ -Hecke orbit_ of a moduli point
$[(A,\lambda)]\in{\mathcal{A}}_{g}(\mathbb{R})$ as the set of
$[(B,\mu)]\in{\mathcal{A}}_{g}(\mathbb{R})$ admitting an isogeny $A\to B$
preserving the polarizations up to a product of powers of $a$ and $b$ (see
Definition 3.6). Hecke orbits are well-known to be dense in
${\mathcal{A}}_{g}(\mathbb{C})$ (see [GF22, Lemma 7.14]); we obtain the
following real analogue.

######  Theorem 1.5.

Let $p$ and $q$ be distinct odd prime numbers. The $(p,q)$-Hecke orbit of any
point $x\in{\mathcal{A}}_{g}(\mathbb{R})$ is analytically dense in the
connected component of ${\mathcal{A}}_{g}(\mathbb{R})$ containing $x$.

######  Corollary 1.6.

Let $p$ and $q$ be as above. Every principally polarized abelian threefold
over $\mathbb{R}$ is isogenous, via an isogeny that preserves the
polarizations up to a product of powers of $p$ and $q$, to the Jacobian of a
non-hyperelliptic curve with non-empty real locus.

Reduction step 2 follows because if an odd multiple of $\theta^{2}/2$ is
algebraic for a principally polarized real abelian threefold $(A,\theta)$,
then $\theta^{2}/2$ is algebraic. In the course of proving Theorem 1.3, we
also show that the main theorem of [BGF23] has the following analogue over
$\mathbb{R}$: a principally polarized real abelian variety $(A,\theta)$
satisfies the real integral Hodge conjecture for one-cycles modulo torsion if
and only if $\gamma_{\theta}$ is algebraic (see Theorem 5.2).

Corollary 1.6 turns out to be useful for the general principally polarized
case as well:

######  Theorem 1.7.

Let ${\mathcal{A}}_{3}(\mathbb{R})^{+}$ be a connected component of the moduli
space of principally polarized real abelian threefolds. Assume the real
integral Hodge conjecture holds for every Jacobian $J(C)$ such that
$[J(C)]\in{\mathcal{A}}_{3}(\mathbb{R})^{+}$ and the real locus
$C(\mathbb{R})$ of $C$ is non-empty. Then the real integral Hodge conjecture
holds for every real abelian variety in ${\mathcal{A}}_{3}(\mathbb{R})^{+}$.

In view of Theorems 1.3 and 1.7, Question 1.1 restricted to the principally
polarized case can be rephrased as follows. Let $C$ be a real algebraic curve
of genus three with non-empty real locus. Is the torsion subgroup of
$\textnormal{Hdg}^{4}_{G}(J(C)(\mathbb{C}),\mathbb{Z}(2))_{0}$ algebraic?

Our next result reduces this question further. The theorem concerns torsion
cohomology classes of degree four on real abelian varieties $A$ of any
dimension $g$. Let $F^{\bullet}$ be the filtration on
$\textnormal{Hdg}^{2k}_{G}(A(\mathbb{C}),\mathbb{Z}(k))_{0}\subset{\mathrm{H}}^{2k}_{G}(A(\mathbb{C}),\mathbb{Z}(k))$
induced by the Hochschild-Serre spectral sequence (see Section 2.2).

######  Theorem 1.8.

Let $A$ be an abelian variety over $\mathbb{R}$. The group
$F^{3}\textnormal{Hdg}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}$ is zero and
the group $F^{2}\textnormal{Hdg}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}$ is
algebraic.

The proof of Theorem 1.8 relies on Proposition 3.11 in [BGF23] and an analysis
of the Abel-Jacobi map for zero-cycles.

For an abelian variety $A$ over $\mathbb{R}$, the Hochschild-Serre spectral
sequence degenerates [Kra83]. Define
${\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}(2)))_{0}$ as the
image of the canonical homomorphism

$\displaystyle
F^{1}\textnormal{Hdg}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}\to{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}(2))).$
(3)

In fact, for abelian threefolds $A$ over $\mathbb{R}$, the homomorphism (3) is
surjective unless $\left|\pi_{0}(A(\mathbb{R}))\right|=8$ (Corollary 9.2).
Combining Theorems 1.3 and 1.8 with the compatibility of the cycle class map
(1) and the real Abel-Jacobi map (Theorem 7.2), we obtain:

######  Corollary 1.9.

Let $A$ be an abelian threefold over $\mathbb{R}$. Then $A$ satisfies the real
integral Hodge conjecture if and only if the Abel-Jacobi map

$\textnormal{CH}_{1}(A)_{\textnormal{hom}}\to{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}(2)))_{0}$

is surjective, where $\textnormal{CH}_{1}(A)_{\textnormal{hom}}$ denotes the
kernel of
$\textnormal{CH}_{1}(A)\to{\mathrm{H}}^{4}(A(\mathbb{C}),\mathbb{Z}(2))$.

See Section 7 for the definition of the real Abel-Jacobi map. One can then use
Corollary 1.9 and the Künneth formula to prove the following unconditional
result.

######  Theorem 1.10.

Let $A=B\times E$ be the product of a real abelian surface $B$ and a real
elliptic curve $E$ with $E(\mathbb{R})$ connected. Then $A$ satisfies the real
integral Hodge conjecture.

It follows that there are no topological obstructions to the real integral
Hodge conjecture for those real abelian threefolds $A$ with
$\left|\pi_{0}(A(\mathbb{R}))\right|\neq 8$. In the case $A=E^{3}$ for a real
elliptic curve $E$ with disconnected $E(\mathbb{R})$, we reduce the real
integral Hodge conjecture to the algebraicity of one specific class
$\omega\in{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}(2)))_{0}$,
see Proposition 9.4.

Finally, we establish a connection between the real integral Hodge conjecture
and the Griffiths group of an abelian threefold $A$ over $\mathbb{R}$. Let
$\mathscr{C}^{2}(A)\subset{\mathrm{Griff}}^{2}(A_{\mathbb{C}})^{G}\otimes\mathbb{Z}/2$
be the subgroup generated by pull-backs of Ceresa cycles
$[\Xi]\in{\mathrm{Griff}}^{2}(J(C)_{\mathbb{C}})^{G}\otimes\mathbb{Z}/2$ along
isogenies $\phi\colon A\to J(C)$ to Jacobians of curves of genus three over
$\mathbb{R}$ (c.f. Section 10).

######  Theorem 1.11.

Let $A$ be a real abelian threefold such that $A(\mathbb{R})$ is not
connected. Suppose that the complex abelian threefold $A_{\mathbb{C}}$ is very
general. If $A$ satisfies the real integral Hodge conjecture, then
${\mathscr{C}}^{2}(A)\subsetneq{\mathrm{Griff}}^{2}(A_{\mathbb{C}})^{G}\otimes\mathbb{Z}/2$.

###  1.1 Acknowledgements

This paper is based on results that appear in the last chapter of my PhD
thesis [GF22]. I thank my advisor Olivier Benoist for the great guidance he
has given me over the past three years. In particular, I owe him much for
drawing my attention to the real integral Hodge conjecture, and for the many
discussions we had concerning this project.

I thank Nicolas Tholozan for several discussions on moduli of real abelian
varieties, and in particular for explaining to me how to prove Theorem 1.5.
Moreover, I thank Fabrizio Catanese, François Charles and Javier Fresán for
stimulating conversations, and I thank Olivier Benoist and Olivier Wittenberg
for useful comments on an earlier version of this paper.

This research received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement
No754362 .

##  2 The real integral Hodge conjecture

###  2.1 Generalities

Let $X$ be a smooth projective variety over $\mathbb{R}$. The group

$G=\textnormal{Gal}(\mathbb{C}/\mathbb{R})=\\{\textnormal{id},\sigma\\}$

acts on $X(\mathbb{C})$ via the canonical anti-holomorphic involution
$\sigma\colon X(\mathbb{C})\to X(\mathbb{C})$. For $k\in\mathbb{Z}$, we denote
by $\mathbb{Z}(k)$ the $G$-module that has $\mathbb{Z}$ as underlying
$\mathbb{Z}$-module, on which $G$ acts by $\sigma(1)=(-1)^{k}$. Thus,
$\mathbb{Z}(k)=\mathbb{Z}(q)$ for every $q\in\mathbb{Z}$ with $k\equiv q\bmod
2$. We also denote by $\mathbb{Z}(k)$ the constant $G$-sheaf on
$X(\mathbb{C})$ attached to the $G$-module $\mathbb{Z}(k)$. For
$k,q\in\mathbb{Z}_{\geq 0}$, the $G$-action on the group
${\mathrm{H}}^{k}(X(\mathbb{C}),\mathbb{Z}(q))$ is understood to be the one
induced by the involution ${\mathrm{H}}^{k}(\sigma)\circ F_{\infty}$, where
$F_{\infty}=\sigma^{\ast}$ is the pull-back of the anti-holomorphic involution
$\sigma$ on $X(\mathbb{C})$, and ${\mathrm{H}}^{k}(\sigma)$ is the involution
on cohomology induced by $\sigma\colon\mathbb{Z}(q)\to\mathbb{Z}(q)$.

Let $k\in\mathbb{Z}_{\geq 0}$. Attached to $X$ is also the so-called degree
$2k$ _equivariant cohomology group_ with coefficients in $\mathbb{Z}(k)$, see
[Gro57]. It is denoted by
${\mathrm{H}}_{G}^{2k}(X(\mathbb{C}),\mathbb{Z}(k))$, and relates to singular
cohomology via a canonical homomorphism

$\displaystyle\varphi\colon{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))\to{\mathrm{H}}^{2k}(X(\mathbb{C}),\mathbb{Z}(k))^{G}.$
(4)

A real subvariety $Z\subset X$ of codimension $k$ induces a class
$[Z]\in{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$ whose image
$\varphi([Z])$ in ${\mathrm{H}}^{2k}(X(\mathbb{C}),\mathbb{Z}(k))^{G}$ is the
class $[Z_{\mathbb{C}}]$ attached to the subvariety $Z_{\mathbb{C}}\subset
X_{\mathbb{C}}$.

It turns out that such algebraic cycle classes satisfy an additional
condition, discovered by Kahn and Krasnov [Kah87, Kra94], and defined purely
in terms of the structure of $X(\mathbb{C})$ as topological $G$-space. For
$i\in\\{0,\dotsc,2k\\}$, define

$\displaystyle\phi_{i}\colon{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))\to{\mathrm{H}}^{i}(X(\mathbb{R}),\mathbb{Z}/2)$

as the composition

$\displaystyle{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$
$\displaystyle\xrightarrow{\bmod
2}{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}/2)$
$\displaystyle\xrightarrow{\text{restriction}}{\mathrm{H}}^{2k}_{G}(X(\mathbb{R}),\mathbb{Z}/2)={\mathrm{H}}^{2k}(X(\mathbb{R})\times
BG,\mathbb{Z}/2)$
$\displaystyle\xrightarrow[\sim]{\text{K\"{u}nneth}}{\mathrm{H}}^{0}(X(\mathbb{R}),\mathbb{Z}/2)\oplus\cdots\oplus{\mathrm{H}}^{2k}(X(\mathbb{R}),\mathbb{Z}/2)$
$\displaystyle\xrightarrow{\text{projection}}{\mathrm{H}}^{i}(X(\mathbb{R}),\mathbb{Z}/2).$

For $\alpha\in{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$, define
$\alpha_{i}=\phi_{i}(\alpha)\in{\mathrm{H}}^{i}(X(\mathbb{R}),\mathbb{Z}/2)$,
and let ${\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))_{0}$ be the group
of classes $\alpha\in{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$ such
that

$\displaystyle(\alpha_{0},\alpha_{1},\dotsc,\alpha_{k},\dotsc,\alpha_{2k})=\left(0,\dotsc,0,\alpha_{k},Sq^{1}(\alpha_{k}),Sq^{2}(\alpha_{k}),\dotsc,Sq^{k}(\alpha_{k})\right).$
(5)

Here, the $Sq^{i}$ are the Steenrod operations

$Sq^{i}:{\mathrm{H}}^{p}(X(\mathbb{R}),\mathbb{Z}/2)\to{\mathrm{H}}^{p+i}(X(\mathbb{R}),\mathbb{Z}/2).$

######  Definition 2.1 (Benoist–Wittenberg).

The subgroup

$\displaystyle\textnormal{Hdg}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))_{0}\subset{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$

is the group of classes
$\alpha\in{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$ that satisfy the
following conditions:

  1. 1.

The class $\alpha$ lies in the subgroup
${\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))_{0}$ of classes satisfying
(5).

  2. 2.

The image of $\alpha$ in ${\mathrm{H}}^{2k}(X(\mathbb{C}),\mathbb{Z}(k))$
under (4) is a Hodge class.

By [BW20, §1.6.4], Definition 2.1 is compatible with cup products, pull-backs
and proper push-forwards. Moreover, by [Kra94] (c.f. [BW20, Theorem 1.18]), we
have that $[Z]\in\textnormal{Hdg}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))_{0}$
for every algebraic subvariety $Z\subset X$. This defines the map (1).

###  2.2 Hochschild-Serre

For a smooth projective variety $X$ over $\mathbb{R}$, the _Hochschild-Serre_
spectral sequence

$\displaystyle
E^{p,q}_{2}={\mathrm{H}}^{p}(G,{\mathrm{H}}^{q}(X(\mathbb{C}),\mathbb{Z}(k)))\implies{\mathrm{H}}^{p+q}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$
(6)

is obtained by viewing ${\mathrm{H}}^{i}_{G}(X(\mathbb{C}),-)$ as the right-
derived functor of the composition of taking global sections and
$G$-invariants on the category of $G$-sheaves on $X(\mathbb{C})$.

Let $A$ be an abelian variety over $\mathbb{R}$. Then (6) degenerates for
$k=0$ by [Kra83, §5.7]. It follows that (6) degenerates for every $k$, as we
see using cup-products with the non-trivial elements in
${\mathrm{H}}^{1}(G,\mathbb{Z}(1))$ and ${\mathrm{H}}^{2}(G,\mathbb{Z})$.
Consequently, there are canonical identifications

$\displaystyle\begin{split}{\mathrm{H}}^{2k}_{G}(A(\mathbb{C}),\mathbb{Z}(k))_{\textnormal{tors}}&={\mathrm{H}}^{2k}_{G}(A(\mathbb{C}),\mathbb{Z}(k))[2]\\\
&=\textnormal{Ker}\left({\mathrm{H}}^{2k}_{G}(A(\mathbb{C}),\mathbb{Z}(k))\to{\mathrm{H}}^{2k}(A(\mathbb{C}),\mathbb{Z}(k))\right)\\\
&=F^{1}{\mathrm{H}}^{2k}_{G}(A(\mathbb{C}),\mathbb{Z}(k)).\end{split}$ (7)

Moreover, these $\mathbb{Z}/2$-modules are (non-canonically) isomorphic to the
$\mathbb{Z}/2$-module

$\bigoplus_{\begin{subarray}{c}p+q=k\\\
p>0\end{subarray}}{\mathrm{H}}^{p}(G,{\mathrm{H}}^{q}(A(\mathbb{C}),\mathbb{Z}(k))).$

###  2.3 The topological condition

We have the following fundamental:

######  Lemma 2.2.

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb{R}$, and
let $k=n-1$. Suppose that $X(\mathbb{C})$ has torsion-free degree $2k$
integral singular cohomology and that the Hochschild-Serre spectral sequence
(6) degenerates. Then each row and each column in the following commutative
diagram is exact:

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0.0pt\lower-1.99997pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
148.14488pt\raise-35.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
148.14488pt\raise-35.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
230.95927pt\raise-30.72583pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\varphi\hskip 2.84526pt}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
251.5701pt\raise-35.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
184.85812pt\raise-59.92778pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\lower-1.99997pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
251.5701pt\raise-35.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{H}}^{2k}(X(\mathbb{C}),\mathbb{Z}(k))^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
347.39957pt\raise-35.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
347.39957pt\raise-35.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-3.0pt\raise-71.70554pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
23.5pt\raise-71.70554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\bigoplus_{p\geq
1}{\mathrm{H}}^{k-2p}(X(\mathbb{R}),\mathbb{Z}/2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 0.0pt\raise
1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern
136.14615pt\raise-71.70554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bigoplus_{p\geq
1}{\mathrm{H}}^{k-2p}(X(\mathbb{R}),\mathbb{Z}/2).}$}}}}}}}{\hbox{\kern
287.48483pt\raise-71.70554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
349.89957pt\raise-71.70554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$ (8)

###### Proof.

By the degeneration of the Hochschild-Serre spectral sequence (6), the map

$\varphi\colon{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))\to{\mathrm{H}}^{2k}(X(\mathbb{C}),\mathbb{Z}(k))^{G}$

is a surjective homomorphism between abelian groups of the same rank. The
target of $\varphi$ is torsion-free, so its kernel is
${\mathrm{H}}^{2k}(X(\mathbb{C}),\mathbb{Z}(k))[2]$, which explains the
horizontal exact sequence in the middle of diagram (8). It follows from the
proof of [BW20, Proposition 1.8] together with [BW20, Equation (1.33) & Remark
1.20.(i)] that the middle vertical sequence in diagram (8) is split exact.
This implies that, in diagram (8), the vertical arrow on the bottom left and
the horizontal map $\varphi$ on the top right are both surjective. ∎

Note that the first and second horizontal sequence in diagram (8) remain exact
after restricting to Hodge classes.

###  2.4 Threefolds

Let $X$ be a smooth projective threefold over $\mathbb{R}$. The topological
condition (5) on degree four classes takes a particularly simple form: a class
$\alpha\in{\mathrm{H}}^{4}_{G}(X(\mathbb{C}),\mathbb{Z}(2))$ lies in
${\mathrm{H}}^{4}_{G}(X(\mathbb{C}),\mathbb{Z}(2))_{0}$ if and only if

$\alpha|_{x}=0\in{\mathrm{H}}^{4}_{G}(\\{x\\},\mathbb{Z}(2))=\mathbb{Z}/2\quad\textnormal{for
any}\quad x\in X(\mathbb{R}).$

Moreover, the conditions $\alpha|_{x}=0$ for $x$ in different connected
components of $X(\mathbb{R})$ are linearly independent over $\mathbb{Z}/2$:
for $n=3$, the middle vertical sequence in (8) is the split exact sequence

$\displaystyle
0\to{\mathrm{H}}^{4}_{G}(X(\mathbb{C}),\mathbb{Z}(2))_{0}\to{\mathrm{H}}^{4}_{G}(X(\mathbb{C}),\mathbb{Z}(2))\xrightarrow{\phi_{0}}{\mathrm{H}}^{0}(X(\mathbb{R}),\mathbb{Z}/2)\to
0.$

Finally, since $X$ satisfies the real integral Hodge conjecture for $i$-cycles
whenever $i\in\\{0,2,3\\}$ (see [BW20, §2.3.1 and §2.3.2]), the real integral
Hodge conjecture for $X$ is equivalent to the surjectivity of the homomorphism

$\textnormal{CH}_{1}(X)\to\textnormal{Hdg}^{4}_{G}(X(\mathbb{C}),\mathbb{Z}(2))_{0}.$

##  3 Density of Hecke orbits

###  3.1 Polarized real abelian varieties

Let $A$ be a real abelian variety, by which we mean an abelian variety over
$\mathbb{R}$. Here, and in the sequel, the dual abelian variety of $A$ is
denoted by ${\widehat{A}}$. Define
$\Lambda={\mathrm{H}}_{1}(A(\mathbb{C}),\mathbb{Z})$. Denote by $\sigma\colon
A(\mathbb{C})\to A(\mathbb{C})$ the canonical anti-holomorphic involution, and
by $F_{\infty}\colon\Lambda\to\Lambda$ its push-forward. There is a canonical
bijection between:

  * •

Symmetric isogenies $\lambda\colon A\to{\widehat{A}}$ such that
$\lambda_{\mathbb{C}}=\varphi_{{\mathcal{L}}}$ is the homomorphism
$\varphi_{{\mathcal{L}}}\colon A_{\mathbb{C}}\to{\widehat{A}}_{\mathbb{C}}$
induced by an ample line bundle ${\mathcal{L}}$ on $A_{\mathbb{C}}$ as in
[Mum08].

  * •

Alternating forms $E\colon\Lambda\times\Lambda\to\mathbb{Z}$ such that
$F_{\infty}^{\ast}(E)=-E$ and such that the following hermitian form is
positive definite:

$H\colon\Lambda_{\mathbb{R}}\times\Lambda_{\mathbb{R}}\to\mathbb{C},\quad
H(x,y)=E(ix,y)+iE(x,y).$

  * •

Classes of ample line bundles
$\theta\in\text{NS}(A_{\mathbb{C}})^{G}=\textnormal{Hdg}^{2}(A(\mathbb{C}),\mathbb{Z}(1))^{G}$.

In the sequel, a _polarization_ on $A$ will be an element in either one of the
three sets above; the context will make clear which structure is meant.

###  3.2 Moduli of real abelian varieties

Let $(A,\lambda)$ be a principally polarized complex abelian variety of
dimension $g$. By Galois descent [Wei56], to give a model of $(A,\lambda)$
over $\mathbb{R}$ is to give an anti-holomorphic involution

$\sigma\colon A(\mathbb{C})\to A(\mathbb{C})\quad\textnormal{ such that
}\quad\sigma(0)=0\quad\textnormal{ and }\quad F_{\infty}^{\ast}(E)=-E.$

By [Sil89, Chapter IV, Theorem (4.1)] (or [GH81, Section 9]), such an anti-
holomorphic involution $\sigma$ exists if and only if the complex principally
polarized abelian variety $(A,\lambda)$ admits a period matrix of the form

$\displaystyle(I_{g},\frac{1}{2}M+{\mathrm{i}}N).$ (9)

Here $N$ is a positive definite real matrix and $M$ is a symmetric $g\times
g$-matrix with integral coefficients such that if $r=\textnormal{rank}(M)\leq
g$, then $M$ is of the form

$\displaystyle\begin{pmatrix}I_{r}&0\\\ 0&0\end{pmatrix}$ (1)

or of the form

$\displaystyle\begin{pmatrix}0&\dots&1&0\\\ \vdots&\iddots&\iddots&0\\\
1&0&\iddots&\vdots\\\ 0&0&\dots&0\end{pmatrix}.$ (2)

######  Definition 3.1 (Silhol).

The _type_ $(r,\alpha)\in\mathbb{Z}^{2}$ of a principally polarized real
abelian variety $(A,\lambda)$ is defined as follows. If
$(I_{g},\frac{1}{2}M+{\mathrm{i}}N)$ is a period matrix for
$(A_{\mathbb{C}},\lambda_{\mathbb{C}})$ as above, then $r=\text{rank}(M)$.
Define $\alpha\in\\{0,1,2\\}$ in the following way:

  * •

If $r$ is odd, then $\alpha=1$. Thus the type of $(A,\lambda)$ is $(r,1)$.

  * •

If $r$ is zero, then $\alpha=0$. Thus the type of $(A,\lambda)$ is $(0,1)$.

  * •

If $r$ is even, but non-zero, then $\alpha=1$ if $M$ is of the form (1) and
$\alpha=2$ if $M$ is of the form (2).

This definition makes sense, because of the following:

######  Proposition 3.2 (Silhol).

The type $(r,\alpha)$ of a principally polarized real abelian variety
$(A,\lambda)$ does not depend on the chosen period matrix (9) for
$(A_{\mathbb{C}},\lambda_{\mathbb{C}})$ nor on the isomorphism class of
$(A,\lambda)$. If $(A,\lambda)$ is of type $(r,\alpha)$, there exists a period
matrix $(I_{g},\frac{1}{2}M+{\mathrm{i}}N)$ for $(A,\lambda)$ such that $M$ is
of the form (1) or (2), according to whether $\alpha$ equals $1$ or $2$.

###### Proof.

See [Sil89, Chapter IV, Corollaries (4.3) and (4.5)]. ∎

######  Definition 3.3.

Let ${\mathscr{T}}(g)$ be the set of types $(r,\alpha)$ of principally
polarized abelian varieties of dimension $g$ over $\mathbb{R}$. For any type
$\tau\in{\mathscr{T}}(g)$, define $M(\tau)$ to be the integral $g\times
g$-matrix (1) or (2) above, according to whether $\alpha$ equals $1$ or $2$.
Then define $\textnormal{GL}_{g}^{\tau}(\mathbb{Z})$ to be the subgroup of
$\textnormal{GL}_{g}(\mathbb{Z})$ of matrices
$T\in\textnormal{GL}_{g}(\mathbb{Z})$ that satisfy

$T^{t}\cdot M(\tau)\cdot T\equiv M(\tau)\mod 2\quad\quad(T^{t}=\textnormal{
transpose of }T).$

Finally, let ${\mathrm{H}}_{g}$ be the set of symmetric positive definite real
matrices of rank $g$.

######  Theorem 3.4 (Silhol).

Let $g$ be a positive integer. For $\tau\in{\mathscr{T}}(g)$, define
$\left|{\mathcal{A}}_{g}(\mathbb{R})\right|^{\tau}$ to be the set of
isomorphism classes of real principally polarized abelian varieties of type
$\tau$. For each type $\tau\in{\mathscr{T}}(g)$, the period map induces a
bijection

$\left|{\mathcal{A}}_{g}(\mathbb{R})\right|^{\tau}=\textnormal{GL}_{g}^{\tau}(\mathbb{Z})\setminus{\mathrm{H}}_{g},$

where $\textnormal{GL}_{g}^{\tau}(\mathbb{Z})$ acts on ${\mathrm{H}}_{g}$ by
$N\mapsto{T^{t}}\cdot N\cdot T$. Therefore,

$\displaystyle\left|{\mathcal{A}}_{g}(\mathbb{R})\right|=\bigsqcup_{\tau\in{\mathscr{T}}(g)}\textnormal{GL}_{g}^{\tau}(\mathbb{Z})\setminus{\mathrm{H}}_{g}.$
(10)

###### Proof.

See [Sil89, Chapter IV, Theorem (4.6)]. ∎

######  Remark 3.5.

By [GF22a, Theorem 8.1], the bijection (10) is a homeomorphism with respect to
the real-analytic topology on $\left|{\mathcal{A}}_{g}(\mathbb{R})\right|$
(see [GF22a, Definition 7.5]).

###  3.3 Density of Hecke orbits over the real numbers

Before we prove Theorem 1.5, let us properly introduce the notion of Hecke
orbits over the real numbers.

######  Definition 3.6.

Let $(A,\lambda)$ be a principally polarized abelian variety of dimension $g$
over $\mathbb{R}$, and let
$x=[(A,\lambda)]\in\left|{\mathcal{A}}_{g}(\mathbb{R})\right|$ the
corresponding moduli point. For a tuple of integers $(a,b)$, the _$(a,b)$
-Hecke orbit of $x$_ is the subset
${\mathcal{G}}_{a,b}(x)\subset\left|{\mathcal{A}}_{g}(\mathbb{R})\right|$ of
isomorphism classes $[(B,\nu)]\in\left|{\mathcal{A}}_{g}(\mathbb{R})\right|$
of principally polarized abelian varieties $(B,\nu)$ of dimension $g$ over
$\mathbb{R}$, for which there exist $n,m\in\mathbb{Z}_{\geq 0}$ and an isogeny

$\phi\colon A\to B\quad\textnormal{ such that }
\quad\phi^{\ast}(\nu)=a^{n}b^{m}\cdot\lambda.$

###### Proof of Theorem 1.5.

Let $p$ and $q$ be distinct odd prime numbers.

  1. Step 1:

_If $x=[(A,\lambda)]\in\left|{\mathcal{A}}_{g}(\mathbb{R})\right|$ and
$\tau\in{\mathscr{T}}(g)$ is the type of $(A,\lambda)$, then
${\mathcal{G}}_{p,q}(x)\subset\left|{\mathcal{A}}_{g}(\mathbb{R})\right|^{\tau}$_.
Indeed, for any $y=[(B,\mu)]\in{\mathcal{G}}_{p,q}(x)$, there exists an
isogeny $\phi\colon A\to B$ such that $\phi^{\ast}(\mu)=n\cdot\lambda$ for
some odd positive integer $n$. Such a map $\phi$ induces an isomorphism

${\mathrm{H}}_{1}(A(\mathbb{C}),\mathbb{Z}/2)\cong{\mathrm{H}}_{1}(B(\mathbb{C}),\mathbb{Z}/2)$

of symplectic spaces with involution. Since
$x\in\left|{\mathcal{A}}_{g}(\mathbb{R})\right|^{\tau}$, this implies that
$y\in\left|{\mathcal{A}}_{g}(\mathbb{R})\right|^{\tau}$ as well, see [GH81,
Section 9].

  2. Step 2:

Define $S=\mathbb{Z}\left[\frac{1}{p},\frac{1}{q}\right]$. The ring
homomorphism $S\to S/2S$ induces a group homomorphism
$\textnormal{GL}_{g}(S)\to\textnormal{GL}_{g}(S/2S).$ For
$\tau\in{\mathscr{T}}(g)$, we define

$\textnormal{GL}_{g}^{\tau}(S)=\\{T\in\textnormal{GL}_{g}(S)\colon T^{t}\cdot
M(\tau)\cdot T\equiv M(\tau)\bmod 2\\}.$

Fix one such $\tau\in{\mathscr{T}}(g)$. Observe that the action of
$\textnormal{GL}_{g}^{\tau}(\mathbb{Z})$ on ${\mathrm{H}}_{g}$ extends to a
transitive action of $\textnormal{GL}_{g}(\mathbb{R})$ on ${\mathrm{H}}_{g}$.
We claim:

_Let
$x=[(A_{x},\lambda_{x})]\in\left|{\mathcal{A}}_{g}(\mathbb{R})\right|^{\tau}$,
and lift $x$ to a point $y\in{\mathrm{H}}_{g}$. Consider the orbit
$\textnormal{GL}_{g}^{\tau}(S)\cdot y\subset{\mathrm{H}}_{g}$ as well as its
image
$\textnormal{GL}_{g}^{\tau}(\mathbb{Z})\setminus\left(\textnormal{GL}_{g}^{\tau}(S)\cdot
y\right)$ in
$\left|{\mathcal{A}}_{g}(\mathbb{R})\right|^{\tau}=\textnormal{GL}_{g}^{\tau}(\mathbb{Z})\setminus{\mathrm{H}}_{g}$.
Then_

$\textnormal{GL}_{g}^{\tau}(\mathbb{Z})\setminus\left(\textnormal{GL}_{g}^{\tau}(S)\cdot
y\right)={\mathcal{G}}_{p,q}(x).$

Indeed, if ${\mathbb{H}}_{g}$ is the genus $g$ Siegel space of symmetric,
complex $g\times g$ matrices $Z=X+{\mathrm{i}}Y$ whose imaginary part $Y$ is
positive definite, then the inclusion

$\rho_{\tau}\colon{\mathrm{H}}_{g}\hookrightarrow{\mathbb{H}}_{g},\quad
N\mapsto\frac{1}{2}\cdot M(\tau)+{\mathrm{i}}N$

is equivariant for the embedding

$f_{\tau}\colon\textnormal{GL}_{g}(\mathbb{R})\hookrightarrow\textnormal{Sp}_{2g}(\mathbb{R}),\quad
T\mapsto\begin{pmatrix}T^{t}&\frac{1}{2}\left(M(\tau)\cdot T^{-1}-T^{t}\cdot
M(\tau)\right)\\\ 0&T^{-1}\end{pmatrix}.$

Moreover, the action of the group $\textnormal{Sp}_{2g}(S)$ on
${\mathbb{H}}_{g}$ has the following geometric meaning: if we consider
${\mathbb{H}}_{g}$ as a moduli space of $g$-dimensional, principally polarized
complex abelian varieties with symplectic basis, then two points $y=[A_{y}]$
and $z=[A_{z}]\in{\mathbb{H}}_{g}$ are in the same
$\textnormal{Sp}_{2g}(S)$-orbit if and only if there exists an isogeny
$\phi\colon A_{y}\to A_{z}$ that preserves the polarizations up to a product
of powers of $p$ and $q$. Since the intersection

$f_{\tau}\left(\textnormal{GL}_{g}(\mathbb{R})\right)\cap\textnormal{Sp}_{2g}(S)=f_{\tau}\left(\textnormal{GL}_{g}^{\tau}(S)\right)$

equals the subgroup of $\textnormal{Sp}_{2g}(S)$ that preserves the locus

$\rho_{\tau}({\mathrm{H}}_{g})=\left\\{\frac{1}{2}\cdot
M(\tau)+{\mathrm{i}}N\right\\}\subset{\mathbb{H}}_{g}$

of real abelian varieties of type $\tau$, this concludes Step 2.

  3. Step 3:

_For any $\tau\in{\mathscr{T}}(g)$, the subgroup
$\textnormal{GL}_{g}^{\tau}(S)\subset\textnormal{GL}_{g}(\mathbb{R})$ is dense
in the analytic topology._

Define

$\textnormal{SL}_{g}^{\tau}(S)=\textnormal{SL}_{g}(S)\cap\textnormal{GL}_{g}^{\tau}(S)=\\{T\in\textnormal{SL}_{g}(S)\colon
T^{t}\cdot M(\tau)\cdot T\equiv M(\tau)\bmod 2\\}.$

We claim that

$\displaystyle\textnormal{SL}_{g}(\mathbb{R})=\overline{\textnormal{SL}_{g}(S)}=\overline{\textnormal{SL}_{g}^{\tau}(S)}\subset\overline{\textnormal{GL}_{g}^{\tau}(S)}\subset\textnormal{GL}_{g}(\mathbb{R}).$
(11)

Indeed, this follows from the following two statements:

    1. (a)

The closure of $\textnormal{SL}_{g}(S)$ in $\textnormal{GL}_{g}(\mathbb{R})$
is $\textnormal{SL}_{g}(\mathbb{R})$.

    2. (b)

The subgroup $\textnormal{SL}_{g}^{\tau}(S)\subset\textnormal{SL}_{g}(S)$ has
finite index.

To prove (a), observe that the subgroup
$\textnormal{SL}_{g}(\mathbb{R})\subset\textnormal{GL}_{g}(\mathbb{R})$ is
closed, which implies that the closure of $\textnormal{SL}_{g}(S)$ in
$\textnormal{GL}_{g}(\mathbb{R})$ equals the closure of
$\textnormal{SL}_{g}(S)$ in $\textnormal{SL}_{g}(\mathbb{R})$. Thus, (a)
follows from the density of $\textnormal{SL}_{g}(S)$ in
$\textnormal{SL}_{g}(\mathbb{R})$, which is true by strong approximation; see
[GF22, Lemma 7.14] for the precise argument. As for (b), the group
$\textnormal{SL}_{g}(S)$ acts on ${\mathrm{M}}_{g}(S/2S)$ via $N\mapsto
T^{t}\cdot N\cdot T\bmod 2$ for $T\in\textnormal{SL}_{g}(S)$, so there is an
injection

$\textnormal{SL}_{g}^{\tau}(S)\setminus\textnormal{SL}_{g}(S)\hookrightarrow{\mathrm{M}}_{g}(S/2S)={\mathrm{M}}_{g}(\mathbb{Z}/2),\quad
T\mapsto T^{t}\cdot M(\tau)\cdot T\mod 2.$

Statement (b) implies that the index of
$\overline{\textnormal{SL}_{g}^{\tau}(S)}\subset\overline{\textnormal{SL}_{g}(S)}$
is finite. e have
$\overline{\textnormal{SL}_{g}(S)}=\textnormal{SL}_{g}(\mathbb{R})$ by (a),
thus
$\overline{\textnormal{SL}_{g}^{\tau}(S)}\subset\textnormal{SL}_{g}(\mathbb{R})$
is a closed subgroup of finite index, hence open. Therefore
$\overline{\textnormal{SL}_{g}^{\tau}(S)}=\textnormal{SL}_{g}(\mathbb{R})$ by
connectivity of $\textnormal{SL}_{g}(\mathbb{R})$, proving Claim (11).

Write $G=\textnormal{GL}_{g}^{\tau}(S)$. If
$H\subset\textnormal{GL}_{g}(\mathbb{R})$ is any Lie subgroup such that
$\textnormal{SL}_{g}(\mathbb{R})\subset H$, then $H=\det^{-1}(\det(H))$.
Consequently, using (11), we obtain:

$\displaystyle\overline{G}={\det}^{-1}(\det(\overline{G}))\subset\textnormal{GL}_{g}(\mathbb{R}).$
(12)

The equality (12) implies that Step 3 reduces to the equality
${\det}^{-1}(\det(\overline{G}))=\textnormal{GL}_{g}(\mathbb{R})$. This
follows from the equality $\det(\overline{G})=\mathbb{R}^{\ast}$, which we now
prove. The morphism
$\det\colon\textnormal{GL}_{g}(\mathbb{R})\to\mathbb{R}^{\ast}$ is open
because its differential at the identity matrix
$I_{g}\in\textnormal{GL}_{g}(\mathbb{R})$ is the trace homomorphism
${\mathrm{M}}_{g}(\mathbb{R})\to\mathbb{R}$. Writing

$\textnormal{GL}_{g}(\mathbb{R})={\det}^{-1}(\mathbb{R}^{\ast})={\det}^{-1}(\det(\overline{G})\sqcup\det(\overline{G})^{c})=\overline{G}\sqcup{\det}^{-1}\left(\det(\overline{G})^{c}\right),$

it follows that $\det(\overline{G})$ is closed in $\mathbb{R}^{\ast}$. We
conclude that
$\overline{\det(G)}\subset\overline{\det(\overline{G})}=\det(\overline{G})$.
Thus, to show that $\det(\overline{G})=\mathbb{R}^{\ast}$, it suffices to show
that $\det(G)$ is dense in $\mathbb{R}^{\ast}$. The homomorphism
$\det\colon\textnormal{GL}_{g}^{\tau}(S)\to S^{\ast}$ is surjective because it
admits the section

$S^{\ast}\to\textnormal{GL}_{g}^{\tau}(S),\quad x\mapsto\begin{pmatrix}x&0\\\
0&I_{g-1}\end{pmatrix}.$

Therefore,

$\det(G)=\det(\textnormal{GL}_{g}^{\tau}(S))=S^{\ast}=\left\\{\pm
p^{n}q^{m}\colon n,m\in\mathbb{Z}\right\\},$

and it remains to prove that the latter is dense in $\mathbb{R}^{\ast}$. This
holds, since $S^{\ast}_{>0}=\left\\{p^{n}q^{m}\colon
n,m\in\mathbb{Z}\right\\}$ is dense in $\mathbb{R}_{>0}$ because
$\log(S^{\ast}_{>0})=\mathbb{Z}\log(p)+\mathbb{Z}\log(q)$ is dense in
$\mathbb{R}$; the latter follows from the fact that $\log(S^{\ast}_{>0})$ is
not a cyclic subgroup of $\mathbb{R}$.

  4. Step 4:

_Finish the proof._ Let
$x=[(A,\lambda)]\in\left|{\mathcal{A}}_{g}(\mathbb{R})\right|$, and let
$\tau\in{\mathscr{T}}(g)$ be the type of the principally polarized real
abelian variety $(A,\lambda)$. Lift $x$ to a point $y\in{\mathrm{H}}_{g}$. By
Step 3, we know that the orbit $\textnormal{GL}_{g}^{\tau}(S)\cdot y$ is dense
in $\textnormal{GL}_{g}(\mathbb{R})\cdot y={\mathrm{H}}_{g}$. Consequently,
the image of $\textnormal{GL}_{g}^{\tau}(S)\cdot y$ under the projection
${\mathrm{H}}_{g}\to\left|{\mathcal{A}}_{g}(\mathbb{R})\right|^{\tau}$ is
dense in $\left|{\mathcal{A}}_{g}(\mathbb{R})\right|^{\tau}$. By Step 2, this
image is precisely ${\mathcal{G}}_{p,q}(x)$. Thus ${\mathcal{G}}_{p,q}(x)$ is
dense in $\left|{\mathcal{A}}_{g}(\mathbb{R})\right|^{\tau}$ as desired.

∎

##  4 Principal polarizations in real isogeny classes

The goal of this section is to prove the following:

######  Theorem 4.1.

Let $(A,\lambda_{A})$ be a polarized abelian variety over $\mathbb{R}$. Then
there exists a principally polarized abelian variety $(B,\lambda_{B})$ over
$\mathbb{R}$ and an isogeny

$\phi\colon A\to B\quad\textnormal{ \emph{such that}
}\quad\phi^{\ast}(\lambda_{B})=\lambda_{A}.$

###### Proof.

Let $K\subset A(\mathbb{C})$ be the kernel of the analytified polarization
$\lambda_{A}\colon A(\mathbb{C})\to{\widehat{A}}(\mathbb{C})$. Then $K$ is a
finite group of order $d^{2}$, where $d^{2}$ is the degree of $\lambda_{A}$,
such that the real structure $\sigma\colon A(\mathbb{C})\to A(\mathbb{C})$
restricts to an involution

$\sigma\colon K\to K.$

We may assume that $K\neq(0)$. Let $p$ be any prime number that divides the
order of $K$. We claim that there exists a subgroup $K_{1}\subset K$ of order
$p$ such that $\sigma(K_{1})=K_{1}$. To see this, let $H[p]\subset K$ be the
$p$-torsion subgroup of $K$. Then $H[p]$ is preserved by $\sigma$, so that
$H[p]$ is an $\mathbb{F}_{p}$-vector space of finite rank equipped with a
linear involution $\sigma$. Therefore, $H[p]$ contains a one-dimensional
$\mathbb{F}_{p}$-subspace $K_{1}$ preserved by $\sigma$, which proves our
claim.

The group $K_{1}\subset A(\mathbb{C})$ descends to a finite subgroup scheme
$K_{1}\subset A$ over $\mathbb{R}$; define $A_{1}$ to be the abelian variety
$A/K_{1}$ over $\mathbb{R}$. Let
$\Lambda={\mathrm{H}}_{1}(A(\mathbb{C}),\mathbb{Z})$ and
$M={\mathrm{H}}_{1}(A_{1}(\mathbb{C}),\mathbb{Z})$; the projection $A\to
A_{1}$ induces an exact sequence

$0\to\Lambda\to M\to K_{1}\to 0.$

Let $E\colon\Lambda\times\Lambda\to\mathbb{Z}$ be the alternating form
attached to the polarization $\lambda_{A}$ of $A$. Since
$M/\Lambda=K_{1}\subset K=\Lambda^{\vee}/\Lambda$, we have inclusions

$\Lambda\subset M\subset\Lambda^{\vee},\quad\textnormal{ where
}\quad\Lambda^{\vee}=\left\\{x\in\Lambda\otimes\mathbb{Q}\mid
E(x,\Lambda)\subset\Lambda\right\\}.$

The $\mathbb{Z}$-valued alternating form $E$ on the lattice $\Lambda$ gives
rise to a bilinear form

$\overline{E}\colon\Lambda^{\vee}/\Lambda\times\Lambda^{\vee}/\Lambda\to\mathbb{Q}/\mathbb{Z}$

which vanishes on $M/\Lambda$ because $M/\Lambda\cong\mathbb{Z}/p$ and
$\overline{E}$ is alternating. Thus
$E\colon\Lambda^{\vee}\times\Lambda^{\vee}\to\mathbb{Q}$ restricts to an
integer-valued form $E_{1}$ on $M$. The latter induces a polarization
$\lambda_{A_{1}} \colon A_{1}\to{\widehat{A}}_{1}$ that makes the following
diagram commute:

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{A}}$$\scriptstyle{\pi}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{A_{1}}}$$\textstyle{{\widehat{A}}}$$\textstyle{{\widehat{A}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{{\widehat{\pi}}}$

Here $\pi$ is the quotient map $A\to A_{1}$ and ${\widehat{\pi}}$ its dual.
Since the degree of an isogeny is multiplicative in compositions, and
$\deg(\pi)=p$, we have

$p^{2}\cdot\deg(\lambda_{A_{1}})=\deg(\pi)^{2}\cdot\deg(\lambda_{A_{1}})=\deg\left(\lambda_{A}\right)=d^{2}.$

If $d=p$, we are finished – otherwise, we repeat the above procedure until the
real abelian variety $A_{n}=A_{n-1}/K_{n-1}$ becomes principally polarized. ∎

##  5 Integral Hodge classes modulo torsion

###  5.1 The Fourier transform

Let $A$ be a real abelian variety of dimension $g$, and consider the Poincaré
bundle ${\mathcal{P}}_{A}$ on $A\times{\widehat{A}}$. Let
$a_{1},\dotsc,a_{2g}$ be integers. The Chern character

$\textnormal{ch}({\mathcal{P}}_{A_{\mathbb{C}}})=\exp(c_{1}({\mathcal{P}}_{A_{\mathbb{C}}}))\in{\mathrm{H}}^{2\bullet}(A(\mathbb{C})\times{\widehat{A}}(\mathbb{C}),\mathbb{Z}(\bullet))$

defines the _Fourier transform_

$\displaystyle{\mathscr{F}}_{A}\colon\bigoplus_{i\in\mathbb{Z}_{\geq
0}}{\mathrm{H}}^{i}(A(\mathbb{C}),\mathbb{Z}(a_{i}))\to\bigoplus_{i\in\mathbb{Z}_{\geq
0}}{\mathrm{H}}^{i}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(a_{2g-i}-g+i)).$ (13)

It is defined as
${\mathscr{F}}_{A}(x)=\pi_{2,\ast}(\textnormal{ch}({\mathcal{P}}_{A})\cdot\pi_{1}^{\ast}(x))$,
where $\pi_{i}$ is the projection of $A\times{\widehat{A}}$ onto the $i$-th
factor. By [Bea83], the map (13) is an isomorphism, inducing isomorphisms

$\displaystyle{\mathscr{F}}_{A}\colon{\mathrm{H}}^{i}(A(\mathbb{C}),\mathbb{Z}(a_{i}))\xrightarrow{\sim}{\mathrm{H}}^{2g-i}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(a_{i}+g-i)).$
(14)

Since $\textnormal{ch}({\mathcal{P}}_{A_{\mathbb{C}}})$ is fixed by $G$, these
maps are isomorphisms of $G$-modules.

###  5.2 Divisors

Let $A$ be an abelian variety over $\mathbb{R}$. Both homomorphisms in the
following composition are surjective:

$\displaystyle\textnormal{CH}^{1}(A)\to\textnormal{Hdg}_{G}^{2}(A(\mathbb{C}),\mathbb{Z}(1))\to\textnormal{Hdg}^{2}(A(\mathbb{C}),\mathbb{Z}(1))^{G}.$
(15)

Indeed, the first map is surjective by the real integral Hodge conjecture for
divisors (see [BW20, Proposition 2.8]), and the second by the degeneration of
the Hochschild-Serre spectral sequence (see Section 2.2).

###  5.3 The real integral Hodge conjecture for one-cycles modulo torsion

The goal of this section is to provide an application of Theorems 1.5 and 4.1
combined: the real integral Hodge conjecture for one-cycles modulo torsion
follows, in some cases, from the real integral Hodge conjecture for divisors
modulo torsion (c.f. Section 5.2). For this, we use Fourier transforms on
Betti cohomology of real abelian varieties (c.f. Section 5.1) and their lifts
to integral Chow groups, see [BGF23, MP10]. The theory in [BGF23, Section 3]
was developed for abelian varieties over a general field $k$ – to apply it, we
take $k=\mathbb{R}$.

######  Definition 5.1.

Let $A$ be a real abelian variety, and $k$ a non-negative integer. An element
$\alpha\in\textnormal{Hdg}^{2k}(A(\mathbb{C}),\mathbb{Z}(k))^{G}$ is called
_algebraic_ if it is in the image of

$\textnormal{CH}^{k}(A)\to\textnormal{Hdg}^{2k}(A(\mathbb{C}),\mathbb{Z}(k))^{G}.$

By the main theorem of [BGF23], the integral Hodge conjecture for one-cycles
on a fixed principally polarized abelian variety $(A,\theta)$ over
$\mathbb{C}$ is equivalent to the algebraicity of the minimal class
$\gamma_{\theta}=\theta^{g-1}/(g-1)!$ on $A$ (see _loc.cit._ , Theorem 1.1).
Moreover, Grabowski reduced the integral Hodge conjecture for one-cycles for
every complex abelian variety of dimension $g$ to the algebraicity of
$\gamma_{\theta}$ for every principally polarized abelian variety of dimension
$g$, see [Gra04]. We have the following real analogue of these results:

######  Theorem 5.2.

Fix a positive integer $g$. Let $(A,\theta)$ be a principally polarized
abelian variety of dimension $g$ over $\mathbb{R}$. The following are
equivalent:

  1. 1.

The real abelian variety $A$ satisfies the real integral Hodge conjecture for
one-cycles modulo torsion.

  2. 2.

The minimal class

$\gamma_{\theta}=\frac{\theta^{g-1}}{(g-1)!}\in\textnormal{Hdg}^{2g-2}(A(\mathbb{C}),\mathbb{Z}(g-1))^{G}\quad\text{
is algebraic.}$

  3. 3.

The Chern character

$\textnormal{ch}({\mathcal{P}}_{A_{\mathbb{C}}})=\exp(c_{1}({\mathcal{P}}_{A_{\mathbb{C}}}))\in\textnormal{Hdg}^{2\bullet}(A(\mathbb{C})\times{\widehat{A}}(\mathbb{C}),\mathbb{Z}(\bullet))^{G}\quad\text{
is algebraic.}$

Moreover, if the real integral Hodge conjecture for one-cycles modulo torsion
holds for every principally polarized abelian variety of dimension $g$ over
$\mathbb{R}$, then it holds for every abelian variety of dimension $g$ over
$\mathbb{R}$.

###### Proof.

The direction 1 $\implies$2 is trivial; let us assume that 2 holds. Let
$(A,\theta)$ be a principally polarized abelian variety of dimension $g$ over
$\mathbb{R}$, and suppose that $\gamma_{\theta}$ is algebraic. By [BGF23,
Proposition 3.11], the abelian variety $A$ admits a motivic integral Fourier
transform up to homology, see [BGF23, Definition 3.1]. This means the
following. Let $\ell$ be a prime number. There exists a cycle

$\Gamma\in\textnormal{CH}(A\times{\widehat{A}})\quad\textnormal{ such that
}\quad[\Gamma_{\mathbb{C}}]=\textnormal{ch}({\mathcal{P}}_{A_{\mathbb{C}}})\quad\in\quad{\mathrm{H}}^{2\bullet}_{\textnormal{\'{e}t}}(A_{\mathbb{C}}\times{\widehat{A}}_{\mathbb{C}},\mathbb{Z}_{\ell}(\bullet)).$

As a consequence,
$[\Gamma_{\mathbb{C}}]=\textnormal{ch}({\mathcal{P}}_{A_{\mathbb{C}}})\in{\mathrm{H}}^{2\bullet}(A(\mathbb{C})\times{\widehat{A}}(\mathbb{C}),\mathbb{Z}(\bullet))^{G}$,
which proves 3.

Let us now assume that 3 holds, and let
$\Gamma\in\textnormal{CH}(A\times{\widehat{A}})$ be a cycle that induces the
class $\textnormal{ch}({\mathcal{P}}_{A_{\mathbb{C}}})$ in Betti cohomology.
The correspondence $\Gamma$ defines a group homomorphism $\Gamma_{\ast}
\colon\textnormal{CH}^{\bullet}(A)\to\textnormal{CH}^{\bullet}({\widehat{A}})$
such that the following diagram commutes:

Here $k=g-1$, the composition on the bottom row is an isomorphism (see (14) in
Section 5.1), and the left vertical arrow is surjective (see (15) in Section
5.2). Therefore, the right vertical arrow is surjective, which implies 1.

Next, suppose that 1 holds for every principally polarized real abelian
variety of dimension $g$, and let $A$ be any real abelian variety of dimension
$g$. We would like to show that $A$ satisfies the real integral Hodge
conjecture for one-cycles modulo torsion. The isomorphism (14) induces an
isomorphism

${\mathscr{F}}_{A}
\colon\textnormal{Hdg}^{2}(A(\mathbb{C}),\mathbb{Z}(1))^{G}\xrightarrow{\sim}\textnormal{Hdg}^{2g-2}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(g-1))^{G}.$

Since $\textnormal{Hdg}^{2}(A(\mathbb{C}),\mathbb{Z}(1))^{G}$ is algebraic by
Section 5.2, it suffices to show that ${\mathscr{F}}_{A}([L_{\mathbb{C}}])$ is
algebraic for every line bundle $L$ on $A$. By [Har77, II, Exercice 7.5],
there is an ample line bundle $M$ on $A$ such that $L\otimes M^{\otimes n}$ is
ample for $n\gg 0$; we may thus assume that $L$ is ample. By Theorem 4.1,
there is a principally polarized abelian variety $(B,\lambda)$, and an isogeny

$\phi\colon A\to B$

such that, if
$\theta\in\textnormal{NS}(B_{\mathbb{C}})^{G}=\textnormal{Hdg}^{2}(B(\mathbb{C}),\mathbb{Z}(1))^{G}$
is the class corresponding to the principal polarization $\lambda\colon
B\to{\widehat{B}}$, then

$\phi^{\ast}(\theta)=[L_{\mathbb{C}}]\in\textnormal{Hdg}^{2}(A(\mathbb{C}),\mathbb{Z}(1))^{G}.$

On the other hand, the following diagram commutes by [Bea83, Proposition 3]:

$\textstyle{\textnormal{Hdg}^{2}(A(\mathbb{C}),\mathbb{Z}(1))^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathscr{F}}_{A}}$$\textstyle{\textnormal{Hdg}^{2g-2}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(g-1))^{G}}$$\textstyle{\textnormal{Hdg}^{2}(B(\mathbb{C}),\mathbb{Z}(1))^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathscr{F}}_{B}}$$\scriptstyle{\phi^{\ast}}$$\textstyle{\textnormal{Hdg}^{2g-2}({\widehat{B}}(\mathbb{C}),\mathbb{Z}(g-1))^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\hat{\phi}_{\ast}}$

Moreover, by [Bea83, Proposition 5], we have

${\mathscr{F}}_{B}(\theta)=(-1)^{g-1}\cdot\frac{\hat{\theta}^{g-1}}{(g-1)!}\in{\mathrm{H}}^{2g-2}({\widehat{B}}(\mathbb{C}),\mathbb{Z}(g-1))^{G},$

where
$\hat{\theta}\in\textnormal{Hdg}^{2}({\widehat{B}}(\mathbb{C}),\mathbb{Z}(1))^{G}$
denotes the dual polarization class. We conclude that
${\mathscr{F}}_{B}(\theta)$ is algebraic, so that
${\mathscr{F}}_{A}([L_{\mathbb{C}}])={\widehat{\phi}}_{\ast}({\mathscr{F}}_{B}(\theta))$
is algebraic as well. ∎

######  Corollary 5.3.

Let $C_{1},\dotsc,C_{n}$ be smooth projective geometrically integral curves
over $\mathbb{R}$ such that $C_{i}(\mathbb{R})\neq\emptyset$ for each $i$. The
real abelian variety $A=J(C_{1})\times\cdots\times J(C_{n})$ satisfies the
real integral Hodge conjecture for one-cycles modulo torsion.

###### Proof.

The minimal class on a product of principally polarized abelian varieties over
$\mathbb{R}$ is the sum of the pull-backs of the minimal classes on the
factors, so by Theorem 5.2, it suffices to treat the case $n=1$. For a real
algebraic curve $C$ whose real locus is non-empty, any Abel-Jacobi map gives
an embedding of real varieties $\iota\colon C\hookrightarrow J(C)$. By
Poincaré’s formula, one has

$[\iota(C)_{\mathbb{C}}]=\frac{\theta^{g-1}}{(g-1)!}\in\textnormal{Hdg}^{2g-2}(J(C)(\mathbb{C}),\mathbb{Z}(g-1))^{G},$

where the class on the right hand side of the equality is the minimal
cohomology class $\gamma_{\theta}$ of $J(C)$. Thus $\gamma_{\theta}$ is
algebraic, so we are done by Theorem 5.2. ∎

###  5.4 Integral Hodge classes modulo torsion on real abelian threefolds

We define a _real algebraic curve_ as a smooth projective geometrically
connected curve over $\mathbb{R}$. Let ${\mathcal{M}}_{3}(\mathbb{R})$ be the
moduli space of real algebraic curves of genus three (see [SS89] and [GF22a,
Theorem 8.2]), and consider the Torelli map

$\displaystyle
t\colon{\mathcal{M}}_{3}(\mathbb{R})\to{\mathcal{A}}_{3}(\mathbb{R}).$

Let ${\mathcal{N}}_{3}(\mathbb{R})\subset{\mathcal{M}}_{3}(\mathbb{R})$ be the
non-hyperelliptic locus.

######  Lemma 5.4.

The subset $t({\mathcal{N}}_{3}(\mathbb{R}))$ is open in
${\mathcal{A}}_{3}(\mathbb{R})$.

###### Proof.

On the level of stacks,
${\mathcal{T}}\colon{\mathcal{M}}_{3}\to{\mathcal{A}}_{3}$ is an open
immersion when restricted to the non-hyperelliptic locus
${\mathcal{N}}_{3}\subset{\mathcal{M}}_{3}$. Moreover, for any algebraic stack
${\mathscr{X}}$ of finite type over $\mathbb{R}$, the set
$|{\mathscr{X}}(\mathbb{R})|$ of isomorphism classes of $\mathbb{R}$-points of
${\mathscr{X}}$ admits a _real-analytic topology_ (see [GF22a, Definition
7.5]). For this topology, the subset
$|{\mathcal{T}}({\mathcal{N}}_{3})(\mathbb{R})|\subset|{\mathcal{A}}_{3}(\mathbb{R})|$
is open by [GF22, Corollary 2.8.2]. By Remark 3.5, we are done. ∎

######  Lemma 5.5.

Every connected component of the moduli space ${\mathcal{A}}_{3}(\mathbb{R})$
contains a non-empty open subset of non-hyperelliptic curves of genus three
with non-empty real locus.

###### Proof.

Let ${\mathcal{A}}_{3}(\mathbb{R})^{\tau}$ be a connected component of
${\mathcal{A}}_{3}(\mathbb{R})$. By [GH81, page 182], there is a (unique)
connected component ${\mathcal{M}}_{3}(\mathbb{R})^{\tau}$ of
${\mathcal{M}}_{3}(\mathbb{R})$ that satisfies the following two conditions:
$C(\mathbb{R})$ is not empty for each
$[C]\in{\mathcal{M}}_{3}(\mathbb{R})^{\tau}$, and

$t\left({\mathcal{M}}_{3}(\mathbb{R})^{\tau}\right)\subset{\mathcal{A}}_{3}(\mathbb{R})^{\tau}.$

Now ${\mathcal{M}}_{3}(\mathbb{R})^{\tau}$ contains a component
${\mathcal{N}}_{3}(\mathbb{R})^{\tau}$ of ${\mathcal{N}}_{3}(\mathbb{R})$ by
[GH81, Proposition 3.1] and the table on page 174 of _loc.cit._ Moreover,
${\mathcal{N}}_{3}(\mathbb{R})^{\tau}$ is open in
${\mathcal{A}}_{3}(\mathbb{R})$ by Lemma 5.4. ∎

###### Proof of Corollary 1.6.

This follows directly from Theorem 1.5 and Lemma 5.5.∎

We are now in the position to prove Theorem 1.3.

###### Proof of Theorem 1.3.

By Theorem 5.2, it suffices to show that for any principally polarized abelian
threefold $(A,\theta)$ over $\mathbb{R}$, the class
$\gamma_{\theta}=\theta^{2}/2\in\textnormal{Hdg}^{4}(A(\mathbb{C}),\mathbb{Z}(2))^{G}$
is algebraic. Let us prove this. Let $p$ and $q$ be two distinct odd prime
numbers. By Corollary 1.6, there exists a real algebraic curve $C$ of genus
three such that $C(\mathbb{R})\neq\emptyset$ together with an isogeny

$\phi\colon A\to J(C)$

such that $\phi^{\ast}(\lambda_{C})=p^{n}q^{m}\cdot\lambda_{A}$ for some non-
negative integers $n$ and $m$. Here $\lambda_{C}$ denotes the canonical
polarization on $J(C)$. Let
$\theta_{C}\in\textnormal{Hdg}^{2}(J(C)(\mathbb{C}),\mathbb{Z}(1))^{G}$ be the
corresponding cohomology class; then
$\phi^{\ast}(\theta_{C})=p^{n}q^{m}\cdot\theta$. Since
$C(\mathbb{R})\neq\emptyset$, the minimal class $\theta^{2}_{C}/2$ on $J(C)$
is algebraic; see Corollary 5.3. Therefore, the class

$p^{2n}q^{2m}\cdot\theta^{2}/2=\phi^{\ast}(\theta^{2}_{C}/2)\in\textnormal{Hdg}^{4}(A(\mathbb{C}),\mathbb{Z}(2))^{G}$

is algebraic. We are done because $\theta^{2}$ is algebraic and the integer
$p^{n}q^{m}$ is odd. ∎

###### Proof of Corollary 1.4.

For each $i\in\\{1,\dotsc,6\\}$, cup-product defines a canonical isomorphism
of $G$-modules:

$\bigwedge^{i}{\mathrm{H}}^{1}(A(\mathbb{C}),\mathbb{Z})\xrightarrow{\sim}{\mathrm{H}}^{i}(A(\mathbb{C}),\mathbb{Z}).$

This allows us to calculate the $G$-module structure on
${\mathrm{H}}^{i}(A(\mathbb{C}),\mathbb{Z})$, for
${\mathrm{H}}^{1}(A(\mathbb{C}),\mathbb{Z})\cong\mathbb{Z}[G]^{3}$ as
$G$-modules [Ham97, Example 3.1]. It turns out that the group
${\mathrm{H}}^{p}(G,{\mathrm{H}}^{q}(A(\mathbb{C}),\mathbb{Z}))$ vanishes
whenever $p+q=4$ and $p>0$. Therefore,

$\textnormal{Hdg}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}=\textnormal{Hdg}^{4}(A(\mathbb{C}),\mathbb{Z}(2))^{G}$

by Section 2.2 and Lemma 2.2. By Theorem 1.3, we are done. ∎

##  6 Torsion cohomology classes on real abelian varieties

###  6.1 Isogenies and torsion cohomology classes

Let $X$ be a smooth projective variety over $\mathbb{R}$, and let $k$ be any
non-negative integer.

######  Definition 6.1.

For a subgroup $K\subset{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$,
let $K_{0}=K\cap{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))_{0}$. In
case the Hochschild-Serre spectral sequence (6) degenerates, we define, for
$p>0$,

$\displaystyle{\mathrm{H}}^{p}(G,{\mathrm{H}}^{2k-p}(X(\mathbb{C}),\mathbb{Z}(k)))_{0}=\textnormal{Im}(F^{p}{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))_{0}\to{\mathrm{H}}^{p}(G,{\mathrm{H}}^{2k-p}(X(\mathbb{C}),\mathbb{Z}(k)))).$

A subgroup $K\subset{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$ is
_algebraic_ if every element of $K$ is in the image of the cycle class map
(1). If the Hochschild-Serre spectral sequence (6) degenerates, we call a
subgroup
$L\subset{\mathrm{H}}^{p}(G,{\mathrm{H}}^{2k-p}(X(\mathbb{C}),\mathbb{Z}(k)))$
_algebraic_ if $L$ is the image of an algebraic subgroup $K\subset
F^{p}{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$ under
$F^{p}{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))\to{\mathrm{H}}^{p}(G,{\mathrm{H}}^{2k-p}(X(\mathbb{C}),\mathbb{Z}(k)))$.

The following lemma provides the key to our proof of Theorem 1.7.

######  Lemma 6.2.

Let $A$ and $B$ be real abelian varieties, and $\phi$ an isogeny $A\to B$ of
odd degree. For all integers $p,k,i$ with $p>0$, the pull-back $\phi^{\ast}$
defines an isomorphism

$\displaystyle
F^{p}{\mathrm{H}}^{k}_{G}(B(\mathbb{C}),\mathbb{Z}(i))_{0}\xrightarrow{\sim}F^{p}{\mathrm{H}}^{k}_{G}(A(\mathbb{C}),\mathbb{Z}(i))_{0}.$
(16)

If $k\in 2\mathbb{Z}_{\geq 0}$ and $i=k/2$, then $\phi^{\ast}$ restricts to an
isomorphism between the subgroups of classes that satisfy the topological
condition (5). In this case, the left-hand side of (16) is algebraic (in the
sense of Definition 6.1) if and only if the right-hand side of (16) is
algebraic.

###### Proof.

Let us first prove the lemma in the case $A=B$ and $\phi=[d]$ for some integer
$d\in\mathbb{Z}$ with $d\not\equiv 0\bmod 2$. Write
${\mathrm{H}}={\mathrm{H}}^{k}_{G}(A(\mathbb{C}),\mathbb{Z}(i))$. The
Hochschild-Serre spectral sequence (6) degenerates, so
$F^{1}{\mathrm{H}}={\mathrm{H}}[2]$ and for each $p\in\\{0,\dotsc,k\\}$ there
is an exact sequence

$0\to F^{p+1}{\mathrm{H}}\to
F^{p}{\mathrm{H}}\to{\mathrm{H}}^{p}(G,{\mathrm{H}}^{k-p}(A(\mathbb{C}),\mathbb{Z}(i)))\to
0.$

We have $F^{k+1}{\mathrm{H}}=(0)$, and for each $p\in\\{1,\dotsc,k\\}$, the
morphism

$[d]^{\ast}\colon{\mathrm{H}}^{k-p}(A(\mathbb{C}),\mathbb{Z}(i))\to{\mathrm{H}}^{k-p}(A(\mathbb{C}),\mathbb{Z}(i))$

induces the identity on
${\mathrm{H}}^{p}(G,{\mathrm{H}}^{k-p}(A(\mathbb{C}),\mathbb{Z}(i)))$. By the
snake lemma and descending induction on $p$, we find that
$[d]^{\ast}\colon{\mathrm{H}}\to{\mathrm{H}}$ restricts to an isomorphism on
$F^{p}{\mathrm{H}}$ for each $p>0$.

Write ${\mathrm{M}}={\mathrm{H}}^{2k}_{G}(A(\mathbb{C}),\mathbb{Z}(k))$. By
[BW20, §1.6.4], the class $[d]^{\ast}(\alpha)$ satisfies the topological
condition (5) whenever $\alpha\in{\mathrm{M}}$ the topological condition (5),
thus $[d]^{\ast}$ induces an embedding $F^{p}{\mathrm{M}}_{0}\to
F^{p}{\mathrm{M}}_{0}$, which is an isomorphism since $F^{p}{\mathrm{M}}_{0}$
is finite. For similar reasons, $[d]^{\ast}\colon
F^{p}{\mathrm{M}}_{{\textnormal{alg}}}\to
F^{p}{\mathrm{M}}_{{\textnormal{alg}}}$ is an isomorphism, where
$F^{p}{\mathrm{M}}_{{\textnormal{alg}}}\subset F^{p}{\mathrm{M}}$ is the
subgroup of algebraic elements. In the general case, let $\psi\colon B\to A$
be an isogeny such that $\psi\circ\phi=[d]_{A}$ and $\phi\circ\psi=[d]_{B}$,
where $d$ is the degree of the isogeny $\phi$. The equalities

$\psi^{\ast}\circ\phi^{\ast}=[d]_{A}^{\ast}\quad\textnormal{ and
}\quad\psi^{\ast}\circ\phi^{\ast}=[d]_{B}^{\ast}$

together with the conclusion of the lemma for $[d]_{A}$ and $[d]_{B}$ readily
imply the result. ∎

###  6.2 Reduction to the Jacobian case

###### Proof of Theorem 1.7.

Let ${\mathcal{A}}_{3}(\mathbb{R})^{+}$ be a connected component of
${\mathcal{A}}_{3}(\mathbb{R})$, and consider a point
$x=[(A,\lambda)]\in{\mathcal{A}}_{3}(\mathbb{R})^{+}$. By Corollary 1.6, there
is a real algebraic curve $C$ with non-empty real locus together with an
isogeny

$\phi\colon A\to J(C)$

that preserves the polarizations up to an odd positive integer. By [GH81, page
180], we have $[J(C)]\in{\mathcal{A}}_{3}(\mathbb{R})^{+}$ for the moduli
point $[J(C)]$ of the principally polarized Jacobian $J(C)$ of $C$. The
following sequence is exact (see Lemma 2.2):

$0\to{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}[2]\to\textnormal{Hdg}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}\to\textnormal{Hdg}^{4}(A(\mathbb{C}),\mathbb{Z}(2))^{G}\to
0.$

By Theorem 1.3, we know that
$\textnormal{Hdg}^{4}(A(\mathbb{C}),\mathbb{Z}(2))^{G}$ is generated by
classes of one-cycles on $A$. Therefore, $A$ satisfies the real integral Hodge
conjecture if and only if the group
${\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}[2]$ is algebraic. By
Lemma 6.2, this is equivalent to the algebraicity of
${\mathrm{H}}^{4}_{G}(J(C)(\mathbb{C}),\mathbb{Z}(2))_{0}[2]$. ∎

###  6.3 Analysis of the Hochschild-Serre filtration

Let $A$ be an abelian variety over $\mathbb{R}$, define
${\mathrm{H}}={\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))$, and consider
the Hochschild-Serre filtration (c.f. Section 2.2):

$\displaystyle
0\subset{\mathrm{H}}^{4}(G,{\mathrm{H}}^{0}(A(\mathbb{C}),\mathbb{Z}(2)))=F^{4}{\mathrm{H}}\subset
F^{3}{\mathrm{H}}\subset F^{2}{\mathrm{H}}\subset
F^{1}{\mathrm{H}}={\mathrm{H}}[2]\subset{\mathrm{H}}.$ (17)

The pull-back $\pi^{\ast}$ along the structural morphism $\pi\colon
A\to\textnormal{Spec}(\mathbb{R})$ defines a section

$\mathbb{Z}/2={\mathrm{H}}^{4}_{G}(\\{x\\},\mathbb{Z}(2))={\mathrm{H}}^{4}(G,{\mathrm{H}}^{0}(\\{x\\},\mathbb{Z}(2)))\to{\mathrm{H}}^{4}(G,{\mathrm{H}}^{0}(A(\mathbb{C}),\mathbb{Z}(2)))\subset{\mathrm{H}}$

of the restriction
${\mathrm{H}}\to{\mathrm{H}}^{4}_{G}(\\{x\\},\mathbb{Z}(2))=\mathbb{Z}/2$ to
the equivariant cohomology of any $\mathbb{R}$-point $x\in X(\mathbb{R})$. By
Section 2.4, this implies that $F^{4}{\mathrm{H}}_{0}=(0)$, so that the
intersection of (17) with the group of classes satisfying the topological
condition (5) becomes

$\displaystyle 0\subset F^{3}{\mathrm{H}}_{0}\subset
F^{2}{\mathrm{H}}_{0}\subset F^{1}{\mathrm{H}}_{0}\subset{\mathrm{H}}_{0}.$
(18)

Continue to consider our abelian variety $A$ over $\mathbb{R}$. _Assume that
there exists a cycle_

$\Gamma\in\textnormal{CH}(A\times{\widehat{A}})\quad\textnormal{ \emph{such
that}
}\quad[\Gamma_{\mathbb{C}}]=\textnormal{ch}({\mathcal{P}}_{A_{\mathbb{C}}})\in{\mathrm{H}}^{2\bullet}(A(\mathbb{C})\times{\widehat{A}}(\mathbb{C}),\mathbb{Z}(\bullet))^{G}.$

Under this assumtion, we may define a homomorphism $\Gamma_{\ast}$ as in the
following diagram:

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3.0pt}}}}}}\ignorespaces{\hbox{\kern 96.91118pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
96.91118pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{H}}^{2\bullet}_{G}(A(\mathbb{C})\times{\widehat{A}}(\mathbb{C}),\mathbb{Z}(\bullet))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
147.71126pt\raise-20.56888pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\left([\Gamma]\cdot-\right)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 147.71126pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{H}}^{2\bullet}_{G}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(\bullet))}$}}}}}}}{\hbox{\kern
95.5223pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{H}}^{2\bullet}_{G}(A(\mathbb{C})\times{\widehat{A}}(\mathbb{C}),\mathbb{Z}(\bullet))\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
66.305pt\raise-46.67802pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.47362pt\hbox{$\scriptstyle{\pi_{2,\ast}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
34.55008pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\end{split}$
(19)

Then $\Gamma_{\ast}$ preserves the classes that satisfy the topological
condition (5) by [BW20, Theorem 1.21]. Define

$\displaystyle\Gamma_{\ast}^{i,j}\colon{\mathrm{H}}^{2i}_{G}(A(\mathbb{C}),\mathbb{Z}(i))\to{\mathrm{H}}^{2j}_{G}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(j))
\quad(i,j\in\mathbb{Z}_{\geq 0})$ (20)

as the composition of $\Gamma_{\ast}$ with the natural inclusion and
projection morphisms.

With these definitions in place, we are in a position to prove:

######  Proposition 6.3.

Let $A$ be an abelian variety over $\mathbb{R}$. Then

$F^{3}{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}=(0).$

###### Proof.

Let $g=\dim(A)$. By [Ham97, Chapter IV, Example 3.1], we have that
$A(\mathbb{C})$ is $G$-equivariantly homeomorphic to the $g$-fold product of
copies of the torus ${\mathbb{S}}\times{\mathbb{S}}$, where
${\mathbb{S}}\subset\mathbb{C}^{\ast}$ is the unit circle and where $G$ acts
on each copy ${\mathbb{S}}\times{\mathbb{S}}$ in one of the following ways:
either $\sigma(x,y)=(x,\bar{y})$ for $(x,y)\in{\mathbb{S}}\times{\mathbb{S}}$,
or $\sigma(x,y)=(y,x)$ for $(x,y)\in{\mathbb{S}}\times{\mathbb{S}}$. In
particular, there exist $g$ elliptic curves $E_{i}$ over $\mathbb{R}$ and a
$G$-equivariant homeomorphism

$\displaystyle A(\mathbb{C})\cong E_{1}(\mathbb{C})\times\cdots\times
E_{g}(\mathbb{C}).$ (21)

Since the conclusion of Proposition 6.3 is a statement that only depends on
the structure of $A(\mathbb{C})$ as a topological $G$-space, we may therefore
assume that $A$ is principally polarized by
$\theta\in\textnormal{Hdg}^{2}(A(\mathbb{C}),\mathbb{Z}(1))^{G}$ and that the
following class is algebraic:

$\frac{\theta^{g-1}}{(g-1)!}\in\textnormal{Hdg}^{2g-2}(A(\mathbb{C}),\mathbb{Z}(g-1))^{G}.$

By Theorem 5.2, this means that the Chern character
$\textnormal{ch}({\mathcal{P}}_{A_{\mathbb{C}}})$ is algebraic, which allows
us to define a homomorphism
$\Gamma_{\ast}\colon{\mathrm{H}}^{2\bullet}_{G}(A(\mathbb{C}),\mathbb{Z}(\bullet))\to{\mathrm{H}}^{2\bullet}_{G}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(\bullet))$
as in (19), and homomorphisms $\Gamma_{\ast}^{ij}$ as in (20). We have a
commutative diagram:

Here
${\mathscr{F}}_{A}\colon{\mathrm{H}}^{1}(A(\mathbb{C}),\mathbb{Z}(2))\to{\mathrm{H}}^{2g-1}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(g+1))$
is the Fourier transform considered in Section 5.1. This map is an
isomorphism, with inverse

$(-1)^{1+g}\cdot{\mathscr{F}}_{{\widehat{A}}}\colon{\mathrm{H}}^{2g-1}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(g+1))\to{\mathrm{H}}^{1}(A(\mathbb{C}),\mathbb{Z}(2)),$

see [Huy06, Corollary 9.24]. Consequently,
${\mathrm{H}}^{3}(G,{\mathscr{F}}_{A})$ is an isomorphism, with inverse
${\mathrm{H}}^{3}(G,{\mathscr{F}}_{{\widehat{A}}})$. By the compatibility of
$\Gamma_{\ast}$ with the topological condition (5), we obtain an isomorphism

$\displaystyle{\mathrm{H}}^{3}(G,{\mathscr{F}}_{A})\colon{\mathrm{H}}^{3}(G,{\mathrm{H}}^{1}(A(\mathbb{C}),\mathbb{Z}(2)))_{0}\xrightarrow{\sim}{\mathrm{H}}^{3}(G,{\mathrm{H}}^{2g-1}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(g+1)))_{0}.$
(22)

Because
${\mathrm{H}}^{2g+2}_{G}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(g+1))_{0}=(0)$
by [BW20, §2.3.2], the group on the right of equation (22) vanishes.
Therefore, the group on the left must be zero as well. ∎

In fact, the argument used in the above proposition can be generalized to
prove the following lemma, which we record for later use:

######  Lemma 6.4.

Let $A$ be an abelian variety of dimension $g$ over $\mathbb{R}$. Let $p$ and
$q$ be non-negative integers such that $p+q=2k\in 2\mathbb{Z}_{\geq 0}$. The
isomorphism on group cohomology

${\mathrm{H}}^{p}(G,{\mathscr{F}}_{A})\colon{\mathrm{H}}^{p}(G,{\mathrm{H}}^{q}(A(\mathbb{C}),\mathbb{Z}(k)))\to{\mathrm{H}}^{p}(G,{\mathrm{H}}^{2g-q}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(g+k-q)))$

identifies
${\mathrm{H}}^{p}(G,{\mathrm{H}}^{q}(A(\mathbb{C}),\mathbb{Z}(k)))_{0}$ with
${\mathrm{H}}^{p}(G,{\mathrm{H}}^{2g-q}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(g+k-q)))_{0}$.

###### Proof.

This is a topological statement, so we may and do assume that $A$ is a product
of elliptic curves (see (21)). In this case, we may lift
${\mathrm{H}}^{p}(G,{\mathscr{F}}_{A})$ to an algebraic homomorphism

$\Gamma_{\ast}^{2k,2g+2k-2q}\colon
F^{p}{\mathrm{H}}^{2k}_{G}(A(\mathbb{C}),\mathbb{Z}(k))\to
F^{p}{\mathrm{H}}^{2g+2k-2q}_{G}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(g+k-q))$

as above (see Section 6.3). We can do the same thing for
${\mathrm{H}}^{p}(G,{\mathscr{F}}_{{\widehat{A}}})$; the lemma follows from
arguments similar to those used to prove Proposition 6.3. ∎

##  7 Abel-Jacobi maps over the real numbers

Let $X$ be a smooth projective variety over $\mathbb{R}$. The goal of this
section is to prove that the equivariant cycle class map and the real Abel-
Jacobi map for $X$ are compatible; see Theorem 7.2 below for the precise
statement. To prove this, we need Deligne cohomology and Deligne cycle class
maps in the equivariant setting. Recall that Jannsen introduced
$G$-equivariant Deligne cycle class maps for varieties over $\mathbb{R}$ in
[Jan88]. Since his results do not seem to directly imply Theorem 7.2, we will
take a somewhat different approach.

Let $\sigma\colon X(\mathbb{C})\to X(\mathbb{C})$ be the canonical anti-
holomorphic involution. Fix an integer $k$ with $0\leq k\leq\dim(X)$, and
consider the $k$-th intermediate Jacobian of $X_{\mathbb{C}}$ [Voi02, PS08]:

$\displaystyle\begin{split}J^{2k-1}(X_{\mathbb{C}})&={\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{C})/\left(\Lambda+F^{k}{\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{C})\right),\\\
\Lambda&={\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{Z}(k))/(\textnormal{torsion}).\end{split}$
(23)

Observe that $\sigma$ induces an anti-holomorphic involution
$J^{2k-1}(X_{\mathbb{C}})\to J^{2k-1}(X_{\mathbb{C}})$. Consider the Abel-
Jacobi map
${\mathrm{AJ}}_{\mathbb{C}}\colon\textnormal{CH}^{k}({X_{\mathbb{C}}})_{{\textnormal{hom}}}\to
J^{2k-1}(X_{\mathbb{C}})$ introduced by Griffiths [Gri69]. By [EV88, §7], the
Abel-Jacobi map fits naturally in the following commutative diagram:

(28)

######  Lemma 7.1.

With respect to the canonical $G$-actions, each arrow in (28) is
$G$-equivariant.

###### Proof.

The $G$-equivariance of the horizontal homomorphisms in (28) is
straightforward, and the fact that the vertical cycle class map on the right
is $G$-equivariant is classical. To prove the lemma, it suffices to show that
Deligne cycle class map $cl_{\mathbb{C},D}$ is $G$-equivariant. For this,
consider a codimension $k$ subvariety $Y\subset X_{\mathbb{C}}$. By [PS08,
p.172], Deligne cohomology with supports in $Y(\mathbb{C})$ can be expressed
as a fibre product

$\displaystyle{\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}_{D}(k))={\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}(k))\times_{{\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{C})}{\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),F^{k}\Omega^{\bullet}_{X(\mathbb{C})}).$

There is a unique class
$\tau_{{\textnormal{Del}}}(Y)\in{\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}_{D}(k))$
that lifts both the Thom class $\tau(Y)$ in
${\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}(k))$ and the Thom-
Hodge class
$\tau_{\textnormal{Hdg}}(Y)\in{\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),F^{k}\Omega^{\bullet}_{X(\mathbb{C})})$,
see [PS08, Proposition 7.19]. The equality
$\sigma^{\ast}(\tau(Y))=\tau(Y^{\sigma})$ is classical. It is also clear that
$\sigma^{\ast}(\tau_{\textnormal{Hdg}}(Y))=\tau_{\textnormal{Hdg}}(Y^{\sigma})$;
see the construction of $\tau_{\textnormal{Hdg}}(Y)$ in the proof of Theorem
2.38 in [PS08]. The lemma follows. ∎

By Lemma 7.1, we obtain a _real Abel-Jacobi map_

$\displaystyle{\textnormal{AJ}}\colon\;\textnormal{CH}^{k}(X)_{{\textnormal{hom}}}\to
J^{2k-1}(X_{\mathbb{C}})^{G}.$ (29)

The boundary map in group cohomology induced by the exact sequence of
$G$-modules

$\displaystyle
0\to\Lambda\to{\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{R}(k))\to
J^{2k-1}(X_{\mathbb{C}})\to 0$ (30)

induces an isomorphism between $\pi_{0}(J^{2k-1}(X_{\mathbb{C}})^{G})$, the
group of connected components of the real locus of $J^{2k-1}(X_{\mathbb{C}})$,
and ${\mathrm{H}}^{1}(G,\Lambda)$. The spectral sequence (6) gives a
filtration $F^{\bullet}$ on
${\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$, together with natural
maps
$F^{1}{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))\to{\mathrm{H}}^{1}(G,\Lambda)$
and
${\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))\to{\mathrm{H}}^{2k}(X(\mathbb{C}),\mathbb{Z}(k))^{G}$,
such that
$cl\colon\textnormal{CH}^{k}(X)_{{\textnormal{hom}}}\to{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$
factors through
$F^{1}{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))\subset{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$.

######  Theorem 7.2.

Let $X$ be a smooth projective variety over $\mathbb{R}$. Let $k$ be an
integer with $0\leq k\leq\dim(X)$, and
$\Lambda={\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{Z}(k))/(\textnormal{torsion})$.
The following diagram commutes:

$\textstyle{\textnormal{CH}^{k}(X)_{{\textnormal{hom}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$AJ$\scriptstyle{cl\hskip
22.76219pt}$$\textstyle{F^{1}{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J^{2k-1}(X_{\mathbb{C}})^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{1}(G,\Lambda).}$

To prove Theorem 7.2, we need some notation. For a subring
$R\subset\mathbb{R}$ and an integer $d\in\mathbb{Z}_{\geq 0}$, define
$R_{D}(d)$ as the complex

$\displaystyle 0\to
R(d)\to\mathcal{O}_{X(\mathbb{C})}\to\Omega_{X(\mathbb{C})}^{1}\to\cdots\to\Omega_{X(\mathbb{C})}^{d-1}\to
0.$ (31)

By definition,
${\mathrm{H}}^{m}_{D}(X(\mathbb{C}),R(d))\coloneqq{\mathrm{H}}^{m}(X(\mathbb{C}),R_{D}(d))$
is the degree $m$ Deligne cohomology group with coefficients in $R(d)$ [Voi02,
PS08]. Note that $G$ acts on the complex (31).

######  Definition 7.3.

For an integer $m\in\mathbb{Z}_{\geq 0}$, define the _$G$ -equivariant Deligne
cohomology group of degree $m$ with coefficients in $R(d)$_ as
${\mathrm{H}}^{m}_{G,D}(X(\mathbb{C}),R(d))={\mathrm{H}}^{m}_{G}(X(\mathbb{C}),R_{D}(d))$.
Define the _$k$ -th equivariant intermediate Jacobian_ as
$J^{2k-1}_{G}(X)={\mathrm{H}}^{2k-1}_{G}(X(\mathbb{C}),\mathbb{R}/\mathbb{Z}(k)).$

The exact sequence of $G$-complexes $0\to\Omega_{X(\mathbb{C})}^{\leq
k-1}[-1]\to\mathbb{Z}_{D}(k)\to\mathbb{Z}(k)\to 0$ induces a natural
homomorphism
${\mathrm{H}}^{2k}_{G,D}(X(\mathbb{C}),\mathbb{Z}(k))\to{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$,
and we have:

######  Proposition 7.4.

There is a natural cycle class map
$\varphi\colon{\mathcal{Z}}^{k}(X)\to{\mathrm{H}}^{2k}_{G,D}(X(\mathbb{C}),\mathbb{Z}(k))$
such that the following diagram commutes:

$\textstyle{{\mathcal{Z}}^{k}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\scriptstyle{\varphi\hskip
25.60747pt}$$\textstyle{{\mathrm{H}}^{2k}_{G,D}(X(\mathbb{C}),\mathbb{Z}(k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}}^{k}(X_{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl_{\mathbb{C}}}$$\textstyle{{\mathrm{H}}^{2k}_{D}(X(\mathbb{C}),\mathbb{Z}(k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{2k}(X(\mathbb{C}),\mathbb{Z}(k)).}$

###### Proof.

We adapt the proof of Proposition 7.19 in [PS08]. One has a canonical quasi-
isomorphism
$R_{D}(k)\simeq{\textnormal{Cone}}^{\bullet}\left(\epsilon^{k}-\iota^{k}\colon
R(k)\oplus
F^{k}\Omega_{X(\mathbb{C})}^{\bullet}\to\Omega_{X(\mathbb{C})}^{\bullet}\right)[-1]$
for any ring $R\subset\mathbb{R}$, see [PS08, Lemma-Definition 7.13]. Thus, by
[PS08, (A-12)], there is an exact sequence

$\displaystyle\Omega_{X(\mathbb{C})}^{\bullet}\to\mathbb{Z}_{D}(k)[1]\to\mathbb{Z}(k)[1]\oplus
F^{k}\Omega_{X(\mathbb{C})}^{\bullet}[1].$ (32)

The complexes and morphisms in (32) are all $G$-equivariant. Let $Y\subset X$
be a codimension $k$ subvariety over $\mathbb{R}$. Taking $G$-equivariant
cohomology with supports in $Y$ of the sequence (32) gives a long exact
sequence

$\displaystyle\cdots\to$
$\displaystyle{\mathrm{H}}^{2k-1}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\Omega_{X(\mathbb{C})}^{\bullet})\to{\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}_{D}(k))$
$\displaystyle\to$
$\displaystyle{\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}(k))\oplus{\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),F^{k}\Omega_{X(\mathbb{C})}^{\bullet})\to{\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\Omega_{X(\mathbb{C})}^{\bullet})\to\cdots$

which is canonically isomorphic to the long exact sequence

$\displaystyle\cdots\to$
$\displaystyle{\mathrm{H}}^{2k-1}_{Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{C})^{G}\to{\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}_{D}(k))$
$\displaystyle\to$
$\displaystyle{\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}(k))\oplus{\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),F^{k}\Omega_{X(\mathbb{C})}^{\bullet})^{G}\to{\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{C})^{G}\to\cdots.$

The group ${\mathrm{H}}^{2k-1}_{Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{C})$
vanishes because by Poincaré-Lefschetz duality (see [Ful98, Section 19.1] or
[PS08, Theorem B.28]), one has
${\mathrm{H}}^{2k-1}_{Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{C})\cong{\mathrm{H}}^{{\textnormal{BM}}}_{2n-(2k-1)}(Y(\mathbb{C}),\mathbb{C})=(0).$
We conclude that
${\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}_{D}(k))$ can be
expressed as a fibre product

${\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}_{D}(k))={\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}(k))\times_{{\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{C})^{G}}{\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),F^{k}\Omega_{X(\mathbb{C})}^{\bullet})^{G}.$

Let $[Y]\in{\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}(k))$
be the image of $1\in\mathbb{Z}$ under the canonical isomorphism
${\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}(k))=\mathbb{Z}$,
see [BW20, equation (1.54)]. Then $[Y]$ maps to the Thom class $\tau(Y)$ under
${\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}(k))\to{\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{C})={\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{C})^{G},$
so there is a unique class
$[Y]_{D}\in{\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}_{D}(k))$
that maps to $[Y]$ in the group
${\mathrm{H}}^{2k}_{G,Y(\mathbb{C})}(X(\mathbb{C}),\mathbb{Z}(k))$ and to
$\tau_{{\textnormal{Hdg}}}(Y)$ in the group
${\mathrm{H}}^{2k}_{Y(\mathbb{C})}(X(\mathbb{C}),F^{k}\Omega^{\bullet}_{X(\mathbb{C})})$.
Upon forgetting the support, we obtain an equivariant Deligne cycle class
$[Y]_{D}\in{\mathrm{H}}^{2k}_{G,D}(X(\mathbb{C}),\mathbb{Z}(k)$. ∎

The natural isomorphism
$J^{2k-1}(X_{\mathbb{C}})={\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{R}/\mathbb{Z}(k))$
of $G$-modules gives a homomorphism $J^{2k-1}_{G}(X)\to
J^{2k-1}(X_{\mathbb{C}})^{G}$. Consider the short exact sequence of
$G$-complexes

$\displaystyle
0\to\mathbb{Z}_{D}(k)\to\mathbb{R}_{D}(k)\to\mathbb{R}/\mathbb{Z}\to 0.$ (33)

######  Lemma 7.5.

Sequence (33) induces the following commutative diagram with exact rows:

(42)

###### Proof.

The morphism
${\mathrm{H}}^{2k}_{D}(X(\mathbb{C}),\mathbb{Z}(k))\to{\mathrm{H}}^{2k}_{D}(X(\mathbb{C}),\mathbb{R}(k))$
factors through an embedding
$\textnormal{Hdg}^{2k}(X(\mathbb{C}),\mathbb{Z}(k))\hookrightarrow{\mathrm{H}}^{2k}_{D}(X(\mathbb{C}),\mathbb{R}(k))$,
which gives the triangle on the bottom right of (42). Moreover, the vertical
arrow on the right of (42) is an isomorphism because $\mathbb{R}_{D}(k)$ is a
$G$-complex of $\mathbb{R}$-vector spaces.

Next, we show that the boundary map
$J_{G}^{2k-1}(X)\to{\mathrm{H}}^{2k}_{G,D}(X(\mathbb{C}),\mathbb{Z}(k))$
induced by (33) is injective. This follows from the fact that the canonical
homomorphism

$f\colon{\mathrm{H}}^{2k-1}_{G,D}(X(\mathbb{C}),\mathbb{R}(k))\to
J^{2k-1}_{G}(X)$

is zero. To prove this, note that $f$ factors as

$\textstyle{{\mathrm{H}}^{2k-1}_{G,D}(X(\mathbb{C}),\mathbb{R}(k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\wr}$$\textstyle{{\mathrm{H}}^{2k-1}_{G}(X(\mathbb{C}),\mathbb{R}(k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{{\mathrm{H}}^{2k-1}_{G}(X(\mathbb{C}),\mathbb{R}/\mathbb{Z}(k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J^{2k-1}_{G}(X)}$$\textstyle{{\mathrm{H}}^{2k-1}_{D}(X(\mathbb{C}),\mathbb{R}(k))^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{{\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{R}(k))^{G},}$

so that it suffices to show that the map
$g\colon{\mathrm{H}}^{2k-1}_{D}(X(\mathbb{C}),\mathbb{R}(k))\to{\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{R}(k))$
is zero. This holds, because the exact sequence
$0\to\Omega_{X(\mathbb{C})}^{\leq
k-1}[-1]\to\mathbb{R}_{D}(k)\to\mathbb{R}(k)\to 0$ induces an exact sequence

$\cdots\to{\mathrm{H}}^{2k-1}_{D}(X(\mathbb{C}),\mathbb{R}(k))\xrightarrow{g}{\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{R}(k))\to{\mathbb{H}}^{2k-1}(X(\mathbb{C}),\Omega_{X(\mathbb{C})}^{\leq
k-1})\to\cdots$

in which

$\displaystyle{\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{R}(k))={\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{C})/F^{k}{\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{C})={\mathbb{H}}^{2k-1}(X(\mathbb{C}),\Omega_{X(\mathbb{C})}^{\leq
k-1}).$ (43)

See the proof of Proposition 12.26 in [Voi02] for the second isomorphism in
(43).

Finally, the map
$J^{2k-1}_{G}(X)\to{\mathrm{H}}^{2k}_{G,D}(X(\mathbb{C}),\mathbb{Z}(k))$ is
the boundary map in cohomology induced by an exact sequence of $G$-complexes,
thus shifts the Hochschild-Serre filtrations on both sides by one degree. This
gives a diagram in which both squares commute:

$\textstyle{J^{2k-1}_{G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{1}{\mathrm{H}}^{2k}_{G,D}(X(\mathbb{C}),\mathbb{Z}(k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{1}{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J^{2k-1}(X_{\mathbb{C}})^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{1}(G,{\mathrm{H}}^{2k-1}(X(\mathbb{C}),\mathbb{Z}_{D}(k)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{1}(G,\Lambda).}$

Consequently, the outer square commutes as well, and we are done. ∎

By Proposition 7.4 and Lemma 7.5, we obtain a commutative diagram with exact
rows:

$\displaystyle\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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(44)

######  Lemma 7.6.

The composition of the induced morphism
${\mathcal{Z}}^{k}(X)_{{\textnormal{hom}}}\to J_{G}^{2k-1}(X)$ with
$J^{2k-1}_{G}(X)\to J^{2k-1}(X_{\mathbb{C}})$ coincides with
${\mathcal{Z}}^{k}(X)_{{\textnormal{hom}}}\to{\mathcal{Z}}^{k}(X_{\mathbb{C}})_{{\textnormal{hom}}}\to
J^{2k-1}(X_{\mathbb{C}})$.

###### Proof.

Consider the following diagram:

$\textstyle{{\mathcal{Z}}^{k}(X)_{{\textnormal{hom}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}}^{k}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}}^{k}(X_{\mathbb{C}})_{{\textnormal{hom}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\textnormal{AJ}}_{\mathbb{C}}}$$\textstyle{{\mathcal{Z}}^{k}(X_{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J^{2k-1}_{G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{2k}_{G,D}(X(\mathbb{C}),\mathbb{Z}(k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J^{2k-1}(X_{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{2k}_{D}(X(\mathbb{C}),\mathbb{Z}(k)).}$

By Proposition 7.4 and Lemma 7.5, each square in this diagram, except possibly
the square on the left, commutes. By injectivity of the two horizontal arrows
in the bottom square, the square on the left commutes as well. ∎

###### Proof of Theorem 7.2.

The vertical composition on the left of the diagram (44) coincides with the
cycle class map $cl\colon{\mathcal{Z}}^{k}(X)_{{\textnormal{hom}}}\to
F^{1}{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))$ because the analogous
statement for the vertical composition in the middle of (44) holds, see
Proposition 7.4. We are done because by Lemma’s 7.5 and 7.6, each square in
the following diagram commutes:

$\textstyle{{\mathcal{Z}}^{k}(X)_{{\textnormal{hom}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J^{2k-1}_{G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{1}{\mathrm{H}}^{2k}_{G}(X(\mathbb{C}),\mathbb{Z}(k))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}}^{k}(X_{\mathbb{C}})^{G}_{{\textnormal{hom}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J^{2k-1}(X_{\mathbb{C}})^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{1}(G,\Lambda).}$

∎

##  8 Codimension-two cycles on real abelian varieties

We begin with the proof of Theorem 1.8. The proof consists of two steps: first
we prove the theorem for the Jacobian of a real algebraic curve with non-empty
real locus, and then we reduce the general case to this particular case.

###### Proof of Theorem 1.8 (Jacobian case).

Let us prove Theorem 1.8 in the case where $A$ is the Jacobian $J=J(C)$ of a
real algebraic curve $C$ of genus $g\in\mathbb{Z}_{\geq 1}$ such that
$C(\mathbb{R})\neq\emptyset$. By Proposition 6.3, one needs to show that
$F^{2}{\mathrm{H}}^{4}_{G}(J(\mathbb{C}),\mathbb{Z}(2))_{0}$ is algebraic. By
Theorem 5.2, the Chern character of the Poincaré bundle of $J$ in cohomology
${\mathrm{H}}^{2\bullet}(J(\mathbb{C})\times{\widehat{J}}(\mathbb{C}),\mathbb{Z}(\bullet))^{G}$
is algebraic. By Section 6.3 and [Huy06, Corollary 9.24], we obtain a
commutative diagram

(49)

such that the composition on the bottom row of (49) is the identity.
Therefore, the composition on the top row of (49) is the identity. Since $J$
satisfies the real integral Hodge conjecture for zero-cycles by [BW20,
Proposition 2.10], the group
$F^{2}{\mathrm{H}}^{2g}_{G}(J(\mathbb{C}),\mathbb{Z}(g))_{0}$ – and hence also
the group $F^{2}{\mathrm{H}}^{4}_{G}(J(\mathbb{C}),\mathbb{Z}(2))_{0}$ – is
algebraic. ∎

The following notation will be used throughout the remaining part of Section
8.

######  Definition 8.1.

For a smooth projective variety $X$ of dimension $n$ over $\mathbb{R}$, let
$\textnormal{CH}_{0}(X)_{0}$ be the group of zero-cycles of degree zero.
Observe that
$\textnormal{CH}_{0}(X)_{0}=\textnormal{CH}_{0}(X)_{{\textnormal{hom}}}$.

######  Lemma 8.2.

Let $f\colon J\to A$ be a surjective homomorphism of real abelian varieties.

  1. 1.

The induced homomorphism $f_{\mathbb{R}}^{0}\colon J(\mathbb{R})^{0}\to
A(\mathbb{R})^{0}$ is surjective.

  2. 2.

If $f_{\mathbb{R}}\colon J(\mathbb{R})\to A(\mathbb{R})$ is surjective, then
$\phi=f_{\ast}\colon\textnormal{CH}_{0}(J)_{0}\to\textnormal{CH}_{0}(A)_{0}$
is surjective.

###### Proof.

(i). Let ${\mathrm{Nm}}_{X}\colon X(\mathbb{C})\to X(\mathbb{R})^{0},x\mapsto
x+\sigma(x)$ be the norm homomorphism of a real abelian variety $X$. One has
that $f_{\mathbb{R}}^{0}\circ{\mathrm{Nm}}_{J}={\mathrm{Nm}}_{A}\circ
f_{\mathbb{C}}$. By [GH81, Proposition 1.1], the homomorphism
${\mathrm{Nm}}_{A}$ is surjective. Thus $f_{\mathbb{R}}^{0}$ is surjective.

(ii). Indeed, for an abelian variety $X$ over a field $k$, the group
$\textnormal{CH}_{0}(X)_{0}$ is generated by zero-cycles of the form
$[x]-\deg(k(x_{i})/k)\cdot[0]$ for closed points $x$ on $X$. ∎

###### Proof of Theorem 1.8 (general case).

To prove that $F^{2}{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}$ is
algebraic for a real abelian variety $A$, we would like to reduce to the
Jacobian case. The fact that this can be done rests on the following
proposition in combination with Lemma 8.5 below.

######  Proposition 8.3.

Let $0\to B\to J\xrightarrow{f}A\to 0$ be an exact sequence of abelian
varieties over $\mathbb{R}$ such that $f_{\mathbb{R}}\colon J(\mathbb{R})\to
A(\mathbb{R})$ is surjective. Let $d=\dim(J)$ and $g=\dim(A)$. Then

$\displaystyle f_{\ast}\colon
F^{2}{\mathrm{H}}^{2d}_{G}(J(C)(\mathbb{C}),\mathbb{Z}(d))_{0}\to
F^{2}{\mathrm{H}}^{2g}_{G}(A(\mathbb{C}),\mathbb{Z}(g))_{0}\quad\quad\text{
and }$ $\displaystyle\hat{f}^{\ast}\colon
F^{2}{\mathrm{H}}^{4}_{G}({\widehat{J}}(C)(\mathbb{C}),\mathbb{Z}(2))_{0}\to
F^{2}{\mathrm{H}}^{4}_{G}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(2))_{0}$

are surjective, where ${\widehat{f}}\colon{\widehat{A}}\to{\widehat{J}}$ is
the dual homomorphism of $f\colon J\to A$.

###### Proof of Proposition 8.3.

We will need some notation.

######  Definition 8.4.

For an abelian variety $A$ of dimension $g$ over $\mathbb{R}$, define

$\displaystyle F^{2}\textnormal{CH}_{0}(A)_{0}$
$\displaystyle=\textnormal{Ker}\left(\textnormal{CH}_{0}(A)_{0}\to{\mathrm{H}}^{1}(G,{\mathrm{H}}^{2g-1}(A(\mathbb{C}),\mathbb{Z}(g)))\right)$
$\displaystyle=\left\\{\alpha\in\textnormal{CH}_{0}(A)_{0}\mid\textnormal{AJ}(\alpha)\in
A(\mathbb{R})^{0}\right\\}.$

For the second equality in Definition 8.4, see Theorem 7.2. For an abelian
variety $X$ of dimension $n$ over $\mathbb{R}$, define
$\Lambda_{X}={\mathrm{H}}^{2n-1}(X(\mathbb{C}),\mathbb{Z}(n))$. Poincaré
duality identifies $\Lambda_{X}$ with
${\mathrm{H}}_{1}(X(\mathbb{C}),\mathbb{Z})$ [Man17, Corollaire 3.1.9], and we
have $\pi_{0}(X(\mathbb{R}))={\mathrm{H}}^{1}(G,\Lambda_{X})$ (see (30)).

  1. Step 1:

_The kernel of the push-forward
$\phi\colon\textnormal{CH}_{0}(J)_{0}\to\textnormal{CH}_{0}(A)_{0}$ surjects
onto the kernel of the homomorphism
$f\colon\pi_{0}(J(\mathbb{R}))\to\pi_{0}(A(\mathbb{R}))$._ It suffices to show
that the composition

$\displaystyle\textnormal{CH}_{0}(B)_{0}\to\textnormal{Ker}(\phi)\to\textnormal{Ker}(f)$
(50)

is surjective. Now (50) coincides with the composition
$\textnormal{CH}_{0}(B)_{0}\to\pi_{0}(B(\mathbb{R}))\to\textnormal{Ker}(f)$ of
surjections; the map $\pi_{0}(B(\mathbb{R}))\to\textnormal{Ker}(f)$ is
surjective by the exact sequence

$\displaystyle
0\to\Lambda_{B}^{G}\to\Lambda_{J}^{G}\to\Lambda_{A}^{G}\to{\mathrm{H}}^{1}(G,\Lambda_{B})\to{\mathrm{H}}^{1}(G,\Lambda_{J})\to{\mathrm{H}}^{1}(G,\Lambda_{A})\to
0$

arising from the short exact sequence of $G$-modules
$0\to\Lambda_{B}\to\Lambda_{J}\to\Lambda_{A}\to 0$.

  2. Step 2:

_Surjectivity of $f_{\ast}$_: Consider the following commutative diagram:

$\textstyle{F^{2}\textnormal{CH}_{0}(J)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi|_{F^{2}}}$$\textstyle{F^{2}\textnormal{CH}_{0}(A)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{2}{\mathrm{H}}^{2d}_{G}(J(\mathbb{C}),\mathbb{Z}(d))_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\ast}}$$\textstyle{F^{2}{\mathrm{H}}^{2g}_{G}(A(\mathbb{C}),\mathbb{Z}(g))_{0}.}$

Its vertical arrows are surjective by the real integral Hodge conjecture for
zero-cycles and Definition 8.4. To prove the surjectivity of $f_{\ast}$ on the
bottom row, it thus suffices to prove the surjectivity of $\phi|_{F^{2}}$ on
the top row. By Definition 8.4, the rows in the following commutative diagram
are exact:

Note that $\phi$ is surjective by Lemma 8.2. Since $J(\mathbb{R})\to
A(\mathbb{R})$ is surjective we have that the map
$f\colon\pi_{0}(J(\mathbb{R}))\to\pi_{0}(A(\mathbb{R}))$ is surjective. By the
snake lemma, to prove that $\phi|_{F^{2}}$ is surjective, it suffices to show
that $\textnormal{Ker}(\phi)\to\textnormal{Ker}(f)$ is surjective, which is
Step 1.

  3. Step 3:

_The maps $f_{\ast}$ and $\hat{f}^{\ast}$ fit into the following commutative
diagram, where the arrows $\twoheadrightarrow$ are surjective and the arrows
$\xrightarrow{\sim}$ are isomorphisms_:

(59)

Indeed, this follows from the functoriality of Fourier transforms (see [MP10,
(3.7.1)]), together with Proposition 6.3 and Lemma 6.4.

  4. Step 4:

_Surjectivity of $\hat{f}^{\ast}$_: The surjectivity of $f_{\ast}$ on the left
of (59) was shown in Step 2. By the commutativity on the left of diagram (59),
the morphism ${\mathrm{H}}^{2}(G,f_{\ast})$ is surjective. By the
commutativity of the square on the front, this implies that
${\mathrm{H}}^{2}(G,\hat{f}^{\ast})$ is surjective which, by the commutativity
on the right hand side of the diagram, implies that the morphism

$\hat{f}^{\ast}\colon
F^{2}{\mathrm{H}}^{4}_{G}({\widehat{J}}(\mathbb{C}),\mathbb{Z}(2))_{0}\to
F^{2}{\mathrm{H}}^{4}({\widehat{A}}(\mathbb{C}),\mathbb{Z}(2))_{0}$

is surjective.

This finishes the proof of Proposition 8.3. ∎

######  Lemma 8.5.

Let $A$ be an abelian variety over $\mathbb{R}$. Then $A$ contains a smooth,
proper, geometrically connected curve $C$ over $\mathbb{R}$ that passes
through $0\in A(\mathbb{R})$ in such a way that the following two conditions
hold: the induced homomorphism $f\colon J(C)\to A$ is surjective with
connected kernel, and the homomorphism $f_{\mathbb{R}}\colon
J(C)(\mathbb{R})\to A(\mathbb{R})$ is surjective.

###### Proof.

Let $S\subset A(\mathbb{R})$ be a finite set of points containing $0\in
A(\mathbb{R})$ and at least one point of each connected component of
$A(\mathbb{R})$. Since $\mathbb{R}$ is infinite, Bertini’s theorem can be
applied: there exists a smooth and geometrically connected hyperplane section
$Z\subset A$ passing through $S$ [Har77, II, Theorem 8.18], [Deb01, Footnote
12, page 32]. Let $g=\dim(A)$. By taking $g-1$ general such hyperplane
sections, we get a smooth, geometrically connected curve $C$ in $A$ that
contains $S$. We claim that $C$ satisfies the requirements stated in the
proposition. Write $J=J(C)$ and consider the map $f\colon J\to A$ arising from
the inclusion $(C,0)\hookrightarrow(A,0)$. By the proof of Theorem 10.1 in
[Mil86], the homomorphism $f$ is surjective. Moreover, the kernel of $f\colon
J_{\mathbb{C}}\to A_{\mathbb{C}}$ is connected. Indeed, this follows from the
fact that
${\mathrm{H}}_{1}(C(\mathbb{C}),\mathbb{Z})\to{\mathrm{H}}_{1}(A(\mathbb{C}),\mathbb{Z})$
is surjective by the Lefschetz hyperplane theorem; alternatively, see [Gab01,
Proposition 2.4] for an algebraic argument. The morphism
$f_{\mathbb{R}}^{0}\colon J(\mathbb{R})^{0}\to A(\mathbb{R})^{0}$ is
surjective by Lemma 8.2. Since $S$ is contained in the image of
$f_{\mathbb{R}}$, we conclude that $f_{\mathbb{R}}$ is surjective and we are
done. ∎

We can now finish the proof of Theorem 1.8. Let $A$ be an abelian variety over
$\mathbb{R}$. By Proposition 6.3, it remains to prove that
$F^{2}{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}$ is algebraic. Let
$C\subset{\widehat{A}}$ be a real algebraic curve that satisfies the
conditions of Lemma 8.5. By Proposition 8.3, the pull-back
$\hat{f}^{\ast}\colon
F^{2}{\mathrm{H}}^{4}_{G}({\widehat{J}}(C)(\mathbb{C}),\mathbb{Z}(2))_{0}\to
F^{2}{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}$ is surjective,
where $f\colon J(C)\to{\widehat{A}}$ is the homomorphism induced by the
inclusion of $(C,0)$ in $({\widehat{A}},0)$. By the proof of Theorem 1.8 in
the Jacobian case, we know that
$F^{2}{\mathrm{H}}^{4}_{G}({\widehat{J}}(C)(\mathbb{C}),\mathbb{Z}(2))_{0}$ is
algebraic. Therefore
$F^{2}{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}$ is algebraic. ∎

###### Proof of Corollary 1.9.

Consider the filtration (18). By Lemma 2.2, $A$ satisfies the real integral
Hodge conjecture if and only if
$F^{1}{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}$ and
$\textnormal{Hdg}^{4}(A(\mathbb{C}),\mathbb{Z}(2))^{G}$ are algebraic. Thus,
the corollary follows from Theorems 1.3, 1.8 and 7.2. ∎

##  9 One-cycles on real abelian threefolds that split as a product

###  9.1 An elliptic curve with connected real locus times an abelian surface

######  Proposition 9.1.

Let $B$ be a real abelian surface, and let $E$ be a real elliptic curve whose
real locus $E(\mathbb{R})$ is connected. The Abel-Jacobi homomorphism

$\displaystyle{\mathrm{AJ}}\;\colon\textnormal{CH}^{2}(B\times
E)_{\hom}\to{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(B(\mathbb{C})\times
E(\mathbb{C}),\mathbb{Z}(2))).$ (60)

is surjective. In particular, Theorem 1.10 holds.

###### Proof.

We have
${\mathrm{H}}^{1}(G,{\mathrm{H}}^{2}(B(\mathbb{C}),\mathbb{Z})\otimes{\mathrm{H}}^{1}(E(\mathbb{C}),\mathbb{Z}))=(0)$
because ${\mathrm{H}}^{1}(E(\mathbb{C}),\mathbb{Z})\cong\mathbb{Z}[G]$; the
latter follows from the fact that the real locus $E(\mathbb{R})$ of $E$ is
connected [Ham97, Chapter IV, Example 3.1]. By the Künneth formula, the
canonical morphism

$\displaystyle\begin{split}{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(B(\mathbb{C}),\mathbb{Z}(2))\;\;&\bigoplus\;\;{\mathrm{H}}^{1}(G,{\mathrm{H}}^{1}(B(\mathbb{C}),\mathbb{Z}(1))\otimes{\mathrm{H}}^{2}(E(\mathbb{C}),\mathbb{Z}(1)))\\\
&\to\;\;{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(B(\mathbb{C})\times
E(\mathbb{C}),\mathbb{Z}(2)))\end{split}$ (61)

is therefore an isomorphism. Since
${\mathrm{H}}^{2}(E(\mathbb{C}),\mathbb{Z}(1))\cong\mathbb{Z}$ as $G$-modules,
the canonical map

$\displaystyle\begin{split}{\mathrm{H}}^{1}(G,{\mathrm{H}}^{1}(B(\mathbb{C}),\mathbb{Z}(1)))\;\;&\bigotimes\;\;{\mathrm{H}}^{0}(G,{\mathrm{H}}^{2}(E(\mathbb{C}),\mathbb{Z}(1)))\\\
&\to\;\;{\mathrm{H}}^{1}(G,{\mathrm{H}}^{1}(B(\mathbb{C}),\mathbb{Z}(1))\otimes{\mathrm{H}}^{2}(E(\mathbb{C}),\mathbb{Z}(1)))\end{split}$
(62)

is an isomorphism as well. To simplify notation, for any abelian variety $X$
over $\mathbb{R}$, define

$\displaystyle{\mathrm{H}}^{i}_{X}(j)={\mathrm{H}}^{i}(X(\mathbb{C}),\mathbb{Z}(j)).$
(63)

Consider the following commutative diagram:

$\displaystyle\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
109.44698pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\textnormal{CH}^{2}(B)_{0}\bigoplus\left(\textnormal{CH}^{1}(B)_{\hom}\otimes\textnormal{CH}^{1}(E)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
153.46574pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-17.24998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
153.46574pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\textnormal{CH}^{2}(B\times
E)_{\hom}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
185.33109pt\raise-17.24998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{F^{1}{\mathrm{H}}^{4}_{G}(B(\mathbb{C}),\mathbb{Z}(2))\bigoplus\left(F^{1}{\mathrm{H}}^{2}_{G}(B(\mathbb{C}),\mathbb{Z}(1))\otimes{\mathrm{H}}^{2}_{G}(E(\mathbb{C}),\mathbb{Z}(1))\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
133.44698pt\raise-28.02396pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-45.27394pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
133.44698pt\raise-28.02396pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{F^{1}{\mathrm{H}}^{4}_{G}(B(\mathbb{C})\times
E(\mathbb{C}),\mathbb{Z}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
185.33109pt\raise-73.2979pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}_{B}(2))\bigoplus\left({\mathrm{H}}^{1}(G,{\mathrm{H}}^{1}_{B}(1))\otimes{\mathrm{H}}^{0}(G,{\mathrm{H}}^{2}_{E}(1))\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-70.05989pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.57501pt\hbox{$\scriptstyle{\wr}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-73.2979pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
182.33109pt\raise-56.04791pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-68.62631pt\raise-84.07187pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}_{B}(2))\bigoplus{\mathrm{H}}^{1}(G,{\mathrm{H}}^{1}_{B}(1)\otimes{\mathrm{H}}^{2}_{E}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
72.98917pt\raise-79.91621pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.15565pt\hbox{$\scriptstyle{\hskip 28.45274pt\sim}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
157.77295pt\raise-84.07187pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
157.77295pt\raise-84.07187pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}_{B\times
E}(2)).}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$

The indicated isomorphisms $\xrightarrow{\sim}$ in this diagram arise from
(61) and (62) above. The map
$\textnormal{CH}^{2}(B)_{0}\to{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}_{B}(2))$ is
surjective because by Theorem 7.2 it factors as

$\textnormal{CH}^{2}(B)_{0}=\textnormal{CH}_{0}(B)_{0}\twoheadrightarrow
B(\mathbb{R})\twoheadrightarrow{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(B(\mathbb{C}),\mathbb{Z}(2))).$

The map
$\textnormal{CH}^{1}(B)_{\hom}\to{\mathrm{H}}^{1}(G,{\mathrm{H}}^{1}_{B}(1))$
is surjective by the real integral Hodge conjecture for divisors (see Section
5.2) and the fact that the topological condition for degree two cohomology
classes is trivial [BW20, §2.3.1]. Thus, (60) is surjective. By Corollary 1.9,
the abelian variety $A=B\times E$ satisfies the real integral Hodge
conjecture. ∎

The topological condition (5) does not appear on the right hand side of (60).
This has the following corollary. For a real abelian variety $A$, one has
$\left|\pi_{0}(A(\mathbb{R}))\right|\leq 2^{\dim(A)}$ (see [GH81, §1]).
Moreover, for every non-negative pair of integers $(i,g)$ with $i\leq g$,
there is a real abelian variety $A$ of dimension $g$ such that
$\left|\pi_{0}(A(\mathbb{R}))\right|=2^{i}$ (take a suitable product of
elliptic curves). In particular, the map
${\mathcal{A}}_{3}(\mathbb{R})\to\mathbb{Z}$,
$[A]\mapsto\left|\pi_{0}(A(\mathbb{R}))\right|$ induces a bijection
$\pi_{0}({\mathcal{A}}_{3}(\mathbb{R}))\cong\\{1,2,4,8\\}$.

######  Corollary 9.2.

Let $A$ be a real abelian threefold, and suppose that
$\left|\pi_{0}(A(\mathbb{R}))\right|\neq 8$. The canonical map

$\displaystyle
F^{1}{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}\to{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}(2)))$
(64)

is surjective. Thus, $A$ satisfies the real integral Hodge conjecture if and
only if the real Abel-Jacobi map
$\textnormal{CH}_{1}(A)_{\hom}\to{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}(2)))$
is surjective.

###### Proof.

To prove the surjectivity of (64), we may replace $A(\mathbb{C})$ by any
differentiable $G$-manifold which is $G$-equivariantly diffeomorphic to
$A(\mathbb{C})$. In particular, we may assume that $A=B\times E$, where $B$ is
an abelian surface and $E$ an elliptic curve whose real locus is connected
(see (21)). The surjectivity of (64) follows then from the fact that the group
${\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(B(\mathbb{C})\times
E(\mathbb{C}),\mathbb{Z}(2)))$ is algebraic by Proposition 9.1. ∎

###  9.2 The product of three real elliptic curves

Recall that $\left|\pi_{0}(A(\mathbb{R}))\right|\in\\{1,2,4,8\\}$ for an
abelian threefold $A$ over $\mathbb{R}$. It follows from Theorem 1.10 that for
each $m\in\\{1,2,4\\}$, there is a real abelian threefold $A$ with
$\left|\pi_{0}(A(\mathbb{R}))\right|=m$ such that $A$ satisfies the real
integral Hodge conjecture.

######  Question 9.3.

Does there exist an abelian threefold $A$ over $\mathbb{R}$ with
$\left|\pi_{0}(A(\mathbb{R}))\right|=8$ such that $A$ satisfies the real
integral Hodge conjecture?

In this section, we consider this question for the third self-product of a
real elliptic curve.

Let $E$ be an elliptic curve over $\mathbb{R}$ with disconnected real locus.
Choose a symplectic basis $\\{x,y\\}$ of
${\mathrm{H}}^{1}(E(\mathbb{C}),\mathbb{Z})$ with $\sigma^{\ast}(x)=x$ and
$\sigma^{\ast}(y)=-y$ (see e.g. [GH81, §9]). Let $A=E^{3}$, and define
$\pi_{i}\colon A(\mathbb{C})\to E(\mathbb{C})$ as the projection onto the
$i$-th factor. We consider the basis element
$\gamma\coloneqq\pi_{1}^{\ast}(x)\cdot\pi_{2}^{\ast}(y)\cdot\pi_{3}^{\ast}(x)\in{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z})$,
as well as the Galois cohomology class
$[\gamma]=[\pi_{1}^{\ast}(x)\cdot\pi_{2}^{\ast}(y)\cdot\pi_{3}^{\ast}(x)]\in{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}))$
that $\gamma$ defines. Let $\beta\in
F^{1}{\mathrm{H}}^{2}_{G}(E(\mathbb{C}),\mathbb{Z})$ be any element that lifts
$[y]$. Let $\alpha\in{\mathrm{H}}^{1}_{G}(E(\mathbb{C}),\mathbb{Z})$ be the
unique element that maps to
$x\in{\mathrm{H}}^{1}(E(\mathbb{C}),\mathbb{Z})^{G}$. Finally, consider the
canonical map

$\displaystyle c\;\colon{\mathrm{H}}^{1}_{G}(E(\mathbb{C}),\mathbb{Z})\otimes
F^{1}{\mathrm{H}}^{2}_{G}(E(\mathbb{C}),\mathbb{Z})\otimes{\mathrm{H}}^{1}_{G}(E(\mathbb{C}),\mathbb{Z})\to
F^{1}{\mathrm{H}}_{G}^{4}(A(\mathbb{C}),\mathbb{Z}).$

######  Proposition 9.4.

The element $\omega=c(\alpha\otimes\beta\otimes\alpha)\in
F^{1}{\mathrm{H}}_{G}^{4}(A(\mathbb{C}),\mathbb{Z}(2))$ satisfies the
topological condition (5), and every class in
$F^{1}{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))_{0}$ is a
$\mathbb{Z}/2$-linear combination of $\omega$ and algebraic elements. In
particular, $A$ satisfies the real integral Hodge conjecture if and only if
$\omega$ is algebraic if and only if
$[\gamma]\in{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}(2)))$
is algebraic.

######  Lemma 9.5.

Let $X$ be a paracompact Hausdorff topological $G$-space and $F$ a $G$-sheaf
on $X$. For $m>\dim(X)$, the canonical map
${\mathrm{H}}^{m}_{G}(X,F)\to{\mathrm{H}}^{m}_{G}(X^{G},F|_{X^{G}})$ is an
isomorphism.

###### Proof.

Define $Z=X^{G}$ and let $j\colon U=X\setminus Z\to X$ and $\iota\colon Z\to
X$ be the canonical inclusions. Write $F_{U}=j_{!}(F|_{U})$ and
$F_{Z}=\iota_{\ast}(F|_{Z})$. The canonical homomorphism
${\mathrm{H}}^{i}_{G}(Z,F|_{Z})\to{\mathrm{H}}^{i}_{G}(X,F_{Z})$ is an
isomorphism, see [Gro57, Propositions 2.2.1 & 3.1.4], which gives an exact
sequence
${\mathrm{H}}^{m}_{G}(X,F_{U})\to{\mathrm{H}}^{m}_{G}(X,F)\to{\mathrm{H}}^{m}_{G}(Z,F|_{Z})\to{\mathrm{H}}^{m+1}_{G}(X,F_{U}).$
By [Gro57, Theorems 5.2.1 & 5.3.1], we have
${\mathrm{H}}^{i}(X/G,{\mathscr{H}}^{0}(G,F_{U}))={\mathrm{H}}^{i}_{G}(X,F_{U})$
for each positive $i$. Moreover, $\dim(X/G)\leq\dim(X)$ by [Pea75, Chapter 6,
Proposition 4.11]. On paracompact Hausdorff spaces, sheaf cohomology coincides
with Čech cohomology, see [God58, Théorème 5.10.1]. In particular,
${\mathrm{H}}^{i}(X/G,{\mathscr{H}}^{0}(G,F_{U}))=(0)$ for $i>\dim(X)$. ∎

###### Proof of Proposition 9.4.

Let $p=(p_{1},p_{2},p_{3})\in A(\mathbb{R})$. Since
${\mathrm{H}}^{1}_{G}(\\{p_{i}\\},\mathbb{Z})=(0)$, we have $\omega|_{p}=0$,
hence $\omega$ satisfies the topological condition. By Theorem 1.8, it
suffices to show that
${\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}(2)))_{0}$ is
generated by $[\gamma]$ and algebraic elements. We have

${\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}))\cong\bigoplus_{i+j+k=3}{\mathrm{H}}^{1}\left(G,{\mathrm{H}}^{i}(E(\mathbb{C}),\mathbb{Z})\otimes{\mathrm{H}}^{j}(E(\mathbb{C}),\mathbb{Z})\otimes{\mathrm{H}}^{k}(E(\mathbb{C}),\mathbb{Z})\right).$

Define ${\mathrm{H}}^{i}_{E}(j)$ as in (63). The map
${\mathrm{H}}^{2}_{E}(1)^{G}\otimes{\mathrm{H}}^{1}(G,{\mathrm{H}}^{1}_{E}(1))\to{\mathrm{H}}^{1}(G,{\mathrm{H}}^{2}_{E}(1)\otimes{\mathrm{H}}_{E}^{1}(1))$
is an isomorphism, and lifts to
$F^{0}{\mathrm{H}}^{2}_{G}(E(\mathbb{C}),\mathbb{Z}(1))\otimes
F^{1}{\mathrm{H}}^{2}_{G}(E(\mathbb{C}),\mathbb{Z}(1))\to
F^{1}{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))$. Consequently, the
group
${\mathrm{H}}^{1}(G,{\mathrm{H}}^{i}_{E}\otimes{\mathrm{H}}^{j}_{E}\otimes{\mathrm{H}}^{k}_{E})\subset{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}))$
is algebraic for each distinct $i,j,k$ such that $i+j+k=3$.

######  Claim 9.6.

Let
$\xi=[\pi_{1}^{\ast}(y)\cdot\pi_{2}^{\ast}(y)\cdot\pi_{3}^{\ast}(y)]\in{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}(2)))$.
There exists no lift $\tau\in
F^{1}{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z}(2))$ of $\xi$ that
satisfies topological condition (5).

Let
${\mathbb{S}}=\left\\{z\in\mathbb{C}\mid\left|z\right|=1\right\\}\subset\mathbb{C}$.
Define ${\mathbb{S}}^{+}$ to be the differentiable $G$-manifold ${\mathbb{S}}$
with trivial $G$-action, and let ${\mathbb{S}}^{-}$ be the differentiable
$G$-manifold ${\mathbb{S}}$ such that $\sigma(z)=\overline{z}$. Define
$Y={\mathbb{S}}^{-}\times{\mathbb{S}}^{-}\times{\mathbb{S}}^{-}$. Any
$p\in{\mathbb{S}}^{+}$ defines a section $s\colon Y\to A(\mathbb{C})$ of the
projection $q\colon A(\mathbb{C})\to Y.$ For a generator
$\eta\in{\mathrm{H}}^{3}(Y,\mathbb{Z})$, one has $q^{\ast}([\eta])=\xi$ with
respect to
$q^{\ast}\colon{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(Y,\mathbb{Z}))\to{\mathrm{H}}^{1}(G,{\mathrm{H}}^{3}(A(\mathbb{C}),\mathbb{Z}))$.
We have $\dim_{\mathbb{Z}/2}{\mathrm{H}}^{4}_{G}(Y,\mathbb{Z})=8$ because the
Hochschild-Serre spectral sequence for $Y$ degenerates (since $Y$ is a direct
factor of an abelian variety). The map
${\mathrm{H}}^{4}_{G}(Y,\mathbb{Z})\to{\mathrm{H}}^{0}(Y^{G},\mathbb{Z}/2)$ is
surjective by Lemma 9.5 and [BW20, §1.2.2]; because the $\mathbb{Z}/2$-ranks
on both sides agree, it is an isomorphism. If $\tau\in
F^{1}{\mathrm{H}}^{4}_{G}(A(\mathbb{C}),\mathbb{Z})_{0}$ lifts
$\xi=q^{\ast}([\eta])$, then $s^{\ast}(\gamma)$ is a lift of
$s^{\ast}q^{\ast}([\eta])=[\eta]$ such that
$s^{\ast}(\gamma)|_{p}=0\in{\mathrm{H}}^{4}_{G}(\\{p\\},\mathbb{Z})$ for $p\in
Y^{G}$. This is absurd, and Claim 9.6 follows. Let
$\Lambda={\mathrm{H}}^{1}(E(\mathbb{C}),\mathbb{Z})$, and consider the basis

$\displaystyle\left\\{[x\otimes x\otimes y],[x\otimes y\otimes x],[y\otimes
x\otimes x],[y\otimes y\otimes
y]\right\\}\subset{\mathrm{H}}^{1}(G,\Lambda\otimes\Lambda\otimes\Lambda).$

To finish the proof, it suffices to show that $[x\otimes x\otimes y]-[x\otimes