\section{Introduction}

\subsection{Overview} For a smooth projective variety $X$ over $\mathbb{C}$ and an integer $p \ge 0$ there is a cycle class map $\operatorname{CH}^{p}(X) \to \rH^{2p}(X, \mathbb{Q})$ from the space of codimension $p$ cycles on $X$ to the $2p$-th Betti cohomology of $X$. This cycle class map lands in $\operatorname{Hdg}^{p}(X,\mathbb{Q}):=\rH^{2p}(X,\mathbb{Q}) \cap \rH^{p,p}(X, \mathbb{C})$, where $\rH^{p,p}(X, \mathbb{C}) \subset \rH^{2p}(X, \mathbb{C})$ is part of the Hodge decomposition. The \emph{Hodge conjecture} for $X$ asserts that the cycle class map surjects onto $\operatorname{Hdg}^{p}(X,\mathbb{Q})$ for all $p$, see e.g. \cite{DeligneHodge}, \cite{HodgeHodge}. The goal of this note is to explain the following result of \cite{OAI}.
\begin{mainThm} \label{Thm:IntroMain}
If $X$ is an abelian variety admitting complex multiplication, then the Hodge conjecture holds for $X$.
\end{mainThm}
For previous work on the Hodge conjecture for abelian varieties, see \cite{gordon1997surveyhodgeconjectureabelian} and the references therein for a survey of work done up to 1997. For subsequent work, see e.g. \cite{MilneLefschetzClasses},\cite{MoonenZarhin},\cite{AbdulaliV},\cite{AbdulaliIV},\cite{Abdulali},\cite{Hazama},\cite{HazamaIII}, \cite{AbdulaliIII},\cite{MR2099754},\cite{HazamaII},\cite{MR2384535},\cite{OSullivan},\cite{AbdulaliVI}, \cite{LombardoSchutt},\cite{AbdulaliVII},\cite{AbdulaliVIII},\cite{MilneHodgeClasses},
\cite{Ancona},\cite{VarescoVoisin},\cite{vGeemen2022},\cite{vGeemen2023},\cite{MarkmanKummer}, \cite{floccari2025k3surfacesassociatedvarieties}, \cite{Agugliaro2025},\cite{markman2025cyclesabelian2nfoldsweil}, \cite{vangeemen2026hyperkahlersixfoldsabelianfourfolds}, \cite{floccari2026hodgeconjectureweilfourfolds}, \cite{markman2026secantsheavesweilclasses} for an incomplete list.

\subsubsection{Consequences for CM abelian varieties} By \cite[Main result]{AbdulaliGHC}, Theorem \ref{Thm:IntroMain} implies the generalized Hodge conjecture for CM abelian varieties (cf. \cite[Theorem 8.3]{Hazama}). Since the Mumford--Tate conjecture is known for CM abelian varieties by work of Pohlmann \cite{Pohlmann}, Theorem \ref{Thm:IntroMain} implies the Tate conjecture for CM abelian varieties over finitely generated fields of characteristic zero. 

\subsubsection{Consequences for all abelian varieties} Let $k$ be an algebraically closed field. By \cite[Theorem 4.6.(a)]{MilneHodgeStandard}, Theorem \ref{Thm:IntroMain} implies that numerical equivalence coincides with $\ell$-adic homological equivalence for abelian varieties over $k$ with $\ell$ coprime to the characteristic of $k$. By \cite[Theorem 4.6.(b)]{MilneHodgeStandard}, Theorem \ref{Thm:IntroMain} implies that the Hodge standard conjecture holds for abelian varieties over $k$ with respect to any good (in the sense of \cite[\S 4]{MilneHodgeStandard}) Weil cohomology theory. 

\subsubsection{Consequences for abelian varieties over finite fields}

By \cite[Theorem 7.1]{MilneTate}, Theorem \ref{Thm:IntroMain} implies the Tate conjecture for abelian varieties over finite fields. This in turn implies, by \cite[Corollary 3.10]{MR2032532}, the Lichtenbaum conjectures \cite{MR756089} for special values of their zeta functions. Moreover, \cite[Theorem 1]{MR2032532} also proves that numerical equivalence coincides with $\ell$-adic homological equivalence for abelian varieties over finite fields.

\subsection{Overview of the proof} The overall structure of our proof is the same as that of \cite{OAI}, see Section \ref{sub:ComparisonOAI} for a more detailed comparison. It follows from work of Hazama, \cite{Hazama}, that it is enough to prove the conjecture in codimension two, i.e., for $p=2$. \smallskip 

Given a CM abelian variety $A$ over $\mathbb{C}$ with CM by a CM algebra $L$, we can write $L=L_1 \times \cdots \times L_n$ as a product of CM fields. We now choose a Galois CM field $E \subset \mathbb{C}$ containing $L_i$ for all $i$ with totally real subfield $F$ and $c$ the nontrivial $F$-automorphism of $E$. Let $\Sigma=\operatorname{Hom}(L,E)$ be the set of ring homomorphisms from $L$ to $E$, equipped with its action of $G=\operatorname{Gal}(E/\mathbb{Q})$, and let $\Phi \subset \Sigma$ denote the CM type of $A$. We may write 
\begin{align}
    \rH^1(A, E) = \bigoplus_{s \in \Sigma} \rH^1(A, E)_s
\end{align}
as a direct sum (one-dimensional) of $s$-isotypic components. We may thus identify
\begin{align}
      \rH^r(A,E) = \bigoplus_{\Delta} \rH^r(A,E)_{\Delta},
\end{align}
where $\Delta$ runs over size $r$ subsets of $\Sigma$. A result of Pohlmann, see \cite[Theorem 1]{Pohlmann} and \cite[Theorem 3.1]{gao2025hodgecyclesquadraticrelations} for the nonsimple case, describes the Hodge classes
as
\begin{align} \label{eq:IntroDegreeTwoHodgeClasses}
    \operatorname{Hdg}^2(A, \mathbb{Q}) \otimes E = \bigoplus_{\substack{\Delta \subset \Sigma, |\Delta|=4 \\ \forall g \in G |(g \Delta \cap \Phi)| =2 }} \rH^{4}(A,E)_{\Delta}.
\end{align}
For $\Delta$ as above, we can thus ponder the $\Delta$-part of the (codimension $2$) Hodge conjecture for $A$. A key new idea of \cite{OAI} is to give a \emph{separate} argument for the $\Delta$-part of the Hodge conjecture for each $\Delta$.

\subsubsection{Auxiliary abelian varieties with CM} Writing $\Delta=(s_1, s_2, c \circ s_3, c \circ s_4)$, we construct CM types $\Phi_1, \cdots, \Phi_4 \subset \operatorname{Hom}(E,E)$ for the CM field $E$, defined by 
\begin{align}
     \Phi_i:= \ \{ g \in G=\operatorname{Hom}(E,E) : g \circ s_i \in \Phi\}.
\end{align}
Let $B_1, \cdots, B_4$ be abelian varieties over $\mathbb{C}$ with CM by $E$ and CM types $\Phi_1, \cdots, \Phi_4$, and let $B=B_1 \times \cdots \times B_4$. This construction of the auxiliary CM abelian varieties $B_i$ is a special case of a construction occuring in Deligne's famous proof that Hodge cycles are absolute Hodge \cite{MR654325}; it also occurs in \cite{Andre} where it is used to prove that every Hodge class on $A$ is a sum of pullbacks of Weil classes on auxiliary abelian varieties. \smallskip 

In \cite{OAI}, the auxiliary abelian varieties $B_i$ are used to make the elementary observation that the $\Delta$-part of the Hodge conjecture for $A$ follows for the Hodge conjecture for
\begin{align} \label{eq:IntroClass}
    \bigotimes_i H^1(B_i, E)_{\lambda_i} \subset H^4(B,E),
\end{align}
where $\lambda=(\lambda_1, \lambda_2, \lambda_3, \lambda_4)=(\operatorname{Id}, \operatorname{Id}, c , c)$, see Lemma \ref{Lem:AbelianVarietyTransfer}. The benefit of having four CM abelian varieties instead of one, is that we now have an action of four copies of $E$. We then have the following proposition.
\begin{Prop} \label{Prop:IntroSurfaceReduction}
If there is a smooth projective surface $S$ over $\mathbb{C}$ together with maps of $\mathbb{Q}$-Hodge structures 
\begin{align}
    h_i:\rH^1(B_i, \mathbb{Q}) \to \rH^1(S, \mathbb{Q})
\end{align}
for $i=1,2,3,4$, and nonzero vectors $\alpha_i \in  \rH^1(B_i, E)_{\lambda_i}$ such that 
\begin{align}
    h_1(\alpha_1) \cup h_2(\alpha_2) \cup h_3(\alpha_3) \cup h_4(\alpha_4) \not = 0
\end{align}
then the line \eqref{eq:IntroClass} lies in the image of the $E$-linear cycle class map
\end{Prop}
To prove Proposition \ref{Prop:IntroSurfaceReduction}, we note that the $h_i$ induce a morphism $h:S \to B$ and $h_{\ast}[S]$ gives a cycle $Z$ on $B$ in dimension two which pairs nontrivially with the line \eqref{eq:IntroClass}. Next, we use the action of $E$ on $B_i$ and the algebraicity of the K\"unneth projectors for each $B_i$ to get an idempotent endomorphism $e$ of the $E$-linear motive associated to $B$ projecting onto the line \eqref{eq:IntroClass}. Then $e^t Z$ is a cycle pairing nontrivially with the line \eqref{eq:IntroClass}, and so by equality of homological and numerical equivalence on $B$, there must be an $E$-linear algebraic cycle in codimension two whose cycle class map spans \eqref{eq:IntroClass}.

\subsubsection{} Of course it is not at all clear how Proposition \ref{Prop:IntroSurfaceReduction} is useful. Another key idea of \cite{OAI} is to take for $S$ a Picard modular surface associated to $E$. More precisely, we take a Shimura variety associated to a Hermitian space $V$ over $E$ with signature $(2,1)$ at the distinguished embedding $E \subset \mathbb{C}$ and $(3,0)$ at all other places. The first cohomology of these two-dimensional Shimura varieties can be completely understood using theta lifts, see Theorem \ref{Thm:HodgeStructureComputation}, which we explain following \cite{Liu}. It turns out to be essentially immediate from this that the Hodge structures of CM abelian varieties with CM by $E$ appear in this first cohomology, see Corollary \ref{Cor:CMAVOccurs}.

\subsubsection{} To carry out the cup product computation, we produce explicit cohomology classes with complex coefficients. More precisely, the work of Bergeron--Millson--Moeglin \cite{BMM} shows that the first cohomology of $S$ is spanned by theta lifts, see Proposition \ref{Prop:ThetaRealization}. Theorem \ref{Thm:IntroMain} thus reduces to showing that there are theta lifts whose cup-product is nonzero. To do this, we consider the fourfold cup-product as a functional on a Schwartz space, and then argue that it is enough to show that this functional is nonzero. This we finally do using the doubling method in Section \ref{subsec: doubling}. 

\begin{Rem}
We recently learned that Wei Zhang had considered the rough
strategy followed in this paper and in \cite{OAI} many years ago, and also tried to use
this to prove the Hodge conjecture for CM abelian varieties. A key point missing to him was a nonvanishing
result for the surface period analogous to Theorem \ref{Thm:ExistenceCohomologyClasses}.
\end{Rem}

\subsection{Comparison to [Ope26]} \label{sub:ComparisonOAI} Our proof of Theorem \ref{Thm:IntroMain} is heavily inspired by the proof given in \cite{OAI}. Nevertheless, we do not claim to understand (or even to have read) the details of the arguments written in \cite{OAI}. The goal of this note is to give a variant of the proofs in \cite{OAI} that relies as much as possible on the written literature. This has the benefit of making the proofs shorter, and possibly also easier to understand. To the best of the authors' knowledge, the argument for Theorem \ref{Thm:ExistenceCohomologyClasses} is a departure from the argument for the analogous statement in \cite{OAI}. We leave the task of a careful exegesis of the original proof to others. \smallskip 

\subsection{Tool use and methodology} This article is entirely human written. That said, we made extensive use of ChatGPT Pro during the preparation of this argument.  \smallskip

We asked the model to give a version of the arguments in \cite[Section 2]{OAI} that relies on \cite{Hazama}, and this formed the starting point of our Section \ref{Sec:Reduction}. Moreover, Lemma \ref{Lem:AbelianVarietyTransfer} and its proof were suggested to us by that model. We asked the model for a more direct proof of \cite[Proposition 3.1]{OAI}, which lead our proof of Theorem \ref{Thm:ReductionToSurface}. When we asked the model for a reference to compute the first Betti cohomology of Picard modular surfaces, it suggested \cite{Liu}, and we wrote Section \ref{Sec:PMS} without further assistance except for the proof of Lemma \ref{Lem:ExistenceTriple} and Corollary \ref{Cor:CMAVOccurs}, which were also suggested to us by the model. Section \ref{Sec:CupProduct} was written after extensive back-and-forth with the model, in which the authors insisted that the corresponding calculation in \cite{OAI} should be done instead using the doubling method and recourse to the Siegel--Weil formula. In particular, Lemma \ref{lem: cst term} was eventually output after this back and forth with the model, the current proof was then worked out after a further extensive back and forth. The basic idea of varying the test function as is carried out in Lemma \ref{lem: wiggle} was also suggested by the model. In addition we used the model in more quotidian ways: to proofread the document and check the arguments once the paper was assembled, and to do a literature search for the introduction. 

\subsection{Acknowledgments} We thank Yifeng Liu for very helpful discussions and comments on a draft.