\section{What is proved and what is open}\label{sec:status}


Table~\ref{tab:status} lists the statements. Three remarks on what the table says.

The two open statements are not independent problems that happen to add up to the conjecture; they are two halves of one phenomenon. Both say that the algebraic locus of a flat Hodge class, a countable union of closed strata through a dense set of points of complex multiplication, is not meagre. For \Fii{} the dense set is supplied by explicit cycles at CM points and the first instance is a single class on an abelian fourteenfold or a single Ext group on an eightfold; for \Fiii{} the dense set is supplied by the K3 surfaces with complex multiplication and by the chains of Section~\ref{sec:f3prime}, and the first instance is a cubic real multiplication on one K3 surface of Picard number $13$, which Theorem~\ref{thm:zeta9} supplies for the fields $\QQ(\zeta_{7})^{+}$ and $\QQ(\zeta_{9})^{+}$ with the algebra ramified at two real places, and which stays open for every other pair of a cubic field and an algebra.

Each construction of this paper ends at a question with a definite answer. Whether some convolution of the pieces of Theorem~\ref{thm:product} has $\dim\Ext^{2}=57$ is a finite computation in the Ext algebra of two sheaves on a fourfold; at the point $E_{0}^{6}$ of complex multiplication every group entering it is written down in Lemma~\ref{lem:cmext}, Proposition~\ref{prop:separated} answers it in the negative for every convolution with three powers of $L$, Proposition~\ref{prop:e06count} answers it in the negative for two threefolds at one shift and for every placement of the thirty-five pieces, Proposition~\ref{prop:chainfree} with Corollary~\ref{cor:chainfree} for every further piece of positive twist within one shift of the thirty-two, Theorem~\ref{thm:fivebundles} for at most five line bundles at any twists and every polarisation, the enumeration of Remark~\ref{rem:sevenbundles} for six and seven and for eight outside thirty-one arrangements, Corollary~\ref{cor:threefolds} and Theorem~\ref{thm:sixtyfour} for every convolution of threefold pieces, of one family or of both, at any twists and shifts, and for the configurations left, eight or more line bundles with or without threefolds, it is the rank of the products along the chains of Proposition~\ref{prop:twostep} and along the cycles through the line bundles, the support of the object being constrained by Theorem~\ref{thm:locality} and Propositions~\ref{prop:rest} and \ref{prop:ci}. Whether the link $\xi$ is algebraic is a question about one Hodge class on one abelian variety, equivalent by Theorem~\ref{thm:link}(v) to the exceptional classes together with one more cycle. Whether a correspondence acting on $T(S_{\lambda})$ otherwise than by a similarity exists is a question about one K3 surface; for $F=\QQ(\zeta_{7})^{+}$ or $\QQ(\zeta_{9})^{+}$ and $D=D_{\infty}$ Theorem~\ref{thm:zeta9} answers it, and for every other pair $(F,D)$ it is open, with Proposition~\ref{prop:cubicstop} marking the three places where the present mechanism fails. A negative answer to the first closes that route, as the analogous question on $E_{0}^{6}$ closed; a positive answer to any of them proves the corresponding statement at its first instance.

\subsection{Specialisation, degeneration and blow-ups}\label{ssec:bypass}

Three devices are often proposed for passing from the members where a class is known to be algebraic to the others, and we record exactly what each one proves.

\begin{proposition}\label{prop:bypass}
Let $\pi:\mathcal{X}\to B$ be a smooth projective family over an irreducible base and $\xi$ a flat Hodge class on its fibres.
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.4em,itemsep=2pt]
\item If $\xi_{t}$ is algebraic at the generic point of a curve $C\subseteq B$ then it is algebraic at every point of $C$. The converse implication, from one point of $C$ to its generic point, holds when $\xi_{t_{0}}$ is the Chern character of a perfect complex whose semiregularity map is injective, and in no greater generality is it known.
\item Let $\tilde X$ be the blow-up of a smooth projective variety $X$ along a smooth closed subvariety $Z$ of codimension $c$. Then $\Hdg^{k}(\tilde X)/\Alg^{k}(\tilde X)\cong\Hdg^{k}(X)/\Alg^{k}(X)\oplus\bigoplus_{i=1}^{c-1}\Hdg^{k-i}(Z)/\Alg^{k-i}(Z)$. In particular the Hodge conjecture for $\tilde X$ is equivalent to the Hodge conjecture for $X$ and for $Z$, and no sequence of blow-ups and blow-downs reduces the defect of a variety.
\item For the Weil classes of $\cD_{n,n}$ and the exceptional class of a Mumford fourfold, every degeneration to a fibre with a semistable model on which the limit class is algebraic has its lift to the generic fibre as its missing step, and for the Mumford fourfold no degeneration keeping the class of Hodge type exists, since the Hodge locus of the class is a compact Shimura curve.
\end{enumerate}
\end{proposition}

\begin{proof}
(i) Specialisation of cycles along a curve \cite[Ch.~20]{Ful98} sends an algebraic cycle on the generic fibre to one on each special fibre with the same flat cohomology class; this is the direction used by Schoen's degeneration and by Proposition~\ref{prop:weil-dense}. In the other direction the algebraic locus is a countable union of closed sets by Lemma~\ref{lem:dichotomy}, and a point in one stratum says nothing about the stratum being the whole base; the one proved lift is the semiregularity theorem of Bloch, Buchweitz and Flenner \cite{Blo72,BF03}, which is the content of Theorem~\ref{thm:criterion}. (ii) Manin's decomposition of the motive of a blow-up \cite{Man68}, $h(\tilde X)=h(X)\oplus\bigoplus_{i=1}^{c-1}h(Z)(-i)$, is realised by the algebraic correspondences $\pi^{*}$ and $j_{*}\circ p^{*}\circ(\text{powers of the relative hyperplane class})$, which are compatible with Hodge classes and with algebraic classes; so both the Hodge classes and the algebraic classes of $\tilde X$ decompose accordingly, and the quotients add. A birational map between smooth projective varieties factors through blow-ups and blow-downs along smooth centres only in special cases, but every birational map is dominated by a sequence of blow-ups, and (ii) shows that the defect of the top variety is at least that of the bottom one, so the defect cannot be removed on the way. (iii) A flat class has a limit in the limit mixed Hodge structure of a semistable fibre, and on a totally degenerate fibre the graded pieces are combinatorial and every class is algebraic on the components; the return to the generic fibre is the lift of (i), with the same unproved step. For a Mumford fourfold the Hodge locus of the exceptional class is the compact Shimura curve of \cite[Thm.~21.7]{BMB26}, which has no boundary, so no fibre at which the class is still of Hodge type degenerates.
\end{proof}

\begin{figure*}[t]
\centering
\includegraphics[width=0.92\textwidth]{figures/fig_surgery}
\caption{Where the defect goes under surgery. A blow-up along $Z$ carries the Hodge classes of $X$ to $\tilde X$ and adds those of $Z$, so the quotient $\Hdg/\Alg$ grows by the defect of the centre; a dominating map $Y\to X$ carries the defect of $X$ into $Y$; a Fourier--Mukai equivalence permutes degrees and preserves the total defect; a degeneration to $X_{0}$ loses the class only in the special fibre, from which no cycle lifts.}
\label{fig:surgery}
\end{figure*}

\subsection{Dominating maps, birational models and derived equivalences}\label{ssec:surgery}

Write $\delta^{p}(X)=\dim\Hdg^{p}(X)-\dim\Alg^{p}(X)$ for the defect of the conjecture in degree $2p$, $\delta_{\mathrm{ab}}^{p}(X)=\dim\Hdg^{p}(X)-\dim\Ab^{p}(X)$ for the defect of \Fiii{}, and $\delta(X)=\sum_{p}\delta^{p}(X)$. The conjecture for $X$ is $\delta(X)=0$. The following proposition records how the surgeries of birational geometry move these numbers; Figure~\ref{fig:surgery} draws it.

\begin{proposition}[Defects under surgery]\label{prop:surgery}
Let $X$, $Y$ be smooth complex projective varieties.
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.4em,itemsep=2pt]
\item If $f\colon Y\to X$ is surjective and generically finite, then $\delta^{p}(Y)\ge\delta^{p}(X)$ and $\delta^{p}_{\mathrm{ab}}(Y)\ge\delta^{p}_{\mathrm{ab}}(X)$ for every $p$. In particular no blow-up along a singular centre, weighted blow-up, resolution of a modification or finite cover of $X$ has a smaller defect than $X$.
\item If $X$ and $Y$ are birational, then $\delta^{2}(X)=\delta^{2}(Y)$ and $\delta^{d-1}(X)=\delta^{d-1}(Y)$, $d=\dim X$; for $3\le p\le d-2$ the difference $\delta^{p}(X)-\delta^{p}(Y)$ is a signed sum of defects $\delta^{p-j}(Z)$, $j\ge1$, of smooth projective varieties $Z$ of dimension less than $d$.
\item If $\Gamma\in\mathrm{CH}^{*}(X\times Y)_{\QQ}$ induces an isomorphism from $H^{*}(X,\QQ)$ onto $H^{*}(Y,\QQ)$ whose inverse is induced by a cycle, for instance the kernel of an equivalence between the derived categories of $X$ and $Y$, then $\delta(X)=\delta(Y)$ and $\delta_{\mathrm{ab}}(X)=\delta_{\mathrm{ab}}(Y)$.
\item Let $\mathcal{X}\to\Delta$ be a projective family, smooth over the punctured disc, with a flat Hodge class $\xi$ on the smooth fibres that is algebraic on a crepant resolution $\hat X$ of the special fibre $X_{0}$, or on the components of a semistable model of $X_{0}$. Then $\xi$ is algebraic on $X_{t}$ for $t\ne0$ if and only if the specialisation $H^{2p}(X_{0})\to H^{2p}(X_{t})$ sends an algebraic class to an algebraic class, which is the statement (V) for the family; the monodromy plays no role, since it fixes $\xi$.
\end{enumerate}
\end{proposition}

\begin{proof}
(i) $f_{*}f^{*}$ is multiplication by the degree, so $f^{*}$ is an injective morphism of Hodge structures and $f^{*}\Hdg^{p}(X)\subseteq\Hdg^{p}(Y)$. If $f^{*}\alpha$ is algebraic then so is $\deg(f)\,\alpha=f_{*}f^{*}\alpha$; hence $\Alg^{p}(Y)\cap f^{*}\Hdg^{p}(X)=f^{*}\Alg^{p}(X)$ and $f^{*}$ embeds $\Hdg^{p}(X)/\Alg^{p}(X)$ into $\Hdg^{p}(Y)/\Alg^{p}(Y)$. For $\Ab$ the same argument applies because $f_{*}$ of a class $\Gamma_{*}\gamma$ reached from an abelian variety is $(\Gamma_{f}\circ\Gamma)_{*}\gamma$, reached from the same one. The coarse space of a weighted blow-up has quotient singularities and $f_{*}f^{*}=\mathrm{id}$ with rational coefficients, so the argument is unchanged there.

(ii) By weak factorisation \cite{AKMW02} a birational map between smooth projective varieties is a composite of blow-ups and blow-downs along smooth centres with smooth projective intermediate varieties, and Proposition~\ref{prop:bypass}(ii) gives the change of $\delta^{p}$ at each step as $\pm\sum_{j=1}^{c-1}\delta^{p-j}(Z)$. For $p=2$ the terms are $\delta^{1}(Z)$ and $\delta^{0}(Z)$, which vanish by the Lefschetz $(1,1)$ theorem; for $p=d-1$ they are $\delta^{d-1-j}(Z)$ with $\dim Z\le d-2$, hence $\delta^{q}(Z)$ with $q\ge\dim Z-1$ after $j\ge1$, and these vanish by hard Lefschetz and Lefschetz $(1,1)$ on $Z$.

(iii) By the projection formula $\Gamma_{*}$ sends $\bigoplus_{p}\Hdg^{p}(X)$ into $\bigoplus_{p}\Hdg^{p}(Y)$, since $[\Gamma]$ is a sum of classes of type $(i,i)$ and cup product and push-forward are morphisms of Hodge structures up to Tate twists; it sends algebraic classes to algebraic classes because $[\Gamma]\cup\mathrm{pr}_{X}^{*}[Z]$ is the class of a cycle and the push-forward of a cycle is a cycle; and it sends $\Ab$ to $\Ab$ by composing correspondences. The inverse does the same in the other direction, so the two total quotients are isomorphic. The kernel of a Fourier--Mukai equivalence and the kernel of its inverse are cycles by Mukai's and Orlov's description of the inverse functor \cite{Orl97}, which is the equivalence with kernel the dual of the original one shifted by the dimension.

(iv) $\xi$ is monodromy invariant, so by the local invariant cycle theorem it is the specialisation of a class in $H^{2p}(X_{0})$, which pulls back to $\hat X$ or to the components of the semistable model; the hypothesis says that the pulled back class is algebraic there, and push-forward keeps it algebraic on $X_{0}$. Cycles specialise from the generic fibre to the special fibre \cite[Ch.~20]{Ful98} and not the other way, so the remaining step is exactly the one in the statement. If the limit cycle on $X_{0}$ lifts to the family, its restriction to $X_{t}$ has class $\xi_{t}$; conversely if $\xi_{t}$ is algebraic the specialisation of a representing cycle is a cycle on $X_{0}$ with class the limit of $\xi$, and the specialisation map sends that class to $\xi_{t}$. The vanishing cycles, on which the monodromy acts, are orthogonal to the invariant part, and $\xi$ is invariant, so no condition on them enters.
\end{proof}

Items (i) to (iii) say that every surgery available in birational and derived geometry carries the defect of $X$ into its output as long as the open class survives there, and loses the class only in outputs of lower dimension, the centres, where it has become a defect of those varieties. Both open statements of this paper sit in degree four, \Fii{} for the Weil classes and the exceptional classes, \Fiii{} for the classes of $H^{4}(S\times S)$ on a K3 surface, and in degree four the defect is a strict birational invariant by (ii) and a derived invariant by (iii); so the open classes are present, and not algebraic, on every birational model and every Fourier--Mukai partner of the varieties that carry them. Item (iv) is the reason that degenerations to products of elliptic curves, where every Hodge class is algebraic \cite{MZ99}, do not close \Fii{}: the class returns to the smooth fibre only through (V). Flows of K\"ahler metrics change no complex structure between their surgeries, and the surgeries themselves are the contractions and flips of the minimal model programme, covered by (i) and (ii); the surgery of Perelman has no counterpart for Hodge structures, since it cuts a manifold along spheres and a Hodge structure has no local pieces to cut.

The devices therefore relocate the two open statements without weakening them. The one true reduction in shape is Theorem~\ref{thm:propagation}: the whole of \Fii{} is equivalent to a lift from CM points along polarised abelian schemes, and the lift along each scheme is a question about one semiregular object at one point.

\subsection{Residues and mass-minimising currents}\label{ssec:analytic}

Two analytic descriptions of algebraic classes are exact, and we record what each reduces the conjecture to. The first is Griffiths' residue calculus on hypersurfaces \cite{Gri69}. For a smooth hypersurface $X=\{F=0\}\subset\PP^{n+1}$ of even dimension $n=2m$ and degree $d$, with Jacobian ring $R(F)$ and $\Omega=\sum_{i}(-1)^{i}x_{i}\,dx_{0}\wedge\dots\widehat{dx_{i}}\dots\wedge dx_{n+1}$, the residues of the forms $P\Omega/F^{q+1}$ identify $H^{n-q,q}_{\mathrm{prim}}(X)$ with $R(F)_{(q+1)d-n-2}$.

\begin{proposition}[Residues of complete intersections]\label{prop:residues}
Let $X=\{F=0\}$ be as above, and suppose that $F=\sum_{i=1}^{m+1}f_{i}g_{i}$ with $Z=\{f_{1}=\dots=f_{m+1}=0\}$ a complete intersection of codimension $m+1$ in $\PP^{n+1}$ whose ideal is generated by the $f_{i}$.
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.4em,itemsep=2pt]
\item Let $\theta$ be the hyperplane class and $J_{H}$ the Jacobian determinant of $(f_{1},g_{1},\dots,f_{m+1},g_{m+1})$, a polynomial of degree $(m+1)(d-2)$. Then
\[
[Z]=\frac{\deg Z}{d}\,\theta^{m}-\frac{m!}{d}\,\operatorname{res}\Bigl(\frac{J_{H}\,\Omega}{F^{m+1}}\Bigr)^{m,m}
\]
\cite[Thm.~1.1]{VL22}.
\item Let $J(F)\subseteq R(F)_{(m+1)(d-2)}$ be the span of the classes of the determinants $J_{H}$ over all such decompositions of $F$. If the rational span of the residues of $J(F)$ and $\theta^{m}$ contains $\Hdg^{m}(X)$, the Hodge conjecture holds for $X$.
\end{enumerate}
\end{proposition}

\begin{proof}
(i) is the cited theorem; the degree of the determinant is $\sum_{i}(\deg f_{i}+\deg g_{i})-(n+2)=(m+1)d-2m-2$. (ii) By (i) every residue of an element of $J(F)$ is a rational combination of $\theta^{m}$ and of classes of complete intersections in $X$, so it is algebraic. The Hodge conjecture for a hypersurface holds outside the middle degree by the Lefschetz hyperplane theorem, and in the middle degree it is the statement that $\Hdg^{m}(X)$ consists of algebraic classes.
\end{proof}

The proposition replaces a cycle by a polynomial, and the open step becomes the statement that the polynomials of $J(F)$ have enough rational residues. Deciding whether a given element of $R(F)$ has a rational residue requires its periods, so the residue calculus moves the conjecture from cycles to transcendental numbers without closing it. On a Fermat hypersurface $\sum_{i}x_{i}^{d}=0$ the linear subspaces cut out by $x_{2i}=\zeta_{i}x_{2i+1}$, $\zeta_{i}^{d}=-1$, are of this form, since $x_{2i}^{d}+x_{2i+1}^{d}$ factors into linear forms; whether their residues and those of the other decompositions exhaust $\Hdg^{m}(X)$ is again a question about periods.

The second description uses currents and is the analogue for cycles of the flows of geometric analysis. Let $X$ be a smooth complex projective variety of dimension $N$, $h$ an ample class with K\"ahler form $\omega$, and $0<p<N$. For $\beta\in H_{2(N-p)}(X,\ZZ)$ let $M(\beta)$ be the infimum of the masses of the integral currents in $\beta$, put $\mathrm{vol}(\beta)=\int_{\beta}\omega^{N-p}/(N-p)!$, and call $g(\beta)=M(\beta)-\mathrm{vol}(\beta)$ the gap of $\beta$. We identify $H^{2p}(X,\ZZ)$ with $H_{2(N-p)}(X,\ZZ)$ by Poincar\'e duality.

\begin{proposition}[Mass-minimising currents]\label{prop:mass}
\leavevmode
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.4em,itemsep=2pt]
\item Every class $\beta$ contains an integral current of mass $M(\beta)$, and $g(\beta)\ge0$, with equality if and only if $\beta$ is the class of an effective algebraic cycle with integer coefficients.
\item Let $\alpha\in H^{2p}(X,\ZZ)$. The image of $\alpha$ in $H^{2p}(X,\QQ)$ is algebraic if and only if $g(e\alpha+kh^{p})=0$ for some integers $e\ge1$ and $k\ge0$.
\item There are $X$ and $\alpha$ with $g(\alpha+kh^{p})>0$ for every $k\ge0$ although $g(e\alpha)=0$ for some $e\ge1$: for a very general hypersurface $X\subset\PP^{4}$ of degree $D$ divisible by $\ell^{3}$ for an integer $\ell>1$ prime to $6$, $p=2$, and the generator $\alpha$ of $H^{4}(X,\ZZ)$.
\end{enumerate}
\end{proposition}

\begin{proof}
(i) By the compactness theorem of Federer and Fleming \cite{FF60} the integral currents in $\beta$ of mass at most $M(\beta)+1$ form a compact set for the flat topology, and mass is lower semicontinuous, so the infimum is attained. For an integral current $T$ in $\beta$, Wirtinger's inequality, that $\omega^{N-p}/(N-p)!$ has comass one with equality exactly on the complex $(N-p)$-planes with their canonical orientation, gives $M(T)\ge T\bigl(\omega^{N-p}/(N-p)!\bigr)=\mathrm{vol}(\beta)$, the last value depending only on $\beta$ because $\omega$ is closed \cite{HL82}. If equality holds, $T$ is calibrated by $\omega^{N-p}/(N-p)!$, hence a closed positive current of bidimension $(N-p,N-p)$ \cite{HL82}, and by King's theorem \cite{Kin71} a closed positive integral current of that bidimension is $\sum_{i}n_{i}[V_{i}]$ with $n_{i}$ positive integers and $V_{i}$ irreducible analytic subvarieties, which are algebraic by Chow's theorem. Conversely, for an effective cycle $Z=\sum n_{i}V_{i}$ the tangent planes of the smooth locus of each $V_{i}$ are complex, so the current $[Z]$ has mass $\mathrm{vol}([Z])$ and $g([Z])=0$.

(ii) If $g(e\alpha+kh^{p})=0$, then $e\alpha+kh^{p}$ is the class of an effective cycle by (i), and $\alpha=\bigl([Z]-kh^{p}\bigr)/e$ in $H^{2p}(X,\QQ)$. Conversely let $\alpha$ be rationally algebraic, so that $e_{0}\alpha-[Z^{+}]+[Z^{-}]$ is a torsion class for an integer $e_{0}\ge1$ and effective cycles $Z^{\pm}$; multiplying by its order, $e\alpha=[Z^{+}]-[Z^{-}]$ in $H^{2p}(X,\ZZ)$. Write $Z^{-}=\sum_{i}c_{i}V_{i}$. For $r$ large the ideal sheaf of each $V_{i}$ twisted by $\mathcal{O}(rh)$ is globally generated, so $p$ general members of $|rh|$ containing $V_{i}$ meet in a complete intersection $Y_{i}$ of codimension $p$ containing $V_{i}$, and $[Y_{i}]=r^{p}h^{p}$; the cycle $Z'_{i}$ of $Y_{i}$ minus $V_{i}$ is effective. Then $e\alpha+\sum_{i}c_{i}r^{p}h^{p}=[Z^{+}+\sum_{i}c_{i}Z'_{i}]$, effective, and (i) gives $g=0$ with $k=\sum_{i}c_{i}r^{p}$.

(iii) By the Lefschetz hyperplane theorem $H^{4}(X,\ZZ)\cong\ZZ\alpha$ with $\int_{X}\alpha\cdot h=1$, and $h^{2}=D\alpha$, the class of a complete intersection, so $g(D\alpha)=0$. By Koll\'ar's theorem \cite{Kol92}, in the form of \cite[Thm.~2]{SV05}, every curve on $X$ has degree divisible by $\ell$. The class $\alpha+kh^{2}$ has degree $1+kD$, prime to $\ell$, so it is not the class of an effective cycle, and $g(\alpha+kh^{2})>0$ by (i).
\end{proof}

The analytic part of the variational approach is therefore complete. A minimiser exists in every class without any flow, so a mass-decreasing flow of a cycle, with whatever surgeries along its singularities, can at best reach it; what such a flow cannot provide is the equality case, and the equality case is the whole conjecture: by (ii) the Hodge conjecture for a class $\alpha$ is the vanishing of the gap of $e\alpha+kh^{p}$ for one pair $(e,k)$. Part (iii) shows that the multiplier $e$ cannot be omitted, since integral minimisers of a class that is rationally algebraic need not be calibrated for any $k$, and so no argument by integral currents alone, including one modelled on Ricci flow with surgery, proves the conjecture without passing to rational coefficients. For the Weil classes of $\cD_{n,n}$ and the exceptional classes of a Mumford fourfold the gap of $e\alpha+kh^{p}$ is not known to vanish for any $(e,k)$.

Nothing in the paper assumes an open statement, and no section depends on a computation that is not also proved by hand in the text. The machine checks of Appendix~\ref{app:checks} recompute the finite quantities; they do not carry any proof.