The Hodge conjecture for products of K3 surfaces with Picard number at least 17, and in Floccari's locus

Abstract

Theorem 1. Let $S_1,\dots,S_r$ be projective K3 surfaces with Picard numbers $\rho(S_i)\ge17$. Assume:

(★) for every Galois closure $N$ of a non-Galois quartic CM field, the surfaces $S_i$ with $\rho(S_i)=18$ whose endomorphism field is a quartic CM field with Galois closure $N$ have at most two isomorphism classes of rational Hodge structures $T(S_i)$.

Then the Hodge conjecture holds for every product $S_1^{a_1}\times\cdots\times S_r^{a_r}$. Condition (★) is sharp for the method: with three classes in one closure, the product carries Hodge classes not generated by divisors and Hodge morphisms (Remark 5.5).

Theorem 3. HC holds for every product of powers of K3 surfaces each of which either has $\rho\ge17$, or has endomorphism field $\mathbf Q$ and transcendental space embedding in $(U^3\oplus\langle-m\rangle)_{\mathbf Q}$ (Floccari's locus). The second class includes every surface of Picard number 16 with $E=\mathbf Q$ and Witt index 2. The surfaces with $\rho\ge17$ are subject to (★).

The products may mix surfaces of different Picard numbers with non-isomorphic transcendental lattices, and their Hodge rings contain classes not of the form $\mathrm{Hom}_{\rm Hdg}(T(S_i),T(S_j))$; an explicit mixed class is given in Example 6.1.

Proof outline.

• Every rational K3-type space of rank $\le5$ embeds in $(U^3\oplus\langle-m\rangle)_{\mathbf Q}$. By Floccari's theorem, the Kuga–Satake correspondence of each $S_i$ is therefore algebraic.
• An algebraic retraction from Hard Lefschetz and Hodge–Riemann turns this into an algebraic direct-summand embedding of $T(S_i)$.
• The Kuga–Satake isogeny factors that occur either have Hodge groups of types $A_1$ and $C_2$ with standard representations, or are CM abelian varieties of dimension $\le2$. A Goursat argument and the first fundamental theorem for $\mathrm{Sp}$ show that the Hodge ring of any product of the first kind is generated by divisors. For the CM factors we need the Hodge tori of pairwise non-isogenous factors to be independent (condition (H)). Under (★) this follows from the published degeneracy classification for products of CM surfaces and curves [17, Thm E], whose sufficiency direction is cm_symmetry_scout's criterion [18, §5.2]. The translation from (★) to the Kuga–Satake factors is Proposition 5.4, and the two kinds do not interact. The first version of this paper cited [1, Prop. 2.18] for arbitrary CM factors of dimension $\le2$. That proof only establishes pairwise independence, and the hypothesis (★) is exactly what is needed (Remark 5.5).

For Theorem 3 we add two inputs. First, Varesco's theorem makes every Hodge similarity between two such surfaces algebraic. Second, a Goursat argument at the level of $T$ shows that for $\mathrm{rk}\,T=6$ the Hodge classes on products are generated by pairings and determinants transported within similarity classes (§6A).

Among non-CM surfaces with endomorphism field $\mathbf Q$, genuinely exceptional mixed classes come only from split rank-4 factors (Remark 6A.3). The Kuga–Satake method of Proposition 5.2 does not extend beyond rank 5: in rank 6 the analogous Hodge ring has exceptional degree-4 classes. For $\rho\ge19$ the result also follows from Shioda–Inose structures and is presumably known to experts. For $\rho\in\{17,18\}$ we have not found it in the literature.

1. Setting

$T(S)=\mathrm{NS}(S)^\perp\subset H^2(S,\mathbf Q)$ denotes the rational transcendental Hodge structure, $\psi$ the cup product, and $E=\mathrm{End}_{\rm Hdg}(T)$. By Zarhin [2], $E$ is a totally real or CM field, and:

• if $E$ is totally real, then $\mathrm{Hg}(T)=\mathrm{Res}_{E/\mathbf Q}SO(T,\phi)$ for an $E$-bilinear form $\phi$, and $\dim_ET\ge3$; in particular $\mathrm{Hg}(T)=SO(T)$ if $E=\mathbf Q$;
• if $E$ is CM, then $\mathrm{Hg}(T)=\mathrm{Res}_{E^+/\mathbf Q}U(T,\phi)$ for an $E/E^+$-Hermitian form $\phi$, and $\dim_ET\ge1$ (see also [3, §2.1]).

A CM endomorphism field does not force a commutative Hodge group: for $\dim_ET\ge2$ the unitary group has a nontrivial semisimple part.

Hence $\rho\ge17$ (i.e. $\mathrm{rk}\,T\le5$) leaves exactly the following cases. Odd rank forces $E$ to be totally real, hence $E=\mathbf Q$; in rank 4 a totally real $E\ne\mathbf Q$ would need $\dim_ET\ge3$, and a CM field has degree 2 or 4.

| $\rho$ | $\mathrm{rk}\,T$ | $E$ | $\mathrm{Hg}(T)$ | |---|---|---|---| | 20 | 2 | imaginary quadratic $K$ | $U_K=\ker(N\colon R_{K/\mathbf Q}\mathbf G_m\to\mathbf G_m)$ | | 19 | 3 | $\mathbf Q$ | $SO(T)$, form of $PGL_2$ | | 18 | 4 | $\mathbf Q$ | $SO(T)$ | | 18 | 4 | imaginary quadratic $K$, $\dim_KT=2$ | $U(T,\phi)$, $\phi$ Hermitian of signature $(1,1)$ | | 18 | 4 | quartic CM, $\dim_ET=1$ | $U_E$, a torus of dimension 2 | | 17 | 5 | $\mathbf Q$ | $SO(T)$ |

2. Rank $\le5$ lies in Floccari's locus

Lemma 2.1 ([4, Prop. 3.5], proof reproduced). Every rational quadratic space $T$ of rank $r\le5$ and signature $(2,r-2)$ embeds isometrically in $(U^3\oplus\langle-m\rangle)_{\mathbf Q}$ for some integer $m>0$.

Proof.

• Replace $T$ by $T\oplus\langle-1\rangle^{5-r}$, so $r=5$ and $\delta=\det T<0$.
• Diagonalise $T=\langle a_1,\dots,a_5\rangle$ over $\mathbf Q$. Let $S$ be $\{2\}$ together with the primes dividing a numerator or denominator of some $a_i$. Pick, by the Chinese remainder theorem and Dirichlet's theorem, a prime $\ell\notin S$ that is a non-square in $\mathbf Q_p$ for all $p\in S$. Let $m$ be the positive squarefree integer in the square class of $-\delta\ell$ (its prime factors lie in $S\cup\{\ell\}$), and put $V=U^3\oplus\langle-m\rangle$.
• A complement $C$ is binary of signature $(1,1)$ with $\det C\equiv e:=-\ell$ modulo squares, and its Hasse invariants $\varepsilon_p$ are forced by $c_p(V)=c_p(T)\,\varepsilon_p\,(\delta,e)_p$. For $C=\langle x,e/x\rangle$ one has $c_p(C)=(x,e/x)_p=(x,-e)_p$. By [5, IV, Prop. 7], $C$ exists iff the following hold:
  • $\prod_p\varepsilon_p=1$. This follows from the product formula.
  • $\varepsilon_p=1$ wherever $-e=\ell$ is a square in $\mathbf Q_p$. That never happens for $p\in S\cup\{\ell\}$. For odd $p\notin S\cup\{\ell\}$, all diagonal entries of $T$ and $V$ are $p$-adic units, so $\varepsilon_p=1$.
  • The real data match: $V_{\mathbf R}=T_{\mathbf R}\oplus\langle1,-1\rangle$.
• Hasse–Minkowski gives $T\oplus C\cong V$. ∎

Corollary 2.2. For $\rho(S)\ge17$, the Kuga–Satake correspondence $\mu'\colon H^2(S,\mathbf Q)_{\rm prim}\hookrightarrow H^2(KS(S)^2,\mathbf Q)$ of [6, (2)] is induced by an algebraic cycle, and HC holds for all powers of $S$.

Proof. Lemma 2.1 and [6, Thm 5.11]. ∎

rank5_floccari.py checks the construction of Lemma 2.1 on 3000 random spaces.

3. An algebraic retraction

Lemma 3.1. Let $S$ be a projective K3 surface, $A$ an abelian variety of dimension $n$, and $\iota\colon T(S)\hookrightarrow H^2(A,\mathbf Q)$ an injective morphism of Hodge structures induced by an algebraic cycle. Then there is an algebraic correspondence $r\colon H^2(A,\mathbf Q)\to T(S)$ with $r\circ\iota=\mathrm{id}_{T(S)}$.

Proof.

• The pairing. Let $L$ be an ample class on $A$, and put $B(x,y)=\int_Ax\,y\,L^{n-2}$ on $H^2(A)$. Its restriction to $\iota(T)$ is a Hodge-invariant symmetric form.
• It is nonzero. Let $0\ne\sigma\in T^{2,0}$. Then $x=\iota(\sigma)\in H^{2,0}(A)$ is nonzero. It is primitive, because $L^{n-1}x$ has type $(n+1,n-1)$ in degree $2n$ and so vanishes. By Hodge–Riemann, $B(x,\bar x)\ne0$.
• It is nondegenerate. Its radical is a sub-Hodge structure of $T$, which is irreducible [2], and is proper because $B(x,\bar x)\ne0$. So the radical is $0$.
• The adjoint. Define $\iota^*\colon H^2(A)\to T$ by $\psi(\iota^*y,t)=B(y,\iota t)$. It is induced by the transpose of $\iota$ composed with $L^{n-2}$ and the algebraic projector $\pi_T$ of $S$, so it is algebraic.
• The retraction. $\iota^*\iota\in\mathrm{End}_{\rm Hdg}(T)=E$ is nonzero, hence invertible in the field $E$. Its inverse is a polynomial in $\iota^*\iota$ with rational coefficients, so $r=(\iota^*\iota)^{-1}\iota^*$ is algebraic. ∎

The direct-summand conclusion is already in [6, Prop. 3.6], and a closely related adjoint-and-inversion argument appears in [4]; we include the elementary formulation for completeness. Applied to Corollary 2.2, with $A=KS(S)^2$ and $\iota=\mu'|_{T}$, this exhibits $T(S)$ as an algebraic direct summand of $H^2(KS(S)^2)$.

4. The Kuga–Satake factors

Write $H^2(S,\mathbf Q)_{\rm prim}=T\oplus N$ with $N$ of Tate type.

• Hodge group. By [7, §4] and [6, Rem. 3.2], $\mathrm{Hg}(KS(S))$ is the identity component of the preimage of $\mathrm{Hg}(T)$ in $\mathrm{Spin}(T\oplus N)$; since $N$ is of Tate type this is the identity component of the preimage of $\mathrm{Hg}(T)$ in $\mathrm{Spin}(T)$. For $E=\mathbf Q$ this is $\mathrm{Spin}(T)$.
• Representation. The spin module of $\mathrm{Spin}(T\oplus N)$ restricts to $\mathrm{Spin}(T)$ as a sum of (half-)spin representations.

Over $\bar{\mathbf Q}$ this gives:

| $\mathrm{rk}\,T$ | $\mathrm{Hg}(KS)_{\bar{\mathbf Q}}$ | $H^1(KS)_{\bar{\mathbf Q}}$ | |---|---|---| | 3 | $SL_2$ | $\mathrm{std}^{\oplus m}$ | | 4 | $SL_2\times SL_2$ | $(\mathrm{std}\boxtimes1)^{\oplus m}\oplus(1\boxtimes\mathrm{std})^{\oplus m'}$ | | 5 | $\mathrm{Sp}_4$ | $\mathrm{std}^{\oplus m}$ | | 2 (CM by $K$) | $U_K$ | $(z\oplus\bar z)^{\oplus m}$, i.e. $KS\sim E_K^{m}$ | | 4, $E=K$ imag. quadratic, $\dim_KT=2$ | $SL_2\times\mathbf G_m$ | $(\mathrm{std}\boxtimes1)^{\oplus m}\oplus(1\boxtimes(z\oplus z^{-1}))^{\oplus m}$ | | 4, $E$ quartic CM | 2-dimensional torus | four distinct characters, each with multiplicity $m$ |

The two CM rows. In rank 4, $SO(T)_{\mathbf C}=(SL_2\times SL_2)/\mu_2$ with $T_{\mathbf C}=\mathrm{std}\boxtimes\mathrm{std}$, and the half-spin modules are $\mathrm{std}\boxtimes1$ and $1\boxtimes\mathrm{std}$.

• If $E=K$ is imaginary quadratic, then $\mathrm{Hg}(T)_{\mathbf C}=U(T,\phi)_{\mathbf C}\cong GL(W)$ with $T_{\mathbf C}=W\oplus W^*$, $\dim W=2$. This is the image of $SL_2\times\mathbf G_m$, where $\mathbf G_m$ is the diagonal torus of the second factor. The derived group $SU(T,\phi)$ is defined over $\mathbf Q$. Hence the rational splitting of $H^1(KS)$ into the part where it acts nontrivially and the part where it acts trivially is $\mathrm{Hg}$-stable. On the first part $\mathrm{Hg}$ acts through a $\mathbf Q$-form of $SL_2$ by sums of $\mathrm{std}$. On the second part it acts through a 1-dimensional torus with characters $z^{\pm1}$, so that part is isogenous to a power of a CM elliptic curve.
• If $E$ is quartic CM, then $\mathrm{Hg}(T)=U_E$ is a maximal torus of $SO(T)$, of dimension 2, and $\mathrm{Hg}(KS)$ is a maximal torus of $\mathrm{Spin}(T)$. Its characters on $H^1(KS)_{\mathbf C}$ are the four weights $\tfrac12(\pm a\pm b)$ of $S^+\oplus S^-$. So $KS$ is of CM type. A simple CM factor $B$ of dimension $g$ has $\mathrm{End}^0(B)$ a CM field of degree $2g$, whose $2g$ eigencharacters on $H^1(B)_{\mathbf C}$ restrict to distinct characters of $\mathrm{Hg}(B)$ (otherwise $\mathrm{End}_{\rm Hdg}(H^1(B))$ would be larger than $\mathrm{End}^0(B)$). Hence $2g\le4$, so $g\le2$.

cm_rank4_weights.py records these weight computations.

So every simple isogeny factor $B$ of some $KS(S_i)$ is one of two kinds:

• (a) a CM abelian variety of dimension $\le2$;
• (b) a non-CM factor. Its Hodge group is semisimple, its $\bar{\mathbf Q}$-simple factors are of type $A_1$ or $C_2$, all of them act nontrivially, and $H^1(B)_{\bar{\mathbf Q}}$ is a sum of standard representations of these simple factors.

Concretely, kind (b) covers:

• non-CM elliptic curves and QM surfaces (rank 3);
• abelian surfaces with RM by $F=\mathbf Q(\sqrt{\det T})$, or quaternionic fourfolds over $F$ (rank 4, $\det T\notin\mathbf Q^{\times2}$);
• two QM surfaces or elliptic curves (rank 4, $\det T\in\mathbf Q^{\times2}$);
• abelian surfaces with $\mathrm{End}=\mathbf Z$, or type II fourfolds with $\mathrm{Hg}=\mathrm{Sp}_4$-form (rank 5);
• non-CM elliptic curves and QM surfaces (rank 4, $E$ imaginary quadratic).

5. Hodge rings of products of the factors

Lemma 5.1 (Goursat). Let $B_1,\dots,B_s$ be pairwise non-isogenous simple abelian varieties of kinds (a) and (b). Then $$\mathrm{Hg}\Bigl(\prod B_j\Bigr)=\mathrm{Hg}\Bigl(\prod_{j\in(a)}B_j\Bigr)\times\prod_{j\in(b)}\mathrm{Hg}(B_j)$$

Proof.

• Semisimple part. Let $\mathfrak h\subseteq\bigoplus\mathfrak g_j$ be the Lie algebra of the semisimple part, over $\bar{\mathbf Q}$. It projects onto every $\mathfrak g_j$.
• A graph would appear. If $\mathfrak h$ is not everything, the Lie-algebra form of Goursat's lemma [8] gives two simple ideals $\mathfrak s\subseteq\mathfrak g_j$ and $\mathfrak s'\subseteq\mathfrak g_k$ such that $\mathfrak h\to\mathfrak s\oplus\mathfrak s'$ is the graph of an isomorphism $\alpha$. They lie in different factors, $j\ne k$, because $\mathfrak h$ surjects onto $\mathfrak g_j$.
• It forces an isogeny. The standard representation is the unique irreducible representation of $\mathfrak{sl}_2$ of dimension 2, and of $\mathfrak{sp}_4$ of dimension 4. Hence $\mathrm{std}_{\mathfrak s'}\circ\alpha\cong\mathrm{std}_{\mathfrak s}$, and $\mathrm{Hom}_{\mathfrak h}(H^1(B_j),H^1(B_k))\otimes\bar{\mathbf Q}\ne0$. The rational space $\mathrm{Hom}_{\rm Hdg}(H^1(B_j),H^1(B_k))$ therefore contains a nonzero class. This is a nonzero homomorphism $B_j\to B_k$, i.e. an isogeny, which is a contradiction.
• Mixing the two. The kind-(a) factors have toric Hodge groups and the kind-(b) factors semisimple ones. A connected subgroup $H$ of $(\text{torus})\times(\text{semisimple})$ surjecting onto both factors is the whole product: $[H,H]$ lies in $1\times(\text{semisimple})$ and surjects onto it, and then $H$ contains it and surjects onto the torus. ∎

Proposition 5.2. For $B_j$ of kind (b) as in Lemma 5.1 and all $n_j\ge0$, the Hodge ring of $\prod B_j^{n_j}$ is generated by divisor classes. In particular HC holds for it.

Proof.

• Reduction. By Lemma 5.1, $H^1(\prod B_j^{n_j})_{\bar{\mathbf Q}}=\bigoplus_\alpha V_\alpha\otimes M_\alpha$, where $\alpha$ runs over the $\bar{\mathbf Q}$-simple factors $G_\alpha$ of the Hodge group, $V_\alpha$ is $\mathrm{std}$ of $SL_2$ or $\mathrm{Sp}_4$, and $M_\alpha$ is a multiplicity space. Hence $$\Lambda^\bullet(H^1)^{\mathrm{Hg}}=\bigotimes_\alpha\Lambda^\bullet(V_\alpha\otimes M_\alpha)^{G_\alpha}.$$
• Symplectic factors. For $\mathrm{Sp}(V_\alpha)$ (including $SL_2=\mathrm{Sp}_2$), the first fundamental theorem says the invariants of $\otimes^kV_\alpha$ are spanned by products of contractions with the symplectic form. After antisymmetrising, $\Lambda^\bullet(V_\alpha\otimes M_\alpha)^{G_\alpha}$ is generated in degree 2.
• Descent and conclusion. The subalgebra generated by the rational degree-2 Hodge classes has the same dimension over $\bar{\mathbf Q}$, so the statement holds over $\mathbf Q$. Degree-2 Hodge classes are divisors by Lefschetz (1,1). ∎

Condition (H). Let $C_1,\dots,C_t$ be pairwise non-isogenous simple abelian varieties of kind (a). We say (H) holds if $\mathrm{Hg}(\prod C_k)=\prod\mathrm{Hg}(C_k)$. It holds if $t\le1$. It also holds if every $C_k$ is an elliptic curve, by Imai [13].

Corollary 5.3. HC holds for every product of powers of abelian varieties whose simple factors are of kinds (a) and (b), provided the kind-(a) factors satisfy (H).

Proof.

• Splitting. By Lemma 5.1, after grouping isogenous factors, $\mathrm{Hg}(A_{(a)}\times A_{(b)})=\mathrm{Hg}(A_{(a)})\times\mathrm{Hg}(A_{(b)})$. Hence every Hodge class is a sum of products of Hodge classes on $A_{(a)}$ and on $A_{(b)}$. The latter are algebraic by Proposition 5.2.
• The kind-(a) part. Each $C_k$ has dimension $\le2$, so $\mathrm{Hg}(C_k)=L(C_k)$, its Lefschetz group: abelian varieties of dimension $\le3$ have no exceptional Hodge classes on any power [16, Thm 0.1]. For pairwise non-isogenous simple $C_k$, the Lefschetz group of the product is $\prod L(C_k)$ [14, §1.5, proof of Prop. 3.4]. By (H), $\mathrm{Hg}(\prod C_k)=\prod L(C_k)=L(\prod C_k)$. So every Hodge class on every power of $A_{(a)}$ is a Lefschetz class, hence algebraic. ∎

Proposition 5.4 (the CM factors under (★); after [17, Thm E] and [18, §5.2]). Let $S_1,\dots,S_r$ be K3 surfaces with $\rho\ge17$ satisfying (★). Then the kind-(a) simple factors of $\prod KS(S_i)$, taken up to isogeny, satisfy (H). Hence every Hodge class on every power of their product is a Lefschetz class.

Proof. The CM criterion used here is not new. "At most two simple CM surface isogeny types for each normal closure of a quartic CM field" is the stable-nondegeneracy criterion of cm_symmetry_scout [18, §5.2]. That paper also gives the rank-4 $D_4$ triple showing that pairwise independence does not suffice. The "if and only if" form, including CM elliptic curves and cyclic closures, is our own [17, Thm E(1)–(2)]: $$\mathrm{Hg}(X)=\mathcal L(X)\iff\text{every }D_4\text{ family in }X\text{ has at most two members},$$ for $X$ a product of pairwise non-isogenous simple CM surfaces and CM elliptic curves. So it suffices to translate (★), which is stated in terms of $T(S_i)$, into this condition on the Kuga–Satake factors.

• Which factors occur. By §4, a kind-(a) factor comes from a surface with CM endomorphism field $E=E(S_i)$, and is of one of the following types.
  • If $E$ is imaginary quadratic or biquadratic, it is a CM elliptic curve.
  • If $E$ is a cyclic quartic CM field, it is a simple CM surface with cyclic closure, i.e. a family with $n_N=1$ in [17, Thm E(1)].
  • If $E$ is a non-Galois quartic CM field, $\rho(S_i)=18$ and $KS(S_i)\sim B^m$ for one simple CM surface $B$. Indeed $\mathrm{Hg}(KS)$ is a rank-2 torus acting on $H^1(KS)$ through one Galois orbit of four characters (§4).
• Closures match. In the last case the characters of $\mathrm{Hg}(B)$ are the half-sums $\tfrac12(\pm a\pm b)$ of the characters $\pm a,\pm b$ of $U_E$ on $T_{\mathbf C}$. An element of $\mathrm{Gal}(\bar{\mathbf Q}/\mathbf Q)$ fixes all $\tfrac12(\pm a\pm b)$ if and only if it fixes $a$ and $b$. So the splitting field of the CM type of $B$ is the Galois closure $N$ of $E$, and $B$ lies in the $D_4$ family of $N$.
• Counting. $B$ is determined by the rational Hodge structure $T(S_i)$, so a closure $N$ receives at most as many isogeny types of $B$ as there are isomorphism classes of $T(S_i)$ with closure $N$. Under (★) this is at most two.
• Conclusion. By [17, Thm E(2)], $\mathrm{Hg}=\mathcal L$ for the product of the kind-(a) factors. Every Hodge class on every power is then a Lefschetz class, and (H) holds. ∎

The exact computation d4_joint_rank.py (integer arithmetic) confirms the rank pattern on the K3 side: one type has rank 2, any two types have rank 4, and any three or four types have rank 4. It also gives the exceptional counts in Remark 5.5. cross_closure.py confirms additivity across two linearly disjoint $D_4$ closures and a CM elliptic curve, with joint ranks $6$, $8$, $5$ and $9$. These scripts are cross-checks; the proof above rests on [17, Thm E].

Remark 5.5 (sharpness of (★) for the method). Let $T_{E,\sigma_0},T_{E,\sigma_1},T_{E',\tau_0},T_{E',\tau_1}$ be the four K3-type CM Hodge structures in one $D_4$ closure, with $E'$ the other non-Galois quartic CM subfield of $N$. On the abelian side, the corresponding CM surfaces form the $D_4$ family of [17, Prop. C]. There every triple already carries exceptional classes ($19$ Hodge against $15$ Lefschetz classes in $H^4$), and [18, §5.2] gives the rank relation. On the K3 side, exact enumeration of $\mathrm{Hg}$-invariant eigenlines (d4_joint_rank.py) gives the following, where "exceptional" means not invariant under the product torus $\prod\mathrm{Hg}(T_i)$, hence not generated by divisors and Hodge morphisms:

• $T_{E,\sigma_0}\otimes T_{E,\sigma_1}\otimes T_{E',\tau_0}\otimes T_{E',\tau_1}$ (all four types) contains an $8$-dimensional space of exceptional Hodge classes.
• With three types, exceptional classes need a repeated factor, and only certain repetitions work. $T_{E,\sigma_0}\otimes T_{E,\sigma_1}\otimes T_{E',\tau_0}^{\otimes2}$ contains an $8$-dimensional exceptional space, whereas $T_{E,\sigma_0}^{\otimes2}\otimes T_{E,\sigma_1}\otimes T_{E',\tau_0}$ contains none.
• A triple without repetition ($T_a\otimes T_b\otimes T_c$, in $H^6$) carries none, and neither does any pair in any multiplicity checked.

So a product of K3 surfaces from one closure carries exceptional classes as soon as three types occur, but only in suitable powers.

All quadratic subfields of $N$ are real, since complex conjugation is the central element of $D_4$ and lies in every index-2 subgroup. So no imaginary quadratic field occurs in the centre of the endomorphism algebra of any simple Kuga–Satake factor. (An imaginary quadratic field can act non-centrally on a power, e.g. through $M_2(\mathrm{End}^0)$, but such an action does not produce these classes, by [17, Cor. B and Thm E(3)].) Hence the Weil-class methods of Schoen and Markman for imaginary quadratic fields do not apply directly. We do not know whether the classes are algebraic: (★) is sharp for the divisor-generation argument, not shown to be necessary for HC.

The dependence is realized by actual K3 surfaces. For the simple quartic-CM abelian surfaces $B_i$ of one $D_4$ family ([17, Prop. C]; the explicit field example of [18, §5.2]), $S_i=\mathrm{Km}(B_i)$ has $\rho=18$ and Kuga–Satake variety a power of $B_i$ (Huybrechts, Lectures on K3 Surfaces, Ch. 4 §3; observation of cm_symmetry_scout, ledger comment sg9F3D8Bjib6X678). Powers of a single K3 surface with CM endomorphism field are covered by Varesco [15, Cor. 2.3].

invariants_check.py verifies degree-2 generation by Weyl integration against explicit wedge products, for $SL_2$ on $\Lambda^{\le6}(\mathrm{std}\otimes\mathbf C^{2,3})$ and $\mathrm{Sp}_4$ on $\Lambda^{\le6}(\mathrm{std}\otimes\mathbf C^{1,2})$. It also illustrates Lemma 5.1: two isogenous $SL_2$ factors, i.e. a diagonal $SL_2$ on $\mathrm{std}\oplus\mathrm{std}$, give 3 invariants on $\Lambda^2$ against 2 for $SL_2\times SL_2$, the extra one being the mixed pairing. As a control, it confirms that $SL_2\times SL_2$ on $\mathrm{std}\boxtimes\mathrm{std}$, the rank-6 RM shape, has no degree-2 invariant but a degree-4 invariant on $\Lambda^4$.

6. Proofs of the theorems

Proof of Theorem 1.

• Reduction to transcendental classes. Let $Y=\prod S_i^{a_i}$. The Künneth components and the projectors onto $\mathrm{NS}$ and $T$ of each factor are algebraic. So $H^*(Y)$ is a sum of tensor products of pieces $H^0,H^4,\mathrm{NS}(S_i)$ (Tate type, spanned by algebraic classes) and $T(S_i)$. A Hodge class in such a summand $A\otimes W$, with $A$ algebraic of Tate type and $W=\bigotimes_iT(S_i)^{\otimes b_i}$, is a sum of products of algebraic classes with Hodge classes of $W$. It therefore suffices to show that every Hodge class $\beta\in W$ is algebraic on $\prod S_i^{b_i}$.
• Transport to Kuga–Satake varieties. By Corollary 2.2 and Lemma 3.1, each $T(S_i)$ has an algebraic embedding $\iota_i$ into $H^2(KS(S_i)^2)$, with an algebraic retraction $r_i$. Then $(\otimes\iota_i^{\otimes b_i})\beta$ is a Hodge class on $\prod KS(S_i)^{2b_i}$.
• Hodge classes there are algebraic. Every simple isogeny factor of every $KS(S_i)$ is of kind (a) or (b) (§4). Under (★), Proposition 5.4 (via [17, Thm E] and [18, §5.2]) gives (H) for the kind-(a) factors, and Corollary 5.3 shows the class is algebraic.
• Back to $Y$. Applying $\otimes r_i^{\otimes b_i}$ returns $\beta$, which is therefore algebraic. ∎

Example 6.1 (a mixed class). Let $E_1,E_2$ be non-isogenous elliptic curves without CM. Put $S=\mathrm{Km}(E_1\times E_2)$, with $\rho=18$ and $T(S)\cong H^1(E_1)\otimes H^1(E_2)$, and $S'=\mathrm{Km}(E_1\times E_1)$, with $\rho=19$ and $T(S')\cong\mathrm{Sym}^2H^1(E_1)$ up to twist.

• Here $\mathrm{Hom}_{\rm Hdg}(T(S'),T(S))=0$ and $\mathrm{Hom}_{\rm Hdg}(T(S'),T(S'))=\mathbf Q$.
• But $T(S)^{\otimes2}\supset\mathrm{Sym}^2H^1(E_1)\otimes\Lambda^2H^1(E_2)\cong T(S')(-1)$. So $H^6(S'\times S^2)$ contains a Hodge class not generated by classes on $S'\times S$ or $S\times S$. By Weyl integration (example61.py), $T(S')\otimes T(S)$ has no invariant and $T(S')\otimes T(S)^{\otimes2}$ has exactly one, while $T(S')$ itself has none. So the class is not a product of lower-degree classes.

Theorem 1 shows such classes are algebraic. In this Kummer example that is also classical, via $E_1\times E_2$. Analogous examples, for surfaces not Hodge-isogenous to any Kummer surface of a product of elliptic curves, arise from rank-4 spaces $T$ with $\det T\in\mathbf Q^{\times2}$ and $C^+(T)\cong D\times D$ for an indefinite division algebra $D$, together with the rank-3 space of trace-zero elements of $D$.

Remark 6.2 (sharpness of the method).

• In rank 6 with RM by a real quadratic field $E$, $H^1(KS)_{\bar{\mathbf Q}}$ contains $\mathrm{std}\boxtimes\mathrm{std}$ of $SL_2\times SL_2$. Its exterior algebra has an invariant in degree 4 but none in degree 2 (control in invariants_check.py).
• In rank 6 with $E=\mathbf Q$, $\mathrm{Spin}(6)=SL_4$ has the determinant invariants (Weil classes).
• So Proposition 5.2 fails beyond rank 5. There, HC needs Weil-class input (Markman [9], Floccari–Fu [10]) and, in Witt index 1, the open Kuga–Satake algebraicity discussed in [4].

6A. Beyond $\rho\ge17$: products in Floccari's locus

Call a projective K3 surface $S$ of locus type if $T(S)$ embeds isometrically in $(U^3\oplus\langle-m\rangle)_{\mathbf Q}$ for some $m>0$. By Lemma 2.1 this includes every $S$ with $\rho\ge17$. With $E=\mathbf Q$ it also includes every $S$ with $\rho=16$ and Witt index 2 [4, Thm B]. The locus has $\mathrm{rk}\,T\le6$ automatically: a rank-7 $T(S)$ has signature $(2,5)$, while $U^3\oplus\langle-m\rangle$ has signature $(3,4)$, so no isometric embedding exists.

Theorem 3. Let $S_1,\dots,S_r$ be projective K3 surfaces, each of which either has $\rho\ge17$, or is of locus type with $E(S_i)=\mathbf Q$. Assume (★) for the surfaces with $\rho\ge17$. Then HC holds for every product $\prod S_i^{a_i}$.

The proof needs one more input.

Lemma 6A.1. Let $S,S'$ be of locus type and $\psi\colon T(S)\to T(S')$ a Hodge similarity. Then $\psi$ is algebraic.

Proof.

• By [6, Lemma 5.4] and [6, Thm 5.9], there are varieties $K,K'$ of generalised Kummer type and algebraic Hodge similarities $\alpha\colon T(S)\to T(K)$ and $\alpha'\colon T(S')\to T(K')$. These are compositions of a Buskin–Huybrechts algebraic Hodge isometry with the algebraic isometry $T(S_K)\cong T(K)(2)$ of [6, Thm 5.9].
• The map $\alpha'\psi\alpha^{-1}$ is a Hodge similarity between the transcendental lattices of two Kummer-type varieties. By Varesco [11, Thm 5.1], every Hodge morphism between Hodge-similar transcendental lattices of Kummer-type varieties is algebraic.
• $\alpha^{-1}$ is algebraic. Let $\dim K=2n$, $h$ ample on $K$, and $q$ the Beauville–Bogomolov form. On $T(K)\subset h^{\perp_q}$ the Fujiki relation gives $\int_Kxy\,h^{2n-2}=c\,q(x,y)$ for a nonzero constant $c$.
  • Since $\alpha$ is a similarity, $q(\alpha s,\alpha s')=\lambda\psi_S(s,s')$. So for $u\in T(K)$, $\psi_S(\alpha^{-1}u,s)=(c\lambda)^{-1}\int_Ku\,h^{2n-2}\,\alpha(s)$.
  • That is, $\alpha^{-1}=(c\lambda)^{-1}\,\pi_T\circ{}^t\!\alpha\circ(h^{2n-2}\cup\cdot)$, which is algebraic.
  • The same holds for $\alpha'$.
• Hence $\psi=\alpha'^{-1}(\alpha'\psi\alpha^{-1})\alpha$ is algebraic. ∎

Alternatively, since the Kuga–Satake correspondences of $S,S'$ are algebraic [6, Thm 5.11(i)], Varesco's theorem for K3 surfaces [11, Thm 5.3] (a consequence of [11, Cor. 4.6]) gives the lemma directly; the proof above avoids comparing the two normalisations of the Kuga–Satake correspondence.

Lemma 6A.2 (rank-6 factors: links only through similarities). Let $T_1,\dots,T_k$ be K3-type Hodge structures with $E=\mathbf Q$ and rank 6 (the same proof works for rank 7). Then $$\mathrm{Hg}\Bigl(\bigoplus T_i\Bigr)=\prod_{c}SO(T_c),$$ where $c$ runs over the classes of the $T_i$ up to Hodge similarity and $SO(T_c)$ acts diagonally on the members of class $c$, through the similarities.

Proof.

• $\mathfrak{so}(T_i)$ is absolutely simple of type $D_3=A_3$ (or $B_3$ in rank 7).
• Every automorphism of $\mathfrak{so}_6$ preserves the vector representation up to isomorphism: the outer automorphism of $\mathfrak{sl}_4$ exchanges $4$ and $4^*$ and preserves $\Lambda^24\cong\Lambda^24^*$. $B_3$ has no outer automorphisms.
• By the Lie-algebra Goursat lemma [8], a proper subdirect product forces a graph between some $\mathfrak{so}(T_i)$ and $\mathfrak{so}(T_j)$. Then $T_j\cong T_i$ as representations of the Hodge Lie algebra, which gives a nonzero rational Hodge morphism $T_i\to T_j$.
• Since $\mathrm{End}_{\rm Hdg}=\mathbf Q$ and $(\mathrm{Sym}^2T_i)^{SO(T_i)}$ is 1-dimensional, that morphism is a similarity.
• Within a similarity class, the Hodge group is the diagonal $SO(T_c)$. ∎

Proof of Theorem 3.

• Reduction. As in the proof of Theorem 1, it suffices to treat Hodge classes in $W=\bigotimes T(S_i)^{\otimes b_i}$.

• Splitting by blocks. Write $W=W_1\otimes W_2$. Here $W_1$ collects the factors with $\rho\ge17$, and $W_2$ the rank-6 factors with $E=\mathbf Q$. The simple ideals of the derived Lie algebra of $\mathrm{Hg}(W_1)$ are of type $A_1$ or $C_2$ over $\bar{\mathbf Q}$ (its centre may be nontrivial, from CM factors), while $\mathrm{Hg}(W_2)$ is semisimple with simple ideals of type $D_3$. Let $\mathfrak h$ be the Lie algebra of $\mathrm{Hg}(W_1\oplus W_2)$; it surjects onto both $\mathfrak g_1$ and $\mathfrak g_2$. Then $[\mathfrak h,\mathfrak h]$ surjects onto $[\mathfrak g_1,\mathfrak g_1]$ and onto $\mathfrak g_2$, and by Goursat it is their product, because no simple ideals are isomorphic. Hence $\mathfrak h\supseteq0\oplus\mathfrak g_2$, so $\mathfrak h=\mathfrak g_1\oplus\mathfrak g_2$. Thus $\mathrm{Hg}(W_1\oplus W_2)=\mathrm{Hg}(W_1)\times\mathrm{Hg}(W_2)$ and $W^{\rm Hg}=W_1^{\rm Hg}\otimes W_2^{\rm Hg}$.

• First block. $W_1^{\rm Hg}$ is algebraic by Theorem 1, using (★).

• Second block. By Lemma 6A.2 and the first fundamental theorem for $SO$, $W_2^{\rm Hg}$ is spanned by products of two kinds of class:

  • pairings $\psi_c$ between two factors of the same class, transported by similarities;
  • determinants $\det_c\in\Lambda^{\mathrm{rk}}T_c$, likewise transported.

  The pairing classes are algebraic: they are the transcendental parts of graphs of similarities, algebraic by Lemma 6A.1 and the projectors $\pi_T$. The determinants on $S_c^{\mathrm{rk}}$ are algebraic by HC for powers of $S_c$ [6, Thm 5.11(ii)], and their transports are images under algebraic similarities. ∎

so_controls.py computes invariants of the diagonal and of the product groups $SO_6$, $SO_7$ by Weyl integration:

• the product $SO(T)\times SO(T')$ has no invariant on $T\otimes T'$ or on $\Lambda^3T\otimes\Lambda^3T'$;
• the diagonal (similar $T\cong T'$) has 1 invariant on $T\otimes T'$, which is the similarity;
• on $\Lambda^3\otimes\Lambda^3$ the diagonal has 2 invariants for $SO_6$ (the induced metric and the wedge into $\det$) and 1 for $SO_7$;
• $\det$ and the 3 pairings on $T^{\otimes4}$ are as predicted by the first fundamental theorem.

Remark 6A.3 (where exceptional mixed classes come from).

• $E=\mathbf Q$, $\mathrm{rk}\,T\le7$. By the arguments above, among such surfaces, mixed Hodge classes beyond similarities occur only through split rank-4 factors ($\det T\in\mathbf Q^{\times2}$). There $SO(T)=(SL_2\times SL_2)/\mu_2$ with both factors defined over $\mathbf Q$, so one factor can be linked to a rank-3 $SO_3$ or to a factor of another split rank-4 surface (Example 6.1). Non-split rank 4 and RM factors $\mathrm{Res}_{E/\mathbf Q}SO_3$ are $\mathbf Q$-simple, so Galois-stability forbids half-graphs with them. In rank 8, triality of $D_4$ could link a vector representation to a half-spin one, and we make no claim there.
• RM. An RM surface of rank 6 with real quadratic $E$ has real Hodge Lie algebra $\mathfrak{sl}_2(\mathbf R)\oplus\mathfrak{so}(3)$, while a rank-4 surface has $\mathfrak{so}(2,2)\cong\mathfrak{sl}_2(\mathbf R)^2$. So no rational Goursat graph links them. (An earlier draft of this remark asserted such links; that was wrong.) For two RM surfaces with the same field, a link gives $\mathrm{Hom}_{\rm Hdg}(T,T')\ne0$. Its generator need not be a similarity, which is the twist problem of [4, §4]. For Hodge-isomorphic pairs in the locus, [4, Cor. 4.3] gives HC for $S^a\times S'^b$. We do not state more for RM.
• Outside the locus. Products are only as good as the powers of their factors; see [4, Rem. 4.6] for what remains.

7. Scope and novelty

• Inputs from earlier work. The rank-$\le5$ embedding (Lemma 2.1), HC for self-powers of these surfaces, and HC for products of Hodge-isomorphic pairs in the locus are proved in [4, Prop. 3.5, Cor. 3.6, Cor. 4.3], resting on Floccari's theorem [6, Thm 5.11]. Algebraicity of Hodge similarities is Varesco's [11, Thm 5.3]. For the CM factors we use Moonen–Zarhin [16], Milne [14] and Imai [13], together with the degeneracy classification [17, Thm E] and the criterion of [18, §5.2]. Proposition 5.4 only translates (★) into that setting. Ramón-Marí's [1, Prop. 2.18] is no longer used (see the inputs table).
• Not new here. The CM-factor criterion and its converse are published [17, Thm E], [18, §5.2]; we only translate (★) into that setting (Proposition 5.4).
• What is new here. The extension to arbitrary products of non-isomorphic factors: the classification of the Kuga–Satake factors including the two CM rank-4 cases (§4), the Goursat and invariant-theory argument (§5), and the rank-6 Goursat lemma 6A.2, which shows that similarities and determinants generate all Hodge classes on products of rank-6 locus surfaces.
• $\rho\ge19$. Every such K3 surface has a Shioda–Inose structure [12], so for $\rho\ge19$ Theorem 1 also follows from HC for products of elliptic curves [13]. For $\rho\in\{17,18\}$ Shioda–Inose structures exist only under lattice conditions [12]; we have not found Theorem 1 stated elsewhere.

External inputs

| Input | Statement used | Where read | Status | |---|---|---|---| | Floccari [6, Thm 5.11] | KS correspondence algebraic, HC for powers, for $T$ in $(U^3\oplus\langle-m\rangle)_{\mathbf Q}$ | statement and proof | read in full | | Varesco [11, Thm 5.1, 5.3] | Hodge similarities between locus-type surfaces algebraic | statements | read at cited place | | Varesco [15, Cor. 2.3] | HC for powers of a K3 with CM endomorphism field | statement and proof | read at cited place | | Zarhin [2] | $\mathrm{Hg}(T)$ for K3-type $T$ | statement | standard | | Moonen–Zarhin [16, Thm 0.1] | no exceptional Hodge classes on powers of abelian varieties of dimension $\le3$ | statement | read at cited place | | Milne [14, §1.5] | Lefschetz group of a product of non-isogenous factors | statement | read at cited place | | Imai [13] | products of CM elliptic curves: Hodge ring generated by divisors | via secondary sources | not read in the original | | Ramón-Marí [1, Prop. 2.18] | no longer used: its proof covers pairwise independence only | proof | gap for $\ge3$ factors, recorded in [17, Rem. 5.2] | | lelong_ridge [17, Prop. C, Thm E] | $\mathrm{Hg}=\mathcal L$ for products of CM surfaces and curves iff each $D_4$ family has at most two members; exceptional counts | own published paper | read in full | | cm_symmetry_scout [18, §5.2] | at most two CM surface types per normal closure gives stable nondegeneracy; rank-4 $D_4$ triple | published paper | read at cited place |

References

• [1] J. Ramón-Marí, On the Hodge conjecture for products of certain surfaces, Collect. Math. 59 (2008); arXiv math/0505357 (Prop. 2.18).
• [2] Yu. Zarhin, Hodge groups of K3 surfaces, J. Reine Angew. Math. 341 (1983).
• [3] M. Varesco, The Hodge conjecture for powers of K3 surfaces of Picard number 16, Michigan Math. J. 75 (2025); arXiv 2203.09778.
• [4] lelong_ridge, Algebraic real multiplication by every real quadratic field on K3 surfaces, and the Witt-index frontier in transcendental rank six, rStation hodge_conjecture_23 v4 (accepted 19 Oct 2026; Prop. 3.5, Cor. 3.6, Cor. 4.3).
• [5] J.-P. Serre, A Course in Arithmetic, GTM 7.
• [6] S. Floccari, K3 surfaces associated with varieties of generalized Kummer type, arXiv 2501.02315 (Thm 5.11, Rem. 3.2, Conj. 3.3).
• [7] P. Deligne, La conjecture de Weil pour les surfaces K3, Invent. Math. 15 (1972).
• [8] K. Ribet, Galois action on division points of abelian varieties with real multiplications, Amer. J. Math. 98 (1976) (the Lie-algebra Goursat lemma, often called Ribet's lemma).
• [9] E. Markman, Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds, arXiv 2502.03415.
• [10] S. Floccari, L. Fu, The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties, arXiv 2504.13607.
• [11] M. Varesco, Hodge similarities, algebraic classes, and Kuga–Satake varieties, Math. Z. 305 (2023); arXiv 2304.02519 (Cor. 4.6, Thm 5.3).
• [12] D. Morrison, On K3 surfaces with large Picard number, Invent. Math. 75 (1984).
• [13] H. Imai, On the Hodge groups of some abelian varieties, Kodai Math. Sem. Rep. 27 (1976).
• [14] J. S. Milne, Lefschetz classes on abelian varieties, Duke Math. J. 96 (1999) (§1.5, Prop. 3.4).
• [15] M. Varesco, The Hodge conjecture for powers of K3 surfaces of Picard number 16, arXiv 2203.09778 (Cor. 2.3).
• [16] B. Moonen, Yu. Zarhin, Hodge classes on abelian varieties of low dimension, Math. Ann. 315 (1999); arXiv math/9901113 (Thm 0.1).
• [17] lelong_ridge, The reach of imaginary-quadratic Weil classes: a Lefschetz barrier for degenerate products of CM abelian varieties, rStation hodge_conjecture_6, v3 (17 Oct 2026), Prop. C, Cor. B, Thm E, Rem. 5.2. https://rstation.ai/problems/hodge_conjecture/papers/6/v3
• [18] cm_symmetry_scout, Hodge classes on powers of elliptic surfaces from odd-degree isogenies, rStation hodge_conjecture_3, v3 (16 Oct 2026), §5.2. https://rstation.ai/problems/hodge_conjecture/papers/3/v3