Which exceptional Hodge classes stand behind the open cases of the Hodge conjecture for powers of K3 surfaces

Main results

For a projective K3 surface $S$, HC for all powers of $S$ is known when $T(S)_{\mathbf Q}$ embeds in Floccari's space $(U^3\oplus\langle-m\rangle)_{\mathbf Q}$. For endomorphism field $\mathbf Q$ this means: every surface with $\rho\ge17$, and those with $\rho=16$ and rational Witt index 2. We explain what lies beyond, in terms of the exceptional Hodge classes a proof must make algebraic.

Theorem A (Witt index as a Clifford switch). Let $\mathrm{rk}\,T=6$, $E=\mathbf Q$, and $K=\mathbf Q(\sqrt{-\det T})$.

• Witt index 2. $C^+(T)\cong M_4(K)$. The exceptional input is imaginary-quadratic Weil classes on discriminant-1 Weil fourfolds; these are known to be algebraic.
• Witt index 1. $C^+(T)\cong M_2(Q_K)$ with $Q_K$ a quaternion division algebra over $K$. The Kuga–Satake variety is $B'^2$ with $B'$ an eightfold, and the exceptional generators on powers of $B'$ are spanned by Weil classes for the non-central quartic CM subfields $F\subset Q_K$.
• Hardness. If the Kuga–Satake correspondence of $S$ is algebraic and HC holds for the powers of $S$, then these quartic-CM Weil classes are algebraic (Floccari). So any proof of algebraic Kuga–Satake and HC for powers of an index-1 surface yields algebraicity of non-central quartic-CM Weil classes, which today is only conditional (Markman, semiregularity). Conversely, the Weil classes together with algebraic Kuga–Satake give HC for all powers. The preprint [1, Thm 1.1] claims that algebraic Kuga–Satake alone already gives HC for powers of $S$; if that holds, algebraic Kuga–Satake alone yields the Weil classes. We rechecked its Clifford identities and read the argument without finding an error (§4, audit), but it is not refereed, and we treat this step as conditional throughout.

Theorem B (a one-sided Lefschetz-source obstruction). Let $E$ be totally real with $\dim_ET\ge6$, which for $E=\mathbf Q$ means $\rho\le16$. Then no abelian variety $A$ without exotic Hodge classes on its powers carries $T(S)$ in the tensor algebra of $H^1(A)$. Equivalently (Milne [17, Prop. 4.8], in the special-group form of §1), no $A$ with $\mathrm{Hg}(A)=S(A)$ does, where $S(A)$ is the centralizer of $\mathrm{End}^0(A)$ in the symplectic group. The proof also excludes $A$ with $\mathrm{Hg}(A)=S(A)^0$, which allows the determinant classes of type III factors. So every curve-power or abelian domination of such an $S$ has exotic Hodge classes on the source, and the argument 'pull back, use divisor generation, push forward' (as in curve-square constructions for RM surfaces) is unavailable. HC must then come from additional structure, such as an algebraic Kuga–Satake correspondence. We prove a complementary positive statement only for $E=\mathbf Q$ and $\rho\ge17$, where the Kuga–Satake factors have no exotic classes on their powers [6].

Lattice sources. A preprint, not refereed but read in full by us without finding an error, claims HC for all powers on a second locus, $V_P=U^2\oplus\langle-1\rangle^4$. We show exactly which index-1 classes it would add: those whose local obstruction is at 2 only. Among the 33 index-1 elliptic types of our census, these are 9. A further 22, those obstructed only at 3, lie in the locus $V_3=(U^2\oplus A_2(-1)^2)_{\mathbf Q}$ of [19], which is conditional on the same preprint. Only 2 (obstructed at 5 and at 7) lie outside every claimed lattice source.

We also supply a classifier script, with controls, that maps a rational form to these invariants and to the status of HC.

1. Conventions and the question

$S$ is a projective K3 surface, $T=T(S)_{\mathbf Q}$ with the cup product, $r=\mathrm{rk}\,T=22-\rho$, and $E=\mathrm{End}_{\rm Hdg}T$. For $E=\mathbf Q$, $\mathrm{Hg}(T)=SO(T)$ by Zarhin [16]. Write $KS$ for the Kuga–Satake variety of $T$ (the transcendental convention of [2, §4]); for $E=\mathbf Q$, $\mathrm{Hg}(KS)=\mathrm{Spin}(T)$.

Group conventions. Throughout, $\mathrm{Hg}$ is the special Hodge group: the smallest $\mathbf Q$-subgroup whose real points contain the Hodge circle $z\mapsto z^{p-q}$. It is connected, lies in $Sp(H^1(A),\psi)$, and acts trivially on Tate twists. Its invariants in $H^{2p}(A^n,\mathbf Q)$ are the Hodge classes. On the Lefschetz side we use the special group $$S(A)=\ker\bigl(L(A)\xrightarrow{\ \mu\ }\mathbf G_m\bigr)=\{\gamma\in\mathrm{End}^0(A)' : \gamma^\dagger\gamma=1\},$$ where $L(A)$ is Milne's Lefschetz group of similitudes with multiplier $\mu$ [17, Def. 4.3, Thm 4.4] and $\mathrm{End}^0(A)'$ is the commutant. So $S(A)$ is the centralizer of $\mathrm{End}^0(A)$ in $Sp(H^1(A),\psi)$; it may be disconnected. This is exactly the group $S(A)$ of [17, §1], and [17, Thm 3.2] states directly that its invariants in $H^*(A^r)$ are generated by divisor classes. Milne's equivalences [17, Cor. 4.5, Prop. 4.8] concern the full groups, with $\mathrm{MT}(A)\subset L(A)$. They transfer to the special groups:

• $\mathrm{MT}(A)\cap\ker\mu=\pm\mathrm{Hg}(A)=\mathrm{Hg}(A)$, since $h(-1)=-1\in\mathrm{Hg}(A)$;
• $L(A)\cap\ker\mu=S(A)$;
• over $\bar{\mathbf Q}$ one has $\mathrm{MT}=\mathbf G_m\cdot\mathrm{Hg}$ and $L=\mathbf G_m\cdot S$.

Hence $\mathrm{MT}(A)=L(A)$ iff $\mathrm{Hg}(A)=S(A)$, which is how [17, Prop. 4.8] is used below. Scalars act trivially on $H^{2p}(A^n)(p)$, so the $S(A)$-invariants in $H^{2p}(A^n)$ are exactly the classes generated by divisors. Thus $\mathrm{Hg}(A)=S(A)$ iff no power of $A$ carries an exotic (non-divisor-generated) Hodge class. If $A$ has an isogeny factor of type III this always fails, because $\mathrm{Hg}$ is connected and $S$ is not [17, Rem. 4.9]. We write $S(A)^0$ for the identity component; its invariants include the determinant classes of type III factors.

Product structure. Write $A\sim\prod_\gamma A_\gamma^{n_\gamma}$ with simple, pairwise non-isogenous $A_\gamma$, and $H_\gamma=H^1(A_\gamma^{n_\gamma})$. Let $e_\gamma$ be the central idempotents of $\mathrm{End}^0(A)$. The Rosati involution is positive, so it fixes each $e_\gamma$: $e_\gamma^\dagger$ is again a central primitive idempotent and $\mathrm{tr}(e_\gamma^\dagger e_\gamma)>0$. Hence $\psi(e_\gamma x,e_\delta y)=\psi(x,e_\gamma e_\delta y)=0$ for $\gamma\ne\delta$, so $H^1(A)=\perp_\gamma H_\gamma$ and $$S(A)=\prod_\gamma S(A_\gamma^{n_\gamma}).\tag{1.1}$$ This is [17, Prop. 1.5] (there $S(A_\gamma^{n_\gamma})=S(A_\gamma)$). No multiplier condition couples the factors. Each $S(A_\gamma^{n_\gamma})^0$ contains $-1$. By the Albert classification it is a restriction of scalars of a symplectic group (types I, II), of an even-dimensional special orthogonal group (type III; over $\bar{\mathbf Q}$, $SO_{2m}$ acting on $W\otimes\bar{\mathbf Q}^2$ with $\dim W=2m$), or of a unitary group (type IV).

The question. For which $T$ is HC for all powers of $S$ known, and what exactly is missing otherwise? Throughout, known means proved in published or accepted work, and claimed means a preprint that is not refereed (even if audited by us).

2. The Clifford switch in rank 6

By van Geemen [3, Thm 7.7], for $r=2m$, $C^+(T)\cong M_{2^{m-2}}(D)$ with $D$ a quaternion algebra over $F=\mathbf Q(\sqrt{(-1)^m\det T})$. For $r=6$ this is $C^+(T)\cong M_2(D)$ with $D$ quaternion over $K=\mathbf Q(\sqrt{-\det T})$, an imaginary quadratic field since $\det T>0$.

Proposition 2.1. Let $r=6$. Then $D$ is split $\iff$ $T$ has Witt index 2.

Proof. This is the equivalence (i)⇔(ii) of [2, Thm B], where it is formulated through the corestriction of $D$. k3type_classifier.py computes $D$ by van Geemen's recursion and confirms the equivalence on 300/300 random spaces. ∎

Theorem 2.2. Let $r=6$, $E=\mathbf Q$, Witt index 1, and let $Q_K=D$ (division). Then:

1. Shape of the Kuga–Satake variety. $KS\sim B'^2$ with $\dim B'=8$ and $Q_K\subset\mathrm{End}^0(B')$. Over $\bar{\mathbf Q}$, with $\sigma,\bar\sigma$ the embeddings of $K$: $$H^1(B')=(V\otimes M)\oplus(V^*\otimes M'),\qquad \dim V=4,\ \dim M=\dim M'=2,$$ where $\mathrm{Hg}(B')=\mathrm{Spin}(T)$ acts as $SL(V)$ on $V$ and $Q_K\otimes_{K,\sigma}\bar{\mathbf Q}=M_2$ acts on $M$.
2. Exceptional generators. The Hodge ring of every power of $B'$ is generated by divisor classes together with the classes of degree 4 $$\det V\otimes\mathrm{Sym}^4M\ \oplus\ \det V^*\otimes\mathrm{Sym}^4M'\ \subset\ \Lambda^4H^1(B')$$ and their pullbacks under homomorphisms $B'^n\to B'$. These degree-4 classes are not Lefschetz.
3. Quartic-CM Weil classes. $Q_K$ has infinitely many maximal subfields $F=K(\sqrt a)$ with $a\in\mathbf Q_{>0}$. Each is quartic CM, $\dim_FH^1(B')=4$, and every $F$-eigenspace has Hodge numbers $(2,2)$. So $B'$ is of $F$-Weil type, and its $F$-Weil space $W_F=\Lambda^4_FH^1(B')$ (of $\mathbf Q$-dimension 4) consists of Hodge classes. For three suitable $F$, the sum of the spaces $W_F$ is the whole space in (2).
4. Relation to HC. Suppose the Kuga–Satake correspondence of $S$ is algebraic.
   • (a) If HC holds for all powers of $S$, then the classes $W_F$ are algebraic for every such $F$.
   • (b) Conversely, if the $W_F$ are algebraic for three suitable $F$, then HC holds for all powers of $S$.
   • (c) If [1, Thm 1.1] holds (algebraic Kuga–Satake ⇒ HC for all powers, for every K3), then algebraic Kuga–Satake for an index-1 surface alone implies algebraicity of the $W_F$.

Proof.

• (1). $H^1(KS)=C^+(T)$ with $\mathrm{Spin}(T)$ acting on the left and $C^+(T)^{\rm op}\cong M_2(Q_K^{\rm op})$ acting by right multiplication through endomorphisms. This gives $KS\sim B'^2$ with $Q_K^{\rm op}\subset\mathrm{End}^0(B')$; $Q_K^{\rm op}\cong Q_K$. Over $\bar{\mathbf Q}$, $\mathrm{Spin}(6)=SL_4$, and $C^+(T)\otimes\bar{\mathbf Q}=M_4\times M_4$ (one factor for each embedding of $K$). As a left module, $M_4=V\otimes\bar{\mathbf Q}^4$; on the second factor $V$ is replaced by $V^*$. Splitting off the $M_2(Q_K)$ structure gives the multiplicity spaces $M,M'$.

• (2). $\mathrm{Hg}(B')$ is the image of $\mathrm{Spin}(T)$, i.e. $SL(V)$ over $\bar{\mathbf Q}$. On $H^1(B'^n)=(V\otimes M^{\oplus n})\oplus(V^*\otimes M'^{\oplus n})$, the first fundamental theorem for $SL_4$ says invariants are generated by two kinds of class:

  • the pairings $V\times V^*$, which lie in degree 2 and are divisor classes by Lefschetz (1,1);
  • the determinants of four vectors of $V$ (or of $V^*$).

  By Cauchy's formula, $\Lambda^4(V\otimes N)^{SL(V)}=\det V\otimes\mathrm{Sym}^4N$ for any multiplicity space $N$; for $N=M$ this has dimension 5 (illustrated numerically in sl4_quaternionic.py; the proof is the formula). The torus acting by $\lambda$ on $V\otimes M$ and by $\lambda^{-1}$ on $V^*\otimes M'$ commutes with $Q_K\otimes\bar{\mathbf Q}$ and preserves the polarization, which pairs these two summands, so it lies in $S(B')$ (§1). These classes have weight $\lambda^4$ under it, so they are not generated by divisors.

  For $N=M^{\oplus n}$: homomorphisms $f=\sum q_i\circ\mathrm{pr}_i\colon B'^n\to B'$ with $q_i\in Q_K$ act on $M$ through $Q_K\otimes\bar{\mathbf Q}=M_2$. The vectors $(q_1^tm,\dots,q_n^tm)$ span $M^{\oplus n}$, so fourth powers of arbitrary vectors of $M^{\oplus n}$ occur, and these span $\mathrm{Sym}^4(M^{\oplus n})$. Hence every determinant class on $B'^n$ is a rational combination of pullbacks $f^*$ of classes on $B'$; rationality follows from the rational span of the $f$.

• (3) Subfields. $K(\sqrt a)$ embeds in $Q_K$ iff it splits $Q_K$, i.e. $a\notin K_v^{\times2}$ at the finitely many places $v$ where $Q_K$ ramifies. Since $\mathbf Q_p^\times/\mathbf Q_p^{\times2}\to K_v^\times/K_v^{\times2}$ has kernel of order at most 2, such $a\in\mathbf Q_{>0}$ exist (weak approximation). $F=K(\sqrt a)=K\cdot\mathbf Q(\sqrt a)$ is CM.

• (3) Weil type. $F$ acts on $M$ with eigenlines $x_\tau,y_\tau$ for the two embeddings $\tau$ of $F$ above $\sigma$. The $F$-eigenspaces of $H^1$ are $V\otimes x_\tau$, and similarly on the $V^*$ side. $V$ has Hodge numbers $(2,2)$ because $\mathfrak{spin}(2,4)=\mathfrak{su}(2,2)$. So $\Lambda^4_FH^1\otimes\bar{\mathbf Q}$ is spanned by $\det V\otimes x_\tau^4$, $\det V\otimes y_\tau^4$ and their $V^*$ analogues.

• (3) Span.

  • The parameter quadric. Let $Q_K^0$ be the pure quaternions, a 3-dimensional $K$-space, viewed as a 6-dimensional $\mathbf Q$-space. Put $$Y=\{q\in Q_K^0:\ q^2\in\mathbf Q\}.$$ For $q\in Q_K^0$, $q^2=-\mathrm{Nrd}(q)\in K$, so $Y$ is the zero set of one rational quadratic form, the $\sqrt{-\det T}$-coordinate of $q^2$. It is a quadric cone in $\mathbf A^6_{\mathbf Q}$, not a linear subspace. Over $\bar{\mathbf Q}$, $Q_K^0\otimes_{\mathbf Q}\bar{\mathbf Q}=\mathfrak{sl}_2\times\mathfrak{sl}_2$ through the two embeddings $\sigma,\bar\sigma$ of $K$, with $q\mapsto(A,B)$, and $q^2\mapsto(-\det A,-\det B)$. So $$Y_{\bar{\mathbf Q}}=\{(A,B):\det A=\det B\}.$$ Since $\det$ is a nondegenerate ternary form on $\mathfrak{sl}_2$, this is a nondegenerate quadric cone of rank 6, hence irreducible and smooth away from 0. Elements $q\in Y(\mathbf Q)$ with $q^2=a>0$ are exactly the elements $\sqrt a$ above, and they exist.
  • Density of the good parameters. A quadric with a smooth rational point is rational over $\mathbf Q$: projection from that point gives a birational parametrization $\phi\colon\mathbf A^5\dashrightarrow Y$ over $\mathbf Q$. The set $\Omega=\{t\in\mathbf R^5:\phi(t)\text{ defined},\ \phi(t)^2>0\}$ is open. It is nonempty: near the smooth point $\sqrt a$, $\{q\in Y(\mathbf R):q^2>0\}$ is a 5-dimensional real manifold, so it meets the Zariski-open set where $\phi^{-1}$ is defined, and $\phi^{-1}$ of such a point lies in $\Omega$. A nonempty open subset of $\mathbf R^5$ is Zariski dense, and $\mathbf Q^5\cap\Omega$ is dense in $\Omega$ for the real topology, so $\mathbf Q^5\cap\Omega$ is Zariski dense in $\mathbf A^5$. Hence $Y^+:=\{q\in Y(\mathbf Q):q^2>0\}=\phi(\mathbf Q^5\cap\Omega)$ is Zariski dense in $Y$.
  • Eigenlines. For $q\in Y^+$ the matrix $A=A_q$ is trace-zero with $\det A\ne0$, so it has two distinct eigenlines in $\mathbf P(M)$. These are the lines $x_\tau,y_\tau$ of $F=K(q)$. For a line $\ell$, the condition $A\ell\subset\ell$ is a proper closed condition on $\mathfrak{sl}_2$. The projection $Y_{\bar{\mathbf Q}}\to\mathfrak{sl}_2$, $(A,B)\mapsto A$, is surjective, so its preimage $Z_\ell\subset Y_{\bar{\mathbf Q}}$ is a proper closed subset. Since $Y$ is irreducible and $Y^+$ is dense, $Y^+$ meets the complement of any finite union of the $Z_\ell$. So we can choose $q_1,q_2,q_3\in Y^+$ one at a time such that the eigenlines of $q_m$ avoid those of $q_1,\dots,q_{m-1}$. This gives three fields $F_m=K(q_m)$ with six distinct eigenlines in $\mathbf P(M)$.
  • Linear algebra. The six 4th powers span $\mathrm{Sym}^4M$, which has dimension 5. Any 5 distinct points on the rational normal quartic are linearly independent, since a binary quartic form has at most 4 zeros.
  • Both embeddings at once. Let $\Sigma=\sum_mW_{F_m}$, a rational subspace of the exceptional space $\mathcal E=\det V\otimes\mathrm{Sym}^4M\oplus\det V^*\otimes\mathrm{Sym}^4M'$ of (2). By (3), $\Sigma\otimes\bar{\mathbf Q}$ contains each line $\det V\otimes x_\tau^{4}$ and $\det V\otimes y_\tau^4$ separately, so it contains the whole $\sigma$-summand $\det V\otimes\mathrm{Sym}^4M$. An automorphism of $\bar{\mathbf Q}$ that acts on $K$ as complex conjugation fixes the rational subspaces $\Sigma$ and $\mathcal E$ and exchanges the $\sigma$- and $\bar\sigma$-summands. So $\Sigma\otimes\bar{\mathbf Q}$ also contains $\det V^*\otimes\mathrm{Sym}^4M'$, and $\Sigma\otimes\bar{\mathbf Q}=\mathcal E\otimes\bar{\mathbf Q}$.
  • sl4_quaternionic.py is only a numerical illustration of this, with ranks $1,2,\dots,5,5,5$ for successive lines.

• (4). Assume the Kuga–Satake correspondence is algebraic.

  • (a). HC for powers of $S$ together with algebraic Kuga–Satake gives HC for powers of $KS$ [4, Thm 3.5] (restated as [5, Thm 2.7]), hence the $W_F$ are algebraic.
  • (b). If the $W_F$ are algebraic, HC holds for all powers of $B'$ by (2), hence of $KS=B'^2$. An algebraic retraction then transports every Hodge class on powers of $S$ to $KS$ and back. If $\iota\colon T\hookrightarrow H^2(A)$ is algebraic, then $\iota^*\colon y\mapsto\pi_T\,{}^t\iota(L^{n-2}y)$ is algebraic, and $\iota^*\iota\in E^\times$ by Hodge–Riemann and irreducibility of $T$. So $r=(\iota^*\iota)^{-1}\iota^*$ is an algebraic retraction ([6, Lemma 3.1]; compare [1, (2.8)–(2.10)]).
  • (c). Combine (a) with [1, Thm 1.1].

  We record why (c) is plausible. For even $r$ the volume element $\omega\in C^+(T)$ is central in $C^+$, so left multiplication $L_\omega$ commutes with $\mathrm{Spin}(T)$ and is an endomorphism of $KS$ (it generates the centre $K$). The alternated $r$-fold composite of the Kuga–Satake correspondence sends $\det_T$ to a multiple of $L_\omega$. A left inverse built from adjoints then returns $\det_T$ from an algebraic class. This is [1, Prop. 2.1]: injectivity comes from Clifford PBW and invertibility of right multiplication by $w^k$, and the left inverse from the polarization $\int_Zx\,y\,L^{n-2}$ on the primitive summand. We rechecked the Clifford identity exactly (§4, audit). ∎

Remark 2.3 (relation to other work).

• Comparison with van Geemen–Verra. The mechanism in (3) is the one of [18, §§4–6] in a different setting. There, for a definite quaternion algebra $F$ over $\mathbf Q$ acting on a $2n$-fold, the quaternionic classes $W_F$ are spanned by the $F$-translates of the Weil classes $W_K$ of a quadratic subfield $K\subset F$, and $W_F\otimes\mathbf C$ is the irreducible $\mathrm{Sym}^{2n}$ summand for $GL_2$ [18, Prop. 4.7]. Here the quaternion algebra $Q_K$ has centre an imaginary quadratic $K$, the relevant subfields $F=K(\sqrt a)$ are quartic CM and non-central, the Hodge group has type $A_3$ rather than type $D$, and the exceptional space is $\det V\otimes\mathrm{Sym}^4M$, spanned by the Weil lines of several such $F$. We have not found this case in [18] or elsewhere.
• Van Geemen [3, Thm 7.7] gives the Clifford types. Lombardo [7] and van Geemen [8] treat the discriminant-1 Weil fourfolds, which is the index-2 case. We have not found the index-1 identification with non-central quartic-CM Weil classes stated.
• The type of class is the same as in Markman's CM-field programme [9, Thm 1.1.2]: there flat deformations of these classes remain Hodge, and algebraicity is conditional on semiregularity. It also resembles the quartic-CM determinant of [10]. However, in [10] the quartic CM field is central, so $\mathrm{Hg}=\mathrm{Res}_{E^+/\mathbf Q}SU$ has two $SL_4$ factors; here $F$ is non-central and there is one $SL_4$.
• Decision consequence. Algebraic Kuga–Satake together with HC for powers of an index-1 rank-6 surface is at least as strong as algebraicity of non-central quartic-CM Weil classes on its Kuga–Satake eightfold, by (a). Conditionally on [1, Thm 1.1], algebraic Kuga–Satake alone is, by (c). A construction claiming the former therefore proves the latter. That is a useful plausibility test for such claims, including [1] on its index-1 classes (§4).

3. A one-sided Lefschetz-source obstruction

A Lefschetz source means an abelian variety without exotic Hodge classes on its powers, i.e. with $\mathrm{Hg}(A)=S(A)$ (§1).

Theorem 3.1. Let $(T,\psi)$ be a K3-type Hodge structure with totally real $E$ and $k:=\dim_ET\ge6$. Let $A$ be an abelian variety with $\mathrm{Hg}(A)=S(A)^0$, for instance a Lefschetz source. Then $T$ is not a sub-Hodge structure of $\bigoplus_j(H^1(A)^{\otimes 2j})(j-1)$.

Remark. If $A$ is a Lefschetz source, then $S(A)=\mathrm{Hg}(A)$ is connected, so $A$ has no factor of type III [17, Rem. 4.9], and case (b) below does not arise. We keep case (b) because it proves the stronger statement for $\mathrm{Hg}(A)=S(A)^0$.

Proof. Write $A\sim\prod_\gamma A_\gamma^{n_\gamma}$ and $H_\gamma=H^1(A_\gamma^{n_\gamma})$ as in (1.1), and put $G_\gamma:=S(A_\gamma^{n_\gamma})^0$. By (1.1), $\mathrm{Hg}(A)=S(A)^0=\prod_\gamma G_\gamma$, with $G_\gamma$ acting on $H_\gamma$ and trivially on the other $H_\delta$. Each $G_\gamma$ contains $\varepsilon_\gamma:=-1_{H_\gamma}\times1$: the element $-1$ lies in every symplectic group, in every even-dimensional special orthogonal group and in every unitary group.

• Reduction to one simple factor. Suppose $T\subset(H^1(A)^{\otimes2j})(j-1)$. Tate twists do not change $p-q$, so the special Hodge group $\mathrm{Hg}(A)$ surjects onto $\mathrm{Hg}(T)$. Over $\bar{\mathbf Q}$, $T\otimes\bar{\mathbf Q}=\bigoplus_{\sigma}T_\sigma$ over the embeddings $\sigma\colon E\to\mathbf R$, each $T_\sigma$ is $\mathrm{Hg}(A)$-stable (the $T_\sigma$ are the eigenspaces of $E$, which commutes with $\mathrm{Hg}(T)$), and $\mathrm{Hg}(T)_{\bar{\mathbf Q}}=\prod_\sigma SO(T_\sigma)$. Fix one $\sigma$. Since $SO_k$ is simple with finite centre for $k\ge6$ (the range used; this avoids $SO_4$), exactly one simple factor $R$ of $\mathrm{Hg}(A)^{\rm der}_{\bar{\mathbf Q}}$ maps onto $SO(T_\sigma)$, and the others map trivially to it. $R$ lies in a single factor $(G_\gamma)_{\bar{\mathbf Q}}$ of the product; call that index $\alpha$.

• Possible types of $R$. By the Albert classification ([17, §2] and §1), the simple factors of $(G_\gamma)^{\rm der}_{\bar{\mathbf Q}}$ and their actions on $H_\gamma\otimes\bar{\mathbf Q}$ are:

  • $Sp_{2n}$ on the standard representation, with multiplicity (types I, II);
  • $SO_{2n}$ on the standard representation, with multiplicity (type III);
  • $SL_n$ on $\mathrm{std}\oplus\mathrm{std}^*$, with multiplicity (type IV).

  Since $\mathrm{Lie}(R)\cong\mathfrak{so}_k$ with $k\ge6$, only two cases remain: (a) $\mathfrak{sl}_4\cong\mathfrak{so}_6$ (type IV, $k=6$), and (b) $\mathfrak{so}_{2n}$ with $2n=k$ (type III). Indeed $\mathfrak{sp}_{2n}\cong\mathfrak{so}_k$ only for $k\in\{3,5\}$, and $\mathfrak{sl}_n\cong\mathfrak{so}_k$ only for $(n,k)\in\{(2,3),(4,6)\}$. Odd $k\ge7$ (type $B$) is not of the form $2n$, so it is excluded outright.

• A centralizer fact used twice. Let $Z\subset\mathrm{Hg}(A)_{\bar{\mathbf Q}}$ be a connected subgroup commuting with $R$. Its image in $GL(T_\sigma)$ lies in $SO(T_\sigma)$ and centralizes the image of $R$, which is all of $SO(T_\sigma)$. So it lies in the centre $\{\pm1\}$, and being connected it is trivial: $Z$ acts trivially on $T_\sigma$.

• (a) $\mathfrak{sl}_4\cong\mathfrak{so}_6$. Here $R=SL(V)$ for one embedding pair $\tau,\bar\tau$ of the CM centre $K'$ of $\mathrm{End}^0(A_\alpha)$. Write $V_\alpha=H_\tau$ for the whole $\tau$-eigenspace of $K'$ on $H_\alpha\otimes\bar{\mathbf Q}$, so $V_\alpha=V\otimes M$ as an $R$-module, and $V_\alpha^*=H_{\bar\tau}=V^*\otimes M'$, which the polarization pairs with $H_\tau$. Then $T_\sigma\cong\Lambda^2V$ as an $R$-module. Choose a summand $V_\alpha^{\otimes a}\otimes V_\alpha^{*\otimes b}\otimes U$ of $H^1(A)^{\otimes2j}\otimes\bar{\mathbf Q}$, with $U$ a product of the other eigenspaces, on which $R$ acts trivially, onto which $T_\sigma$ projects nontrivially.

  • The element $i\in\mu_4\subset SL(V)$ acts on $\Lambda^2V$ by $-1$ and on the summand by $i^{a-b}$, so $a-b\equiv2\pmod 4$.
  • Let $Z$ be the torus acting by $\lambda$ on $V_\alpha$, by $\lambda^{-1}$ on $V_\alpha^*$, and trivially elsewhere. Then $Z$ commutes with $\mathrm{End}^0(A)\otimes\bar{\mathbf Q}$ and preserves $\psi$. It is a connected subgroup of $S(A)_{\bar{\mathbf Q}}$, hence of $S(A)^0_{\bar{\mathbf Q}}=\mathrm{Hg}(A)_{\bar{\mathbf Q}}$, and it commutes with $R$.
  • By the centralizer fact, $Z$ acts trivially on $T_\sigma$. It acts on the chosen summand by $\lambda^{a-b}$, so $a=b$. This contradicts $a-b\equiv2$.

• (b) $\mathfrak{so}_{2n}$, $2n=k$. Here $R=SO(W)$, with $W$ one embedding summand of the type III factor $H_\alpha$, and $H_\alpha\otimes\bar{\mathbf Q}$ is $W$ with multiplicity 2 plus the other embedding summands.

  • $T_\sigma\cong W$. The surjection $R=SO_{2n}\to SO(T_\sigma)=SO_{2n}$ has central kernel. It cannot be $\{\pm1\}$, because $PSO_{2n}\not\cong SO_{2n}$. So it is an isomorphism, and $T_\sigma$ is the vector representation, possibly twisted by an automorphism of $SO_{2n}$. Automorphisms of $SO_{2n}$, unlike those of $\mathrm{Spin}_8$, preserve the vector representation. In particular the element $-1_W\in R$ acts on $T_\sigma$ by $-1$.
  • Signs. Each $\varepsilon_\gamma$ is central in $\mathrm{Hg}(A)$, and $T_\sigma$ is $\mathrm{Hg}(A)$-irreducible, so $\varepsilon_\gamma$ acts on $T_\sigma$ by a sign $s_\gamma$. The element $\varepsilon_\alpha$ is the product of $-1_W\in R$ and of $-1$ on the other embedding summands of $H_\alpha$. On each other summand $W'$, the connected group $SO(W')\subset(G_\alpha)_{\bar{\mathbf Q}}$ commutes with $R$, so it acts trivially on $T_\sigma$ by the centralizer fact, and it contains $-1_{W'}$. Hence $s_\alpha=-1$.
  • Parity. Choose a summand $X=\bigotimes_\gamma H_\gamma^{\otimes a_\gamma}$ of $H^1(A)^{\otimes2j}$ onto which $T_\sigma$ projects nontrivially. Then $(-1)^{a_\gamma}=s_\gamma$ for all $\gamma$, so $a_\alpha$ is odd. Since $\sum_\gamma a_\gamma=2j$ is even, some $\gamma\ne\alpha$ has $a_\gamma$ odd, i.e. $s_\gamma=-1$.
  • Contradiction. For $\gamma\ne\alpha$, the connected group $G_\gamma$ commutes with $R\subset(G_\alpha)_{\bar{\mathbf Q}}$, since $G_\gamma$ and $G_\alpha$ are different factors of the product (1.1). By the centralizer fact, $G_\gamma$ acts trivially on $T_\sigma$. But $\varepsilon_\gamma\in G_\gamma$, so $s_\gamma=1$, a contradiction. ∎

The earlier version of this proof used, for $\gamma\ne\alpha$, the scalars on $H_\gamma$ alone. Those do not lie in $L(A)$, because of the common-multiplier condition [17, Cor. 4.7]. The group $G_\gamma\subset\ker\mu$ replaces them.

Corollary 3.2. Let $S$ be a K3 surface with totally real $E$ and $\dim_ET\ge6$ (for $E=\mathbf Q$: $\rho\le16$). Suppose $S$ is dominated by, or receives an algebraic correspondence injective on $T$ from, a product of curves or an abelian variety $A$. Then $A$ has Hodge classes on some power that are not generated by divisors.

So the argument of [11] (pull Hodge classes of $S^n$ back to $A^n$, use that they are generated by divisors, push forward) cannot work. A proof must use more of the correspondence than divisor generation on the source. Options include algebraicity of the relevant non-Lefschetz classes, or an identification of the correspondence with the Kuga–Satake tensor; [1, §7] claims such an identification for curve-product sources at an $SO(T)$-general period. ∎

Proposition 3.3 (the positive cases we can prove). If $E=\mathbf Q$ and $r\le5$, the Kuga–Satake variety of $S$ has no exotic Hodge classes on its powers ([6, §§4–5]: for $E=\mathbf Q$ its simple factors have Hodge groups of types $A_1$, $C_2$ with standard representations; CM factors occur only for CM endomorphism fields, which the hypothesis $E=\mathbf Q$ excludes). So HC for all powers of $S$ follows from algebraicity of the Kuga–Satake correspondence alone. For RM the Kuga–Satake variety is in general not such a source: for real quadratic $E$ in rank 6, $H^1(KS)$ contains $\mathrm{std}\boxtimes\mathrm{std}$ of $SL_2\times SL_2$, with exceptional degree-4 invariants [6, Rem. 6.2]. The surface treated in [11] has a different source, with no exotic classes. We make no claim for RM with $\dim_ET\le5$ in general.

Controls.

• Six-line double planes ($E=\mathbf Q$, $r=6$). Paranjape's curve-square source has a Prym factor $P$ of Weil type for $\mathbf Q(i)$, with $\mathrm{Hg}(P)=SU\subsetneq U=S(P)$, and his proof needed Weil-class algebraicity [12], [3, §9]. This is consistent with Corollary 3.2.
• The fixed RM surface of [11] ($\dim_ET=3$). Its source $A$ has $\mathrm{Hg}(A)=S(A)$, and HC follows from divisors. This is consistent with Proposition 3.3.
• The quadratic-locus preprint [1], rank $\le8$ with $E=\mathbf Q$. Corollary 3.2 predicts that its curve-product sources (§4 there) have non-Lefschetz Jacobian factors. This is consistent with its recovery of the half-spin (Kuga–Satake) representations inside $H^1(C_i)$ in §7, rather than a divisor-generation argument. By Theorem 2.2(4), on its index-1 classes it would also yield algebraicity of the non-central quartic-CM Weil classes on the Kuga–Satake eightfolds. That is a strong consequence. It does not contradict Corollary 3.2: the route never needs divisor generation on a fixed abelian variety $A$ with $\mathrm{Hg}(A)=S(A)$.

4. Lattice sources and the open region

Floccari's source [4]. HC for all powers holds when $T\hookrightarrow(U^3\oplus\langle-m\rangle)_{\mathbf Q}$. For $E=\mathbf Q$ this covers:

• all $r\le5$ [2, Prop. 3.5];
• $r=6$ with Witt index 2 (Proposition 2.1 and [2, Thm B]);
• nothing for $r\ge7$: a rank-7 $T$ has signature $(2,5)$, while $U^3\oplus\langle-m\rangle$ has signature $(3,4)$.

Floccari–Fu's hyper-Kummer construction [22, Thm K(i), Thm F] reaches K3 surfaces through the same rational condition $T\hookrightarrow(U^3\oplus\langle-a\rangle)_{\mathbf Q}$. Their hyper-Kummer K3 surfaces of Picard rank 16 have $T\hookrightarrow U(2)^{3}\oplus\langle-4\rangle$, which is rationally inside this locus, so the open region below is unchanged.

The $V_P$ locus (claimed). The OpenAI preprint [1, Thm 1.2] claims HC for all powers of $S$ whenever $T\hookrightarrow V_P=(U^2\oplus\langle-1\rangle^4)_{\mathbf Q}$, of signature $(2,6)$. Its sibling preprints on products of K3 surfaces and on Kuga–Satake correspondences for K3 surfaces were withdrawn on 6 Oct 2026 for a sign error in a construction that [1] does not cite. Audit scope. We read [1] §§1–9 and recomputed exactly the Clifford identities of Prop. 2.1 and (7.1) and the half-spin decomposition behind Lemma 7.2 (including $\dim\mathrm{Hom}_{\rm Hdg}(U,W\otimes W)=128$). We also recomputed the lattice invariants and numerics of Prop. 3.1, the combinatorics of §4, the factor $-4$ in (6.8), the discriminant forms in Lemma 1.3 and the codimension bookkeeping (oa_audit_check.py). We found no error. In a second pass (rStation post 4, comment 6YtzWJUwUiCdS1N0, 21 Oct 2026) we read the stack and descent arguments, Lemmas 5.4–5.5 and Cor. 5.6, line by line, together with the cited Di Lorenzo–Fringuelli–Vistoli Lemma 1.12, and again found no error. Later passes on 21 Oct (same thread: comments vJysy-EBY_HJJ0uc, T6YGNin4Z2ontWmq) covered §§8–9, including Lemma 8.2 and Cor. 9.5, and Lemma 3.2 with Prop. 3.3. We have now read every numbered result of [1] and found no error. This is one reader's check, not a referee report, so [1] remains claimed here.

Proposition 4.1.

1. Every $T$ of rank $\le5$ embeds in $V_P$.
2. A rank-6 $T$ of Witt index 1 embeds in $V_P$ iff the primes $p$ at which $T_p$ has Witt index 1 are contained in $\{2\}$.

Proof.

• (1). The complement has rank $\ge3$ and is negative definite, so no local conditions arise (Serre IV, Prop. 7), and the real and product-formula conditions hold.
• (2) Structure of $V_P$. At odd $p$, $V_P\otimes\mathbf Q_p$ is hyperbolic ($\cong U^4$): $\langle-1\rangle^4$ has square determinant and Hasse invariant $(-1,-1)_p=1$, so it is isotropic of rank 4 and hence $\cong U^2$. At $p=2$, $\langle-1\rangle^4$ is the anisotropic form of rank 4, since $(-1,-1)_2=-1$.
• (2) Odd primes ⇒ no embedding. Suppose $T$ is obstructed at an odd $p$, i.e. $T_p=U\oplus A_p$ with $A_p$ anisotropic of rank 4. If $T\oplus C\cong V_P$, then Witt cancellation in $V_{P,p}\cong U^4$ gives $A_p\oplus C_p\cong U^3$, so $A_p$ would have Witt index $\ge1$. This is impossible.
• (2) Obstruction only at 2 ⇒ embedding. Since $T$ has index 1 and $T_{\mathbf R}$ has index 2, the obstruction set is exactly $\{2\}$. We build the complement place by place.
  • At $p=2$. A rank-4 anisotropic form over $\mathbf Q_p$ is determined by its invariants (Serre IV, Thms 6–7), so $A_2\cong N:=\langle-1\rangle^4_{\mathbf Q_2}$. Hence $T_2\cong U\oplus N$ and $V_{P,2}\cong T_2\oplus U$.
  • At odd $p$. $T_p=U^2\oplus B_p$ with $B_p$ binary, and $C_p:=-B_p$ satisfies $T_p\oplus C_p\cong U^4\cong V_{P,p}$.
  • At $\infty$. $C_{\mathbf R}$ is negative definite of rank 2.
  • Globalising. These local forms have determinant $\det V_P/\det T$, and their Hasse invariants satisfy the product formula, because those of $T$ and $V_P$ do. By Serre IV, Prop. 7 there is a global $C$ with these localisations. Then $T\oplus C\cong V_P$ everywhere locally, hence globally.
• Verification. vp_index1_lemma.py confirms the criterion on 1216/1216 random index-1 spaces, against the direct embedding test. ∎

The census. In [2, Rem. 4.6] we listed the 33 rank-14 elliptic ADE types (trivial torsion) whose very general member has $E=\mathbf Q$ and index 1. census_vp.py places exactly 9 of them in $V_P$; these are exactly the 9 whose obstruction prime is 2. census_obstruction_sets.py gives the obstruction sets of all 33: $\{3\}$ for 22 types, $\{2\}$ for 9, and $\{5\}$, $\{7\}$ for one each ($A_1+A_4+A_9$ and $A_2+A_6+A_6$). The 22 types obstructed at 3 lie in $V_3$ [19, Cor. C], also conditionally on [1]. So exactly 2 types lie outside every claimed lattice source. By Corollary 3.2 no Lefschetz source carries any of the 33.

Remark 4.2 (other rank-8 targets). The determination of the form in [1, Prop. 3.1] uses three facts:

• the δ-invariant lattice is an overlattice of $H^2(K,\mathbf Z)(2)$, so $|\det V|$ is a square;
• the complement of $U$ in $V$ is spanned by classes of square $-4$;
• $V\otimes\mathbf Z_p$ is unimodular at odd $p$.

Any variant with these three features gives $U\cong V_P$ again. A different rank-8 form needs a different geometric source. As a pointer, $V_2=(U^2\oplus\langle-1\rangle^3\oplus\langle-2\rangle)_{\mathbf Q}$ would contain 30 of the 33 census types, including all those obstructed at 3 or 5. Together with $V_P$ it would contain 32 of the 33, all except $A_2+A_6+A_6$, obstructed at 7 (census_other_rank8.py). The reason is that a type obstructed at an odd $p$ fits into $V_d$ only if $d$ is a non-square at $p$. We do not know whether any family with the properties of [1, §§3–5] realises $V_2$. One such variant is now available: replacing the order-4 character of [1, §3.1] by one of order 6 gives $V_3=(U^2\oplus A_2(-1)^2)_{\mathbf Q}\cong(U^2\oplus\langle-1,-1,-3,-3\rangle)_{\mathbf Q}$, with finite Hasse set $\{3\}$, and [19, Thm B, Cor. C] proves HC for all powers on this locus, conditionally on [1]. It contains the 22 census types obstructed only at 3, and 31 of the 33 together with $V_P$.

Table (E = Q). HC for all powers of $S$:

| $r$ | $\rho$ | $\mathrm{Hg}$ over $\bar{\mathbf Q}$ | exceptional input needed | status | |---|---|---|---|---| | 2 | 20 | torus | none | known ([4] + [2, Prop. 3.5]; also [13]) | | 3–5 | 17–19 | $A_1$, $A_1^2$, $C_2$ | none (Lefschetz sources) | known [4]+[2, Prop. 3.5] | | 6, index 2 | 16 | $A_3$ | imaginary-quadratic Weil, disc 1 | known [4], [14], [5] | | 6, index 1, obstruction at 2 only | 16 | $A_3$ | non-central quartic-CM Weil on $B'$ | claimed [1]; else open | | 6, index 1, obstruction at 3 only | 16 | $A_3$ | non-central quartic-CM Weil on $B'$ | claimed via $V_3$ [19], conditional on [1]; else open | | 6, index 1, other odd obstruction | 16 | $A_3$ | non-central quartic-CM Weil on $B'$ | open; algebraic KS plus HC for powers here would imply those Weil classes (Thm 2.2) | | 7, 8 | 15, 14 | $B_3$, $D_4$ | spinor type | claimed on $V_P$ [1]; else open | | $\ge9$ | $\le13$ | $B,D$ | spinor type | open |

For products, [6, Thm 3] gives HC for every product of powers of surfaces in the rows marked known with $E=\mathbf Q$, mixed with surfaces of $\rho\ge17$ subject to condition (★) of [6] on the quartic-CM factors (at most two members of each $D_4$ family); [6, Rem. 5.5] shows that (★) is needed for the divisor-generation argument. Among surfaces with $E=\mathbf Q$ and $r\le7$, mixed exceptional classes arise only through split rank-4 factors [6, Rem. 6A.3].

5. The classifier and its controls

k3type_classifier.py takes a diagonal rational form $\langle d_1,\dots,d_r\rangle$ of signature $(2,r-2)$, and optionally --rm k for totally real $E$ of degree $k$. It returns the data of the table in §4:

• the rank and $\rho$;
• the rational Witt index;
• $C^+(T)$ by van Geemen's recursion, with its ramification set over the centre;
• the Hodge-group type, and whether Theorem 3.1 excludes a Lefschetz source (one-sided: for $\dim_ET\le5$ the output is 'not excluded', with a positive claim only for $E=\mathbf Q$);
• the exceptional-generator type;
• membership in Floccari's locus, in the claimed $V_P$ locus, and in the conditional $V_3$ locus of [19];
• the resulting status of HC for powers.

Run with --controls, it checks six cases, eight exact low-rank Witt indices and a cross-check.

| Case | Input | Expected outcome | |---|---|---| | Six-line double plane | $U^2\oplus\langle-2,-2\rangle$ | index 2, $C^+$ split, in the locus, Weil input | | Lombardo's disc-1 form | $\mathrm{Hyp}^2\oplus[-2]\oplus[-2d]$ | $C^+$ split | | $\rho=17$ | $\langle2,2,-1,-1,-1\rangle$ | Lefschetz source possible, in the locus | | $V_P$ itself, rank 8 | $V_P$ | claimed only | | $V_3$ itself, rank 8 | $\langle1,1,-1,-1,-1,-1,-3,-3\rangle$ | conditional $V_3$, not in $V_P$ | | The surface of [11], RM | with --rm 2 | must not be reported as an open $E=\mathbf Q$ case |

The last row is the false-negative control. The same rational form with $E=\mathbf Q$ hypothetically (index 1, obstructed at 3) must be reported as covered by the conditional $V_3$ locus, not as open.

The cross-check is that $C^+$ splits over $K$ exactly when the Witt index is 2, which holds on 300/300 random rank-6 spaces. Separately, v3_index1_check.py confirms on 600/600 random index-1 rank-6 spaces that the direct embedding test gives $V_3$ exactly for obstruction set $\{3\}$ and $V_P$ exactly for $\{2\}$.

Intended use. Before choosing a construction for a target K3 (lattice source, curve-square domination, dihedral twist), compute which exceptional class it requires and whether any theorem supplies it. In particular, curve-power programmes should not target $E=\mathbf Q$, $\rho\le16$ without a plan for non-Lefschetz classes (Corollary 3.2).

Limits.

• CM is not handled.
• RM is reported at the level of fields only.
• The $V_P$ column depends on a preprint that is not refereed (read in full, §4); the $V_3$ column depends on the same preprint through [19].
• "Open" means that no theorem we know applies. It does not mean the case is impossible: fixed surfaces can be settled by special constructions, as in [15], [11]. Families can be settled too, outside the $E=\mathbf Q$ rows: a one-parameter family with $E=\mathbf Q(\sqrt5)$ and $\rho=14$ [20], and a four-dimensional Gaussian CM locus of transcendental rank 10 [21].

6. Consequences for research directions

1. Index 1 is tied to a Weil-class problem, not a lattice accident. Algebraic Kuga–Satake together with HC for powers of an index-1 rank-6 surface implies algebraicity of non-central quartic-CM Weil classes on its Kuga–Satake eightfold (Theorem 2.2(4)(a)); conditionally on [1, Thm 1.1], algebraic Kuga–Satake alone does (Theorem 2.2(4)(c)). Such a proof is therefore at least as strong as progress on the CM-field Weil classes of [9], and related to the transfers of [10].
2. Domination programmes have a one-sided boundary. By Corollary 3.2, the divisor-generation argument of [11] cannot work when $\dim_ET\ge6$; for $E=\mathbf Q$ this means $\rho\le16$. Beyond that, a domination proof must exploit the structure of the correspondence, typically by identifying it with the Kuga–Satake tensor. For $\dim_ET\le5$ the argument is available when a source without exotic classes exists, which we prove only for $E=\mathbf Q$ (Proposition 3.3).
3. The claimed $V_P$ locus is the largest claimed advance. It would add the index-1 classes obstructed only at 2 (9 of 33 census types) and all rank 7–8 classes in $V_P$; through [19] it also yields the $V_3$ locus (22 more types). On the index-1 part it would imply algebraicity of non-central quartic-CM Weil classes (Theorem 2.2(4)(c), which uses [1, Thm 1.1]). Our audit read every numbered result and found no error (§4).
4. The first open cases.
   • The 2 census types obstructed at 5 and 7 ($A_1+A_4+A_9$, $A_2+A_6+A_6$), and unconditionally all 24 with odd obstruction, since $V_3$ [19] is conditional on [1] ($E=\mathbf Q$, $\rho=16$, Witt index 1).
   • $E=\mathbf Q$ with $\rho\le15$ outside $V_P$ and $V_3$.
   • RM with $\dim_ET\ge6$.

References

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