Full-spin domination and generalized Hodge support for K3 surfaces

1. Results and relation to the Hodge conjecture

Let $T$ be a simple polarized rational Hodge structure of K3 type: it has types $(2,0),(1,1),(0,2)$, $h^{2,0}=h^{0,2}=1$, and no proper nonzero rational Hodge substructure. Let $A=\operatorname{KS}(T)$ denote the Kuga–Satake abelian variety defined by the left regular even-Clifford Hodge structure $C^+(T)$. We retain this full source, including both complex half-spin representations when $\dim T$ is even. For a pure rational Hodge structure $U$, its level is

$$ \ell(U)=\max\{|p-q|:U^{p,q}\ne0\}. $$

An effective weight-$\ell$ Hodge structure with a nonzero $(\ell,0)$ component is called fully twisted.

Theorem 1.1 (full-spin self-domination). Let $U\subset H^j(A^n,\mathbb Q)$ be an irreducible rational Hodge substructure, and put $\ell=\ell(U)$. Then

$$ U\left(\frac{j-\ell}{2}\right) \quad\text{is isomorphic to a direct summand of}\quad H^\ell(A^\ell,\mathbb Q). \tag{1.1} $$

For $\ell=0$ use $A^0=\operatorname{Spec}\mathbb C$. The summand is fully twisted. The theorem holds for CM and totally real endomorphism fields, including compact real factors and arbitrary rational Schur indices.

Thus powers of the actual full Kuga–Satake variety are dominated, in the Hodge-theoretic sense, by powers of that same variety. No algebraicity assertion is needed for Theorem 1.1. Its geometric consequence uses ordinary Hodge classes to construct supports.

Theorem 1.2 (ordinary-to-generalized criterion). Let $S$ be a smooth complex projective K3 surface. Suppose:

1. its usual specified Kuga–Satake correspondence is algebraic;
2. the ordinary rational Hodge conjecture holds for every power of $S$.

Then the generalized rational Hodge conjecture holds for every

$$ S^r\times\operatorname{KS}(S)^n,\qquad r,n\ge0. \tag{1.2} $$

The same conclusion holds after adjoining abelian isogeny factors of powers of $\operatorname{KS}(S)$. Either the primitive or the transcendental convention for $\operatorname{KS}(S)$ may be used. All products are attached to one fixed surface $S$. In hypothesis 1, the specified correspondence means the standard embedding of the transcendental Hodge structure $T(S)$ into $H^2(\operatorname{KS}(S)^2,\mathbb Q)$ obtained from Clifford multiplication and a polarization.

Here generalized HC means that a rational Hodge substructure $U\subset H^j(X,\mathbb Q)$ of level at most $j-2p$ is supported on an algebraic subset of codimension at least $p$. We prove the support assertion by an actual algebraic correspondence in Section 6; we do not identify Hodge level with coniveau without constructing that correspondence.

There are unconditional applications.

Corollary 1.3. The conclusion of Theorem 1.2 holds if $S$ has a nonsymplectic automorphism of order three whose fixed locus contains a curve. In particular it holds throughout the two nine-dimensional components with that property and their smooth specialization strata.

Corollary 1.4. The conclusion holds for every projective K3 surface $S$ for which there is a rational isometric embedding

$$ T(S)_{\mathbb Q}\hookrightarrow \left(U^{\oplus3}\oplus\langle-m\rangle\right)_{\mathbb Q} \quad\text{for some integer }m>0, \tag{1.3} $$

where $U$ is the hyperbolic plane. It also holds for the families satisfying the precise hypotheses of Varesco's Theorem 0.2 [1].

The new step is the full-spin domination criterion and its resulting generalized support. We do not claim a new Kuga–Satake correspondence for either application. The general rational Hodge conjecture, ordinary HC for arbitrary K3 powers, and algebraicity of a generic orthogonal Kuga–Satake correspondence remain open.

2. Prior work and what is added

Zarhin determines the Hodge groups and endomorphism fields of simple K3-type structures [2]. The Kuga–Satake construction and its Clifford decomposition are classical; van Geemen gives a detailed account [3]. We use both results as inputs.

The strategy of realizing a fully twisted Tate twist and then using ordinary HC to obtain generalized HC goes back to Grothendieck; Abdulali develops domination systematically [4, Section 3]. His semisimple PEL theorem [5, Theorem 5.1], in the formulation without type-III factors in [4, Theorem 4.1], proves self-domination in that range. A full KS source need not be of PEL type, and its Hodge group need not be semisimple. His $SU(p,1)$ theorem gives domination by suitable abelian varieties [4, Theorem 6.6], using the constructions in [6]. It does not identify those sources with powers of a given full KS variety, nor does its stated hypothesis include arbitrary compact real factors. The spin augmentation in [4, Theorem 7.2] is an important low-rank precedent: it supplies an additional spin abelian variety for a particular type-III PEL problem. Here the source is already the full KS variety, and we construct every required representation in its $\ell$th power, in both endomorphism field cases.

We use the corrected formulations of these results. In particular, [4, Appendix B] retracts the rational product decomposition asserted in [5, Proposition 4.4.1] and corrects several type-III and type-IV statements. Our argument does not decompose a rational irreducible into rational factors: it aligns all complex factors first, computes the level of the full rational orbit, and then descends an equivariant Hom space. The distinction matters when Galois acts on several factors.

Floccari proves that, once the specified KS correspondence is algebraic, ordinary HC for all K3 powers is equivalent to ordinary HC for all KS powers [7, Theorem 3.5]. This is the ordinary-HC input to Theorem 1.2, not a new result here. Varesco establishes ordinary HC for all powers of a K3 with CM endomorphism field, and for the specific Picard-sixteen families in his Theorem 0.2 [1]. Bolognesi and Laterveer establish the KS Hodge conjecture for the order-three families used in Corollary 1.3 [8, Theorem 4.4 and Propositions 6.2,6.4]. Floccari proves both hypotheses of Theorem 1.2 under (1.3) [7, Theorem 5.11].

A previous Gaussian rank-ten construction gave a full-spin rectangle argument when the CM endomorphism field contains a fixed imaginary quadratic field [9, Section 10.9]. That argument allowed all smaller Hodge groups at its special points but used simultaneous conjugation of its CM factors. Section 4 below removes the imaginary-quadratic subfield restriction by tracking independent dualizations. Section 5 adds the entirely orthogonal case through dominant minuscule paths. Together they give a criterion for every full KS source of simple K3 type.

The related half-twist decomposition for an imaginary-quadratic endomorphism field is due to van Geemen [10, Theorem 3.10]. Our theorem does not replace that decomposition: it uses the complete regular spin source to control every irreducible representation, every central character, and the rational Hodge level.

3. Hodge groups, full spin, and rational descent

Let $F=\operatorname{End}_{\mathrm{Hdg}}(T)$. It is a totally real or CM field [2]. Put $G=\operatorname{Hg}(T)$, and let $\widetilde G$ be its inverse image in $\operatorname{Spin}(T)$. We will verify in each case that this inverse image is connected.

The weight-one KS Hodge circle lies in $\operatorname{Spin}(T)$ and projects onto the weight-two circle of $T$. Therefore $\operatorname{Hg}(A)\subseteq\widetilde G$, and its projection is $G$: the projected group contains the circle and is contained in its smallest rational algebraic group. The kernel is finite, so both groups have the same dimension. Once $\widetilde G$ is connected, it follows that

$$ \operatorname{Hg}(A)=\widetilde G. \tag{3.1} $$

This concerns Hodge groups, not the additional weight-scalar torus in the Mumford–Tate group.

The complex left regular even Clifford module contains the spin module when $\dim T$ is odd, and both half-spin modules when $\dim T$ is even, with positive multiplicities [3, Sections 6–8]. We will only need their occurrence. A chosen minimal rational spin module or one half-spin factor is not substituted for this regular source. Rational splitting of a Clifford algebra is not assumed.

We use the following rational-descent observation repeatedly. Suppose a rational irreducible Hodge structure $U$ has level $\ell$ and weight $j$, and a complex irreducible constituent of $U$ occurs in $H^\ell(A^\ell,\mathbb C)$ as a $\widetilde G$ representation. Then

$$ \operatorname{Hom}_{\widetilde G} \left(H^\ell(A^\ell,\mathbb Q),U((j-\ell)/2)\right) \otimes_{\mathbb Q}\mathbb C\ne0. \tag{3.2} $$

Here equivariant Hom commutes with scalar extension. A nonzero rational map is surjective because $U$ is irreducible. The two sides have equal weight, and equivariance gives equality of Hodge-circle actions; hence the map is a Hodge morphism. The category of polarizable rational Hodge structures is semisimple, so the quotient splits. The parity of $j-\ell$ follows from the Hodge decomposition. This avoids descending a single complex spin constituent and handles rational multiplicities and Schur indices.

Consequently it suffices to exhibit each complex irreducible in degree equal to the level of its rational orbit. Its rational summand is then effective and contains an extreme Hodge component; complex conjugation supplies the other extreme if necessary.

4. CM endomorphism fields: signed rectangles

Write $F^+$ for the maximal totally real subfield, $d=[F^+:\mathbb Q]$, and $m=\dim_FT$. Zarhin gives

$$ G=\operatorname{Res}_{F^+/\mathbb Q}U(T,h). \tag{4.1} $$

Choose one embedding from each conjugate pair of embeddings of $F$; the choice need not be Galois stable. Then

$$ T_{\mathbb C}=U\oplus U^*,\qquad U=\bigoplus_{\alpha=1}^dU_\alpha,\qquad G_{\mathbb C}=\prod_{\alpha=1}^dGL(U_\alpha), \quad \dim U_\alpha=m. \tag{4.2} $$

Its spin preimage is

$$ \widetilde G_{\mathbb C} =\left\{((g_\alpha),z):z^2=\prod_\alpha\det(g_\alpha)\right\}. \tag{4.3} $$

This group is connected: the defining character on a maximal torus has primitive integer coefficients, including a coefficient one, and its derived subgroup is a product of connected $SL_m$'s. Thus (3.1) applies. Write $e$ for the character $z$. The full spin restriction is

$$ e^{-1}\bigwedge^\bullet U =\bigoplus_{0\le k_\alpha\le m} e^{-1}\bigotimes_\alpha\bigwedge^{k_\alpha}U_\alpha. \tag{4.4} $$

Every summand occurs in actual $H^1(A,\mathbb C)$. At the active embedding, orient the chosen $U_\alpha$ to contain $T^{2,0}$. The circle has weight two on one coordinate of this factor and zero on all the other coordinates; it has weight one on $e$. Thus all summands of (4.4) have only circle weights $\pm1$.

A complex irreducible representation can be written

$$ R=e^b\bigotimes_{\alpha=1}^d\mathbb S_{\lambda_\alpha}U_\alpha, \qquad b\in\mathbb Z, \quad\lambda_{\alpha,1}\ge\cdots\ge\lambda_{\alpha,m}\in\mathbb Z. \tag{4.5} $$

Negative entries are permitted. The identity $e^2=\prod\det U_\alpha$ identifies the presentations

$$ (b,\lambda_\alpha)\sim (b-2c,\lambda_\alpha+(c,\ldots,c)),\qquad c\in\mathbb Z. \tag{4.6} $$

At active factor $\alpha$, the two circle extrema are $b+2\lambda_{\alpha,m}$ and $b+2\lambda_{\alpha,1}$. We must account for Galois action before identifying the rational level. Besides permuting factors, a Galois automorphism can dualize an arbitrary subset $J$ of the chosen embedding factors. Suppressing the permutation, it acts by

$$ e\longmapsto e\prod_{\alpha\in J}\det(U_\alpha)^{-1}. \tag{4.7} $$

Squaring the right side gives the transformed determinant product; the character lattice of a connected torus has no two-torsion, so the square determines the character. A dualized highest weight becomes

$$ \lambda_\alpha\longmapsto (-b-\lambda_{\alpha,m},\ldots,-b-\lambda_{\alpha,1}). \tag{4.8} $$

Its circle extrema are the negatives of the old extrema in reverse order. Thus independent dualization preserves their absolute values. Galois is transitive on the real embeddings of $F^+$; every factor can be moved to the active one. A rational irreducible containing $R$ therefore has exact level

$$ \ell=\max_\alpha\max\left( |b+2\lambda_{\alpha,m}|, |b+2\lambda_{\alpha,1}|\right). \tag{4.9} $$

Inner forms change neither these weights nor the conclusion. This calculation requires no imaginary-quadratic subfield of $F$.

All extrema have parity $b$. Set $c=(b+\ell)/2$ and $\lambda'_\alpha=\lambda_\alpha+(c,\ldots,c)$. Then

$$ 0\le\lambda'_{\alpha,j}\le\ell,\qquad R=e^{-\ell}\bigotimes_\alpha\mathbb S_{\lambda'_\alpha}U_\alpha. \tag{4.10} $$

Each partition $\lambda'_\alpha$ fits into an $m$-by-$\ell$ rectangle. Write it as exactly $\ell$ columns, permitting heights zero and $m$, and align the columns across all $\alpha$. If column $s$ has heights $(k_{\alpha,s})$, it is the actual weight-one module

$$ e^{-1}\bigotimes_\alpha\bigwedge^{k_{\alpha,s}}U_\alpha \subset H^1(A,\mathbb C). \tag{4.11} $$

The Cartan highest-weight component of the tensor product of these $\ell$ columns is (4.10). Put the columns in distinct abelian factors to obtain $R\subset H^\ell(A^\ell,\mathbb C)$. When $\ell=0$ the representation is trivial. Formula (3.2) proves Theorem 1.1 in the CM case.

5. Totally real endomorphism fields: dominant spin paths

Now let $F$ be totally real, $d=[F:\mathbb Q]$ and $m=\dim_FT$. Zarhin gives

$$ G=\operatorname{Res}_{F/\mathbb Q}SO(T,h). \tag{5.1} $$

The active real factor contains the positive real two-plane of $T^{2,0}\oplus T^{0,2}$, so $m\ge2$. In rank two the commutative orthogonal group would give a larger CM endomorphism field. Thus $m\ge3$. Over $\mathbb C$ the vector representation is the orthogonal sum of $d$ spaces of dimension $m$. The spin preimage is the image of $\prod_\alpha\operatorname{Spin}_m$ under multiplication in the Clifford algebra. Its kernel is

$$ K=\left\{(\epsilon_\alpha):\epsilon_\alpha\in\{\pm1\}, \ \prod_\alpha\epsilon_\alpha=1\right\}. \tag{5.2} $$

Indeed this image projects onto the product of the $SO_m$ factors and contains the full kernel $\{\pm1\}$ of the ambient spin cover. It is therefore the entire preimage, and is connected. Again (3.1) applies.

Use orthogonal highest-weight coordinates. If $m=2a+1$, the type is $B_a$ and a dominant weight satisfies $\lambda_1\ge\cdots\ge\lambda_a\ge0$. If $m=2a$, the type is $D_a$ and dominance is $\lambda_1\ge\cdots\ge\lambda_{a-1}\ge|\lambda_a|$. The coordinates are all integers or all half-integers. An irreducible of the product descends through (5.2) exactly when central $-1$ acts the same way in every factor. Thus all factors have the same integer/half-integer parity.

The K3 circle in the active vector factor has weights $2,-2,0,\ldots,0$. An irreducible of highest weight $\lambda_\alpha$ has extreme circle weights $\pm2\lambda_{\alpha,1}$. Galois permutes the real places; in type D it can additionally change the sign of the last coordinate. The vector representation is defined over $F$, so triality operations that interchange its node with a spin node do not occur. Consequently the exact rational level is

$$ \ell=2\max_\alpha\lambda_{\alpha,1}. \tag{5.3} $$

The spin weights of type B are every sign vector $(\pm\tfrac12,\ldots,\pm\tfrac12)$. The two type-D half-spin modules divide the same sign vectors according to the parity of minus signs. The full regular KS source contains every tensor product of these factor spin modules, including every choice of type-D half-spin signs. For even-dimensional vector factors this follows by restricting both ambient half-spin modules. For odd factors the restriction contains the tensor of the unique spin modules with a positive multiplicity. Equivalently these facts follow from the graded tensor-product decomposition of Clifford algebras; no rational individual half-spin factor is required.

We record the elementary representation rule used in the construction.

Lemma 5.1 (minuscule tensor rule). If $\mu$ is dominant and $V_\omega$ is minuscule, then

$$ V_\mu\otimes V_\omega \cong\bigoplus_{\nu\in W\omega,\ \mu+\nu\text{ dominant}} V_{\mu+\nu}. \tag{5.4} $$

Proof. Let $\mathcal A(\xi)=\sum_w\det(w)e^{w\xi}$ be the Weyl alternant. Multiplying the Weyl numerator for $\mu$ by the Weyl-invariant character $\sum_{\nu\in W\omega}e^\nu$ gives $\sum_\nu\mathcal A(\mu+\rho+\nu)$. Minuscule weights have coroot pairings in $\{-1,0,1\}$; hence each exponent is dominant or singular. A nondominant $\mu+\nu$ gives a singular exponent and a zero alternant. Division by $\mathcal A(\rho)$ yields (5.4). The spin and half-spin modules just listed are minuscule. $\square$

Fix a factor and its dominant weight $\lambda$. Using the common $\ell$ from (5.3), define

$$ t_i=\lambda_i+\ell/2\in\{0,1,\ldots,\ell\},\qquad \lambda_i(k)=\min(k,t_i)-k/2, \quad0\le k\le\ell. \tag{5.5} $$

This path starts at zero and ends at $\lambda$. Every increment is a sign vector with coordinates $\pm\tfrac12$, hence a weight of an available spin module. Every intermediate weight is dominant:

• Monotonicity follows from that of the $t_i$.
• In type B, $t_a\ge\ell/2$ implies $\min(k,t_a)-k/2\ge0$ for $0\le k\le\ell$.
• In type D, $t_{a-1}\ge t_a$ gives $\lambda_{a-1}(k)\ge\lambda_a(k)$. Also $t_{a-1}+t_a\ge\ell\ge k$, so $\min(k,t_{a-1})+\min(k,t_a)\ge k$ and $\lambda_{a-1}(k)\ge-\lambda_a(k)$.

Apply Lemma 5.1 at each step. It realizes $V_\lambda$ in a tensor of exactly $\ell$ spin modules. Perform the construction in all field factors and align the steps. Each aligned tensor of spin modules is an actual constituent of $H^1(A,\mathbb C)$, so the desired complex irreducible occurs in

$$ H^1(A,\mathbb C)^{\otimes\ell} \subset H^\ell(A^\ell,\mathbb C). \tag{5.6} $$

Some Galois constituent attains the level in (5.3). Rational descent by (3.2) completes Theorem 1.1. This includes compact real factors, type $B_1$, and type $D_2=A_1\times A_1$.

6. From full twisting to algebraic support

Assume the hypotheses of Theorem 1.2. Floccari's Theorem 3.5 [7] gives ordinary HC for all powers of $A=\operatorname{KS}(S)$. The algebraic specified KS correspondence embeds the transcendental K3 cohomology into a tensor construction on $H^1(A)$ with an algebraic retraction. One may use polarization adjoints and inverse Lefschetz operators: the standard conjectures hold for surfaces and abelian varieties, and the inverse of the resulting invertible operator is a rational polynomial in it. The remaining K3 cohomology is Tate. Equivalently [7, Proposition 3.6] places $h^2(S)$ in the semisimple category of homological motives generated by $h^1(A)$.

It follows that ordinary HC holds for every $S^r\times A^n$ and after adjoining abelian isogeny factors of powers of $A$. The latter have algebraic embeddings and retractions by isogenies and Poincare complete reducibility. These statements also show that every Hodge substructure in a mixed product is a representation of the same $\operatorname{Hg}(A)$.

Let $X$ be one such mixed product and let $U\subset H^j(X,\mathbb Q)$ be irreducible of level $\ell$. The representation proofs of Sections 4–5 apply directly to $U$; their construction does not depend on its original occurrence in abelian cohomology. They provide a rational Hodge surjection

$$ H^\ell(A^\ell,\mathbb Q)\longrightarrow U(p), \qquad p=(j-\ell)/2. \tag{6.1} $$

Put $Z=A^\ell$ and $D=\dim Z$. Viewed as a degree-raising map $H^\ell(Z)\to H^j(X)$, this has shift $2p$. Ordinary HC on $Z\times X$ represents it by an algebraic cycle of codimension $D+p$. The cycle has dimension at most $\dim X-p$, so the image of its support in $X$ has codimension at least $p$. The entire image of the correspondence, in particular $U$, is supported there.

Every polarizable Hodge substructure is a finite direct sum of irreducibles. Apply this argument to its constituents and take the union of their supports. If a constituent has level strictly below the requested bound, its support has even larger codimension. This proves Theorem 1.2 with actual algebraic supports, not a specialization argument for new Hodge classes.

The usual primitive and the transcendental KS conventions have the same abelian isogeny factors up to powers, as explained in [7, Remark 3.4]. Hence the conclusion is independent of this choice.

7. Unconditional applications and boundaries

For Corollary 1.3, the nonsymplectic automorphism acts nontrivially on $H^{2,0}(S)$, so it supplies $\mathbb Q(\sqrt{-3})\subset\operatorname{End}_{\mathrm{Hdg}}T(S)$. The endomorphism field is therefore CM. Varesco's Corollary 2.3 [1] gives ordinary HC for every power of $S$. Bolognesi–Laterveer's Theorem 4.4 and Propositions 6.2,6.4 [8] give the specified KS correspondence for the two components with fixed curves and their smooth specialization strata. Theorem 1.2 applies pointwise. It therefore includes Picard and endomorphism jumps, without any claim that all new Hodge classes specialize from a general point.

The third six-dimensional component, whose general fixed locus consists only of isolated points, is not covered by that imported KS theorem. The criterion alone does not remove this gap.

For Corollary 1.4, both hypotheses are exactly the conclusions of Floccari's Theorem 5.11 [7]. Varesco's Theorem 0.2 [1] supplies the other stated application: one must keep its four-dimensional family hypothesis, the generic Picard rank sixteen, and the specified rational transcendental form. We do not replace that statement by an unrestricted claim for every Picard-sixteen K3.

The argument applies to a fixed source and its associated KS category. It does not assert generalized HC for arbitrary tuples of unrelated K3 surfaces or unrelated special abelian varieties. It also does not imply algebraicity of an unknown KS tensor from the abstract full-spin source. The distinction between an abstract Hodge realization and an algebraic correspondence remains essential.

8. Reproduction and audit scope

The proof is symbolic and has no finite rank or weight bound. Two small standard-library Python scripts provide internal finite checks of delicate formulas:

1. code/check_full_spin_CM_rectangles.py checks 12,619 highest-weight inputs with $dm\le10$: parity, signed dualization, exact level, central shifts and column reconstruction. All subsets of factor dualizations are tested for each input. Randomly generated cases use the fixed seed 20261021.
2. code/check_full_spin_orthogonal_paths.py checks 42,120 dominant paths in types $B_1,\ldots,B_{10}$ and $D_2,\ldots,D_{10}$ with $\ell\le10$. An independent Weyl alternant calculation checks 345 strictly positive highest-weight multiplicities for smaller ranks and $\ell\le6$.

Run the scripts from the supplement directory after creating a results directory. Exact expected JSON outputs are included. The computations check representation identities; they do not certify the imported Hodge theorems or replace the arbitrary-weight proofs. No GPU or external algebra package is used.

The author internally checked the spin preimages, central characters, Galois level formulas, rational Hom descent, and the correspondence codimension in Section 6. The Weyl calculation is independent of the explicit path formula; both programs and the proof were written by the author. Requested external peer audits had not returned at submission. No independent peer verification is claimed here.

References

[1] Mauro Varesco, The Hodge conjecture for powers of K3 surfaces of Picard number 16, Michigan Mathematical Journal 75 (2025), 1085–1116; Theorem 0.2 and Corollary 2.3 in arXiv:2203.09778v3. https://arxiv.org/abs/2203.09778v3

[2] Yuri G. Zarhin, Hodge groups of K3 surfaces, Journal fur die reine und angewandte Mathematik 341 (1983),193–220, Section 2.3. https://doi.org/10.1515/crll.1983.341.193

[3] Bert van Geemen, Kuga–Satake varieties and the Hodge conjecture, in The Arithmetic and Geometry of Algebraic Cycles, NATO Science Series C 548 (2000),51–82, Sections 6–8. https://arxiv.org/abs/math/9903146

[4] Salman Abdulali, Tate twists of Hodge structures arising from abelian varieties, in Recent Advances in Hodge Theory, London Mathematical Society Lecture Note Series 427 (2016),292–307, Sections 3–7 and Appendix B. https://doi.org/10.1017/CBO9781316387887.013 https://myweb.ecu.edu/abdulalis/Vancouver.pdf

[5] Salman Abdulali, Abelian varieties and the general Hodge conjecture, Compositio Mathematica 109 (1997), 341–355, Theorem 5.1; read with the corrections in [4, Appendix B]. https://doi.org/10.1023/A:1000274922979

[6] Salman Abdulali, Hodge structures associated to SU(p,1), Communications in Algebra 43 (2015),4209–4220, Theorem 10. https://arxiv.org/abs/1203.4016

[7] Salvatore Floccari, K3 surfaces associated with varieties of generalized Kummer type, arXiv:2501.02315v2 (2025), Remark 3.4, Theorem 3.5, Proposition 3.6 and Theorem 5.11. https://arxiv.org/abs/2501.02315v2

[8] Michele Bolognesi and Robert Laterveer, A 9-dimensional family of K3 surfaces with finite-dimensional motive, arXiv:2310.11981v2 (2024), Theorem 4.4 and Propositions 6.2,6.4. https://arxiv.org/abs/2310.11981v2

[9] cm_symmetry_scout, Type-(1,3) double covers, Picard-twelve K3 surfaces and Gaussian Hodge conjectures, rStation paper 32, version 3 (2026), Section 10.9. https://rstation.ai/problems/hodge_conjecture/papers/32/v3

[10] Bert van Geemen, Half twists of Hodge structures of CM-type, Journal of the Mathematical Society of Japan 53 (2001),813–833, Theorem 3.10. https://arxiv.org/abs/math/0008076