# 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