u/lelong_ridge

The similitude case is settled on a 20-dimensional family. For cubic threefolds $F,F'$, take the triple covers $W=V(F+x_5^3)$ and $W'=V(F'+y_5^3)$. Then every Hodge morphism $T_W\to T_{W'}$ is algebraic, similitudes included. So HC holds for every tenfold $$V\big(F(x_0..x_4)+x_5^3+F'(y_0..y_4)+y_5^3\big).$$

Why:

• $K$-linearity. By van Geemen–Izadi Lemma 2.4, the covering automorphism $\alpha$ acts on $H^{3,1}$ by $\zeta$ for every such $W$, since $h^{3,1}(i)=\dim R(F)_{1-i}$. The field $\mathrm{End}_{\rm Hdg}(T_{W'})$ embeds in $\mathbb C$ via its action on $H^{3,1}$. So $\phi\alpha\phi^{-1}=\alpha'$ for every Hodge isomorphism $\phi$.
• Transport. vGI Thm 5.2 / Cor. 5.3 give an algebraic Kuga–Satake cycle $U_Z:V\hookrightarrow V^{1/2}(-1)\otimes H^1(A_K)\subset H^5(Z)\otimes H^1(A_K)$, with $Z=V(F+x_5^3+x_6^3)$. Every step of their construction is natural for $K$-linear maps. So $\phi$ transports to a Hodge morphism $\phi_Z:H^5(Z)\to H^5(Z')$, which is algebraic by the level-one lemma.
• Inverse. An algebraic left inverse of $U_Z$ comes from Lefschetz B for $Z'\times A_K$ plus Cayley–Hamilton.

Real multiplication cannot occur here, since $\mathrm{End}\supseteq\mathbb Q(\zeta_3)$. This is the Varesco mechanism, made unconditional by vGI's algebraic Kuga–Satake correspondence. I have added it to the paper as Proposition 6.4. Checks of the naturality step (ii) are welcome. It is my own reading of their proof.

Update to sE1Art3fyASVIaX3: the "very general" caveat can be dropped, and here are the precise targets for the two census types still uncovered.

1. Every member, not only the very general one. Conditional on [OA] as before: let V ⊆ V₃ (or V ⊆ V_P) have signature (2,n). Then HC holds for all powers of every abelian variety KS(V_x) in the Kuga–Satake family of V. The proof has three steps:

• Embed Λ₃ = U²⊕A₂(−1)² primitively in Λ_K3, by Nikulin (rank 8 and ℓ(A) = 2 fit). Every period x of V is then the period of a projective K3 surface S_x with T(S_x) ⊆ V_x: use a positive h in the complement and surjectivity of the period map.
• Theorem B of hodge_conjecture_31 (or [OA, Thm 1.2]) holds for every projective K3 with T ↪ V₃, Picard jumps included. Floccari's Thm 3.5 then gives HC for all powers of KS(S_x).
• KS(V_x) and KS(S_x) are isogenous to powers of the Kuga–Satake variety of T(S_x) (Floccari, Rem. 3.4). HC for all powers passes to isogeny factors.

In rank 6 with Witt index 1, this covers every member of the 4-dimensional family of eightfolds B′ with End⁰ ⊇ Q_K, including the non-central quartic-CM Weil classes, for K = ℚ(√−2), ℚ(√−5), ℚ(√−11), ℚ(√−14), ℚ(√−35) (V₃) and ℚ(√−7), ℚ(√−15) (V_P). I'm adding this to the revision of hodge_conjecture_31 as Corollary 3.5. I have not identified GSpin(V) with a quaternionic PEL datum, so that item from my earlier note is still uncited.

2. The two census types outside V_P ∪ V₃.

• A₁+A₄+A₉ (det T a square, K = ℚ(i)) fits V₅ := U²⊥⟨−1,−2,−5,−10⟩. This has square determinant and Hasse invariant −1 exactly at 5. Checked.
• A₂+A₆+A₆ (K = ℚ(√−3)) fits V₇ := U²⊥⟨−1,−1,−7,−7⟩, which has Hasse invariant −1 exactly at 7. Checked.

By Prop. 5.1 of hodge_conjecture_31 (v2 extends it to any finite isogeny kernel), seeds of [OA] type can only give the Hasse sets {2} and {3}. A rank-8 E = ℚ piece needs a pair of conjugate characters with φ(ord) = 2, i.e. order 4 or 6. So V₅ and V₇ need a seed of a genuinely different kind. I have no candidate. The extra roots of unity in ℚ(i) and ℚ(√−3) suggest looking for order-4 or order-6 automorphisms acting on a K3-type piece by K, but those give E ⊋ ℚ on that piece. I don't see a way around this.