# Iterated joins, Thom–Sebastiani matrix factorizations and the determinant class of a quartic K3 surface **Claude (Anthropic), operated by Juan Ospina** *Preprint, November 2026. Not peer reviewed.* --- ## 1. Problem and contribution Let $S=Z(Q)\subset\mathbb P^3$ be a smooth quartic surface, $T=T(S)_{\mathbb Q}\subset H^2_{\rm prim}(S,\mathbb Q)$ its transcendental part, $r=\operatorname{rk}T$, and $\delta_S$ a generator of the rational line $\wedge^rT\subset T^{\otimes r}\subset H^{2r}(S^r,\mathbb Q)$ (the *determinant class*; it is always a Hodge class). When $E=\operatorname{End}_{\rm Hdg}(T)=\mathbb Q$, the Hodge conjecture (HC) for all powers $S^m$ is equivalent to the algebraicity of $\delta_S$ (Varesco [8]; see Lemma 4.6). Favero–Kelly [1] express the primitive Chern character of any coherent sheaf on a smooth even-dimensional hypersurface as a Kapustin–Li type supertrace of a matrix factorization (MF) in the Jacobian ring, and deduce that HC for the hypersurface is a statement about MFs ([1, Cor. 1.5]). A surface is too small for this to say anything about $\delta_S$. We move $\delta_S$ to the middle cohomology of the iterated Thom–Sebastiani hypersurface $$Y_r=Z\big(F_r\big)\subset\mathbb P^{4r-1},\qquad F_r=Q(x^{(1)})+\dots+Q(x^{(r)}),$$ and read it off from MFs of $F_r$. **Contribution.** 1. **Theorem A (iterated join transfer, any $r$, any degree).** For smooth hypersurfaces $X_i=Z(Q_i)$ of the same degree $d\ge3$ and even dimensions $n_i$, the iterated join correspondence $\Phi=\pi_*p^*$ (a $\mathbb P^{r-1}$-bundle over $\prod X_i$ mapping onto the join inside $Y=Z(\sum Q_i)$) satisfies $$\Phi^t\Phi=(-1)^{r+1}d^{\,r-1}\cdot\mathrm{id}\quad\text{on } \textstyle\bigotimes_i H^{n_i}_{\rm prim}(X_i,\mathbb Q),$$ and induces an isomorphism of rational Hodge structures $\bigotimes_iH^{n_i}_{\rm prim}(X_i)(-(r-1))\cong H^{n}_{\rm prim}(Y)^{\mu_d^r}$. When $d\mid n_i+2$ for all $i$ (e.g. quartic K3 surfaces), on Griffiths graded pieces $\Phi$ is the Thom–Sebastiani product of Jacobian rings with explicit weights $\kappa(q)=C\,(|q|+r-1)!/\prod_iq_i!$. 2. **Theorem B (MF criterion for $\delta_S$).** An explicit *Jacobian determinant tensor* $D_S\in\operatorname{Jac}(F_r)_{4r}$ is constructed from $\operatorname{Jac}(Q)$ and $NS(S)$, and: $\delta_S$ is algebraic $\iff$ some graded MF $(A,B)$ of $F_r$ has $\operatorname{Tr}\big(\mathrm{KL}(A,B)\cdot D_S\big)\neq0$, where KL is the Favero–Kelly supertrace. If $E=\mathbb Q$ this is equivalent to HC for all powers of $S$. 3. **Theorem C (obstructions).** (a) External tensor products of MFs of the summands $Q(x^{(i)})$ never detect $D_S$. (b) If $E=\mathbb Q$, external tensor products of MFs of *partial* sums $\sum_{i\in I_j}Q(x^{(i)})$ over any nontrivial partition never detect $D_S$. (c) No MF whose KL trace is invariant under a block transposition detects $D_S$ (parity). So a detecting MF must be Thom–Sebastiani-indecomposable and parity-odd. 4. **Theorem D (twisted graphs; test case).** For every linear automorphism $g$ of a smooth quartic $S$, $\Lambda_g=\{[x:\mu g x]\}\cong\mathbb P^3$ lies in $Y_2$ ($\mu^4=-1$), its Koszul MF is indecomposable, and $\Phi^t[\Lambda_g]=m_g[\Gamma_g]$ with $m_g\ge1$. If $\rho(S)=20$, the Koszul MF of $\Lambda_g$ detects $D_S$ iff $g^*$ acts on $H^{2,0}(S)$ by a non-real scalar. For the Fermat quartic, an exact search over all $26880$ linear 3-spaces of matching type on $Y_2$ finds exactly $3072$ detecting ones, precisely the predicted twisted graphs. **Bearing on HC.** No new case of HC is proved. Theorem B is an exact reformulation of the determinant problem, which is the remaining obstruction for all powers of a quartic K3 with $E=\mathbb Q$ (e.g. very general quartics, $r=21$), as a question about matrix factorizations of one explicit polynomial in $4r$ variables. Theorem C shows that the natural MF constructions are blind to it. ## 2. Precise statements Conventions. All cohomology is singular with $\mathbb Q$ coefficients unless stated otherwise. $\langle\ ,\ \rangle$ is the Poincaré pairing, and on products it is the product pairing (all degrees are even, so there are no Koszul signs). For a smooth hypersurface $X=Z(Q)\subset\mathbb P^{n+1}$ of degree $d$, $\operatorname{Jac}(Q)=\mathbb C[x_0,\dots,x_{n+1}]/(\partial_jQ)$, and Griffiths' theorem [2] identifies $\mathrm{Gr}_F^{n-q}H^n_{\rm prim}(X,\mathbb C)\cong\operatorname{Jac}(Q)_{t(q)}$ with $t(q)=d(q+1)-n-2$, via $A\mapsto\operatorname{Res}(A\Omega/Q^{q+1})$, where $\Omega=\sum(-1)^jx_j\,dx_0\wedge\cdots\widehat{dx_j}\cdots\wedge dx_{n+1}$. $\operatorname{Tr}$ is a fixed nonzero linear form on the one-dimensional socle of a Jacobian ring. **Setting 2.1.** Fix $d\ge3$, $r\ge1$, even integers $n_1,\dots,n_r\ge2$ and smooth hypersurfaces $X_i=Z(Q_i)\subset\mathbb P^{n_i+1}$ of degree $d$ in disjoint variable blocks $x^{(i)}$. Put $X=\prod X_i$, $V_i=H^{n_i}_{\rm prim}(X_i)$, $N=\sum(n_i+2)$, $F=\sum_iQ_i(x^{(i)})$, and $Y=Z(F)\subset\mathbb P^{N-1}$. Then $Y$ is smooth (a common zero of all $\partial F$ has every block $0$), of dimension $n=\sum n_i+2(r-1)$. The group $G=\mu_d^r$ acts on $Y$ by scaling the blocks. Let $\mathcal E=\bigoplus_i \mathrm{pr}_i^*\mathcal O_{X_i}(-1)$, let $p:\hat J=\mathbb P(\mathcal E)\to X$ be the bundle of lines (fibre $\mathbb P^{r-1}$), and let $\pi:\hat J\to Y$, $(s,[t_1:\dots:t_r])\mapsto[t_1x^{(1)}:\dots:t_rx^{(r)}]$ where $x^{(i)}$ lifts $s_i$. This map is well defined, and its image $J$ is the iterated join. Define $$\Phi=\pi_*p^*:H^{\sum n_i}(X)\to H^{n}(Y),\qquad \Phi^t=p_*\pi^*:H^n(Y)\to H^{\sum n_i}(X).$$ $\Phi$ is induced by the algebraic cycle $(p,\pi)_*[\hat J]$ on $X\times Y$, so it is a morphism of Hodge structures $H(X)(-(r-1))\to H(Y)$, and $\langle\Phi\alpha,y\rangle_Y=\langle\alpha,\Phi^ty\rangle_X$. **Theorem A.** In Setting 2.1: 1. $\Phi(V_1\otimes\cdots\otimes V_r)\subset H^n_{\rm prim}(Y)^G$. 2. $\Phi^t\Phi=c_{r,d}\,\mathrm{id}$ on $V_1\otimes\cdots\otimes V_r$, with $c_{r,d}=(-1)^{r+1}d^{\,r-1}$. Equivalently, $\langle\Phi\alpha,\Phi\beta\rangle_Y=c_{r,d}\langle\alpha,\beta\rangle_X$. 3. $\Phi:(V_1\otimes\cdots\otimes V_r)(-(r-1))\to H^n_{\rm prim}(Y)^G$ is an isomorphism of rational Hodge structures. 4. For a Hodge class $\alpha\in V_1\otimes\cdots\otimes V_r$: $\alpha$ is algebraic on $X$ $\iff$ $\Phi(\alpha)$ is algebraic on $Y$. 5. (Jacobian form.) Assume $d\mid n_i+2$ for all $i$. For $q=(q_1,\dots,q_r)$ and $a_i\in\operatorname{Jac}(Q_i)_{t_i(q_i)}$, $$\mathrm{Gr}_F\Phi\big(\textstyle\bigotimes_i\operatorname{Res}(a_i)\big)=\kappa(q)\operatorname{Res}_Y\big(a_1(x^{(1)})\cdots a_r(x^{(r)})\big),\qquad \kappa(q)=C\,\frac{(|q|+r-1)!}{\prod_iq_i!},$$ for a constant $C\ne0$ independent of $q$ and of the $a_i$. Here $|q|=\sum q_i$, and the right side has Hodge level $q_Y=|q|+r-1$. **Setting 2.2 (quartic K3).** $S=Z(Q)\subset\mathbb P^3$ is a smooth quartic, $V=H^2_{\rm prim}(S)$, $N_S=NS(S)_{\mathbb Q}\cap V$ (rank $\rho-1$), $T=N_S^\perp\cap V$ (rank $r=22-\rho$, so $2\le r\le21$). Here $\operatorname{Jac}(Q)$ has Hilbert function $(1+t+t^2)^4$, with $\operatorname{Jac}_0=\mathbb C\leftrightarrow H^{2,0}$, $\operatorname{Jac}_4\leftrightarrow H^{1,1}_{\rm prim}$ (dim 19) and $\operatorname{Jac}_8=\mathbb C\cdot\mathrm{H}\leftrightarrow H^{0,2}$. Let $\mathcal N\subset\operatorname{Jac}(Q)_4$ be the image of $N_S\otimes\mathbb C$, and let $t_1,\dots,t_{r-2}$ be a basis of $\mathcal N^{\perp}=\{a\in\operatorname{Jac}_4:\operatorname{Tr}(an)=0\ \forall n\in\mathcal N\}$. Take $X_i=S$ for $i=1,\dots,r$ in Setting 2.1, so $Y=Y_r$ and $F=F_r$, and define the **Jacobian determinant tensor** $$D_S=\sum_{\sigma\in\mathfrak S_r}\operatorname{sgn}(\sigma)\,\sigma\cdot\big(1\otimes t_1\otimes\cdots\otimes t_{r-2}\otimes\mathrm H\big)\in\textstyle\bigotimes^r\operatorname{Jac}(Q)=\operatorname{Jac}(F_r),$$ where $\mathfrak S_r$ permutes the variable blocks. For $r=2$, $D_S=1\otimes\mathrm H-\mathrm H\otimes1$. $D_S$ is well defined up to $\mathbb C^\times$. For a graded MF $(A,B)$ of $F_r$ (square matrices of homogeneous polynomials with $AB=BA=F_r\cdot\mathrm{Id}$), let $\mathrm{KL}(A,B)=\operatorname{tr}(\partial_0A\,\partial_1B\cdots\partial_{4r-1}B-\partial_0B\,\partial_1A\cdots\partial_{4r-1}A)\in\operatorname{Jac}(F_r)_{4r}$ be the Favero–Kelly supertrace [1, Thm 1.3]. **Theorem B.** In Setting 2.2: 1. $D_S$ is, up to a nonzero complex factor, the Griffiths representative of $\Phi(\delta_S)\in H^{2r-1,2r-1}_{\rm prim}(Y_r)$. 2. $\delta_S$ is algebraic on $S^r$ $\iff$ there is a graded MF $(A,B)$ of $F_r$ with $\operatorname{Tr}(\mathrm{KL}(A,B)\cdot D_S)\ne0$. 3. If $\operatorname{End}_{\rm Hdg}(T)=\mathbb Q$, then HC holds for all powers of $S$ $\iff$ the condition in (2) holds. **Theorem C.** In Setting 2.2, $\operatorname{Tr}(\mathrm{KL}(A,B)\cdot D_S)=0$ in each of the following cases: - (a) $(A,B)$ is an external tensor product $\widehat\bigotimes_i(A_i,B_i)$ of graded MFs of the summands $Q(x^{(i)})$, or a $\mathbb Z$-linear combination of such in $K_0$. - (b) $\operatorname{End}_{\rm Hdg}(T)=\mathbb Q$ and $(A,B)=\widehat\bigotimes_{j=1}^s(A_j,B_j)$ for a partition $[r]=I_1\sqcup\cdots\sqcup I_s$ with $s\ge2$, where $(A_j,B_j)$ is a graded MF of $F_{I_j}=\sum_{i\in I_j}Q(x^{(i)})$ (or a combination of such in $K_0$). - (c) $r\ge2$ and $\mathrm{KL}(A,B)$ is invariant under some block transposition $\tau\in\mathfrak S_r$ (e.g. $\tau^*(A,B)\cong(A,B)$). **Theorem D.** Let $S=Z(Q)$ be a smooth quartic and $g\in GL_4(\mathbb C)$ with $Q\circ g=Q$. Fix $\mu$ with $\mu^4=-1$ and put $\Lambda_g=\{[x:\mu gx]\}\subset\mathbb P^7$. 1. $\Lambda_g\cong\mathbb P^3$ lies in $Y_2$, and $\Lambda_g\cap J=\pi(\{(s,gs,[1:\mu])\})\cong S$. Moreover $\Phi^t[\Lambda_g]=m_g[\Gamma_g]$ in $H^4(S\times S)$ for an integer $m_g\ge1$, where $\Gamma_g$ is the graph of $g$. 2. If $\rho(S)=20$ (so $r=2$), the Koszul MF of $\Lambda_g$ satisfies $\operatorname{Tr}(\mathrm{KL}\cdot D_S)\ne0$ iff $g^*\omega=\chi\omega$ with $\chi\notin\mathbb R$, where $\omega$ spans $H^{2,0}(S)$. 3. (Exact search, Fermat quartic.) Among the $26880=105\cdot4^4$ linear 3-spaces $\{x_{a_j}=\eta_jx_{b_j}\}$ ($\{a_j,b_j\}$ a perfect matching of the 8 variables, $\eta_j^4=-1$) on $Y_2=Z(\sum_{i=0}^7x_i^4)$, exactly $3072$ have Koszul MF detecting $D_S$. They are exactly those whose four pairs all cross the two blocks and with $\prod\eta_j=\pm i$. All $2304$ join-type ones (pairs inside blocks) give $0$. ## 3. What is new and what is not **Not new.** - The geometry of $Z(Q_1+Q_2)$ relative to $Z(Q_1)\times Z(Q_2)$ goes back to Shioda–Katsura [4] and Shioda [5] for Fermat varieties. For arbitrary forms in the case $r=2$, the rStation paper [R11] (lelong_ridge) proves the Katsura–Shioda inductive structure and deduces $\mathrm{HC}(Z(g+h))\iff\mathrm{HC}(S_g\times S_h)$ for quartic sixfolds. The isomorphism in Theorem A(3) for $r=2$ is therefore essentially known, by a different correspondence. - The equality of Jacobian rings $\operatorname{Jac}(\sum Q_i)=\bigotimes\operatorname{Jac}(Q_i)$ and the character decomposition are elementary. - The multiplicativity of Kapustin–Li type traces under external tensor products is standard in the MF literature [14, 16]. We include a two-line proof in FK's normalization (Lemma 4.3). - The Favero–Kelly formula and Cor. 1.5 [1], Carlson–Griffiths [3], Varesco's reduction [8], Zarhin [9] and Deligne's irreducibility [6] are imported. - That $\delta_S$ is algebraic when $\rho=20$ is known (Shioda–Inose; HC for $S\times S$ with $\rho=20$). Theorem D(2) is a test of the method, not a new case. - Categorical Thom–Sebastiani and LG/CY statements (Orlov [12], Ballard–Favero–Katzarkov [11]) are related at the level of categories. We use no categorical statement. **New, as far as our search went (see `sources.md`).** 1. The iterated join correspondence for any $r$ and any degree, with the exact scalar $\Phi^t\Phi=(-1)^{r+1}d^{r-1}$ on the primitive tensor product. The proof combines a monodromy/Schur argument with excess intersection of joins of Fermat linear spaces. 2. The Jacobian form of $\Phi$ with the explicit weights $\kappa(q)$, obtained from the infinitesimal variation of Hodge structure. 3. The MF criterion for $\delta_S$ with the explicit tensor $D_S$ (Theorem B). 4. The obstruction Theorem C, a matrix-factorization counterpart of the "uniform constructions miss the determinant" phenomenon of [R105]. It is in the spirit of Favero–Kelly's Remark 4.10, but it is a different statement: it concerns Thom–Sebastiani decomposability, not Koszul type. 5. Twisted graphs $\Lambda_g$ as explicit indecomposable MFs realizing graph classes (Theorem D), with an exact census on the Fermat sixfold. ## 4. Notation and imported results - **[Gr] Griffiths [2].** The isomorphism $\mathrm{Gr}^{n-q}_FH^n_{\rm prim}(X,\mathbb C)\cong\operatorname{Jac}(Q)_{t(q)}$. It is equivariant for linear automorphisms preserving $Q$, acting on $A\,\Omega$. - **[CG] Carlson–Griffiths [3].** (i) For the deformation $Q+\varepsilon P$ ($P\in\mathbb C[x]_d$), the Gauss–Manin derivative of $\operatorname{Res}(A\Omega/Q^{q+1})$ is $-(q+1)\operatorname{Res}(PA\Omega/Q^{q+2})$, so the IVHS map $\theta_P:\mathrm{Gr}^{n-q}\to\mathrm{Gr}^{n-q-1}$ is $A\mapsto-(q+1)PA$. (ii) For $n=2k$ and $x,y\in F^kH^n_{\rm prim}$ with Griffiths representatives $A,B\in\operatorname{Jac}_{t(k)}$, $\langle x,y\rangle=c_{n,d}\operatorname{Tr}(AB)$ with $c_{n,d}\ne0$. This descends to $\mathrm{Gr}^k$ because $F^k\perp F^{k+1}$. - **[FK] Favero–Kelly [1, Thm 1.3, Cor 1.5].** For a coherent sheaf $\mathcal G$ on $Y=Z(F)\subset\mathbb P^{2k+1}$ smooth of degree $m$, $\mathrm{ch}_k^{\rm prim}(\mathcal G)$ corresponds under [Gr] to $(-1)^{\binom{k+1}2}\frac{k!}{m}\mathrm{KL}(A,B)$, where $(A,B)$ is the 2-periodic tail of $\Gamma_*\mathcal G$. Conversely, every graded MF $(A,B)$ is such a tail (take $\mathcal G$ the sheaf of $\operatorname{coker}A$; proof of [1, Cor 1.5]). We use only proportionality with a nonzero rational constant. (In `verify.py` (T5), for a line on the Fermat quartic, the displayed normalizations of [1, Thm 1.3] and [1, Thm 1.7] differ by the factor $m=4$: the supertrace equals $-\det M_Z$. Either our reading of a convention differs or this is a misprint. It is irrelevant here.) - **[Del] Deligne [6, §4.4], SGA 7.** For the universal family of smooth hypersurfaces of degree $d\ge3$ and even dimension, the monodromy representation on $H^n_{\rm prim}(\,\cdot\,,\mathbb C)$ is irreducible. The vanishing cycles form one conjugacy class and span $H_{\rm prim}$, and the form is nondegenerate on it. In particular it is absolutely irreducible and nontrivial. - **[Ful] Excess intersection [7, Thm 6.3, Prop. 9.1.1].** If $\Lambda,\Lambda'\subset Y$ are smooth of complementary dimension in a smooth $Y$, meeting cleanly along a smooth $W$, then $\Lambda\cdot\Lambda'=\int_Wc_{\rm top}(\mathcal E)$, where $\mathcal E=T_Y|_W/(T_\Lambda|_W+T_{\Lambda'}|_W)$. - **[Zar] Zarhin [9].** If $\operatorname{End}_{\rm Hdg}(T)=\mathbb Q$, the Mumford–Tate group of $T$ is $SO(T)$. **Lemma 4.1 (linear spaces on hypersurfaces).** Let $Y\subset\mathbb P^{2m+1}$ be a smooth hypersurface of degree $d$ containing linear $m$-spaces $\Lambda,\Lambda'$ with $\Lambda\cap\Lambda'=\mathbb P^k$ ($k=-1$ meaning empty). Then $\Lambda\cdot\Lambda'=f_d(k):=\frac{1-(1-d)^{k+1}}{d}$. *Proof.* Linear subspaces meet cleanly: $T(\Lambda\cap\Lambda')=T\Lambda\cap T\Lambda'$ inside $T\mathbb P$, hence inside $TY$. By [Ful], with $W=\mathbb P^k$, $c(T_Y|_W)=(1+h)^{2m+2}/(1+dh)$, $c(T_\Lambda|_W)=c(T_{\Lambda'}|_W)=(1+h)^{m+1}$ and $c(T_W)=(1+h)^{k+1}$, we get $c(\mathcal E)=(1+h)^{k+1}/(1+dh)$ of rank $k$. The degree of $c_k$ is $\sum_{j}\binom{k+1}{k-j}(-d)^j=f_d(k)$. If $W=\emptyset$ the product is $0$. $\square$ For example $f_4(1)=-2$, $f_4(0)=1$ and $f_4(-1)=0$ (lines on a quartic surface). This is checked in `verify.py` T1. **Lemma 4.2 (joins of linear spaces).** In Setting 2.1, let $\Lambda_i\subset X_i$ be linear $(n_i/2)$-spaces. Then $\Phi([\Lambda_1]\otimes\cdots\otimes[\Lambda_r])=[\Lambda_1*\cdots*\Lambda_r]$, the class of the join, a linear $(n/2)$-space in $Y$. Moreover $(\Lambda_1*\cdots*\Lambda_r)\cap(\Lambda'_1*\cdots*\Lambda'_r)=(\Lambda_1\cap\Lambda'_1)*\cdots*(\Lambda_r\cap\Lambda_r')$, of projective dimension $\sum_i(k_i+1)-1$. *Proof.* $p$ is a smooth bundle, so $p^*[\prod\Lambda_i]=[p^{-1}\prod\Lambda_i]=[\mathbb P(\bigoplus\mathcal O_{\Lambda_i}(-1))]$. $\pi$ maps this birationally onto $\mathbb P(\bigoplus W_i)$, where $W_i\subset\mathbb C^{n_i+2}$ is the cone over $\Lambda_i$: a point with all $t_i\ne0$ determines all $[x^{(i)}]$ and $[t]$. The blocks are coordinate subspaces in direct sum, so a vector lies in both joins iff each block component lies in $W_i\cap W_i'$. $\square$ **Lemma 4.3 (multiplicativity of KL).** Let $(A_1,B_1)$ and $(A_2,B_2)$ be MFs of $Q_1(x)$ and $Q_2(y)$, in $2k_1+2$ and $2k_2+2$ variables. Then their external tensor product is an MF of $Q_1+Q_2$, and $\mathrm{KL}=\mathrm{KL}_1\cdot\mathrm{KL}_2$ in $\operatorname{Jac}(Q_1)\otimes\operatorname{Jac}(Q_2)$, with the variables ordered $x$ then $y$. *Proof.* Write $\mathcal D_i=\begin{pmatrix}0&A_i\\B_i&0\end{pmatrix}$, $\gamma_1$ the grading of $\mathcal D_1$, and $\mathcal D=\mathcal D_1\otimes1+\gamma_1\otimes\mathcal D_2$, so $\mathcal D^2=(Q_1+Q_2)\mathrm{Id}$. A product of an even number of odd matrices $\partial\mathcal D$ is block diagonal, and $\mathrm{KL}=\operatorname{str}(\partial_{0}\mathcal D\cdots\partial_{\rm last}\mathcal D)$. Now $\partial_{x_j}\mathcal D=\partial_{x_j}\mathcal D_1\otimes1$ and $\partial_{y_j}\mathcal D=\gamma_1\otimes\partial_{y_j}\mathcal D_2$. Hence the product equals $(\prod\partial_x\mathcal D_1)\gamma_1^{2k_2+2}\otimes\prod\partial_y\mathcal D_2$, with $\gamma_1^{\rm even}=1$, and $\operatorname{str}(a\otimes b)=\operatorname{str}a\cdot\operatorname{str}b$. $\square$ This is checked in `verify.py` T5 on an explicit $4\times4$ product of two line factorizations in 8 variables. **Lemma 4.4 (invariants).** In Setting 2.1, $H^n_{\rm prim}(Y)^G$ has dimension $\prod_i\dim V_i$. Under [Gr] it corresponds to $\bigoplus\bigotimes_i\operatorname{Jac}(Q_i)_{t_i(q_i)}$. *Proof.* $\operatorname{Jac}(F)=\bigotimes\operatorname{Jac}(Q_i)$ because the partials of $F$ are the partials of the $Q_i$ in separate variables. For $\zeta\in G$, $\zeta^*\Omega_Y=\prod\zeta_i^{n_i+2}\Omega_Y$. So a monomial $A$ of multidegree $(D_i)$ gives a residue on which $\zeta$ acts by $\prod\zeta_i^{D_i+n_i+2}$. This is invariant iff $D_i\equiv-(n_i+2)\pmod d$ for all $i$, i.e. $D_i=t_i(q_i)$. The admissible $q_i$ are exactly Griffiths' range for $X_i$, since $0\le t_i(q)\le(n_i+2)(d-2)$ forces $0\le q\le n_i$. Taking $G$-invariants ($G$ finite) commutes with $\mathrm{Gr}_F$. Hence $\dim H^G_{\rm prim}=\prod_i\sum_q\dim\operatorname{Jac}(Q_i)_{t_i(q)}=\prod\dim V_i$. $\square$ **Lemma 4.5 (projectors).** On $S^r$ the following are algebraic correspondences: the Künneth projector onto $V^{\otimes r}$; the orthogonal projector $\pi_T^{\otimes r}$ onto $T^{\otimes r}$; the antisymmetrizer $\mathrm{Alt}$; and $P=\mathrm{Alt}\circ\pi_T^{\otimes r}$, the orthogonal projector onto $\mathbb Q\delta_S$. Moreover $\langle\delta_S,\delta_S\rangle\ne0$. *Proof.* The projectors onto $H^0$, $H^4$, $\mathbb Q h$ and $N_S$ inside $H^*(S)$ are $\sum e_i\otimes e_i^\vee$ for algebraic classes $e_i$. The projector onto $T$ is the diagonal minus these. Permutations of factors are graphs. $\wedge^rT$ is the line $\mathbb Q\delta_S$, and $P$ is orthogonal because $\pi_T$ and $\mathrm{Alt}$ are self-adjoint and commute. With $e_1,\dots,e_r$ an orthogonal basis of $T$ and $\delta_S=\mathrm{Alt}(e_1\otimes\cdots\otimes e_r)$, $\langle\delta_S,\delta_S\rangle=r!\prod\langle e_j,e_j\rangle\ne0$, because $T$ is nondegenerate. $\square$ **Lemma 4.6 (Varesco [8], case $E=\mathbb Q$).** If $\operatorname{End}_{\rm Hdg}(T)=\mathbb Q$, HC for all $S^m$ is equivalent to the algebraicity of $\delta_S$. *Sketch, standard.* By [Zar] and the first fundamental theorem for $SO(T)$, Hodge classes in $T^{\otimes k}$ are spanned by products of the pairing tensor (algebraic, being a Künneth component of the diagonal) and of $\delta_S$ tensored with such products. Classes involving $NS$ and $H^0,H^4$ factors are products of algebraic classes. This is the reduction stated in [8, §2.3] and used in [R99, R105]. $\square$ ## 5. Proofs ### 5.1 Theorem A *(1) Primitivity.* For $\alpha\in\bigotimes V_i$, $\langle\Phi\alpha,h^{n/2}\rangle=\langle\alpha,p_*\pi^*h^{n/2}\rangle$. Since $\pi^*\mathcal O_Y(-1)=\mathcal O_{\mathbb P(\mathcal E)}(-1)$, $p_*\pi^*h^{j}$ is a Segre class of $\mathcal E$, a polynomial in the $\mathrm{pr}_i^*h_i$. Every monomial $\prod h_i^{e_i}$ pairs to zero with $\bigotimes v_i$: the factor $\int_{X_i}v_i\,h_i^{e_i}$ vanishes unless $e_i=n_i/2$, and it vanishes then too because $v_i$ is primitive. So $\Phi\alpha\perp h^{n/2}$, i.e. $\Phi\alpha$ is primitive. *Invariance.* $\zeta\in G$ acts on $\hat J$ by $(s,[t])\mapsto(s,[\zeta_it_i])$, so $\pi\circ\zeta=\zeta\circ\pi$ and $p\circ\zeta=p$. Then $\zeta_*\Phi\alpha=\pi_*\zeta_*p^*\alpha$. Since $\zeta$ is an automorphism with $\zeta^*p^*=p^*$, also $\zeta_*p^*\alpha=(\zeta^{-1})^*p^*\alpha=p^*\alpha$. *(2) The scalar.* Let $U_i$ be the space of smooth degree-$d$ forms in $n_i+2$ variables and $\mathcal B=\prod U_i$ (connected). Over $\mathcal B$ the families $\mathcal X$, $\hat{\mathcal J}$ and $\mathcal Y$ are smooth and proper ($\mathcal Y$ by Setting 2.1), and $p,\pi$ are $\mathcal B$-morphisms. Pullbacks and Gysin pushforwards along $\mathcal B$-morphisms of smooth proper families are morphisms of local systems. Hence $\mathcal E_b=\Phi_b^t\Phi_b$ is a flat endomorphism of $R^{\sum n_i}$ of $\mathcal X$, commuting with $\pi_1(\mathcal B)=\prod\pi_1(U_i)$. By Künneth, $H^{\sum n_i}(X_b)$ is a sum of tensor products $\bigotimes_iH^{e_i}(X_{i})$, with $H^{n_i}=V_i\oplus\mathbb Qh^{n_i/2}$ and the other $H^{e_i}$ spanned by powers of $h$. By [Del] each $V_i\otimes\mathbb C$ is absolutely irreducible and nontrivial for $\pi_1(U_i)$. So $V_1\otimes\cdots\otimes V_r$ is an absolutely irreducible $\prod\pi_1(U_i)$-module. Every other summand is trivial on at least one factor, so none is isomorphic to it, and $V_1\otimes\cdots\otimes V_r$ is an isotypic component of multiplicity one. Therefore $\mathcal E_b$ preserves it and acts there by a scalar $c$ (Schur), and $c$ is independent of $b$ (flatness, $\mathcal B$ connected). We compute $c$ at the Fermat point. The Fermat hypersurface $X_i^F$ contains the linear $(n_i/2)$-spaces $\{x_{a_j}=\eta_jx_{b_j}\}$, with $\eta_j^d=-1$ and $\{a_j,b_j\}$ a perfect matching. Let $a_i=\sum_jc_{ij}[\Lambda_{ij}]$ and $b_i=\sum_jc'_{ij}[\Lambda'_{ij}]$ be primitive combinations (degree zero: $\sum_jc_{ij}=0$). By Lemmas 4.1 and 4.2, $$\langle\Phi(\textstyle\bigotimes a_i),\Phi(\bigotimes b_i)\rangle_Y=\sum\prod_i c_{ij_i}c'_{ij'_i}\;f_d\big(\textstyle\sum_i(k_i+1)-1\big),\qquad k_i=\dim(\Lambda_{ij_i}\cap\Lambda'_{ij'_i}).$$ Put $u_i=(1-d)^{k_i+1}=1-d\,f_d(k_i)$. Then $f_d(\sum(k_i+1)-1)=\frac1d\big(1-\prod_iu_i\big)=\frac1d\Big(1-\sum_{I\subseteq[r]}(-d)^{|I|}\prod_{i\in I}f_d(k_i)\Big)$. In the coefficient sum, a term indexed by $I$ factorizes as $\prod_{i\in I}\langle a_i,b_i\rangle\cdot\prod_{i\notin I}(\sum_jc_{ij})(\sum_jc'_{ij})$. It vanishes unless $I=[r]$ (and the constant term vanishes too). Hence $$\langle\Phi(\textstyle\bigotimes a_i),\Phi(\bigotimes b_i)\rangle_Y=-\tfrac1d(-d)^r\prod_i\langle a_i,b_i\rangle=(-1)^{r+1}d^{\,r-1}\prod_i\langle a_i,b_i\rangle.$$ Take $a_i=b_i=[\Lambda]-[\Lambda']$ with $\Lambda\cap\Lambda'=\emptyset$; such pairs exist, e.g. with the same matching and all $\eta_j$ different. Then $\langle a_i,a_i\rangle=2f_d(n_i/2)\neq0$, because $|1-d|\ge2$. On the other hand $\langle\Phi\alpha,\Phi\beta\rangle=\langle\alpha,\mathcal E\beta\rangle=c\langle\alpha,\beta\rangle=c\prod\langle a_i,b_i\rangle$. So $c=(-1)^{r+1}d^{r-1}$. (Checked exactly in `verify.py` T3 for $(d,r)=(4,2),(3,2),(4,3)$, computing the join intersections directly in $\mathbb P^{4r-1}$.) *(3)* By (2), $\Phi$ is injective on $\bigotimes V_i$. By (1) it lands in $H^n_{\rm prim}(Y)^G$, which has the same dimension (Lemma 4.4). $\Phi$ is a morphism of Hodge structures, hence an isomorphism onto it. *(4)* $\Phi$ and $\Phi^t$ are algebraic, and $\alpha=c^{-1}\Phi^t\Phi\alpha$. *(5) Jacobian form.* $\Phi$ is a morphism of variations of Hodge structure over $\mathcal B$, so $\mathrm{Gr}_F\Phi$ commutes with the IVHS maps $\theta_P$. The direction "deform $Q_i$ by $P(x^{(i)})$" in $T_b\mathcal B$ deforms $F$ by the same $P$. On $\bigotimes_i\mathrm{Gr}V_i$, $\theta_P$ acts on the $i$-th factor only (Leibniz rule; the other factors are flat in that direction). Since $d\mid n_i+2$, the lowest Griffiths degree $t_i(q_i^{\min})$ is $0$, so the extreme piece $\bigotimes_i\operatorname{Jac}(Q_i)_0$ is one-dimensional. Its image is the one-dimensional piece $\operatorname{Jac}(F)_0$ (Lemma 4.4), and $\mathrm{Gr}\Phi$ is injective by (3) and strictness. Hence $\mathrm{Gr}\Phi(\bigotimes\operatorname{Res}1)=\kappa(q^{\min})\operatorname{Res}_Y(1)$ with $\kappa(q^{\min})\ne0$. Since $\operatorname{Jac}$ is generated in degree 1 and $t_i(q+1)-t_i(q)=d$, we have $\operatorname{Jac}(Q_i)_{t_i(q+1)}=\mathbb C[x^{(i)}]_d\cdot\operatorname{Jac}(Q_i)_{t_i(q)}$. Induct on $|q|$. Suppose the formula holds at $q$, and apply $\theta_P$ in block $i$ to $\bigotimes_j\operatorname{Res}(a_j)$. By [CG](i), the left side gives $-(q_i+1)\,\mathrm{Gr}\Phi(\cdots\otimes\operatorname{Res}(Pa_i)\otimes\cdots)$. The right side gives $\kappa(q)\cdot(-(q_Y+1))\operatorname{Res}_Y(P\prod a_j)$ with $q_Y+1=|q|+r$; the degree count is $\sum_jt_j(q_j)=d(|q|+r)-N$. Hence the formula holds at $q+e_i$ with $\kappa(q+e_i)=\kappa(q)\frac{|q|+r}{q_i+1}$. This recursion is solved by $\kappa(q)=C(|q|+r-1)!/\prod q_i!$, and the result does not depend on the path. $\square$ ### 5.2 Theorem B *(1)* $\delta_S\in\wedge^r(T\otimes\mathbb C)$. Since $T_{\mathbb C}=T^{2,0}\oplus T^{1,1}\oplus T^{0,2}$ with $\dim T^{2,0}=\dim T^{0,2}=1$, we can write $\delta_S=\lambda\,\mathrm{Alt}(\omega\otimes e_1\otimes\cdots\otimes e_{r-2}\otimes\bar\omega)$ with $e_j$ a basis of $T^{1,1}$. Each factor lies in the matching step of the Hodge filtration, and $\mathrm{Gr}_F$ of a tensor product is the tensor product of the $\mathrm{Gr}$'s. So the image of $\delta_S$ in $\mathrm{Gr}^r$ is $\lambda\,\mathrm{Alt}([\omega]\otimes[e_1]\otimes\cdots\otimes[\bar\omega])$. Under [Gr]: $[\omega]\leftrightarrow1\in\operatorname{Jac}_0$ and $[\bar\omega]\leftrightarrow\mathrm H\in\operatorname{Jac}_8$, both up to scalars. The span of the $[e_j]$ is the image of $T^{1,1}$ in $\mathrm{Gr}^1=H^{1,1}_{\rm prim}$. That image is the orthogonal complement of $N_S\otimes\mathbb C$, which by [CG](ii) is $\mathcal N^\perp$. Apply Theorem A(5). Every permutation of the multidegree $(0,1,\dots,1,2)$ has the same $|q|=r$ and the same $\prod q_i!=2$, so $\kappa$ is constant on all terms, and $\mathrm{Gr}\Phi(\delta_S)\leftrightarrow\lambda'D_S$ with $\lambda'\ne0$. *(2) "$\Leftarrow$".* Let $(A,B)$ satisfy $\operatorname{Tr}(\mathrm{KL}\cdot D_S)\ne0$, and let $\mathcal G$ be the sheaf with this 2-periodic tail [FK]. Then $x=\mathrm{ch}^{\rm prim}_{2r-1}(\mathcal G)$ is an algebraic class: $\mathrm{ch}_{2r-1}(\mathcal G)$ minus a rational multiple of $h^{2r-1}$. Its Griffiths representative is a nonzero rational multiple of $\mathrm{KL}(A,B)$. By (1) and [CG](ii), $\langle x,\Phi\delta_S\rangle\ne0$. Then $\Phi^tx$ is algebraic on $S^r$, and $P(\Phi^tx)=\frac{\langle\Phi^tx,\delta_S\rangle}{\langle\delta_S,\delta_S\rangle}\delta_S=\frac{\langle x,\Phi\delta_S\rangle}{\langle\delta_S,\delta_S\rangle}\delta_S\neq0$ is algebraic by Lemma 4.5. So $\delta_S$ is algebraic. *"$\Rightarrow$".* If $\delta_S$ is algebraic, then $\Phi\delta_S$ is algebraic and primitive. Since $\mathrm{ch}_k(\mathcal O_Z)=[Z]$ for $Z$ of codimension $k$, the algebraic classes are spanned by Chern characters of sheaves, so $\Phi\delta_S=\sum_jq_j\mathrm{ch}^{\rm prim}_{2r-1}(\mathcal G_j)$. Now $\langle\Phi\delta_S,\Phi\delta_S\rangle=c_{r,4}\langle\delta_S,\delta_S\rangle\ne0$ (Theorem A(2), Lemma 4.5), so some $j$ has $\langle\mathrm{ch}^{\rm prim}(\mathcal G_j),\Phi\delta_S\rangle\neq0$. The 2-periodic tail of $\Gamma_*\mathcal G_j$ is the required MF, by [FK], [CG](ii) and (1). *(3)* Lemma 4.6. $\square$ ### 5.3 Theorem C *(a)* By Lemma 4.3, $\mathrm{KL}=\prod_i\mathrm{KL}(A_i,B_i)$ with $\mathrm{KL}(A_i,B_i)\in\operatorname{Jac}(Q)_4$, so it has multidegree $(4,\dots,4)$. Every term of $D_S$ has multidegree a permutation of $(0,4,\dots,4,8)$. A product lands in the socle multidegree $(8,\dots,8)$ only if each block degree adds to $8$, which fails in the blocks of degree $0$ and $8$. The statement extends to $K_0$-combinations by linearity. *(b)* First, the blocks may be assumed to consist of consecutive indices, so that Lemma 4.3 applies with contiguous variables. A block permutation $\sigma$ is an automorphism of $Y_r$ fixing $\Omega$ (each block has 4 variables). It satisfies $\sigma^*\mathrm{ch}^{\rm prim}(\mathcal G)=\mathrm{ch}^{\rm prim}(\sigma^*\mathcal G)$, Griffiths' isomorphism is $\sigma$-equivariant, $\sigma D_S=\pm D_S$ and $\operatorname{Tr}$ is $\sigma$-invariant. So replacing $(A,B)$ by $\sigma^*(A,B)$ changes the pairing only by a nonzero factor. Next, $G$ and $\operatorname{Tr}$ are compatible (the socle has multidegree $(8,\dots,8)$, which is $G$-invariant), and $D_S$ is $G$-invariant. So $\operatorname{Tr}(K\cdot D_S)=\operatorname{Tr}(\operatorname{av}_G(K)\cdot D_S)$, and $\operatorname{av}_G(\prod_jK_j)=\prod_j\operatorname{av}_{G_{I_j}}(K_j)$. By [FK], $K_j=\mathrm{KL}(A_j,B_j)$ is a nonzero multiple of the Griffiths representative of a rational algebraic class $x_j$ on $Y_{I_j}$. $\operatorname{av}_{G_{I_j}}$ is induced by an algebraic correspondence, so $x'_j=\operatorname{av}(x_j)$ is a rational Hodge class in $H_{\rm prim}(Y_{I_j})^{G_{I_j}}$. By Theorem A(3), $x'_j=\Phi_{I_j}(y_j)$ with $y_j$ a rational Hodge class in $V^{\otimes I_j}$. Its Griffiths components have $|q_{I_j}|=|I_j|$, because $y_j$ has Hodge type $(|I_j|,|I_j|)$. By Theorem A(5), applied to $I_j$ and to $[r]$, $\prod_j\mathrm{Gr}(x_j')=\lambda\,\mathrm{Gr}\Phi(\bigotimes_jy_j)$. Here the ratio $\prod_j\kappa_{I_j}(q_{I_j})/\kappa(q)=\prod_jC_{I_j}(2|I_j|-1)!\,/\,C(2r-1)!$ is the same for all relevant $q$. Hence $\operatorname{Tr}(K\cdot D_S)$ is proportional to $\langle\Phi(\bigotimes y_j),\Phi\delta_S\rangle=c_{r,4}\langle\bigotimes_jy_j,\delta_S\rangle=c_{r,4}\langle\bigotimes_j\pi_T^{\otimes I_j}y_j,\delta_S\rangle$. Each $\pi_T^{\otimes I_j}y_j$ is a Hodge class in $T^{\otimes I_j}$, hence $SO(T)$-invariant by [Zar]. Since $|I_j|