Proof. For an even lattice \(L\), write \(A_L=L^*/L\) and \(q_L:A_L\to\mathbb Q/2\mathbb Z\) for its discriminant form. The forms of \(D_8(-1)\) and \(D_4(-1)\) are respectively the split and anisotropic forms \(u,v\) on \((\mathbb Z/2)^2\). Their values are integral modulo \(2\mathbb Z\), so each is isomorphic to its negative. Also \(q_{\mathbb U(2)}=u\). Hence \[(A_P,q_P)\simeq u\oplus v\simeq(A_{T_P},-q_{T_P}).\] An anti-isometry glues \(P\oplus T_P\) to an even unimodular lattice of signature \((3,19)\), hence to \(\Lambda_{\mathrm{K3}}\), with both summands primitive. This gives the asserted embedding. For another embedding, Witt’s extension theorem [30] applied to the two copies of the nondegenerate rational space \(P_{\mathbb Q}\) gives an isometry of \(\Lambda_{\mathrm{K3},\mathbb Q}\) carrying one copy to the other. It carries their rational perpendiculars to each other. Finally rescaling one isotropic basis vector identifies \(\mathbb U(2)_{\mathbb Q}\) with \(\mathbb U_{\mathbb Q}\), and the coordinate description of \(D_4\) gives \(D_4(-1)_{\mathbb Q}\simeq\langle-1\rangle^4\). ◻

Proof. Choose an ample divisor class \(\ell\) on \(A\), and use the polarization \[e(x,y)=\int_Ax\cup y\cup\ell^{g-1},\qquad g=\dim A.\] Changing the rational polarization in the hypothesis preserves algebraicity: the change on the identified dual factor is a rational Hodge automorphism of \(H^1(A,\mathbb Q)\), and hence comes from an abelian homomorphism with rational coefficients. The elementary argument for this fact is given in the proof of Lemma 36. Here \(n=2g\); put \[B_W=H^1(A,\mathbb Q)\otimes H^1(A,\mathbb Q)\subset H^2(Z,\mathbb Q).\] The polarization identifies \(\mathop{\mathrm{End}}(W)\) with \(B_W(1)\).

The Clifford calculation. Define \(F_k\) by alternating the composition of \(k\) copies of \(j_w\). Associativity gives \[j_w(v_1)\cdots j_w(v_k)(a) =v_1\cdots v_k\,a\,w^k.\] Thus, if \(c_k:\bigwedge^kT\to C(T,q)\) is the antisymmetrized Clifford multiplication, then \[ F_k(\alpha)(1)=c_k(\alpha)w^k. \tag{7}\] The Clifford PBW decomposition identifies \(c_k\) with the degree-\(k\) exterior summand, so it is injective. Moreover \(w^{-1}=w/q(w,w)\), and right multiplication by \(w^k\) is invertible. Equation (7) proves injectivity in every degree under consideration. For odd \(k\), the left and right Clifford multiplications are both odd and their composite still preserves \(C^+(T,q)\).

The map \(j_w\) is a Hodge morphism for the left spin action; tensor products, composition and alternation preserve this property. Consequently \[ F_k:\bigwedge^kT\longrightarrow B_W(1-k) \tag{8}\] is the corresponding untwisted Hodge morphism. In particular, the construction uses the exact multiplication in (6); an unspecified algebraic copy of \(T\) in \(H^2(A^2,\mathbb Q)\) would not imply (7).

Algebraic realization and degree. Represent the tensor form of \(j_w\) by the cycle in the hypothesis, projected by \(\pi_T\) and \(\pi_1^A\boxtimes\pi_1^A\). Take its \(k\)-fold exterior product and project the input by \(\pi_{\wedge,k}\). Composition in \(\mathop{\mathrm{End}}(W)\) contracts \(k-1\) pairs of \(W\)-slots by \(e\), in the appropriate tensor order. Each such contraction is algebraic: the cycle \[(\Delta_A)_*(\ell^{g-1})\in\mathop{\mathrm{CH}}^{2g-1}(A^2)_{\mathbb Q},\] regarded as a correspondence to a point, realizes \(W\otimes W\to\mathbb Q(-1)\). It lowers cohomological degree by two. The two uncontracted slots give \(Z\); any signs arising from permuting degree-one cohomology factors are included in the cycle coefficients. Hence the resulting correspondence, denoted by \(\Gamma_k\), has \[ \Gamma_k\in\mathop{\mathrm{CH}}^{k+1}(S^k\times Z)_{\mathbb Q},\qquad (\Gamma_k)_*|_{\wedge^kT}=F_k. \tag{9}\] Indeed its degree shift is \(2-2k\), which agrees with \(2((k+1)-\dim S^k)\). Its output twist is \(1-k\), as in (8).

An algebraic left inverse. Let \(L=\operatorname{pr}_1^*\ell+\operatorname{pr}_2^*\ell\) on \(Z\). The summand \(B_W\) is primitive in degree two for \(L\). To see this, expand \(L^{n-1}\): every term acting on \(B_W\) has odd degree in each abelian factor, whereas the only possible top bidegree on \(A^2\) is \((2g,2g)\). Thus \(L^{n-1}B_W=0\). The Hodge–Riemann bilinear relations imply that \[Q_Z(x,y)=\int_Zx\cup y\cup L^{n-2}\] is a polarization on \(B_W\), up to a fixed sign. Retwisting (8) gives a morphism of weight-two Hodge structures \((\bigwedge^kT)(k-1)\to B_W\). Thus its image is a Hodge substructure of \(B_W\). The restriction of \(Q_Z\) to this image is nondegenerate: the associated definite Hermitian form restricts to a definite form there.

Multiplication by \(L^{n-2}\), followed by the transpose of \(\Gamma_k\) and by \(\pi_{\wedge,k}\), defines an algebraic map \[ G_k:H^2(Z,\mathbb Q)\longrightarrow\bigwedge^kT,\qquad G_k\in\mathop{\mathrm{CH}}^{n+k-1}(Z\times S^k)_{\mathbb Q}. \tag{10}\] Here the cohomological degrees are \[H^2(Z)\xrightarrow{\,L^{n-2}\,}H^{2n-2}(Z) \xrightarrow{\,{}^t\Gamma_k\,}H^{2k}(S^k).\] If \(Q_k\) denotes middle intersection on \(\bigwedge^kT\), self-adjointness of \(\pi_{\wedge,k}\) gives \[ Q_k(G_kF_kx,y)=Q_Z(F_kx,F_ky). \tag{11}\] The pairing on the right is nondegenerate, and \(Q_k\) is nondegenerate by the discussion of (4). Therefore \(D_k=G_kF_k\) is invertible on \(\bigwedge^kT\).

Its characteristic polynomial has nonzero constant term, so Cayley–Hamilton supplies a polynomial \(P_k\in\mathbb Q[X]\) such that \(P_k(D_k)D_k=\mathop{\mathrm{id}}\) on that summand. Composition of cycles realizes \(P_k(D_k)\) in codimension \(2k\) on \(S^k\times S^k\), using \(\pi_{\wedge,k}\) as the identity of the summand. The correspondence \[ R_k=P_k(D_k)G_k\in\mathop{\mathrm{CH}}^{n+k-1}(Z\times S^k)_{\mathbb Q} \tag{12}\] satisfies \(R_kF_k=\mathop{\mathrm{id}}\); in weight-zero notation it maps \(H^2(Z,\mathbb Q)(1)\) to \((\bigwedge^kT)(k)\). This construction uses multiplication by an ample class, transposition and a polynomial inverse on a finite-dimensional summand. It requires no algebraic inverse to a Lefschetz operator. The composite \(\Gamma_k\circ R_k\), of codimension \(n\) on \(Z\times Z\), is the corresponding cohomological projector onto \(F_k(\bigwedge^kT)\subset H^2(Z,\mathbb Q)\).

Finally, let \(\alpha\in(\bigwedge^kV)^{0,0}_{\mathbb Q}\). Its image is a rational \((1,1)\) class in \(B_W\), so Lefschetz’s \((1,1)\) theorem gives a divisor class on \(Z\) representing it. Applying \(R_k\) yields \(\alpha\). The resulting cycle has codimension \(1+(n+k-1)-n=k\) on \(S^k\), as asserted. ◻

Proof. Choose a finite cycle representative for \(C\). Each component has dimension \(D-p\), so the union \(B\) of its proper images in \(Y\) has dimension at most \(D-p\). The restriction of the correspondence to \(X\times(Y\setminus B)\) is zero. Compatibility of its action with restriction proves the assertion, including for rational linear combinations of components. ◻

Proof. Retain \(Z=A^2\) and \(n=\dim Z\) from Proposition 4. Equation (12) gives \[R_k\in\mathop{\mathrm{CH}}^{n+k-1}(Z\times S^k)_\mathbb Q, \qquad R_kF_k=\mathop{\mathrm{id}}\quad\text{on }\bigwedge^kT.\] The support of this cycle has dimension \(n+2k-(n+k-1)=k+1\). Lemma 5 therefore puts its entire image from \(H^2(Z,\mathbb Q)\) in coniveau \(k-1\), and that image contains \(\bigwedge^kT\).

The only possible types of \(\bigwedge^kT\) are \((k+1,k-1)\), \((k,k)\) and \((k-1,k+1)\), since the extreme Hodge spaces of \(T\) are one-dimensional. Thus a sub-Hodge structure requiring more than coniveau \(k-1\) is necessarily of type \((k,k)\). Every rational vector in such a substructure is algebraic by Proposition 4. Choosing a finite rational basis and taking the union of the supports of its codimension-\(k\) cycles proves coniveau \(k\). This also applies to \(\bigwedge^tT\), whose only type is \((t,t)\). These are all possible coniveau requirements. ◻

Proof. Over \(\mathbb C\), choose orthogonal torus coordinates so that the Hodge cocharacter has vector weights \(1,0,\ldots,0,-1\). The integral highest weights of tensor representations of \(\mathop{\mathrm{SO}}_t\) have the form \[\begin{cases} \lambda_1\ge\cdots\ge\lambda_l\ge0,&t=2l+1,\\ \lambda_1\ge\cdots\ge\lambda_{l-1}\ge|\lambda_l|,&t=2l. \end{cases}\] Only integral tensor weights occur here; half-integral spin weights are not representations in tensor powers of the standard module. For an irreducible constituent of highest weight \(\lambda\), the largest value of the Hodge cocharacter on its weights is \(\lambda_1\): the cocharacter is dominant, and all weights lie in the convex hull of the Weyl orbit of \(\lambda\). The largest value is attained at \(\lambda\). The orthogonal Weyl group also gives the value \(-\lambda_1\). Hence this constituent has Hodge level exactly \(2\lambda_1\).

Put \(c=\lambda_1\). In odd dimension take the columns of the partition \((\lambda_1,\ldots,\lambda_l)\). Their number is \(c\). The tensor product of their exterior highest-weight vectors is a highest-weight vector of weight \(\lambda\). Complete reducibility therefore places the desired irreducible representation in the tensor product of these \(c\) exterior powers. In even dimension use the columns of \((\lambda_1,\ldots,\lambda_{l-1},|\lambda_l|)\). For columns of height \(l\), use the highest-weight vector of the appropriate sign in \(\bigwedge^lV_\mathbb C\); the two available weights are \((1,\ldots,1,1)\) and \((1,\ldots,1,-1)\). Their sign is chosen to agree with \(\lambda_l\). Shorter columns are unchanged. The same highest-weight argument applies. We keep the full exterior powers, including both middle-exterior constituents, as the carriers.

Choose a complex irreducible constituent \(H_\lambda\) of \(H_\mathbb C\). It has \(c=\lambda_1\le b\). The preceding construction defines a rational representation, for \(\mathbf k=(k_1,\ldots,k_c)\), \(C_{\mathbf k}=\bigotimes_i\bigwedge^{k_i}V\) whose complexification shares this constituent with \(H_\mathbb C\). Consequently \[\mathop{\mathrm{Hom}}_{\mathop{\mathrm{SO}}(V,q)}(C_{\mathbf k},H)\otimes_\mathbb Q\mathbb C =\mathop{\mathrm{Hom}}_{\mathop{\mathrm{SO}}(V_\mathbb C,q)}((C_{\mathbf k})_\mathbb C,H_\mathbb C)\ne0.\] This scalar-extension identity follows by taking invariants in \((C_{\mathbf k})^*\otimes H\). Any nonzero rational map in this space is surjective by rational simplicity of \(H\), and it is a Hodge map because the Hodge cocharacter lies in \(\mathop{\mathrm{SO}}(V,q)\). This proves the required rational assertion without descending an individually chosen complex projector.

For \(t=3\) the argument is the single-coordinate case, with columns of height one. For \(t=4\) the coordinates are \(\lambda_1\ge|\lambda_2|\), and columns have heights one and two; the two signs in the latter give the same construction. Thus the small orthogonal ranks cause no exception. ◻

Proof. Put \(E=\mathop{\mathrm{End}}_{\mathrm{Hdg}}(T)\) and \(a=\dim_E T\). We use the full groups in (13) and (14). Choose an algebraic closure of \(\mathbb Q\) inside \(\mathbb C\) over which these groups split. Highest-weight constituents can be described over that closure and then extended to \(\mathbb C\). The multiset of constituents of \(H\) is invariant under the absolute Galois group because \(H\) is rational. We shall use this rationality before applying any Hodge-level bound.

The possible orthogonal ranks. In the totally real case, \(a\ge3\). Indeed \(\mathop{\mathrm{End}}_G(V)=\mathop{\mathrm{End}}_{\mathrm{Hdg}}(V)=E\), because commuting with the Hodge circle is equivalent to commuting with its rational Zariski closure. This equality remains true after scalar extension, by scalar extension of invariants. Put \(e=[E:\mathbb Q]\). If \(a=1\), the split orthogonal group is trivial, so its commutant on \(V\) has dimension \(e^2>e\); here \(e=\dim_\mathbb QV\ge2\) because the two extreme K3 Hodge lines are nonzero. If \(a=2\), each split standard orthogonal module is the sum of two distinct characters of \(\mathop{\mathrm{SO}}_2\), so the commutant has dimension at least \(2e>e\). Both contradict the dimension \(e\) of \(E\). Thus the orthogonal factors below have the ranks covered by Lemma 7.

Totally real widths and columns. Suppose first that \(E\) is totally real. Over \(\mathbb C\) the group is \(\prod_{\sigma:E\hookrightarrow\mathbb C}\mathop{\mathrm{SO}}(V_\sigma,q_\sigma)\). There is one distinguished embedding \(\sigma_0\) for which \(V_{\sigma_0}\) has the types \((1,-1)\) and \((-1,1)\); the other summands have type \((0,0)\). Accordingly the Hodge cocharacter is trivial on the other factors and has vector weights \(1,0,\ldots,0,-1\) on the distinguished factor.

Fix one complex irreducible constituent of \(H_\mathbb C\), with tuple of orthogonal highest weights \(\lambda_\sigma\), and write \(w_\sigma=\lambda_{\sigma,1}\) in the coordinates of Lemma 7. Galois acts transitively on the embeddings of \(E\) and transports the actual standard eigenspace of each orthogonal factor to the actual standard eigenspace at the conjugate embedding. The conjugacy class of cocharacters with weights \(1,-1,0,\ldots,0\) on that standard module is therefore preserved. The maximum of this class on a constituent is its first highest-weight coordinate, so Galois transports \(w_\sigma\) to the corresponding width at \(\sigma_0\). This argument also covers type \(D_4\): a triality automorphism exchanging the vector and a half-spin module does not preserve the actual standard module and cannot arise from this rational eigenspace transport. That conjugate constituent still occurs in \(H_\mathbb C\). The Hodge-level calculation in Lemma 7 consequently gives \(w_\sigma\le b\) for every \(\sigma\).

Set \(c=\max_\sigma w_\sigma\). For each factor form its signed orthogonal columns as in that lemma, with heights \(h_{\sigma j}\) for \(1\le j\le c\), padding with zeros. In the even case the sign of a middle column is retained. Set \(k_j=\sum_\sigma h_{\sigma j}\). The exterior decomposition of a direct sum includes \[\bigotimes_\sigma\bigwedge^{h_{\sigma j}}V_\sigma \ \subset\ \bigwedge^{k_j}V_\mathbb C.\] Taking the Cartan components over all columns shows that \(\bigotimes_j\bigwedge^{k_j}V_\mathbb C\) contains the chosen tuple of irreducibles. Columns from different embeddings thus occupy the same exterior carrier; their widths are maximized, not added. The ranks three and four are included by Lemma 7.

CM widths and signed columns. Now suppose that \(E\) is CM. Choose one embedding \(\eta\) from each conjugate pair and write \[G_\mathbb C=\prod_\eta\mathop{\mathrm{GL}}(W_\eta),\qquad V_\mathbb C=\bigoplus_\eta(W_\eta\oplus W_\eta^*), \qquad \dim W_\eta=a.\] Choose the representatives so that the distinguished embedding containing \(V^{1,-1}\) is represented by a standard module. On this factor the Hodge cocharacter is conjugate to \(\operatorname{diag}(z,1,\ldots,1)\); on the other factors it is trivial. Its action on the distinguished dual is the inverse. A complex irreducible constituent of \(H_\mathbb C\) has, on each factor, an integral highest weight \[\lambda_{\eta,1}\ge\cdots\ge\lambda_{\eta,a}.\] These are highest weights for the full general linear group: negative entries and central determinant characters are retained. The extrema of the distinguished cocharacter on a factor of highest weight \(\lambda_\eta\) are \(\lambda_{\eta,1}\) and \(\lambda_{\eta,a}\), by the weight-polytope description and the permutation Weyl group.

Galois acts transitively on the conjugate embedding pairs. Since the decomposition of \(V\) comes from its rational \(E\)-action, this action transports standard modules to standard modules or their duals. On highest weights the latter possibility replaces \((\lambda_1,\ldots,\lambda_a)\) by \((-\lambda_a,\ldots,-\lambda_1)\); it introduces no additional central twist. Transporting each factor to the distinguished one, rationality of \(H\) and its level bound therefore imply \[w_\eta:=\max\{\lambda_{\eta,1}, -\lambda_{\eta,a},0\}\le b.\] In particular this bound controls signed central weights as well as differences of adjacent weights.

Set \(c=\max_\eta w_\eta\). For \(1\le j\le c\) put \[h^+_{\eta j}=\#\{i:\lambda_{\eta,i}\ge j\},\qquad h^-_{\eta j}=\#\{i:\lambda_{\eta,i}\le-j\},\qquad k_j=\sum_\eta(h^+_{\eta j}+h^-_{\eta j}).\] The positive and negative columns can be placed in the same exterior power, because its direct-sum decomposition contains \[\bigotimes_\eta \left(\bigwedge^{h^+_{\eta j}}W_\eta \otimes\bigwedge^{h^-_{\eta j}}W_\eta^*\right) \ \subset\ \bigwedge^{k_j}V_\mathbb C.\] The highest weight of \(\bigwedge^h W\) has \(h\) initial entries equal to \(1\), and that of \(\bigwedge^h W^*\) has \(h\) final entries equal to \(-1\). Summing these weights over \(j\) gives exactly \(\lambda_\eta\). Their tensor highest-weight vector therefore produces the desired Cartan component, including its central character. Thus the product of these \(c\) full exterior carriers contains the chosen constituent. The cost is the maximum of the positive and negative widths, not their sum.

Rational descent of the carrier. In both cases \(c=0\) would make the chosen constituent trivial. A rational simple representation with a trivial complex constituent is trivial, by scalar extension of invariants, so our nontriviality assumption gives \(c\ge1\). A factor attaining the maximum width has a nonempty column for each \(1\le j\le c\); therefore \(k_j\ge1\). The displayed direct-sum exterior decompositions also give \(k_j\le\dim_\mathbb QV\).

Finally, for \(\mathbf k=(k_1,\ldots,k_c)\), keep the rational carrier \(C_{\mathbf k}=\bigotimes_j\bigwedge^{k_j}V\), rather than selecting any of its complex summands. Sharing a complex constituent yields \[\mathop{\mathrm{Hom}}_G(C_{\mathbf k},H)\otimes_\mathbb Q\mathbb C =\mathop{\mathrm{Hom}}_{G_\mathbb C}((C_{\mathbf k})_\mathbb C,H_\mathbb C)\ne0.\] The equality follows by scalar extension of invariants in \((C_{\mathbf k})^*\otimes H\). A nonzero rational equivariant map is surjective because \(H\) is rationally simple, and it is a Hodge map because \(G\) contains the Hodge cocharacter. This handles nonsplit forms, Galois-conjugate constituents and their multiplicities without assuming any complex projector rational. ◻

Proof of the generalized Hodge assertion in Theorem 1. Put \(V=T(1)\) and \(G=\operatorname{Hg}(V)\), with the group descriptions (13) and (14). The ordinary Hodge assertion just proved gives rational HC for every power of \(S\) under the same exact-map hypothesis. We use that conditional conclusion in both endomorphism-field cases.

Fix a rational sub-Hodge structure \(H\subset H^{2d}(S^m,\mathbb Q)\) of Hodge coniveau at least \(p\), where \(0\le p\le d\). By semisimplicity it suffices to treat \(H\) simple. The K3 Künneth decomposition shows that \(H^{2d}(S^m,\mathbb Q)(d)\) is a direct sum of tensor powers of \(V\): its other factors, coming from \(H^0\), the Néron–Severi part of \(H^2\), and \(H^4\), are Tate and become trivial after the total normalization. Thus \(H(d)\) is a rational simple representation of \(G\), of Hodge level at most \(2(d-p)\).

If \(H(d)\) is trivial, \(H\) has type \((d,d)\). Ordinary HC and a finite basis of cycle classes put \(H\) in \(N^dH^{2d}(S^m,\mathbb Q)\subset N^pH^{2d}(S^m,\mathbb Q)\). Otherwise Lemma 8 supplies a tuple \(\mathbf k=(k_1,\ldots,k_c)\) and a surjective rational Hodge map \[f:C_{\mathbf k}(s)\longrightarrow H(d),\qquad C_{\mathbf k}=\bigotimes_{i=1}^c\bigwedge^{k_i}T \subset H^{2s}(S^s,\mathbb Q),\quad s=\sum_i k_i,\quad c\le d-p.\] Extend \(f\) by zero on a rational Hodge complement of \(C_{\mathbf k}(s)\) in \(H^{2s}(S^s,\mathbb Q)(s)\), which exists by polarizability. Poincaré duality identifies this extended map to \(H(d)\) with a rational Hodge class of codimension \(s+d\) on \(S^s\times S^m\). Ordinary HC for \(S^{s+m}\) therefore gives \[C_f\in\mathop{\mathrm{CH}}^{s+d}(S^s\times S^m)_\mathbb Q\] realizing it.

Retain \(Z=A^2\) and \(n=\dim Z\) from Proposition 4. For each column take the return \(R_{k_i}\) of (12). Their exterior product, with factors reordered, gives \[R_{\mathbf k}\in\mathop{\mathrm{CH}}^{nc+s-c}(Z^c\times S^s)_\mathbb Q, \qquad (R_{\mathbf k})_*:H^{2c}(Z^c,\mathbb Q)\longrightarrow H^{2s}(S^s,\mathbb Q),\] whose image contains \(C_{\mathbf k}\). Indeed the tensor product of the \(F_{k_i}\) is a right inverse on \(C_{\mathbf k}\). Composition gives \[C_f\circ R_{\mathbf k}\in\mathop{\mathrm{CH}}^{nc+d-c}(Z^c\times S^m)_\mathbb Q,\] since \((nc+s-c)+(s+d)-2s=nc+d-c\). Its image on \(H^{2c}(Z^c,\mathbb Q)\) contains \(H\), and its support has dimension \[nc+2m-(nc+d-c)=2m-d+c.\] Lemma 5 yields \[H\subset N^{d-c}H^{2d}(S^m,\mathbb Q) \subset N^pH^{2d}(S^m,\mathbb Q).\] For a nonsimple substructure, take the finite union of the supports obtained for its simple summands. This treats every actual occurrence, including multiplicities and substructures embedded diagonally across isomorphic Künneth summands.

For the converse, put \(X=S^m\) and let \(B\subset X\) be any closed support of codimension at least \(p\), including a singular or non-equidimensional support. For each irreducible component \(B_i\) choose a smooth projective resolution and let \(q_i:\widetilde B_i\to X\) be its proper map; put \(c_i=\operatorname{codim}_X B_i\ge p\). Deligne’s exactness theorem for a proper smooth variety and a closed support, using the disjoint union of these resolutions as a proper smooth surjective source [13], gives \[\ker\bigl(H^j(X,\mathbb Q)\longrightarrow H^j(X\setminus B,\mathbb Q)\bigr) =\sum_i(q_i)_*H^{j-2c_i}(\widetilde B_i,\mathbb Q).\] Each proper Gysin map is a Hodge morphism \(H^{j-2c_i}(\widetilde B_i,\mathbb Q)(-c_i)\to H^j(X,\mathbb Q)\). The source before twisting has nonnegative Hodge indices, since \(\widetilde B_i\) is smooth projective; after twisting both indices are at least \(c_i\ge p\). Taking sums over \(B\) therefore proves that geometric coniveau \(p\) has Hodge coniveau at least \(p\).

Thus the two filtrations agree. If \(p>d\), the same reverse inclusion forces \(N^pH^{2d}(S^m,\mathbb Q)=0\), since a pure structure of weight \(2d\) cannot have both indices at least \(p\). Cases \(p\le0\) are immediate. Odd cohomology of \(S^m\) vanishes. The result therefore holds in every degree, not only in middle cohomology or the pure transcendental tensor summand. ◻

Proof. We first determine the fibre signs of the normalized involution on \(L\). Write \(B=\mathbb C^2/\Lambda\) and represent \(\chi\) by a character \(c:\Lambda\to\mu_4\) of exact order four. Then \(B'=\mathbb C^2/\Lambda'\), where \(\Lambda'=\ker(c^2)\). Choose \(\lambda\in\Lambda\setminus\Lambda'\) and put \(k=\lambda\bmod\Lambda'\in\ker\pi\). Translation by \(k\) pairs the sixteen points of \(B'[2]\). The fibre sign of \(\pi^*M\) is the same at \(x\) and \(x+k\), since these points have the same image in \(B\). For the flat factor, representing \(x\) by \(v\) with \(2v\in\Lambda'\) shows that its sign changes by \[\frac{c(2v+2\lambda)}{c(2v)}=c(2\lambda)=-1.\] Thus the signs of \(L\) in each pair are opposite: eight are positive and eight negative.

The branch curve is absent near these sixteen points. At a positive-sign point, \(j\) preserves each local sheet of \(X\) and acts by \((u,v)\mapsto(-u,-v)\). The quotient has two ordinary double points, whose resolutions give two disjoint \((-2)\)-curves over the corresponding exceptional curve of \(K\). At a negative-sign point, \(j\) exchanges the local sheets. The quotient is smooth, and its map to \(B'/\{\pm1\}\) is locally \(\mathbb C^2\to\mathbb C^2/\{\pm1\}\). Blowing up this smooth source point resolves the map to \(K\) and gives a double cover branched along the exceptional curve of \(K\). These local models construct the smooth double cover \(Y\to K\).

Set \(D'=\pi^{-1}D\). It is smooth and connected, since it is ample, and \((D')^2=16\). Adjunction gives \(g(D')=9\). Inversion acts freely on \(D'\), so its quotient has genus five. The branch divisor of \(Y\to K\) is the disjoint union of that quotient and the eight exceptional curves belonging to the negative signs. Therefore \[e(Y)=2e(K)-\bigl((2-2\cdot5)+8\cdot2\bigr)=48-8=40.\] The cover algebra, Serre duality, and ample vanishing give \(H^1(B',L^{-1})=0\) and hence \(H^1(X,\mathbb Q)=f^*H^1(B',\mathbb Q)\). Inversion is \(-1\) on this space. Taking quotient invariants, and using invariance of \(H^1\) under the intervening blowups, gives \(q(Y)=0\). For holomorphic two-forms, \[H^{2,0}(X)=H^{2,0}(B')\oplus H^0(B',L),\qquad H^0(B',L)=H^0(B,M\chi)\oplus H^0(B,M\chi^{-1}).\] The first summand is invariant under both involutions. Inversion replaces \(\chi\) by \(\chi^{-1}\); the trivialization of \(\pi^*\chi^2\) used to identify the pullbacks has the nontrivial deck character, so it exchanges the two one-dimensional summands in \(H^0(B',L)\). They are sheet-anti-invariant. Thus \(p_g(Y)=2\). Consequently \(b_2(Y)=38\), and Hodge index gives signature \((5,33)\) on \(H^2(Y)\).

The sheet-invariant rational cohomology is the pullback of \(H^2(K,\mathbb Q)\), with pairing multiplied by two, and has signature \((3,19)\). Its perpendicular \(V_\mathbb Q\) has signature \((2,14)\). The canonical class \(K_Y\) is invariant. For \(v\in V\) we have \(v\cdot K_Y=0\), and Wu’s characteristic parity formula [27] \[v^2\equiv v\cdot K_Y\pmod2\] shows that \(V\) is even.

For the integral assertions, let \(\mathcal H=H^2(Y,\mathbb Z)/\mathrm{torsion}\) and let \(\mathcal H^+\) be its invariant lattice. Integral Poincaré duality makes \(\mathcal H\) unimodular. Both eigensublattices are primitive, and \[2x=(x+\delta x)+(x-\delta x) \quad\text{for every }x\in\mathcal H.\] The quotient \(\mathcal H/(\mathcal H^+\oplus V)\) is therefore killed by two. Unimodular gluing identifies this quotient with each of \(A_{\mathcal H^+}\) and \(A_V\), proving that their discriminant groups are \(2\)-elementary. For fixed \(B'\), pullback from the fixed Kummer surface makes the invariant rational cohomology pointwise constant. Its integral sublattice and its discriminant group are consequently fixed as well. The gluing identification is equivariant, so monodromy acts trivially on \(A_V\).

At a positive-sign point, denote the two exceptional curves on \(Y\) by \(E_i^+\) and \(E_i^-\). They are exchanged by \(\delta\), and \[a_i=[E_i^+]-[E_i^-],\qquad a_i^2=-4.\] The eight differences are mutually orthogonal. Parallel transport can permute their indices or exchange the two curves, but preserves their span, so its perpendicular \(U\) is flat. Near a degeneration at which the branch passes through a positive-sign point, the two labels are governed by a scalar square root; the local action on \(a_i\) changes its sign.

The same local models identify \(U\) with \(U_X\). Resolving the two quotient nodes at a positive-sign point adds precisely one sheet-anti-invariant exceptional direction, namely \(a_i\); the blowup at a negative-sign point adds only a sheet-invariant direction. On a common resolution of \(X\dashrightarrow Y\), pullback of a class in \(U\) is orthogonal to all exceptional classes over \(X\), and hence comes from \(H^2(X,\mathbb Q)\). Its image is exactly the \((j,\delta)=(1,-1)\) eigenspace. The degree-two projection formula gives \[(\beta x,\beta y)_X=2(x,y)_Y.\] In particular, \(U\) has signature \((2,6)\). We retain its \(Y\)-pairing throughout the rational quadratic-form and K3 arguments.

The correspondence (19) is the product of the commuting projectors \((1+j)/2\) and \((1-\delta)/2\). The sheet involution fixes \(H^0\) and \(H^4\), and the calculation of \(H^1(X,\mathbb Q)\) shows that it fixes \(H^1\); by duality it fixes \(H^3\) as well. Thus \(e\) vanishes outside degree two and has image exactly \(U_X\) there.

We now determine the rational form on \(U\) with its \(Y\)-pairing. The invariant lattice \(\mathcal H^+\) contains the pullback lattice \(H^2(K,\mathbb Z)(2)\) with finite index, necessarily a power of two. The latter has absolute determinant \(2^{22}\), so \(|\det\mathcal H^+|\) is a square in \(\mathbb Q^*/\mathbb Q^{*2}\). Unimodular gluing gives the same absolute determinant for \(V\). Since the negative index of \(V\) is fourteen, its determinant is positive. Splitting off the eight orthogonal lines of square \(-4\) therefore gives \[\det(U)=1\quad\text{in }\mathbb Q^*/\mathbb Q^{*2}.\] For an odd prime \(p\), the projectors \((1\pm\delta)/2\) split \(\mathcal H\otimes\mathbb Z_p\) orthogonally, so \(V\otimes\mathbb Z_p\) is unimodular. The \(a_i\) span a unimodular \(\mathbb Z_p\)-lattice, since their Gram determinant is \((-4)^8\). Orthogonal projection using its inverse Gram matrix is integral over \(\mathbb Z_p\). Its perpendicular thus gives a unimodular \(\mathbb Z_p\)-lattice in \(U\otimes\mathbb Q_p\).

The lattice \(T_P\) has rank eight, signature \((2,6)\), and determinant \((-1)(-4)(4)=16\); it too is unimodular at every odd prime. A unimodular quadratic form over \(\mathbb Z_p\), for odd \(p\), diagonalizes with unit entries, and the Hilbert symbol of two units is \(1\). Hence both rational forms have Hasse invariant \(1\) at every odd prime. Their real Hasse invariants are \((-1)^{\binom62}=-1\). Hilbert reciprocity then gives the same invariant, namely \(-1\), at \(2\). The local classification and the rational local-to-global isometry criterion now give \(U\simeq V_P\). For a symmetric pairing \(q\) we use the quadratic polynomial \(Q(x)=q(x,x)\), whose diagonal coefficients are those of \(q\) itself in the conventions of [30]. The precise Hilbert-symbol, reciprocity, and classification results are [30].

The covers, blowups, quotients, and graph correspondences are all relative after the finite choices and restriction to the stated smooth opens. ◻

Proof. Let \(\alpha:\widetilde B\to B\) be the cyclic degree-four isogeny trivializing \(\chi\), with generator \(g\), and let \(Z\to B\) be the untwisted double cover \(w^2=d\) defined using \(M\). On \(\widetilde X=Z\times_B\widetilde B\), let \(\delta\) change the sign of \(w\), and let \(g\) act through the isogeny. A trivialization of \(\alpha^*\chi\) identifies \[X=\widetilde X/\langle\delta g^2\rangle.\] On sheet-anti-invariant cohomology, invariance under \(\delta g^2\) selects exactly the \(g\)-eigenvalues \(i\) and \(-i\). Inversion conjugates \(g\) to \(g^{-1}\) and exchanges these eigenspaces. Projection to either one therefore identifies their \(j\)-invariant part with that eigenspace over \(\mathbb C\). Choose the generator so that the selected character sheaf is \(\chi\); the other convention replaces \(\chi\) by \(\chi^{-1}\). These finite-group projectors commute with parallel transport.

The eigensheaves can be read directly from the local double-cover equation. If \(D=(x=0)\) and \(w^2=x\), the sheet-odd holomorphic forms are generated over the base by \[w;\qquad w\,dy,\quad w\,\frac{dx}{x}; \qquad w\,\frac{dx}{x}\wedge dy\] in degrees zero, one, and two. The odd generator has the transition rule of \(M^{-1}\). Thus the sheet-odd direct image of holomorphic \(p\)-forms is \(\Omega_B^p(\log D)M^{-1}\); selecting the character tensors this by \(\chi\). Its logarithmic flat differential has local term \(\tfrac12d\log x\), together with the flat connection on \(\chi\). The Hodge decomposition of the smooth projective finite cover, with the character projector applied, gives (20).

Infinitesimal changes of local coordinates for the pair \((B,D)\) give a Čech cocycle of logarithmic vector fields in \(E\). Locally, a logarithmic field \(v\) satisfying \(v(f)=af\) lifts on \(w^2=f\) to \(v+(a/2)w\partial_w\). This lift is unique: an infinitesimal vertical change preserving \(w^2=f\) has coefficient \(b\) with \(2b=0\). Contraction with the sheet-odd two-form computes the Kodaira–Spencer action on the Hodge line. The local generators just displayed identify this contraction with multiplication by \(\theta\) in (21). The bundle isomorphism there is the rank-two determinant identity \[E\otimes\mathcal O_B(D)\simeq\Omega_B^1(\log D), \qquad \mathcal O_B(D)=M^2.\] The chosen finite-order line bundles have unique infinitesimal extensions with their torsion data, so they add no tangent directions.

It remains to identify all tangent directions of our family. Adjunction and the logarithmic normal sequence give \[0\longrightarrow E\longrightarrow T_B \longrightarrow\mathcal O_D(D)=K_D\longrightarrow0.\] An infinitesimal translation preserving \(D\) is zero: its flow would lie in the stabilizer of \(\mathcal O_B(D)\), the finite kernel of the polarization isogeny. Thus \(H^0(E)=0\), and the cohomology sequence yields \[0\longrightarrow \frac{H^0(K_D)}{H^0(T_B)} \longrightarrow H^1(E)\longrightarrow \ker\bigl(H^1(T_B)\longrightarrow H^1(K_D)\bigr) \longrightarrow0.\] Here \(h^0(K_D)=5\), \(h^0(T_B)=2\), and \(h^1(T_B)=4\). The sequence \[0\longrightarrow\mathcal O_B\xrightarrow{\ d\ }M^2 \longrightarrow K_D\longrightarrow0\] identifies \(H^1(K_D)\) with \(H^2(\mathcal O_B)\). Under this identification both normal-sequence boundary maps are contraction with \(c_1(D)\). Indeed, choose local frames \(e_j=g_{ij}e_i\) of \(M^2\) and write \(d=f_i e_i\), so \(f_i=g_{ij}f_j\). Pulling the displayed extension back along \(T_B\to K_D\), the local lift of a vector field \(v\) is \(v(f_i)e_i\). Differences of these lifts, divided by \(d\), are \(v(\log g_{ij})\). The pulled-back extension therefore has class \(c_1(D)\in H^1(\Omega_B^1)\), and its connecting maps in degrees zero and one are the asserted contractions. In degree one the target is the one-dimensional \(H^2(\mathcal O_B)\). This contraction is nonzero and its three-dimensional kernel is exactly the tangent space to polarized deformations of \(B\). Hence \(h^1(E)=3+3=6\).

Extending \(M\) imposes the same condition, since \(c_1(M)=\tfrac12c_1(M^2)\). Among its infinitesimal extensions the symmetric representative is unique: differences form an \(H^1(\mathcal O_B)\)-torsor on which inversion acts by \(-1\). The order-four point \(\chi\) and the associated isogenies extend uniquely because torsion of the relative Picard variety is finite étale. Thus the root and torsion data create no additional infinitesimal condition.

For fixed \(B\), the section directions are \(H^0(M^2)/\langle d\rangle\). Their images in \(H^0(K_D)\) have zero boundary in \(H^1(\mathcal O_B)\). On a translation vector \(v\), that same boundary is \(v\mathbin{\lrcorner}c_1(D)\), and the map \[H^0(T_B)\longrightarrow H^1(\mathcal O_B),\qquad v\longmapsto v\mathbin{\lrcorner}c_1(D)\] is an isomorphism because \(D\) is ample. Section directions therefore identify with the entire quotient \(H^0(K_D)/H^0(T_B)\). Every polarized first-order deformation lifts to our projective bundle of sections, by ample vanishing and base change. The induced map on the right-hand term is the polarized Kodaira–Spencer isomorphism. The two identifications in the exact sequence prove that the full Kodaira–Spencer map is an isomorphism. ◻

Proof. Let \(F=(\theta=0)\). It is a translate of the theta curve of \(B\), hence is a smooth genus-two curve with \(\mathop{\mathrm{Jac}}(F)\simeq B\). For clarity, these facts use indecomposability as follows. An integral singular theta curve would have normalization of genus at most one and could not generate the abelian surface. If the theta divisor is not integral, ampleness and square two force two reduced elliptic components meeting once, and hence the product polarization: each component has positive integral intersection with \(H\) and nonnegative even self-intersection. These possibilities are excluded. Adjunction gives genus two, and the restriction pairing on \(H^1(B,\mathbb Z)\), obtained by cup product with the principal theta class, is unimodular. The induced map on Jacobians, of equal dimension, is therefore an isomorphism.

Set \(K=K_F\) and \(\eta=\chi^{-2}|_F\). Restriction \(\mathop{\mathrm{Pic}}^0(B)\to\mathop{\mathrm{Pic}}^0(F)\) is an isomorphism, so \(\eta\) is nontrivial two-torsion. Adjunction gives \[(M\chi)|_F=K,\qquad M^2|_F=K^{\otimes2}\otimes\eta.\] For a transverse \(D\), multiplication by \(\theta\) yields \[ 0\longrightarrow E\longrightarrow E\otimes M\chi \longrightarrow(E\otimes M\chi)|_F\longrightarrow0. \tag{22}\] We shall prove that the last sheaf has no global sections for general \(d\). By the cohomology sequence and Lemma 10, this implies injectivity of the period differential on all six tangent directions.

Put \(Z=D\cap F\), a reduced divisor of degree four. Restricting the logarithmic normal sequence to \(F\) is exact because the intersection is transverse. After tensoring by \(K\), its map on global sections is \[ A_d:H^0(F,K)^{\oplus2}\longrightarrow H^0\bigl(Z,(K^{\otimes3}\otimes\eta)|_Z\bigr),\qquad v\longmapsto (v\mathbin{\lrcorner}dd)|_Z. \tag{23}\] Its kernel is \(H^0(F,(E\otimes M\chi)|_F)\), and source and target both have dimension four. Differentiating a section at its zero scheme is intrinsic, so \(A_d\) is independent of local bundle frames.

The restriction sequence for \(F\), together with ample vanishing, gives a surjection \[H^0(B,M^2)\longrightarrow H^0(F,K^{\otimes2}\otimes\eta)\] whose kernel is generated by \(\theta\theta'\), where \(0\ne\theta'\in H^0(B,M\chi^{-1})\). Its restriction \(t=\theta'|_F\) is a nonzero section of \(K\otimes\eta\): a section vanishing along \(F\) would give a section of the nontrivial flat bundle \(\chi^{-2}\). The bundle \(K^{\otimes2}\otimes\eta\) is basepoint-free by Riemann–Roch on the genus-two curve. We may therefore fix a general restriction \(s=d|_F\) whose four zeros are simple and avoid the zeros of \(t\).

Replacing \(d\) by \(d+\lambda\theta\theta'\) fixes \(s\) and \(Z\) and replaces \(A_d\) by \(A_d+\lambda A'\). Choose a basis \(v_1,v_2\) of translation-invariant vector fields on \(B\). The normal derivatives \[\theta_i=(v_i\mathbin{\lrcorner}d\theta)|_F,\qquad i=1,2,\] form a basis of \(H^0(F,K)\): the map to this space is injective because the theta divisor has finite translation stabilizer, and both spaces have dimension two. In these coordinates, \[A'(a,b)=t(a\theta_1+b\theta_2)|_Z.\] The canonical pencil of a genus-two curve gives an isomorphism \(\mathop{\mathrm{Sym}}^2H^0(K)\simeq H^0(K^{\otimes2})\). Moreover, \(K^{\otimes2}(-Z)\simeq\eta\) has no section, so restriction to \(Z\) is injective. Since \(t\) does not vanish on \(Z\), we obtain \[\mathop{\mathrm{im}}A'=t\,H^0(K^{\otimes2})|_Z,\qquad \mathop{\mathrm{rank}}A'=3.\] The kernel is spanned by the skew gradient \(k=(\theta_2,-\theta_1)\). The canonical pencil has no base point, so this is a nowhere-zero tangent section of \(T_F\otimes K\simeq\mathcal O_F\).

Under \(T_F\otimes K\simeq\mathcal O_F\), the vector \(k\) is a constant \(c\ne0\). With the adjunction convention \(\operatorname{Res}(\Omega/h)\), where \(\theta=h e\), this constant is \(c=-\Omega(v_1,v_2)\): writing \(\Omega=J\,dh\wedge du\) identifies \(k\) with \(-\Omega(v_1,v_2)\partial_u\otimes du\). Consequently \(A_d(k)=c\,d_Fs|_Z\). Here \(d_Fs|_Z\) is an intrinsic section of \((K^{\otimes3}\otimes\eta)|_Z\). To show that this vector is outside \(\mathop{\mathrm{im}}A'\), consider the sequence \[0\longrightarrow\eta\xrightarrow{\ \cdot s\ }K^{\otimes2} \longrightarrow K^{\otimes2}|_Z\longrightarrow0,\] where \(\eta^{\otimes2}\simeq\mathcal O_F\) is understood. The connecting class of \((d_Fs)/t|_Z\) pairs by Serre duality with \(t\in H^0(K\otimes\eta)\). This pairing is computed by local principal parts. Near \(z\in Z\), choose a coordinate \(u\) and an \(\eta\)-frame \(\epsilon\) with \(\epsilon^2=1\), and write \(s=f(u)(du)^2\epsilon\) and \(t=g(u)du\,\epsilon\). Here \(f(z)=0\), \(f'(z)\ne0\), and \(g(z)\ne0\). A local lift of \((d_Fs)/t|_Z\) is \((f'/g)(du)^2\). Dividing by \(s\) gives the \(\eta\)-valued principal part \([f'/(gf)]\epsilon\); pairing it with \(t\) gives \((f'/f)du\). Thus \[\left\langle\partial\bigl((d_Fs)/t|_Z\bigr),t\right\rangle =\sum_{z\in Z}\operatorname{Res}_z\frac{df}{f}=4,\] up to the fixed nonzero normalization of Serre duality. Each zero is simple. Changing a frame adds a regular differential and leaves the residues unchanged. The formula concerns the four local principal parts; it does not require a global meromorphic differential with those four poles. The connecting class is nonzero, and hence \(A_d(\ker A')\) is not contained in \(\mathop{\mathrm{im}}A'\).

Choose source and target bases in which \(A'\) has an invertible \(3\times3\) block and is zero in the remaining row and column. The coefficient of \(\lambda^3\) in \(\det(A_d+\lambda A')\) is the determinant of that block multiplied by the induced map \(\ker A'\to\mathop{\mathrm{coker}}A'\). It is nonzero by the preceding pairing. Thus \(A_{d+\lambda\theta\theta'}\) is invertible for general \(\lambda\). This construction shows that invertibility is a nonempty open condition in the space of sections. Intersecting with the nonempty open of smooth divisors avoiding \(B[2]\) gives the required general \(d\).

It follows from (22) that \(H^1(E)\to H^1(E\otimes M\chi)\) is injective. By Lemma 10, its source is the entire six-dimensional tangent space. The type-IV period domain of a quadratic space of signature \((2,6)\) has complex dimension six; the period map is therefore a local isomorphism at these points. The rank condition is Zariski open, since it is the rank condition on the algebraic Kodaira–Spencer map between Hodge bundles.

Finally, on a type-IV period domain the Hodge circles rotate its oriented positive two-planes. These rotations generate the identity component of \(\mathop{\mathrm{SO}}(U_\mathbb R)\), and rotations from any nonempty open have the same Zariski closure. Hence the Hodge circles in our open period chart are not all contained in any proper rational algebraic subgroup of \(\mathop{\mathrm{SO}}(U)\).

We recall why the exceptional conditions may be taken complex analytic. A proper rational algebraic subgroup stabilizes a rational line in a tensor representation that is not stabilized by \(\mathop{\mathrm{SO}}(U)\). Indeed, take a finite-degree representation of regular functions containing generators of the subgroup’s defining ideal, intersect it with that ideal, and take the top exterior power. The resulting line has precisely the required stabilizer. Polynomial representations and semisimplicity place this representation in a sum of tensor powers and duals of \(U(1)\). A Hodge circle stabilizing a rational line fixes it pointwise, because it is connected and compact. Thus it forces a specified rational tensor to have type \((0,0)\). Each such condition is a closed complex-analytic Hodge locus, and it is proper unless that tensor is fixed by all the Hodge circles in the domain. There are only countably many rational tensors. Outside their proper loci, \(\mathop{\mathrm{Hg}}(U)=\mathop{\mathrm{SO}}(U)\). The standard representation of \(\mathop{\mathrm{SO}}(U)\) is absolutely irreducible and has no invariants. Simplicity of \(U\) and the absence of nonzero rational \((1,1)\)-classes follow. ◻

Proof. The smooth ample divisor \(D\) is connected: the divisor sequence and \(H^1(B,\mathcal O_B(-D))=0\) give \(H^0(D,\mathcal O_D)=\mathbb C\). Thus it is irreducible. Adjunction gives \(2g(D)-2=D^2=8\). Inversion acts freely because \(D\) avoids \(B[2]\), so Riemann–Hurwitz gives \(g(C)=3\). Connectedness of the double cover makes \(\epsilon\) nontrivial.

The curve \(D\) generates \(B\): containment in a translate of a proper abelian subvariety would make \(D\) an elliptic curve with square zero. Hence \(J(D)\to B\) is surjective and invariant one-forms on \(B\) restrict injectively to \(D\). They are anti-invariant under inversion and span the two-dimensional space \(H^0(C,K_C\otimes\epsilon)\), whose dimension follows from Riemann–Roch. They have no common zero on \(D\), since its tangent line is a nonzero subspace of \(T_0B\). They therefore define the basepoint-free pencil \(|K_C\otimes\epsilon|\), of degree four. Identifying its target with the projective line of tangent directions identifies the composite with the Gauss map.

We justify that these pairs \((C,\epsilon)\) are general. The open projective bundle of smooth curves in \(|2H|\) over the three-dimensional level moduli of principally polarized abelian surfaces has dimension six and maps algebraically to the moduli space \(\mathcal R_3\) of genus-three curves with nonzero two-torsion. Its fibers are countable. In fact, fixing \((C,\epsilon)\) fixes the double cover \(D\); any map of that curve to an abelian surface is a homomorphism from \(J(D)\) followed by a translation. Surjective homomorphisms, up to isomorphism of their targets, form a countable set: the connected kernel is determined by a rational subspace of \(H_1(J(D),\mathbb Q)\), and the further finite isogeny choices are countable. For a fixed homomorphism, requiring inversion on the image to be the given involution on \(D\) imposes an equation \(2b=\text{constant}\) on the translation \(b\), giving finitely many choices. Polarizations are discrete, and \([D]=2[H]\) fixes their numerical class; the remaining symmetric theta and level choices are finite. A countable fiber of a finite-type morphism over \(\mathbb C\) has dimension zero. Since \(\mathcal R_3\) is irreducible of dimension six, the map is dominant on each component of its six-dimensional source. Irreducibility of \(\mathcal R_3\) follows from the transitive symplectic monodromy on nonzero two-torsion. The Torelli theorem for curves, with their integral principal polarization [22], likewise gives zero-dimensional fibers for the Jacobian map from \(\mathcal R_3\) to the moduli space \(\mathcal A_3[\epsilon\ne0]\) of principally polarized abelian threefolds with nonzero two-torsion. This target is irreducible of dimension six: it is a quotient of connected Siegel space, and the symplectic group is transitive on nonzero vectors modulo two. Hence the Jacobian map is dominant.

Choose a theta characteristic \(\kappa\) on a genus-three curve and set \[\Theta_\kappa= \{\alpha\in J(C):h^0(C,\kappa\otimes\alpha)>0\}.\] The Riemann theta theorem identifies \(\Theta_\kappa\) with a symmetric theta divisor; see [28]. An all-even member of \(|K_C\otimes\epsilon|\) exists precisely when some \(\alpha\) with \(2\alpha=\epsilon\) lies on \(\Theta_\kappa\): an effective divisor \(E\) with \(2E\in|K_C\otimes\epsilon|\) corresponds to \(\alpha=\mathcal O_C(E)\kappa^{-1}\). This condition is independent of \(\kappa\): translating a symmetric theta divisor by two-torsion permutes the set of all sixty-four halves. The condition is closed on \(\mathcal A_3[\epsilon\ne0]\): after a finite theta-level cover, the theta divisor and the finitely many half sections are defined, and incidence with each section is closed. Their finite union descends to a closed subset.

It is a proper condition. On a product of three elliptic curves, choose \(\epsilon\) nonzero on every factor. Every half then has coordinates of order four, whereas a symmetric product theta divisor is a union of coordinate divisors at two-torsion points. No half lies on that divisor. The resulting nonempty open in the moduli of principally polarized threefolds meets the dominant Jacobian locus, and hence pulls back to a nonempty open in our parameter space. This proves the assertion about fibers. The order-four choice \(\chi\) on \(B\) is independent of \(D\), \(C\), and \(\epsilon\). Every connected component of its finite étale torsion/level cover surjects onto the connected base, so the same open remains nonempty in each component.

Finally, a translation preserving \(D\) belongs to the finite kernel \(K(M^2)=B[2]\). Each nonzero element of that kernel acts nontrivially on \(\mathbf PH^0(B,M^2)\): otherwise its commutator with all theta-group elements would be trivial, contrary to the nondegenerate theta-group pairing [4]. Its fixed locus is proper. Avoiding the union of these finitely many loci gives a nonempty open on each section fiber. Its intersection with the preceding open proves the lemma. ◻

Proof. For a fixed regular fiber, a coherent assignment is a function \(h\) on its sixteen selections satisfying Equation (24). Choosing one selection \(s_0\) and one half \(b\) of its sum determines it uniquely by \[h(s)=b-\sum_{p\in s_0\setminus s}p.\] Thus coherent assignments form a torsor under the constant group \(B[2]\). This torsor is algebraic and finite étale over \(R\): start with the cover of pairs \((s,b)\) satisfying \(2b=\sum_{p\in s}p\), and identify \((s,b)\) with \((s[p],b-p)\). After étale-locally labeling the four pairs, the flips commute, are involutions, and their equivalence classes have sixteen elements; the resulting finite étale quotient is precisely the assignment torsor. Parallel transport commutes with the constant \(B[2]\)-action, so its monodromy acts by translations, despite possible permutations of the selections.

Consider a small disc about any omitted value, including infinity if it is omitted. At the central fiber choose one point from each pair of opposite points of \(D\), and near each chosen point take the entire cluster specializing to it. Their union is a monodromy-invariant selection on the punctured disc. Its sum extends holomorphically to the disc. Because multiplication by two on \(B\) is étale, that sum has a holomorphic half on the simply connected disc. The corresponding coherent assignment is fixed by local monodromy. A translation of a torsor with a fixed point is the identity. Hence the assignment torsor has trivial monodromy about every puncture, extends étale over \(\mathbf P^1\), and is trivial. Any of its global sections gives the algebraic map \(h\).

Two selections have the same parity if they differ by an even number of flips. The two parity classes define a degree-two étale cover of \(R\). The invariant selection over each punctured disc fixes its parity class, hence both classes. This cover also extends étale over \(\mathbf P^1\) and is trivial. Labeling its two sections gives the global covers \(\mathcal S_\pm\). Properness of \(B\) extends their maps uniquely to the smooth projective completions. Finally, differentiation of Equation (24), using the invariant tangent space of \(B\), gives locally \[ da_\pm=\tfrac12\sum_{i=1}^4\sigma_i\,dx_i, \qquad \sigma_i\in\{1,-1\}. \tag{25}\] All the \(dx_i\) lie in the same Gauss direction. Their sum, whenever nonzero, lies in that direction as well. Continuity gives the same conclusion at the omitted values wherever the extended map has nonzero differential. ◻

Proof. A one-term relation is impossible because a branch of \(D\) is nonconstant. A two-term relation identifies open subsets of \(D\) and a translate of \(D\) or \(-D\). Since \(D=-D\), triviality of its translation stabilizer forces the translate to be zero; this would identify two of the distinct opposite pairs, also impossible.

Suppose there is a three-term relation. The monodromy on the four unsigned pairs is transitive, since \(C\) is connected. Analytic continuation therefore supplies four constant relations, one omitting each index. Their coefficient matrix has zero diagonal and entries \(\pm1\) off the diagonal. Let \(J\) denote the all-ones matrix. Modulo two the coefficient matrix is \(J-I\) in dimension four, whose square is the identity over \(\mathbb F_2\). Its determinant is therefore odd, in particular nonzero. The corresponding homomorphism \(B^4\to B^4\) is an isogeny. The four moving branches lie in a fixed finite fiber of this isogeny and must all be constant, a contradiction.

Two four-term relations whose sign vectors are neither equal nor opposite differ in between one and three positions. Subtracting them makes twice a nonempty proper signed partial sum constant. That partial sum is itself locally constant, since \([2]\) has finite fibers, contrary to the cases just proved. Thus a four-term relation, if it exists, determines its sign selection uniquely up to complement. Monodromy must preserve that selection or exchange it with its complement. Every local monodromy has a fixed signed point: a partition of four which is not all-even is one of \(1+1+1+1\), \(2+1+1\), or \(3+1\), and there is a singleton in each case. The étale cover \(D\to C\) gives two fixed signed points above that singleton. The selection therefore cannot be exchanged with its opposite. The local monodromies generate the full monodromy, so the four chosen signed points form a global invariant subset of the eight sheets of \(D\to\mathbf P^1\). This contradicts connectedness of \(D\).

A constant component of \(a_\pm\) would give a constant four-term relation by Equation (24). Suppose image components from opposite parities satisfy \(Z_+=b-Z_-\). Choose a general smooth point \(z_+\in Z_+\) and its partner \(z_-=b-z_+\in Z_-\), and lift them along local inverse branches of \(a_+\) and \(a_-\) away from ramification and the exceptional Gauss parameters. The tangent directions at these paired image points agree. Lemma 13 identifies both directions with their Gauss parameters, so the corresponding local selections \(s_+,s_-\) lie over the same parameter and satisfy \(h(s_+)+h(s_-)=b\). Opposite parities have either one or three agreed signed points. Equation (26) then gives one of the excluded partial relations. ◻

Proof. Take the open of Lemma 12 and the common coherent assignment of Lemma 13.

Image classes and finiteness. For either parity, its eight sign vectors satisfy \[\sum_\sigma\sigma_i\sigma_j=8\delta_{ij}.\] If \(\omega\) is an invariant real two-form on \(B\), Equation (25) therefore gives, over a regular-value disc, \[\sum_{s\in\mathcal S_\pm}h(s)^*\omega =2\sum_{i=1}^4x_i^*\omega.\] The right side is exactly the sum over the eight branches \(\pm x_i\) of \(D\), since inversion fixes invariant two-forms. Integrating over the regular-value open, whose omitted points have measure zero, proves \[ (a_+)_*[C_+]=(a_-)_*[C_-]=[D] \quad\text{in }H_2(B,\mathbb R). \tag{27}\] Invariant forms represent all of \(H^2(B,\mathbb R)\), so this equality of cycle classes includes the degrees of all image maps and every connected component of both sources.

Consider addition \[F:C_+\times C_-\longrightarrow B,\qquad (p,q)\longmapsto a_+(p)+a_-(q).\] It is finite on every product of connected components. Otherwise a positive-dimensional fiber would contain a curve. Neither projection of that curve could be constant, since each factor map is nonconstant and hence finite onto its image. Both projections would therefore be surjective, forcing the two image curves to be opposite translates, contrary to Lemma 14. Properness now gives finiteness. Each component product is two-dimensional and its finite image is all of \(B\). Its total degree, summed over the component products, is, by Equation (27), \[ \deg F=[D]^2=(2H)^2=8. \tag{28}\] One can obtain this formula directly by intersecting one image cycle with a general translate of the negative of the other; the intersection multiplicities count the fibers of addition with the image-map degrees.

The pullback divisor. Fix a signed point \(x\) of a regular Gauss fiber. For each selection \(s\in\mathcal S_+\) containing \(x\), let \(\tau=s^c[-x]\). It differs from \(s\) in three signs, so \(\tau\in\mathcal S_-\), and Equation (26) gives \[ h(s)+h(\tau)=x. \tag{29}\] There are exactly four such selections \(s\). The resulting four points of \(C_+\times C_-\) are distinct, since their first selections differ.

As \(x\) varies locally on \(D\), each construction defines a smooth local curve in the source, parametrized by the common regular Gauss parameter and mapped isomorphically to an open subset of \(D\). At its points both factor differentials have image in the tangent line of \(D\) at \(x\), so \(dF\) has rank at most one. Its restriction to the constructed local curve is nonzero. Choose target coordinates \((r,s)\) with \(D=(s=0)\) and source coordinates \(u=r\circ F,v\), with the local curve given by \(v=0\). Then \[F(u,v)=(u,G(u,v)),\qquad G(u,0)=0,\qquad G_v(u,0)=0.\] Thus the multiplicity of \(F^*D\) along each constructed local curve is at least two.

Write \(F^*D=\sum_\alpha e_\alpha E_\alpha\), with \(E_\alpha\) running through the prime divisors on all component products. Every \(E_\alpha\) dominates \(D\), since \(F\) is finite. Put \(f_\alpha=[\mathbb C(E_\alpha):\mathbb C(D)]\). The valuation degree formula and Equation (28) give \[\sum_\alpha e_\alpha f_\alpha=8.\] Above a general complex point of \(D\), the normalization of \(E_\alpha\) has \(f_\alpha\) distinct local branches. The four branches constructed above already contribute at least \(4\cdot2\) to this sum. They exhaust it, and each has multiplicity exactly two. Several of these local branches may belong to the same prime divisor. This proves, on every component product, the Cartier divisor identity \[ F^*D=2E_0 \tag{30}\] for an effective integral divisor \(E_0\); the smooth source makes \(E_0\) Cartier.

Lifting to the double cover. Choose connected components \(C_1^0\subset C_+\) and \(C_2^0\subset C_-\). Pull back \(\pi:B'\to B\) separately along their maps and choose a connected component of each pullback. This gives finite étale surjective maps \(C_i'\to C_i^0\) and lifts \(a_i':C_i'\to B'\). Their sum \[F'=a_1'+a_2':C_1'\times C_2'\longrightarrow B'\] is finite and surjective: its composite with \(\pi\) is the finite surjective map obtained from \(F\) by the factor covers. Pullback of Equation (30) gives \(F'^*(\pi^*D)=2E\) on the smooth product \(S'=C_1'\times C_2'\).

Let \(s_E\) be the canonical section of \(\mathcal O_{S'}(E)\) and put \[N=F'^*L\otimes\mathcal O_{S'}(-E),\qquad u=F'^*(\pi^*d)/s_E^2\in H^0(S',N^2).\] The section \(u\) is everywhere nonzero. It defines an étale double cover \[\widetilde S'=\operatorname{Spec}_{S'} (\mathcal O_{S'}\oplus N^{-1})\longrightarrow S'.\] This is the normalization of \(S'\times_{B'}X\) and has an everywhere-defined morphism to that fiber product. Explicitly, where \(E\) has local equation \(f=0\), the pulled-back equation of \(X\) is \(z^2=uf^2\). Its normalization is \(y^2=u\), and the morphism back is \(z=fy\). This formula also applies at singular points and intersections of \(E\) and glues by the definition of \(N\).

The étale double cover \(\widetilde S'\to S'\) corresponds to a character \[\rho:\pi_1(C_1')\times\pi_1(C_2')\longrightarrow\mathbb Z/2.\] It is the sum of its two factor restrictions \(\rho_1,\rho_2\). Let \(C_i\to C_i'\) be the connected finite étale cover corresponding to \(\ker\rho_i\), using the identity cover if \(\rho_i=0\). These covers are algebraic and projective. The pullback of \(\widetilde S'\) to \(C_1\times C_2\) is trivial, so choosing either section and composing the preceding morphisms gives \(g:C_1\times C_2\to X\). Its composite to \(B'\) is finite surjective, so its image has dimension two. The surface \(X\) is irreducible: its nonempty reduced branch divisor makes its function-field square class nontrivial. Hence \(g\) is dominant and generically finite.

The cohomological correspondence. The projection formula gives \(g_*g^*=(\deg g)\mathop{\mathrm{id}}\) on rational cohomology, so \(g^*\) is injective. Suppose \(U_X\) has no nonzero rational \((1,1)\)-class. Then it has no nonzero Hodge map to \(\mathbb Q(-1)\): dualizing such a map with a polarization would give a rational \((1,1)\)-class in \(U_X\). Thus the two outer projections of \(g^*U_X\) in the decomposition \[\begin{split} H^2(C_1\times C_2,\mathbb Q)={}& \bigl(H^2(C_1)\otimes H^0(C_2)\bigr) \oplus\bigl(H^1(C_1)\otimes H^1(C_2)\bigr)\\ &\oplus\bigl(H^0(C_1)\otimes H^2(C_2)\bigr) \end{split}\] vanish. Choose \(p_i\in C_i\) and the algebraic projectors \[\pi_1^{C_i}=\Delta_{C_i}-[p_i\times C_i]-[C_i\times p_i].\] With \(e\) from Equation (19), the correspondence \[\bigl(\pi_1^{C_1}\boxtimes\pi_1^{C_2}\bigr) \circ{}^t\Gamma_g\circ e \in\mathop{\mathrm{CH}}^2(X\times C_1\times C_2)_\mathbb Q\] induces the asserted injection on the whole rational space \(U_X\). Proposition 11 supplies the additional Hodge hypothesis at a very general parameter. ◻

Proof. Start with any nonempty regular period chart. The simultaneous geometric open of Propositions 11 and 15 is dense, so choose a datum \((a_*,[d_*])\) in its intersection with this chart. In the same section fiber, choose an independent section \(d_\infty\) whose divisor is smooth and avoids two-torsion. Trivialize the section bundle, choose \(\ell(d_*)=1\) and \(\ell(d_\infty)=0\), extend these choices in a local frame, and shrink the abelian base. Smoothness and avoidance of two-torsion are open conditions, so \(\mathbf P(\ker\ell)\) contains a good section over every point of this base.

Use the normalization \(\ell(d)=1\) of Section [geom:section-charts] and choose a product polydisc \(A\times Q\) within the original chart and the simultaneous geometric open. Its period map is a local isomorphism. These choices precede any actual pencil. Affine lines in each section chart are the pencils \(d_t=d_0+t d_1\) on the original affine \(t\)-line, with \(\ell(d_0)=1\) and \(\ell(d_1)=0\). The good directions form a nonempty dense Zariski-open subset of \(\mathbf P(\ker\ell)\). Since the direction map on lines meeting \(Q\) is a surjective submersion, pencils with a smooth leading section avoiding two-torsion form a dense open in every nonempty admissible pencil open. Every pencil meeting \(Q\) lies in the geometric open at all but finitely many parameters. Its period map is nonconstant, since the period differential is injective on section directions.

Pull back the prescribed analytic conditions to \(H_m\subset A\times Q\). For each \(m\), the set \[A_m=\{a\in A:\{a\}\times Q\subset H_m\}\] is closed: it is the intersection, over \(q\in Q\), of the closed sets \(\{a:(a,q)\in H_m\}\). It has empty interior, since otherwise \(H_m\) would contain an open product and hence the connected chart by the identity theorem. The complement of \(\bigcup_m A_m\) is therefore residual in \(A\). For every \(a\) in this complement, each \(H_{m,a}\) is a proper closed analytic subset of \(Q\).

Fix such \(a\). In the open space of affine lines meeting \(Q\), put \[\mathcal L_{m,a}=\{\Lambda:\Lambda\cap Q\subset H_{m,a}\},\] where \(\Lambda\) denotes an affine line. This set is closed: every point of a limiting line inside \(Q\) is a limit of points on the approximating lines inside \(Q\). It has empty interior, since an open set of lines meeting \(Q\) sweeps out an open subset of \(Q\); equivalently, evaluation from the incidence space of lines and points is a submersion. Excluding these countably many closed nowhere-dense sets leaves a residual set in every nonempty open of admissible pencils. For any remaining line, \(\Lambda\cap Q\) is connected, and each condition restricts to a proper analytic subset of this one-dimensional domain, hence to a discrete and countable set. Their union is countable.

For the exceptional Hodge-tensor conditions on the connected smooth open of a whole pencil, enlarge the initial list to include every proper Hodge locus of a rational tensor in the marked local system. Continue markings on the universal cover. A tensor condition holding identically on any open would, by analytic continuation, hold on the selected pencil disc for one of the marked rational tensors already excluded there. Thus every such condition has discrete zeros on marked pencil charts. There are only countably many rational tensors, continuation choices, and charts, so only countably many exceptional pencil parameters.

Intersect the residual pencil choices with the dense open of good directions. The geometric bad parameters are finite, while the local analytic exceptions and the continued Hodge-tensor exceptions are countable. Any further pencil-specific countable set can be added after fixing that pencil. This proves the stated order and conclusions. ◻

Proof. Put \(A_V=V^*/V\), and denote its discriminant quadratic and bilinear forms by \(q:A_V\to\mathbb Q/2\mathbb Z\) and \(b:A_V\times A_V\to\mathbb Q/\mathbb Z\). On \(A_V\simeq\mathbf F_2^n\), the form \(\beta=2b\), valued in \(\mathbb Z/2\mathbb Z\), is nondegenerate and symmetric. Let \(E=\ker(x\mapsto\beta(x,x))\), so \(\dim E\geq n-1\). The restriction of \(\beta\) to \(E\) is alternating. The inverse image in \(E\) of a Lagrangian in its symplectic quotient by the radical is a totally isotropic subspace of dimension \[\frac{\dim E+\dim\operatorname{rad}(\beta|_E)}2 \ \geq\ \left\lfloor\frac{n-1}{2}\right\rfloor.\] On this subspace, \(b(x,x)=0\) implies that \(q(x)\) is integral modulo \(2\mathbb Z\); moreover \(q(x+y)-q(x)-q(y)=2b(x,y)=0\) modulo \(2\mathbb Z\). Thus \(q\) restricts to a linear functional with values in \(\mathbf F_2\). If \(n\geq5\), the subspace has dimension at least two, and this functional has a nonzero isotropic vector.

Adjoin the corresponding isotropic element to the lattice. The new lattice is even and has discriminant form \(H^\perp/H\), where \(H\) is the order-two subgroup generated by that vector. This reduces the discriminant length by two and leaves the rational quadratic space unchanged. Repetition gives an even overlattice \(M\) with \(\ell(A_M)\leq4\). Each enlargement is \(G\)-stable, since \(G\) acts trivially on the preceding discriminant group; its action on the new discriminant group is again trivial.

The sufficient primitive-embedding criterion of Nikulin [22] applies, since \[3>2,\qquad 19>14,\qquad 22-16=6\geq\ell(A_M)+2.\] Choose such an embedding. Its complement has signature \((1,5)\), and therefore contains a primitive positive integral vector \(h\). The complement is primitive in the unimodular K3 lattice, so \(h\) is also primitive in \(\Lambda_{\mathrm{K3}}\). Unimodular gluing identifies the discriminant forms of \(M\) and \(M^\perp\) with opposite signs. Consequently an isometry of \(M\) trivial on \(A_M\), together with the identity on \(M^\perp\), extends uniquely to \(\Lambda_{\mathrm{K3}}\). These extensions respect composition.

In particular, a root \(r\in V\), with \(r^2=-2\), remains an integral root throughout the construction. Its reflection is \[s_r(x)=x+(x,r)r.\] It is integral on every even overlattice under consideration and on \(\Lambda_{\mathrm{K3}}\), and is the identity on \(M^\perp\). Whenever a monodromy operator on \(V\) is a product of these reflections, its extension is the same product of K3 root reflections. ◻

Proof. The possible singular members. The morphism defined by \(|2H|\) factors through inversion and realizes the quotient as the classical quartic Kummer surface. A general line in the dual projective space meets its discriminant either at a simple Lefschetz contact in the smooth locus or at a hyperplane through one ordinary Kummer node. Here is the transversality assertion being used. The hyperplanes containing each Kummer node form a plane in the dual space. A general point of such a plane avoids every other node and is transverse to the quadratic tangent cone at that node. For tangencies in the smooth locus, the conormal incidence has generically separable projection to the dual variety. If its image has dimension less than two, a general line avoids it. If it has dimension two, its general smooth point represents a unique contact, because that contact point is recovered as the tangent hyperplane to the dual variety. Separability makes the quadratic contact nondegenerate. The smooth-contact dual component cannot be a node plane, by biduality. The higher contacts, intersections of these discriminant components, and their singular loci have codimension at least two in the dual projective space. A general line avoids them and meets each remaining component transversely. Upstairs on \(B\), the first event gives two ordinary nodes exchanged by inversion. At a two-torsion point in the second event, the invariant local equation has a nondegenerate quadratic term and its constant term vanishes to first order in \(t\). Thus the branch has an ordinary node with transverse smoothing. The base points of a general pencil are smooth on every member, since the differentials of \(d_0\) and \(d_1\) there are independent. Pulling back along the isogeny to \(B'\) gives disjoint copies of these events. We examine their inversion quotients.

An exchanged pair. At an exchanged pair away from the fixed points, the branch curve on \(K=\mathop{\mathrm{Km}}(B')\) acquires one ordinary node. The local double surface is \(z^2=xy+t\). Its vanishing sphere has square \(-2\), and the deck involution \(\delta\) negates it. It therefore gives an integral root of \(V\) and its Picard–Lefschetz reflection.

A positive-sign fixed point. At a fixed point with nonbranch exceptional curve on \(K\), choose coordinates in which the upstairs model is \[z^2=x^2+y^2+t,\qquad j(x,y,z)=(-x,-y,z).\] These coordinates can be chosen equivariantly: the branch equation is even in \((x,y)\), and the holomorphic Morse lemma, applied to its nondegenerate even quadratic-coefficient matrix, preserves that parity. The parameter is replaced locally by its constant term, which is a coordinate because the discriminant is met transversely. On the chart \(a=x^2,\ v=y/x\) of the resolved base quotient, the double surface has equation \[z^2=a(1+v^2)+t.\] At \(t=0\), its branch is the exceptional curve \(a=0\) together with a residual curve meeting it at \(v=i,-i\). At each crossing the equation is \(z^2=ab+t\). There are consequently two disjoint ordinary node smoothings, and two orthogonal integral anti-invariant roots in \(V\). The local monodromy is their product. The difference of the two lifts of the nonbranch exceptional curve, of square \(-4\), need not be fixed by this monodromy; it is not being used as one of the individual roots.

A negative-sign fixed point. At a fixed point with branch exceptional curve, the involution in the upstairs quadratic model is instead \[j(x,y,z)=(-x,-y,-z).\] It acts antipodally on the real vanishing sphere \(S^2\), and hence by \(-1\) on its degree-two class. There is no vanishing class in the \(j\)-invariant quotient cohomology. To check the resolution contribution, for \(t\ne0\) the points \((0,0,\pm\sqrt t)\) are exchanged by \(j\). Their quotient is one smooth point. On the chart represented near one chosen square root, the induced sheet involution is \((x,y)\mapsto(-x,-y)\). Blowing up this point therefore gives a \(\mathbf P^1\) fixed pointwise by \(\delta\); its cohomology class is \(\delta\)-invariant. More explicitly, on the Kummer-resolution chart \(a=x^2,\ v=y/x\), put \(\eta=xz\). The finite double-cover model is \[\eta^2=a\bigl(a(1+v^2)+t\bigr),\qquad \delta(\eta)=-\eta.\] For \(t\ne0\), the preimage of \(a=0\) is that ramification curve; its square is \(-1\), because the base exceptional curve has square \(-2\). Its class is the only added resolution class at this point. Thus this event contributes no negative eigenspace to \(V\).

The full monodromy operator. All the displayed roots are integral classes on the nearby smooth \(Y\). They are mutually orthogonal because their Milnor neighborhoods are disjoint, and the local monodromy negates each of them. To see that there is no further monodromy, use the orthogonal decomposition of variations in Proposition 9, \[V_\mathbb Q=U\oplus\langle a_1,\ldots,a_8\rangle_\mathbb Q, \qquad U\simeq U_X.\] On \(X\), the complement of the disjoint node neighborhoods is locally topologically trivial, so Picard–Lefschetz gives the product of the corresponding commuting reflections. An exchanged pair has a one-dimensional \(j\)-invariant vanishing space. At a positive fixed point, \(j\) has degree \(+1\) on the vanishing sphere and contributes one dimension to \(U_X\). At a negative fixed point it has degree \(-1\) and contributes none. On the difference lines \(\mathbb Qa_i\), the two labels are exchanged by \(\sqrt t\mapsto-\sqrt t\) exactly at the affected positive fixed points; all other difference lines are fixed. Consequently the full operator on \(V_\mathbb Q\) is an involution whose negative eigenspace has one dimension per exchanged pair and two per positive fixed point. This is precisely the number of mutually orthogonal roots found above. Those roots span that negative eigenspace, proving that the entire operator is their product of reflections. The same argument includes all disjoint lifts of a single event through the isogeny.

The limiting periods. After a quadratic cover at each ordinary node, the upstairs surface family has a simultaneous local analytic resolution. In the \(A_1\) case used here, write \(xy=(z-v)(z+v)\) and blow up the non-Cartier ideal \((x,z-v)\). On the chart \(z-v=xa\), the equations are \(z=v+xa\) and \(y=a(2v+xa)\), so \((x,a,v)\) are smooth coordinates. On the other chart \(x=(z-v)b\), they are \(z=by-v\) and \(x=b(by-2v)\), with smooth coordinates \((b,y,v)\). Both projections to \(v\) are smooth and their central fibers resolve the node. The relative two-form \(dx\wedge dz/x\) becomes \(dx\wedge da\) on the first chart, while the equal expression \(-dy\wedge dz/y\) becomes \(db\wedge dy\) on the second. Thus the resolution is crepant. These are the local \(A_1\) charts of the classical simultaneous-resolution constructions [7, 3]. They glue to the unchanged locus because the exceptional neighborhoods are disjoint.

Choose a nonzero \(\theta_+\in H^0(B',L)^{j=+1}\). Using an invariant volume form on \(B'\), the Hodge line of \(U_X\) has frame \[\frac{\theta_+\,dx\wedge dy}{z}.\] Crepantness extends it regularly after the local base change and resolution. It remains a nonzero global form on the resolved central surface: test at a point of \(B'\) outside the branch divisor and the zero divisor of \(\theta_+\). At a minus-sign node its coefficient is odd and vanishes at that node, but this does not make the global Hodge-line frame vanish over the parameter.

The chosen simultaneous resolution need not carry the involutions. Their flat actions on cohomology nevertheless extend after the base change, preserve the limiting Hodge filtration by continuity, and retain the prescribed eigenspaces. The nonzero limiting holomorphic form on the projective resolved central surface has positive norm. Thus \(U_X\), and hence \(U\), has an interior limiting period. The section just used was single-valued before the local base change, so inertia fixes its limiting vector. In particular the negative eigenspace of local monodromy is perpendicular to that vector. The additional exceptional-difference classes remain algebraic, giving the same conclusion for the roots in \(V\).

Infinity. Near infinity write \(q_\infty=t^{-1}\). The equation is \(z^2=q_\infty^{-1}(d_1+q_\infty d_0)\). The leading member \(d_1\) is smooth and avoids the fixed points. After adjoining \(\sqrt{q_\infty}\) and rescaling \(z\), the family extends smoothly. The only twist around \(q_\infty=0\) is the change of sign of this square root. This is \(\delta\), which acts as \(-\mathop{\mathrm{id}}\) on \(V\). ◻

Proof. The existence and factorization of \(q\) are [16]. We use the analytic form of that morphism. Its hypotheses are primitive ample polarization classes on ADE K3 surfaces, primitive big and nef classes on smooth K3 surfaces, and the full integral period stack over \(\mathbb C\). The stable group and the representable, surjective étale period map of the quasi-polarized stack are described in [16]. Taking one component of the period domain with its stabilizer gives the equivalent stack used here. We now prove the two stated compatibilities.

Let \(\mathscr K_h\) be the smooth quasi-polarized K3 stack. Over a local étale period chart its universal smooth family contracts to the pullback of the ADE family. The latter is flat with Gorenstein fibers, so its relative dualizing sheaf is invertible. Off the exceptional curves, contraction identifies the relative two-forms. In the total smooth family over this full period chart the exceptional locus has codimension at least two: the curves occur only over root walls. The identification of the two line bundles therefore extends over the exceptional locus. On each fiber it is the usual crepant identification for resolution of rational double points. This gives the claimed comparison of Hodge lines after pullback to \(\mathscr K_h\).

For descent, consider \(\mathscr K_h\times_{\mathscr P_h}\mathscr K_h\). It is locally smooth over the period domain, and the ample locus is dense in every component. There each period arrow comes from an actual polarized isomorphism by Torelli, so the two induced Hodge-line identifications agree. Equality extends over this reduced relation space. Hence the comparison descends with its entire \(\Gamma_h\)-linearization, including on the root mirrors.

Fix a root \(r\in h^\perp\) and consider its mirror. Its reflection \(s_r\) fixes the period vector, so its image under \(q\) acts trivially on the Hodge line. At a very general point of the mirror, the resolution has rational Picard space \(\mathbb Qh\oplus\mathbb Qr\). The polarized automorphism induced by \(q(s_r)\) lifts uniquely to that resolution. It fixes \(h\), and preservation of effectivity forces it to fix the effective choice of \(r\). It is symplectic. The transcendental Hodge structure is simple, being polarizable of K3 type with no nonzero rational \((1,1)\)-classes. A Hodge endomorphism fixing its \((2,0)\)-line pointwise is the identity: the kernel of the difference from the identity is a nonzero Hodge substructure. The lifted automorphism thus acts identically on all second cohomology and is the identity by Torelli.

On the whole mirror, \(q(s_r)\) and the identity are sections of a separated isomorphism space. Their equality on the dense locus just considered extends everywhere. Since root reflections generate \(W_\sigma\), this group is killed even at points with several or persistent contracted roots. This agrees with the generic root-reflection kernel identified in [16].

Conversely, an integral Hodge isometry fixing \(h\) takes the ample chamber of a chosen resolution to another chamber whose closure contains \(h\). The Weyl group generated by the algebraic roots perpendicular to \(h\) acts transitively on these chambers: only walls through \(h\) separate chambers incident to \(h\). After composition with an element of \(W_\sigma\), strong Torelli realizes the isometry by an automorphism of the resolution. By the factorization defining \(q\), its image is the induced automorphism of the contraction. If the image is the identity, its lift is the identity on the dense complement of the exceptional curves and hence everywhere. The original isometry then belongs to \(W_\sigma\). ◻

Proof. The family and its algebraization. The torsor of markings of \((\mathcal L,h)\), with a component chosen by \(\mathcal F\), and the equivariant period map define a holomorphic map \(C^\circ\to\mathscr P_h\). Its monodromy lies in \(\Gamma_h\): it fixes \(h\), has trivial restricted discriminant action by unimodular gluing, and preserves the period component by continuity. Pullback along \(q\) in Lemma 20 gives an actual analytic family of polarized ADE K3 surfaces on \(C^\circ\), with its compatible descent data.

Choose a finite Galois cover \(\widetilde C\to C\) whose ramification indices at the points above \(D\) are divisible by the local monodromy orders. Such a cover exists: choose a common multiple \(N\) of these orders, use Riemann–Roch with sufficiently large poles away from \(D\) to prescribe a simple zero at each point of \(D\), and normalize the cyclic cover obtained by adjoining an \(N\)-th root of that function. Additional branch points lie over points where the original family is already defined. If \(D\) is empty the identity cover suffices. The pulled-back local system and period extend across the punctures: in a flat trivialization the latter is the holomorphic interior extension in hypothesis (iii). They define a map \(\widetilde C\to\mathscr P_h\), and hence a proper analytic family \(\widetilde{\mathcal Z}\to\widetilde C\) of polarized ADE K3 surfaces. The original monodromy data equip it, functorially through \(q\), with an action of the Galois group. All identifications and cocycles are provided by this stack morphism; no choices of smooth simultaneous resolutions have to glue.

This compact analytic family is algebraic and normal. For normality, use flatness and the normal ADE fibers over the smooth base. To check the projectivity needed for GAGA, represent its relative polarizing class by line bundles on small open sets of the base. A common sufficiently divisible power is relatively ample. On an overlap, the ratio of two such representatives is trivial on each fiber. Its direct image is a line bundle, and evaluation is an isomorphism, by cohomology and base change and \(H^1(\mathcal O)=0\) on the fibers. Thus the ratio comes from the base. After local trivializations, the remaining gluing obstruction is a scalar cocycle in \(H^2(\widetilde C,\mathcal O_{\widetilde C}^*)\). The exponential sequence makes this group zero, since \[H^2(\widetilde C,\mathcal O_{\widetilde C})=0, \qquad H^3(\widetilde C,\mathbb Z)=0.\] The polarizing power consequently glues to a relatively ample line bundle. For a sufficiently high power its direct image is a vector bundle, and the family embeds into the associated projective bundle on \(\widetilde C\). That vector bundle and the closed embedded family are algebraic by GAGA [36]. More explicitly, first algebraize the vector bundle on the projective curve, then algebraize the coherent ideal of the family in its projective bundle. The finite group action algebraizes by the same fully faithful comparison. Twist the relative ample bundle by a sufficiently positive base bundle to make it ample on the total space. The tensor product of its translates by the Galois group is ample and has its natural linearization; invariant sections of a sufficiently high power give a projective finite quotient. That quotient is a normal projective model \(\mathcal Z\to C\). Over \(C^\circ\) its construction is descent of the pullback of the initial family, also at any auxiliary branch points. This proves the first assertion, with no cover remaining in its conclusion.

Descent at a Weyl puncture. Assume [pencil:rootcondition]. On a disc killing the monodromy, write the map to the original disc as \(t=w^m\). The cyclic inertia group acts on the central polarized surface via the homomorphism on stabilizers induced by \(q\). Its image is the identity by Lemma 20. Thus the group acts pointwise on the whole central surface as the identity, including its singular points.

Here is the corresponding scheme-theoretic descent calculation. Let \(A\) be a local completed ring of \(\widetilde{\mathcal Z}\) along that fiber. Flatness makes \(w\) a nonzerodivisor. If \(g\) generates inertia, then \(g(w)=\zeta w\), with \(\zeta\) primitive of order \(m\), and \(g\) is the identity on \(A/(w)\). Set \(B=A^{\langle g\rangle}\). Reynolds averaging makes \(B\to A/(w)\) surjective. If \(a=wb\) is invariant, then \(g(b)=\zeta^{-1}b\). For \(m>1\), reduction modulo \(w\) forces \(b\) to be divisible by \(w\). At each subsequent division the character changes by \(\zeta^{-1}\); until the \(m\)-th division it is nontrivial and the reduction must again vanish. This gives, also trivially when \(m=1\), \[B\cap wA=w^mB,\qquad B/(t)\simeq A/(w).\] There is also an exact base-change statement. Write \(A_j=\{a:g(a)=\zeta^j a\}\), for \(0\leq j<m\). Successive division by \(w\) gives \(A_j=w^jB\); character projection and the fact that \(w\) is a nonzerodivisor then give \[A=\bigoplus_{j=0}^{m-1}w^jB, \qquad B\otimes_{\mathbb C[[t]]}\mathbb C[[w]]\xrightarrow{\ \sim\ }A, \quad t\longmapsto w^m.\] These assertions pass from completed rings to the local rings by faithful flatness of completion. Thus the quotient central fiber is exactly the original reduced ADE surface, and the upstairs family is its flat pullback. The quotient is flat over the parameter disc, since \(t\) is a nonzerodivisor in \(B\), a torsion-free module over the discrete valuation ring of the disc. This proves the assertion about all fibers and rules out a multiple central fiber.

The underlying ADE family has now descended. To retain its primitive polarization, choose on a smaller invariant analytic \(w\)-disc a line bundle \(L\) representing its relative Picard class; the preceding scalar-cocycle argument applies on the disc as well. The same base-change argument used above shows that \(g^*L\otimes L^{-1}\) comes from the disc and that automorphisms of \(L\) are base units; the base line is trivial after shrinking. For a \(g\)-semilinear lift \(T\), write \(T^m=c\,\mathop{\mathrm{id}}\), where commutation with \(T\) makes \(c=c(t)\) an invariant base unit. Replacing \(T\) by \(aT\) replaces \(c\) by \(c\prod_{i=0}^{m-1}g^i(a)\). Every unit in \(\mathbb C\{t\}\) has an \(m\)-th root after shrinking, whose norm as an invariant upstairs unit is its \(m\)-th power. Taking such a root of \(c^{-1}\) therefore arranges \(T^m=\mathop{\mathrm{id}}\). On the proper connected reduced central fiber, \(T\) is one scalar \(\chi\in\mu_m\). Replacing \(T\) by \(\chi^{-1}T\) makes it the identity on \(L/(w)L\), without changing the relative Picard class or the separately prescribed Hodge-line linearization.

At any central point, including an ADE point, work with the actual analytic local rings and let \(s\) be a local generator of \(L\). Then \(s_*:=m^{-1}\sum_{i=0}^{m-1}T^i(s)\) is invariant and congruent to \(s\) modulo \(w\), hence is still a generator. The invariant local modules are consequently free of rank one over the invariant local rings and pull back to \(L\). Their invariant transition units glue them to an actual invertible sheaf on the analytic quotient. Under \(B/(t)\simeq A/(w)\) its central restriction is exactly \(L_0\), so the primitive ample class, not a power, descends. These local classes agree with the original one off the punctures and glue on \(C\); the preceding scalar-cocycle argument glues their primitive representatives, which GAGA algebraizes on the projective quotient.

The Hodge line must descend with its prescribed character, separately from the scalar adjustment of the polarizing bundle. Its limiting inertia action is trivial: root reflections fix a vector perpendicular to their roots. Averaging a local frame which is nonzero at the center gives an invariant frame with that same central value, and hence a nowhere-vanishing frame on a smaller disc. Consequently the extended Hodge line descends to a line bundle \(\lambda\) on \(C\). By Lemma 20, it is the Hodge line of the descended family, with precisely the linearization prescribed by \(\mathcal L\). No character twist has been discarded.

Every ADE K3 fiber is Gorenstein with trivial dualizing sheaf. A flat family with these Gorenstein fibers over a smooth curve is a Gorenstein morphism, so \(\omega_{\mathcal Z/C}\) is invertible. Its direct image is a line bundle and formation commutes with base change, since every fiber has a one-dimensional space of canonical forms. Evaluation is an isomorphism on each fiber and therefore everywhere. The identification of this direct image with \(\lambda\) proves (31). For the subsequent intersection argument, its restriction to codimension one would already suffice.

A smooth family on an open of the original curve. This part uses the normal contraction family from the first assertion, not the additional Weyl condition. We resolve its generic fiber over \(k=\mathbb C(C)\), then recover \(\mathcal U\) by comparing local period lifts through the cohomology of that same contraction. The generic polarized ADE surface has a projective minimal resolution over \(k\). One way to see the field assertion is to resolve its rational double points over a finite Galois extension. Every Galois translate is a minimal resolution of the same surface; the unique isomorphism over that surface supplies a descent datum and automatically satisfies its cocycle. A norm of a relatively ample bundle makes this descent projective. The descended resolution is defined over \(k\) itself. Spreading the morphism and its projective embedding, and then shrinking an algebraic open of \(C^\circ\), gives a smooth projective K3 family \(\mathcal S^0\to C_0\) resolving the fixed contraction family. All fibers and exceptional configurations can be assumed to be of the generic type on this open.

Here are the details of the cohomological comparison on \(C_0\). For the minimal resolution \(\pi:S\to Z\) of a complex ADE K3 surface, let \(E_Z\subset H^2(S,\mathbb Q)\) be spanned by its exceptional curves. The rational cohomology sequence for the resolution gives \[\pi^*:H^2(Z,\mathbb Q)\hookrightarrow H^2(S,\mathbb Q), \qquad \mathop{\mathrm{im}}(\pi^*)=E_Z^\perp.\] Indeed, the exceptional fibers are trees of rational curves, with \(H^1=0\), and the restriction map in degree two records intersection with their components. It is onto on their span because the ADE intersection matrix is negative definite. This proves both the injectivity and the stated image. In a resolving family these pullbacks are maps of higher direct-image sheaves. Shrinking \(C_0\) once more makes the sheaves local systems, by constructibility for the algebraic proper morphisms. They preserve Hodge structures and the cup-product forms on the indicated orthogonal complements.

Locally lift the marked period map through the étale map \(\mathscr K_h\to\mathscr P_h\). Its K3 family has an identification of its contraction with the fixed family obtained from \(q\). The exceptional root lattice is locally constant on \(C_0\); all its roots are therefore persistent in the sense of the hypothesis, and are perpendicular to \(\mathcal U\). The marked copy of \(\mathcal U\) in this local lift consequently lies in the image of \(H^2\) of the contraction. The displayed pullback description gives a flat, isometric map from it to \(R^2\mathcal S^0_*\mathbb Q\). For two such local lifts, compare their minimal resolutions over the identity of their common contraction on each fiber. The resulting marking discrepancy has trivial image under \(q\), so belongs to the Weyl group in Lemma 20. This group fixes \(\mathcal U\) pointwise. Thus the maps just constructed agree on overlaps, including their flat structures, and glue over \(C_0\). Nondegeneracy of \(\mathcal U\) makes its image an orthogonal Hodge summand. This proves the final assertion on the original curve.

For clarity about the shrinking in this argument, the mirrors of the \(h\)-perpendicular roots are locally finite in \(\mathscr D_h\). On a compact set of periods, a root orthogonal to \(\langle h,\Re\sigma,\Im\sigma\rangle\) has bounded norm in a fixed positive definite metric, since its square is \(-2\). Only finitely many lattice vectors can occur. A finite collection of such local period charts covers the compact finite cover used above. In each chart a nonpersistent mirror has only isolated intersections with the period curve. Local finiteness and compactness therefore leave only finitely many such intersection parameters. Removing their images, as well as the exceptional points from spreading, gives the open used here. The argument uses constant root type there; it requires no uniqueness assertion for simultaneous resolutions across jumps in the root configuration. ◻

Proof. Let \(p:\widetilde{\mathcal Z}\to\mathcal Z\) be a smooth projective resolution, and let \(A\) be sufficiently ample on \(\mathcal Z\). Since \(\mathcal Z\) is normal and Gorenstein, the difference \(K_{\widetilde{\mathcal Z}}-p^*K_{\mathcal Z}\) is supported on the exceptional divisors. Every such divisor has zero intersection with \((p^*A)^2\): its image has codimension at least two, and the projection formula applies. Geometrically, two general members of \(|A|\) have a complete-intersection curve avoiding all these images. Equation (31) consequently gives \[K_{\widetilde{\mathcal Z}}\cdot(p^*A)^2 =\deg(K_C\otimes\lambda)\,(A|_{\mathcal Z_c})^2<0.\] Here \(\mathcal Z_c\) is any general fiber, and its displayed self-intersection is positive by ampleness. The divisor \(p^*A\) is nef, so every effective divisor, and by continuity every pseudo-effective divisor, has nonnegative intersection with \((p^*A)^2\). Thus \(K_{\widetilde{\mathcal Z}}\) is not pseudo-effective. The theorem of Boucksom–Demailly–Păun–Peternell [6] applies to this smooth projective threefold and gives uniruledness. This is a birational property, proving the assertion for every smooth projective model. ◻

Proof of Proposition 17. Realization over the \(t\)-line. By Proposition 9, the monodromy of \(V\) is trivial on its discriminant group. Apply Lemma 18, extending that monodromy to the K3 lattice and choosing its fixed positive class \(h\). The lattice embedding preserves the form inherited from \(Y\). Give this lattice the period line of \(V\), which is also that of \(U\). Over \(\mathbb Q\) its orthogonal decomposition is \[\Lambda_{\mathrm{K3},\mathbb Q}=U\oplus\langle a_1,\ldots,a_8\rangle_\mathbb Q \oplus M_\mathbb Q^\perp.\] The last two summands, of dimensions eight and six, are of type \((1,1)\), so the complement of \(U\) has dimension fourteen. For the choices specified in Lemma 16, and a very general parameter, \(U\) has no rational \((1,1)\)-classes by Proposition 11. If a locally flat rational root remains algebraic on a nonempty open, its projection to \(U\) in the displayed orthogonal decomposition is a rational flat class of type \((1,1)\) there. Analytic continuation keeps it of that type wherever its flat continuation is defined. At a very general point this projection is zero, so it is identically zero. Every persistent root is therefore perpendicular to \(U\). Lemmas 19 and 21 now give, over a nonempty open of the original line, the asserted smooth projective K3 family with the specified \(U\)-variation. Its orthogonal complement has dimension fourteen and is entirely of type \((1,1)\). Lefschetz’s \((1,1)\)-Theorem and the absence of rational \((1,1)\)-classes in \(U\) show that its very general transcendental part is exactly \(U\). This determines the rational Hodge summand and its pairing; the integral Néron–Severi lattice of the auxiliary K3 surface is not specified by this construction.

The quadratic substitution. Let \(\Delta\) be the finite union of the critical values and \(\mathbf A^1_t\setminus O\). For every \(s\notin\Delta\), make the substitution \(t=s+u^2\). Its finite branch value is \(t=s\), a point of the smooth open \(O\). It is unramified over every finite critical value, so those monodromies remain the products in Lemma 19. At any other removed finite value the original period data extend, and the local monodromy is the identity. At infinity the ramification index is two and kills the square-root twist. Thus the full extended K3 lattice has trivial monodromy there: the square is the identity on \(M\), and the action on \(M^\perp\) was already the identity. Every remaining finite monodromy is a product of integral root reflections fixing \(h\) and the limiting period vector, by Lemma 18. All local monodromies therefore satisfy the Weyl condition in Lemma 21. Apply that lemma on \(\mathbf P^1_u\) to obtain a flat projective ADE K3 family \(f:\mathcal Z\to\mathbf P^1_u\) and its Hodge line \(\lambda\).

The Hodge line and uniruledness of the ADE model. We first prove uniruledness for this model using its Hodge line, whose degree is exactly one. We then identify its function field with the square pullback of the fixed \(S^0\)-family.

Indeed, the compactified upstairs double cover is defined over \(B'\times\mathbf P^1_u\) using \(L\boxtimes\mathcal O_{\mathbf P^1}(1)\). In homogeneous coordinates \([u_0:u_1]\) its branch section is \((d_0+s d_1)u_0^2+d_1u_1^2\), with pullback to \(B'\) understood. On the affine chart \(u=u_1/u_0\) its equation is \[z^2=d_0+(s+u^2)d_1.\] The double-cover formula for the relative dualizing sheaf, with \(K_{B'}\simeq\mathcal O_{B'}\), gives the anti-sheet part of the direct image of relative canonical forms as \[H^0(B',L)\otimes\mathcal O_{\mathbf P^1}(1).\] Taking \(j=+1\) picks out the fixed one-dimensional eigenspace of \(H^0(B',L)\) from Proposition 9, hence gives \(\mathcal O_{\mathbf P^1}(1)\). Choose a nonzero \(\theta_+\in H^0(B',L)^{j=+1}\). For completeness, with \(v=1/u\) and \(w=z/u\) the equation becomes \[w^2=d_1+(d_0+s d_1)v^2,\] which is smooth at infinity. Local relative forms transform as \[\frac{\theta_+\,dx\wedge dy}{z} =v\,\frac{\theta_+\,dx\wedge dy}{w},\] the transition for \(\mathcal O(1)\). In particular the displayed frame on the finite chart has a simple zero, measured against the nonzero frame on the infinity chart, at \(v=0\). At every finite nodal fiber, Lemma 19 shows that the frame extends after the local resolving cover and is a nonzero global form on the central resolution. There is therefore no additional integral zero or pole at a finite parameter. The limiting inertia acts trivially on that form: its vanishing roots are perpendicular to the limiting period vector. The invariant-frame argument in Lemma 21 consequently introduces no fractional local weight either. The Hodge-line identification in Lemma 20 includes these exact characters, so the period realization transfers this extension to \(\lambda\). We obtain \(\lambda\simeq\mathcal O_{\mathbf P^1}(1)\), as line bundles with the stated local extensions.

The canonical bundle of this normal polarized contraction model is \[f^*(\mathcal O_{\mathbf P^1}(-2)\otimes\lambda) =f^*\mathcal O_{\mathbf P^1}(-1).\] Thus \(\deg(K_{\mathbf P^1}\otimes\lambda)=-2+1=-1\), and Corollary 22 shows that every smooth projective model of its function field is uniruled.

The algebraic square-pullback function field. We use a common finite cover for this construction and the fixed construction over the \(t\)-line. Put \(K=\mathbb C(t)\), \(K_s=\mathbb C(u)\) with \(t=s+u^2\), and \(\widetilde K=\mathbb C(\widetilde C)\), where \(\widetilde C\to\mathbf P^1_t\) is the finite Galois cover chosen in Lemma 21. Choose the fields in a common algebraic closure of \(K\), put \(\widetilde K_s=\widetilde K K_s\), and let \(\widetilde C_s\) be the normalization of \(\mathbf P^1_u\) in \(\widetilde K_s\). This is the normalization of an irreducible component of \(\widetilde C\times_{\mathbf P^1_t}\mathbf P^1_u\). Since \(\widetilde K/K\) is Galois, so is \(\widetilde K_s/K_s\); its group \(G_s\) is identified by restriction with a subgroup of \(G=\operatorname{Gal}(\widetilde K/K)\).

This common cover is admissible in the second application of Lemma 21. Indeed, at a point where the original cover has ramification index \(e\), the substitution has index \(d\), and the original local monodromy has order \(m\mid e\), normalization of the fiber product gives index \(e/\gcd(e,d)\). The pulled-back monodromy has order \(m/\gcd(m,d)\), which divides that index. This also covers auxiliary branch points, where \(m=1\). Thus all pulled-back local monodromies are killed. We take the extended period map on \(\widetilde C_s\) to be the pullback of the previous extended map, with the restricted equivariance data.

Write \(\widetilde{\mathcal Z}_t\to\widetilde C\) for the algebraized family in the first construction. Functoriality of the stack morphism \(q\) identifies the analytic family on \(\widetilde C_s\), together with its \(G_s\)-action, with the analytification of the cartesian algebraic pullback \[\widetilde{\mathcal Z}_t\times_{\widetilde C}\widetilde C_s \longrightarrow \widetilde C_s.\] This pullback is already flat and projective with ADE K3 fibers. If the new analytic family is algebraized separately as in the lemma, the fully faithful GAGA comparison used there, applied to the projective total spaces over \(\widetilde C_s\), algebraizes the displayed analytic identification and its compatible group action. We may therefore use this literal algebraic pullback, with the induced \(G_s\)-action, in forming the second quotient.

Let \(Z_K\) be the generic fiber of the original contraction over \(K\). Original descent identifies \(\widetilde{\mathcal Z}_{t,\widetilde K}\) with \(Z_K\times_K\widetilde K\), including its \(G\)-descent datum. After extending scalars to \(\widetilde K_s\) and restricting to \(G_s\), this is exactly the descent datum of \(Z_K\times_K K_s\). Uniqueness of effective finite Galois descent therefore identifies the generic fiber of the second quotient with \(Z_K\times_K K_s\). The fixed smooth generic K3 surface \(S^0_K\) is a resolution of \(Z_K\); its base change is still birational to \(Z_K\times_K K_s\), since geometric integrality ensures that the base change retains the dense open on which the resolution is an isomorphism. Consequently the second model \(\mathcal Z\) has the function field of the actual algebraic square pullback of the fixed \(S^0\)-family. The uniruledness just proved therefore applies to every smooth projective model of that fixed pulled-back family, as required. ◻

Proof. Uniruledness gives a generically finite dominant rational map \(T\times\mathbf P^1\dashrightarrow Z\) for a smooth projective surface \(T\). Indeed, choose a two-dimensional parameter space of rational curves dominating \(Z\) and, if necessary, make a generically finite change of its parameter surface to trivialize the generic rational curve. Principalize the base ideal of the rational map in the smooth product, using projective blowups with smooth centers [42]. This gives a morphism \(a:\widetilde{T\times\mathbf P^1}\to Z\). The identity \(a_*a^*=(\deg a)\mathop{\mathrm{id}}\) gives an algebraic splitting of \(H^3(Z)\) from the third cohomology of the source.

The product formula and the blowup formula give \[H^3(T\times\mathbf P^1)=H^3(T)\oplus H^1(T)(-1),\] and each blowup along a curve adds its \(H^1(-)(-1)\); a point adds nothing in degree three. The product projections are algebraic, and the full blowup decompositions are given by algebraic projectors [26]; see also [19] for the Chow blowup formalism. To see the sign in the only new summand, let \(\pi:\widetilde Z\to Z\) be a blowup along a smooth curve \(C\), with exceptional divisor \(j:E\hookrightarrow\widetilde Z\) and \(q:E\to C\). Its inclusion is \(j_*q^*\) and its retraction is \(-q_*j^*\), since \[q_*j^*j_*q^*y =q_*\bigl(c_1(\mathcal O_E(-1))\smile q^*y\bigr)=-y.\] The other summand has inclusion \(\pi^*\) and projection \(\pi_*\). These maps and the product projections provide the required algebraic splittings at every step of the resolution.

For completeness, the needed splittings for a surface use only divisor classes. Choose a smooth ample curve \(i:C\hookrightarrow T\). Hard Lefschetz says that multiplication by \([C]\) is an isomorphism \(H^1(T)(-1)\to H^3(T)\). Since this is \(i_*i^*\), restriction \(i^*:H^1(T)\to H^1(C)\) is injective and \(i_*:H^1(C)(-1)\to H^3(T)\) is surjective. Polarizability gives a Hodge retraction \(r\) of \(i^*\) and a Hodge section \(s\) of \(i_*\). Poincaré duality identifies their tensors with rational classes of type \((0,0)\) in, respectively, \[H^1(C)\otimes H^1(T)(1) \quad\hbox{and}\quad H^1(T)\otimes H^1(C)(1).\] The underlying untwisted classes have type \((1,1)\) on \(C\times T\) and \(T\times C\), so the Lefschetz \((1,1)\) theorem represents them by rational divisors. Their actions on the indicated degrees are \(r\) and \(s\). Together with restriction and Gysin, they split both surface summands through \(H^1(C)(-1)\). Taking products with the point and fundamental classes of \(\mathbf P^1\) realizes these splittings in the product decomposition above. Composing with \(a^*\) and \((\deg a)^{-1}a_*\) proves the first claim.

Finally, a Hodge map between two finite sums of \(H^1(C_i)(-1)\) is given by divisor correspondences on products of the curves. Composing with the splittings gives a cycle in \(\mathop{\mathrm{CH}}^3(Z\times Z')_\mathbb Q\) with exactly the prescribed action on \(H^3\). ◻

Proof. An affine complex curve has the homotopy type of a one-dimensional CW complex. For local systems \(L,M\), \[\operatorname{Ext}^r_B(L,M) =H^r(B,L^\vee\otimes M)=0\qquad(r>1).\] For two distinct nonzero even degrees \(i<j\), the obstruction to splitting the corresponding truncation triangle lies in an \(\operatorname{Ext}^{j-i+1}\) group, and the ambiguity lies in an \(\operatorname{Ext}^{j-i}\) group. Their degrees are at least three and two, respectively, so both vanish. Induction proves (i), including uniqueness with the specified action on cohomology. In (ii), the hypercohomology spectral sequence has only columns zero and one; its degeneration and functoriality give the assertion. ◻

Proof. Put \(R=E^S\times_I E^X\), and resolve its closure in \(Z^S\times_{\overline I}Z^X\), preserving \(R\). This gives a smooth projective fivefold \(\overline R\) with proper maps \(\pi_S:\overline R\to Z^S\) and \(\pi_X:\overline R\to Z^X\). On each fiber, Poincaré duality represents \(a\) by the unique tensor in the \(H^2(E^S_t)\otimes H^2(E^X_t)\) summand of \(H^4(R_t)\) with that correspondence action. The tensors form a flat rational section \(\alpha\) of \(R^4(R/I)_*\mathbb Q\) and have type \((2,2)\).

The global invariant cycle theorem [11] identifies the image of \(H^4(\overline R)\to H^4(R_t)\) with the monodromy invariants. Restriction is a morphism of polarizable pure Hodge structures. Its image therefore contains \(\alpha_t\), and semisimplicity gives a rational lift \(\widetilde\alpha\in H^4(\overline R)\) of type \((2,2)\). For example, split restriction onto its image as a Hodge map and apply the splitting to \(\alpha_t\). The restriction of this lift to every fiber is \(\alpha\), because both define flat sections. Define \[c(z)=(\pi_X)_* \bigl(\widetilde\alpha\smile\pi_S^*z\bigr).\] Cup product raises degree by four and the proper pushforward lowers degree by four, so this is a rational Hodge map from \(H^3\) to \(H^3\).

Its restriction to \(I\) is the relative cohomological correspondence defined by \(\widetilde\alpha|_R\). Pullback, cup product, and integration along the smooth proper relative surface projection give a morphism \(Rh_{S*}\mathbb Q\to Rh_{X*}\mathbb Q\); on degree-two cohomology sheaves it is \(a\). For the source, \(R^3h_{S*}\mathbb Q=0\), and for the projected target, \(eR^3h_{X*}\mathbb Q=0\). Lemma 25(ii), with \(n=3\), therefore identifies the entire source and projected target \(H^3\) with their canonical \(H^1(R^2)\) terms. Functoriality shows that the projected open action of \(c\) is exactly \(H^1(I,a)\), as asserted. The proof uses no claim that \(\widetilde\alpha\) is algebraic. ◻

Proof. Choose a smooth projective compactification \(\overline{\mathcal Y}\to\mathbf P^1_t\) whose inverse image over \(O\) is \(\mathcal Y\). It is obtained by taking a projective closure of the family and resolving away from its smooth open. Take the component containing the given open. Its source is smooth and integral, and its map to the nonsingular curve is flat. The generic fiber is geometrically integral: a smooth geometrically connected surface is geometrically integral. Hence the base changes below have the dense unchanged open and the same function field as the indicated pullback. The map \[C=\mathbf P^1_u\times\mathbf A^1_s\longrightarrow\mathbf P^1_t, \qquad t=s+u^2,\] is a morphism: in homogeneous coordinates on \(\mathbf P^1_u\), with \(u=U_0/U_1\), it is \([U_0:U_1]\mapsto[U_0^2+sU_1^2:U_1^2]\). Set \(W=\overline{\mathcal Y}\times_{\mathbf P^1_t}C\). Choose a projective resolution \(Z\to W\) as a sequence of blowups, all supported over \(t\notin O\). The flat base change is generically smooth and therefore reduced, with no component supported only over the boundary; its open over \(O\) is smooth. Resolution is performed on each component if needed. Strong projective resolution in characteristic zero, preserving the regular locus, permits this construction [42]. The map \(Z\to\mathbf A^1_s\) is projective, and is smooth after deleting finitely many values of \(s\). Fix this entire construction, and take \(G\) inside its smooth locus and inside \(O\).

Let \(B_b\) be the normalization of \(\mathbf P^1_t\times\mathbf P^1_v\) in the field \[\mathbb C(t,v)\bigl(\sqrt{t-b-v^2}\bigr).\] It is a projective surface finite over the product. Its finite-\(v\) open is \(\mathbf P^1_u\times\mathbf A^1_v\), with \(t=b+u^2+v^2\). Its open over \(O\) is \(Q_b\): the latter is smooth and the map \[Q_b\longrightarrow O\times\mathbf P^1_v, \qquad (t,[u:v:w])\longmapsto(t,[v:w])\] has no base point: \(v=w=0\) would force \(u=0\). It is proper and has finite fibers, hence is finite of degree two, with the required function field. The same verification on the finite-\(v\) open uses \(\mathbf P^1_u\times\mathbf A^1_v\) and the morphism \([U_0:U_1]\mapsto[U_0^2+(b+v^2)U_1^2:U_1^2]\). These assertions identify both opens by the universal property of normalization. Smoothness of \(Q_b\) follows in the affine chart \(w=1\) from the derivative in \(t\), and at infinity from the derivatives in \(u,v\).

Start with the projective fourfold \(W_b=\overline{\mathcal Y}\times_{\mathbf P^1_t}B_b\). On its finite-\(v\) open, pull back the chosen blowup sequence \(Z\to W\) by \(s=b+v^2\). This map of affine lines is finite flat, so blowup commutes with the base change. Each blowup extends to the whole current projective model without altering its open over \(O\). Explicitly, if \(j:U\hookrightarrow W_b^{(r)}\) is its finite-\(v\) open and \(\mathcal I\subset\mathcal O_U\) is the next center ideal, extend it by \[\mathcal J=\ker\bigl(\mathcal O_{W_b^{(r)}} \longrightarrow j_*(\mathcal O_U/\mathcal I)\bigr).\] This is a coherent ideal: the schemes are noetherian, the open immersion is quasi-compact, and the kernel is a quasi-coherent submodule of \(\mathcal O_{W_b^{(r)}}\). It restricts to \(\mathcal I\) and is the unit ideal off the closed inverse image of \(\mathbf P^1_t\setminus O\). Thus its blowup has the asserted properties.

After extending the finite sequence, both opens in the statement are present and smooth. A final projective resolution which is an isomorphism on the regular locus preserves their union. It gives \(M_b\) and the two morphisms to the parameter lines. The construction uses one fixed set of discarded \(s\)-values, independent of \(b\). ◻

Proof of Theorem 23. We retain the notation \(\mathscr U\), \(p\), and \(\varphi\) above.

Proof. The variation, the projector, and the identification are supplied by [geom:structure,pencil:realization]. 17 gives the uniruled threefold models on the K3 side. For \(X\), after \(t=s+u^2\) the generic fiber over \(B'\) is the conic \[z^2-\pi^*d_1\,u^2=\pi^*(d_0+s d_1)\] over \(\mathbb C(B')\). A finite extension splits this conic, so a generically finite surface cover of \(B'\) yields a dominant rational map from a product of a surface and \(\mathbf P^1\). Every smooth projective model of the threefold is therefore uniruled. Theorem 23 applies. It applies separately to every fixed pencil with these properties; its countable set of excluded \(b\) is chosen only after the pencil, the degree-three correspondence, and the good \(s\)-open have been fixed. ◻

Proof. Let \(\Theta_b\in\mathop{\mathrm{CH}}^2(S_b^0\times X_b)_\mathbb Q\) be the correspondence of Corollary 28; its restriction to \(U_b\) is the specified identification \(\varphi_b:U_b\to U_{X_b}\). At these periods \(U_{X_b}\) has no nonzero rational \((1,1)\)-class by Proposition 11. Proposition 15 therefore supplies curves \(C_1,C_2\), a morphism \(g:C_1\times C_2\to X_b\), and the algebraic correspondence \[Z_X=\bigl(\pi_1^{C_1}\boxtimes\pi_1^{C_2}\bigr) \circ{}^t\Gamma_g\circ e \in\mathop{\mathrm{CH}}^2(X_b\times C_1\times C_2)_\mathbb Q,\] where the curve projectors are those constructed there. Its action on \(U_{X_b}\) is the injection \(g^*\), with image entirely in \(H^1(C_1,\mathbb Q)\otimes H^1(C_2,\mathbb Q)\). Set \[Z=Z_X\circ\Theta_b.\] Its action on \(U_b\) is \(f=g^*\varphi_b\) and is injective. The composition passes through the surface \(X_b\), so its codimension is \(2+2-2=2\). The equality \(U_b=T_{S_b^0}\) and the full Hodge group follow from Propositions 11 and 17 at these selected periods. ◻

Proof. We prove surjectivity of the rational map \(\Phi\) by spanning both complex half-spin tensor orientations. We first isolate the rational orthogonal Lie algebra in the common Hodge group. Weight-one effectivity will restrict the nontrivial constituents of \(W_1,W_2\) to half-spins and identify their rational intertwiners with Hodge maps to \(W\). This allows the tensor-product calculation to be made over \(\mathbb C\) and its surjectivity to be descended to \(\mathbb Q\).

The rational Hodge group. Let \[G=\mathop{\mathrm{Hg}}(W_1\oplus W_2\oplus W).\] The Hodge group is the connected rational algebraic group generated by the Hodge circle. Any rational invariant subspace of the polarized pure weight-one structure \(W_1\oplus W_2\oplus W\) has a Hodge complement, obtained from its polarization. The complement is again \(G\)-invariant. Its faithful representation is therefore completely reducible, so \(G\) is reductive.

The embedding in Equation (48) realizes \(U(1)\) in \(\mathop{\mathrm{End}}(W)\); Tate twists do not change the Hodge-circle action. This gives a representation of \(G\) on \(U\) preserving \(q\). Since \(G\) is connected, its image lies in \(\mathop{\mathrm{SO}}(U,q)\). It contains the Hodge circle of \(U\) and therefore \(\mathop{\mathrm{Hg}}(U)=\mathop{\mathrm{SO}}(U,q)\), so the map is surjective. Write \(\mathfrak g=\operatorname{Lie}(G)\). The derived Lie algebra of \(\mathfrak g\) is a direct sum of rational simple ideals. Since \(\mathfrak{so}(U,q)\) is absolutely simple, exactly one of those ideals maps nontrivially to it: two such images would be commuting copies of the same nonabelian simple algebra. The nonzero map is an isomorphism, and the remaining ideals and the center form its kernel. We obtain rational Lie ideals \[ \mathfrak g=\mathfrak s\oplus\mathfrak k, \qquad \mathfrak s\xrightarrow{\sim}\mathfrak{so}(U,q), \qquad \mathfrak k=\ker\bigl(\mathfrak g\to\mathfrak{so}(U,q)\bigr). \tag{50}\] The ideal \(\mathfrak s\) integrates to a connected rational normal semisimple subgroup. Finite central kernels are harmless here; we do not identify a spin representation with a representation of \(\mathop{\mathrm{SO}}(U,q)\).

The allowed weight-one constituents. Normalize the complexified infinitesimal Hodge-circle generator \(D\) so that its eigenvalues on weight one are \(1,-1\). Its image on \(U\) has eigenvalues \(2,0,\ldots,0,-2\). Write \(D=D_{\mathfrak s}+D_{\mathfrak k}\) according to Equation (50). Each \(\mathfrak s_\mathbb C\)-isotypic summand of \(W_{i,\mathbb C}\) has the form \(R_\lambda\otimes M\), with \(R_\lambda\) irreducible. The commuting operator \(D_{\mathfrak k}\) acts only on \(M\). The two projections of the semisimple element \(D\) are semisimple, so their eigenvalues add on simultaneous eigenspaces.

Identify the root system of \(\mathfrak s_{\mathbb C}\) with \(D_4\), with roots \(\pm e_i\pm e_j\), using a Cartan subalgebra in which \(D_{\mathfrak s}\) pairs with weights as \(2e_1\). A dominant highest weight has \[\lambda_1\geq\lambda_2\geq\lambda_3\geq|\lambda_4|,\] with coordinates either all integral or all half-integral. Its Weyl orbit contains weights giving the eigenvalues \(2\lambda_1\) and \(-2\lambda_1\) for \(D_{\mathfrak s}\). Translation by any eigenvalue of the commuting factor cannot reduce this spread. Effectivity of weight one therefore gives \(4\lambda_1\leq2\). The only possibilities are the trivial representation and the two half-spin representations \(S_+,S_-\), with highest weights \(\tfrac12(e_1+e_2+e_3+e_4)\) and \(\tfrac12(e_1+e_2+e_3-e_4)\). Their weights are all sign choices \(\tfrac12(\pm e_1\pm e_2\pm e_3\pm e_4)\) of the respective parities. Each half-spin therefore has both \(D_{\mathfrak s}\)-eigenvalues \(1,-1\). If \(\mu\) is an eigenvalue of \(D_{\mathfrak k}\) on its multiplicity space, both \(\mu+1\) and \(\mu-1\) must belong to \(\{1,-1\}\); hence \(\mu=0\). Semisimplicity now shows that \(D_{\mathfrak k}\) is zero on that multiplicity space.

The rational decompositions \[W_i=W_i^{\mathfrak s}\oplus W_i^{\mathrm{nt}}, \qquad W_i^{\mathrm{nt}}=\mathfrak s W_i\] are \(G\)-stable. On \(W_i^{\mathrm{nt}}\) the entire Hodge-circle generator consequently acts through \(\mathfrak s\). The same assertion holds on \(W\). Its Hodge group lies in the rational spin group acting by left multiplication on \(C^+(U,q)\) and maps onto \(\mathop{\mathrm{SO}}(U,q)\). The spin covering has finite kernel, so the Lie algebra of this image on \(W\) maps isomorphically to \(\mathfrak{so}(U,q)\); in particular \(\mathfrak k\) acts trivially on \(W\).

For completeness, write \(U_\mathbb C=F\oplus F^\vee\) with \(F\) maximal isotropic of dimension four. Exterior multiplication and contraction give the Clifford action on \(\bigwedge^\bullet F\). These operators generate its matrix units, and the even algebra preserves parity. Thus, with \(S_+=\bigwedge^{\mathrm{even}}F\) and \(S_-=\bigwedge^{\mathrm{odd}}F\), one has \[C^+(U)_{\mathbb C}\simeq\mathop{\mathrm{End}}(S_+)\oplus\mathop{\mathrm{End}}(S_-).\] Since \(\dim S_+=\dim S_-=8\), left multiplication gives eight copies of each half-spin. In particular \(W\) has no \(\mathfrak s\)-invariant summand.

Rational Hodge maps and complex multiplicities. A rational \(\mathfrak s\)-intertwiner from \(W_i^{\mathrm{nt}}\) to \(W\) commutes with \(D\) and hence with the connected Hodge circle. Extended by zero on \(W_i^{\mathfrak s}\), it is therefore a Hodge map. Conversely every rational Hodge map commutes with \(G\) and hence with \(\mathfrak s\). We have proved \[ \begin{aligned} \mathop{\mathrm{Hom}}_{\mathrm{HS}}(W_i,W) &=\mathop{\mathrm{Hom}}_{\mathfrak s}(W_i^{\mathrm{nt}},W),\\ \mathop{\mathrm{Hom}}_{\mathrm{HS}}(U,W\otimes W) &=\mathop{\mathrm{Hom}}_{\mathfrak s}(U,W\otimes W). \end{aligned} \tag{51}\] In the second equality, both Hodge structures have weight two and their Hodge-circle actions factor through \(\mathfrak s\), so the same commutation argument applies. For any two rational \(\mathfrak s\)-modules \(V_1,V_2\) the intertwiner space is the kernel of the rational linear map \[\mathop{\mathrm{Hom}}_\mathbb Q(V_1,V_2)\longrightarrow \mathop{\mathrm{Hom}}_\mathbb Q(\mathfrak s\otimes V_1,V_2),\qquad a\longmapsto\bigl((x,v)\mapsto a(xv)-x\,a(v)\bigr).\] Extension of scalars is exact. Hence \(\mathop{\mathrm{Hom}}_{\mathfrak s}(V_1,V_2)\otimes_\mathbb Q\mathbb C =\mathop{\mathrm{Hom}}_{\mathfrak s_\mathbb C}(V_{1,\mathbb C},V_{2,\mathbb C})\). In particular, after complexifying the rational Hodge Hom spaces in Equation (51), all complex maps between the half-spin multiplicity spaces are available. This assertion does not require either half-spin type to be defined separately over \(\mathbb Q\).

Spanning both tensor orientations. Write \[\begin{aligned} W_{i,\mathbb C}^{\mathrm{nt}} &=(S_+\otimes M_{i,+})\oplus(S_-\otimes M_{i,-}),\\ W_{\mathbb C}&=(S_+\otimes N_+)\oplus(S_-\otimes N_-), \qquad \dim N_+=\dim N_-=8. \end{aligned}\] Both half-spins are self-dual: the Weyl group contains simultaneous negation of all four coordinates, preserving the two weight parities. Clifford antisymmetrization identifies \(C(U)_\mathbb C\) with \(\bigwedge^\bullet U_\mathbb C\) as a module for conjugation. The even part consists of the diagonal endomorphism blocks of \(S_+\oplus S_-\), and the odd part of the two off-diagonal blocks. Choose a normalized volume element \(\omega\) with \(\omega^2=1\). Left multiplication by \(\omega\) exchanges exterior degrees \(k\) and \(8-k\) and distinguishes the two blocks of each parity. Each block therefore contains one copy of each paired degree; degree four splits into its two \(35\)-dimensional eigenspaces. This gives \[ \begin{aligned} S_+\otimes S_-&\simeq U_{\mathbb C}\oplus\bigwedge^3U_{\mathbb C},\\ S_\pm\otimes S_\pm&\simeq \mathbb C\oplus\bigwedge^2U_{\mathbb C}\oplus\bigwedge^4_\pm U_{\mathbb C}. \end{aligned} \tag{52}\] The choice of the signs on \(\bigwedge^4_\pm U_\mathbb C\) follows that of \(S_\pm\). To check the needed multiplicity, \(\bigwedge^3U_\mathbb C\) has no standard summand. Indeed, an equivariant map from \(U_\mathbb C\) to this exterior power would send a unit vector \(u\) to an invariant of its stabilizer \(\mathop{\mathrm{SO}}(u^\perp)\simeq\mathop{\mathrm{SO}}_7\). Neither \(\bigwedge^3u^\perp\) nor \(u\wedge\bigwedge^2u^\perp\) has such an invariant: determinant-one sign changes in an orthonormal basis kill every exterior basis coefficient. Thus the map vanishes on the orbit of \(u\), which spans \(U_\mathbb C\). The standard summand in each opposite-sign product has multiplicity one. There is none in a same-sign product, since \(-\mathop{\mathrm{id}}\in\mathop{\mathrm{SO}}(U_\mathbb C)\) acts trivially on all even exterior powers and by \(-1\) on \(U_\mathbb C\).

Choosing a nonzero equivariant map from \(U_\mathbb C\) into each opposite-sign product identifies the complex target space with \[\mathop{\mathrm{Hom}}_{\mathfrak s_\mathbb C}(U_\mathbb C,W_\mathbb C\otimes W_\mathbb C) \simeq (N_+\otimes N_-)\oplus(N_-\otimes N_+).\] The ordinary flip interchanges these two summands. In particular, Equation (51) and base change give \[\dim_\mathbb Q\mathop{\mathrm{Hom}}_{\mathrm{HS}}(U,W\otimes W)=8^2+8^2=128.\] It remains to generate both multiplicity spaces from the one map \(f\).

A tensor involving a trivial \(\mathfrak s\)-constituent of either \(W_i\) is a sum of trivial or half-spin representations, so it also contains no standard summand. Therefore \(f_{\mathbb C}\) is described by two multiplicity tensors \[t_{+-}\in M_{1,+}\otimes M_{2,-}, \qquad t_{-+}\in M_{1,-}\otimes M_{2,+},\] at least one of which is nonzero.

Suppose, for example, that \(t_{+-}\ne0\). Choose linear functionals \(\ell_1,\ell_2\) with \((\ell_1\otimes\ell_2)(t_{+-})\ne0\). For arbitrary \(n_+\in N_+\) and \(n_-\in N_-\), the rank-one maps \(m\mapsto\ell_1(m)n_+\) and \(m\mapsto\ell_2(m)n_-\) send this component to a nonzero scalar multiple of \(n_+\otimes n_-\). Taking the maps zero on the other signs isolates the component. Such decomposable tensors span \(N_+\otimes N_-\), and the independent second copy of \(\mathcal H\) in \(\Phi\) supplies the other orientation by \(\tau\). The case \(t_{-+}\ne0\) is identical. Hence \(\Phi_{\mathbb C}\) is surjective. Its rational cokernel has zero complexification and is therefore zero, proving surjectivity of \(\Phi\). This descent accounts for arbitrary rational forms and Schur multiplicities; neither half-spin type need be defined separately over \(\mathbb Q\).

Algebraic realization. Let \(a:H^1(C_i,\mathbb Q)\to H^1(A,\mathbb Q)\) be a rational Hodge map. Poincaré duality on \(C_i\) identifies it with a unique rational class \(\alpha_a\) in the middle Künneth summand \(H^1(C_i,\mathbb Q)\otimes H^1(A,\mathbb Q)\subset H^2(C_i\times A,\mathbb Q)\), normalized by \[(p_A)_*(\alpha_a\smile p_{C_i}^*x)=a(x).\] The Hodge condition says precisely that \(\alpha_a\) has type \((1,1)\). After clearing denominators, the exponential sequence realizes it as the first Chern class of a holomorphic line bundle: the obstruction is its zero image in \(H^2(\mathcal O_{C_i\times A})\). Projective GAGA makes this line bundle algebraic. Dividing by the same denominator gives a divisor \(D_a\in\mathop{\mathrm{CH}}^1(C_i\times A)_\mathbb Q\) with \(\mathop{\mathrm{cl}}(D_a)=\alpha_a\). Thus its cohomology class has exactly the required middle component.

For maps \(a:H^1(C_1)\to W\) and \(b:H^1(C_2)\to W\), the exterior product \(D_a\boxtimes D_b\), with its factors ordered as \((C_1\times C_2)\times(A\times A)\), has codimension two and acts on \(H^1(C_1)\otimes H^1(C_2)\) by \(a\otimes b\). If \(Z\) is the given cycle from a surface, its composition with this product has codimension \(2+2-2=2\) and has image in \(W\otimes W\).

Let \(s:A\times A\to A\times A\) interchange the factors. For \(u,v\in H^1(A,\mathbb Q)\), \[s_*\bigl(p_1^*u\smile p_2^*v\bigr) =p_2^*u\smile p_1^*v =-p_1^*v\smile p_2^*u.\] Consequently \(-\Gamma_s\) induces the ordinary tensor flip \(\tau\) on this summand. Given a target map \(h\), choose a rational preimage under \(\Phi\) and write it as a finite sum \[h=\sum_\nu c_\nu(a_\nu\otimes b_\nu)f +\sum_\mu d_\mu\tau(a'_\mu\otimes b'_\mu)f.\] It is realized exactly by the rational cycle \[\sum_\nu c_\nu(D_{a_\nu}\boxtimes D_{b_\nu})\circ Z -\sum_\mu d_\mu\Gamma_s\circ (D_{a'_\mu}\boxtimes D_{b'_\mu})\circ Z.\] Composition with \(\Gamma_s\), whose codimension is \(\dim(A\times A)\), also preserves codimension two. On the specified source subspace the resulting action equals \(h\) in the full \(H^2(A\times A,\mathbb Q)\). ◻

Proof. The assumed \(r_T\) uses the form on \(U_b\) inherited from \(Y\), which the auxiliary K3 realizes isometrically. The factor-two pullback to \(X_b\) enters only the curve correspondence of Lemma 29 and is not part of this isometry.

Choose markings of the full second cohomology spaces by \(\Lambda_{\mathrm{K3},\mathbb Q}\). They turn \(r_T\) into an isometry between two nondegenerate rational subspaces of this one quadratic space. Witt’s Extension Theorem [30] extends it to an isometry of \(\Lambda_{\mathrm{K3},\mathbb Q}\). Returning through the markings gives \[r:H^2(S,\mathbb Q)\xrightarrow{\sim}H^2(S_b^0,\mathbb Q).\] The isometry \(r\) maps the orthogonal complement of \(T_S\) to the orthogonal complement of \(T_{S_b^0}\). These are exactly \(\mathop{\mathrm{NS}}(S)_\mathbb Q\) and \(\mathop{\mathrm{NS}}(S_b^0)_\mathbb Q\), both of type \((1,1)\). Since its restriction \(r_T\) is Hodge, \(r\) is therefore a rational Hodge isometry on the full second cohomology. Buskin’s Theorem [9], applied to these smooth projective complex K3 surfaces, gives a cycle \[R\in\mathop{\mathrm{CH}}^2(S\times S_b^0)_\mathbb Q,\qquad R_*|_{H^2(S,\mathbb Q)}=r.\] The hypotheses here are precisely full rational Hodge isometry and projectivity; the integral Néron–Severi lattice of \(S_b^0\) is not required to equal \(P\).

Compose \(R\) with the cycle \(Z\) of Lemma 29. The resulting codimension-two correspondence induces the injection \[f_S:T_S\xrightarrow{r_T}U_b \xrightarrow{f}H^1(C_1,\mathbb Q)\otimes H^1(C_2,\mathbb Q).\] At these periods \(\mathop{\mathrm{Hg}}(T_S)=\mathop{\mathrm{SO}}(T_S)\). Apply Lemma 30 to \(f_S\) and to the given full even-Clifford model \(W=H^1(A,\mathbb Q)\) of dimension \(128\). The prescribed map \(\iota_{w,E}\) of Equation (49) belongs to the target of \(\Phi\). A rational preimage under \(\Phi\), realized by the divisor products and signed interchange in that lemma, supplies \(\Gamma_S\) with exactly the asserted action. In particular the equality is on all of \(T_S\) and has no extra component in the other Künneth summands of \(H^2(A\times A,\mathbb Q)\).

The choices of abelian data, pencil, and final parameter retain the successive order of Lemma 16 and Corollary 28. We have used the full Hodge group only at these construction periods. Proposition 33 will realize one fixed standard tensor throughout a target period ball, including Picard jumps. ◻

Proof. Embedded parameters over an analytic base. Let \(R\) be the Hilbert polynomial of the embedded fibres of \(f\); it is constant because the family is flat. For any fixed Hilbert polynomial \(Q\), write \(K_Q=\operatorname{Hilb}^{Q}(\mathbf P^N)\). This is a projective scheme with a flat universal subscheme, by the projective Hilbert-scheme theorem [21]. We use the incidence space \[H_Q=\{(b,z)\in B\times K_Q^{\mathrm{an}}: \mathcal U_z\subset\mathcal Z_b\}.\] It is closed analytic, with its natural incidence structure. To see this directly, let \(\mathcal J\) be the ideal sheaf of \(\mathcal Z\) in \(\mathbf P^N\times B\). The bounded regularity for the fixed polynomials \(R\) and \(Q\) supplies an integer \(k\) such that \(\mathcal J(k)\) is relatively generated, its direct image is locally free with base change, and the universal subscheme for \(Q\) has the corresponding vanishing and base-change properties. On \(B\times K_Q^{\mathrm{an}}\), restriction defines a holomorphic vector-bundle map from sections of \(\mathcal J(k)\) to sections of \(\mathcal O_{\mathcal U}(k)\). Its zero scheme is exactly \(H_Q\): vanishing of these generating sections is equivalent to containment of subschemes. Since \(K_Q^{\mathrm{an}}\) is compact, \(H_Q\to B\) is proper. Its universal subscheme is flat over \(H_Q\), being the pullback of the universal Hilbert family. No algebraicity assumption on \(B\) or on \(f\) is required.

The ambient Betti class on singular parameters. For polynomials of degree \(n-r\), associate to each universal fibre its \((n-r)\)-dimensional fundamental cycle, with the generic lengths of its structure sheaf as multiplicities. We claim that its class in the ambient \(H^{2r}(\mathcal Z_b,\mathbb Q)\) is locally constant on \(H_Q\), including at singular or nonreduced Hilbert parameters.

Here are details that justify this assertion in the analytic setting. The same high-degree ideal construction gives a holomorphic classifying map \(B\to\operatorname{Hilb}^{R}(\mathbf P^N)^{\mathrm{an}}\): the bundle of degree-\(k\) ideals defines the map to the Hilbert scheme’s Grassmannian realization. Its image is in the open locus \(K_R^{\mathrm{sm}}\) of smooth, pure \(n\)-dimensional fibres. Over this algebraic locus let \(\mathfrak Z\to K_R^{\mathrm{sm}}\) be the universal smooth family, and let \(\mathfrak H_Q\) be its relative Hilbert scheme. Thus \(H_Q\) and its universal pair are analytic pullbacks of this algebraic universal pair.

On \(\mathfrak Z\times_{K_R^{\mathrm{sm}}}\mathfrak H_Q\), the universal structure sheaf \(\mathcal F\) is flat over \(\mathfrak H_Q\) and is a perfect complex on the ambient space. For this last assertion, take successive locally free surjections onto \(\mathcal F\) and their kernels on the quasi-projective ambient space. All the kernels remain flat over \(\mathfrak H_Q\). The fibres of the ambient family are regular of dimension \(n\), so after at most \(n\) syzygies the fibrewise kernels are locally free. The local flatness criterion then makes this kernel locally free on the ambient space. This gives a bounded vector-bundle resolution even when \(\mathfrak H_Q\) is singular. Flatness over the parameter space also shows that the resolution restricts to a resolution of the ordinary structure sheaf on every ambient fibre.

The topological Chern character of this complex, pulled back to the analytic universal pair, consequently satisfies \[ \operatorname{ch}_r(\mathcal F)|_{\mathcal Z_b} =\mathop{\mathrm{cl}}\bigl([\mathcal U_z]_{n-r}\bigr) \quad\text{in }H^{2r}(\mathcal Z_b,\mathbb Q). \tag{54}\] This is the leading Chern-character identity for a coherent sheaf supported in codimension at least \(r\). It follows from Grothendieck–Riemann–Roch (GRR) and dévissage, followed by the comparison of algebraic and topological Chern classes [19]. For the precise form needed here, remove a subset of codimension at least \(r+1\): dévissage then reduces to the generic lengths on smooth codimension-\(r\) supports, where the regular-embedding case of GRR gives their fundamental classes as the leading terms. The removed subset contributes nothing in degree \(2r\), since its cohomology with supports vanishes below degree \(2r+2\) [19]. This also explains why lower-dimensional embedded components cause no problem.

A class on the total space of a pulled-back smooth proper family restricts to a flat section of its fibre cohomology. Indeed, locally trivialize the original differentiable family over \(B\) and pull that trivialization back to the parameter space; inclusions of fibres along a path are homotopic. Complex analytic spaces are locally path connected, and their nilpotents do not change their topology. Equation (54) proves the claimed constancy on all connected components, without assuming that the universal subschemes themselves are smooth.

Successful components and their images. Take every finite list \(Q_1,\ldots,Q_s\) of polynomials of degree \(n-r\) and every list \(c_1,\ldots,c_s\in\mathbb Q\), and form \[H_{\boldsymbol Q}=H_{Q_1}\times_B\cdots\times_B H_{Q_s}, \qquad \mathcal Z_{\boldsymbol c,z} =\sum_{i=1}^s c_i[\mathcal U_{i,z}]_{n-r}.\] The empty list represents the zero cycle. These spaces are proper over \(B\), and the displayed universal combinations have constant ambient Betti class on each connected component. Because \(B\) is simply connected, both \(L\) and \(\tau\) are constant in local-system trivializations. Hence (53) holds either everywhere or nowhere on each component.

Retain exactly the successful components. There are countably many polynomial and coefficient lists and countably many components: all the parameter spaces are second countable and locally connected, so their connected components are open and closed. Each retained component is a closed analytic subspace, proper over \(B\). Its image is closed analytic by Remmert’s proper mapping theorem; for a primary statement, see [20], applied to the reduction of the component. Reduction changes neither its points nor its image. For a textbook statement of the proper mapping theorem, see [15].

Every rational algebraic cycle on a fibre is a finite rational sum of integral subvarieties and therefore occurs in this list. Conversely, every point of any retained image has an actual universal fibre combination above it. Their supports in the projective fibre are algebraic by Chow’s theorem, so their fundamental cycles are algebraic. We have proved that the successful set is exactly the stated countable union, with the asserted cycle-theoretic meaning at every image point.

A proper closed analytic subset of a connected ball is nowhere dense. If the union contains a nonempty open set, Baire’s theorem forces one image to have interior; the analytic identity principle makes that image all of \(B\). If instead the union escapes every countable union of proper closed analytic subsets, the same conclusion follows: otherwise the successful images would themselves be such a union. ◻

Proof. Cartan–Dieudonné expresses each real orthogonal transformation as a finite product of reflections in nonisotropic real vectors. The reflection \[r_v(x)=x-\frac{2q(x,v)}{q(v,v)}v\] depends continuously on \(v\) on the nonisotropic open and is rational for rational \(v\). Approximating the finitely many vectors proves density. The full real orthogonal group acts transitively on the oriented positive two-planes, which are the period-domain points; it also interchanges the two components. Density proves the orbit assertion. We use the standard reflection theorem in [30]. ◻

Proof of Proposition 33. The fixed local family and tensor. Choose a marking of the target and an integral ample class \(h\in P\). The local K3 period theorem gives a six-dimensional marked deformation ball \(B\) through \(S\) on which \(P\) remains algebraic, identified with an open subset of the period domain of \(U_P\). The class \(h\) lifts locally to a line bundle; shrink \(B\) so it is relatively ample and a power embeds the family \(\mathcal S\to B\) projectively. Only \(P\), not the target’s whole Picard group, is kept algebraic. Thus \(U_P\) is a fixed nondegenerate rational Hodge subvariation even at jumps. These local assertions follow from local Torelli and the deformation theory of polarized K3 surfaces [22].

Fix any rational \(w_0\in U_P\) with \(q(w_0,w_0)\ne0\) and put \(W_U=C^+(U_P,q)\). There is one rational alternating form polarizing the left Clifford Hodge structures on the chosen period component. Here is a construction that also fixes its characters. Let \(x^*\) be reversal and \(B_C(x,y)=\operatorname{scal}(x^*y)\). Choose two orthogonal positive rational vectors with oriented product \(\beta\) in that component and set, up to an overall sign, \[E_0(x,y)=B_C(x,y\beta).\] Reversal sends \(\beta\) to \(-\beta\), and cyclic invariance of scalar part proves alternation. Left spin multiplication preserves \(E_0\) because \(g^*g=1\). At the chosen plane write \(\beta=cJ\), \(c>0\). For an even orthogonal monomial containing \(a\) of the two positive generators, \(B_C(x,x)\) has sign \((-1)^a\) and \(Jx\beta=(-1)^{a+1}cx\). Thus \(E_0(x,Jx)\) is definite after one overall choice of sign. Transitivity of the connected real spin group on the component proves positivity throughout it. This is the standard Kuga–Satake polarization [24].

The \((1,0)\) space of \(W_U\) is \(\ker L_\sigma\) for a period vector \(\sigma\), and varies holomorphically. A rational symplectic basis for \(E_0\) and its self-dual integral lattice give, over a smaller \(B\), a polarized holomorphic abelian family \(\mathcal A\to B\) with this fixed rational \(H^1\). The usual torus construction over Siegel space and a relative theta bundle give a relative projective embedding after shrinking. The tensor \(\tau=\iota_{w_0,E_0}\) is a fixed rational linear map \(U_P\to W_U\otimes W_U\) and a Hodge map at every period, by spin equivariance of \(j_{w_0}\).

A geometric period chart in the target ball. Choose a geometric datum in the simultaneous regular-period and domination open of Sections 3 and 4. On a local marking fix a rational isometry \(\theta:U\simeq U_P\) for the form inherited from the quotient surface \(Y\); this is possible by Proposition 9 and Lemma 3. Lemma 35 gives a rational \(g\in O(U_P)(\mathbb Q)\) moving its period into \(B\). Fix \(g\) and \(\theta\), and shrink the original normalized geometric product chart so that its period map followed by \(g\theta\) is a submersion into \(B\). The change \(g\theta\) acts only on rational period coordinates. Pencils subsequently selected still have the form \(d_0+td_1\) in the normalized algebraic section chart, with directions in the original good open at infinity.

Proper images for the full action. Apply Lemma 34 to \(\mathcal Z=\mathcal S\times_B\mathcal A^2\), of relative dimension \(130\), with \(r=2\) and \[\mathcal V=\mathop{\mathrm{Hom}}_{\mathbb Q}\bigl(U_P,R^2(\mathcal A^2/B)_*\mathbb Q\bigr), \qquad L_b(\gamma)(x)=(p_{AA})_*(\gamma\smile p_S^*x).\] The success condition is equality to \(\tau\) on all of \(U_P\) in the full target \(H^2(A_b^2)\), not only after a projection or on the holomorphic line. It is a flat local-system condition. The successful set is therefore a countable union of closed analytic images \(Y_i\) of proper cycle-parameter spaces, with actual successful cycles over every point of each image.

A seed outside every proposed proper image. Suppose every \(Y_i\) is proper. In the regular construction period chart, pull back the \(Y_i\) by the fixed map \(g\theta\) and also pull back the extra-Picard hyperplanes in \(B\). These are countably many proper closed analytic loci: the period submersion is open, and an extra integral class has a nonzero \(U_P\) projection because \(\Lambda_{\mathrm{K3}}\cap P_{\mathbb Q}=P\). Its period equation is therefore nontrivial. Include also the proper marked Hodge-tensor conditions used in Lemma 29.

Apply Lemma 16 to this prescribed list, using the admissible pencil open of the construction. It selects abelian data, then an actual admissible section pencil with good leading section. Once that pencil, its first models, and its relative correspondence are fixed, intersect its algebraic good open with that of Corollary 28 and add the corollary’s countable exceptional parameters. The intersection is still a nonempty algebraic open of the pencil. The last choice in the lemma supplies a point in this open outside those exceptions and every prescribed period locus. In particular, the full special orthogonal Hodge group and the curve source of Lemma 29 hold at this construction point.

The target fiber at its moved period has \(\mathop{\mathrm{NS}}=P\), so its full \(U_P\) is transcendental. The auxiliary K3 has transcendental space \(U\). The rational period isometry \(g\theta\) identifies these transcendental Hodge structures with their cup forms. Proposition 31 therefore supplies on the target fiber a cycle realizing exactly the fixed tensor \(\tau\). That period lies in a successful image, contrary to its choice. Thus some image is all of \(B\), and its fiber over the original target supplies \(\Gamma_U\).

The rank-eight source and its full action have remained fixed throughout the ball. Neither a global moduli cycle nor specialization of a smooth curve presentation is required. ◻

Proof. In the ordering \(\mathop{\mathrm{End}}(W_U)=W_U\otimes W_U^\vee\), tensor conversion is \(\mathop{\mathrm{id}}\otimes(E_U^\flat)^{-1}\) after the common Tate twist. Hence the left side of (55) is \[\bigl(p\otimes(E_T^\flat)^{-1}i^\vee(-1)\bigr)j,\] which is exactly the tensor conversion of \(pji\). All maps are rational Hodge maps. For completeness, if \(f:H^1(B,\mathbb Q)\to H^1(A,\mathbb Q)\) is any such map, its dual on homology commutes with complex structures. For some positive integer \(n\), \(nf^\vee\) is integral and induces a holomorphic homomorphism \(\phi:A\to B\). Projectivity and GAGA make \(\phi\) algebraic, and \(f=\phi^*/n\). Thus \([\Gamma_\phi^t]/n\in\mathop{\mathrm{CH}}^{\dim B}(B\times A)_{\mathbb Q}\) induces \(f\) on \(H^1\). No equality of dimensions or isogeny assumption is needed. ◻

Proof. Write \(U=U_P\) and \(M=U\cap\mathop{\mathrm{NS}}(S)_{\mathbb Q}\). Since \(P\) contains an ample class, Hodge index makes \(M\) negative definite and \[U=T_S\perp M.\] The source projection to \(T_S\) is algebraic: choose divisor bases for \(\mathop{\mathrm{NS}}(S)_{\mathbb Q}\) and use their inverse intersection matrix together with the diagonal. Proposition 33 gives one exact tensor \(\iota_{w_0,E_0}\) on all of \(U\).

For the desired \(w\in T_S\), put \(r=w_0^{-1}w\in C^+(U,q)\). It is invertible. Right multiplication \(R_r\) commutes with the left Hodge complex structure, so is a rational Hodge automorphism, and associativity gives \[j_w=R_rj_{w_0},\qquad \iota_{w,E_0}=(R_r\otimes\mathop{\mathrm{id}})\iota_{w_0,E_0}.\] This is a change of the first, output tensor slot.

The natural Clifford inclusion \(i:W_T=C^+(T_S,q)\hookrightarrow W_U=C^+(U,q)\) is injective by PBW and is Hodge: the left period bivector lies in \(C^+(T_S,q)\) on both spaces. Polarizability gives a rational Hodge retraction \(p\). Apply Lemma 36 with the desired \(E\) on \(W_T\). For \(v,w\in T_S\), \(j_w(v)\) preserves \(W_T\) and restricts there to the standard operator \(a\mapsto vaw\). Therefore \[\bigl(pR_r\otimes(E^\flat)^{-1}i^\vee(-1)E_0^\flat\bigr) \iota_{w_0,E_0} =\iota^{T_S}_{w,E}.\] The order matters: \(R_r\) itself need not preserve \(W_T\) when \(w_0\notin T_S\), but the resulting operator \(j_w\) does.

Both tensor-slot maps are algebraic on abelian \(H^1\) by the lemma. After clearing denominators, the single product homomorphism on the two abelian factors and its normalized transpose graph realize their tensor product, with no unrecorded permutation sign. The normalized transpose graph has codimension \(2\dim A_U=\dim A_U^2\), so postcomposition preserves codimension two. Precompose with the algebraic source projector and postcompose with algebraic degree-one projectors on the abelian factors. The full degree-two action is then precisely the displayed tensor. The same argument with the prescribed rational Hodge identification transports it to any stipulated isogenous model. ◻

Proof. Use the primitive embedding in Lemma 3 and an exact-\(P\) surface supplied there. Choose an integral ample \(h\in P\) and a small marked \(P\)-polarized period patch \(\Omega\) about that surface, shrinking so that \(h\) stays ample. This is an actual open image in the period domain of \(V_P\), and includes its Picard-jump fibers.

The line \(z=\iota(H^{2,0}(S))\) is a point of this period domain. Lemma 35 gives \(g\in O(V_P)(\mathbb Q)\) with \(gz\in\Omega\). Choose the fiber \(R\) with period exactly \(gz\). Then \(g\iota:T_S\to H^2(R,\mathbb Q)\) is an isometry taking the holomorphic line to that of \(R\). The two full second-cohomology spaces have the rational K3 form. Witt’s extension theorem extends this map to a full rational isometry \(\Phi\). It preserves the holomorphic line, its conjugate by rationality, and their orthogonal complement, which is the \((1,1)\) space. Hence \(\Phi\) is Hodge. Lefschetz \((1,1)\) gives \(\Phi(\mathop{\mathrm{NS}}(S)_{\mathbb Q})=\mathop{\mathrm{NS}}(R)_{\mathbb Q}\) and therefore \(\Phi(T_S)=T_R\). Buskin’s theorem [9] algebraizes \(\Phi\) and its inverse. It requires neither integrality nor preservation of an ample class. Density was used only to enter the patch; the comparison uses the exact realized period. ◻

Proof. Choose \(R,\Phi\) by Lemma 38 and set \(\phi=\Phi|_{T_S}:T_S\simeq T_R\). The isometry induces a unital Clifford algebra isomorphism \[c_\phi:C^+(T_S,q)\xrightarrow{\sim}C^+(T_R,q), \qquad c_\phi(vaw)=\phi(v)c_\phi(a)\phi(w).\] It is a weight-one Hodge isomorphism because it takes the oriented period bivector defining left multiplication to the corresponding bivector for \(R\). For the broader functoriality under Hodge similarities, see [40]; the multiplier here is one, with the prescribed right factor transported as in the displayed identity. Proposition 37 gives the standard tensor on \(T_R\) for \(\phi(w)\) and the polarization transported from \(E\). Precompose its cycle with the algebraic correspondence for \(\Phi\) and transport both abelian slots by \(c_\phi^{-1}\). These rational Hodge maps on \(H^1\) are algebraic by Lemma 36. The displayed Clifford identity proves equality with the specified tensor, not merely with an abstract Hodge embedding. Source and degree-one target projectors isolate its full action. All operations are degree-preserving on their intermediate factors, so the resulting cycle has codimension two. The same algebraic transport treats the stipulated isogenous model. ◻

Proof of Theorem 2. Proposition 39 proves the exact standard-tensor assertion under the stated rational isometric-embedding hypothesis, with every prescribed right vector, polarization, and transported model. Choose any one such tensor and apply Theorem 1. This gives rational HC and GHC in every degree of every self-power. For an ample \(P\)-polarized surface, the actual transcendental space is a nondegenerate subspace of its fixed perpendicular \(U_P\simeq V_P\) by Lemma 3, so this case is included as asserted. ◻

Proof. The assertion for zero-dimensional \(M\) is immediate. Write \(h(-)\) for the Chow motive with rational coefficients. The decomposition of de Cataldo–Migliorini [10] expresses \(h(S^{[n]})\) as a sum of Tate twists of summands of motives of powers of \(S\); the summands corresponding to symmetric products are cut out by rational averaging projectors. For either stated moduli space, Bülles [8] likewise makes \(h(M)\) a direct summand of a finite sum of Tate twists of motives \(h(S^k)\). Tensoring these split correspondences gives analogous splittings for every self-power, again through Tate twists of powers of \(S\). Thus these varieties are motivated by \(S\) in the sense of Arapura. Theorem 2 supplies HC and GHC for every power of \(S\), so Arapura’s transfer lemma [2], which retains geometric coniveau with its Tate shifts, gives both conjectures for every power of each indicated variety. Finally, Rieß’s specialization correspondence for birational smooth projective hyperkähler varieties [32] gives an isomorphism of Chow motives, as formulated in [8]. This proves the last assertion. ◻