Proof. Every projective K3 surface has an ample integral class. Divide it to primitivity in \(H^2(S,\mathbb Z)\); the resulting integral \((1,1)\) class is the first Chern class of a line bundle, a positive power of which is ample. Hence it is itself ample, and its square is a positive even integer. Apply Theorem 2 in its component. ◻

Proof. Polarized local Torelli identifies such a chart with an open subset of the domain of oriented positive two-planes in the fixed real quadratic space \(V_{\mathbb R}\); see [10]. The infinitesimal generators of the Hodge circles over any such open subset span \(\mathfrak{so}(V_{\mathbb R},q)\). Indeed, under the natural identification \(\mathfrak{so}(V_{\mathbb R},q)\simeq\Lambda^2V_{\mathbb R}\), each generator is a nonzero multiple of the bivector of its positive plane. A linear functional vanishing on all these bivectors is an alternating bilinear form vanishing on an open set of ordered pairs of vectors. It therefore vanishes identically.

The Hodge group is a connected rational algebraic subgroup of \(\operatorname{SO}(V,q)\). Connectedness follows because its rational identity component contains the connected Hodge circle and hence, by minimality, is the whole group. If this subgroup is proper, its Lie algebra is a nonzero proper rational subspace \(L\subset\mathfrak{so}(V,q)\). Set \(r=\dim L\), and choose a rational generator \(\tau_L\) of \(\Lambda^rL\). Conjugation by the circle preserves \(L\) and acts trivially on its real determinant line: a continuous character from a compact connected circle to \(\mathbb R^\times\) is trivial. Thus \(\tau_L\), viewed in \(\Lambda^r\operatorname{End}(V)\), has Hodge type \((0,0)\).

There are only countably many possible rational subspaces \(L\). For each one, the condition that \(\tau_L\) have type \((0,0)\) is closed analytic in the period chart. In fact the tensor has weight zero and is real, so this condition is equivalent to \(\tau_L\in F^0\), a holomorphic condition. The locus is proper. Otherwise \(L_{\mathbb R}\) would be preserved by every circle in the chart, hence by the adjoint actions of their infinitesimal generators. The spanning assertion above would make \(L_{\mathbb C}\) an ideal of the simple Lie algebra \(\mathfrak{so}_{21}(\mathbb C)\), a contradiction.

Finally, the standard representation of \(\operatorname{SO}(V,q)\) is absolutely irreducible and has no nonzero invariants. Consequently \(V\) has no proper nonzero rational Hodge substructure and no nonzero rational vector of type \((1,1)\). Since the primitive part of \(\operatorname{NS}(S)_{\mathbb Q}\) consists of such vectors, \(\operatorname{NS}(S)_{\mathbb Q}=\mathbb Qh\) and \(T(S)=V\). ◻

Proof. The group \(\mathcal H\) is connected by the same circle argument as in Proposition 11. It is reductive. To see this directly, each of the four pure summands is semisimple as a rational \(\mathcal H\)-module: an invariant rational subspace is a Hodge substructure, and a polarization provides a rational Hodge complement. This is the usual semisimplicity argument for polarizable Hodge structures; compare [2]. Their direct sum is a faithful semisimple module. The unipotent radical has nonzero invariants in each irreducible constituent; normality makes those invariants \(\mathcal H\)-stable, so it acts trivially on every constituent. Faithfulness forces the radical to be trivial.

Projection to \(V\) maps \(\mathcal H\) onto \(\operatorname{Hg}(V)=\operatorname{SO}(V,q)\). Indeed, its closed rational image contains the projected circle and hence contains \(\operatorname{Hg}(V)\). Conversely, the preimage of \(\operatorname{Hg}(V)\) contains the joint circle, so minimality places \(\mathcal H\) inside that preimage. The reductive rational Lie algebra \(\mathfrak h\) is the direct sum of its center and its rational simple ideals. Exactly one simple ideal maps nontrivially to the absolutely simple quotient \(\mathfrak{so}(V,q)\): its image is the whole quotient, and the commuting images of all other ideals must vanish. The nonzero map from that simple ideal is an isomorphism. This gives the asserted rational decomposition.

On \(V\oplus W^+\oplus W^-\), the image of \(\mathcal H\) is contained in the rational algebraic image of \(\operatorname{Spin}(V,q)\) acting on \(V\) through its orthogonal quotient and on \(W^\pm\) by left multiplication. That image contains the projected Hodge circle, so the containment again follows by minimality. The differential of \(\operatorname{Spin}(V,q)\to\operatorname{SO}(V,q)\) is an isomorphism. Its kernel on Lie algebras is therefore zero, proving both remaining assertions. ◻

Proof. Let \(D\) be the infinitesimal Hodge-circle generator divided by \(i\); its eigenvalue on type \((p,r)\) is \(p-r\). In the decomposition of Lemma 12, write \(D=D_{\mathfrak g}+D'\). On \(V_{\mathbb C}\) the eigenvalues are \(2,-2,0,\ldots,0\). In the standard Cartan coordinates for type \(B_{10}\), in which the vector weights are \(\pm e_1,\ldots,\pm e_{10},0\), we may therefore take \(D_{\mathfrak g}=2e_1^\vee\).

The element \(D_{\mathfrak g}\) is semisimple on every finite-dimensional \(\mathfrak g_{\mathbb C}\)-module. It commutes with \(D\), so \(D'=D-D_{\mathfrak g}\) is also semisimple on \(U_{\mathbb C}\). Each \(D'\)-eigenspace is \(\mathfrak g_{\mathbb C}\)-stable. Consider an irreducible constituent in such an eigenspace, with highest weight \(\lambda=(\lambda_1,\ldots,\lambda_{10})\). The highest-weight description for \(B_{10}\) gives \[\lambda_1\ge\cdots\ge\lambda_{10}\ge0, \qquad \lambda_i\in\mathbb Z\ \text{for every }i, \quad\text{or}\quad \lambda_i\in\tfrac12+\mathbb Z\ \text{for every }i.\] The largest and smallest eigenvalues of \(2e_1^\vee\) are respectively \(2\lambda_1\) and \(-2\lambda_1\). Indeed \(e_1^\vee\) is dominant, and the Weyl group permits every coordinate sign change. These elementary highest-weight and spin facts may be found in the orthogonal-group chapters of [5].

Adding the scalar by which \(D'\) acts does not change the difference \(4\lambda_1\) between the extreme eigenvalues. Since \(U\) has Hodge types \((1,0)\) and \((0,1)\), the eigenvalues of \(D\) on \(U_{\mathbb C}\) belong to \(\{1,-1\}\). Hence \(4\lambda_1\le2\). The only possibilities are the trivial representation and \(\lambda=(\tfrac12,\ldots,\tfrac12)\), which is \(\Delta\). On a spin constituent the extreme eigenvalues already are \(1,-1\); the only scalar shift preserving this pair is zero. Thus \[ D'=0\quad\text{on }\Delta\otimes M. \tag{17}\] This proves the assertion about \(U\). The complex Clifford identity \(C^+(V_{\mathbb C},q)\simeq\operatorname{End}_{\mathbb C}(\Delta)\), with left multiplication, gives the assertion about \(W^+\). An invertible odd right multiplier identifies the two parity spaces for the left spin action and gives the assertion about \(W^-\). Lemma 12 shows that \(D'=0\) on both of them.

Now a rational \(\mathfrak g\)-map \(f:U\to W^\pm\) vanishes on \(M_0\) after complexification. The summand \(M_0\) is \(D\)-stable. On the spin summand, Equation (17) shows that \(f\) intertwines \(D\). It therefore intertwines \(D\) on all of \(U\), hence the Hodge circle, and is a Hodge map. Conversely, every rational Hodge map intertwines \(\mathcal H\) and therefore \(\mathfrak g\).

Finally, rational \(\mathfrak g\)-intertwiners are the kernel of the finite system of rational linear equations \[f\rho_U(X_a)=\rho_{W^\pm}(X_a)f,\] where \(\{X_a\}\) is a rational basis of \(\mathfrak g\). Taking that kernel commutes with extension of scalars. Schur’s Lemma over \(\mathbb C\) and Equation (15) now give Equation (16). ◻

Proof. The spin representation in type \(B_{10}\) is self-dual, since the longest Weyl element is \(-1\). Under the conjugation action, the Clifford antisymmetrization map gives \[ \Delta\otimes\Delta\simeq\operatorname{End}(\Delta) \simeq C^+(V_{\mathbb C},q) \simeq\bigoplus_{k\ \mathrm{even}}\Lambda^kV_{\mathbb C}. \tag{18}\] Here the action is conjugation, in contrast to the left action used in Equation (15).

For completeness, the vector multiplicity in Equation (18) can be read off without decomposing every exterior power. Choose an orthonormal basis \(v_1,\ldots,v_{21}\). An equivariant map \(F:V_{\mathbb C}\to\Lambda^kV_{\mathbb C}\) is determined by \(F(v_1)\), because the orthogonal orbit of \(v_1\) spans \(V_{\mathbb C}\). Its value is fixed by all determinant-one diagonal sign changes on \(v_2,\ldots,v_{21}\). A basis exterior monomial fixed by all these sign changes contains either none or all of these twenty vectors. For even \(k\), only \(k=0\) and \(k=20\) can therefore occur. The degree-zero map vanishes, since a determinant-one orthogonal transformation sending \(v_1\) to \(-v_1\) forces \(F(v_1)=F(-v_1)=-F(v_1)\). For \(k=20\) the possible value is a multiple of \(v_2\wedge\cdots\wedge v_{21}\), and volume duality \(\Lambda^{20}V_{\mathbb C}\simeq V_{\mathbb C}\) supplies a nonzero equivariant map. The multiplicity is exactly one. The last assertion follows from irreducibility and the distinct dimensions \(21\), \(2^{10}\), and \(1\). ◻

Proof. Pull back \(\alpha\) along addition \(B\times B\to B\) and apply the algebraic projector to \(U\otimes U\subset H^2(B\times B,\mathbb Q)\) from Lemma 5. The resulting codimension-two cycle induces \[\alpha':V\longrightarrow U\otimes U.\] It is nonzero: on \(H^2(B,\mathbb Q)=\Lambda^2U\), the mixed component of addition pullback is the injective map \(u\wedge v\mapsto u\otimes v-v\otimes u\).

Fix a nonzero \(\mathfrak g_{\mathbb C}\)-equivariant map \(\gamma:V_{\mathbb C}\to\Delta\otimes\Delta\). The map \(\alpha'\) is rational and Hodge, hence \(\mathfrak g\)-equivariant. Lemmas 13 and 14 give \[ \alpha'_{\mathbb C}=\gamma\otimes t, \qquad 0\ne t\in M\otimes M. \tag{19}\] In this formula the tensor factors have been regrouped so that the two spin factors come first. The other summands of \(U_{\mathbb C}\otimes U_{\mathbb C}\) contain no vector copy.

Put \(N_j=N_{\epsilon_j}\) for \(j=1,2\). The independent operations \[a\in\operatorname{Hom}_{\mathbb C}(M,N_1),\qquad b\in\operatorname{Hom}_{\mathbb C}(M,N_2)\] send \(t\) to \((a\otimes b)t\), and these images span \(N_1\otimes N_2\). To check this, choose \(\mu,\nu\in M^\vee\) with \(c=(\mu\otimes\nu)(t)\ne0\). For arbitrary \(n_1\in N_1\) and \(n_2\in N_2\), the maps \[a(m)=\mu(m)n_1,\qquad b(m)=\nu(m)n_2\] send \(t\) to \(c\,n_1\otimes n_2\). Thus any symmetry imposed on \(t\) by the original exterior square causes no restriction: the two operations are chosen independently.

To implement these operations over \(\mathbb Q\), choose rational bases \[f_1,\ldots,f_r\quad\text{of } \operatorname{Hom}_{\mathfrak g}(U,W^{\epsilon_1}),\qquad g_1,\ldots,g_s\quad\text{of } \operatorname{Hom}_{\mathfrak g}(U,W^{\epsilon_2}).\] By Lemma 13, all these maps are Hodge, and their complex spans are the full spaces of the independent operations just used. Lemma 4 realizes each by a rational abelian homomorphism acting contravariantly on \(H^1\); this is the classical weight-one equivalence, also stated in [13]. Products of their transpose-graph correspondences, followed by the mixed projector if necessary, therefore realize every map \[ \Phi_{ij}=(f_i\otimes g_j)\alpha' \tag{20}\] by a rational algebraic cycle \(\Gamma_{ij}\) on \(S\times A_1\times A_2\). These cycles have codimension two: the target correspondences have degree shift zero, and composition preserves the degree shift of the original surface correspondence.

Lemma 14, Equation (19), and the spanning calculation show that the complex span of the maps in Equation (20) is \[\operatorname{Hom}_{\mathfrak g_{\mathbb C}} \bigl(V_{\mathbb C}, W^{\epsilon_1}_{\mathbb C}\otimes W^{\epsilon_2}_{\mathbb C}\bigr).\] The prescribed rational Hodge map \(\beta\) lies in this space. Consequently the finite system \[\beta=\sum_{i=1}^r\sum_{j=1}^s c_{ij}\Phi_{ij}\] has a solution over \(\mathbb C\). All its coefficients and its right-hand side are rational, so it has a solution with \(c_{ij}\in\mathbb Q\) by Lemma 6. The cycle \(\Gamma=\sum_{i,j}c_{ij}\Gamma_{ij}\) has precisely the required action.

Finally, Clifford multiplication is the rational Hodge map \[V(1)\longrightarrow\operatorname{Hom}(W^+,W^-), \qquad x\longmapsto L_x.\] Using a weight-one polarization to identify \((W^+)^\vee(-1)\) with \(W^+\) makes it a rational Hodge map \(V\to W^-\otimes W^+\). It is therefore one of the maps just realized. The fixed odd right multiplier and prescribed weight-one duality then give exactly the specified even-Clifford tensor, by Lemma 10. No choice of a single spin copy or undetermined scalar remains. ◻

Proof. The embedded fibers have a common Hilbert polynomial \(P\). Indeed, Riemann–Roch expresses \(\chi(\mathcal X_t,\mathcal O_{\mathcal X_t}(k))\) in terms of the fiber integrals of the Chern classes of the relative tangent bundle and the hyperplane bundle. These integrals are locally constant in a smooth proper family, and \(D\) is connected.

Let \(\operatorname{Hilb}^{P}(\mathbb P^N)\) be the algebraic Hilbert scheme. Fixed-polynomial boundedness supplies one sufficiently large integer \(k\) for which every member \(Y\) has \[\dim \operatorname{im}\bigl(H^0(\mathbb P^N,\mathcal O(k)) \longrightarrow H^0(Y,\mathcal O_Y(k))\bigr)=P(k),\] and its degree-\(k\) ideal space defines the Hilbert scheme’s closed embedding in the corresponding Grassmannian. These uniform statements precede the construction of \(\eta\); see [8].

We check that \(t\mapsto H^0(\mathbb P^N,\mathcal I_{\mathcal X_t}(k))\) is holomorphic. Fix \(t_0\in D\). Since \(\mathcal X_{t_0}\) is reduced, point evaluations separate the restrictions of degree-\(k\) polynomials. Choose finitely many points whose evaluations have rank \(P(k)\), and extend these points to local holomorphic sections of \(p\), using its submersion property. After locally trivializing \(\mathcal O(k)\) along these sections, evaluation is a holomorphic matrix. A nonzero \(P(k)\)-minor remains nonzero near \(t_0\). Its rank cannot exceed \(P(k)\), because evaluation factors through restriction to \(\mathcal X_t\). Consequently its kernel equals the degree-\(k\) ideal space, and these kernels give the required holomorphic Grassmannian map.

Pointwise this map lies in the Hilbert locus. Since \(D\) is reduced, the defining equations of that locus vanish on \(D\), so the map factors holomorphically through the Hilbert scheme. Take \(H\) to be the reduction of the open locus parametrizing smooth pure \(n\)-dimensional subschemes, and let \(\mathcal U\) be the restricted universal family. Flatness of the universal family and smoothness of all its fibers imply that \(\mathcal U\to H\) is smooth. Its pullback to \(D\) and \(\mathcal X\) are reduced closed analytic subspaces of \(D\times\mathbb P^N\) with the same fibers, hence the same underlying set. They therefore coincide. ◻

Proof. The total space \(M\) is smooth. Flatness and the dimension formula give \(\dim Z=n+b-2\), so its top cycle has a class \[\zeta=\operatorname{cl}([Z]_{n+b-2})\in H^4(M,\mathbb Q).\] Here the usual cycle class on a smooth variety takes values in ordinary cohomology; completeness of \(M\) is unnecessary. We claim that, for the inclusion \(i_r:M_r\hookrightarrow M\), \[ i_r^*\zeta=\operatorname{cl}([Z_r]_{n-2}). \tag{21}\]

Choose local algebraic coordinates by an étale map from a neighborhood \(U\subset R\) to \(\mathbb A^b\) taking \(r\) to \(0\). Shrink \(U\) to remove the other points over \(0\). The coordinate functions then cut out a smooth flag \[U=U_0\supset U_1\supset\cdots\supset U_b=\{r\}.\] Put \(Z_j=Z\times_R U_j\). It remains flat over \(U_j\) and has dimension at most \(n+b-2-j\). At each step the next coordinate is a nonzerodivisor on \(U_j\) and hence on the entire scheme \(Z_j\). The Cartier intersection formula therefore gives \[[Z_j]_{n+b-2-j}\longmapsto[Z_{j+1}]_{n+b-3-j},\] with the scheme multiplicities [15]. Applying the formula successively yields \[i_r^![Z]_{n+b-2}=[Z_r]_{n-2}.\] The formula applies to the whole possibly nonpure or nonreduced scheme at each cut; no componentwise flatness assertion is needed. Compatibility of Betti cycle classes with these regular pullbacks proves (21).

Finally, the restrictions of a single cohomology class on the total space of a smooth proper family form a section of its cohomology local system. This follows directly from local differentiable triviality. Apply it to \(\zeta\). ◻

Proof. Fix a desired point \(0\in D\) and trivialize \(R^4p_*\mathbb Q\) over \(D\). Use Lemma 17 to write the family as the pullback of \(\mathcal U\to H\) along \(\eta:D\to H^{\mathrm{an}}\). For each possible Hilbert polynomial \(Q\) of degree \(n-2\), consider \[T_Q=\operatorname{Hilb}^{Q}(\mathcal U/H).\] This is projective over \(H\) and has a flat universal subscheme. It has finitely many reduced irreducible components. Resolve each component by a projective birational morphism, obtaining a finite collection of smooth irreducible varieties \(R\to (T_Q)_{\mathrm{red}}\) whose images cover \((T_Q)_{\mathrm{red}}\); use [9]. Thus each composite \(R\to H\) is proper. Pullback gives a smooth projective ambient family \(\mathcal U_R\to R\) and a flat closed subscheme \(Z_R\subset\mathcal U_R\). By Lemma 18, the classes of the top cycles of its fibers are sections of \(R^4(\mathcal U_R/R)_*\mathbb Q\).

Form the analytic spaces \[K_R=D\times_{H^{\mathrm{an}}}R^{\mathrm{an}}.\] They are proper over \(D\). Their pulled-back ambient families agree with the pullback of \(\mathcal X\), so the preceding cycle classes become locally constant functions on \(K_R\) with values in the fixed marked space \(H^4(\mathcal X_0,\mathbb Q)\). The spaces \(K_R\) may be singular or nonreduced: this causes no difficulty, since local constancy was established on the smooth parameter \(R\) before base change.

There are countably many \(Q\) and finitely many choices of \(R\) for each \(Q\). Each \(K_R\) is second countable and has a locally finite family of closed irreducible analytic components. It consequently has at most countably many such components. Every component is proper over \(D\); Remmert’s Proper Mapping Theorem makes its image a closed analytic subset of \(D\) [7]. Let \(E\) be the union of the images of all components whose image is not all of \(D\). Thus \(E\) is a countable union of proper closed analytic subsets.

Choose \(t\in D\) outside \(E\) and outside the exceptional sets in the hypothesis. Such a point exists by the Baire Category Theorem. Choose an actual cycle representing \(\theta_t\), and write it as a finite sum \[\gamma_t=\sum_{j=1}^{s}a_j[Y_{j,t}], \qquad a_j\in\mathbb Q,\] where each \(Y_{j,t}\subset\mathcal X_t\) is an integral subvariety of codimension two. Its Hilbert point lifts to one of the parameters \(R\), and hence to a point \(u_j\in K_R\) above \(t\). Choose an irreducible component \(C_j\) containing \(u_j\). Because \(t\notin E\), the map \(C_j\to D\) is surjective. In particular, there is a point \(v_j\in C_j\) above \(0\).

An irreducible complex analytic space is connected. The marked cycle class is therefore constant on \(C_j\), and the fiber of the universal subscheme at \(v_j\) has top cycle \(\delta_{j,0}\) satisfying \[\operatorname{cl}(\delta_{j,0}) =\operatorname{cl}([Y_{j,t}]) \quad\text{under the marking.}\] These are algebraic cycles on \(\mathcal X_0\): the fibers of the universal subscheme are algebraic in \(\mathbb P^N\), and the embedded fiber identification is algebraic by GAGA [14]. Consequently \[\gamma_0=\sum_{j=1}^{s}a_j\delta_{j,0} \quad\text{satisfies}\quad \operatorname{cl}(\gamma_0)=\theta_0.\] The argument applies to any chosen point of \(D\). ◻

Proof. Polarized deformation theory and local Torelli give the smooth K3 family over a \(19\)-dimensional polydisc on which the period map is a local isomorphism; see [10]. Since \(D\) is contractible, its cohomology local systems are marked. The class \(h\) remains of type \((1,1)\). We recall why it is represented by a line bundle on the total family. Local differentiable triviality over the contractible base extends \(h\) to \(H^2(\mathcal S,\mathbb Z)\). The polydisc is Stein, and coherent Leray and Cartan’s Theorem B identify \[H^2(\mathcal S,\mathcal O_{\mathcal S}) =H^0(D,R^2p_*\mathcal O_{\mathcal S}), \qquad p:\mathcal S\to D.\] The direct image on the right is locally free and commutes with base change. The image of \(h\) in it vanishes fiberwise, hence vanishes. The exponential sequence now gives a holomorphic line bundle \(\mathcal L\) with first Chern class \(h\).

A high power of \(\mathcal L\) is very ample on \(S_0\) and has vanishing higher cohomology there. Proper cohomology and base change extend a basis of its sections over a neighborhood of \(0\). They give a relative embedding after shrinking: basepoint freeness and immersion persist near the central fiber, and properness excludes pairs of distinct points with the same image. For the latter assertion, any sequence of such pairs approaching the central fiber either limits to distinct points, contrary to its embedding, or to a single point, contrary to the local embedding furnished by the immersion. This proves local relative projectivity, and also that \(h\) stays ample. The analytic tools used here are the proper direct-image and base-change theorems and the Stein vanishing theorem; see [7].

On the fixed Clifford space \(W\), the Hodge filtration and rational alternating polarization are those constructed in Proposition 9. To recall the holomorphic dependence, if \(\omega_t\) spans the period line, then \[F^1W_{\mathbb C} =\operatorname{im}\bigl(L_{\omega_t}:C^-(V_{\mathbb C},q) \longrightarrow C^+(V_{\mathbb C},q)\bigr).\] This has constant half dimension and varies holomorphically. The polarization can be fixed on a sufficiently small chart. Choose a rational symplectic basis and let its integral span be a lattice \(\Lambda\) in \(W\). It is unimodular for the chosen symplectic form and does not change the rational Hodge structure. Since \(W\) is cohomology, the associated torus is explicitly \[\mathcal A_t=(F^1W_t)^\vee/\operatorname{image}(\Lambda^\vee),\] where a lattice functional is restricted to \(F^1W_t\). Thus \(H_1(\mathcal A_t,\mathbb Z)=\Lambda^\vee\), \(H^1(\mathcal A_t,\mathbb Z)=\Lambda\), and \(H^{1,0}(\mathcal A_t)=F^1W_t\), compatibly with the markings. The polarization supplies the Riemann form on the dual lattice. Equivalently, the marked filtrations give a holomorphic map to Siegel space and \(\mathcal A\to D\) is the pullback of its marked torus family. The full markings supply, in particular, level-\(3\) data. The spin action is defined on the fixed Clifford space over the polydisc, so there is no monodromy lifting problem on this base.

For completeness, in symplectic period coordinates this torus family is \[\mathbb C^g/(\mathbb Z^g+\tau(t)\mathbb Z^g), \qquad \tau(t)^{\mathsf t}=\tau(t),\quad \operatorname{Im}\tau(t)>0.\] The factors \[\exp\bigl(-\pi i\,n^{\mathsf t}\tau(t)n -2\pi i\,n^{\mathsf t}z\bigr), \qquad m+\tau(t)n\in\mathbb Z^g+\tau(t)\mathbb Z^g,\] define a holomorphic theta line bundle, ample on every fiber. The preceding extension-of-sections argument gives a relative projective embedding after shrinking \(D\).

Fix a rational nonisotropic vector \(z_0\in V\) and the polarization identification \(j:W^\vee(-1)\xrightarrow{\sim}W\). The maps \[V(1)\longrightarrow\operatorname{End}(W), \qquad x\longmapsto\bigl(w\longmapsto xwz_0\bigr)\] give \(\mu_t\) after untwisting and applying \(j\) to the dual slot. Clifford multiplication, \(z_0\), and \(j\) are constant rational data in the markings. Thus \(\mu_t\) is flat as a rational tensor and is a Hodge map in every fiber. A Segre embedding combines the relative embeddings of \(\mathcal S\) and the two copies of \(\mathcal A\). ◻

Proof. Take the families in Lemma 20 about \((S_0,h)\). Under the markings, extend \(\mu_t\) by zero on \(\mathbb Qh\) and on the other degrees of \(H^*(\mathcal S_t,\mathbb Q)\). Poincaré duality and the Künneth decomposition give a unique flat class \[\theta\in H^0(D,R^4(\mathcal X/D)_*\mathbb Q)\] with this action. More explicitly, for a rational basis \((v_i)\) of \(V\), let \((q^{ij})\) be the inverse cup-pairing matrix. Then \[ \theta_t=\sum_{i,j}q^{ij}v_i\otimes\mu_t(v_j) \ \in\ V\otimes \bigl(H^1(\mathcal A_t)\otimes H^1(\mathcal A_t)\bigr) \ \subset\ H^4(\mathcal X_t,\mathbb Q). \tag{22}\]

By Proposition 11, after excluding countably many proper closed analytic subsets we have \(\operatorname{Hg}(V_t)=\operatorname{SO}(V,q)\). At a point satisfying this condition and the assumed input, Theorem 15 produces a cycle inducing \(\mu_t\) on \(V_t\). Its full class can be made exactly \(\theta_t\). Indeed, on \(\mathcal S_t\) the algebraic correspondence \[ \pi_{V,t}=[\Delta_{\mathcal S_t}] -[o_t]\times\mathcal S_t-\mathcal S_t\times[o_t] -\frac{h\times h}{\int_{\mathcal S_t}h^2}, \qquad o_t\in\mathcal S_t, \tag{23}\] acts identically on \(V_t\) and kills \(H^0\), \(H^4\), and \(\mathbb Qh\). There is no odd K3 cohomology. Precompose the chosen cycle with \(\pi_{V,t}\) and postcompose with the algebraic projector onto \(H^1(\mathcal A_t)\otimes H^1(\mathcal A_t)\), as in Lemma 5. These operations preserve codimension two. The resulting cycle has the prescribed action in every source degree, so Poincaré duality and Künneth identify its entire class with (22). The point \(o_t\) is needed only on this fiber; its cohomology class, and hence the projector’s cohomological action, is canonical.

If the hypothesis is stated in algebraic moduli, its exceptional loci pull back to proper closed analytic subsets of \(D\). They remain proper because the full-dimensional polarized period chart has open image locally in that component, up to a finite automorphism quotient. Thus \(\theta_t\) is algebraic very generally on \(D\) in either version of the hypothesis. Proposition 19 produces an actual rational codimension-two cycle with class \(\theta_0\). Its action on \(V\) is exactly \(\mu_0\). Undoing the fixed invertible right multiplier on the output Hom slot, using Lemma 10, also algebraically realizes \(m^q\) itself. ◻