A Conditional Reduction for Algebraic Kuga–Satake Correspondences

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 1.2 in its component.□

Proof. Dualize a rational Hodge map to a map on first homology. After multiplication by a positive integer it preserves the integral lattices. The Hodge condition says that its real extension commutes with the complex structures on the universal covers. It therefore induces a holomorphic group homomorphism Q\to P, which is algebraic by projectivity. Conversely pullback by every such homomorphism is a Hodge map. This is the usual equivalence between abelian varieties up to isogeny and polarizable rational Hodge structures of weight one, written contravariantly [13, Theorem 6.8 and Corollary 6.9]. Transpose graphs realize the pullbacks. Products of two such correspondences, and their rational linear combinations, act independently on the two tensor factors. Preservation of the chosen polarizations is not required.□

Proof. For an abelian variety B of dimension b, multiplication by 2 acts on H^j(B) by 2^j. Thus the rational polynomial

applied to the correspondence [2]^* is the projector onto H^j(B). Apply these polynomials separately to each factor of a product. Finally m^*u=p_1^*u+p_2^*u on H^1; multiplication in the exterior algebra, with its usual signs, gives the displayed formula.□

Proof. The cycle class map and restriction of its action are rational linear maps, so E_{\mathrm{alg}} is a rational subspace. If a rational target lies in the scalar-extended span of finitely many rational cycle actions, write these actions as columns in a rational basis of E. The linear system with that target as right-hand side is consistent over K, and row reduction over \mathbb Q gives a rational solution. Only finitely many cycles are used.□

Proof. Suppose first that f:Y\to S satisfies (1.3). Let D=\operatorname{Alb}(Y), let a:Y\to D be an Albanese map, and put b=\dim D. Albanese pullback identifies holomorphic one-forms [3, Chapter VI, §9.2, pp. 315–316]. Hence there is

In particular b\ge2 and y\in H^{2,0}(S)\subset V_{\mathbb C}. The surface correspondence (a,f)_*[Y] has codimension b on D\times S. Its transpose acts by

This action is nonzero on V: the projection formula gives

We give an algebraic degree adjustment. For an ample line bundle L on D, written additively, set

where m is addition. The class of E is purely mixed. Put U=H^1(D,\mathbb Q) and write this class as \sum_i u_i\otimes v_i, choosing a basis (u_i) of U. Its matrix

is invertible: it is the alternating tensor associated with c_1(L), whose top exterior power is nonzero since L is ample. Intrinsically it is a Hodge map M:U^\vee\to U(1); in the matrix calculation we use the underlying rational spaces. With the Koszul convention for Künneth products,

Poincaré duality identifies H^{2b-2}(D) with (\bigwedge^2U)^\vee on underlying rational spaces, by integration against degree-two classes. Consequently (E^2)_* is -2\bigwedge^2M after this identification, and is an isomorphism

The twists can also be checked from H^{2b-2}(D)(b-2)\simeq H^2(D)^\vee(-2) and \bigwedge^2M:H^2(D)^\vee\to H^2(D)(2). Composing with (f,a)_*[Y] yields a cycle of codimension b+2-b=2 on S\times D, nonzero on V by (2.2). This uses a square of an explicit divisor; it assumes no algebraic inverse Lefschetz operator for a general variety.

Conversely suppose that \alpha_*:V\to H^2(B) is nonzero and \operatorname{Hg}(V)=\operatorname{SO}(V,q). The standard representation is irreducible, so \alpha_*|_V is injective. For 0\ne\sigma\in H^{2,0}(S), put \eta=\alpha_*\sigma\ne0 in H^{2,0}(B). Write c=\dim B\ge2 and let \ell be an ample divisor class on B, replaced by a very ample multiple if necessary. The rational surface cycle

satisfies

Here rationality gives \alpha_*\overline\sigma=\overline\eta, and the last pairing is definite up to a fixed nonzero sign on H^{2,0}(B) by the Hodge–Riemann bilinear relations. An actual representative of C is a finite rational combination of integral surfaces; for example use general hyperplanes to represent the indicated intersections. At least one component has nonzero pairing in (2.4). Resolve that component projectively and call the resulting smooth connected surface Y. Its projections give morphisms f:Y\to S and a':Y\to B with

Thus f^*\sigma\ne0, which forces f to have two-dimensional image. It is therefore dominant and generically finite. The projection formula also gives f_*a'^*\eta\ne0. Since \eta is a sum of wedges of holomorphic one-forms on B, its pullback lies in the wedge image in (1.3). This proves the converse.□

Proof. The Clifford relations give

Moreover, J commutes with the orthogonal complement of the period plane. Thus g_\theta\in\operatorname{Spin}(H,Q)(\mathbb R), and its conjugation action on H is the prescribed weight-two Hodge circle. Since

the map in (3.2) is a Hodge map. It is injective because L_x(1)=x. Choose a rational nonisotropic vector z\in H. Right multiplication R_z exchanges the parities, has inverse R_{z/Q(z,z)}, and commutes with the left circle. The same observation applies to any invertible odd right multiplier.

For holomorphic variation, put \omega=e_1+i e_2. The relation J\omega=i\omega shows that the image of L_\omega from the opposite parity is contained in the i-eigenspace of L_J. Conversely \omega J=-i\omega, so this eigenspace is contained in \ker L_\omega. Finally

shows that \ker L_\omega=\operatorname{im}L_\omega, with the appropriate parities understood. Hence F^1P_{H,\mathbb C}^\pm=\operatorname{im}L_\omega; this is a holomorphic subbundle as the isotropic line varies.

Here is a fixed rational polarization. Write u\mapsto u^* for reversal in the Clifford algebra and \operatorname{scal} for its scalar part. Choose rational orthogonal vectors f_1,f_2 of sign \epsilon whose oriented plane is sufficiently close to the given period plane. Rational approximation followed by rational orthogonalization supplies them. Put a=f_1f_2 and, on C^+(H,Q),

The scalar part is cyclic and invariant under reversal, as is checked on an orthogonal monomial basis. Since a^*=-a, the form s is alternating. It is nondegenerate because a is invertible and (u,v)\mapsto\operatorname{scal}(uv^*) is nondegenerate. Reversal gives g_\theta^*g_\theta=1, so s is invariant under the left circle at every period.

At the reference plane spanned by f_1,f_2, write a=cJ with c\in\mathbb R^\times. Then

To see definiteness, extend a normalized basis of this plane to an orthogonal basis with squared lengths \epsilon,\epsilon,-\epsilon,\ldots,-\epsilon. The last pairing is diagonal on even monomials e_I. If r=|I| and k entries of I belong to the period plane, its diagonal sign is

because r is even. Up to the overall sign required by the weight-one polarization convention, s is therefore a polarization. Circle invariance makes s(u,Jv) symmetric and nondegenerate at every period. Its signature is constant on a connected period chart containing both the reference plane and the original period, so the chosen sign of s polarizes throughout that chart. Transport by R_z supplies a fixed polarization on C^-(H,Q) as well.□

Proof. Under (3.4), the two operations are precisely

Both factors are rational weight-one Hodge maps, so Lemma 2.1 realizes them by abelian correspondences. Their product acts independently on the two factors and preserves the codimension of the correspondence from the surface. Changing j_P likewise composes with a weight-one Hodge isomorphism. The factor-switching isomorphism is algebraic; its Künneth sign can be included in the rational coefficient. The projectors of Lemma 2.2 remove all other Künneth components when necessary.□

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, Chapter 6]. 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 4.1. 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, Proposition 3.6]. 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 4.2, 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

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

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 4.2 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 (4.4) 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

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 (4.2) now give Equation (4.3).□

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

Here the action is conjugation, in contrast to the left action used in Equation (4.2).

For completeness, the vector multiplicity in Equation (4.5) 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 2.2. The resulting codimension-two cycle induces

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 4.3 and 4.4 give

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

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

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

By Lemma 4.3, all these maps are Hodge, and their complex spans are the full spaces of the independent operations just used. Lemma 2.1 realizes each by a rational abelian homomorphism acting contravariantly on H^1; this is the classical weight-one equivalence, also stated in [13, Theorem 6.8 and Corollary 6.9]. Products of their transpose-graph correspondences, followed by the mixed projector if necessary, therefore realize every map

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 4.4, Equation (4.6), and the spanning calculation show that the complex span of the maps in Equation (4.7) is

The prescribed rational Hodge map \beta lies in this space. Consequently the finite system

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 2.3. The cycle \Gamma=\sum_{i,j}c_{ij}\Gamma_{ij} has precisely the required action.

Finally, Clifford multiplication is the rational Hodge map

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 3.2. 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

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, Theorem 3.2 and Proposition 3.8].

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

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,

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

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

with the scheme multiplicities [15, Lemma 42.29.5, Tag 02TB]. Applying the formula successively yields

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 (5.1).

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 5.1 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

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 5.2, the classes of the top cycles of its fibers are sections of R^4(\mathcal U_R/R)_*\mathbb Q.

Form the analytic spaces

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, §7, Satz 1]. 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

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

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

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, Chapter 6, Section 2]. 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

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, §6, Hauptsatz I; §7, Sätze 3,5; §2, Satz 3].

On the fixed Clifford space W, the Hodge filtration and rational alternating polarization are those constructed in Proposition 3.1. To recall the holomorphic dependence, if \omega_t spans the period line, then

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

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

The factors

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

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 5.4 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

with this action. More explicitly, for a rational basis (v_i) of V, let (q^{ij}) be the inverse cup-pairing matrix. Then

By Proposition 4.1, 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 4.5 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

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 2.2. 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 (5.2). 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 5.3 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 3.2, also algebraically realizes m^q itself.□

Proof. The orthogonal decomposition V=T\oplus T^\perp is a decomposition of rational Hodge structures, and T contains the period plane. By the Clifford Basis Theorem there are inclusions

They are weight-one Hodge inclusions: the element J defining the left Hodge circle already belongs to C^+(T_{\mathbb R},q_0). Let i:C^+(T,q_0)\to C^+(V,q) be the inclusion and choose a rational Hodge retraction p:C^-(V,q)\to C^-(T,q_0). Such a retraction exists by polarizability: the orthogonal complement for a weight-one polarization is a rational Hodge complement. For every x\in T,

because xC^+(T,q_0)\subset C^-(T,q_0) and p restricts to the identity there. Lemma 3.2 realizes the two operations on the abelian factors. Restricting the resulting cycle action to T gives the asserted equality; no operation on the K3 factor is needed.□

Proof. Step 1: square-root isometries. Write

The two form isomorphisms from T to its twisted dual show that P is an invertible rational Hodge endomorphism. Symmetry of the forms makes P self-adjoint for q_0. Let K be +1 on the real period plane and -1 on the real (1,1)-part, and let \epsilon_0 be the sign of q_0 on the period plane. Then

is positive definite. Since P commutes with K, it is also self-adjoint for this positive definite real form. Hence P is semisimple with real nonzero eigenvalues. Its distinct eigenvalues \lambda_1,\ldots,\lambda_r and spectral projectors p_1,\ldots,p_r are defined over \overline{\mathbb Q}, the latter being polynomials in P.

Choose \nu_i\in\overline{\mathbb Q} with \nu_i^2=\lambda_i. For each s=(s_1,\ldots,s_r)\in\{1,-1\}^r, put

These are invertible self-adjoint operators with R_s^2=P. Consequently

Thus R_s:(T,q_1)_{\overline{\mathbb Q}}\to (T,q_0)_{\overline{\mathbb Q}} is an isometry and extends to a parity-preserving Clifford algebra isomorphism

Step 2: weight-one Hodge maps. We show that the parity restrictions of F_s and their inverses lie in the \overline{\mathbb Q}-span of rational weight-one Hodge maps.

Let \mathcal J be the joint rational Hodge group of T and the four weight-one spaces P_j^\pm=C^\pm(T,q_j), j=0,1. Here the joint Hodge group is the smallest rational algebraic subgroup containing their simultaneous real Hodge circle. Its geometric identity component is defined over \mathbb Q and already contains that connected circle. Minimality therefore gives \mathcal J=\mathcal J^\circ, so \mathcal J is geometrically connected. Write \rho_T for its action on T. Since P is a rational Hodge endomorphism, it commutes with \rho_T(\mathcal J); therefore every R_s does as well after scalar extension.

For each j, the projection of \mathcal J to the actions on (T,P_j^+,P_j^-) lies in the image of

using the orthogonal action on T and left multiplication on the parity spaces. Indeed, this rational algebraic image contains the projected circle by (3.1). The representation is faithful, since the left action on C^+ recovers a spin element by evaluation at 1. Thus it identifies the spin group with its algebraic image, and gives algebraic homomorphisms

covering the same action \rho_T.

The isometry R_s induces an isomorphism of spin groups \widehat R_s:\operatorname{Spin}(T,q_1)_{\overline{\mathbb Q}} \to\operatorname{Spin}(T,q_0)_{\overline{\mathbb Q}}. Since R_s commutes with \rho_T, both \widehat R_s\rho_1 and \rho_0 lift the same orthogonal representation of \mathcal J_{\overline{\mathbb Q}}. Their pointwise quotient therefore takes values in the central kernel \mu_2 of the spin covering. This quotient is a morphism from a connected group to a finite group, and is the identity at the identity. Hence

Because F_s is an algebra isomorphism, (6.4) says precisely that F_s intertwines the left \mathcal J-actions on both parities.

For finite-dimensional rational representations, equivariant Hom-spaces commute with field extension: equivariance is a set of rational linear equations on the matrix of a map. Rational maps intertwine \mathcal J exactly when they intertwine its Hodge circle. Thus

expresses F_s^\pm, and likewise (F_s^\pm)^{-1}, as finite linear combinations of rational weight-one Hodge maps.

Step 3: recovery and rational descent. Fix abelian realizations and rational dualities for the q_1 target, and let \mathcal A be the rational subspace of maps

obtained by restricting rational codimension-two cycle actions from S and projecting to this Künneth component. It is a subspace of a finite-dimensional rational Hom-space. By Lemma 3.2 and (6.5), conjugating the algebraically realized q_0 multiplication on its Hom slots gives elements of \mathcal A\otimes\overline{\mathbb Q}. The exact Clifford identity is

Here the conjugation is expanded using rational Hodge maps on the two abelian factors; it does not apply R_s^{-1} to the cohomology of S by a correspondence.

An elementary sign average recovers the unmodified source:

Indeed R_s^{-1}=\sum_j s_j\nu_j^{-1}p_j and \sum_s s_i s_j=2^r\delta_{ij}. Equations (6.6) and (6.7) therefore place the tensor realization of x\mapsto L_x^{q_1} in \mathcal A\otimes\overline{\mathbb Q}. This map is rational, so Lemma 2.3 places it in \mathcal A itself. A finite rational linear combination of actual cycles realizes it.□

Proof of Theorem 1.1. Proposition 4.1 gives T=V. Proposition 2.4 proves the equivalence of conditions (i) and (ii). From condition (i), Theorem 4.5 realizes the full primitive Clifford multiplication. Proposition 6.2 and Lemma 3.2 then give condition (iii) with the stipulated data. Conversely, condition (iii) gives condition (i) by taking B=A_S\times A_S and \alpha=\Gamma_S.□

Proof of Theorem 1.2. By Proposition 2.4, either assumed input gives the nonzero correspondence very generally. Corollary 5.5, followed by the inverse right multiplier in Lemma 3.2, algebraically realizes m^q on V at the desired point. Proposition 6.1 gives m^{q|_T} on T. Proposition 6.2 and Lemma 3.2 then give the prescribed quadratic form, parity identification, duality, and isogenous realization, hence (1.2). The operations on the abelian factors preserve the degree shift, so the resulting cycle has codimension two.□