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On the standard conjecture for compactifications of Néron models of 4-dimensional Abelian varieties S. G. Tankeev Vladimir State University
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    IntroductionLet $H$ be an ample divisor on a smooth complex projective $d$-dimensional variety $X$. Then, for every positive integer $i\leqslant d$, the map
$$
\begin{equation*}
L^{d-i}\colon H^i(X,\mathbb{Q}) \xrightarrow{\smile\,\operatorname{cl}_X(H)^{\smile\, d-i}} H^{2d-i}(X,\mathbb{Q})
\end{equation*}
\notag
$$
is an isomorphism by the strong Lefschetz theorem. The Grothendieck standard conjecture $B(X)$ of Lefschetz type [1] asserts that there exists an algebraic $\mathbb{Q}$-cycle $Z$ on the Cartesian product $X\times X$ which determines the inverse algebraic isomorphism
$$
\begin{equation*}
H^{2d-i}(X,\mathbb{Q})\underset{\widetilde{\qquad}}{\xrightarrow{x\,{\mapsto} \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast x\,{\smile}\operatorname{cl}_{X\times X}(Z))}} H^i(X,\mathbb{Q}).
\end{equation*}
\notag
$$
It is known that the Lefschetz theorem on $(1,1)$-classes yields the existence of an algebraic isomorphism $H^{2d-1}(X,\mathbb{Q})\,\widetilde{\to}\,H^1(X,\mathbb{Q})$. We write $^{\mathrm{c}}\Lambda$ for the dual operator for $L$ in the classical Hodge theory. It is well known that the conjecture $B(X)$ is equivalent to the algebraicity of the operator $^{\mathrm{c}} \Lambda$ ([2], Proposition 2.3). The standard conjecture $B(X)$ is equivalent to the coincidence of the numerical and homological equivalences of algebraic cycles on $X\times X$ ([3], (1.11)). Moreover, by Proposition 1.7 in [4], the conjecture $B(X)$ is equivalent to the semisimplicity of the $\mathbb{Q}$-algebra $\mathcal A(X)=\operatorname{cl}_{X\times X}(\operatorname{CH}^\ast(X\times X))\otimes_\mathbb{Z}\mathbb{Q}$ of algebraic self-correspondences on $X$ with the bilinear composition law ([2], § 1.3.1)
$$
\begin{equation*}
g\circ f=\operatorname{pr}_{13\ast}(\operatorname{pr}_{12}^\ast(f)\smile \operatorname{pr}_{23}^\ast(g))
\end{equation*}
\notag
$$
and $B(X)\Rightarrow C(X)$, where the standard conjecture $C(X)$ of Künneth type states that the Künneth components of the class of the diagonal $\Delta_X\hookrightarrow X\times X$ are algebraic ([2], Lemma 2.4). The conjecture $B(X)$ is compatible with passages to Cartesian products ([2], Corollary 2.5), hyperplane sections ([2], Theorem 2.13) and specialization ([2], Introduction). Furthermore, it is compatible with monoidal transformations along smooth centres ([5], Theorem 4.3). By definition, a $d$-dimensional elliptic variety is birationally equivalent to a variety containing a smooth family of elliptic curves parameterized by some affine variety of dimension $d- 1$. It is known that the standard conjecture $B(X)$ holds for all smooth complex projective curves, surfaces, Abelian varieties [6] and threefolds of Kodaira dimension $\varkappa(X)<3$ [7] (in particular, it holds for all complex elliptic threefolds and for the compactifications of Néron minimal models of Abelian surfaces over fields of algebraic functions of one variable with field of constants $\mathbb{C}$). Moreover, $B(X)$ holds for Hilbert schemes of points on surfaces ([8], Corollary 7.5), for hyperkähler varieties deformation equivalent to Hilbert schemes of points of $K3$-surfaces [9], and for the fibre product $X_1\times_CX_2$ of projective non-isotrivial smooth families $\pi_k\colon X_k\to C$ ($k=1,2$) of regular surfaces with geometric genus $1$ over a smooth projective curve $C$ provided that the ranks of the lattices of transcendental cycles on generic geometric fibres $X_{ks}$ ($k=1,2$) are distinct odd primes (see [10], [11]). Let $S$ be a $K3$ surface or an Abelian surface, $H$ an ample line bundle on $S$, and $X$ the Gieseker–Maruyama–Simpson moduli space of $H$-stable torsion-free sheaves of rank $r$ on $S$ with fixed Chern classes $\operatorname{c}_1$, $\operatorname{c}_2$. Then the standard conjecture of Lefschetz type holds for $X$ provided that $X$ is projective ([8], Theorem 7.8, Corollary 7.9). The standard conjecture also holds for the Altman–Kleiman compactification $X$ of the relative Jacobian of a family $\mathcal C\to\mathbb P^2$ of hyperelliptic curves of genus $2$ with weak degeneracies provided that the canonical projection $X\to\mathbb P^2$ is a Lagrangian fibration [12]. The conjecture $B(X)$ holds for any smooth projective model of a complex projective threefold $X$ provided that $\operatorname{End}_\mathbb{C}(\operatorname{Pic}^0(X_s))\,\widetilde{\to}\,\mathbb{Z}$ for some smooth fibre $X_s$ of the structure morphism $\pi\colon X\to S$ of $X$ onto a surface $S$, and the rank of the corresponding Kodaira–Spencer map of the smooth part $X'\to S'$ of the morphism $\pi$ equals $1$ on some non-empty open subset of $S'$ (the condition on the endomorphism ring $\operatorname{End}_\mathbb{C}(\operatorname{Pic}^0(X_s))$ may be omitted if the genus of the generic fibre of $\pi$ equals $2$) [13]. Furthermore, the standard conjecture $B(X)$ holds for a smooth complex projective fourfold $X$ fibred by Abelian varieties (possibly, with degeneracies) over a smooth projective curve if the endomorphism ring $\operatorname{End}_{\overline{\kappa(\eta)}\,}(X_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)})$ of the generic geometric fibre is not an order of an imaginary quadratic field. This condition holds automatically when the reduction of the generic scheme fibre $X_\eta$ at some place of the curve is semi-stable in Grothendieck’s sense and has an odd toric rank or the generic geometric fibre is not a simple Abelian variety [14]. Let $R$ be a Dedekind domain with fraction field $K$ and let $A_\eta$ be an Abelian variety over $\eta=\operatorname{Spec} K$ with semi-stable reductions in Grothendieck’s sense. Künnemann ([15], § 5.8) showed that there is a finite extension $K'$ of $K$ such that the Abelian variety $A_\eta\otimes_K K'$ has a flat projective regular model $P'$ (not necessarily unique) over the integral closure $R'$ of the ring $R$ in the field $K'$. This model $P'$ has strict semi-stable reductions over each localization of the ring $R'$ (in particular, every special fibre $P'_s$ is a union of smooth divisors of multiplicity $1$ with normal crossings; see [16], § 1.9) and the scheme $P'$ contains the Néron minimal model $\mathcal A'$ of the variety $A_\eta\otimes_K K'$ in the case when all residue fields of the scheme $\operatorname{Spec} R'$ are perfect ([16], §§ 4.1, 4.2, 4.4, 4.5, Theorem 4.6). Consider the Néron minimal model $\mathcal M\to C$ of an Abelian variety $\mathcal M_\eta$ over the field $\kappa(\eta)$ of rational functions on a smooth projective curve $C$. After the base change determined by an appropriate ramified covering $\widetilde{C}\to C$, we may assume in view of Künnemann’s results that there is a smooth compactification $X$ of $\mathcal M$ such that $X$ is flat and projective over the curve $C$ and the following conditions hold: (i) the model $X/C$ has strictly semi-stable reductions (in particular, all fibres of the structure morphism $\pi\colon X\to C$ are unions of smooth irreducible components of multiplicity $1$ with normal crossings); (ii) the variety $X$ contains the variety $\mathcal M$ as an open dense subscheme; (iii) the restriction $\pi|_{\mathcal M}\colon \mathcal M\to C$ coincides with the structure morphism of the Néron model; (iv) the connected component $\mathcal M^0_s$ of the neutral element of any fibre $\mathcal M_s$ ($s\in C$) is an extension of an Abelian variety $A_s$ by a linear torus of dimension $r_s$ (the number $r_s$ is called the toric rank in what follows); (v) the $C$-group law $\mathcal M^0{\times_C}\mathcal M^0{\to}\mathcal M^0$ extends to a $C$-group action $\mathcal M^0{\times_C}X{\to}X$. These compactifications of the Néron model are referred to as Künnemann compactifications. By definition, the Abelian variety $\mathcal M_\eta$ has trivial trace if, for any finite ramified covering $\widetilde{C}\to C$, the group scheme $\mathcal M\times_C\widetilde{C}\to\widetilde{C}$ has no non-trivial constant Abelian subscheme. In this article we prove the following main result. Theorem. Let $\mathcal M\to C$ be the Néron minimal model of a $4$-dimensional principally polarized Abelian variety $\mathcal M_\eta$ with trivial trace over the field $\kappa(\eta)$ of rational functions on a smooth projective curve $C$. Suppose that the variety $\mathcal M_\eta$ is isomorphic to the product of absolutely simple Abelian varieties over $\kappa(\eta)$ and, for any simple Abelian subvariety $I_{\overline\eta}\subset \mathcal M_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)}$ of dimension greater than $2$, the centre of the ring $\operatorname{End}_{\overline{\kappa(\eta)}\,}(I_{\overline\eta})\otimes_\mathbb{Z}\mathbb{Q}$ is not a purely imaginary quadratic extension of a totally real number field (a $CM$-field) and the ring $\operatorname{End}_{\overline{\kappa(\eta)}\,}(I_{\overline\eta})\otimes_\mathbb{Z}\mathbb{Q}$ is not a definite quaternion division algebra over the field $\mathbb{Q}$. Then there is a finite ramified covering $\widetilde{C}\to C$ such that for any Künnemann compactification $\widetilde{X}$ of the Néron minimal model of the Abelian variety $\mathcal M_\eta\otimes_{\kappa(\eta)}\kappa(\widetilde{\eta})$ there are algebraic isomorphisms
$$
\begin{equation*}
H^8(\widetilde{X},\mathbb{Q})\,\widetilde{\to}\,H^2(\widetilde{X},\mathbb{Q}),\qquad H^7(\widetilde{X},\mathbb{Q})\,\widetilde{\to}\,H^3(\widetilde{X},\mathbb{Q}).
\end{equation*}
\notag
$$
Moreover, assume that all bad reductions of the Abelian variety $\mathcal M_\eta$ are semi-stable with toric rank $1$ and, for any places $\delta,\delta'\in C$ of bad reductions, the Hodge conjecture on algebraic cycles holds for the product $A_\delta\times A_{\delta'}$ of the Abelian varieties $A_\delta$, $A_{\delta'}$ which are the quotients of the connected components of neutral elements in special fibres of the Néron minimal model modulo toric parts. Then the standard conjecture $B(\widetilde{X})$ holds.Corollary. In the case under consideration, it follows from $B(\widetilde{X})$ that the conjecture $D(X)$ on the coincidence of the numerical and homological equivalences of algebraic cycles on $X$ holds as well. Indeed, $B(\widetilde{X})\Rightarrow D(\widetilde{X})$ ([2], Corollary 2.2). On the other hand, the validity of the conjecture $D(\widetilde{X})$ is preserved under monoidal transformations of the $5$-dimensional variety $\widetilde{X}$ along smooth centres by Corollary 2.4 in [5]. Therefore a resolution of indeterminacies of the rational dominant map $\widetilde{X}-\to X$ yields the implication $D(\widetilde{X})\Rightarrow D(X)$ ([3], Theorem 1.5). Remark. The condition of algebraicity of Hodge cycles on the variety $A_\delta\times A_{\delta'}$ holds automatically if the Abelian varieties $A_\delta$, $A_{\delta'}$ are isogenous to products of elliptic curves. § 1. Representations of semisimple Lie groups and reduction to a construction of certain algebraic isomorphisms1.1.We may assume that $\operatorname{End}_{\kappa(\eta)} (X_\eta)=\operatorname{End}_{\overline{\kappa(\eta)}\,} (X_{\overline\eta})$ and the Abelian variety $X_\eta$ over the field $\kappa(\eta)$ of rational functions on the curve $C$ is the product of absolutely simple principally polarized Abelian $\kappa(\eta)$-varieties. Moreover, the following assertion holds in the case of semi-stable reductions. For every finite ramified covering $\widetilde{C}\to C$, the connected component of the neutral element in the fibre of the Néron model $\widetilde{\mathcal M}\to \widetilde{C}$ over a point $\widetilde{s}\in\widetilde{C}$ lying over the point $s\in C$, is isomorphic to the connected component $\mathcal M_s^0$ of the neutral element in the fibre of the Néron model $\mathcal M\to C$ ([17], Corollary 3.3, Corollary 3.9). In particular, the toric rank $r_s$ and the Abelian variety $A_s$ are preserved under the base change $\widetilde{C}\to C$. By the Grothendieck theorem on semi-stable reductions of Abelian varieties, we may assume that all fibres of the structure morphism $\mathcal M^0\to C$ are extensions of Abelian varieties by linear tori ([17], Theorem 3.6). Moreover, by [15], § 5.8 and [16], §§ 4.1, 4.2, 4.4, 4.5, Theorem 4.6, we may assume that there is a Künnemann compactification $X$ of the Néron minimal model $\mathcal M$ such that every singular fibre $X_\delta$ is a union of smooth irreducible components of multiplicity $1$ with normal crossings, the closure $G$ of the image of the global monodromy $\pi_1(C',s)\to\operatorname{GL}(H^1(X_s,\mathbb{Q}))$ (associated with a smooth part $\pi'\colon X'\to C'=C\setminus\Delta$ of the structure morphism $\pi\colon X\to C$) is a connected $\mathbb{Q}$-group and the local monodromies (Picard–Lefschetz transformations) are unipotent. We may also assume that
$$
\begin{equation*}
\{s\in C\mid \text{ the fibre }\mathcal M_s\text{ is non-compact}\} =\Delta := \{\delta\in C\mid \operatorname{Sing}(X_\delta)\neq\varnothing\}.
\end{equation*}
\notag
$$
Let $C'\stackrel{j}{\hookrightarrow} C$ be the canonical embedding. Consider canonical diagrams of fibre products Let $\iota\colon X\times_CX\hookrightarrow X\times X$ be the canonical embedding, $\sigma\colon Y\to X\times_CX$ a resolution of singularities of the variety $X\times_CX$. We may assume that $\sigma$ induces an isomorphism over $C'$. In particular, $Y$ may be regarded as a smooth projective compactification of the fibre product $X'\times_{C'}X'$. Using the existence of a Künnemann model ([15], § 5.8, [16], §§ 4.1, 4.2, 4.4, 4.5, Theorem 4.6) of the generic scheme fibre of the Abelian scheme $X'\times_{C'}X'\to C'$ (after the base change determined by some ramified covering $\widetilde{C}\to C$) or the Mumford theorem on semi-stable reductions ([18], pp. 53, 54) or, alternatively, Consani’s method ([19], § 4, § 5, Lemma 5.2, Remark 5.4) of resolution of singularities of the fibre product $X\times_CX$, we may assume that the fibre $Y_s$ is a union of smooth irreducible components of multiplicity $1$ with normal crossings for all points $s\in C$.1.2.Consider the normalization $f\colon Z\to\pi^{-1}(\Delta)$ of the scheme $\pi^{-1}(\Delta)$. Then $Z$ is a disjoint union of smooth irreducible components of the divisor $\pi^{-1}(\Delta)$. Since $f$ is a resolution of singularities of the closed subscheme $i_\Delta\colon \pi^{-1}(\Delta)\hookrightarrow X$, there is a canonical exact sequence of mixed Hodge $\mathbb{Q}$-structures ([20], Corollary 8.2.8)
$$
\begin{equation*}
H^{n-2}(Z,\mathbb{Q})\xrightarrow{(i_\Delta f)_\ast} H^n(X,\mathbb{Q})\xrightarrow{\varphi_n} H^n(X',\mathbb{Q}),
\end{equation*}
\notag
$$
where $(i_\Delta f)_\ast$ is a morphism of bidegree $(1,1)$ of pure Hodge structures and $\varphi_n$ is the restriction morphism. In particular,
$$
\begin{equation}
(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q}) =\operatorname{Ker}[H^n(X,\mathbb{Q})\xrightarrow{\varphi_n} H^n(X',\mathbb{Q})].
\end{equation}
\tag{1.1}
$$
1.3.We may assume that
$$
\begin{equation}
H^0(C',R^1\pi'_\ast\mathbb{Q})=H^0(C',R^3\pi'_\ast\mathbb{Q})=H^0(C',R^5\pi'_\ast\mathbb{Q}) =H^0(C',R^7\pi'_\ast\mathbb{Q})=0,
\end{equation}
\tag{1.2}
$$
$$
\begin{equation}
H^2(C,R^1\pi_\ast\mathbb{Q})=H^2(C,R^3\pi_\ast\mathbb{Q})=H^2(C,R^5\pi_\ast\mathbb{Q})=0.
\end{equation}
\tag{1.3}
$$
Indeed, since $R^1\pi'_\ast\mathbb{Q}$ is a polarizable family of Hodge $\mathbb{Q}$-structures of weight $1$, there is an isomorphism $R_1\pi'_\ast\mathbb{Q}=[R^1\pi'_\ast\mathbb{Q}]^\vee\,\widetilde{\to}\, R^1\pi'_\ast\mathbb{Q}(1)$ of families of Hodge $\mathbb{Q}$-structures ([21], § 4.2.3). We have $H^0(C',R^1\pi'_\ast\mathbb{Q})=0$ because otherwise the results of Deligne, Grothendieck and Katz ([21], § 4.4.3, Corollary 4.1.2, (4.1.3.2), (4.1.3.3)) yield the existence of a non-trivial constant Hodge substructure $\mathcal H_\mathbb{Z}\subset R_1\pi'_\ast\mathbb{Z} := [R^1\pi'_\ast\mathbb{Z}]^\vee$ of type $(-1,0)+(0,-1)$ on $C'$, corresponding to a non-trivial constant Abelian subscheme of the Abelian scheme $\pi'\colon X'\to C'$ ([21], § 4.4.3) contrary to the hypothesis that the Abelian variety $X_\eta=\mathcal M_\eta$ has trivial trace. The Poincaré duality on the fibres of the morphism $\pi'\colon X'\to C'$ yields an isomorphism $R^7\pi'_\ast\mathbb{Q}\,\widetilde{\to}\,R^1\pi'_\ast\mathbb{Q}$ of local systems. Therefore,
$$
\begin{equation*}
H^0(C',R^7\pi'_\ast\mathbb{Q})\,\widetilde{\to}\,H^0(C',R^1\pi'_\ast\mathbb{Q})=0.
\end{equation*}
\notag
$$
On the other hand, the natural $\smile$-products (together with a polarization of the local system $R^n\pi'_\ast\mathbb{Q}$) determine a non-degenerate pairing ([22], Proposition 10.5)
$$
\begin{equation*}
H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})\times H^2(C,j_\ast R^n\pi'_\ast\mathbb{Q})\to H^2(C,\mathbb{Q})\,\widetilde{\to}\,\mathbb{Q}.
\end{equation*}
\notag
$$
By the theorem on local invariant cycles, the canonical map $R^n\pi_\ast\mathbb{Q} \to j_\ast R^n\pi'_\ast\mathbb{Q}$ is surjective and its kernel is concentrated on a finite set $\Delta$ ([23], § 3.7, [22], Proposition 15.12). Therefore
$$
\begin{equation}
H^2(C,R^n\pi_\ast\mathbb{Q})=H^2(C,j_\ast R^n\pi'_\ast\mathbb{Q})\,\widetilde{\to}\, H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})^\vee=H^0(C',R^n\pi'_\ast\mathbb{Q})^\vee.
\end{equation}
\tag{1.4}
$$
Thus (1.3) follows from (1.2). Since there is an isomorphism
$$
\begin{equation*}
H^0(C',R^5\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\, H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}})^\vee,
\end{equation*}
\notag
$$
it remains to prove the equality For every non-trivial Abelian subscheme $J'\subset X'\to C'$ there is a countable subset $\Delta_{\text{countable}}\subset C'$ such that, for any point $s\in C'\setminus \Delta_{\text{countable}}$, the closure of the image of the global monodromy representation $\pi_1(C',s)\to\operatorname{GL}(H^1(J'_s,\mathbb{Q}))$ in the Zariski topology of the group $\operatorname{GL}(H^1(J'_s,\mathbb{Q}))$ is a connected semisimple ([21], Corollary 4.2.9) normal ([24], Theorem 7.3) subgroup of the Hodge group ([25], Definition B.51) $\operatorname{Hg}(J'_s) := \operatorname{Hg}(H^1(J'_s\mathbb{Q}))$ of the Abelian variety $J'_s$. By Mumford’s results, the reductive Hodge group $\operatorname{Hg}(J'_s)$ is commutative (and, therefore, it is a linear $\mathbb{Q}$-torus) if and only if $J'_s$ is an Abelian variety of CM-type [26]. Since $X_\eta$ has trivial trace, it follows that the generic scheme fibre $J'_\eta$ cannot be an Abelian variety of CM-type (because the closure of the image of its monodromy representation is a non-trivial connected semisimple group). For the same reason, $J'_s$ cannot be an Abelian variety of CM-type for $s\in C'\setminus \Delta_{\text{countable}}$. We first assume that the Abelian variety $X_\eta$ is the product of four elliptic curves. Then standard arguments ([27], § 1.3) show that (1.5) follows easily from the Clebsch–Gordan formula ([28], Ch. VIII, § 9, n$^0$ 4) for representations of simple Lie algebras of type $A_1$. Assume that the Abelian variety $X_\eta$ is the product of two elliptic curves $E_{1\eta}$, $E_{2\eta}$ and an absolutely simple Abelian surface $F_\eta$. Then $E_{1\eta}$, $E_{2\eta}$ and $F_\eta$ cannot be Abelian varieties of CM-type (see above). This is also true for the fibres $E_{is}$ ($i=1,2$), $F_s$ of their Néron models $E_i\to C$, $F\to C$ over a point $s\in C'\setminus \Delta_{\text{countable}}$. In the case under consideration, the Hodge group $\operatorname{Hg}(E_{is})$ is a $\mathbb{Q}$-simple group with Lie algebra $\operatorname{Lie}\operatorname{Hg}(E_{is})(\overline{\mathbb{Q}})$ of type $A_1$ ([29], § 2.1) and the Hodge group $\operatorname{Hg}(F_s)$ is a $\mathbb{Q}$-simple group ([29], Proposition 2.4) with Lie algebra $\operatorname{Lie}\operatorname{Hg}(F_s)(\overline{\mathbb{Q}})$ of type $C_2$ when $\operatorname{End}_\mathbb{C}(F_s)=\mathbb{Z}$, of type $A_1\times A_1$ when $\operatorname{End}_\mathbb{C}(F_s)$ is an order of a real quadratic field, and of type $A_1$ when $\operatorname{End}_\mathbb{C}(F_s)\otimes_\mathbb{Z}\mathbb{Q}$ is a quaternion division algebra over $\mathbb{Q}$ split at the place $\infty$ ([29], § 2.2). Since the Lie algebra $\operatorname{Lie}G$ is an ideal of the Lie algebra $\operatorname{Lie}\operatorname{Hg}(X_s) \subset \operatorname{Lie}\operatorname{Hg}(E_{1s}) \times\operatorname{Lie}\operatorname{Hg}(E_{2s}) \times\operatorname{Lie}\operatorname{Hg}(F_s)$ with surjective projections of $\operatorname{Lie} G$ onto $\operatorname{Lie}\operatorname{Hg}(E_{is})$ and $\operatorname{Lie}\operatorname{Hg}(F_s)$ (because the groups $\operatorname{Hg}(E_{is})$, $\operatorname{Hg}(F_s)$ are $\mathbb{Q}$-simple and the Abelian variety $X_\eta$ has trivial trace), the pair
$$
\begin{equation*}
\bigl(\text{type of} \operatorname{Lie} G(\overline{\mathbb{Q}}),\, H^1(X_s,\overline{\mathbb{Q}})\bigr)
\end{equation*}
\notag
$$
takes one of the following values:
$$
\begin{equation}
\bigl(A_1\times C_2,E(\omega_1^{(1)})+E(\omega_1^{(1)})+E(\omega_1^{(2)})\bigr),
\end{equation}
\tag{1.6}
$$
$$
\begin{equation}
\bigl(A_1\times A_1\times C_2,E(\omega_1^{(1)})+E(\omega_1^{(2)})+E(\omega_1^{(3)})\bigr),
\end{equation}
\tag{1.7}
$$
$$
\begin{equation}
\bigl(A_1\times A_1\times A_1, E(\omega_1^{(1)})+E(\omega_1^{(1)})+E(\omega_1^{(2)})+E(\omega_1^{(3)})\bigr),
\end{equation}
\tag{1.8}
$$
$$
\begin{equation}
\bigl(A_1\times A_1\times A_1\times A_1, E(\omega_1^{(1)})+E(\omega_1^{(2)})+E(\omega_1^{(3)})+E(\omega_1^{(4)})\bigr),
\end{equation}
\tag{1.9}
$$
$$
\begin{equation}
\bigl(A_1\times A_1, E(\omega_1^{(1)})+E(\omega_1^{(1)})+E(\omega_1^{(1)})+E(\omega_1^{(2)})\bigr),
\end{equation}
\tag{1.10}
$$
$$
\begin{equation}
\bigl(A_1\times A_1, E(\omega_1^{(1)})+E(\omega_1^{(2)})+E(\omega_1^{(1)})+E(\omega_1^{(2)})\bigr),
\end{equation}
\tag{1.11}
$$
$$
\begin{equation}
\bigl(A_1,E(\omega_1^{(1)})+E(\omega_1^{(1)})+E(\omega_1^{(1)})+E(\omega_1^{(1)})\bigr),
\end{equation}
\tag{1.12}
$$
where $E(\omega_1^{(k)})$ is the standard irreducible representation of the $k$th simple factor of the semisimple Lie algebra $\operatorname{Lie} G(\overline{\mathbb{Q}})$ in Bourbaki’s notation [28]. In case (1.6) we have $\wedge^2E(\omega_1^{(1)})=E(0)$, $\wedge^2E(\omega_1^{(2)})=E(\omega_2^{(2)})+E(0)$ ([28], Ch. VIII, § 13, n$^0$ 3, Lemma 2) and $\wedge^3E(\omega_1^{(2)})=E(\omega_1^{(2)})^\vee= E(\omega_1^{(2)})$ since the symplectic representation $E(\omega_1^{(2)})$ of degree $4$ is self-dual ([28], Ch. VIII, Table 1). Therefore the Clebsch–Gordan formula $E(\omega^{(1)}_1)\otimes E(\omega^{(1)}_1)=E(2\omega^{(1)}_1)+ E(0)$ ([28], Ch. VIII, § 9, n$^0$ 4) yields that
$$
\begin{equation*}
\begin{aligned} \, H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}}) &\,\widetilde{\to}\, \bigl[\wedge^3\bigl(E(\omega_1^{(1)})^{\oplus 2}+E(\omega_1^{(2)})\bigr)\bigr]^{\operatorname{Lie} G(\overline{\mathbb{Q}})} \\ &\,\widetilde{\to}\,\bigl[E(\omega_1^{(1)})^{\oplus 2}+E(\omega_1^{(2)})^{\oplus 3}+ E(2\omega_1^{(1)}+\omega_1^{(2)}) \\ &\qquad+E(\omega_1^{(1)})^{\oplus 2}\otimes[E(\omega_2^{(2)})+E(0)]+ E(\omega_1^{(2)})\bigr]^{\operatorname{Lie} G(\overline{\mathbb{Q}})}=0. \end{aligned}
\end{equation*}
\notag
$$
In case (1.7) we have
$$
\begin{equation*}
\begin{aligned} \, &H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}}) \,\widetilde{\to}\, \bigl[E(\omega_1^{(1)})^{\oplus 2} +E(\omega_1^{(2)})^{\oplus 2} +E(\omega_1^{(3)})^{\oplus 3} \\ &\qquad+E(\omega_1^{(1)}+\omega_1^{(2)}+\omega_1^{(3)}) +E(\omega_1^{(1)}+\omega_2^{(3)}) + E(\omega_1^{(2)}+\omega_2^{(3)})\bigr]^{\operatorname{Lie}G(\overline{\mathbb{Q}})}=0. \end{aligned}
\end{equation*}
\notag
$$
In case (1.8), the Clebsch–Gordan formula shows that
$$
\begin{equation*}
\begin{aligned} \, &H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\, \bigl[E(\omega_1^{(1)})^{\oplus 6} + E(\omega_1^{(2)})^{\oplus 4} +E(\omega_1^{(3)})^{\oplus 4} \\ &\qquad+E(2\omega_1^{(1)})\otimes [E(\omega_1^{(2)})+E(\omega_1^{(3)})] + E(\omega_1^{(1)}+\omega_1^{(2)} +\omega_1^{(3)})^{\oplus 2}\bigr]^{\operatorname{Lie}G(\overline{\mathbb{Q}})}=0. \end{aligned}
\end{equation*}
\notag
$$
In case (1.9) we similarly have
$$
\begin{equation*}
\begin{aligned} \, &H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\, \bigl[E(\omega_1^{(1)})^{\oplus 3} + E(\omega_1^{(2)})^{\oplus 3} + E(\omega_1^{(3)})^{\oplus 3} + E(\omega_1^{(4)})^{\oplus 3} \\ &\qquad+E(\omega_1^{(1)}+\omega_1^{(2)}+\omega_1^{(3)}) +E(\omega_1^{(1)}+\omega_1^{(2)}+\omega_1^{(4)}) \\ &\qquad+E(\omega_1^{(1)}+\omega_1^{(3)}+\omega_1^{(4)}) +E(\omega_1^{(2)}+\omega_1^{(3)} +\omega_1^{(4)})\bigr]^{\operatorname{Lie} G(\overline{\mathbb{Q}})}=0. \end{aligned}
\end{equation*}
\notag
$$
In case (1.10), the Clebsch–Gordan formula yields the equalities
$$
\begin{equation*}
E(\omega^{(1)}_1)\otimes E(\omega^{(1)}_1)\otimes E(\omega^{(1)}_1)=[E(2\omega^{(1)}_1)+ E(0)]\otimes E(\omega^{(1)}_1)=E(3\omega^{(1)}_1)+E(\omega^{(1)}_1)^{\oplus 2}
\end{equation*}
\notag
$$
because
$$
\begin{equation*}
\begin{aligned} \, &H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\, [E(\omega_1^{(1)})^{\oplus 11} \\ &\qquad+[E(0)^{\oplus 6} + E(2\omega_1^{(1)})^{\oplus 3}]\otimes E(\omega_1^{(2)}) +E(3\omega_1^{(1)})]^{\operatorname{Lie} G(\overline{\mathbb{Q}})} =0. \end{aligned}
\end{equation*}
\notag
$$
In case (1.11) we have
$$
\begin{equation*}
\begin{aligned} \, &H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\, \bigl[E(\omega_1^{(1)})^{\oplus 8}+E(\omega_1^{(2)})^{\oplus 8} \\ &\qquad + E(2\omega_1^{(1)}+\omega_1^{(2)})^{\oplus 2} +E(\omega_1^{(1)}+2\omega_1^{(2)})^{\oplus 2} \bigr]^{\operatorname{Lie} G(\overline{\mathbb{Q}})}=0. \end{aligned}
\end{equation*}
\notag
$$
Finally, the following relations hold in case (1.12):
$$
\begin{equation*}
H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\,\bigl[ E(\omega_1^{(1)})^{\oplus 20}+ E(3\omega_1^{(1)})^{\oplus 4}\bigr]^{\operatorname{Lie}G(\overline{\mathbb{Q}})}=0.
\end{equation*}
\notag
$$
Assume that the Abelian variety $X_\eta$ is the product of an elliptic curve $E_\eta$ and an absolutely simple $3$-dimensional Abelian variety $F_\eta$. Then $F_s$ cannot be an Abelian variety of CM-type for $s\in C'\setminus \Delta_{\text{countable}}$. Moreover, it follows from the equality $\dim_{\kappa(\eta)}F_\eta=3$ and triviality of trace that there is a canonical isomorphism ([21], Corollary 4.4.13)
$$
\begin{equation*}
\operatorname{End}_{C'}(F')\,\widetilde{\to}\,\operatorname{End}_{\pi_1(C',s)}H^1(F_s,\mathbb{Z}).
\end{equation*}
\notag
$$
Using the existence of canonical embeddings
$$
\begin{equation*}
\operatorname{Im}[\pi_1(C',s)\,{\to}\, \operatorname{GL}(H^1(F_s,\mathbb{Q}))]\,{\subset}\, G_F := \overline{\operatorname{Im}[\pi_1(C',s){\to}\operatorname{GL}(H^1(F_s,\mathbb{Q}))]} \,{\subset}\operatorname{Hg}(F_s),
\end{equation*}
\notag
$$
which are determined by a choice of $s\in C'\setminus\Delta_{\text{countable}}$, and the well-known equality $\operatorname{End}_{\operatorname{Hg}(F_s)}H^1(F_s,\mathbb{Q}) =\operatorname{End}_\mathbb{C}(F_s)\otimes_\mathbb{Z}\mathbb{Q}$ ([25], Lemma B.60), we see that this isomorphism determines the canonical maps
$$
\begin{equation}
\begin{aligned} \, &\operatorname{End}_{\kappa(\eta)}(F_\eta)\otimes_\mathbb{Z}\mathbb{Q}\,\widetilde{\to} \operatorname{End}_{\pi_1(C',s)}H^1(F_s,\mathbb{Q}) \nonumber \\ &\qquad\hookleftarrow \operatorname{End}_{\operatorname{Hg}(F_s)}H^1(F_s,\mathbb{Q}) =\operatorname{End}_\mathbb{C}(F_s)\otimes_\mathbb{Z}\mathbb{Q}. \end{aligned}
\end{equation}
\tag{1.13}
$$
The restriction map $\operatorname{End}_{\kappa(\eta)}(F_\eta)\otimes_\mathbb{Z}\mathbb{Q} \to \operatorname{End}_\mathbb{C}(F_s)\otimes_\mathbb{Z}\mathbb{Q}$ is injective. Hence it follows from (1.13) that $\operatorname{End}_{\kappa(\eta)}(F_\eta)\otimes_\mathbb{Z}\mathbb{Q}\,\widetilde{\to}\, \operatorname{End}_\mathbb{C}(F_s)\otimes_\mathbb{Z}\mathbb{Q}$. In particular, the Abelian variety $F_s$ is simple. By hypothesis of the theorem, the $\mathbb{Q}$-algebra $\operatorname{End}_{\kappa(\eta)}(F_\eta)\otimes_\mathbb{Z}\mathbb{Q}$ is not a purely imaginary quadratic extension of a totally real field. (This holds automatically if there is a point $t\in C$ such that the algebraic group $\mathcal M_t$ has an odd toric rank. Indeed, in this case, the $\mathbb{Q}$-algebra $\operatorname{End}_{\kappa(\eta)}(F_\eta)\otimes_\mathbb{Z}\mathbb{Q}$ is a totally real field by Theorem 1 in [30].) Hence the field $\operatorname{End}_\mathbb{C}(F_s)\otimes_\mathbb{Z}\mathbb{Q}$ is either $\mathbb{Q}$ (and then the Lie algebra $\operatorname{Lie}\operatorname{Hg}(F_s)(\overline{\mathbb{Q}})$ has type $C_3$; see [29], § 2.3, Type I(1)) or a totally real cubic extension of $\mathbb{Q}$; see [29], § 2.3, Type I(3) (and then the Hodge group $\operatorname{Hg}(F_s)$ is a $\mathbb{Q}$-simple algebraic group by Proposition 2.4, (iii) in [29] and the Lie algebra $\operatorname{Lie}\operatorname{Hg}(F_s)(\overline{\mathbb{Q}})$ has type $A_1\times A_1\times A_1$). Since the group $\operatorname{Hg}(F_s)$ is $\mathbb{Q}$-simple and contains a connected normal subgroup $G_F$ while the Abelian variety $F_\eta$ has trivial trace, we have $\operatorname{Hg}(F_s)=G_F$. Therefore the pair $(\text{type of}\operatorname{Lie}G(\overline{\mathbb{Q}}),\,H^1(X_s,\overline{\mathbb{Q}}))$ takes either the values (1.8), (1.9) or the value
$$
\begin{equation}
\bigl(A_1\times C_3,\,E(\omega_1^{(1)})+E(\omega_1^{(2)})\bigr).
\end{equation}
\tag{1.14}
$$
In case (1.14) we have
$$
\begin{equation*}
\wedge^3E(\omega_1^{(2)})=E(\omega_3^{(2)})+E(\omega_1^{(2)}), \qquad \wedge^2E(\omega_1^{(2)})=E(\omega_2^{(2)})+E(0)
\end{equation*}
\notag
$$
([28], Ch. VIII, § 13, n$^0$ 3, Lemma 2). Therefore,
$$
\begin{equation*}
\begin{aligned} \, &H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\, \bigl[\wedge^3E(\omega_1^{(1)})+E(\omega_1^{(2)})\bigr]^{\operatorname{Lie} G(\overline{\mathbb{Q}})} \\ &\qquad=\bigl[E(\omega_1^{(2)})+E(\omega_1^{(1)}) \otimes [E(\omega_2^{(2)})+E(0)]+ E(\omega_3^{(2)})+ E(\omega_1^{(2)})\bigr]^{\operatorname{Lie} G(\overline{\mathbb{Q}})}=0. \end{aligned}
\end{equation*}
\notag
$$
Assume that the Abelian variety $X_\eta$ is the product of two absolutely simple Abelian surfaces $F_{i\eta}$. Then the varieties $F_{is}$ cannot be Abelian varieties of CM-type for $s\in C'\setminus \Delta_{\text{countable}}$. Moreover, the Hodge group $\operatorname{Hg}(F_{is})$ is $\mathbb{Q}$-simple by Proposition 2.4 in [29] and its Lie algebra $\operatorname{Lie}\operatorname{Hg}(F_{is})(\overline{\mathbb{Q}})$ is either of type $C_2$ (when $\operatorname{End}_\mathbb{C}(F_{is})=\mathbb{Z}$), of type $A_1\times A_1$ (when $\operatorname{End}_\mathbb{C}(F_{is})$ is an order of a real quadratic field), or of type $A_1$ (when $\operatorname{End}_\mathbb{C}(F_{is})\otimes_\mathbb{Z}\mathbb{Q}$ is a quaternion division algebra over $\mathbb{Q}$ split at the place $\infty$); see [29], § 2.2. Since the Lie algebra $\operatorname{Lie}G$ is an ideal of the Lie algebra $\operatorname{Lie}\operatorname{Hg}(X_s)\subset\operatorname{Lie}\operatorname{Hg}(F_{1s}) \times\operatorname{Lie}\operatorname{Hg}(F_{2s})$ with surjective projections of $\operatorname{Lie}G$ onto $\operatorname{Lie}\operatorname{Hg}(F_{is})$ (because the groups $\operatorname{Hg}(F_{is})$ are $\mathbb{Q}$-simple and the Abelian variety $X_\eta$ has trivial trace), the pair $(\text{type of } \operatorname{Lie} G(\overline{\mathbb{Q}}),\, H^1(X_s,\overline{\mathbb{Q}}))$ takes either the values (1.6)–(1.12) or one of the values
$$
\begin{equation}
(C_2\times C_2,E(\omega_1^{(1)})+E(\omega_1^{(2)})).
\end{equation}
\tag{1.16}
$$
In case (1.15) we have
$$
\begin{equation*}
\wedge^2E(\omega_1^{(1)})=E(\omega_2^{(1)})+E(0),\qquad \wedge^3E(\omega_1^{(1)})=E(\omega_1^{(1)})^\vee= E(\omega_1^{(1)})
\end{equation*}
\notag
$$
since the symplectic representation $E(\omega_1^{(1)})$ of degree $4$ is self-dual ([28], Ch. VIII, § 13, n$^0$ 3, Lemma 2, Table 1). Therefore, by the Schur lemma,
$$
\begin{equation*}
\begin{aligned} \, &H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\, \bigl[E(\omega_1^{(1)})+[E(\omega_2^{(1)})+E(0)]\otimes E(\omega_1^{(1)}) \\ &\qquad+E(\omega_1^{(1)})\otimes [E(\omega_2^{(1)})+E(0)]+ E(\omega_1^{(1)})\bigr]^{\operatorname{Lie}G(\overline{\mathbb{Q}})}=0. \end{aligned}
\end{equation*}
\notag
$$
In case (1.16) we similarly have
$$
\begin{equation*}
\begin{aligned} \, &H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\, \bigl[E(\omega_1^{(1)})+[E(\omega_2^{(1)})+E(0)]\otimes E(\omega_1^{(2)}) \\ &\qquad+E(\omega_1^{(1)})\otimes [E(\omega_2^{(2)})+E(0)]+ E(\omega_1^{(2)})\bigr]^{\operatorname{Lie}G(\overline{\mathbb{Q}})}=0. \end{aligned}
\end{equation*}
\notag
$$
We finally assume that the Abelian variety $X_\eta$ is absolutely simple. By hypothesis of the theorem, the centre of the ring $\operatorname{End}_{\overline{\kappa(\eta)}\,}(X_\eta\otimes_{\kappa(\eta)} \overline{\kappa(\eta)})\otimes_\mathbb{Z}\mathbb{Q}$ is not a purely imaginary quadratic extension of a totally real field. Therefore, by triviality of trace, there is a canonical isomorphism $\operatorname{End}_{C'}(X')\,\widetilde{\to}\,\operatorname{End}(R^1\pi'_\ast\mathbb{Z})$ ([31], § 4, Remark 1 on the derivation of Theorem 4.1 from Lemmas 1–3). Using the canonical inclusions
$$
\begin{equation*}
\operatorname{Im}[\pi_1(C',s)\to \operatorname{GL}(H^1(X_s,\mathbb{Q}))] \subset G\subset \operatorname{Hg}(X_s),
\end{equation*}
\notag
$$
which are determined by a choice of $s\in C'\setminus\Delta_{\text{countable}}$, and the well-known equality $\operatorname{End}_{\operatorname{Hg}(X_s)}H^1(X_s,\mathbb{Q}) =\operatorname{End}_\mathbb{C}(X_s)\otimes_\mathbb{Z}\mathbb{Q}$ ([25], Lemma B.60), we see that this isomorphism determines the canonical maps
$$
\begin{equation}
\begin{aligned} \, &\operatorname{End}_{\kappa(\eta)}(X_\eta)\otimes_\mathbb{Z}\mathbb{Q}\,\widetilde{\to}\, \operatorname{End}_{\pi_1(C',s)}H^1(X_s,\mathbb{Q})\hookleftarrow \operatorname{End}_{\operatorname{Hg}(X_s)}H^1(X_s,\mathbb{Q}) \nonumber \\ &\qquad=\operatorname{End}_\mathbb{C}(X_s)\otimes_\mathbb{Z}\mathbb{Q}. \end{aligned}
\end{equation}
\tag{1.17}
$$
The restriction map $\operatorname{End}_{\kappa(\eta)}(X_\eta) \otimes_\mathbb{Z}\mathbb{Q} \to\operatorname{End}_\mathbb{C} (X_s)\otimes_\mathbb{Z}\mathbb{Q}$ is injective. Hence it follows from (1.17) that $\operatorname{End}_{\kappa(\eta)}(X_\eta)\otimes_\mathbb{Z}\mathbb{Q}\, \widetilde{\to}\, \operatorname{End}_\mathbb{C}(X_s)\otimes_\mathbb{Z}\mathbb{Q}$. Thus the 4-dimensional Abelian variety $X_s$ is simple, the centre of the division $\mathbb{Q}$-algebra $\operatorname{End}_\mathbb{C}(X_s)\otimes_\mathbb{Z}\mathbb{Q}$ is a totally real field, so the Hodge group $\operatorname{Hg}(X_s)$ is $\mathbb{Q}$-simple ([31], § 4, Remark 1 on the derivation of Theorem 4.1 from Lemmas 1–3) and $\operatorname{Hg}(X_s)=G$ (because the Abelian variety $X_\eta$ has trivial trace and there is a non-trivial connected normal subgroup $G\subset\operatorname{Hg}(X_s)$; see [24], Theorem 7.3). Therefore it follows from Albert’s classification ([25], § B37) of division $\mathbb{Q}$-algebras $\operatorname{End}_\mathbb{C}(X_s)\otimes_\mathbb{Z}\mathbb{Q}$ that the pair $({\text{type of }}\operatorname{Lie}G(\overline{\mathbb{Q}}),\,H^1(X_s,\overline{\mathbb{Q}}))$ takes either the values (1.9), (1.11), (1.15), (1.16) or one of the following values ([31], § 4, Table before the statement of Theorem 4.1, [25], §§ B92, B94–B97):
$$
\begin{equation}
\bigl(A_1\times A_1\times A_1,E(\omega_1^{(1)}) \otimes E(\omega_1^{(2)}) \otimes E(\omega_1^{(3)})\bigr),
\end{equation}
\tag{1.19}
$$
$$
\begin{equation}
(D_2,E(\omega_1)+E(\omega_1))=\bigl(A_1\times A_1,E(\omega_1^{(1)})\otimes E(\omega_1^{(2)})+ E(\omega_1^{(1)})\otimes E(\omega_1^{(2)})\bigr).
\end{equation}
\tag{1.20}
$$
In case (1.18) we have a decomposition $\wedge^3E(\omega_1)=E(\omega_3)+E(\omega_1)$ ([28], Ch. VIII, § 13, n$^0$ 3, Lemma 2). Therefore,
$$
\begin{equation*}
H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\, [\wedge^3E(\omega_1)]^{\operatorname{Lie}G(\overline{\mathbb{Q}})} =[E(\omega_3)+E(\omega_1)]^{\operatorname{Lie}G(\overline{\mathbb{Q}})}=0.
\end{equation*}
\notag
$$
In case (1.19) we write $[\operatorname{Lie}G(\overline{\mathbb{Q}})]^{(1)}$ for the Lie algebra of type $A_1$ which is the first simple factor of the decomposition of the algebra $\operatorname{Lie}G(\overline{\mathbb{Q}})$ into the product of simple Lie algebras. We clearly have an isomorphism of $[\operatorname{Lie}G(\overline{\mathbb{Q}})]^{(1)}$-modules
$$
\begin{equation*}
\wedge^3\bigl(E(\omega_1^{(1)}) \otimes E(\omega_1^{(2)}) \otimes E(\omega_1^{(3)})\bigr)\,\widetilde{\to}\, \wedge^3\bigl(E(\omega_1^{(1)})^{\oplus 4}\bigr).
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\begin{aligned} \, &H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\, \bigl[\wedge^3\bigl(E(\omega_1^{(1)}) \otimes E(\omega_1^{(2)}) \otimes E(\omega_1^{(3)})\bigr)\bigr]^{\operatorname{Lie}G(\overline{\mathbb{Q}})} \\ &\qquad \hookrightarrow \bigl[\wedge^3\bigl(E(\omega_1^{(1)})^{\oplus 4}\bigr)\bigr]^{[\operatorname{Lie}G(\overline{\mathbb{Q}})]^{(1)}}= [E(\omega_1^{(1)})^{\oplus 20}+ E(3\omega_1^{(1)})^{\oplus 4}]^{[\operatorname{Lie}G(\overline{\mathbb{Q}})]^{(1)}}=0. \end{aligned}
\end{equation*}
\notag
$$
The case (1.20) can be studied in a similar way. The formulae (1.2), (1.3), (1.5) are proved. 1.4.By hypothesis of the theorem, the generic scheme fibre $\mathcal M_\eta$ of the Néron model is a principally polarized Abelian variety. Hence. for every point $s\in C'$, the Abelian variety $X_s$ has a principal polarization determined by a certain ample divisor $H_s$ on $X_s$. It is known that the Poincaré bundle $\mathcal P'_s$ on the variety $X_s\times \overset\vee{X}_s$ is uniquely determined (up to an isomorphism) by the following properties ([32], Ch. 2, § 5): a) $\mathcal P'_s|_{X_s\times\{L_s\}}\,\widetilde{\to}\,L_s$ for all $L_s\in\overset\vee{X}_s=\operatorname{Pic}(X_s)$; b) $\mathcal P'_s\Big|_{\{0\}\times\overset\vee{X}_s}\,\widetilde{\to}\,\mathcal O_{\overset\vee{X}_s}$. Since $X_s$ is a principally polarized Abelian variety, we have $X_s=\operatorname{Pic}^0(X_s)=\overset\vee{X}_s$. It follows from the properties a) and b) that the element $\operatorname{c}_1(\mathcal P'_s)\in H^2(X_s\times X_s,\mathbb{Q})$ has Künneth type $(1,1)$ ([32], Ch. 14, Lemma 14.1.9). Hence
$$
\begin{equation*}
\begin{aligned} \, &\operatorname{c}_1(\mathcal P'_s)\in [H^1(X_s,\mathbb{Q})\otimes_\mathbb{Q} H^1(X_s,\mathbb{Q})] \cap H^{1,1}(X_s\times X_s,\mathbb{C}) \\ &\qquad=[H^1(X_s,\mathbb{Q}) \otimes_\mathbb{Q} H^1(X_s,\mathbb{Q})]^{\operatorname{Hg}(X_s)}. \end{aligned}
\end{equation*}
\notag
$$
The correspondence $\operatorname{c}_1(\mathcal P'_s)$ induces an algebraic isomorphism ([2], § 2A1(ii), Theorem 2A9, [32], Ch. 16, § 16.4, p. 532)
$$
\begin{equation*}
H^7(X_s,\mathbb{Q})\underset{\widetilde{\qquad}}{\xrightarrow{x\,{\mapsto} \operatorname{pr}_{2s\ast} (\operatorname{pr}^\ast_{1s}(x)\smile \operatorname{c}_1(\mathcal P'_s))}} H^1(\operatorname{Pic}^0(X_s),\mathbb{Q})=H^1(X_s,\mathbb{Q}).
\end{equation*}
\notag
$$
For every point $s\in C'$ outside a countable subset $\Delta_{\text{countable}}$, the group $G$ (defined in § 1.1) is a normal subgroup of the Hodge group $\operatorname{Hg}(X_s)=\operatorname{Hg}(H^1(X_s,\mathbb{Q}))$ of the rational Hodge structure $H^1(X_s,\mathbb{Q})$ ([24], Theorem 7.3). We fix such a point $s$. It follows from the existence of an inclusion $G\hookrightarrow\operatorname{Hg}(X_s)$ that the correspondence $\operatorname{c}_1(\mathcal P'_s)$ determines a section
$$
\begin{equation*}
\Lambda'_{1,1}\in H^0(C',R^1\pi'_\ast\mathbb{Q} \otimes_\mathbb{Q} R^1\pi'_\ast\mathbb{Q})\,\widetilde{\to}\, [H^1(X_s,\mathbb{Q})\otimes_\mathbb{Q} H^1(X_s,\mathbb{Q})]^{\pi_1(C',s)}
\end{equation*}
\notag
$$
of type $(1,1)$ of the local system of Hodge structures $R^1\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^1\pi'_\ast\mathbb{Q}$ inducing the correspondence $\operatorname{c}_1(\mathcal P'_t)$ for any point $t\in C'$. By Deligne’s theorem, the canonical morphism $H^2(Y,\mathbb{Q})\to H^0(C',R^2\tau'_\ast\mathbb{Q})$ is a surjective morphism of Hodge $\mathbb{Q}$-structures ([21], Theorem 4.1.1, proof of Corollary 4.1.2). Since the element $\Lambda'_{1,1}\in H^0(C',R^1\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^1\pi'_\ast\mathbb{Q}) \subset H^0(C',R^2\tau'_\ast\mathbb{Q})$ has Hodge type $(1,1)$, the Lefschetz theorem on divisors yields an algebraic $\mathbb{Q}$-cycle $D^{(1)}$ on $Y$ such that the image of the class $\operatorname{cl}_Y(D^{(1)})\in H^2(Y,\mathbb{Q})\cap H^{1,1}(Y,\mathbb{C})$ under the canonical surjective morphism $H^2(Y,\mathbb{Q})\to H^0(C',R^2\tau'_\ast\mathbb{Q})$ coincides with the section $\Lambda'_{1,1}$. 1.5.Lemma. The $\mathbb{Q}$-space $H^0(C',R^2\pi'_\ast\mathbb{Q})$ of invariant cycles is a Hodge $\mathbb{Q}$-structure of type $(1,1)$. Proof. We begin by assuming that the generic scheme fibre of the Abelian scheme $\pi'\colon X'\to C'$ is an absolutely simple $4$-dimensional Abelian variety. By construction, it has trivial trace. Since the centre of the $\mathbb{Q}$-algebra $\operatorname{End}_{C'}(X')\otimes_\mathbb{Z}\mathbb{Q}$ is a totally real field by hypothesis of the theorem, there is a canonical isomorphism
$$
\begin{equation*}
\operatorname{End}_{C'}(X')\,\widetilde{\to}\,\operatorname{End}_{C'}(R^1\pi'_\ast\mathbb{Z})
\end{equation*}
\notag
$$
([31], Remark 1 on the derivation of Theorem 4.1 from Lemmas 1–3). On the other hand, a polarization of the Abelian scheme $\pi'\colon X'\to C'$ determines an isomorphism $[R^1\pi'_\ast\mathbb{Q}]^\vee\,\widetilde{\to}\,R^1\pi'_\ast\mathbb{Q}(1)$ of families of rational Hodge structures ([21], proof of Lemma 4.2.3). Since there is an embedding of families of rational Hodge structures $R^2\pi'_\ast\mathbb{Q}=\wedge^2R^1\pi'_\ast\mathbb{Q}\hookrightarrow R^1\pi'_\ast\mathbb{Q} \otimes_\mathbb{Q} R^1\pi'_\ast\mathbb{Q}$ and the $\mathbb{Q}$-space $\operatorname{End}_{C'}(X')\otimes_\mathbb{Z}\mathbb{Q}$ coincides with the component of type $(0,0)$ of the rational Hodge structure $\operatorname{End}_{C'}(R^1\pi'_\ast\mathbb{Q})$ ([21], § 4.4.6), it follows that the Hodge $\mathbb{Q}$-structure $H^0(C',R^2\pi'_\ast\mathbb{Q})$ has type $(1,1)$.
We need only consider the case when the Abelian scheme $\pi'\colon X'\to C'$ is the fibre product of two Abelian schemes $\pi'_k\colon X'_k\to C'$ of relative dimension at most $3$ with trivial traces. In this situation there are canonical isomorphisms
$$
\begin{equation*}
\begin{aligned} \, \operatorname{End}_{C'}(X'_k) &\,\widetilde{\to}\, \operatorname{End}_{C'}(R^1\pi'_{k\ast}\mathbb{Z}), \\ \operatorname{Hom}_{C'}(X'_1,X'_2) &\,\widetilde{\to}\, \operatorname{Hom}_{C'}(R_1\pi'_{1\ast}\mathbb{Z},R_1\pi'_{2\ast}\mathbb{Z}) \end{aligned}
\end{equation*}
\notag
$$
([21], Corollary 4.4.13). Therefore, using the existence of a canonical embedding of families of rational Hodge structures
$$
\begin{equation*}
R^2\pi'_\ast\mathbb{Q}\hookrightarrow R^2\pi'_{1\ast}\mathbb{Q}\oplus [R^1\pi'_{1\ast}\mathbb{Q} \otimes_\mathbb{Q} R^1\pi'_{2\ast}\mathbb{Q}]\oplus R^2\pi'_{2\ast}\mathbb{Q}
\end{equation*}
\notag
$$
and the coincidence of the $\mathbb{Q}$-space $\operatorname{Hom}_{C'}(X'_1,X'_2)\otimes_\mathbb{Z}\mathbb{Q}$ with the component of type $(0,0)$ of the rational Hodge structure $\operatorname{Hom}_{C'}(R_1\pi'_{1\ast}\mathbb{Q},R_1\pi'_{2\ast}\mathbb{Q})$ ([21], § 4.4.6), we easily see that the Hodge $\mathbb{Q}$-structure $H^0(C',R^2\pi'_\ast\mathbb{Q})$ has type $(1,1)$. $\Box$ 1.6.Since the standard conjecture holds for the Abelian variety $X_{\overline\eta}$, it follows from Lemma 1.5 that there is an algebraic isomorphism
$$
\begin{equation*}
H^8(X,\mathbb{Q})\,\widetilde{\to}\,H^2(X,\mathbb{Q})
\end{equation*}
\notag
$$
(the proof coincides verbatim with that of Theorem 1.2 in [13], where the case of families of surfaces $X'\to C'$ was studied). Therefore, by Theorem 2.9 in [2], in order to prove the conjecture $B(X)$ it suffices to construct algebraic isomorphisms
$$
\begin{equation*}
H^7(X,\mathbb{Q})\,\widetilde{\to}\,H^3(X,\mathbb{Q}), \qquad H^6(X,\mathbb{Q})\,\widetilde{\to}\,H^4(X,\mathbb{Q}).
\end{equation*}
\notag
$$
§ 2. Canonical decompositions of rational cohomology of odd degree and algebraic isomorphisms2.1.From now on we write
$$
\begin{equation*}
K_{nX}\,\,:=\,\,\operatorname{Ker}[H^n(X,\mathbb{Q})\to H^0(C,R^n\pi_\ast\mathbb{Q})]
\end{equation*}
\notag
$$
for the kernel of the edge map of the Leray spectral sequence $E_2^{p,q}(\pi)$ for the structure morphism $\pi\colon X\to C$. We also put
$$
\begin{equation*}
K_{nY}:=\operatorname{Ker}[H^n(Y,\mathbb{Q})\to H^0(C,R^n(\tau\sigma)_\ast\mathbb{Q})].
\end{equation*}
\notag
$$
Note that the Leray spectral sequences $E_2^{p,q}(\pi)=H^p(C,R^q\pi_\ast\mathbb{Q})$ and $E_2^{p,q}(\tau\sigma) =H^p(C,R^q(\tau\sigma)_\ast\mathbb{Q})$ degenerate: $E_2^{p,q}=E_\infty^{p,q}$ ([22], Corollary 15.15). Hence for every positive integer $n$ there are exact sequences of Hodge $\mathbb{Q}$-structures
$$
\begin{equation}
0\to H^2(C,R^{n-2}\pi_\ast\mathbb{Q})\to K_{nX} \xrightarrow{\alpha_{nX}} H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\to 0,
\end{equation}
\tag{2.1}
$$
$$
\begin{equation}
0\to H^2(C,R^{n-2}(\tau\sigma)_\ast\mathbb{Q})\to K_{nY} \xrightarrow{\alpha_{nY}} H^1(C,R^{n-1}(\tau\sigma)_\ast\mathbb{Q})\to 0
\end{equation}
\tag{2.2}
$$
([33], (2.4)), Therefore, by (1.3), the sequence (2.1) yields identifications
$$
\begin{equation}
K_{nX}=H^1(C,R^{n-1}\pi_\ast\mathbb{Q}) \quad \forall\, n\in\{3,5,7\}.
\end{equation}
\tag{2.3}
$$
2.2.We claim that, for every odd positive integer $n$,
$$
\begin{equation}
H^0(C',R^n\tau'_\ast\mathbb{Q})=H^2(C,R^n(\tau\sigma)_\ast\mathbb{Q})=0.
\end{equation}
\tag{2.4}
$$
Indeed, the natural $\smile$-products (together with a polarization of the local system $R^n\tau'_\ast\mathbb{Q}$) determine a non-degenerate pairing ([22], Proposition 10.5)
$$
\begin{equation*}
H^0(C,j_\ast R^n\tau'_\ast\mathbb{Q})\times H^2(C,j_\ast R^n\tau'_\ast\mathbb{Q})\to H^2(C,\mathbb{Q})\,\widetilde{\to}\,\mathbb{Q}.
\end{equation*}
\notag
$$
By the theorem on local invariant cycles, the canonical map $R^n(\tau\sigma)_\ast\mathbb{Q}$ $\to j_\ast R^n\tau'_\ast\mathbb{Q}$ is surjective and its kernel is concentrated on a finite set $\Delta$ ([23], § 3.7, [22], Proposition 15.12). Therefore
$$
\begin{equation*}
H^2(C,R^n(\tau\sigma)_\ast\mathbb{Q})=H^2(C,j_\ast R^n\tau'_\ast\mathbb{Q})\,\widetilde{\to}\, H^0(C,j_\ast R^n\tau'_\ast\mathbb{Q})^\vee =H^0(C',R^n\tau'_\ast\mathbb{Q})^\vee.
\end{equation*}
\notag
$$
Hence it suffices to prove that $H^0(C',R^n\tau'_\ast\mathbb{Q})=0$. For every point $s\in C'\setminus \Delta_{\text{countable}}$, a polarization of $X_s$ determines an isomorphism $H^a(X_s,\mathbb{Q})^\vee\,\widetilde{\to}\,H^a(X_s,\mathbb{Q})(a)$ of rational Hodge structures and $G$-modules ([20], § 4.2.3). Therefore, by the Künneth formula, we have isomorphisms
$$
\begin{equation*}
H^0(C',R^n\tau'_\ast\mathbb{Q})\,\widetilde{\to}\, H^n(X_s\times X_s,\mathbb{Q})^G\,\widetilde{\to}\,\bigoplus_{a=0}^8 \operatorname{Hom}_G\bigl(H^a(X_s,\mathbb{Q}),H^{n-a}(X_s,\mathbb{Q})\bigr).
\end{equation*}
\notag
$$
On the other hand, the Poincaré duality on the smooth fibre $X_s$ yields an isomorphism $H^q(X_s,\mathbb{Q})^\vee\,\widetilde{\to}\,H^{8-q}(X_s,\mathbb{Q})$ of $G$-modules. Hence, by (1.2), it suffices to check that
$$
\begin{equation*}
\begin{aligned} \, &\operatorname{Hom}_{\operatorname{Lie}G(\overline{\mathbb{Q}})} \bigl(H^1(X_s,\overline{\mathbb{Q}}),H^2(X_s,\overline{\mathbb{Q}})\bigr) =\operatorname{Hom}_{\operatorname{Lie}G(\overline{\mathbb{Q}})} \bigl(H^1(X_s,\overline{\mathbb{Q}}),H^4(X_s,\overline{\mathbb{Q}})\bigr) \\ &\qquad=\operatorname{Hom}_{\operatorname{Lie}G(\overline{\mathbb{Q}})} \bigl(H^2(X_s,\overline{\mathbb{Q}}),H^3(X_s,\overline{\mathbb{Q}})\bigr)=0. \end{aligned}
\end{equation*}
\notag
$$
We first assume that the Abelian variety $X_\eta$ is the product of four elliptic curves. Then the desired equalities follow easily by standard arguments ([27], § 1.3) from the Clebsch–Gordan formula ([28], Ch. VIII, § 9, n$^0$ 4) for representations of simple Lie algebras of type $A_1$. In case (1.6), the Lie algebra $\operatorname{Lie}G(\overline{\mathbb{Q}})$ has type $A_1\times C_2$:
$$
\begin{equation*}
\begin{aligned} \, H^1(X_s,\overline{\mathbb{Q}}) &=E(\omega_1^{(1)})+E(\omega_1^{(1)})+E(\omega_1^{(2)}), \\ H^2(X_s,\overline{\mathbb{Q}}) &= E(0)^{\oplus 4}+E(2\omega_1^{(1)}) +E(\omega_1^{(1)} +\omega_1^{(2)})^{\oplus 2}+E(\omega_2^{(2)}), \\ H^3(X_s,\overline{\mathbb{Q}}) &=E(\omega_1^{(1)})^{\oplus 4}+E(\omega_1^{(2)})^{\oplus 4}+ E(2\omega_1^{(1)}+\omega_1^{(2)})+E(\omega_1^{(1)}+\omega_2^{(2)})^{\oplus 2}, \\ H^4(X_s,\overline{\mathbb{Q}}) &=E(0)^{\oplus 5}\,{+}\,E(\omega_1^{(1)}\,{+}\,\omega_1^{(2)})^{\oplus 4}\,{+}\,E(\omega_2^{(2)})^{\oplus 3}\,{+}\,E(2\omega_1^{(1)}\,{+}\,\omega_2^{(2)})\,{+}\,E(2\omega_1^{(1)}). \end{aligned}
\end{equation*}
\notag
$$
Therefore, (2.4) follows from the Schur lemma. In the remaining cases (1.7)–(1.12), (1.14)–(1.16), (1.18)–(1.20), one can easily verify the equalities (2.4) by using direct computations and the decompositions obtained in § 1.3. 2.3.It is known that for every point $s\in C'$ the algebraic correspondence $\operatorname{c}_1(\mathcal P'_s)^{\smile\,n}$ $(1\leqslant n\leqslant 3)$ yields an algebraic isomorphism $H^{8-n}(X_s,\mathbb{Q})\,\widetilde{\to}\,H^n(X_s,\mathbb{Q})$ ([2], Lemma 2A12, Remark 2A13, [32], Ch. 16, § 16.4, p. 532). Therefore the section ${\Lambda'_{1,1}}^{\smile\,n}$ yields an isomorphism of local systems $R^{8-n}\pi'_\ast\mathbb{Q}\,\widetilde{\to}\,R^n\pi'_\ast\mathbb{Q}$, determined by the composite map
$$
\begin{equation}
R^{8-n}\pi'_\ast\mathbb{Q}\xrightarrow{(p'_1)^\ast} R^{8-n}\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q}{\pi'_\ast\mathbb{Q}} \xrightarrow{\smile\,{\Lambda'_{1,1}}^{\smile \,n}} R^8\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q}{R^n\pi'_\ast\mathbb{Q}} \xrightarrow{(p'_2)_\ast} R^n\pi'_\ast\mathbb{Q}.
\end{equation}
\tag{2.5}
$$
The compatibility of the $\smile$-products with Leray’s spectral sequence $E_2^{p,q}(\tau\sigma)=H^p(C,R^q(\tau\sigma)_\ast\mathbb{Q})$ ([34], Vol. II, Ch. 4, § 4.2.1, (4.5), Lemma 4.13) and a standard algorithm ([35], § 2.3, the construction of (2.10)) enables us to extend (2.5) to a sequence of maps
$$
\begin{equation}
R^{8-n}\pi_\ast\mathbb{Q}\xrightarrow{(p_1\sigma)^\ast}R^{8-n}(\tau\sigma)_\ast\mathbb{Q} \xrightarrow{{\smile}\,\operatorname{cl}_Y(D^{(1)})^{\smile\,n}} R^{8+n}(\tau\sigma)_\ast\mathbb{Q}\xrightarrow{(p_2\sigma)_\ast}R^n\pi_\ast\mathbb{Q}
\end{equation}
\tag{2.6}
$$
whose composite is an isomorphism outside a finite set $\Delta$. In its turn, (2.6) yields a sequence of canonical maps of cohomology groups
$$
\begin{equation*}
\begin{aligned} \, &H^1(C,R^{8-n}\pi_\ast\mathbb{Q})\xrightarrow{[(p_1\sigma)^\ast]_1} H^1(C,R^{8-n}(\tau\sigma)_\ast\mathbb{Q}) \\ &\qquad\xrightarrow{[{\smile}\,\operatorname{cl}_Y(D^{(1)})^{\smile\,n}]_1} H^1(C,R^{8+n}(\tau\sigma)_\ast\mathbb{Q}) \xrightarrow{[(p_2\sigma)_\ast]_1} H^1(C,R^n\pi_\ast\mathbb{Q}). \end{aligned}
\end{equation*}
\notag
$$
By functoriality ([36], § 2.4, [37], Ch. II, Theorem 3.11), the composite of these maps coincides with the canonical map
$$
\begin{equation}
H^1(C,R^{8-n}\pi_\ast\mathbb{Q})\xrightarrow{[x\,{\mapsto}\, (p_2\sigma)_\ast((p_1\sigma)^\ast x\,{\smile}\operatorname{cl}_Y(D^{(1)})^{\smile\,n})]_1} H^1(C,R^n\pi_\ast\mathbb{Q})
\end{equation}
\tag{2.7}
$$
corresponding to the morphism of sheaves
$$
\begin{equation}
R^{8-n}\pi_\ast\mathbb{Q}\xrightarrow{x\,{\mapsto}\, (p_2\sigma)_\ast((p_1\sigma)^\ast x \,{\smile}\operatorname{cl}_Y(D^{(1)})^{\smile\,n})} R^n\pi_\ast\mathbb{Q}.
\end{equation}
\tag{2.8}
$$
Since the kernel and cokernel of (2.8) are concentrated on $\Delta$, their higher cohomology groups vanish. Hence the map (2.7) is surjective. On the other hand, in the notation of § 1.1, the strong Lefschetz theorem on the fibres of the smooth morphism $\pi'$ determines an isomorphism of sheaves
$$
\begin{equation}
j_\ast R^n\pi'_\ast\mathbb{Q}\,\widetilde{\to}\,j_\ast R^{8-n}\pi'_\ast\mathbb{Q}.
\end{equation}
\tag{2.9}
$$
By the theorem on local invariant cycles ([23], § 3.7; [22], Proposition 15.12), the canonical map $R^p\pi_\ast\mathbb{Q}\to j_\ast R^p\pi'_\ast\mathbb{Q}$ is surjective and its kernel is concentrated on $\Delta$. Hence there is a canonical isomorphism $H^1(C,R^p\pi_\ast\mathbb{Q})\,\widetilde{\to}\,H^1(C,j_\ast R^p\pi'_\ast\mathbb{Q})$. Therefore, by (2.7) and (2.9), the surjective map (2.7) is an isomorphism
$$
\begin{equation}
H^1(C,R^{8-n}\pi_\ast\mathbb{Q})\underset{\widetilde{\qquad}}{\xrightarrow{[x\,{\mapsto}\, (p_2\sigma)_\ast((p_1\sigma)^\ast x\smile \operatorname{cl}_Y(D^{(1)})^{\smile\,n})]_1}} H^1(C,R^n\pi_\ast\mathbb{Q})
\end{equation}
\tag{2.10}
$$
of bidegree $(n-4,n-4)$ of rational Hodge structures. 2.4.For every point $s\in C$ let $\iota_{X_s/X}\colon X_s\hookrightarrow X$ be the canonical embedding. Since the morphism $\pi$ is proper, the fibre of the sheaf $R^n\pi_\ast\mathbb{Q}$ over the point $s\in C$ coincides with the space $H^n(X_s,\mathbb{Q})$ ([38], Ch. II, § 4, Remark 4.17.1). Hence the restriction map $\iota_{X_s/X}^\ast$ coincides with the composite ([34], Vol. II, Ch. 4, § 4.3.1, [39], Ch. 9, the beginning of § 5)
$$
\begin{equation*}
H^n(X,\mathbb{Q})\to E_\infty^{0,n}(\pi)\to E_2^{0,n}(\pi)=H^0(C,R^n\pi_\ast\mathbb{Q})\to H^n(X_s,\mathbb{Q}).
\end{equation*}
\notag
$$
Thus $\iota_{X_s/X}^\ast$ is the composite of the canonical maps
$$
\begin{equation*}
H^n(X,\mathbb{Q})\to H^0(C,R^n\pi_\ast\mathbb{Q})\hookrightarrow \prod_{s\in C}H^n(X_s,\mathbb{Q})\to H^n(X_s,\mathbb{Q}),
\end{equation*}
\notag
$$
where the $\mathbb{Q}$-space $\prod_{s\in C}H^n(X_s,\mathbb{Q})$ is identified with the $\mathbb{Q}$-space of discontinuous global sections of the sheaf $R^n\pi_\ast\mathbb{Q}$ ([38], Ch. II, § 4, § 4.4.4). It is clear that
$$
\begin{equation}
\omega\in K_{nX}\quad\Longleftrightarrow\quad (\forall\, s\in C)\quad \iota_{X_s/X}^\ast(\omega)=0.
\end{equation}
\tag{2.11}
$$
Therefore for all $\omega\in K_{nX}$ we see from [40], Ch. 2, § 8, (5) that
$$
\begin{equation*}
\iota_{X_s/X}^\ast(\operatorname{cl}_X(H)\,{\smile}\,\omega) =\iota_{X_s/X}^\ast(\operatorname{cl}_X(H))\,{\smile}\, \iota_{X_s/X}^\ast(\omega)=0.
\end{equation*}
\notag
$$
Hence,
$$
\begin{equation}
\operatorname{cl}_X(H)\,{\smile}\,K_{nX}\subset K_{(n+2)X}.
\end{equation}
\tag{2.12}
$$
Moreover, there is a canonical embedding
$$
\begin{equation}
(p_k\sigma)^\ast|_{K_{nX}}\colon K_{nX}\hookrightarrow K_{nY}
\end{equation}
\tag{2.13}
$$
induced (since the morphism $p_k\sigma\colon Y\to X$ is surjective) by the canonical injection $(p_k\sigma)^\ast\colon H^n(X,\mathbb{Q})\hookrightarrow H^n(Y,\mathbb{Q})$ ([2], Proposition 1.2.4). Indeed, put $p_{ks}=p_k|_{X_s\times X_s}$, $\sigma_s=\sigma|_{Y_s}$ and consider the commutative diagram of morphisms which in turn yields the commutative diagram of canonical maps Hence it follows from (2.11) that $\iota_{Y_s/Y}^\ast(p_k\sigma)^\ast(\omega)=0$ for every $s\in C$ and every $\omega\in K_{nX}$. Therefore (2.13) follows from the equalities $\iota_{Y_s/Y}^\ast(p_k\sigma)^\ast(\omega)=0$ ($s\in C$) and from (2.11).2.5.It follows from the theorem on local invariant cycles and from the strong Lefschetz theorem for the fibres of the smooth morphism $\pi'$ that
$$
\begin{equation*}
\begin{aligned} \, &\operatorname{cl}_X(H)^{\smile\,2}\,{\smile}\, H^1(C,R^2\pi_\ast\mathbb{Q})= \operatorname{cl}_X(H)^{\smile\,2}\,{\smile}\, H^1(C,j_\ast R^2\pi'_\ast\mathbb{Q}) \\ &\qquad=H^1(C,j_\ast R^6\pi'_\ast\mathbb{Q})=H^1(C,R^6\pi_\ast\mathbb{Q}). \end{aligned}
\end{equation*}
\notag
$$
Hence we obtain the following equality from (2.3) and (2.11);
$$
\begin{equation}
K_{7X}=\operatorname{cl}_X(H)^{\smile\,2}\,{\smile}\,K_{3X}.
\end{equation}
\tag{2.14}
$$
Using the arguments in § 1.2 of [14] along with (2.3) and (2.14), we see that there are canonical splittings (independent of the choice of an ample divisor $H$ on $X$) of Hodge $\mathbb{Q}$-structures where
$$
\begin{equation}
\begin{aligned} \, K_{3X}^\perp &= \{x\in H^3(X,\mathbb{Q})\mid x\,{\smile}\,y\,{\smile} \operatorname{cl}_X(H)^{\smile\,2}=0\ \forall\, y\in K_{3X}\} \nonumber \\ &=\{x\in H^3(X,\mathbb{Q})\mid x\,{\smile}\,y=0\ \forall\, y\in K_{7X}\} \end{aligned}
\end{equation}
\tag{2.17}
$$
is the orthogonal complement of the subspace $K_{3X}\hookrightarrow H^3(X,\mathbb{Q})$ with respect to the non-degenerate ([2], § 1.2.A) bilinear form
$$
\begin{equation*}
\Phi\colon H^3(X,\mathbb{Q})\times H^3(X,\mathbb{Q})\xrightarrow{x\times y\,{\mapsto}\,x\,{\smile}\,y\,{\smile}\operatorname{cl}_X(H)^{\smile\,2}} H^{10}(X,\mathbb{Q})=\mathbb{Q}(-5)\underset{\widetilde{\qquad}}{\xrightarrow{(2\pi i)^5}}\mathbb{Q}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
K_{7X}^\perp=\operatorname{cl}_X(H)^{\smile\,2}\,{\smile}\,K_{3X}^\perp,\qquad K_{3X}\,{\smile}\,K^\perp_{7X}=K_{7X}\,{\smile}\,K^\perp_{3X}=0
\end{equation}
\tag{2.18}
$$
since the restriction of $\Phi$ to the subspace $K_{3X}\hookrightarrow H^3(X,\mathbb{Q})$ is non-degenerate by the identifications (2.3), the theorem on local invariant cycles (which enables us to identify the $\mathbb{Q}$-spaces $H^1(C',j_\ast R^n\pi'_\ast\mathbb{Q})$ and $H^1(C,R^n\pi_\ast\mathbb{Q})$) and the non-degeneracy ([22], Proposition 10.5) of the canonical pairing
$$
\begin{equation*}
\begin{aligned} \, &H^1(C,R^2\pi_\ast\mathbb{Q})\times H^1(C,R^2\pi_\ast\mathbb{Q}) \\ &\qquad\xrightarrow{x\times y\,{\mapsto}\,x\,{\smile}\,y\,\smile\,\operatorname{cl}_X(H)^{\smile\,2}} H^2(C,R^8\pi_\ast\mathbb{Q})=H^{10}(X,\mathbb{Q}). \end{aligned}
\end{equation*}
\notag
$$
2.6.Lemma. We have Proof. If $\Delta=\varnothing$, then $H^0(C,R^3\pi_\ast\mathbb{Q})=0$ by (1.2). Hence there is an algebraic isomorphism $H^7(X,\mathbb{Q})\,\widetilde{\to}\,H^3(X,\mathbb{Q})$ ([3], Theorem 10.1).
From now on we assume that $\Delta\neq\varnothing$. Let $D^\ast(\delta)$ be a small punctured disc on $C$ centred at a point $\delta\in\Delta$. The Leray spectral sequence for the embedding $j\colon C' \subset C$ yields an exact sequence of mixed Hodge structures ([22], proof of Proposition 12.5, Corollary 13.10, Remark 14.5)
$$
\begin{equation}
0\to H^1(C,j_\ast R^2\pi'_\ast\mathbb{Q})\to H^1(C',R^2\pi'_\ast\mathbb{Q}) \to H^0(C,R^1j_\ast R^2\pi'_\ast\mathbb{Q}),
\end{equation}
\tag{2.19}
$$
where
$$
\begin{equation*}
H^0(C,R^1j_\ast R^2\pi'_\ast\mathbb{Q})= \bigoplus_{\delta\in\Delta} H^1(D^\ast(\delta),R^2\pi'_\ast\mathbb{Q}) \,\widetilde{\to}\, \bigoplus_{\delta\in\Delta} H^2(X_s,\mathbb{Q})/N_\delta H^2(X_s,\mathbb{Q}),
\end{equation*}
\notag
$$
the space $H^2(X_s,\mathbb{Q})$ ($s\in C'$) has the limiting mixed Hodge structure associated with the local monodromy $\gamma_\delta$ around the point $\delta\in C$ (the Picard–Lefschetz transformation) and $N_\delta=\log \gamma_\delta$. By the theorem on local invariant cycles, the sequence (2.19) yields an exact sequence
$$
\begin{equation}
0\to H^1(C,R^2\pi_\ast\mathbb{Q})\to H^1(C',R^2\pi'_\ast\mathbb{Q}) \to \bigoplus_{\delta\in\Delta} H^2(X_s,\mathbb{Q})/N_\delta H^2(X_s,\mathbb{Q}).
\end{equation}
\tag{2.20}
$$
By the well-known property of functoriality of the Leray spectral sequence, the canonical embedding $\iota_{X'/X}\colon X'\hookrightarrow X$ induces homomorphisms $E_2^{p,q}(\pi)\to E_2^{p,q}(\pi')$ which are compatible with the differentials and filtrations ([36], § 2.4, [34], Vol. II, Proposition 4.8). Using the equality $H^2(C',R^1\pi'_\ast\mathbb{Q})=0$ (since the cohomology dimension of the affine curve $C'$ equals $1$; see [41], Ch. VI, § 7, Theorem 7.2), the surjectivity of the edge maps $H^3(X,\mathbb{Q})\to H^0(C,R^3\pi_\ast\mathbb{Q})$ ([22], Corollary 15.14) and $H^3(X',\mathbb{Q})\to H^0(C',R^3\pi'_\ast\mathbb{Q})$ ([21], proof of Theorem 4.1.1), the formulae (2.20), (2.1), the commutative diagram (15.1) in [22], the exactness of the canonical sequence
$$
\begin{equation*}
\begin{aligned} \, 0 &\to H^2(C',R^1\pi'_\ast\mathbb{Q})\to \operatorname{Ker}[H^3(X',\mathbb{Q})\to H^0(C',R^3\pi'_\ast\mathbb{Q})] \\ &\xrightarrow{\alpha'_{3X}} H^1(C',R^2\pi'_\ast\mathbb{Q})\to 0 \end{aligned}
\end{equation*}
\notag
$$
and the commutativity of the diagram of morphisms we obtain a commutative diagram of mixed Hodge $\mathbb{Q}$-structures with exact rows In view of (1.2), (2.3), (2.15), this diagram takes the form The corresponding exact sequence of Hodge $\mathbb{Q}$-structures ([35], § 2.3) of a snake-like diagram ([42], § 1, Proposition 2) and (1.1) yield that $(i_\Delta f)_\ast\,H^1(Z,\mathbb{Q})=K_{3X}^\perp$. $\Box$ 2.7.Lemma. There is a canonical embedding $(p_2\sigma)_\ast(K_{11X})\subset K_{3X}$. Proof. Given any irreducible smooth projective variety $W$, we denote the orientation isomorphism of Weil cohomology ([2], § 1.2.A) by $\langle\ \rangle\colon H^{2\dim_\mathbb{C} W}(W,\mathbb{Q})\,\widetilde{\to}\,\mathbb{Q}$. It is determined by the choice of an element $\sqrt{-1}\in\mathbb{C}$.
Since
$$
\begin{equation*}
K_{3X}=\{x\in H^3(X,\mathbb{Q}) \mid x\,{\smile}\,K_{3X}^\perp\,{\smile} \operatorname{cl}_X(H)^{\smile\,2}=0\},
\end{equation*}
\notag
$$
it suffices to verify the equality
$$
\begin{equation*}
(p_2\sigma)_\ast(K_{11Y})\,{\smile}\,K^\perp_{3X}\,{\smile}\operatorname{cl}_X(H)^{\smile\,2}=0,
\end{equation*}
\notag
$$
which is equivalent by the formula
$$
\begin{equation*}
\langle(p_2\sigma)_\ast(K_{11Y})\,{\smile}\, K^\perp_{3X}\,{\smile} \operatorname{cl}_X(H)^{\smile\,2}\rangle = \bigl\langle K_{11Y}\,{\smile}\, (p_2\sigma)^\ast\bigl(K^\perp_{3X}\,{\smile}\operatorname{cl}_X(H)^{\smile\,2}\bigr)\bigr\rangle
\end{equation*}
\notag
$$
([2], § 1.2.A, [41], Ch. VI, § 11, Remark 11.6) to the equality
$$
\begin{equation*}
K_{11Y}\,{\smile}\, (p_2\sigma)^\ast\bigl(K^\perp_{3X}\,{\smile} \operatorname{cl}_X(H)^{\smile\,2}\bigr)=0.
\end{equation*}
\notag
$$
Therefore, by Lemma 2.6, it suffices to prove that
$$
\begin{equation}
K_{11Y}\,{\smile}\, (p_2\sigma)^\ast\bigl((i_\Delta f)_\ast\,H^1(Z,\mathbb{Q})\,{\smile} \operatorname{cl}_X(H)^{\smile\,2}\bigr)=0.
\end{equation}
\tag{2.21}
$$
Let $Z_\delta$ be the normalization of the divisor $\pi^{-1}(\delta)=X_\delta$. The irreducible components of the smooth variety $Z$ can naturally be identified with the irreducible components $X_{\delta i}$ of the divisor $\pi^{-1}(\Delta)=\sum_{\delta\in\Delta}X_{\delta}$. Let $\iota_{X_{\delta i}/X}\colon X_{\delta i}\hookrightarrow X$, $\iota_{X_{\delta i}/Z}\colon X_{\delta i}\hookrightarrow Z$, and $\iota_{Z_{\delta}/Z}\colon Z_{\delta}\hookrightarrow Z$ be the canonical embeddings. Since the diagram of canonical morphisms is commutative, we have
$$
\begin{equation}
(i_\Delta f)_\ast(\iota_{X_{\delta i}/Z})_\ast|_{H^k(X_{\delta i},\mathbb{Q})}= \iota_{X_{\delta i}/X\ast}|_{H^k(X_{\delta i},\mathbb{Q})}.
\end{equation}
\tag{2.22}
$$
It is known that the Gysin map $\iota_{X_{\delta i}/X\ast}\colon H^k(X_{\delta i},\mathbb{Q})\to H^{k+2}(X,\mathbb{Q})$ is of the following form ([43], (4.20), [14], (3.37)):
$$
\begin{equation}
\alpha\,{\mapsto}\,\alpha\,{\smile}\operatorname{cl}_X(X_{\delta i}).
\end{equation}
\tag{2.23}
$$
On the other hand, by the strong Lefschetz theorem for $X_{\delta i}$, there is an embedding
$$
\begin{equation*}
H^1(X_{\delta i},\mathbb{Q})\,{\smile}\,\iota^\ast_{X_{\delta i}/X}\operatorname{cl}_X(H)^{\smile\,2}\subset H^5(X_{\delta i},\mathbb{Q}).
\end{equation*}
\notag
$$
Hence the projection formula ([2], § 1.2.A) and (2.23) yield an embedding
$$
\begin{equation*}
\iota_{X_{\delta i}/X\ast}H^1(X_{\delta i},\mathbb{Q})\,{\smile} \operatorname{cl}_X(H)^{\smile\,2}\subset \iota_{X_{\delta i}/X\ast}H^5(X_{\delta i},\mathbb{Q})=H^5(X_{\delta i},\mathbb{Q})\,{\smile}\operatorname{cl}_X(X_{\delta i})
\end{equation*}
\notag
$$
and, therefore, (2.22) yields the existence of a canonical embedding
$$
\begin{equation}
(i_\Delta f)_\ast H^1(Z,\mathbb{Q})\smile\,\operatorname{cl}_X(H)^{\smile\,2}\subset \sum_{\delta,i}H^5(X_{\delta i},\mathbb{Q})\,{\smile}\operatorname{cl}_X(X_{\delta i}).
\end{equation}
\tag{2.24}
$$
By definition ([34], Vol. II, Ch. 4, § 4.2.1), for every point $s\in C'$, the $\smile$-multiplication by the class $\operatorname{cl}_X(X_{\delta i})\in H^2(X,\mathbb{Q})$ acts on the fibre $H^q(X_s,\mathbb{Q})=[j_\ast R^q\pi'_\ast\mathbb{Q}]_s$ of the sheaf $j_\ast R^q\pi'_\ast\mathbb{Q}$ as the $\smile$-multiplication by the class $\iota^\ast_{X_s/X}(\operatorname{cl}_X(X_{\delta i}))$. The obvious equality
$$
\begin{equation*}
\iota^\ast_{X_s/X}(\operatorname{cl}_X(X_{\delta i}))=0
\end{equation*}
\notag
$$
yields that
$$
\begin{equation}
j_\ast R^q\pi'_\ast\mathbb{Q}\,{\smile}\operatorname{cl}_X(X_{\delta i})=0.
\end{equation}
\tag{2.25}
$$
Using (2.2), (2.4), the theorem on local invariant cycles and the Künneth formula for the fibres of the smooth morphism $\tau'$, we obtain a canonical isomorphism and a splitting
$$
\begin{equation*}
K_{11Y}\underset{\widetilde{\qquad}}{\xrightarrow{\alpha_{11Y}}} H^1(C,j_\ast R^{10}\tau'_\ast\mathbb{Q})=\bigoplus_{p+q=10} H^1(C,j_\ast R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} j_\ast R^q\pi'_\ast\mathbb{Q}).
\end{equation*}
\notag
$$
Hence (2.21) follows from (2.24) and (2.25). $\Box$ 2.8.Using the functoriality of our constructions, the theorem on local invariant cycles, Lemma 2.7 and (2.1)–(2.4), we obtain a commutative diagram where the map $p'_{2\ast}\colon H^1(C,j_\ast R^{10}\tau'_\ast\mathbb{Q})\to H^1(C,j_\ast R^2\pi'_\ast\mathbb{Q})$ of cohomology groups is induced by the canonical map of sheaves $p'_{2\ast}\colon R^{10}\tau'_\ast\mathbb{Q}\to R^2\pi'_\ast\mathbb{Q}$.We also use (2.13) and the commutativity of the diagram where the map $K_{7Y}\xrightarrow{{\smile}\,\operatorname{cl}_Y(D^{(1)})^{\smile\,2}}K_{11Y}$ is determined by (2.11) and the equality
$$
\begin{equation*}
\iota_{Y_s/Y}^\ast\bigl(\operatorname{cl}_Y(D^{(1)})^{\smile\,2}\,{\smile}\,K_{7Y}\bigr) =\iota_{Y_s/Y}^\ast\bigl(\operatorname{cl}_Y(D^{(1)})^{\smile\,2}\bigr)\,{\smile}\, \iota_{Y_s/Y}^\ast(K_{7Y})=0
\end{equation*}
\notag
$$
([40], Ch. 2, § 8, (5)). Combining this with (2.26), we obtain a commutative diagram which is glued from the commutative diagrams and On the other hand, let $\operatorname{pr}_i\colon X\times X\to X$ be the canonical projections of the Cartesian square of $X$. For every $x\in H^7(X,\mathbb{Q})$, the projection formula ([2], § 1.2.A) yields that
$$
\begin{equation*}
\begin{aligned} \, &(p_2\sigma)_\ast\bigl((p_1\sigma)^\ast x\,{\smile}\,[\operatorname{cl}_Y(D^{(1)})]^{\smile\,2}\bigr)= [\operatorname{pr}_2\iota\sigma]_\ast \bigl([\operatorname{pr}_1\iota\sigma]^\ast x\,{\smile}\,[\operatorname{cl}_Y(D^{(1)})]^{\smile\,2}\bigr) \\ &\ =\operatorname{pr}_{2\ast}(\iota\sigma)_\ast \bigl((\iota\sigma)^\ast\operatorname{pr}_1^\ast x\,{\smile}\,[\operatorname{cl}_Y(D^{(1)})]^{\smile\,2}\bigr)= \operatorname{pr}_{2\ast}\bigl(\operatorname{pr}_1^\ast x\,{\smile}\,(\iota\sigma)_\ast\bigl[[\operatorname{cl}_Y(D^{(1)})]^{\smile\,2}\bigr]\bigr). \end{aligned}
\end{equation*}
\notag
$$
Since the composite of the maps in the bottom row of the diagram (2.27) coincides with the isomorphism (2.10), the composite of the maps in the top row of (2.27) yields an algebraic isomorphism
$$
\begin{equation}
K_{7X} \underset{\widetilde{\qquad}}{\xrightarrow{x\,{\mapsto} \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast x\,{\smile}\,(\iota\sigma)_\ast[[\operatorname{cl}_Y(D^{(1)})]^{\smile\,2}])}} K_{3X}.
\end{equation}
\tag{2.28}
$$
2.9.Using (2.18) and the standard arguments in §§ 1.2, 3.5 of [43], we can assume that the following equality of Poincaré classes holds:
$$
\begin{equation*}
\wp(H^3(X,\mathbb{Q}))=\wp(K_{3X})+\wp(K_{3X}^\perp),
\end{equation*}
\notag
$$
where the Poincaré class $\wp(K_{3X}^\perp)$ yields an isomorphism
$$
\begin{equation}
K_{7X}^\perp=K_{3X}^\perp\,{\smile}\operatorname{cl}_X(H)^{\smile\,2} \underset{\widetilde{\qquad}}{\xrightarrow{x\,{\mapsto} \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast(x)\,{\smile}\,\wp(K_{3X}^\perp))}} K_{3X}^\perp.
\end{equation}
\tag{2.29}
$$
On the other hand, it follows from the Lefschetz theorem on divisors for the smooth variety $Z\times Z$ and Lemma 2.6 that the correspondence $\wp(K_{3X}^\perp)$ is algebraic ([43], Lemma 3.8). 2.10.Let $u_{3,3}$, $u_{3,3^\perp}$, $u_{3^\perp,3}$, $u_{3^\perp,3^\perp}$, $h$ be the components of the algebraic correspondence $u=(\iota\sigma)_\ast\bigl[[\operatorname{cl}_Y(D^{(1)})]^{\smile\,2}\bigr]$ in the direct summands
$$
\begin{equation*}
K_{3X}\otimes K_{3X}, \quad\dots,\quad K_{3X}^\perp\otimes K_{3X}^\perp,\qquad H:=\bigoplus_{p+q=6,\,p\neq 3} H^p(X,\mathbb{Q})\otimes H^q(X,\mathbb{Q}).
\end{equation*}
\notag
$$
These summands are determined by the canonical decomposition (2.15) and by the Künneth decomposition of the $\mathbb{Q}$-space $H^6(X\times X,\mathbb{Q})$. Using (2.15), (2.16), (2.28), (2.29) and acting by the standard algorithm ([14], §§ 3.5–3.9), we easily verify that the cohomology class $u_{3,3}+u_{3,3^\perp}+h_{10}+\wp(K_{3X}^\perp)$ is algebraic for some $h_{10}\in H$ and we have an isomorphism
$$
\begin{equation*}
H^7(X,\mathbb{Q})\underset{\widetilde{\qquad}}{\xrightarrow{x\,{\mapsto} \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast x\,{\smile}\,(u_{3,3}+u_{3,3^\perp}+h_{10}+\wp(K_{3X}^\perp)))}} H^3(X,\mathbb{Q}).
\end{equation*}
\notag
$$
§ 3. On the structure of rational cohomology of even degree3.1.Lemma. The $\mathbb{Q}$-space $H^0(C',R^4\pi'_\ast\mathbb{Q})$ is generated by the images of the cohomology classes of intersections of divisors on $X$ under the canonical surjective map $H^4(X,\mathbb{Q})\to H^0(C',R^4\pi'_\ast\mathbb{Q})$. Proof. It is well known that the canonical map $H^n(X,\mathbb{Q})\to H^0(C',R^n\pi'_\ast\mathbb{Q})$ is surjective for every positive integer $n$ ([21], Theorem 4.1.1). By construction, any simple Abelian subvariety
$$
\begin{equation*}
I_{\overline\eta}\subset X_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)}
\end{equation*}
\notag
$$
is of the form $I_{\overline\eta}=I_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)}$, where the Abelian variety $I_\eta$ is defined over the field $\kappa(\eta)$, $\operatorname{End}_{\kappa(\eta)}(I_\eta) =\operatorname{End}_{\overline{\kappa(\eta)}\,}(I_{\overline\eta})$, the centre of the ring $\operatorname{End}_{\kappa(\eta)}(I_\eta)\otimes_\mathbb{Z}\mathbb{Q}$ is not a purely imaginary quadratic extension of a totally real number field, and the ring $\operatorname{End}_{\kappa(\eta)}(I_\eta)\otimes_\mathbb{Z}\mathbb{Q}$ is not a definite quaternion division algebra over $\mathbb{Q}$. Hence, by the Moonen–Zarhin theorem ([29], Theorem 0.1), the $\mathbb{Q}$-space of Hodge cycles (invariant cycles of the canonical action of the Hodge group in rational cohomology of even degree) of the generic geometric fibre $X_\eta\otimes_{\kappa(\eta)}\mathbb{C}$ is generated by the classes of intersections of divisors for any embedding of fields $\kappa(\eta)\hookrightarrow \mathbb{C}$.
On the other hand, for any simple Abelian subvariety $I_\eta\subset X_\eta$, the Hodge group of the Abelian variety $I_\eta\otimes_{\kappa(\eta)}\mathbb{C}$ is a $\mathbb{Q}$-simple group ([31], Remark 1 on the derivation of Theorem 4.1 from Lemmas 1–3, [29], Proposition 2.4, Theorem 0.1). Since the Abelian variety $I_\eta$ has trivial trace, the closure $G_{I_\eta}$ of the image of the global monodromy $\pi_1(C',s)\to\operatorname{GL}(H^1(I_s,\mathbb{Q}))$ (which is a non-trivial connected semisimple $\mathbb{Q}$-group by the assumptions above) can naturally be identified with a normal ([24], Theorem 7.3) subgroup of the $\mathbb{Q}$-simple Hodge group of the Abelian variety $I_\eta\otimes_{\kappa(\eta)}\mathbb{C}$. As a result, we obtain an equality
$$
\begin{equation}
G_{I_\eta}=\operatorname{Hg}(I_\eta\otimes_{\kappa(\eta)}\mathbb{C}).
\end{equation}
\tag{3.1}
$$
By construction, there is a $\kappa(\eta)$-isogeny
$$
\begin{equation*}
X_\eta\underset{\operatorname{isogeny}}{\,\sim\,\,}\,X_{1\eta}^{\times m_1}\times \dots\times X_{n\eta}^{\times m_n},
\end{equation*}
\notag
$$
where $X_{k\eta}\subset X_\eta$ ($k=1,\dots,n$) are pairwise non-isogenous absolutely simple Abelian subvarieties over the field $\kappa(\eta)$, $m_k$ is a positive integer.
If the Abelian variety $X_\eta\otimes_{\kappa(\eta)}\mathbb{C}$ is not simple and if it is isogenous to the product $[I_{1\eta}\otimes_{\kappa(\eta)}\mathbb{C}]\times [I_{2\eta}\otimes_{\kappa(\eta)}\mathbb{C}]$ of non-trivial Abelian subvarieties, where $\operatorname{Hom}_\mathbb{C}(I_{1\eta}\otimes_{\kappa(\eta)} \mathbb{C}, I_{2\eta}\otimes_{\kappa(\eta)}\mathbb{C})=0$ and the Abelian variety $I_{2\eta}\otimes_{\kappa(\eta)}\mathbb{C}$ is a power of a simple Abelian variety, then we have $\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C}) =\operatorname{Hg}(I_{1\eta}\otimes_{\kappa(\eta)}\mathbb{C})\times \operatorname{Hg}(I_{2\eta}\otimes_{\kappa(\eta)}\mathbb{C})$ under hypotheses of the theorem (see [29], § 5.4). Therefore,
$$
\begin{equation}
\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C}) =\prod_{k=1}^n\operatorname{Hg}(X_{k\eta}\otimes_{\kappa(\eta)}\mathbb{C}).
\end{equation}
\tag{3.2}
$$
It is clear that $G\subset \prod_{k=1}^n G_{X_{k\eta}}$ and the canonical projections $G\to G_{X_{k\eta}}$ are surjective for all $k$.
We first assume that the Abelian variety $X_\eta\otimes_{\kappa(\eta)}\mathbb{C}$ is isogenous to the product of an elliptic curve $X_{1\eta}\otimes_{\kappa(\eta)}\mathbb{C}$ (without complex multiplication) and a $3$-dimensional simple Abelian variety $X_{2\eta}\otimes_{\kappa(\eta)}\mathbb{C}$. Using §§ 2.1, 2.2 in [29] and (3.1), we see that the Lie algebra $\operatorname{Lie}G_{X_{1\eta}}(\mathbb{C})$ has type $A_1$ and the Lie algebra $\operatorname{Lie}G_{X_{2\eta}}(\mathbb{C})$ has type $C_3$ or $A_1\times A_1\times A_1$. Since the Lie algebra $\operatorname{Lie}G_{X_{2\eta}}$ is $\mathbb{Q}$-simple, the $\mathbb{Q}$-semisimple Lie algebra $\operatorname{Lie}G$ contains a $\mathbb{Q}$-simple Lie subalgebra $L_{X_{2\eta}}$, such that the restriction of the canonical projection $\operatorname{Lie}G\to \operatorname{Lie}G_{X_{2\eta}}$ to $L_{X_{2\eta}}$ is an isomorphism of Lie $\mathbb{Q}$-algebras $L_{X_{2\eta}}\,\widetilde{\to}\,\operatorname{Lie}G_{X_{2\eta}}$. The restriction of the canonical projection $\operatorname{Lie}G\to \operatorname{Lie}G_{X_{1\eta}}$ to $L_{X_{2\eta}}$ is trivial because we obviously have $\dim_\mathbb{Q} L_{X_{2\eta}}>\dim_\mathbb{Q}\operatorname{Lie}G_{X_{1\eta}}$ and the Lie algebras $L_{X_{2\eta}}$, $\operatorname{Lie}G_{X_{1\eta}}$ are $\mathbb{Q}$-simple. Thus, $L_{X_{2\eta}}\neq \operatorname{Lie}G$. Since the Lie algebra $\operatorname{Lie}G/L_{X_{2\eta}}$ is non-trivial and $\mathbb{Q}$-semisimple and possesses the canonical surjection $\operatorname{Lie}G/L_{X_{2\eta}}\to\operatorname{Lie}G_{X_{1\eta}}$, we have a surjection
$$
\begin{equation*}
\operatorname{Lie}G=\operatorname{Lie}G/L_{X_{2\eta}}\times L_{X_{2\eta}}\to\operatorname{Lie}G_{X_{1\eta}}\times L_{X_{2\eta}}=\operatorname{Lie}G_{X_{1\eta}}\times\operatorname{Lie}G_{X_{2\eta}}
\end{equation*}
\notag
$$
because it follows from (3.1), (3.2) and the existence of an inclusion $\operatorname{Lie}G\subset \operatorname{Lie}G_{X_{1\eta}}\times\operatorname{Lie}G_{X_{2\eta}}$ that $G=\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C})$.
If the Abelian variety $X_\eta\otimes_{\kappa(\eta)}\mathbb{C}$ is isogenous to the product of two isogenous simple Abelian surfaces $X_{k\eta}\otimes_{\kappa(\eta)}\mathbb{C}$ ($k=1,2$), then (3.1), (3.2) imply that $\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C}) =\operatorname{Hg}(X_{k\eta}\otimes_{\kappa(\eta)}\mathbb{C})=G_{X_{k\eta}}=G$. Assume that the Abelian variety $X_\eta\otimes_{\kappa(\eta)}\mathbb{C}$ is isogenous to the product of two non-isogenous simple Abelian surfaces $X_{k\eta}\otimes_{\kappa(\eta)}\mathbb{C}$ ($k=1,2$). It follows from § 2.2 in [29] that the Lie algebra $\operatorname{Lie}G_{X_{k\eta}}(\mathbb{C})$ has type $C_2$, $A_1\times A_1$, or $A_1$ (these variants correspond to the cases when $\operatorname{End}_\mathbb{C}(X_{k\eta}\otimes_{\kappa(\eta)}\mathbb{C})=\mathbb{Z}$, $\operatorname{End}_\mathbb{C}(X_{k\eta}\otimes_{\kappa(\eta)}\mathbb{C})$ is an order of a real quadratic field, or $\operatorname{End}_\mathbb{C}(X_{k\eta}\otimes_{\kappa(\eta)}\mathbb{C}) \otimes_\mathbb{Z}\mathbb{Q}$ is a quaternion division algebra over $\mathbb{Q}$ split at the Archimedean place $\infty$ of the field $\mathbb{Q}$). If the types of the Lie algebras $\operatorname{Lie}G_{X_{k\eta}}(\mathbb{C})$ ($k=1,2$) do not coincide, then $G=\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C})$ by the arguments above. In what follows, we assume that the types of the Lie algebras $\operatorname{Lie}G_{X_{k\eta}}(\mathbb{C})$ ($k=1,2$) coincide. First of all, there is a canonical isomorphism ([21], Corollary 4.4.13)
$$
\begin{equation}
\operatorname{Hom}_{C'}(X'_1,X'_2)\,\widetilde{\to}\, \operatorname{Hom}(R_{1\pi'_1\ast}\mathbb{Z}, R_{1\pi'_2\ast}\mathbb{Z}),
\end{equation}
\tag{3.3}
$$
where $\pi'_k\colon X'_k\to C'$ is the Abelian scheme of relative dimension $2$ with generic scheme fibre $X_{k\eta}$. Since the Abelian surfaces $X_{k\eta}\otimes_{\kappa(\eta)}\mathbb{C}$ ($k=1,2$) are non-isogenous, it follows from (3.3) that $\operatorname{Hom}(R_{1\pi'_1\ast}\mathbb{Z}, R_{1\pi'_1\ast}\mathbb{Z})=0$. Therefore,
$$
\begin{equation}
\operatorname{Hom}_{\operatorname{Lie}G(\mathbb{C})}\bigl(H^1(X_{1s},\mathbb{C}), H^1(X_{2s},\mathbb{C})\bigr)=0.
\end{equation}
\tag{3.4}
$$
If the Lie algebra $\operatorname{Lie}G_{X_{k\eta}}(\mathbb{C})$ ($k=1,2$) has type $C_2$, then the Lie algebra $\operatorname{Lie}G(\mathbb{C})$ has type $C_2$ or $C_2\times C_2$. The first case is impossible by (3.4) because the highest weight of the canonical representation of the Lie algebra $\operatorname{Lie}G(\mathbb{C})$ of type $C_2$ in the $\mathbb{C}$-space $H^1(X_{ks},\mathbb{C})$ is a minuscule weight in Bourbaki’s sense ([24], Theorem 0.5.1) and, therefore, this representation is isomorphic to the standard irreducible representation of degree $4$. Hence,
$$
\begin{equation*}
\operatorname{Lie}G(\mathbb{C})=\operatorname{Lie}G_{X_{1\eta}}(\mathbb{C})\times \operatorname{Lie}G_{X_{2\eta}}(\mathbb{C}).
\end{equation*}
\notag
$$
Thus it follows from (3.1), (3.2) that $G=\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C})$.
If the Lie algebra $\operatorname{Lie}G_{X_{k\eta}}(\mathbb{C})$ ($k=1,2$) has type $A_1\times A_1$, then the Lie algebra $\operatorname{Lie}G(\mathbb{C})$ has type $A_1\times A_1$, $A_1\times A_1\times A_1$ or $A_1\times A_1\times A_1\times A_1$. In this situation, the canonical representation of the Lie algebra $\operatorname{Lie}G(\mathbb{C})$ in the $\mathbb{C}$-space $H^1(X_{ks},\mathbb{C})$ factors through a representation of some quotient of type $A_1\times A_1$. This representation is isomorphic to $E(\omega^{(1)}_1)+E(\omega^{(2)}_1)$, where $E(\omega^{(k)}_1)$ is the standard irreducible representation of degree $2$ of the $k$th simple factor of the Lie algebra of type $A_1\times A_1$. Therefore the variants $A_1\times A_1$ and $A_1\times A_1\times A_1$ are excluded by (3.4). Thus $G=\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C})$. If the Lie algebra $\operatorname{Lie}G_{X_{k\eta}}(\mathbb{C})$ ($k=1,2$) has type $A_1$, then the Lie algebra $\operatorname{Lie}G(\mathbb{C})$ has type $A_1$ or $A_1\times A_1$. In this case, the canonical representation of the Lie algebra $\operatorname{Lie}G(\mathbb{C})$ in the $\mathbb{C}$-space $H^1(X_{ks},\mathbb{C})$ factors through a representation of some quotient of type $A_1$. This representation is isomorphic to $E(\omega_1)+ E(\omega_1)$, where $E(\omega_1)$ is the standard irreducible representation of degree $2$ of the Lie algebra of type $A_1$. Hence the variant $A_1$ is excluded by (3.4). Thus $G=\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C})$. We can similarly consider the case when the Abelian variety $X_\eta\otimes_{\kappa(\eta)}\mathbb{C}$ is isogenous to the product of a simple Abelian surface $X_{1\eta}$ and two elliptic curves $X_{2\eta}$, $X_{3\eta}$ without complex multiplication. Here we use the canonical isomorphism $\operatorname{Hom}_{C'}(X'_1,X'_2\times_{C'}X'_3)\,\widetilde{\to}\, \operatorname{Hom}(R_{1\pi'_1\ast}\mathbb{Z}, R_{1\pi'_2\times_{C'}\pi'_3\ast}\mathbb{Z})$ (see [21], Corollary 4.4.13), which in its turn implies that
$$
\begin{equation*}
\operatorname{Hom}_{\operatorname{Lie}G(\mathbb{C})}(H^1(X_{1s},\mathbb{C}), H^1(X_{2s}, \mathbb{C})\oplus H^1(X_{3s},\mathbb{C}))=0.
\end{equation*}
\notag
$$
If the Lie algebra $\operatorname{Lie}G_{X_{1\eta}}(\mathbb{C})$ has type $C_2$, then the Lie algebra $\operatorname{Lie}G(\mathbb{C})$ has type $C_2\times A_1$ when the elliptic curves $X_{2\eta},X_{3\eta}$ are isogenous, or type $C_2\times A_1\times A_1$ otherwise. These variants correspond to the decompositions
$$
\begin{equation*}
\begin{aligned} \, \operatorname{Lie}G(\mathbb{C}) &=\operatorname{Lie}G_{X_{1\eta}}(\mathbb{C})\times \operatorname{Lie}G_{X_{2\eta}}(\mathbb{C}), \\ \operatorname{Lie}G(\mathbb{C}) &=\operatorname{Lie}G_{X_{1\eta}}(\mathbb{C})\times \operatorname{Lie}G_{X_{2\eta}}(\mathbb{C})\times \operatorname{Lie}G_{X_{3\eta}}(\mathbb{C}). \end{aligned}
\end{equation*}
\notag
$$
Hence (3.1), (3.2) imply that $G=\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C})$.
If the Lie algebra $\operatorname{Lie}G_{X_{1\eta}}(\mathbb{C})$ has type $A_1\times A_1$, then the Lie algebra $\operatorname{Lie}G(\mathbb{C})$ has type $A_1\times A_1\times A_1$ when the elliptic curves $X_{2\eta},X_{3\eta}$ are isogenous, or type $A_1\times A_1\times A_1\times A_1$ otherwise because $H^1(X_{1s},\mathbb{C})=E(\omega^{(1)}_1)+E(\omega^{(2)}_1)$ for a Lie algebra of type $A_1\times A_1$. These variants correspond to the decompositions
$$
\begin{equation*}
\begin{aligned} \, \operatorname{Lie}G(\mathbb{C}) &=\operatorname{Lie}G_{X_{1\eta}}(\mathbb{C})\times \operatorname{Lie}G_{X_{2\eta}}(\mathbb{C}), \\ \operatorname{Lie}G(\mathbb{C}) &=\operatorname{Lie}G_{X_{1\eta}}(\mathbb{C})\times \operatorname{Lie}G_{X_{2\eta}}(\mathbb{C})\times \operatorname{Lie}G_{X_{3\eta}}(\mathbb{C}). \end{aligned}
\end{equation*}
\notag
$$
Hence the equality $G=\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C})$ follows from (3.1), (3.2).
If the Lie algebra $\operatorname{Lie}G_{X_{1\eta}}(\mathbb{C})$ has type $A_1$, then the Lie algebra $\operatorname{Lie}G(\mathbb{C})$ has type $A_1\times A_1$ when the elliptic curves $X_{2\eta},X_{3\eta}$ are isogenous, or type $A_1\times A_1\times A_1$ otherwise because $H^1(X_{1s},\mathbb{C})=E(\omega_1)+E(\omega_1)$ for a Lie algebra of type $A_1$. These variants correspond to the decompositions
$$
\begin{equation*}
\begin{aligned} \, \operatorname{Lie}G(\mathbb{C}) &=\operatorname{Lie}G_{X_{1\eta}}(\mathbb{C})\times \operatorname{Lie}G_{X_{2\eta}}(\mathbb{C}), \\ \operatorname{Lie}G(\mathbb{C}) &=\operatorname{Lie}G_{X_{1\eta}}(\mathbb{C})\times \operatorname{Lie}G_{X_{2\eta}}(\mathbb{C})\times \operatorname{Lie}G_{X_{3\eta}}(\mathbb{C}). \end{aligned}
\end{equation*}
\notag
$$
Hence $G=\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C})$.
Furthermore, if the Abelian variety $X_\eta\otimes_{\kappa(\eta)}\mathbb{C}$ is isogenous to the product of four elliptic curves without complex multiplication, then the Lie algebra $\operatorname{Lie}G(\mathbb{C})$ has type $A_1$ (when all these curves are isogenous), type $A_1\times A_1$ (when there are exactly three isogenous elliptic curves or two pairs of isogenous elliptic curves such that the curves in distinct pairs are non-isogenous), type $A_1\times A_1\times A_1$ (when there are exactly two isogeneous elliptic curves), or type $A_1\times A_1\times A_1\times A_1$ (when all the curves are non-isogenous). Therefore $G=\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C})$. In all these cases, $G=\operatorname{Hg}(X_\eta\otimes_{\kappa(\eta)}\mathbb{C})$. Hence, by the Moonen–Zarhin theorem, the $\mathbb{Q}$-space of invariant cycles $H^0(C',R^4\pi'_\ast\mathbb{Q})=H^4(X_s,\mathbb{Q})^{\pi_1(C',s)}$ is generated by the classes of the $\smile$-products of elements in $H^0(C',R^2\pi'_\ast\mathbb{Q})=H^2(X_s,\mathbb{Q})^{\pi_1(C',s)}$. Note that the Hodge $\mathbb{Q}$-structure $H^0(C',R^2\pi'_\ast\mathbb{Q})$ has type $(1,1)$ by Lemma 1.5. Therefore, by the Lefschetz theorem on divisors, it is the image of $\operatorname{NS}(X)\otimes_\mathbb{Z}\mathbb{Q}$ under the canonical surjective map $H^2(X,\mathbb{Q})\to H^0(C',R^2\pi'_\ast\mathbb{Q})$. $\Box$ 3.2.From now on we assume that $\Delta\neq\varnothing$. Let $D^\ast(\delta)$ be a small punctured disc on $C$ centred at a point $\delta\in\Delta$. The Leray spectral sequence for the embedding $j\colon C'\subset C$ yields an exact sequence of mixed Hodge structures ([22], proof of Proposition 12.5, Corollary 13.10, Remark 14.5)
$$
\begin{equation}
0\,{\to}\, H^1(C,j_\ast R^3\pi'_\ast\mathbb{Q})\,{\to}\, H^1(C',R^3\pi'_\ast\mathbb{Q}) \,{\to}\, H^0(C,R^1j_\ast R^3\pi'_\ast\mathbb{Q}) \,{\to}\, H^2(C,j_\ast R^3\pi'_\ast\mathbb{Q}),
\end{equation}
\tag{3.5}
$$
where
$$
\begin{equation*}
H^0(C,R^1j_\ast R^3\pi'_\ast\mathbb{Q})= \bigoplus_{\delta\in\Delta} H^1(D^\ast(\delta),R^3\pi'_\ast\mathbb{Q}) \,\widetilde{\to}\, \bigoplus_{\delta\in\Delta} H^3(X_s,\mathbb{Q})/N_\delta H^3(X_s,\mathbb{Q})
\end{equation*}
\notag
$$
and the space $H^3(X_s,\mathbb{Q})$ ($s\in C'$) has the limiting mixed Hodge structure associated with the local monodromy $\gamma_\delta$ around the point $\delta\in C$ (the Picard–Lefschetz transformation) and $N_\delta=\log \gamma_\delta$. By the theorem on local invariant cycles and (1.3), the sequence (3.5) takes the form
$$
\begin{equation}
0\to H^1(C,R^3\pi_\ast\mathbb{Q})\to H^1(C',R^3\pi'_\ast\mathbb{Q}) \to \bigoplus_{\delta\in\Delta} H^3(X_s,\mathbb{Q})/N_\delta H^3(X_s,\mathbb{Q}) \to 0.
\end{equation}
\tag{3.6}
$$
On the other hand, the following degenerate Leray spectral sequences ([22], Corollary 15.15, [21], Theorem 4.1.1)
$$
\begin{equation*}
E_2^{p,q}(\pi)=H^p(C,R^q\pi_\ast\mathbb{Q}),\qquad E_2^{p,q}(\pi')=H^p(C',R^q\pi'_\ast\mathbb{Q})
\end{equation*}
\notag
$$
yield canonical exact sequences of mixed Hodge $\mathbb{Q}$-structures ([33], (2.4))
$$
\begin{equation}
\begin{aligned} \, 0&\to H^2(C,R^2\pi_\ast\mathbb{Q})\to \operatorname{Ker}[H^4(X,\mathbb{Q})\to H^0(C,R^4\pi_\ast\mathbb{Q})] \\ &\xrightarrow{\alpha_{4X}} H^1(C,R^3\pi_\ast\mathbb{Q})\to 0, \\ 0&\to H^2(C',R^2\pi'_\ast\mathbb{Q})\to \operatorname{Ker}[H^4(X',\mathbb{Q})\to H^0(C',R^4\pi'_\ast\mathbb{Q})] \\ &\xrightarrow{\alpha'_{4X}} H^1(C',R^3\pi'_\ast\mathbb{Q})\to 0 \end{aligned}
\end{equation}
\tag{3.7}
$$
with surjective edge maps $H^4(X,\mathbb{Q})\to H^0(C,R^4\pi_\ast\mathbb{Q})$ ([22], Corollary 15.14) and $H^4(X',\mathbb{Q})\to H^0(C',R^4\pi'_\ast\mathbb{Q})$ ([21], proof of Theorem 4.1.1). By the well-known functoriality of the Leray spectral sequence, the canonical embedding $\iota_{X'/X}\colon X'\hookrightarrow X$ yields homomorphisms $E_2^{p,q}(\pi)\to E_2^{p,q}(\pi')$ which are compatible with differentials and filtrations ([36], § 2.4, [34], Vol. II, Proposition 4.8). Since $H^2(C',R^2\pi'_\ast\mathbb{Q})=0$ (because the cohomology dimension of the affine curve $C'$ equals $1$; see [41], Ch. VI, § 7, Theorem 7.2), we deduce in the case under consideration ([33], § 2.5) that
$$
\begin{equation*}
\begin{gathered} \, H^n(X,\mathbb{Q})=E^n=E^n_0\supset E^n_1\supset E^n_2\supset 0, \\ E^n_2=H^2(C,R^{n-2}\pi_\ast\mathbb{Q})=\operatorname{Ker}[K_{nX}\to H^1(C,R^{n-1}\pi_\ast\mathbb{Q})], \\ E^n_1/E^n_2=H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\qquad E^n_0/E^n_1=H^0(C,R^n\pi_\ast\mathbb{Q}), \\ H^n(X',\mathbb{Q})={E'}^n={E'}^n_0\supset {E'}^n_1\supset 0, \\ {E'}^n_1=H^1(C',R^{n-1}\pi'_\ast\mathbb{Q}), \qquad {E'}^n_0/{E'}^n_1=H^0(C',R^n\pi'_\ast\mathbb{Q}). \end{gathered}
\end{equation*}
\notag
$$
Hence, using the commutative diagram (15.1) in [22], formulae (3.6), (3.7) and the commutativity of the diagram of morphisms we obtain a commutative diagram of mixed Hodge $\mathbb{Q}$-structures with exact rows where the map $\overline{\varphi_4}$ is defined by the formula $x+H^2(C,R^2\pi_\ast\mathbb{Q})\,{\mapsto}\,\varphi_4(x)$. 3.3.For every point $\delta\in\Delta$ we put
$$
\begin{equation*}
\begin{gathered} \, m_\delta :=\operatorname{Card}(\mathcal M_\delta/\mathcal M_\delta^0),\qquad m := \prod_{\delta\in\Delta}m_\delta. \end{gathered}
\end{equation*}
\notag
$$
Fix a prime $p$ not dividing $m$. Write $p^{m!}_{X/C}\colon X-\to X$ for the rational map coinciding on the generic scheme fibre $X_\eta$ of the structure morphism $\pi\colon X\to C$ with the isogeny of multiplication by $p^{m!}$. By the universal property of the Néron model ([17], (1.1.2)), there is a canonical isomorphism
$$
\begin{equation}
\operatorname{End}_C(\mathcal M)\,\widetilde{\to}\, \operatorname{End}_{\kappa(\eta)}(X_\eta).
\end{equation}
\tag{3.9}
$$
Consider the commutative diagram of resolution of indeterminacies of the rational map $p^{m!}_{X/C}$. By Hironaka’s results and (3.9), we may assume that the morphism $\sigma$ is the composite of monoidal transformations along smooth centres and $\sigma|_{\sigma^{-1}(\mathcal M)}\colon \sigma^{-1}(\mathcal M)\to \mathcal M$ is the identity morphism.Let $[p^{m!}_{X/C}]^\ast\colon H^\ast(X,\mathbb{Q}) \xrightarrow{x\,{\mapsto}\,\sigma_\ast\nu^\ast(x)} H^\ast(X,\mathbb{Q})$ be the linear operator determined by the diagram (3.10). The restriction of the map $p^{m!}_{X/C}$ to the Abelian scheme $\pi'\colon X'\to C'$ is clearly a $C'$-isogeny. Hence there is a well-defined linear operator $[p^{m!}_{X'/C'}]^\ast\colon H^4(X',\mathbb{Q})\to H^4(X',\mathbb{Q})$ acting on the finite-dimensional (by (3.6)) subspace
$$
\begin{equation*}
H^1(C',R^3\pi'_\ast\mathbb{Q})\subset H^4(X',\mathbb{Q})
\end{equation*}
\notag
$$
and on the quotient space $H^0(C',R^4\pi'_\ast\mathbb{Q})$ as multiplication by $p^{3m!}$ and $p^{4m!}$ respectively because the isogeny of multiplication by $p^{m!}$ induces multiplication by $p^{m!}$ in the space $H^1(X_s,\mathbb{Q})$ for every fibre $X_s$ of the Abelian scheme $\pi'\colon X'\to C'$ ([2], Lemma 2A3, § 2A11) and $R^n\pi'_\ast\mathbb{Q}=\wedge^nR^1\pi'_\ast\mathbb{Q}$. It follows easily from these properties of the operator $[p^{m!}_{X'/C'}]^\ast$ and from the standard algorithm ([44], Ch. III, § 19, subsection 2) that there exists a canonical splitting
$$
\begin{equation*}
H^4(X',\mathbb{Q})=H^1(C',R^3\pi'_\ast\mathbb{Q})\oplus H^0(C',R^4\pi'_\ast\mathbb{Q}),
\end{equation*}
\notag
$$
which in view of commutativity of the diagram (3.8) yields the canonical splitting
$$
\begin{equation*}
\operatorname{Im}(\varphi_4)=H^1(C,R^3\pi_\ast\mathbb{Q})\oplus H^0(C',R^4\pi'_\ast\mathbb{Q}).
\end{equation*}
\notag
$$
Hence, by (1.1), there is an exact sequence of Hodge $\mathbb{Q}$-structures
$$
\begin{equation}
0\to(i_\Delta f)_\ast\,H^2(Z,\mathbb{Q})\to H^4(X,\mathbb{Q})\to H^1(C,R^3\pi_\ast\mathbb{Q}) \oplus H^0(C',R^4\pi'_\ast\mathbb{Q})\to 0.
\end{equation}
\tag{3.11}
$$
Moreover,
$$
\begin{equation}
H^2(C,R^2\pi_\ast\mathbb{Q})\subset (i_\Delta f)_\ast\,H^2(Z,\mathbb{Q})
\end{equation}
\tag{3.12}
$$
because the restriction map $\varphi_4|_{H^2(C,R^2\pi_\ast\mathbb{Q})}$, which coincides with the canonical map $H^2(C,R^2\pi_\ast\mathbb{Q})\to H^2(C',R^2\pi'_\ast\mathbb{Q})=0$, is trivial. As a result, (3.8) yields a commutative diagram of Hodge $\mathbb{Q}$-structures 3.4.From now on we assume that
$$
\begin{equation*}
\operatorname{End}_{\overline{\kappa(\eta)}\,} \bigl(\mathcal M_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)}\,\bigr) =\mathbb{Z}.
\end{equation*}
\notag
$$
If the Abelian variety $\mathcal M_\eta$ has good reduction at every place $s\in C$, then the standard conjecture $B(X)$ holds ([3], Corollary 11.5). Hence we may assume that $\mathcal M_\eta$ has at least one place of bad reduction, all its bad reductions are semi-stable with toric rank $1$ and the Hodge conjecture on algebraic cycles holds for the product $A_\delta\times A_{\delta'}$ of the Abelian varieties $A_\delta$, $A_{\delta'}$ (the quotients of the connected components of the neutral elements in special fibres of the Néron minimal model modulo the toric parts) for any places $\delta,\delta'\in C$ of bad reductions. (The condition of algebraicity of Hodge cycles holds automatically if the Abelian varieties $A_\delta$, $A_{\delta'}$ are isogenous to products of elliptic curves; see [25], Theorem B.72.) In this case we claim that
$$
\begin{equation}
H^2(C,R^4\pi_\ast\mathbb{Q}) =\operatorname{cl}_X(H)\,{\smile}\, H^2(C,R^2\pi_\ast\mathbb{Q}),
\end{equation}
\tag{3.14}
$$
Indeed, the Hodge group of the generic geometric fibre is a $\mathbb{Q}$-simple algebraic group by Borovoi’s theorem [45] and its Lie algebra has type $C_4$ or $A_1\times A_1\times A_1$ over $\overline{\mathbb{Q}}$. Since the Abelian variety $\mathcal M_\eta$ has trivial trace and the Hodge group of the generic geometric fibre is $\mathbb{Q}$-simple, it follows that invariant cycles on the generic geometric fibre coincide with Hodge cycles and they come from intersections of divisors by Theorem (0.1) in [29]. In particular, the $\mathbb{Q}$-spaces of invariant cycles $H^0(C',R^2\pi'_\ast\mathbb{Q})$ and $H^0(C',R^4\pi'_\ast\mathbb{Q})$ are $1$-dimensional because the classification of the Néron–Severi groups of simple Abelian varieties ([46], § 21) yields an isomorphism $\operatorname{NS}(\mathcal M_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)})\, \widetilde{\to}\,\mathbb{Z}$. As a result, (1.4) implies that the $\mathbb{Q}$-spaces $H^2(C,R^2\pi_\ast\mathbb{Q})$ and $H^2(C,R^4\pi_\ast\mathbb{Q})$ are $1$-dimensional. By (2.1), the strong Lefschetz theorem and the theorem on local invariant cycles ([22], Proposition 15.12, [23], § 3.7), there is an inclusion $\operatorname{cl}_X(H) \smile H^2(C,R^2\pi_\ast\mathbb{Q}) \hookrightarrow H^2(C,R^4\pi_\ast\mathbb{Q})$. This proves (3.14). Finally,
$$
\begin{equation}
\operatorname{cl}_X(H)\,{\smile}\,K_{4X}\subset K_{6X}
\end{equation}
\tag{3.16}
$$
by (2.12). The Poincaré duality on the fibres of the smooth morphism $\pi'\colon X'\to C'$ and the theorem on local invariant cycles yield the equalities
$$
\begin{equation*}
\begin{aligned} \, \dim_\mathbb{Q} H^1(C,R^3\pi_\ast\mathbb{Q}) &=\dim_\mathbb{Q} H^1(C,j_\ast R^3\pi'_\ast\mathbb{Q}) \\ &=\dim_\mathbb{Q} H^1(C,j_\ast R^5\pi'_\ast\mathbb{Q})= \dim_\mathbb{Q} H^1(C,R^5\pi_\ast\mathbb{Q}). \end{aligned}
\end{equation*}
\notag
$$
Therefore, applying (3.16), (2.1) and (3.14), we obtain (3.15). 3.5.The variety $X_{\delta i}$ is the closure of the irreducible component $\mathcal M_{\delta i}$ of the algebraic group $\mathcal M_\delta$ in the Zariski topology of $X$. On the other hand, there is an exact sequence of algebraic groups
$$
\begin{equation}
1\to\operatorname{Gm}\to\mathcal M_\delta^0\xrightarrow{f_\delta} A_\delta\to 0,
\end{equation}
\tag{3.17}
$$
where $A_\delta$ is some Abelian variety of dimension $3$. In what follows we write $\operatorname{alb}_{\delta i}\colon X_{\delta i}\to\operatorname{Alb}(X_{\delta i})$ for the Albanese morphism, which is uniquely determined up to translation on the Abelian variety $\operatorname{Alb}(X_{\delta i})$ ([47], Ch. II, § 3, Theorem 11). It is known ([14], proof of (3.25)) that the morphism $\operatorname{alb}_{\delta i}$ is surjective and
$$
\begin{equation}
\forall\, i \quad\operatorname{Alb}(X_{\delta i})=A_\delta.
\end{equation}
\tag{3.18}
$$
3.6.By property (v) of a Künnemann compactification, the $C$-group law $\mathcal M^0\times_C\mathcal M^0\to \mathcal M^0$ extends to a group $C$-action $\mathcal M^0 \times_C X\to X$. Therefore the group law $\mathcal M^0_\delta\times\mathcal M^0_\delta\to \mathcal M^0_\delta$ extends to a group action $\mathcal M^0_\delta\times X_\delta\to X_\delta$, which endows the irreducible component $X_{\delta i} \subset X_\delta$ with the structure of a contraction product $\mathcal M^0_{\delta}\times^{T_{\mathcal M^0_{\delta}}} Z_{\delta i}$ for some smooth projective toric variety $T_{\mathcal M^0_{\delta}}\hookrightarrow Z_{\delta i}$ ([15], (2)), where $T_{\mathcal M^0_{\delta}}$ is the maximal subtorus of the semi-Abelian variety $\mathcal M^0_{\delta}$ determined by the exact sequence (3.17) of algebraic groups. Recall that $\mathcal M^0_{\delta}$ acts freely on $\mathcal M^0_{\delta}\times Z_{\delta i}$ by the rule $g(w,z)=(gw,g^{-1}z)$, the quotient $[\mathcal M^0_{\delta} \times Z_{\delta i}]/\mathcal M^0_{\delta}$ is a sheaf on $\operatorname{Spec}\mathbb{C}$ with respect to the $fppf$-topology and it is called the contraction product $\mathcal M^0_{\delta}\times^{T_{\mathcal M^0_{\delta}}} Z_{\delta i}$ ([48], Ch. III, § 1, Definition 1.3.1, [16], § 1.19). 3.7.Let $S$ be a scheme, $A$ an Abelian scheme over $S$, $A^\vee$ the dual scheme, and $\mathcal P$ the universal rigid Poincaré line bundle on the scheme $A\times_SA^\vee$. Then $\operatorname{Isom}_{A\times_SA^\vee}(\mathcal O_{A\times_SA^\vee}, \mathcal P)$ is a Poincaré $\operatorname{Gm}$-torsor on $A\times_SA^\vee$ and a bi-extension of the Abelian schemes $A$ and $A^\vee$ by the torus $\operatorname{Gm}$. In particular, over $A^\vee$ it is the universal extension of $A$ by $\operatorname{Gm}$, and over $A$ it is the universal extension of $A^\vee$ by $\operatorname{Gm}$ ([49], § 3). The theory of Poincaré $\operatorname{Gm}$-torsors of Abelian varieties and the Künnemann algorithm ([50], pp. 431, 432) yield an isomorphism
$$
\begin{equation*}
\varphi_{H_l}\colon H^\ast_{\unicode{x00E8}\textrm{t}}(A_\delta,\mathbb{Q}_l)\,\otimes_{\mathbb{Q}_l}\,H^\ast_{\unicode{x00E8}\textrm{t}}(Z_{\delta i},\mathbb{Q}_l)\,\widetilde{\to}\, H^\ast_{\unicode{x00E8}\textrm{t}}(X_{\delta i},\mathbb{Q}_l)
\end{equation*}
\notag
$$
based on constructing a special deformation of the contraction product $X_{\delta i}=\mathcal M^0_{\delta}\times^{T_{\mathcal M^0_{\delta}}}Z_{\delta i}$ to the Cartesian product $A_{\delta}\times Z_{\delta i}$ and on the Künneth decomposition of étale cohomology with coefficients in the field $\mathbb{Q}_l$. Moreover, using the classical Kodaira–Spencer deformation theory [51] instead of the algebraic deformation theory (which was used by Künnemann), we can easily construct an isomorphism of rational Hodge structures
$$
\begin{equation}
\varphi_H\colon H^\ast(A_\delta,\mathbb{Q})\otimes_\mathbb{Q} H^\ast(Z_{\delta i},\mathbb{Q})\,\widetilde{\to}\,H^\ast(X_{\delta i},\mathbb{Q}).
\end{equation}
\tag{3.19}
$$
3.8.There is a commutative diagram of resolution of indeterminacies of the rational map $p^{m!}_{X/C}|_{X_{\delta i}}$, where the morphism $\sigma_{\delta i}$ is the composite of monoidal transformations with smooth centres lying over $X_{\delta i}\setminus\mathcal M_\delta$ ([52], Theorem 0.1.1). Using the following commutative diagram with exact rows of morphisms of algebraic groups ([14], (3.30)) we see that the rational map $p^{m!}_{X/C}|_{Z_{\delta i}}$ is regular on the open subset $\operatorname{Gm}\subset Z_{\delta i}$ of the toric $1$-dimensional variety $Z_{\delta i}=\mathbb P^1$ and it induces multiplication by $p^{m!}$ on the $\mathbb{Q}$-space $H^2(Z_{\delta i},\mathbb{Q})$. Moreover, (3.19) yields a canonical decomposition of rational Hodge structures
$$
\begin{equation}
H^2(A_\delta,\mathbb{Q})\oplus H^2(Z_{\delta i},\mathbb{Q})\,\widetilde{\to}\, H^2(X_{\delta i},\mathbb{Q}).
\end{equation}
\tag{3.20}
$$
We note that this decomposition is compatible with the action of the operator $[p^{m!}_{X/C}|_{X_{\delta i}}]^\ast|_{H^2(X_{\delta i}, \mathbb{Q})}:=[\sigma_{\delta i}]_\ast [\nu_{\delta i}]^\ast|_{H^2(X_{\delta i},\mathbb{Q})}$. This operator acts on the $\mathbb{Q}$-subspace $H^2(A_\delta,\mathbb{Q})\,{=}\,\wedge^2H^1(A_\delta,\mathbb{Q})$ as multiplication by $p^{2m!}$ (in view of the commutative diagram (3.34) in [14]) and on the cohomology $\mathbb{Q}$-subspace $H^2(Z_{\delta i},\mathbb{Q})$ of the toric variety $Z_{\delta i}$ as multiplication by $p^{m!}$ (by the results above). 3.9.By the $p$ we have an equality ([43], § 4.2, [14], § 3.8)
$$
\begin{equation*}
[p^{m!}_{X/C}]^\ast|_{H^2(X,\mathbb{Q})}(\operatorname{cl}_X(X_{\delta i}))= \operatorname{cl}_X(X_{\delta i}).
\end{equation*}
\notag
$$
Therefore, using (2.23) along with [40], Ch. 2, § 8, (5) and the functoriality of our constructions, we obtain
$$
\begin{equation}
[p^{m!}_{X/C}]^\ast(\iota_{X_{\delta i}/X\ast}(\alpha))= [p^{m!}_{X/C}]^\ast(\alpha\,{\smile}\operatorname{cl}_X(X_{\delta i})) =[p^{m!}_{X/C}|_{X_{\delta i}}]^\ast(\alpha)\,{\smile}\operatorname{cl}_X(X_{\delta i}).
\end{equation}
\tag{3.21}
$$
By (1.1), (2.22), (3.20), (3.21) and the results in § 3.8, we have canonical decompositions of Hodge $\mathbb{Q}$-structures and $\mathbb{Q}\bigl[[p^{m!}_{X/C}]^\ast\bigr]$-modules
$$
\begin{equation}
\begin{aligned} \, &(i_\Delta f)_\ast H^2(Z,\mathbb{Q})=\Sigma_1\oplus \Sigma_2 \nonumber \\ &\qquad :=\biggl[\sum_{\delta\in\Delta,\,i} H^2(A_\delta,\mathbb{Q})\,{\smile} \operatorname{cl}_X(X_{\delta i})\biggr]\oplus\biggl[\sum_{\delta\in\Delta,\,i}H^2(Z_{\delta i},\mathbb{Q})\,{\smile}\operatorname{cl}_X(X_{\delta i})\biggr], \end{aligned}
\end{equation}
\tag{3.22}
$$
where the operator $[p^{m!}_{X/C}]^\ast$ acts on the $\mathbb{Q}$-subspace $\Sigma_1$ as multiplication by $p^{2m!}$ and on the $\mathbb{Q}$-subspace $\Sigma_2$ as multiplication by $p^{m!}$. Therefore the results and algorithm in § 3.3, (3.22) and the exact sequence (3.11) yield canonical decompositions of rational Hodge structures and $\mathbb{Q}\bigl[[p^{m!}_{X/C}]^\ast\bigr]$-modules
$$
\begin{equation}
\begin{aligned} \, H^4(X,\mathbb{Q}) &=\Sigma_1\oplus \Sigma_2 \oplus H^1(C,R^3\pi_\ast\mathbb{Q})\oplus H^0(C',R^4\pi'_\ast\mathbb{Q}) \nonumber \\ &=\Sigma_1\oplus \Sigma_2 \oplus H^1(C,R^3\pi_\ast\mathbb{Q})\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q}), \end{aligned}
\end{equation}
\tag{3.23}
$$
where the operator $[p^{m!}_{X/C}]^\ast$ acts canonically on the direct summands as multiplication by $p^{2m!}, p^{m!}, p^{3m!}, p^{4m!}$ respectively. 3.10.Lemma. There is a canonical inclusion Proof. By (1.1), (3.12), (3.22) we have a canonical inclusion
$$
\begin{equation}
H^2(C,R^2\pi_\ast\mathbb{Q})\subset (i_\Delta f)_\ast\,H^2(Z,\mathbb{Q})=\Sigma_1\oplus \Sigma_2.
\end{equation}
\tag{3.24}
$$
On the other hand, the exact sequence (3.7) induces a canonical identification $K_{4X}/H^2(C,R^2\pi_\ast\mathbb{Q})\,\widetilde{\to}\, H^1(C,R^3\pi_\ast\mathbb{Q})$. Using the commutative diagram (3.13) and the decomposition (3.23), we obtain canonical embeddings
$$
\begin{equation*}
\begin{aligned} \, K_{4X} &=\operatorname{Ker}[H^4(X,\mathbb{Q})\to H^0(C,R^4\pi_\ast\mathbb{Q})]\subset \operatorname{Ker}[H^4(X,\mathbb{Q})\to H^0(C',R^4\pi'_\ast\mathbb{Q})] \\ &=\Sigma_1\oplus \Sigma_2 \oplus H^1(C,R^3\pi_\ast\mathbb{Q})=(i_\Delta f)_\ast H^2(Z,\mathbb{Q})\oplus H^1(C,R^3\pi_\ast\mathbb{Q}). \end{aligned}
\end{equation*}
\notag
$$
Therefore in accordance with (3.24) we obtain a canonical decomposition
Since the operator $[p^{m!}_{X/C}]^\ast\colon H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})\to H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})$, acting on the $1$-dimensional (by the results in § 3.4) $\mathbb{Q}$-space $H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})=H^0(C',R^2\pi'_\ast\mathbb{Q})$, is the homothety with coefficient $p^{2m!}$, we obtain the following canonical decompositions from the strong Lefschetz theorem, the theorem on local invariant cycles (saying that the canonical map $R^n\pi_\ast\mathbb{Q}\to j_\ast R^n\pi'_\ast\mathbb{Q}$ is surjective and its kernel is supported on a finite set $\Delta$; see [23], § 3.7, [22], Proposition 15.12) and the decompositions (3.23):
$$
\begin{equation}
\begin{aligned} \, H^6(X,\mathbb{Q}) &=[\Sigma_1\,{\smile}\operatorname{cl}_X(H)]\oplus [\Sigma_2\,{\smile}\operatorname{cl}_X(H)]\oplus H^1(C,R^5\pi_\ast\mathbb{Q}) \oplus H^0(C',R^6\pi'_\ast\mathbb{Q}) \nonumber \\ &=[\Sigma_1\,{\smile}\,H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})]\oplus [\Sigma_2\,{\smile}\,H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})] \nonumber \\ &\qquad\oplus H^1(C,R^5\pi_\ast\mathbb{Q})\oplus H^0(C,j_\ast R^6\pi'_\ast\mathbb{Q}), \end{aligned}
\end{equation}
\tag{3.26}
$$
where the operator $[p^{m!}_{X/C}]^\ast$ acts on the direct summands as multiplication by $p^{4m!}$, $p^{3m!}$, $p^{5m!}$, $p^{6m!}$ respectively in view of the results in § 3.9.
There is an exact sequence of Hodge $\mathbb{Q}$-structures ([33], (2.4))
$$
\begin{equation}
0\to H^2(C,R^{10}(\tau\sigma)_\ast\mathbb{Q})\to K_{12Y} \xrightarrow{\alpha_{12Y}} H^1(C,R^{11}(\tau\sigma)_\ast\mathbb{Q})\to 0.
\end{equation}
\tag{3.27}
$$
By the theorem on local invariant cycles, it takes the form
$$
\begin{equation}
0\to H^2(C,j_\ast R^{10}\tau'_\ast\mathbb{Q})\to K_{12Y} \xrightarrow{\alpha_{12Y}} H^1(C,j_\ast R^{11}\tau'_\ast\mathbb{Q})\to 0.
\end{equation}
\tag{3.28}
$$
Using (3.25), (3.28) and the functoriality of our constructions, we obtain the following commutative diagram of Hodge $\mathbb{Q}$-structures with exact rows: where the cohomology map $[p'_{2\ast}]_k\colon H^k(C,j_\ast R^{8+l}\tau'_\ast\mathbb{Q})\to H^k(C,j_\ast R^l\pi'_\ast\mathbb{Q})$ is induced by the canonical map of sheaves $p'_{2\ast}\colon R^{8+l}\tau'_\ast\mathbb{Q}\to R^l\pi'_\ast\mathbb{Q}$. $\Box$ 3.11.Lemma. The algebraic class $(\iota\sigma)_\ast\bigl[[\operatorname{cl}_Y(D^{(1)})]^{\smile\,3}\bigr]\in H^8(X\times X,\mathbb{Q})$ determines an algebraic isomorphism
$$
\begin{equation}
H^1(C,R^5\pi_\ast\mathbb{Q})\underset{\widetilde{\qquad}}{\xrightarrow{x\,{\mapsto} \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast x\,{\smile}\,(\iota\sigma)_\ast[[\operatorname{cl}_Y(D^{(1)})]^{\smile\,3}])}} H^1(C,R^3\pi_\ast\mathbb{Q}).
\end{equation}
\tag{3.30}
$$
Proof. Using the functoriality of our constructions, (2.13), (3.29) and the commutative diagram where the map $K_{6Y}\xrightarrow{\smile\operatorname{cl}_Y(D^{(1)})^{\smile\,3}}K_{12Y}$ is defined in view of the equivalence (2.11) and the equality
$$
\begin{equation*}
\iota_{Y_s/Y}^\ast(\operatorname{cl}_Y(D^{(1)})^{\smile\,3}\,{\smile}\,K_{6Y}) =\iota_{Y_s/Y}^\ast(\operatorname{cl}_Y(D^{(1)})^{\smile\,3})\,{\smile}\, \iota_{Y_s/Y}^\ast(K_{6Y})=0
\end{equation*}
\notag
$$
([40], Ch. 2, § 8, (5)), we obtain a commutative diagram which is glued from the commutative diagrams and
On the other hand, the following canonical embedding of Hodge $\mathbb{Q}$-structures holds in view of (3.15), (3.25), the strong Lefschetz theorem and the theorem on local invariant cycles:
$$
\begin{equation*}
\begin{aligned} \, H^1(C,R^5\pi_\ast\mathbb{Q}) &=H^1(C,j_\ast R^3\pi'_\ast\mathbb{Q})\,{\smile}\, H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q}) \\ &=H^1(C,R^3\pi_\ast\mathbb{Q})\,{\smile}\operatorname{cl}_X(H)\hookrightarrow K_{6X}. \end{aligned}
\end{equation*}
\notag
$$
Therefore the diagrams (3.31)–(3.33) yield a commutative diagram Since the composite of the maps in the bottom row of (3.34) is the isomorphism (2.10), it is clear that the composite of the maps in the top row of (3.34) is the isomorphism
$$
\begin{equation}
H^1(C,R^5\pi_\ast\mathbb{Q})\underset{\widetilde{\qquad}}{\xrightarrow{x\,{\mapsto}\, (p_2\sigma)_\ast((p_1\sigma)^\ast x\,{\smile}\operatorname{cl}_Y(D^{(1)})^{\smile\,3})}} H^1(C,R^3\pi_\ast\mathbb{Q}).
\end{equation}
\tag{3.35}
$$
For any element $x\in H^6(X,\mathbb{Q})$ the projection formula ([2], § 1.2.A) yields that
$$
\begin{equation*}
\begin{aligned} \, &(p_2\sigma)_\ast\bigl((p_1\sigma)^\ast x\,{\smile}\,[\operatorname{cl}_Y(D^{(1)})]^{\smile\,3}\bigr)= [\operatorname{pr}_2\iota\sigma]_\ast \bigl([\operatorname{pr}_1\iota\sigma]^\ast x\,{\smile}\,[\operatorname{cl}_Y(D^{(1)})]^{\smile\,3}\bigr) \\ &\ =\operatorname{pr}_{2\ast}(\iota\sigma)_\ast \bigl((\iota\sigma)^\ast\operatorname{pr}_1^\ast x\,{\smile}\,[\operatorname{cl}_Y(D^{(1)})]^{\smile\,3}\bigr)= \operatorname{pr}_{2\ast}\bigl(\operatorname{pr}_1^\ast x\,{\smile}\,(\iota\sigma)_\ast\bigl[[\operatorname{cl}_Y(D^{(1)})]^{\smile\,3}\bigr]\bigr). \end{aligned}
\end{equation*}
\notag
$$
Therefore (3.35) takes the form (3.30). $\Box$ § 4. End of the proof of the theorem4.1.Let
$$
\begin{equation*}
\begin{gathered} \, u_{\Sigma_1,\Sigma_1},\quad u_{\Sigma_1,\Sigma_2},\quad u_{\Sigma_1,H^1(C,R^3\pi_\ast\mathbb{Q})},\quad u_{\Sigma_1,H^0(C',R^4\pi'_\ast\mathbb{Q})}, \\ u_{\Sigma_2,\Sigma_1},\quad u_{\Sigma_2,\Sigma_2},\quad u_{\Sigma_2,H^1(C,R^3\pi_\ast\mathbb{Q})},\quad u_{\Sigma_2,H^0(C',R^4\pi'_\ast\mathbb{Q})}, \\ u_{H^1(C,R^3\pi_\ast\mathbb{Q}),\Sigma_1},\quad u_{H^1(C,R^3\pi_\ast\mathbb{Q}),\Sigma_2},\quad u_{H^1(C,R^3\pi_\ast\mathbb{Q}),H^1(C,R^3\pi_\ast\mathbb{Q})}, \\ u_{H^1(C,R^3\pi_\ast\mathbb{Q}),H^0(C',R^4\pi'_\ast\mathbb{Q})},\quad u_{H^0(C',R^4\pi'_\ast\mathbb{Q}),\Sigma_1},\quad u_{H^0(C',R^4\pi'_\ast\mathbb{Q}),\Sigma_2}, \\ u_{H^0(C',R^4\pi'_\ast\mathbb{Q}),H^1(C,R^3\pi_\ast\mathbb{Q})},\quad u_{H^0(C',R^4\pi'_\ast\mathbb{Q}),H^0(C',R^4\pi'_\ast\mathbb{Q})},\quad h \end{gathered}
\end{equation*}
\notag
$$
be the components of the algebraic correspondence $u:=(\iota\sigma)_\ast\bigl[[\operatorname{cl}_Y(D^{(1)})]^{\smile\,3}\bigr]$ in the direct summands
$$
\begin{equation*}
\begin{gathered} \, \Sigma_1\otimes_\mathbb{Q} \Sigma_1,\quad \dots, \quad H^0(C',R^4\pi'_\ast\mathbb{Q})\otimes_\mathbb{Q} H^0(C',R^4\pi'_\ast\mathbb{Q}), \\ H_\mathbb{Q}:=\bigoplus_{p+q=8,\,p\neq 4} H^p(X,\mathbb{Q})\otimes_\mathbb{Q} H^q(X,\mathbb{Q}), \end{gathered}
\end{equation*}
\notag
$$
determined by decompositions (3.23) and by the Künneth decomposition of the $\mathbb{Q}$-space $H^8(X\times X,\mathbb{Q})$. It is clear that the operators
$$
\begin{equation*}
\begin{aligned} \, [p_{X/C}^{m!}]^\ast\otimes_\mathbb{Q}[1_{X/C}]^\ast &=[\sigma_\ast\nu^\ast]\otimes_\mathbb{Q} [1_{X/C}]^\ast, \\ [1_{X/C}]^\ast\otimes_\mathbb{Q}[p_{X/C}^{m!}]^\ast &=[1_{X/C}]^\ast\otimes_\mathbb{Q}[\sigma_\ast\nu^\ast] \end{aligned}
\end{equation*}
\notag
$$
transform the $\mathbb{Q}$-subspace $H_\mathbb{Q}\subset H^8(X\times X,\mathbb{Q})$ to the space $H_\mathbb{Q}$ and send algebraic cohomology classes to algebraic classes ([2], Proposition 1.3.7). By the results in § 3.9, the operator $[p_{X/C}^{m!}]^\ast\colon H^4(X,\mathbb{Q})\to H^4(X,\mathbb{Q})$ acts on the direct summands of the decompositions (3.23) as multiplication by $p^{2m!}$, $p^{m!}$, $p^{3m!}$, $p^{4m!}$ respectively. Acting by the operators $[p_{X/C}^{m!}]^\ast\otimes_\mathbb{Q}[1_{X/C}]^\ast$, $[1_{X/C}]^\ast\otimes_\mathbb{Q}[p_{X/C}^{m!}]^\ast$ on the algebraic class $u$ and using the modified Lieberman method ([2], § 2A11) (adapted to the case when the multiplication by $p^{m!}$ on the generic scheme fibre determines a rational map of $X$), we easily verify that there are elements $h_j\in H$ such that the classes
$$
\begin{equation*}
\begin{gathered} \, u_{\Sigma_1,\Sigma_1}+h_1,\quad u_{\Sigma_1,\Sigma_2}+h_2,\quad u_{\Sigma_1,H^1(C,R^3\pi_\ast\mathbb{Q})}+h_3,\quad u_{\Sigma_1,H^0(C',R^4\pi'_\ast\mathbb{Q})}+h_4, \\ u_{\Sigma_2,\Sigma_1}+h_5,\quad u_{\Sigma_2,\Sigma_2}+h_6,\quad u_{\Sigma_2,H^1(C,R^3\pi_\ast\mathbb{Q})}+h_7,\quad u_{\Sigma_2,H^0(C',R^4\pi'_\ast\mathbb{Q})}+h_8, \\ u_{H^1(C,R^3\pi_\ast\mathbb{Q}),\Sigma_1}+h_9,\quad u_{H^1(C,R^3\pi_\ast\mathbb{Q}),\Sigma_2}+h_{10}, \\ u_{H^1(C,R^3\pi_\ast\mathbb{Q}),H^1(C,R^3\pi_\ast\mathbb{Q})}+h_{11},\quad u_{H^1(C,R^3\pi_\ast\mathbb{Q}),H^0(C',R^4\pi'_\ast\mathbb{Q})}+h_{12}, \\ u_{H^0(C',R^4\pi'_\ast\mathbb{Q}),\Sigma_1}+h_{13},\quad u_{H^0(C',R^4\pi'_\ast\mathbb{Q}),\Sigma_2}+h_{14}, \\ u_{H^0(C',R^4\pi'_\ast\mathbb{Q}),H^1(C,R^3\pi_\ast\mathbb{Q})}+h_{15},\quad u_{H^0(C',R^4\pi'_\ast\mathbb{Q}),H^0(C',R^4\pi'_\ast\mathbb{Q})}+h_{16} \end{gathered}
\end{equation*}
\notag
$$
are algebraic. 4.2.By the definition of the direct image of cohomology we have ([35], (1.2))
$$
\begin{equation*}
\operatorname{pr}_{2\ast}(H^i(X,\mathbb{Q})\otimes_\mathbb{Q} H^\ast(X,\mathbb{Q}))=0 \quad\text{for all } \quad i\neq 10.
\end{equation*}
\notag
$$
Hence the correspondences in the $\mathbb{Q}$-space $H_\mathbb{Q}$ annihilate the $\mathbb{Q}$-space $H^6(X,\mathbb{Q})$. The strong Lefschetz theorem for $X_{\delta i}$ yields the existence of an embedding
$$
\begin{equation*}
H^2(X_{\delta i},\mathbb{Q})\,{\smile}\,\iota^\ast_{X_{\delta i}/X}\operatorname{cl}_X(H) \subset H^4(X_{\delta i},\mathbb{Q}).
\end{equation*}
\notag
$$
Therefore the projection formula ([2], § 1.2.A) and (2.23) yield an inclusion
$$
\begin{equation*}
\iota_{X_{\delta i}/X\ast}H^2(X_{\delta i},\mathbb{Q})\,{\smile} \operatorname{cl}_X(H)\subset \iota_{X_{\delta i}/X\ast}H^4(X_{\delta i},\mathbb{Q}) =H^4(X_{\delta i},\mathbb{Q})\,{\smile}\operatorname{cl}_X(X_{\delta i}),
\end{equation*}
\notag
$$
so that (2.22) implies the existence of a canonical inclusion
$$
\begin{equation}
(i_\Delta f)_\ast H^2(Z,\mathbb{Q})\smile\,\operatorname{cl}_X(H)\subset \sum_{\delta,i} H^4(X_{\delta i},\mathbb{Q})\,{\smile}\operatorname{cl}_X(X_{\delta i}).
\end{equation}
\tag{4.1}
$$
On the other hand, it follows from the theorem on local invariant cycles, (2.22), (2.23) and (2.25) that
$$
\begin{equation*}
\begin{aligned} \, &H^1(C,R^5\pi_\ast\mathbb{Q})\,{\smile}\,(i_\Delta f)_\ast H^2(Z,\mathbb{Q}) \\ &\qquad\subset H^1(C,j_\ast R^5\pi'_\ast\mathbb{Q})\,{\smile}\, \sum_{\delta,\,i}H^2(X_{\delta i},\mathbb{Q})\,{\smile}\operatorname{cl}_X(X_{\delta i})=0. \end{aligned}
\end{equation*}
\notag
$$
Finally,
$$
\begin{equation*}
\begin{aligned} \, &H^1(C,R^5\pi_\ast\mathbb{Q})\,{\smile}\,H^0(C',R^4\pi'_\ast\mathbb{Q}) \\ &\qquad=H^1(C,j_\ast R^5\pi'_\ast\mathbb{Q})\,{\smile}\,H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q})\subset H^1(C,j_\ast R^9\pi'_\ast\mathbb{Q})=0. \end{aligned}
\end{equation*}
\notag
$$
Thus, by (3.22) and Lemma 3.11, the algebraic correspondence
$$
\begin{equation*}
\begin{aligned} \, &u_{H^1(C,R^3\pi_\ast\mathbb{Q}),\Sigma_1}+h_9+u_{H^1(C,R^3\pi_\ast\mathbb{Q}),\Sigma_2}+h_{10} \\ &\qquad+u_{H^1(C,R^3\pi_\ast\mathbb{Q}),H^1(C,R^3\pi_\ast\mathbb{Q})}+h_{11} +u_{H^1(C,R^3\pi_\ast\mathbb{Q}),H^0(C',R^4\pi'_\ast\mathbb{Q})}+h_{12} \end{aligned}
\end{equation*}
\notag
$$
yields an algebraic isomorphism $H^1(C,R^5\pi_\ast\mathbb{Q})\,\widetilde{\to}\,H^1(C,R^3\pi_\ast\mathbb{Q})$. Moreover, this correspondence annihilates the direct summands
$$
\begin{equation*}
\Sigma_1\,{\smile}\operatorname{cl}_X(H), \quad \Sigma_2\,{\smile}\operatorname{cl}_X(H), \quad H^0(C,j_\ast R^6\pi'_\ast\mathbb{Q})
\end{equation*}
\notag
$$
in the decompositions (3.26) of the $\mathbb{Q}$-space $H^6(X,\mathbb{Q})$ because (2.22), (2.23) and (2.25) yield that
$$
\begin{equation*}
\begin{aligned} \, &(i_\Delta f)_\ast H^2(Z,\mathbb{Q})\,{\smile}\operatorname{cl}_X(H) \,{\smile}\,H^1(C,R^3\pi_\ast\mathbb{Q}) \\ &\qquad\subset \sum_{\delta,\,i}H^2(X_{\delta i},\mathbb{Q})\,{\smile}\operatorname{cl}_X(X_{\delta i})\,{\smile}\operatorname{cl}_X(H) \,{\smile}\,H^1(C,j_\ast R^3\pi'_\ast\mathbb{Q})=0, \\ &H^0(C,j_\ast R^6\pi'_\ast\mathbb{Q})\,{\smile}\,H^1(C,R^3\pi_\ast\mathbb{Q}) \\ &\qquad\subset H^0(C,j_\ast R^6\pi'_\ast\mathbb{Q})\,{\smile}\,H^1(C,j_\ast R^3\pi'_\ast\mathbb{Q})\subset H^1(C,j_\ast R^9\pi'_\ast\mathbb{Q})=0. \end{aligned}
\end{equation*}
\notag
$$
As a result, we have the following equality:
$$
\begin{equation}
\bigl[[(i_\Delta f)_\ast H^2(Z,\mathbb{Q})\,{\smile}\operatorname{cl}_X(H)]\oplus H^0(C,j_\ast R^6\pi'_\ast\mathbb{Q})\bigr] \,{\smile}\,H^1(C,R^3\pi_\ast\mathbb{Q})=0.
\end{equation}
\tag{4.2}
$$
Combining these facts with the non-degeneracy ([2], § 1.2.A) of the bilinear form
$$
\begin{equation*}
\Phi\colon H^4(X,\mathbb{Q})\times H^4(X,\mathbb{Q})\xrightarrow{x\times y\,{\mapsto}\,x\,{\smile}\,y\,{\smile}\operatorname{cl}_X(H)} H^{10}(X,\mathbb{Q})=\mathbb{Q}(-5)\underset{\widetilde{\qquad}}{\xrightarrow{(2\pi i)^5}}\mathbb{Q}
\end{equation*}
\notag
$$
and the non-degeneracy ([22], Proposition 10.5) of the restriction
$$
\begin{equation*}
\Psi_1 := \Phi|_{H^1(C,R^3\pi_\ast\mathbb{Q})}\colon H^1(C,R^3\pi_\ast\mathbb{Q})\times H^1(C,R^3\pi_\ast\mathbb{Q})\xrightarrow{x\times y\,{\mapsto}\,\langle x\,{\smile}\,y\,{\smile}\operatorname{cl}_X(H)\rangle}\mathbb{Q},
\end{equation*}
\notag
$$
we obtain the non-degeneracy of the restriction $\Psi_2$ of the form $\Phi$ to the $\mathbb{Q}$-subspace
$$
\begin{equation*}
(i_\Delta f)_\ast H^2(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q}) =\Sigma_1\oplus \Sigma_2\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q}),
\end{equation*}
\notag
$$
where the $\mathbb{Q}$-space $\Sigma_2 \oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q})$ is generated by algebraic classes in view of the definition (3.22) because the $\mathbb{Q}$-space $H^2(Z_{\delta i},\mathbb{Q})$ for the toric variety $Z_{\delta i}$ is generated by algebraic classes ([53], Proposition 10.4) and the $\mathbb{Q}$-space $H^0(C',R^4\pi'_\ast\mathbb{Q})$ is generated by the images of the classes of intersections of divisors on $X$ under the canonical surjective map $H^4(X,\mathbb{Q})\to H^0(C',R^4\pi'_\ast\mathbb{Q})$ by Lemma 3.1. 4.3.Considering, if necessary, the base change $X\times_C\widetilde{C}\to \widetilde{C}$ determined by an appropriate ramified covering $\widetilde{C}\to C$ of sufficiently large degree unramified over points of bad reductions, and taking into account that the canonical projection $X\times_C\widetilde{C}\to X$ is a surjective morphism of smooth projective varieties (and, therefore, $B(X\times_C\widetilde{C})\Rightarrow B(X)$; see [3], Theorem 1.6), we may assume that
$$
\begin{equation*}
\dim_\mathbb{Q}[\Sigma_2\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q})]\geqslant 3.
\end{equation*}
\notag
$$
In this case it follows from the results in § 1.2 of [43] that we have a well-defined algebraic Poincaré class
$$
\begin{equation*}
\wp(\Sigma_2\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q}))
\end{equation*}
\notag
$$
and a well-defined Poincaré class (a priori not necessarily algebraic)
$$
\begin{equation*}
\wp\bigl((i_\Delta f)_\ast H^2(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q})\bigr)
\end{equation*}
\notag
$$
that generates the $1$-dimensional space of invariants of the diagonal action of the group
$$
\begin{equation*}
\operatorname{SO}\bigl(\bigl((i_\Delta f)_\ast H^2(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q})\bigr),\Psi_2\bigr)
\end{equation*}
\notag
$$
on the tensor square of the $\mathbb{Q}$-space $(i_\Delta f)_\ast H^2(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q})$. For the same reasons, we may assume that and, therefore, we have a well-defined Poincaré class
$$
\begin{equation*}
\wp(\Sigma_1)=\wp\biggl(\sum_{\delta\in\Delta,\, i} H^2(A_\delta,\mathbb{Q})\,{\smile}\, \operatorname{cl}_X(X_{\delta i})\biggr).
\end{equation*}
\notag
$$
4.4.Since the Poincaré class
$$
\begin{equation*}
\wp(\Sigma_1)\in\biggl(\sum_{\delta\in\Delta,\, i} H^2(A_\delta,\mathbb{Q})\,{\smile}\, \operatorname{cl}_X(X_{\delta i})\biggr)\otimes_\mathbb{Q}\biggl(\sum_{\delta\in\Delta,\, i} H^2(A_\delta,\mathbb{Q})\,{\smile}\,\operatorname{cl}_X(X_{\delta i})\biggr)
\end{equation*}
\notag
$$
is a Hodge cycle ([43], § 1.2), it is algebraic by the hypothesis of the theorem on the algebraicity of Hodge cycles on the products of type $A_\delta\times A_{\delta'}$. Using the standard arguments ([43], §§ 1.2, 3.5), we may assume that the following equality of Poincaré classes holds:
$$
\begin{equation*}
\wp\bigl((i_\Delta f)_\ast H^2(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q})\bigr)=\wp(\Sigma_1)+\wp\bigl(\Sigma_2\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q})\bigr),
\end{equation*}
\notag
$$
so that all Poincaré classes under consideration are algebraic. On the other hand, the decompositions (3.23) and the non-degeneracy of the symmetric bilinear forms $\Psi_1$, $\Psi_2$ yield a canonical embedding of algebraic groups
$$
\begin{equation*}
\begin{aligned} \, &\operatorname{SO}\bigl(H^1(C,R^3\pi_\ast\mathbb{Q}),\Psi_1\bigr) \times \operatorname{SO}\bigl((i_\Delta f)_\ast H^2(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q}),\Psi_2\bigr) \\ &\qquad \hookrightarrow \operatorname{SO}\bigl(H^4(X,\mathbb{Q}),\Phi\bigr), \end{aligned}
\end{equation*}
\notag
$$
which in turn yields an inclusion
$$
\begin{equation}
\begin{aligned} \, &\mathbb{Q}\cdot\wp\bigl(H^4(X,\mathbb{Q})\bigr)\subset H^1(C,R^3\pi_\ast\mathbb{Q})\otimes_\mathbb{Q} H^1(C,R^3\pi_\ast\mathbb{Q}) \nonumber \\ &\qquad+\mathbb{Q}\cdot\wp\bigl((i_\Delta f)_\ast H^2(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q})\bigr). \end{aligned}
\end{equation}
\tag{4.3}
$$
Since the correspondence $\wp(H^4(X,\mathbb{Q}))$ determines an isomorphism ([13], § 1.2)
$$
\begin{equation*}
H^6(X,\mathbb{Q})\underset{\widetilde{\qquad}}{\xrightarrow{x\,{\mapsto} \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast(x)\,{\smile}\,\wp(H^4(X,\mathbb{Q})))}} H^4(X,\mathbb{Q}),
\end{equation*}
\notag
$$
it follows from (3.26), (4.2), (4.3) that the algebraic Poincaré class
$$
\begin{equation*}
\wp\bigl((i_\Delta f)_\ast H^2(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q})\bigr)
\end{equation*}
\notag
$$
determines an algebraic isomorphism
$$
\begin{equation*}
\begin{aligned} \, &(i_\Delta f)_\ast H^2(Z,\mathbb{Q})\,{\smile}\operatorname{cl}_X(H)\oplus H^0(C,j_\ast R^6\pi'_\ast\mathbb{Q}) \\ &\qquad\widetilde{\to}\, (i_\Delta f)_\ast H^2(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q}). \end{aligned}
\end{equation*}
\notag
$$
Therefore the algebraic correspondence
$$
\begin{equation*}
\begin{aligned} \, &u_{H^1(C,R^3\pi_\ast\mathbb{Q}),\Sigma_1}+h_9+u_{H^1(C,R^3\pi_\ast\mathbb{Q}),\Sigma_2}+h_{10} +u_{H^1(C,R^3\pi_\ast\mathbb{Q}),H^1(C,R^3\pi_\ast\mathbb{Q})}+h_{11} \\ &\ +u_{H^1(C,R^3\pi_\ast\mathbb{Q}),H^0(C',R^4\pi'_\ast\mathbb{Q})}+h_{12} +\wp\bigl((i_\Delta f)_\ast H^2(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^4\pi'_\ast\mathbb{Q})\bigr) \end{aligned}
\end{equation*}
\notag
$$
yields an algebraic isomorphism $H^6(X,\mathbb{Q})\,\widetilde{\to}\,H^4(X,\mathbb{Q})$. The theorem is proved. The author is grateful to the referee, whose suggestions significantly improved the original text.
Citation in format AMSBIB
\Bibitem{Tan22} |
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