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Toric geometry and the standard conjecture for a compactification of the Néron model of Abelian variety over 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 any natural number $i\leqslant d$, 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 defines the inverse algebraic isomorphism
$$
\begin{equation*}
H^{2d-i}(X,\mathbb{Q})\xrightarrow[\widetilde{\qquad}]{x\,{\mapsto} \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast x \smile \operatorname{cl}_{X\times X}(Z))} H^i(X,\mathbb{Q}).
\end{equation*}
\notag
$$
It is known that the Lefschetz theorem on $(1,1)$-classes implies the existence of an algebraic isomorphism $H^{2d-1}(X,\mathbb{Q})\,\widetilde{\to}\, H^1(X,\mathbb{Q})$. Besides, the conjecture $B(X)$ is equivalent to the algebraicity of the Hodge star operator (see [2], Proposition 2.3). Examples of varieties satisfying the standard conjecture and some consequences of the conjecture can be found in [2]–[14]. We will be concerned with the study of the standard conjecture $B(X)$ for compactifications of Néron models of Abelian varieties over the field of rational functions of a smooth projective curve. Recall that, by the André theorem (see Theorem 0.6.2 in [15]), the Hodge conjecture for all complex Abelian varieties is a consequence of the standard conjecture $B(X)$ for all Abelian schemes $\pi\colon X\to C$ over smooth projective curves; on the other hand, the Hodge conjecture for all complex Abelian varieties of CM-type implies the Grothendieck standard conjecture of Hodge type (see [1], Conjecture 2) for Abelian varieties in arbitrary characteristic (see Theorem 3.3 in [16]) and the Tate conjectures on algebraic cycles and poles of Hasse–Weil zeta functions for all Abelian varieties over finite fields (see Theorem 7.1 in [17]). Consider the Néron minimal model $\mathcal M\to C$ of the Abelian variety $\mathcal M_\eta$ over the field $\kappa(\eta)$ of rational functions of a smooth projective curve $C$. After the base change defined by an appropriate ramified covering $\widetilde{C}\to C$, we may assume, in virtue of Künnemann results (see [18], § 5.8, and [19], § § 1.9, 4.1, 4.2, 4.4, 4.5, Theorem 4.6) that, for the Néron minimal model $\mathcal M\to C$, there exists a smooth compactification $X$ of the variety $\mathcal M$ which 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 $\pi\vert_{\mathcal M}\colon \mathcal M\to C$ 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 $r_s$ (in what follows, the number $r_s$ is called the toric rank; by definition, a reduction at the place $s$ has tmultiplicative type if $\mathcal M^0_s$ is a linear torus); (v) the $C$-group law $\mathcal M^0\times_C\mathcal M^0\to \mathcal M^0$ expands to the $C$-group action $\mathcal M^0\times_C X\to X$. Such compactifications of the Néron model will be called Künnemann compactifications. Let $\Delta\,\subset\,C$ be the set of all places where the generic scheme fibre of the Künnemann compactification has bad semi-stable reductions. For a place $\delta\in \Delta$, let $A_\delta$ be the factorgroup of the connected component $\mathcal M_\delta^0$ of the fibre of the minimal Néron model modulo it’s toric part. It is known that
$$
\begin{equation*}
A_\delta=\operatorname{Alb}(\overline{\mathcal M_\delta^0})
\end{equation*}
\notag
$$
is the Albanese variety of the Zariski closure of the subvariety $\mathcal M_\delta^0 \subseteq X$ (see [14], formula (3.25)). We set
$$
\begin{equation*}
\operatorname{End}^0_\mathbb{C}(A_\delta)=\operatorname{End}_\mathbb{C}(A_\delta)\otimes_\mathbb{Z}\mathbb{Q}.
\end{equation*}
\notag
$$
By definition, the Abelian variety $\mathcal M_\eta$ has a 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 paper, we prove the following main result. Theorem. Let $\mathcal M\to C$ be the Néron minimal model of a principally polarized $(d-1)$-dimensional Abelian variety $\mathcal M_\eta$ over the field $\kappa(\eta)$ of rational functions of a smooth projective curve $C$,
$$
\begin{equation*}
\operatorname{End}_{\overline{\kappa(\eta)}} (\mathcal M_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)})=\mathbb Z,
\end{equation*}
\notag
$$
any bad reduction of the Abelian variety $\mathcal M_\eta$ is semi-stable. Assume that the following conditions hold: (A) for any embedding of fields $\kappa(\eta)\,\hookrightarrow\,\mathbb {C}$, the complexification
$$
\begin{equation*}
\operatorname{Lie}\operatorname{Hg}(\mathcal M_\eta\otimes_{\kappa(\eta)}\mathbb{C})\otimes_\mathbb{Q}\mathbb{C}
\end{equation*}
\notag
$$
of the Lie algebra of the Hodge group of the Abelian variety $\mathcal M_\eta\otimes_{\kappa(\eta)}\mathbb {C}$ is a simple Lie algebra of type $C_{d-1}$$;$ this condition holds automatically if
$$
\begin{equation*}
\begin{aligned} \, &d-1\notin \operatorname{Ex}(1) \\ &\quad\stackrel{\mathrm{def}}{=} \biggl\{4^l,\frac12 \begin{pmatrix} 4l+2 \\ 2l+1 \end{pmatrix}^{2m-1}, 2^{8lm+4l-4m-3},4^l(m+1)^{2l+1}\biggm| l,m\in\mathbb{N}^+=\mathbb{N}_{\geqslant 1} \biggr\} \\ &\,\quad =\{4,10,16,32,64,108,126,256,500,512,864,1024,1372,1716,2048,\dots\}; \end{aligned}
\end{equation*}
\notag
$$
(B) for any places $\delta,\delta'$ of bad reductions the $\mathbb Q$-space of Hodge cycles on the product $\operatorname{Alb}(\overline{\mathcal M_\delta^0})\,\times \, \operatorname{Alb}(\overline{\mathcal M_{\delta'}^0})$ of Albanese varieties is generated by classes of algebraic cycles. Then there exists a finite ramified covering $\widetilde{C}\to C$ such that, for any Künnemann compactification $\widetilde{X}$ of the Néron minimal model of the Abelian variety $\mathcal M_\eta\otimes_{\kappa(\eta)}\kappa(\widetilde{\eta})$, the Grothendieck standard conjecture $B(\widetilde{X})$ of Lefschetz type is true. Remark. Condition (B) of the theorem holds automatically in each of the following cases: 1) for any place $\delta$ of bad reduction, the Abelian variety $A_\delta=\operatorname{Alb}(\overline{\mathcal M_\delta^0})$ is stably non-degenerate in Hazama’s sense and contains no simple factors of type IV in Albert’s classification [20] (this holds, for example, if $\operatorname{End}_\mathbb {C}(A_\delta)=\mathbb Z$ and $\operatorname{Lie}\operatorname{Hg}(A_\delta)\otimes_\mathbb Q\mathbb {C}$ is a simple Lie algebra of type $C_{\dim_\mathbb {C} A_\delta}$); 2) every bad reduction of the Abelian variety $\mathcal M_\eta$ has multiplicative type. § 1. Some notation and preliminary results1.1.We first note that if $\operatorname{End}_{\overline{\kappa(\eta)}}(\mathcal M_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)})=\mathbb{Z}$ and $d-1\notin \operatorname{Ex}(1)$, then, for any embedding of fields $\kappa(\eta) \hookrightarrow \mathbb {C}$, the complexification
$$
\begin{equation*}
\operatorname{Lie}\operatorname{Hg}(\mathcal M_\eta\otimes_{\kappa(\eta)}\mathbb{C}) \otimes_\mathbb{Q}\mathbb{C}
\end{equation*}
\notag
$$
of the Lie algebra of the Hodge group of the Abelian variety $\mathcal M_\eta\otimes_{\kappa(\eta)}\mathbb {C}$ is a simple Lie algebra of type $C_{d-1}$ (see Theorem 1.1 in [21]). If the Abelian variety $\mathcal M_\eta$ has good reductions at all places of the curve $C$, then, under conditions of the theorem, the standard conjecture $B(X)$ is true (see [3], Theorem 11.2, Corollary 11.4). Therefore, in accordance with the Grothendieck theorem on semi-stable reductions of Abelian varieties, one may assume that all fibres of the structure morphism $\mathcal M^0\to C$ are extensions of Abelian varieties by linear tori (see Theorem 3.6 in [22]), and there is at least one place of bad reduction. In the case of semi-stable reductions it is known that, for a finite ramified covering $\widetilde{C}\to C$, the connected component of the neutral element of the fibre of the Néron model $\widetilde{\mathcal M}\to \widetilde{C}$ over the 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 of the fibre of the Néron model $\mathcal M\to C$ (see [22], Corollaries 3.3 and 3.9); in particular, the existence of bad semi-stable reduction at the place $\delta$, its toric rank $r_\delta$ and the Abelian variety $A_\delta$ are preserved under the base change $\widetilde{C}\to C$. According to [18], § 5.8 and [19], §§ 4.1, 4.2, 4.4, 4.5, Theorem 4.6, one may assume that there exists a Künnemann compactification $X$ of the Néron minimal model $\mathcal M$ such that any 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, local monodromies (Picard–Lefschetz transformations) are unipotent. The canonical embedding will be denoted by $C'\stackrel{j}{\hookrightarrow} C$. Next, let
$$
\begin{equation*}
K_{nX}\stackrel{\mathrm{def}}{=} \operatorname{Ker}[H^n(X,\mathbb{Q})\to H^0(C,R^n\pi_\ast\mathbb{Q})]
\end{equation*}
\notag
$$
be the kernel of the edge map of the Leray spectral sequence $E_2^{p,q}(\pi)$ of the structure morphism $\pi\colon X\to C$. Besides, for any irreducible smooth projective variety $W$, we denote by $\langle\ \rangle\colon H^\ast(W,\mathbb Q)\,\to\,\mathbb Q$ the degree map defined as zero on $H^n(W,\mathbb Q)$ for $n<2\dim_\mathbb {C} W$ and as the orientation isomorphism (see [2], § 1.2.A) $\langle\ \rangle\colon H^{2\dim_\mathbb {C} W}(W,\mathbb Q)\,\widetilde{\to}\,\mathbb Q$ on $H^{2\dim_\mathbb {C} W}(W,\mathbb Q)$. 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 considered as a smooth projective compactification of the fibre product $X'\times_{C'}X'$. Besides, using the existence of a Künnemann model (see [18], § 5.8; [19], §§ 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 defined by some ramified covering $\widetilde{C}\to C$) or the Mumford theorem on semi-stable reductions (see [23], Ch. II) (or, finally, the Consani method; see [24], §§ 4, 5, Lemma 5.2, Remark 5.4) of a resolution of singularities of the fibre product $X\times_CX$, one may assume that, for all points $s\in C$, the fibre $Y_s$ is a union of smooth irreducible components of multiplicity $1$ with normal crossings.We set $K_{nY}=\operatorname{Ker}[H^n(Y,\mathbb Q)\to H^0(C,R^n(\tau\sigma)_\ast\mathbb Q)]$. 1.2.Consider the normalization $f\colon Z\to\pi^{-1}(\Delta)$ of the scheme $\pi^{-1}(\Delta)$. In this case, $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$, we have, by [25], Corollary (8.2.8),
$$
\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}
$$
where $(i_\Delta f)_\ast$ is a morphism of bidegree $(1,1)$ of pure Hodge structures, and $\varphi_n$ is the restriction morphism. 1.3.It is known that, under the conditions of the theorem, there exist algebraic isomorphisms $H^{2d-2}(X,\mathbb Q)\,\widetilde{\to}\,H^2(X,\mathbb Q)$ and $ H^{2d-3}(X,\mathbb Q)\,\widetilde{\to}\,H^3(X,\mathbb Q)$ (see [26]). Consequently, by Theorem 2.9 in [2], for a proof of the conjecture $B(X)$, it suffices to construct an algebraic isomorphism
$$
\begin{equation*}
H^{2d-r}(X,\mathbb{Q})\,\widetilde{\to}\,H^r(X,\mathbb{Q})
\end{equation*}
\notag
$$
for all $r\in\{4,\dots,d-1\}$. 1.4.For any point $\delta\in\Delta$, we set
$$
\begin{equation*}
m_\delta\stackrel{\mathrm{def}}{=}\operatorname{Card}(\mathcal M_\delta/\mathcal M_\delta^0),\qquad m\stackrel{\mathrm{def}}{=}\prod_{\delta\in\Delta}m_\delta.
\end{equation*}
\notag
$$
We fix a prime number $p$ which does not divide the number $m$. We denote by $p^{m!}_{X/C}\colon X-\to X$ a rational map that coincides on the generic scheme fibre $X_\eta$ of the structure morphism $\pi\colon X\to C$ with the isogeny of the multiplication by the number $p^{m!}$. By the universal property of the Néron model (see [22], formula (1.1.2)), there is the canonical isomorphism
$$
\begin{equation}
\operatorname{End}_C(\mathcal M)\,\widetilde{\to}\,\operatorname{End}_{\kappa(\eta)}(X_\eta).
\end{equation}
\tag{1.2}
$$
Hence the restriction $p^{m!}_{X/C}|_{\mathcal M}\colon \mathcal M\to \mathcal M$ is a regular map. Consider the commutative diagram of a resolution of indeterminacies of the rational map $p^{m!}_{X/C}$. According to Hironaka’s results and (1.2), one may assume that the morphism $\sigma$ is the composite of monoidal transformations with non-singular centres and $\sigma|_{\sigma^{-1}(\mathcal M)}\colon \sigma^{-1}(\mathcal M)\to \mathcal M$ is the identity morphism. Denote by $\mathbb{Q}[[p^{m!}_{X/C}]^\ast]$ the group ring of the linear operator
$$
\begin{equation*}
[p^{m!}_{X/C}]^\ast\colon H^\ast(X,\mathbb{Q})\xrightarrow{x\mapsto \sigma_\ast\nu^\ast(x)} H^\ast(X,\mathbb{Q}),
\end{equation*}
\notag
$$
defined by diagram (1.3). Consider the commutative diagrams with exact rows where $\mathcal M_\delta/\mathcal M^0_\delta$ is a finite group (of order $m_\delta$) of connected components of the algebraic group $\mathcal M_\delta$ (see [22], formula (1.1.5)). The evident surjectivity of canonical maps ${p^{m!}_{X/C}|_{\operatorname{Gm}^{r_\delta}}}$, $A_\delta\xrightarrow{\times p^{m!}} A_\delta$, and the corresponding to diagram (1.4) exact sequence of the snake-like diagram (see [27], § 1, Proposition 2) show that the canonical map ${p^{m!}_{X/C}|_{\mathcal M^0_\delta}}$ is surjective. On the other hand, the multiplication by the invertible in the ring $\mathbb Z/m_\delta\mathbb Z$ element $p\mod m_\delta$ yields a permutation of elements of the finite group $\mathcal M_\delta/\mathcal M^0_\delta$. Consequently, by Lagrange’s theorem, the multiplication by the element $p^{m!}\mod m_\delta$ is the identity bijection of the set $\mathcal M_\delta/\mathcal M^0_\delta$. Therefore, from the commutativity of diagram (1.5), from the exactness of the corresponding sequence of the snake-like diagram and from the surjectivity of the morphism ${p^{m!}_{X/C}|_{\mathcal M^0_\delta}}$ it follows that the morphism $p^{m!}_{X/C}|_{\mathcal M_\delta}$ is surjective, and
$$
\begin{equation}
{p^{m!}_{X/C}}(\mathcal M_{\delta i_\delta})=\mathcal M_{\delta i_\delta} \quad \forall \, i_\delta\in\{1,\dots,m_\delta\},
\end{equation}
\tag{1.6}
$$
where $\mathcal M_{\delta i_\delta}$ is an irreducible component of the variety $\mathcal M_\delta$. 1.5.Let $Z_\delta$ be the normalization of the divisor $\pi^{-1}(\delta)=X_\delta$. Irreducible components of the smooth variety $Z$ are naturally identified with irreducible components $X_{\delta i_\delta}$, $i_\delta\in\{1,\dots,m_\delta\}$, of the divisor By $\iota_{X_s/X}\colon X_s\hookrightarrow X$, $\iota_{X_{\delta i_\delta}/X}\colon X_{\delta i_\delta}\hookrightarrow X$, $\iota_{X_{\delta i_\delta}/Z}\colon X_{\delta i_\delta}\hookrightarrow Z$, $\iota_{Z_{\delta}/Z}\colon Z_{\delta}\hookrightarrow Z$ we denote the canonical embeddings. From the commutativity of the diagram of canonical morphisms we have
$$
\begin{equation}
(i_\Delta f)_\ast(\iota_{X_{\delta i_\delta}/Z})_\ast|_{H^k(X_{\delta i_\delta},\mathbb{Q})}= \iota_{X_{\delta i_\delta}/X\ast}|_{H^k(X_{\delta i_\delta},\mathbb{Q})}.
\end{equation}
\tag{1.7}
$$
The variety $X_{\delta i_\delta}$ is the closure of the irreducible component $\mathcal M_{\delta i_\delta}$ of the algebraic group $\mathcal M_\delta$ in the Zariski topology of the variety $X$. From now on, we denote by $\operatorname{alb}_{\delta i_\delta}\colon X_{\delta i_\delta}\to\operatorname{Alb}(X_{\delta i_\delta})$ the Albanese map, which is defined uniquely up to a translation on the Abelian variety $\operatorname{Alb}(X_{\delta i_\delta})$ (see [28], Chap. II, § 3, Theorem 11). It is known (see [14], the proof of (3.25)]) that the map $\operatorname{alb}_{\delta i_\delta}$ is surjective and
$$
\begin{equation}
\forall \, i_\delta \quad\operatorname{Alb}(X_{\delta i_\delta})=A_\delta.
\end{equation}
\tag{1.8}
$$
§ 2. On rational cohomology of contraction products and toric varieties2.1.By property (v) of a Künnemann compactification, the $C$-group law expands to the group $C$-action $\mathcal M^0\times_C\, X\to X$. Therefore, the group law $\mathcal M^0_\delta\times \mathcal M^0_\delta\to \mathcal M^0_\delta$ expands to the group action which defines, on the irreducible component $X_{\delta i_\delta} \hookrightarrow X_\delta$, the structure of the contraction product $\mathcal M^0_{\delta}\times^{T_{\mathcal M^0_{\delta}}}\,Z_{\delta i_\delta}$ for certain smooth projective toric variety $T_{\mathcal M^0_{\delta}} \hookrightarrow \,Z_{\delta i_\delta}$ (see formula (2) in [18]), where $T_{\mathcal M^0_{\delta}}$ is the maximal subtorus of the semi-Abelian variety $\mathcal M^0_{\delta}$ defined by the exact sequence of algebraic groups
$$
\begin{equation*}
1\to\operatorname{Gm}^{r_\delta}\to\mathcal M^0_\delta\to A_\delta\to 0.
\end{equation*}
\notag
$$
Recall that the torus $T_{\mathcal M^0_{\delta}}$ acts freely on $\mathcal M^0_{\delta}\times Z_{\delta i_\delta}$ by the rule the factor $[\mathcal M^0_{\delta}\times Z_{\delta i_\delta}]/T_{\mathcal M^0_{\delta}}$ by this action of the group $T_{\mathcal M^0_{\delta}}$ (considered as a sheaf on $\operatorname{Spec}\mathbb {C}$ with respect to the $fppf$-topology) is called the contraction product $\mathcal M^0_{\delta}\times^{T_{\mathcal M^0_{\delta}}}\,Z_{\delta i_\delta}$ (see [29], Chap. III, § 1, Definition 1.3.1, and [19], § 1.19). The canonical morphism is denoted by
$$
\begin{equation*}
\mathcal M^0_{\delta}\times Z_{\delta i_\delta}\to[\mathcal M^0_{\delta}\times Z_{\delta i_\delta}]/T_{\mathcal M^0_{\delta}}.
\end{equation*}
\notag
$$
It is evident that there exists a commutative diagram of morphisms of groups so that the canonical surjective morphism of algebraic groups $p^{m!}_{X/C}|_{\mathcal M^0_\delta} \colon \mathcal M^0_\delta \to \mathcal M^0_\delta$ can be considered as a ${p^{m!}_{X/C}|_{T_{\mathcal M^0_\delta}}}$-morphism, because
$$
\begin{equation*}
p^{m!}_{X/C}|_{\mathcal M^0_\delta}(t\cdot w)=p^{m!}_{X/C}|_{T_{\mathcal M^0_\delta}}(t)\cdot p^{m!}_{X/C}|_{\mathcal M^0_\delta}(w).
\end{equation*}
\notag
$$
On the other hand, there is a regular toric endomorphism
$$
\begin{equation*}
p^{m!}_{X/C}|_{Z_{\delta i_\delta}}\colon Z_{\delta i_\delta}\to Z_{\delta i_\delta}
\end{equation*}
\notag
$$
of the smooth toric variety $T_{\mathcal M^0_{\delta}} \hookrightarrow \, Z_{\delta i_\delta}$ (see [30], Theorem 3.3.4, Example 3.3.6), which also may be considered as a ${p^{m!}_{X/C}|_{T_{\mathcal M^0_\delta}}}$-morphism in view of the equality (see [30], formula (3.3.1))
$$
\begin{equation*}
p^{m!}_{X/C}|_{Z_{\delta i_\delta}}(t\cdot z)= p^{m!}_{X/C}|_{T_{\mathcal M^0_\delta}}(t)\cdot p^{m!}_{X/C}|_{Z_{\delta i_\delta}}(z).
\end{equation*}
\notag
$$
Hence, in view of (1.6) and (2.1), it follows from [29], Ch. III, § 1, Proposition 1.3.2, that there exists the canonical commutative diagram of surjective morphisms and, therefore, the restriction $p^{m!}_{X/C}|_{X_{\delta i_\delta}}\colon X_{\delta i_\delta}-\to X_{\delta i_\delta}$ of the rational map $p^{m!}_{X/C}\colon X\,-\,\to X$ to the contraction product $X_{\delta i_\delta}=\mathcal M^0_{\delta}\times^{T_{\mathcal M^0_{\delta}}}\,Z_{\delta i_\delta}$ is a regular endomorphism $p^{m!}_{X/C}|_{X_{\delta i_\delta}}\colon X_{\delta i_\delta}\to X_{\delta i_\delta}$. 2.2.Let $S$ be a scheme, $A$ an Abelian scheme over $S$, $A^\vee$ be the dual Abelian scheme, and $\mathcal P$ be the universal rigid Poincaré line bundle on the scheme $A\times_SA^\vee$. Then
$$
\begin{equation*}
\operatorname{Isom}_{A\times_SA^\vee}(\mathcal O_{A\times_SA^\vee},\mathcal P)
\end{equation*}
\notag
$$
is the Poincaré $\operatorname{Gm}$-torsor on $A\times_SA^\vee$ and a biextension of Abelian schemes $A$ and $A^\vee$ by the torus $\operatorname{Gm}$ (see § 3 in [31]). The theory of the Poincaré $\operatorname{Gm}$-torsors of Abelian varieties and the Künnemann algorithm (see § 2 in [32]) yield the isomorphism
$$
\begin{equation*}
\varphi_{H_l}\colon H^\ast_{\unicode{x00E9}\text{t}}(A_\delta,\mathbb{Q}_l) \otimes_{\mathbb{Q}_l} H^\ast_{\unicode{x00E9}\text{t}}(Z_{\delta i_\delta},\mathbb{Q}_l)\,\widetilde{\to}\, H^\ast_{\unicode{x00E9}\text{t}}(X_{\delta i_\delta},\mathbb{Q}_l),
\end{equation*}
\notag
$$
based on a construction of a special deformation of the contraction product $X_{\delta i_\delta}=\mathcal M^0_{\delta}\times^{T_{\mathcal M^0_{\delta}}} Z_{\delta i_\delta}$ into the Cartesian product $A_{\delta}\times Z_{\delta i_\delta}$ and on the Künneth decomposition of étale cohomology with coefficients in the field $\mathbb Q_l$. Using the classical Kodaira–Spencer deformation theory [33] instead of the algebraic deformation theory (which is used by Künnemann), it is easy to construct an isomorphism of rational Hodge structures
$$
\begin{equation}
\varphi_H\colon H^\ast(A_\delta,\mathbb{Q})\otimes_\mathbb{Q} H^\ast(Z_{\delta i_\delta},\mathbb{Q}) \,\widetilde{\to}\, H^\ast(X_{\delta i_\delta},\mathbb{Q}).
\end{equation}
\tag{2.2}
$$
Taking into account the triviality of rational cohomology of odd degree of the projective toric variety $Z_{\delta i_\delta}$ (see Theorem 10.8 in [34]), for any natural number $q$ we obtain from (2.2) the canonical decompositions of rational Hodge structures
$$
\begin{equation}
\begin{aligned} \, &[H^{2q}(A_\delta,\mathbb{Q})\otimes_\mathbb{Q} H^0(Z_{\delta i_\delta},\mathbb{Q})] \oplus [H^{2q-2}(A_\delta,\mathbb{Q}) \otimes_\mathbb{Q} H^2(Z_{\delta i_\delta},\mathbb{Q})] \nonumber \\ &\qquad\oplus\dots\oplus [H^0(A_\delta,\mathbb{Q}) \otimes_\mathbb{Q} H^{2q}(Z_{\delta i_\delta},\mathbb{Q})] \,\widetilde{\to}\, H^{2q}(X_{\delta i_\delta},\mathbb{Q}), \end{aligned}
\end{equation}
\tag{2.3}
$$
$$
\begin{equation}
\begin{aligned} \, &[H^{2q+1}(A_\delta,\mathbb{Q})\otimes_\mathbb{Q}H^0(Z_{\delta i_\delta},\mathbb{Q})]\oplus [H^{2q-1}(A_\delta,\mathbb{Q})\otimes_\mathbb{Q}H^2(Z_{\delta i_\delta},\mathbb{Q})] \nonumber \\ &\qquad\oplus\dots\oplus [H^1(A_\delta,\mathbb{Q})\otimes_\mathbb{Q}H^{2q}(Z_{\delta i_\delta},\mathbb{Q})] \,\widetilde{\to}\,H^{2q+1}(X_{\delta i_\delta},\mathbb{Q}). \end{aligned}
\end{equation}
\tag{2.4}
$$
2.3.Lemma. The operator $[p^{m!}_{X/C}|_{X_{\delta i_\delta}}]^\ast|_{H^\ast(X_{\delta i_\delta},\mathbb{Q})}$ acts on the $\mathbb Q$-space
$$
\begin{equation*}
H^l(A_\delta,\mathbb{Q})\otimes_\mathbb{Q}H^{2n}(Z_{\delta i_\delta},\mathbb{Q}) =\wedge^lH^1(A_\delta,\mathbb{Q})\otimes_\mathbb{Q}H^{2n}(Z_{\delta i_\delta},\mathbb{Q})
\end{equation*}
\notag
$$
as the multiplication by the number $p^{(l+n)m!}$. Proof. It is well known that the multiplication by the number $p^{m!}$ on the Abelian variety $A_\delta$ induces the multiplication by the number $p^{m!}$ on $H^1(A_\delta,\mathbb Q)$ (see Lemma 2A3 in [2]) and $H^l(A_\delta,\mathbb Q)=\wedge^lH^1(A_\delta,\mathbb Q)$ (see [2], Theorem 2A8). Therefore, by (2.2) it suffices to show that the morphism $p^{m!}_{X/C}|_{X_{\delta i_\delta}}\colon X_{\delta i_\delta} \to X_{\delta i_\delta}$ (defining a regular toric endomorphism of the smooth toric variety $T_{\mathcal M^0_{\delta}} \hookrightarrow \, Z_{\delta i_\delta}$ (see [30], Theorem 3.3.4, Example 3.3.6) induces the multiplication by the number $p^{nm!}$ on $H^{2n}(Z_{\delta i_\delta},\mathbb Q)$.
From now on, we denote by $V_\Sigma\stackrel{\mathrm{def}}{=}Z_{\delta i_\delta}$ a toric variety $Z_{\delta i_\delta}$ with fan $\Sigma$. Let $T_N$ be the maximal subtorus of the variety $V_\Sigma$, $M$ be the group of characters of the torus $T_N$, $N$ be the group of $1$-parameter subgroups of the torus $T_N$, $\sigma$ be a rational polyhedral cone with the semigroup $S_\sigma=\sigma^\vee\cap M$, $U_\sigma=\operatorname{Spec}(\mathbb{C}[S_\sigma])=\operatorname{Spec}(\mathbb{C}[\sigma^\vee\cap M])$ be the corresponding affine toric variety (see [30], Proposition 1.2.17, Theorem 1.2.18). Note that $\sigma^\perp$ is the largest vector subspace of $M_\mathbb{R}$ contained in $\sigma^\vee$, so that $\sigma^\perp \cap M$ is a subgroup of $S_\sigma=\sigma^\vee \cap M$ (see [30], the proof of Lemma 3.2.5). It is known that every cone $\sigma\in\Sigma$ has the distinguished point $\gamma_\sigma\in U_\sigma\,\subset\,V_\Sigma$ and limit points of $1$-parameter subgroups coincide exactly with distinguished points of cones of a fan (see [30], Proposition 3.2.2 and [35], Proposition 1.36). This gives us the toric orbit By the theorem on the orbit-cone correspondence (see [30], Theorem 3.2.6), there is a bijective correspondence
$$
\begin{equation*}
\sigma\in\Sigma\ \leftrightarrow O(\sigma)=T_N\cdot\gamma_\sigma\,\widetilde{\to}\,\operatorname{Hom}_\mathbb{Z}(\sigma^\perp\cap M,\mathbb{C}^\times),
\end{equation*}
\notag
$$
and, for every cone $\sigma\in\Sigma$, we have the equality
$$
\begin{equation}
\dim_\mathbb{C} O(\sigma)=\dim_\mathbb{C} V_{\Sigma}- \dim_\mathbb{Q}\sigma.
\end{equation}
\tag{2.5}
$$
To every cone $\sigma\in\Sigma$ there corresponds a closed subvariety $F_\sigma \hookrightarrow V_\Sigma$ (the closure $\overline{O(\sigma)}$ of the corresponding orbit in the Zariski topology), whose dimension coincides with the codimension of the cone $\sigma$ in the $\mathbb Q$-space $N_\mathbb Q$ (see [34], § 5.7) and is given by (2.5). In order to study $O(\sigma) \hookrightarrow U_\sigma$, we recall how $t\in T_N$ acts on homomorphisms of semigroups. If a point $q\in U_\sigma$ is presented by a homomorphism of semigroups $\gamma\colon S_\sigma\to\mathbb {C}$ (where $\mathbb {C}$ is considered as a semigroup under multiplication (see [30], Proposition 1.3.1), then the point $t\cdot q$ is presented by the homomorphism of semigroups
$$
\begin{equation*}
t\cdot\gamma\colon u\mapsto \chi^u(t)\gamma(u),\qquad u\in M
\end{equation*}
\notag
$$
(see [30], formula (3.2.5)), where $\chi^u\colon T_N\to\mathbb {C}^\times$ is the character corresponding to $u$. Besides, by Lemma 3.2.5 in [30], we have
$$
\begin{equation}
O(\sigma)=\{\gamma\colon S_\sigma\to\mathbb{C}\mid \gamma(u) \neq 0\ \Leftrightarrow\ u\in\sigma^\perp\cap M\}\,\widetilde{\to}\, \operatorname{Hom}_\mathbb{Z}(\sigma^\perp\cap M,\mathbb{C}^\times).
\end{equation}
\tag{2.6}
$$
It is clear that
$$
\begin{equation}
t^{p^{m!}}\cdot\gamma\colon u\mapsto \chi^u(t^{p^{m!}})\gamma(u) =[\chi^u(t)]^{p^{m!}}\gamma(u)=\chi^{p^{m!}u}(t)\gamma(u), \qquad u\in M.
\end{equation}
\tag{2.7}
$$
For each element $v\in N$, we denote by $\mathbb{C}^\times\xrightarrow{z\mapsto \lambda_v(z)} T_N$ the corresponding $1$-parameter subgroup.
From the theorem on toric morphisms (see Theorem 3.3.4 in [30]) and from the results of § 3.3.6 in [30] it follows that the morphism of multiplication extends to the toric morphism $V_\Sigma\xrightarrow{{p^{m!}_{X/C}|_{V_\Sigma}}}V_\Sigma$. It is known that a toric morphism is equivariant (see [30], formula (3.3.1)); in particular,
$$
\begin{equation}
p^{m!}_{X/C}|_{V_\Sigma}(t\cdot\gamma_\sigma)=p^{m!}_{X/C}|_{V_\Sigma}(t)\cdot p^{m!}_{X/C}|_{V_\Sigma}(\gamma_\sigma)= t^{p^{m!}}\cdot p^{m!}_{X/C}|_{V_\Sigma}(\gamma_\sigma)
\end{equation}
\tag{2.8}
$$
by Proposition 1.3.14 in [30]. On the other hand, the relative interior $\operatorname{Relint}(\sigma)$ (see [30], § 1.2, and [36], the proof of Proposition 4.1) of the cone $\sigma$ is characterized as
$$
\begin{equation*}
v\in \operatorname{Relint}(\sigma)\ \Leftrightarrow\ \langle u,v\rangle >0\quad\forall \, u\in\sigma^\vee\setminus \sigma^\perp.
\end{equation*}
\notag
$$
For $v\in\operatorname{Relint}(\sigma)$, we have $\lim_{z\to 0}\lambda_v(z)=\gamma_\sigma$ (see Proposition 3.2.2 in [30]), and, therefore, from (2.6)–(2.8) and from the equality $\lambda_v(z_1z_2)=\lambda_v(z_1)\lambda_v(z_2)$ (see [35], § 1.1.5) it follows that
$$
\begin{equation}
\begin{aligned} \, p^{m!}_{X/C}|_{V_\Sigma}(\gamma_\sigma)&=p^{m!}_{X/C}|_{V_\Sigma}(\lim_{z\to 0}\lambda_v(z))= \lim_{z\to 0}p^{m!}_{X/C}|_{V_\Sigma}(\lambda_v(z)) \nonumber \\ &=\lim_{z\to 0}\lambda_v(z)^{p^{m!}}=\lim_{z\to 0}\lambda_v(z^{p^{m!}})=\lim_{z^{p^{m!}}\to 0}\lambda_v(z^{p^{m!}})= \lim_{z\to 0}\lambda_v(z)=\gamma_\sigma. \end{aligned}
\end{equation}
\tag{2.9}
$$
Since the $n$-dimensional toric variety $F_\sigma$ is normal (see Theorem 3.1.5 in [30]), the set $\operatorname{Sing}(F_\sigma)$ is of codimension at least $2$. Therefore, the exact sequence of cohomology with compact supports (see [37], Chap. III, § 1, Remark 1.30)
$$
\begin{equation*}
\dots\to H^q_{\operatorname{c}}(F_\sigma\setminus \operatorname{Sing}(F_\sigma),\mathbb{Q})\to H^q_{\operatorname{c}}(F_\sigma,\mathbb{Q})\to H^q_{\operatorname{c}}(\operatorname{Sing}(F_\sigma),\mathbb{Q})\to\cdots
\end{equation*}
\notag
$$
and (see [37], Chap. VI, § 1, Theorem 1.1) yield an isomorphism
$$
\begin{equation*}
H^{2n}_{\operatorname{c}}(F_\sigma\setminus \operatorname{Sing}(F_\sigma),\mathbb{Q}) \,\widetilde{\to}\,H^{2n}(F_\sigma,\mathbb{Q}).
\end{equation*}
\notag
$$
By the Poincaré duality, the $1$-dimensional $\mathbb Q$-space
$$
\begin{equation*}
H^{2n}_{\operatorname{c}}(F_\sigma\setminus \operatorname{Sing}(F_\sigma),\mathbb{Q})
\end{equation*}
\notag
$$
is generated by a class of a point (see [37], Chap. VI, § 11, Theorem 11.1), therefore it follows from (2.7)–(2.9) that the degree of the morphism $p^{m!}_{X/C}|_{F_\sigma}\colon F_\sigma\to F_\sigma$ equals $p^{nm!}$ and the map $[p^{m!}_{X/C}|_{F_\sigma}]^\ast\colon H^{2n}(F_\sigma,\mathbb{Q}) \to H^{2n}(F_\sigma,\mathbb{Q})$ of the $1$-dimensional $\mathbb Q$-space $H^{2n}(F_\sigma,\mathbb Q)$ is the multiplication by the number $p^{nm!}$.
On the other hand, there is a sequence of canonical surjections (see [34], Propositions 10.3 and 10.4)
$$
\begin{equation}
\bigoplus_{\substack{\sigma\in\Sigma,\, F_\sigma\hookrightarrow V_\Sigma \\\dim_\mathbb{C} F_\sigma=n}} \operatorname{CH}_n(F_\sigma)\otimes_\mathbb{Z}\mathbb{Q}\to \operatorname{CH}_n(V_\Sigma)\otimes_\mathbb{Z}\mathbb{Q}\to H_{2n}(V_\Sigma,\mathbb{Q})\to 0.
\end{equation}
\tag{2.10}
$$
Finally, for any smooth projective toric variety $V_\Sigma$ and for any natural number $n$, it follows from [38], Theorem 2, Theorem 6, and the comment to it, that there exists the canonical isomorphism
$$
\begin{equation*}
\operatorname{Hom}_\mathbb{Q}(\operatorname{CH}_n(V_\Sigma)\otimes_\mathbb{Z}\mathbb{Q},\mathbb{Q}) \,\widetilde{\to}\, \operatorname{CH}^n(V_\Sigma)\otimes_\mathbb{Z}\mathbb{Q},
\end{equation*}
\notag
$$
and, therefore, the sequence of canonical injections dual to (2.10) takes the form
$$
\begin{equation*}
\begin{aligned} \, \bigoplus_{\substack{\sigma\in\Sigma,\,F_\sigma\hookrightarrow V_\Sigma\\\dim_\mathbb{C} F_\sigma=n}} H^{2n}(F_\sigma,\mathbb{Q}) &\,\widetilde{\leftarrow}\, \bigoplus_{\substack{\sigma\in\Sigma,\,F_\sigma\hookrightarrow V_\Sigma\\\dim_\mathbb{C} F_\sigma=n}} \operatorname{CH}^n(F_\sigma)\otimes_\mathbb{Z}\mathbb{Q} \\ &\hookleftarrow \operatorname{CH}^n(V_\Sigma)\otimes_\mathbb{Z}\mathbb{Q} \hookleftarrow H^{2n}(V_\Sigma,\mathbb{Q}). \end{aligned}
\end{equation*}
\notag
$$
It remains to apply the fact that the map
$$
\begin{equation*}
[p^{m!}_{X/C}|_{F_\sigma}]^\ast\colon H^{2n}(F_\sigma,\mathbb{Q})\to H^{2n}(F_\sigma,\mathbb{Q})
\end{equation*}
\notag
$$
is the multiplication by the number $p^{nm!}$. Lemma 2.3 is proved. 2.4.It is known that the Gysin map $\iota_{X_{\delta i_\delta}/X\ast}\colon H^k(X_{\delta i_\delta},\mathbb Q)\to H^{k+2}(X,\mathbb Q)$ has the form
$$
\begin{equation}
\alpha\mapsto \alpha\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})
\end{equation}
\tag{2.11}
$$
(see [26], the proof of (4.20), and [14], the proof of (3.37)). By the choice of the prime number $p$, we have
$$
\begin{equation}
[p^{m!}_{X/C}]^\ast|_{H^2(X,\mathbb{Q})}(\operatorname{cl}_X(X_{\delta i_\delta}))= \operatorname{cl}_X(X_{\delta i_\delta})
\end{equation}
\tag{2.12}
$$
(see [26], the end of § 4.2, and [14], the end of § 3.8). Therefore, taking into account (2.12), [39], Chap. 2, § 8, formula (5), and the functoriality of constructions under consideration, we obtain
$$
\begin{equation}
[p^{m!}_{X/C}]^\ast(\iota_{X_{\delta i_\delta}/X\ast}(\alpha))= [p^{m!}_{X/C}]^\ast(\alpha\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})) =[p^{m!}_{X/C}|_{X_{\delta i_\delta}}]^\ast(\alpha)\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta}).
\end{equation}
\tag{2.13}
$$
According to (1.1), (1.7), (2.2), (2.3), (2.11)–(2.13), and Lemma 2.3, we obtain the canonical decomposition of Hodge $\mathbb Q$-structures and $\mathbb Q[[p^{m!}_{X/C}]^\ast]$-moduli
$$
\begin{equation}
\begin{aligned} \, (i_\Delta f)_\ast H^{2q}(Z,\mathbb{Q})&=\sum_{\substack{\delta\in\Delta\\ i_\delta\in\{1,\dots,m_\delta\}}}[H^{2q}(A_\delta,\mathbb{Q})\otimes_\mathbb{Q}H^0(Z_{\delta i_\delta},\mathbb{Q}) \,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})] \nonumber \\ &\qquad\qquad\oplus [H^{2q-2}(A_\delta,\mathbb{Q})\otimes_\mathbb{Q} H^2(Z_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})] \nonumber \\ &\qquad\qquad\oplus \dots \oplus [H^0(A_\delta,\mathbb{Q})\otimes_\mathbb{Q}H^{2q}(Z_{\delta i_\delta},\mathbb{Q}) \,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})], \end{aligned}
\end{equation}
\tag{2.14}
$$
where the operator $[p^{m!}_{X/C}]^\ast$ acts on the $\mathbb Q$-subspace
$$
\begin{equation*}
\Sigma_{2q-2r,2r}\stackrel{\mathrm{def}}{=}\sum_{\substack{\delta\in\Delta\\ i_\delta\in\{1,\dots,m_\delta\}}} H^{2q-2r}(A_\delta,\mathbb{Q})\otimes_\mathbb{Q} H^{2r}(Z_{\delta i_\delta},\mathbb{Q}) \,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})
\end{equation*}
\notag
$$
as the multiplication by the number $p^{(2q-r)m!},\quad r=0,1,2,\dots$ . Similarly, taking into account (2.4), we obtain the canonical decomposition
$$
\begin{equation}
\begin{aligned} \, (i_\Delta f)_\ast H^{2q+1}(Z,\mathbb{Q}) &= \sum_{\substack{\delta\in\Delta\\ i_\delta\in\{1,\dots,m_\delta\}}} [H^{2q+1}(A_\delta,\mathbb{Q})\otimes_\mathbb{Q}H^0(Z_{\delta i_\delta},\mathbb{Q}) \,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})] \nonumber \\ &\qquad\qquad\oplus [H^{2q-1}(A_\delta,\mathbb{Q})\otimes_\mathbb{Q}H^2(Z_{\delta i_\delta},\mathbb{Q}) \,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})] \nonumber \\ &\qquad\qquad\oplus\dots\oplus [H^1(A_\delta,\mathbb{Q})\otimes_\mathbb{Q}H^{2q}(Z_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})], \end{aligned}
\end{equation}
\tag{2.15}
$$
where the operator $[p^{m!}_{X/C}]^\ast$ acts on the $\mathbb Q$-subspace
$$
\begin{equation*}
\Sigma_{2q+1-2r,2r}\stackrel{\mathrm{def}}{=}\sum_{\substack{\delta\in\Delta \\ i_\delta\in\{1,\dots,m_\delta\}}} H^{2q+1-2r}(A_\delta,\mathbb{Q})\otimes_\mathbb{Q} H^{2r}(Z_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})
\end{equation*}
\notag
$$
as the multiplication by the number $p^{(2q+1-r)m!},\quad r=0,1,2,\dots$ . 2.5.By functoriality of the Leray spectral sequence the canonical embedding $\iota_{X'/X}\colon X' \hookrightarrow X$ yields the homomorphisms $E_2^{p,q}(\pi)\to E_2^{p,q}(\pi')$, which are compatible with differentials and filtrations (see [40], § 2.4, and [41], Vol. II, Proposition 4.8). Taking into account the degeneracy of the spectral sequences $E_2^{p,q}(\pi)$ (see [42], Corollary (15.15)) and $E_2^{p,q}(\pi')$ (see [43], Theorem 4.1.1), diagram (15.1) in [42], the exact sequence of Hodge $\mathbb Q$-structures (see [44], formula (2.4))
$$
\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{2.16}
$$
the equality $H^2(C',R^{n-2}\pi'_\ast\mathbb Q)=0$ ( since the cohomological dimension of an affine curve $C'$ is $1$ according to [37], Chap. VI, § 7, Theorem 7.2), we obtain the commutative diagram of mixed Hodge $\mathbb Q$-structures with exact rows (see [45], diagram 3.8) where the map $\overline{\varphi_n}$ is defined by $x+H^2(C,R^{n-2}\pi_\ast\mathbb Q)\,\mapsto\,\varphi_n(x)$. Let $D^\ast(\delta)$ be a small punctured disc on the curve $C$ with the centre at the point $\delta\in\Delta$. The Leray spectral sequence for the embedding $j\colon C' \hookrightarrow C$ yields the exact sequence of mixed Hodge structures (see [42], the proof of Proposition (12.5), Corollary (13.10), Remark (14.5))
$$
\begin{equation}
0\to H^1(C,j_\ast R^{n-1}\pi'_\ast\mathbb{Q})\to H^1(C',R^{n-1}\pi'_\ast\mathbb{Q}) \to H^0(C,R^1j_\ast R^{n-1}\pi'_\ast\mathbb{Q}),
\end{equation}
\tag{2.18}
$$
where
$$
\begin{equation*}
\begin{aligned} \, &H^0(C,R^1j_\ast R^{n-1}\pi'_\ast\mathbb{Q}) = \bigoplus_{\delta\in\Delta} H^1(D^\ast(\delta),R^{n-1}\pi'_\ast\mathbb{Q}) \\ &\qquad\,\widetilde{\to}\, \bigoplus_{\delta\in\Delta} H^{n-1}(X_s,\mathbb{Q})/N_\delta H^{n-1}(X_s,\mathbb{Q}), \end{aligned}
\end{equation*}
\notag
$$
the space $H^{n-1}(X_s,\mathbb Q)\quad(s\in C')$ has the limit mixed Hodge structure associated with the local monodromy $\gamma_\delta$ around the point $\delta\in C$ (the Picard–Lefschetz transform) and $N_\delta=\log \gamma_\delta$. By the theorem on local invariant cycles (see [46], § 3, and [42], Proposition (15.12)), sequence (2.18) takes the form
$$
\begin{equation}
0\to H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\to H^1(C',R^{n-1}\pi'_\ast\mathbb{Q}) \to \bigoplus_{\delta\in\Delta} H^{n-1}(X_s,\mathbb{Q})/N_\delta H^{n-1}(X_s,\mathbb{Q}).
\end{equation}
\tag{2.19}
$$
Clearly, the restriction $p^{m!}_{X/C}\circ\iota_{X'/X}\stackrel{\mathrm{def}}{=}p^{m!}_{X'/C'}$ of the rational map $p^{m!}_{X/C}$ to the Abelian scheme $\pi'\colon X'\to C'$ is the $C'$-isogeny defining the linear operator $[p^{m!}_{X'/C'}]^\ast\colon H^n(X',\mathbb{Q})\to H^n(X',\mathbb{Q})$, which acts on the finite-dimensional $\mathbb Q$-space $H^n(X',\mathbb Q)$. This operator acts on the $\mathbb Q$-subspace $H^1(C',R^{n-1}\pi'_\ast\mathbb Q) \hookrightarrow H^n(X',\mathbb Q)$ and on the $\mathbb Q$-factorspace $H^0(C',R^n\pi'_\ast\mathbb Q)$ as multiplications by numbers $p^{(n-1)m!}$ and $p^{nm!}$ respectively because, for any fibre $X_s$ of the Abelian scheme $\pi'\colon X'\to C'$, the isogeny of the multiplication by the number $p^{m!}$ induces the multiplication by $p^{m!}$ in the space $H^1(X_s,\mathbb Q)$ (see [2], Lemma 2A3, Theorem 2A11) and $R^n\pi'_\ast\mathbb Q=\wedge^nR^1\pi'_\ast\mathbb Q$. Therefore, proceeding as in [47], § 3, (3.2), (3.3), we obtain the canonical splitting of mixed Hodge structures
$$
\begin{equation*}
H^n(X',\mathbb{Q})=H^1(C',R^{n-1}\pi'_\ast\mathbb{Q})\oplus H^0(C',R^n\pi'_\ast\mathbb{Q}),
\end{equation*}
\notag
$$
so that the commutative diagram (2.17) yields the commutative diagram of Hodge $\mathbb Q$-structures and the canonical splitting of Hodge $\mathbb Q$-structures and $\mathbb Q[[p^{m!}_{X/C}]^\ast]$-moduli
$$
\begin{equation}
\operatorname{Im}(\varphi_n)=H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\oplus H^0(C',R^n\pi'_\ast\mathbb{Q}),
\end{equation}
\tag{2.21}
$$
where by (1.1) the rational Hodge structure $\operatorname{Im}(\varphi_n)$ is canonically identified with the $\mathbb Q[[p^{m!}_{X/C}]^\ast]$-module $H^n(X,\mathbb Q)/(i_\Delta f)_\ast H^{n-2}(Z,\mathbb Q)$. Now (1.1) and (2.21) follow from the existence of the exact sequence of rational Hodge structures and the $\mathbb Q[[p^{m!}_{X/C}]^\ast]$-moduli
$$
\begin{equation*}
0\to(i_\Delta f)_\ast\,H^{n-2}(Z,\mathbb{Q})\to H^n(X,\mathbb{Q})\to H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\oplus H^0(C',R^n\pi'_\ast\mathbb{Q})\to 0,
\end{equation*}
\notag
$$
so that decompositions (2.14), (2.15) yield the canonical splitting of rational Hodge structures and the $\mathbb Q[[p^{m!}_{X/C}]^\ast]$-moduli
$$
\begin{equation}
\begin{aligned} \, H^{2q}(X,\mathbb{Q}) &=\Sigma_{2q-2,0}\oplus \Sigma_{2q-4,2}\oplus \dots\oplus \Sigma_{0,2q-2} \nonumber \\ &\qquad\oplus H^1(C,R^{2q-1}\pi_\ast\mathbb{Q})\oplus H^0(C',R^{2q}\pi'_\ast\mathbb{Q}), \end{aligned}
\end{equation}
\tag{2.22}
$$
where the operator $[p^{m!}_{X/C}]^\ast$ acts canonically on direct summands as multiplications by the numbers $p^{(2q-2)m!}$, $p^{(2q-3)m!}$, $p^{(2q-4)m!}$, $\dots$, $p^{(q-1)m!}$, $p^{(2q-1)m!}$, $p^{(2q)m!}$, respectively,
$$
\begin{equation}
H^{2q+1}(X,\mathbb{Q}) =\Sigma_{2q-1,0}\oplus \Sigma_{2q-3,2}\oplus \dots\oplus \Sigma_{1,2q-2} \oplus H^1(C,R^{2q}\pi_\ast\mathbb{Q})
\end{equation}
\tag{2.23}
$$
because the Lie algebra $\operatorname{Lie} G\otimes_\mathbb Q\mathbb {C}$ is a non-trivial ideal (see Theorem 7.3 in [48]) of the simple Lie algebra $\operatorname{Lie}\operatorname{Hg}(\mathcal M_\eta\otimes_{\kappa(\eta)}\mathbb {C})\otimes_\mathbb Q\mathbb {C}$ of type $C_{d-1}$, and, therefore,
$$
\begin{equation*}
H^0(C',R^{2q+1}\pi'_\ast\mathbb{Q})=H^{2q+1}(X_s,\mathbb{Q})^{\pi_1(C',s)}=\wedge^{2q+1}H^1(X_s,\mathbb{Q})^G=0
\end{equation*}
\notag
$$
by [50], Chap. VIII, § 13, n$^0$ 3, Lemma 2, and the operator $[p^{m!}_{X/C}]^\ast$ acts canonically on direct summands of decomposition (2.23) as multiplication by the numbers $p^{(2q-1)m!}$, $p^{(2q-2)m!}$, $p^{(2q-3)m!}$, $\dots$, $p^{qm!}$, $p^{(2q)m!}$, respectively. 2.6.Lemma. The canonical inclusion
$$
\begin{equation*}
(p_2\sigma)_\ast(K_{(2d-2+n)Y})\hookrightarrow K_{nX}
\end{equation*}
\notag
$$
holds. Proof. By (1.1), (2.17) and the equality $H^2(C',R^{n-2}\pi'_\ast\mathbb Q)=0$, we have the canonical inclusion
$$
\begin{equation}
H^2(C,R^{n-2}\pi_\ast\mathbb{Q})\hookrightarrow (i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})=\Sigma_{n-2,0}\oplus \Sigma_{n-4,2}\oplus \cdots.
\end{equation}
\tag{2.24}
$$
On the other hand, the exact sequence (2.16) yields the canonical identification
$$
\begin{equation*}
K_{nX}/H^2(C,R^{n-2}\pi_\ast\mathbb{Q})\,\widetilde{\to}\, H^1(C,R^{n-1}\pi_\ast\mathbb{Q}).
\end{equation*}
\notag
$$
Besides, taking into account the commutative diagram (2.20) and decompositions (2.22), (2.23), we obtain the canonical embeddings
$$
\begin{equation*}
\begin{aligned} \, K_{nX} &=\operatorname{Ker}[H^n(X,\mathbb{Q})\to H^0(C,R^n\pi_\ast\mathbb{Q})]\hookrightarrow \operatorname{Ker}[H^n(X,\mathbb{Q})\to H^0(C',R^n\pi'_\ast\mathbb{Q})] \\ &=\Sigma_{n-2,0}\oplus \Sigma_{n-4,2}\oplus \dots \oplus H^1(C,R^n\pi_\ast\mathbb{Q}) \\ &=(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^1(C,R^{n-1}\pi_\ast\mathbb{Q}). \end{aligned}
\end{equation*}
\notag
$$
Hence by (2.24) we obtain the canonical decomposition
The operator $[p^{m!}_{X/C}]^\ast\colon H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})\to H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})$ acting on the $1$-dimensional (see [45], § 3.4) $\mathbb Q$-space $H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})=H^0(C',R^2\pi'_\ast\mathbb{Q})$, is the homothety with coefficient $p^{2m!}$, and hence from the strong Lefschetz theorem, the theorem on local invariant cycles (which asserts the surjectivity of the canonical map $R^n\pi_\ast\mathbb{Q}\to j_\ast R^n\pi'_\ast\mathbb{Q}$ with kernel supported on the finite set $\Delta$ (see [46], § 3, and [42], Proposition (15.12)) and decompositions (2.22), (2.23) we have, for $n\leqslant d$, the canonical decompositions
$$
\begin{equation}
\begin{aligned} \, &H^{2d-n}(X,\mathbb{Q}) =[\Sigma_{n-2,0}\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n}]\oplus [\Sigma_{n-4,2}\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n}] \nonumber \\ &\qquad\qquad\oplus \dots \oplus H^1(C,R^{2d-n-1}\pi_\ast\mathbb{Q}) \oplus H^0(C',R^{2d-n}\pi'_\ast\mathbb{Q}) \nonumber \\ &\qquad=[\Sigma_{n-2,0}\,\smile\,H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})^{\smile\,d-n}]\oplus [\Sigma_{n-4,2} \,\smile\,H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})^{\smile\,d-n}] \nonumber \\ &\qquad\qquad\oplus\dots\oplus H^1(C,R^{2d-n-1}\pi_\ast\mathbb{Q}) \oplus H^0(C,j_\ast R^{2d-n}\pi'_\ast\mathbb{Q}). \end{aligned}
\end{equation}
\tag{2.26}
$$
There is an exact sequence of Hodge $\mathbb Q$-structures
$$
\begin{equation}
\begin{aligned} \, 0 &\to H^2(C,R^{2d-4+n}(\tau\sigma)_\ast\mathbb{Q})\to K_{(2d-2+n)Y} \nonumber \\ &\xrightarrow{\alpha_{(2d-2+n)Y}} H^1(C,R^{(2d-3+n)}(\tau\sigma)_\ast\mathbb{Q})\to 0 \end{aligned}
\end{equation}
\tag{2.27}
$$
(see [44], formula (2.4)). By the theorem on local invariant cycles, this sequence takes the form
$$
\begin{equation}
\begin{aligned} \, 0 &\to H^2(C,j_\ast R^{2d-4+n}\tau'_\ast\mathbb{Q})\to K_{(2d-2+n)Y} \nonumber \\ &\xrightarrow{\alpha_{(2d-2+n)Y}} H^1(C,j_\ast R^{(2d-3+n)}\tau'_\ast\mathbb{Q})\to 0. \end{aligned}
\end{equation}
\tag{2.28}
$$
From the non-degeneracy (see [42], Proposition (10.5)) of the restriction
$$
\begin{equation*}
\!\Phi|_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}\colon H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\times H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\xrightarrow{x\times y\mapsto \langle x\,\smile\,y\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n}\rangle}\mathbb{Q}
\end{equation*}
\notag
$$
of the non-degenerate (see [2], § 1.2.A) bilinear form
$$
\begin{equation*}
\Phi\colon H^n(X,\mathbb{Q})\times H^n(X,\mathbb{Q})\xrightarrow{x\times y\mapsto \langle x\,\smile\,y\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n}\rangle} \mathbb{Q}
\end{equation*}
\notag
$$
we obtain the existence of the decomposition
$$
\begin{equation*}
H^n(X,\mathbb{Q})=H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\oplus H^1(C,R^{n-1}\pi_\ast\mathbb{Q})^\perp
\end{equation*}
\notag
$$
of Hodge $\mathbb Q$-structures (see [49], Chap. IX, § 4, n$^0$ 1, the corollary to Proposition 1), where
$$
\begin{equation*}
H^1(C,R^{n-1}\pi_\ast\mathbb{Q})^\perp {=}\, \{x\in H^n(X,\mathbb{Q}) \mid x\smile H^1(C,R^{n-1}\pi_\ast\mathbb{Q}) \smile \operatorname{cl}_X(H)^{\smile\,d-n}\,{=}\,0\}
\end{equation*}
\notag
$$
is the orthogonal complement of the subspace $H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\hookrightarrow H^n(X,\mathbb{Q})$ with respect to the bilinear form $\Phi$.
We claim that
$$
\begin{equation}
H^1(C,R^{n-1}\pi_\ast\mathbb{Q})^\perp =(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q}) \oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q}).
\end{equation}
\tag{2.29}
$$
Indeed, by definition (see [41], Vol. II, Chap. 4, § 4.2.1, formula (4.5)), for any point $s\in C'$, the $\smile$-product by the class $\operatorname{cl}_X(X_{\delta i_\delta})\in H^2(X,\mathbb Q)$ acts on the fibre $H^q(X_s,\mathbb Q)=[j_\ast R^q\pi'_\ast\mathbb Q]_s$ of the sheaf $j_\ast R^q\pi'_\ast\mathbb Q$ as the $\smile$-product by the class $\iota^\ast_{X_s/X}(\operatorname{cl}_X(X_{\delta i_\delta}))$. It is clear that
$$
\begin{equation*}
\iota^\ast_{X_s/X}(\operatorname{cl}_X(X_{\delta i_\delta}))=0.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
j_\ast R^q\pi'_\ast\mathbb{Q}\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})=0
\end{equation*}
\notag
$$
and, therefore,
$$
\begin{equation}
\operatorname{support}(R^q\pi_\ast\mathbb{Q}\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta}))\subseteq \Delta.
\end{equation}
\tag{2.30}
$$
Suppose that $r\geqslant 1$ and $H^r(C,R^q\pi_\ast\mathbb Q) \subseteq H^\ast(X,\mathbb Q)$. Then by (2.30) the image $\mathcal F$ of the map
$$
\begin{equation*}
R^q\pi_\ast\mathbb{Q}\xrightarrow{\smile\,\operatorname{cl}_X(X_{\delta i_\delta})}R^{q+2}\pi_\ast\mathbb{Q}
\end{equation*}
\notag
$$
is concentrated on a finite set $\Delta$, and, therefore, $H^r(C,\mathcal F)=0$. Hence a sequence of maps of sheaves
$$
\begin{equation*}
R^q\pi_\ast\mathbb{Q}\,\xrightarrow{\smile\,\operatorname{cl}_X(X_{\delta i_\delta})}\,\mathcal F\,\subset R^{q+2}\pi_\ast\mathbb{Q},
\end{equation*}
\notag
$$
and the functoriality of cohomology (see [51], Chap. II, § 3, Theorem 3.11) show that the map
$$
\begin{equation*}
\smile\,\operatorname{cl}_X(X_{\delta i_\delta})\colon H^r(C,R^q\pi_\ast\mathbb{Q})\to H^r(C,R^{q+2}\pi_\ast\mathbb{Q})
\end{equation*}
\notag
$$
is the zero map and by (see [41], Vol. II, Chap. 4, § 4.2.1, formula (4.8)) we obtain the equality
$$
\begin{equation*}
\operatorname{cl}_X(X_{\delta i_\delta})\,\smile\,H^r(C,R^q\pi_\ast\mathbb{Q})=0.
\end{equation*}
\notag
$$
So, by (2.14), (2.15) we have
$$
\begin{equation}
(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\,\smile\,H^r(C,R^q\pi_\ast\mathbb{Q})=0.
\end{equation}
\tag{2.31}
$$
In particular, by (2.25)
$$
\begin{equation}
(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\subseteq H^1(C,R^{n-1}\pi_\ast\mathbb{Q})^\perp,
\end{equation}
\tag{2.32}
$$
$$
\begin{equation}
(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\subseteq K_{nX}^\perp \stackrel{\mathrm{def}}{=}\{x\in H^n(X,\mathbb{Q}) \mid x\,\smile\,K_{nX}\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n}=0\}.
\end{equation}
\tag{2.33}
$$
On the other hand, $R^{2d-1}\pi'_\ast\mathbb Q=0$, and so in the cohomology ring $H^\ast(X,\mathbb Q)$ with usual $\smile$-product we have, by the theorem on local invariant cycles,
$$
\begin{equation*}
\begin{aligned} \, &H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\,\smile\,H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})\,\smile\,\operatorname{cl}_X(H)^{\smile d-n} \\ &\qquad=H^1(C,j_\ast R^{n-1}\pi'_\ast\mathbb{Q})\,\smile\,H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q}) \,\smile\, \operatorname{cl}_X(H)^{\smile d-n} \\ &\qquad\subseteq H^1(C,j_\ast R^{2d-1}\pi'_\ast\mathbb{Q})=0. \end{aligned}
\end{equation*}
\notag
$$
Hence $H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})\subseteq H^1(C,R^{n-1}\pi_\ast\mathbb{Q})^\perp$. Now (2.29) follows from (2.22), (2.23) and (2.32).
From (2.25) and (2.29) it is clear that
$$
\begin{equation}
\begin{aligned} \, &K_{nX}^\perp \subseteq H^1(C,R^{n-1}\pi_\ast\mathbb{Q})^\perp\cap H^2(C,R^{n-2}\pi_\ast\mathbb{Q})^\perp \nonumber \\ &\qquad=[(i_\Delta f)_\ast\,H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})]\cap H^2(C,R^{n-2}\pi_\ast\mathbb{Q})^\perp. \end{aligned}
\end{equation}
\tag{2.34}
$$
Note that by the theorem on local invariant cycles (see [50], Chap. VIII, § 13, n$^0$ 3, Lemma 2), and by the non-degeneracy of the canonical pairing (see [42], Proposition (10.5))
$$
\begin{equation*}
\begin{aligned} \, &H^0(C,j_\ast R^{n-2}\pi'_\ast\mathbb{Q})\times H^2(C,j_\ast R^{2d-n}\pi'_\ast\mathbb{Q}) \\ &\qquad \xrightarrow{\omega\times \omega'\mapsto \omega\,\smile\,\omega'} H^2(C,j_\ast R^{2d-2}\pi'_\ast\mathbb{Q})=H^{2d}(X,\mathbb{Q}) \end{aligned}
\end{equation*}
\notag
$$
we have
$$
\begin{equation}
\begin{aligned} \, \dim_\mathbb{Q} H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q}) &=\dim_\mathbb{Q} H^0(C',R^n\pi'_\ast\mathbb{Q}) \nonumber \\ &= \begin{cases} 1 &\text{if }n\text{ is even and }0\leqslant n\leqslant 2d-2, \\ 0 &\text{otherwise}, \end{cases} \end{aligned}
\end{equation}
\tag{2.35}
$$
$$
\begin{equation}
\begin{aligned} \, \dim_\mathbb{Q} H^2(C,R^{n-2}\pi_\ast\mathbb{Q}) &=\dim_\mathbb{Q} H^0(C',R^{2d-n}\pi'_\ast\mathbb{Q}) \nonumber \\ &=\begin{cases} 1 &\text{if }n\text{ is even and }2\leqslant n\leqslant 2d, \\ 0 &\text{otherwise}. \end{cases} \end{aligned}
\end{equation}
\tag{2.36}
$$
Assume first that $n$ is even and $2\leqslant n\leqslant d-1$. Then
$$
\begin{equation*}
\begin{aligned} \, &H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})\,\smile\,H^2(C,R^{n-2}\pi_\ast\mathbb{Q}) \,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n} \\ &\qquad=H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})\,\smile\,H^2(C,j_\ast R^{n-2}\pi'_\ast\mathbb{Q}) \,\smile\,H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})^{\smile \,d-n} \\ &\qquad=H^2(C,j_\ast R^{2d-2}\pi'_\ast\mathbb{Q})= H^2(C,R^{2d-2}\pi_\ast\mathbb{Q})=H^{2d}(X,\mathbb{Q})=\mathbb{Q}, \end{aligned}
\end{equation*}
\notag
$$
and hence by (2.35)
$$
\begin{equation*}
H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})\cap H^2(C,R^{n-2}\pi_\ast\mathbb{Q})^\perp=0,
\end{equation*}
\notag
$$
and so from (2.34) we have the inclusion
$$
\begin{equation}
K_{nX}^\perp \subseteq (i_\Delta f)_\ast\,H^{n-2}(Z,\mathbb{Q}).
\end{equation}
\tag{2.37}
$$
Since the form $\Phi$ is non-degenerate, we have by [49], Chap. IX, § 1, n$^0$ 6, Corollary 1 to Proposition 4,
$$
\begin{equation}
\begin{aligned} \, \operatorname{codim}_\mathbb{Q} [K^\perp_{nX}\hookrightarrow H^n(X,\mathbb{Q})] &=\dim_\mathbb{Q} K_{nX}, \\ \dim_\mathbb{Q} H^n(X,\mathbb{Q})-\dim_\mathbb{Q} K^\perp_{nX} &=\dim_\mathbb{Q} K_{nX}. \end{aligned}
\end{equation}
\tag{2.38}
$$
Therefore, from (2.22), (2.25), (2.35), (2.36) we obtain
$$
\begin{equation*}
\begin{aligned} \, \dim_\mathbb{Q} K^\perp_{nX} &=\dim_\mathbb{Q} H^n(X,\mathbb{Q})-\dim_\mathbb{Q} K_{nX} \\ &=\dim_\mathbb{Q}[(i_\Delta f)_\ast\,H^{n-2}(Z,\mathbb{Q}) \oplus H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})] \\ &\qquad-\dim_\mathbb{Q} [H^2(C,R^{n-2}\pi_\ast\mathbb{Q})\oplus H^1(C,R^{n-1}\pi_\ast\mathbb{Q})] \\ &=\dim_\mathbb{Q}[(i_\Delta f)_\ast\,H^{n-2}(Z,\mathbb{Q})]. \end{aligned}
\end{equation*}
\notag
$$
Now from (2.37) we have
$$
\begin{equation}
K^\perp_{nX}=(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\quad\text{for even } \ n\in\{2,\dots,d-1\}.
\end{equation}
\tag{2.39}
$$
We finally assume that $n$ is odd. Now from (2.34)–(2.36) it follows that (2.37) holds for odd $n$. Therefore, by (2.23), (2.25), (2.35), (2.36), (2.38), we obtain
$$
\begin{equation*}
\begin{aligned} \, \dim_\mathbb{Q} K^\perp_{nX} &=\dim_\mathbb{Q} H^n(X,\mathbb{Q})-\dim_\mathbb{Q} K_{nX} \\ &=\dim_\mathbb{Q}[(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q}) \oplus H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})] \\ &\qquad-\dim_\mathbb{Q} [H^2(C,R^{n-2}\pi_\ast\mathbb{Q})\oplus H^1(C,R^{n-1}\pi_\ast\mathbb{Q})] \\ &=\dim_\mathbb{Q}[(i_\Delta f)_\ast\,H^{n-2}(Z,\mathbb{Q})]. \end{aligned}
\end{equation*}
\notag
$$
Now from (2.37) we have the equality
$$
\begin{equation}
K^\perp_{nX}=(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\quad\text{for odd } \ n.
\end{equation}
\tag{2.40}
$$
The non-degeneracy of the form $\Phi$ yields the equality $K_{nX}^{\perp\perp}=K_{nX}$ (see [49], Chap. IX, § 1, n$^0$ 6, Corollary 1 to Proposition 4). Therefore,
$$
\begin{equation}
K_{nX}= \{x\in H^n(X,\mathbb{Q})\mid x\,\smile\,K^\perp_{nX}\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n}=0\}.
\end{equation}
\tag{2.41}
$$
By (2.41) it suffices to check the equality
$$
\begin{equation*}
(p_2\sigma)_\ast(K_{(2d-2+n)Y})\,\smile\, K^\perp_{nX}\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n}=0,
\end{equation*}
\notag
$$
which, in turn, is equivalent to the equality
$$
\begin{equation}
K_{(2d-2+n)Y}\,\smile\, (p_2\sigma)^\ast(K^\perp_{nX}\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n})=0,
\end{equation}
\tag{2.42}
$$
because in accordance with [2], § 1.2.A; [37], Chap. VI, § 11, Remark 11.6), we have
$$
\begin{equation*}
\begin{aligned} \, &\langle (p_2\sigma)_\ast(K_{(2d-2+n)Y})\,\smile\, K^\perp_{nX}\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n}\rangle \\ &\qquad=\langle K_{(2d-2+n)Y}\,\smile\, (p_2\sigma)^\ast(K^\perp_{nX}\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n})\rangle. \end{aligned}
\end{equation*}
\notag
$$
On the other hand, if $n\leqslant d-1$, then the strong Lefschetz theorem for the variety $X_{\delta i_\delta}$ yields the existence of the embedding
$$
\begin{equation*}
H^{n-2}(X_{\delta i_\delta},\mathbb{Q})\,\smile\,\iota^\ast_{X_{\delta i_\delta}/X}\operatorname{cl}_X(H) \hookrightarrow H^n(X_{\delta i_\delta},\mathbb{Q}).
\end{equation*}
\notag
$$
Therefore, the projection formula (see [2], § 1.2.A) and the map (2.11) yield the embeddings
$$
\begin{equation}
\begin{aligned} \, &\iota_{X_{\delta i_\delta}/X\ast}H^{n-2}(X_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(H) \hookrightarrow \iota_{X_{\delta i_\delta}/X\ast}H^n(X_{\delta i_\delta},\mathbb{Q}) \nonumber \\ &\qquad=H^n(X_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})\hookrightarrow H^{n+2}(X,\mathbb{Q}). \end{aligned}
\end{equation}
\tag{2.43}
$$
From (1.7) and (2.43) we have the inclusion
$$
\begin{equation}
(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\,\smile\,\operatorname{cl}_X(H)\hookrightarrow \sum_{\substack{\delta\in\Delta\\ i_\delta\in\{1,\dots,m_\delta\}}} H^n(X_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta}).
\end{equation}
\tag{2.44}
$$
Therefore, by (2.39), (2.40), (2.42), (2.44) it suffices to show that
$$
\begin{equation}
K_{(2d-2+n)Y}\,\smile\, \sum_{\substack{\delta\in\Delta\\ i_\delta\in\{1,\dots,m_\delta\}}} (p_2\sigma)^\ast(H^n(X_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta}))=0.
\end{equation}
\tag{2.45}
$$
By the theorem on local invariant cycles and by the Künneth formula on fibres of a smooth morphism $\tau'\colon Y'=X'\times_{C'}X'\to C'$, we have the canonical decompositions
$$
\begin{equation*}
\begin{aligned} \, H^2(C,R^{2d-4+n}(\tau\sigma)_\ast\mathbb{Q}) &=\bigoplus_{p+q=2d-4+n} H^2(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q}R^q\pi'_\ast\mathbb{Q})), \\ H^1(C,R^{2d-3+n}(\tau\sigma)_\ast\mathbb{Q}) &=\bigoplus_{p+q=2d-3+n} H^1(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q}R^q\pi'_\ast\mathbb{Q})). \end{aligned}
\end{equation*}
\notag
$$
On the other hand, there is the section $e\colon C\to X$ of the structure morphism defined by the zero section $C\to\mathcal M$ of the Néron minimal model $\mathcal M\to C$. It yields the section $X\xrightarrow{x_s\mapsto e(s)\times x_s} X\times_CX$ of the canonical projection $p_2\colon X\times_CX\to X$, which identifies $X$ with a subvariety of the fibre product $X\times_CX$ and identifies the $\mathbb Q$-space $H^n(X,\mathbb Q)$ with the $\mathbb Q$-subspace
$$
\begin{equation*}
p_2^\ast(H^n(X,\mathbb{Q}))\hookrightarrow H^n(X\times_CX,\mathbb{Q}).
\end{equation*}
\notag
$$
Since $X_{\delta i_\delta}$ is an irreducible component of the divisor $\pi^{-1}(\Delta)$, it follows from (1.1) that the cohomology class $\operatorname{cl}_X(X_{\delta i_\delta})\in H^2(X,\mathbb{Q})$ vanishes on $X'=X\setminus \pi^{-1}(\Delta)$ in the sense of the theory of coniveau (arithmetic) filtration (see [47], § 1.B). On the other hand, from (2.43) and [41], Vol. II, Chap. 4, § 4.2.1, it follows that, for any point $s\in C'=C\setminus \Delta$, the $\smile$-product by the class
$$
\begin{equation*}
\omega\in\sigma^\ast p_2^\ast(H^n(X_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta}))\hookrightarrow H^{n+2}(Y,\mathbb{Q})
\end{equation*}
\notag
$$
acts on the fibre
$$
\begin{equation*}
H^p(X_s,\mathbb{Q})\otimes_\mathbb{Q} H^q(X_s,\mathbb{Q})=[j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q})]_s
\end{equation*}
\notag
$$
of the sheaf $j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q})$ as the $\smile$-product by the class
$$
\begin{equation*}
\iota^\ast_{X_s\times X_s/Y}(\omega)\in\iota^\ast_{X_s\times X_s/Y} (\sigma^\ast p_2^\ast(H^n(X_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta}))),
\end{equation*}
\notag
$$
where $\iota^\ast_{X_s\times X_s/Y}\colon X_s\times X_s\hookrightarrow Y$ is the canonical embedding. Since $p_2$ and $\sigma$ are $C$-morphisms and we have the equality
$$
\begin{equation*}
\iota^\ast_{X_s\times X_s/Y}(\sigma^\ast p_2^\ast(H^n(X_{\delta i_\delta},\mathbb{Q})\,\smile\, \operatorname{cl}_X(X_{\delta i_\delta})))=0,
\end{equation*}
\notag
$$
and so
$$
\begin{equation*}
j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q})\,\smile\,\sigma^\ast p_2^\ast (H^n(X_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta}))=0.
\end{equation*}
\notag
$$
Proceeding as in the proof of (2.29), for any element
$$
\begin{equation*}
\omega\in\sigma^\ast p_2^\ast (H^n(X_{\delta i_\delta},\mathbb{Q})\,\smile\, \operatorname{cl}_X(X_{\delta i_\delta}))\subseteq H^{n+2}(Y,\mathbb{Q})
\end{equation*}
\notag
$$
we see that the map of sheaves
$$
\begin{equation*}
j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q}) \xrightarrow{f\mapsto f\,\smile\,\omega} j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q})\,\smile\,\omega
\end{equation*}
\notag
$$
is the zero map, inducing (by the functoriality) the zero map of cohomology of degree $r\geqslant 1$ (see [41], Vol. II, Chap. 4, § 4.2.1, Lemma 4.13). Now from (2.27), (2.28) and (2.44) we have
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, &H^2(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q}R^q\pi'_\ast\mathbb{Q}))\,\smile\, (p_2\sigma)^\ast((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\smile\,\operatorname{cl}_X(H)) \\ &\ \qquad \hookrightarrow H^2(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q}R^q\pi'_\ast\mathbb{Q})) \\ &\ \qquad\qquad \smile\,(p_2\sigma)^\ast\Biggl(\sum_{\substack{\delta\in\Delta\\ i_\delta\in\{1,\dots,m_\delta\}}} H^n(X_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})\Biggr) \\ &\ =H^2(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q}R^q\pi'_\ast\mathbb{Q})) \\ &\ \qquad\qquad\smile\, \sigma^\ast p_2^\ast\Biggl(\sum_{\substack{\delta\in\Delta\\ i_\delta\in\{1,\dots,m_\delta\}}} H^n(X_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})\Biggr)=0, \end{aligned} \\ \begin{aligned} \, &H^1(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q}R^q\pi'_\ast\mathbb{Q}))\,\smile\, (p_2\sigma)^\ast((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\smile\,\operatorname{cl}_X(H)) \\ &\ \qquad \hookrightarrow H^1(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q}R^q\pi'_\ast\mathbb{Q})) \\ &\ \qquad\qquad\smile\, \sigma^\ast p_2^\ast\Biggl(\sum_{\substack{\delta\in\Delta\\ i_\delta\in\{1,\dots,m_\delta\}}} H^n(X_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta})\Biggr)=0, \end{aligned} \\ K_{(2d-2+n)Y}\,\smile\, (p_2\sigma)^\ast((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\, \smile\,\operatorname{cl}_X(H))=0. \end{gathered}
\end{equation*}
\notag
$$
This proves (2.45) and (2.42), and, therefore, Lemma 2.6. 2.7.By the condition of the theorem, the generic scheme fibre $\mathcal M_\eta$ of the Néron model is a principally polarized Abelian variety; consequently, for any point $s\in C'$, the Abelian variety $X_s$ has a principal polarization defined by certain ample divisor $H_s$ on the variety $X_s$. It is known that the Poincaré bundle $\mathcal P'_s$ on the variety $X_s\times \overset\vee{X}_s$ is defined (uniquely up to an isomorphism) by the following properties (see [52], Chap. 2, § 5): (a) $\mathcal P'_s|_{X_s\times\{L_s\}}\,\widetilde{\to}\,L_s$ for all $L_s\in\overset\vee{X}_s=\operatorname{Pic}(X_s)$, (b) $\mathcal P'_s|_{\{0\}\times\overset\vee{X}_s}\,\widetilde{\to}\,\mathcal O_{\overset\vee{X}_s}$. Since $X_s$ is an Abelian variety with a principal polarization, there exists an isomorphism $X_s\,\widetilde{\to}\,\operatorname{Pic}^0(X_s)=\overset\vee{X}_s$, which will be considered as an identification. It follows easily from properties (a) and (b) that an element $\operatorname{c}_1(\mathcal P'_s)\in H^2(X_s\times X_s,\mathbb{Q})$ has Künneth type $(1,1)$ (see [52], Chap. 14, Lemma 14.1.9), and so
$$
\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
$$
Besides, for any point $s\in C'$ outside some countable subset $\Delta_{\text{countable}}$, the group $G$ (defined in § 1.1) is a normal subgroup of the Hodge group $\operatorname{Hg}(X_s)=\operatorname{Hg}(H^1(X_s,\mathbb Q))$ of a rational Hodge structure $H^1(X_s,\mathbb Q)$ (see Theorem 7.3 in [48]). We fix such a point $s$. The existence of an inclusion $G \hookrightarrow \operatorname{Hg}(X_s)$ implies that the correspondence $\operatorname{c}_1(\mathcal P'_s)$ defines the section
$$
\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
$$
of type $(1,1)$ of a local system of Hodge structures $R^1\pi'_\ast\mathbb Q\,\otimes_\mathbb Q\, R^1\pi'_\ast\mathbb Q$ inducing the correspondence $\operatorname{c}_1(\mathcal P'_t)$ for any point $t\in C'$. By Deligne’s theorem, the canonical morphism $H^2(Y,\mathbb Q)\to H^0(C',R^2\tau'_\ast\mathbb Q)$ is a surjective morphism of Hodge $\mathbb Q$-structures (see [43], Theorem 4.1.1, the proof of Corollary 4.1.2). Since
$$
\begin{equation*}
\Lambda'_{1,1}\in H^0(C',R^1\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^1\pi'_\ast\mathbb{Q})\hookrightarrow H^0(C',R^2\tau'_\ast\mathbb{Q})
\end{equation*}
\notag
$$
is an element of Hodge type $(1,1)$, it follows from the Lefschetz theorem on divisors that there exists an algebraic $\mathbb Q$-cycle $D^{(1)}$ on the variety $Y$ such that the image of the class $\operatorname{cl}_Y(D^{(1)})\in H^2(Y,\mathbb{Q})\cap H^{1,1}(Y,\mathbb{C})$ with respect to the canonical surjective morphism $H^2(Y,\mathbb{Q})\to H^0(C',R^2\tau'_\ast\mathbb{Q})$ coincides with the section $\Lambda'_{1,1}$. It is known that, for any point $s\in C'$, the algebraic correspondence yields an algebraic isomorphism $H^{2d-n-1}(X_s,\mathbb{Q}) \, \widetilde{\to} \, H^{n-1}(X_s,\mathbb{Q})$ (see [2], Lemma 2A12, Remark 2A13, [52], Chap. 16, § 16.4). Therefore, the element ${\Lambda'_{1,1}}^{\smile\,n-1}$ yields an isomorphism of local systems $R^{2d-n-1}\pi'_\ast\mathbb Q\,\widetilde{\to}\, R^{n-1}\pi'_\ast\mathbb Q$. Using the algorithm of § 2.3 in [45], we obtain the isomorphism of bidegree $(n-d,n-d)$ of pure rational Hodge structures
$$
\begin{equation}
H^1(C,R^{2d-n-1}\pi_\ast\mathbb{Q})\xrightarrow[{\widetilde{\qquad}}]{[x\mapsto (p_2\sigma)_\ast((p_1\sigma)^\ast x\,\smile\,\operatorname{cl}_Y(D^{(1)})^{\smile\,n-1})]_1} H^1(C,R^{n-1}\pi_\ast\mathbb{Q}).
\end{equation}
\tag{2.46}
$$
2.8.Lemma. The algebraic class
$$
\begin{equation*}
u\stackrel{\mathrm{def}}{=} (\iota\sigma)_\ast \bigl[[\operatorname{cl}_Y(D^{(1)})]^{\smile\,n-1}\bigr]\in H^{2n}(X\times X,\mathbb{Q})
\end{equation*}
\notag
$$
defines an algebraic isomorphism
$$
\begin{equation*}
H^1(C,R^{2d-n-1}\pi_\ast\mathbb{Q}) \xrightarrow[\widetilde{\qquad}]{x\mapsto \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast x\,\smile\,u)} H^1(C,R^{n-1}\pi_\ast\mathbb{Q}).
\end{equation*}
\notag
$$
Proof. There is the canonical embedding (see [45], formula (2.13))
$$
\begin{equation}
(p_k\sigma)^\ast|_{K_{nX}}\colon K_{nX}\hookrightarrow K_{nY},
\end{equation}
\tag{2.47}
$$
induced (by surjectivity of the morphism $p_k\sigma\colon Y\to X$) by the canonical injection $(p_k\sigma)^\ast\colon H^n(X,\mathbb{Q})\hookrightarrow H^n(Y,\mathbb{Q})$ (see [2], Proposition 1.2.4).
We have the equivalence
$$
\begin{equation*}
\begin{gathered} \, \omega\in K_{nX}\ \Longleftrightarrow\ \forall \, s\in C\quad \iota_{X_s/X}^\ast(\omega)=0 \end{gathered}
\end{equation*}
\notag
$$
(see [45], formula (2.11)), and hence the equality
$$
\begin{equation*}
\iota_{X_s/X}^\ast(K_{nX}\,\smile\,\operatorname{cl}_X(H))=\iota_{X_s/X}^\ast(K_{nX})\,\smile\, \iota_{X_s/X}^\ast(\operatorname{cl}_X(H))=0
\end{equation*}
\notag
$$
(see [39], Chap. 2, § 8, formula (5)) yields the inclusion
$$
\begin{equation}
K_{nX}\,\smile\,\operatorname{cl}_X(H)\hookrightarrow K_{(n+2)X}.
\end{equation}
\tag{2.48}
$$
A similar argument shows that the map $K_{(2d-n)Y}\xrightarrow{\smile\,\operatorname{cl}_Y(D^{(1)})^{\smile\,n-1}}K_{(2d-2+n)Y}$ is well-defined.
Taking into account the functoriality of constructions under consideration, and using Lemma 2.6, and (2.47), we obtain the commutative diagram glued from the commutative diagrams andOn the other hand, by (2.25), (2.48), the strong Lefschetz theorem, and the theorem on local invariant cycles we have the canonical embeddings of rational Hodge structures
$$
\begin{equation*}
\begin{aligned} \, &H^1(C,R^{2d-n-1}\pi_\ast\mathbb{Q})=H^1(C,j_\ast R^{n-1}\pi'_\ast\mathbb{Q})\,\smile\, H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})^{d-n} \\ &\qquad=H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\,\smile\,\operatorname{cl}_X(H)^{d-n}\hookrightarrow K_{nX}\,\smile\,\operatorname{cl}_X(H)^{d-n}\hookrightarrow K_{(2d-n)X}. \end{aligned}
\end{equation*}
\notag
$$
Therefore, diagrams (2.49)–(2.52) yield the commutative diagram The composition of maps in the low row of diagram (2.53) yields (2.46) by the functoriality of cohomology (see [51], Chap. II, § 3, Theorem 3.11), and so it is clear that the composition of maps in the upper row of diagram (2.53) yields the isomorphism
$$
\begin{equation}
H^1(C,R^{2d-n-1}\pi_\ast\mathbb{Q})\xrightarrow[\widetilde{\qquad}]{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-1}\pi_\ast\mathbb{Q}).
\end{equation}
\tag{2.54}
$$
For any element $x\in H^{2d-n}(X,\mathbb Q)$, from the projection formula (see [2], § 1.2.A) we get the equalities
$$
\begin{equation*}
\begin{aligned} \, (p_2\sigma)_\ast\bigl((p_1\sigma)^\ast x\,\smile\,[\operatorname{cl}_Y(D^{(1)})]^{\smile\,n-1}\bigr) &= [\operatorname{pr}_2\iota\sigma]_\ast \bigl([\operatorname{pr}_1\iota\sigma]^\ast x\,\smile\,[\operatorname{cl}_Y(D^{(1)})]^{\smile\,n-1}\bigr) \\ &=\operatorname{pr}_{2\ast}(\iota\sigma)_\ast \bigl((\iota\sigma)^\ast\operatorname{pr}_1^\ast x\,\smile\,[\operatorname{cl}_Y(D^{(1)})]^{\smile\,n-1}\bigr) \\ &=\operatorname{pr}_{2\ast}\bigl(\operatorname{pr}_1^\ast x\,\smile\,(\iota\sigma)_\ast\bigl[[\operatorname{cl}_Y(D^{(1)})]^{\smile\,n-1}\bigr]\bigr), \end{aligned}
\end{equation*}
\notag
$$
and hence (2.54) assumes the form
$$
\begin{equation*}
\begin{gathered} \, H^1(C,R^{2d-n-1}\pi_\ast\mathbb{Q}) \xrightarrow[\widetilde{\qquad}]{x\mapsto \operatorname{pr}_{2\ast} (\operatorname{pr}_1^\ast x\,\smile\,u)} H^1(C,R^{n-1}\pi_\ast\mathbb{Q}). \end{gathered}
\end{equation*}
\notag
$$
Lemma 2.8 is proved. § 3. Proof of the theorem3.1.For $n\leqslant d-1$, we first assume that $n=2q$ is even. Let
$$
\begin{equation*}
\begin{gathered} \, u_{\Sigma_{n-2,0},\Sigma_{n-2,0}},\ u_{\Sigma_{n-2,0},\Sigma_{n-4,2}},\ \dots,\ u_{\Sigma_{n-2,0},\Sigma_{0,n-2}}, \\ u_{\Sigma_{n-2,0},H^1(C,R^{n-1}\pi_\ast\mathbb{Q})},\ u_{\Sigma_{n-2,0},H^0(C',R^n\pi'_\ast\mathbb{Q})},\ u_{\Sigma_{n-4,2},\Sigma_{n-2,0}},\ u_{\Sigma_{n-4,2},\Sigma_{n-4,2}},\ \dots, \\ u_{\Sigma_{n-4,2},\Sigma_{0,n-2}},\ u_{\Sigma_{n-4,2},H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}, \ u_{\Sigma_{n-4,2},H^0(C',R^n\pi'_\ast\mathbb{Q})},\ \dots, \\ u_{\Sigma_{0,n-2},\Sigma_{n-2,0}},\ u_{\Sigma_{0,n-2},\Sigma_{n-4,2}},\ \dots,\ u_{\Sigma_{0,n-2},\Sigma_{0,n-2}},\ u_{\Sigma_{0,n-2},H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}, \\ u_{\Sigma_{0,n-2},H^0(C',R^n\pi'_\ast\mathbb{Q})},\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-2,0}}, \\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-4,2}},\ \dots,\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{0,n-2}}, \\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),H^1(C,R^{n-1}\pi_\ast\mathbb{Q})},\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),H^0(C',R^n\pi'_\ast\mathbb{Q})}, \\ u_{H^0(C',R^n\pi'_\ast\mathbb{Q}),\Sigma_{n-2,0}},\ u_{H^0(C',R^n\pi'_\ast\mathbb{Q}),\Sigma_{n-4,2}},\ \dots,\ u_{H^0(C',R^n\pi'_\ast\mathbb{Q}),\Sigma_{0,n-2}}, \\ u_{H^0(C',R^n\pi'_\ast\mathbb{Q}),H^1(C,R^{n-1}\pi_\ast\mathbb{Q})},\ u_{H^0(C',R^n\pi'_\ast\mathbb{Q}),H^0(C',R^n\pi'_\ast\mathbb{Q})},\ h \end{gathered}
\end{equation*}
\notag
$$
be the components of the algebraic correspondence $u=(\iota\sigma)_\ast[[\operatorname{cl}_Y(D^{(1)})]^{\smile\,n-1}]$ in direct summands
$$
\begin{equation*}
\begin{gathered} \, \Sigma_{n-2,0}\otimes_\mathbb{Q} \Sigma_{n-2,0}; \ \dots; \ H^0(C',R^n\pi'_\ast\mathbb{Q})\otimes_\mathbb{Q}H^0(C',R^n\pi'_\ast\mathbb{Q}); \\ H_\mathbb{Q} \stackrel{\mathrm{def}}{=} \bigoplus_{p+q=2n,\, p\neq n} H^p(X,\mathbb{Q})\otimes_\mathbb{Q} H^q(X,\mathbb{Q}), \end{gathered}
\end{equation*}
\notag
$$
defined by decomposition (2.22) and by the Künneth decomposition of the $\mathbb Q$-space $H^{2n}(X\times X,\mathbb Q)$. It is clear that the operators
$$
\begin{equation*}
[p_{X/C}^{m!}]^\ast\otimes_\mathbb{Q}[1_{X/C}]^\ast= [\sigma_\ast\nu^\ast]\otimes_\mathbb{Q}[1_{X/C}]^\ast, \qquad [1_{X/C}]^\ast\otimes_\mathbb{Q}[p_{X/C}^{m!}]^\ast=[1_{X/C}]^\ast\otimes_\mathbb{Q}[\sigma_\ast\nu^\ast]
\end{equation*}
\notag
$$
transform the $\mathbb Q$-subspace $H_\mathbb{Q}\subset H^{2n}(X\times X,\mathbb{Q})$ into the space $H_\mathbb Q$, in addition, these operators transform algebraic cohomology classes into algebraic classes (see [2], Proposition 1.3.7). Using a description of the action of the operator $[p_{X/C}^{m!}]^\ast\colon H^n(X,\mathbb{Q})\to H^n(X,\mathbb{Q})$ on direct summands of decomposition (2.22), and acting by the operators
$$
\begin{equation*}
[p_{X/C}^{m!}]^\ast\otimes_\mathbb{Q}[1_{X/C}]^\ast, \qquad [1_{X/C}]^\ast\otimes_\mathbb{Q}[p_{X/C}^{m!}]^\ast
\end{equation*}
\notag
$$
on the algebraic class $u$ by the modified Lieberman method (see [2], Theorem 2A11) adapted to the case when the multiplication by the number $p^{m!}$ on the generic scheme fibre defines a rational map of the variety $X$, it is easy to verify that, for some elements $h_j\in H_\mathbb Q$, the classes
$$
\begin{equation*}
\begin{gathered} \, u_{\Sigma_{n-2,0},\Sigma_{n-2,0}}+h_1,\ u_{\Sigma_{n-2,0},\Sigma_{n-4,2}}+h_2,\ \dots,\ u_{\Sigma_{n-2,0},\Sigma_{0,n-2}}+h_{n-2}, \\ u_{\Sigma_{n-2,0},H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}+h_{n-1},\ u_{\Sigma_{n-2,0},H^0(C',R^n\pi'_\ast\mathbb{Q})}+h_n, \\ u_{\Sigma_{n-4,2},\Sigma_{n-2,0}}+h_{n+1},\ u_{\Sigma_{n-4,2},\Sigma_{n-4,2}}+h_{n+2},\ \dots,\ u_{\Sigma_{n-4,2},\Sigma_{0,n-2}}+h_{2n-2}, \\ u_{\Sigma_{n-4,2},H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}+h_{2n-1},\ u_{\Sigma_{n-4,2},H^0(C',R^n\pi'_\ast\mathbb{Q})}+h_{2n}, \\ \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \\ u_{\Sigma_{0,n-2},\Sigma_{n-2,0}}+h_{n^2-3n+1},\ u_{\Sigma_{0,n-2},\Sigma_{n-4,2}}+h_{n^2-3n+2},\ \dots, \\ u_{\Sigma_{0,n-2},\Sigma_{0,n-2}}+h_{n^2-2n-2},\ u_{\Sigma_{0,n-2},H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}+h_{n^2-2n-1}, \\ u_{\Sigma_{0,n-2},H^0(C',R^n\pi'_\ast\mathbb{Q})}+h_{n^2-2n},\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-2,0}}+h_{n^2-2n+1}, \\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-4,2}}+h_{n^2-2n+2},\ \dots,\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{0,n-2}}+h_{n^2-n-2}, \\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}+h_{n^2-n-1},\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),H^0(C',R^n\pi'_\ast\mathbb{Q})}+h_{n^2-n}, \\ u_{H^0(C',R^n\pi'_\ast\mathbb{Q}),\Sigma_{n-2,0}}+h_{n^2-n+1},\ u_{H^0(C',R^n\pi'_\ast\mathbb{Q}),\Sigma_{n-4,2}}+h_{n^2-n+2},\ \dots, \\ u_{H^0(C',R^n\pi'_\ast\mathbb{Q}),\Sigma_{0,n-2}}+h_{n^2-2},\ u_{H^0(C',R^n\pi'_\ast\mathbb{Q}),H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}+h_{n^2-1}, \\ u_{H^0(C',R^n\pi'_\ast\mathbb{Q}),H^0(C',R^n\pi'_\ast\mathbb{Q})}+h_{n^2} \end{gathered}
\end{equation*}
\notag
$$
are algebraic. The element $v=\alpha\otimes\beta\in H^\ast(X,\mathbb{Q})\otimes_\mathbb{Q} H^\ast(X,\mathbb{Q})=H^\ast(X\times X,\mathbb{Q})$ corresponds to the $\mathbb Q$-linear map
$$
\begin{equation*}
v^\ast\colon H^\ast(X,\mathbb{Q})\to H^\ast(X,\mathbb{Q}),
\end{equation*}
\notag
$$
defined by $v^\ast(\gamma)=\langle\gamma\,\smile\,\alpha\rangle\beta$ (see [2], § 1.3), and so it is clear that
$$
\begin{equation*}
\operatorname{pr}_{2\ast}(H^i(X,\mathbb{Q})\otimes_\mathbb{Q} H^\ast(X,\mathbb{Q}))=0\quad\text{for all } \ i\neq 2d.
\end{equation*}
\notag
$$
Therefore, the $\mathbb Q$-space $H^{2d-n}(X,\mathbb Q)$ is annihilated by the correspondences from the $\mathbb Q$-space $H_\mathbb Q$. From (2.26) and (2.31) we have
$$
\begin{equation*}
H^1(C,R^{2d-n-1}\pi_\ast\mathbb{Q})\,\smile\,(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})=0.
\end{equation*}
\notag
$$
Finally,
$$
\begin{equation*}
\begin{aligned} \, &H^1(C,R^{2d-n-1}\pi_\ast\mathbb{Q})\,\smile\,H^0(C',R^n\pi'_\ast\mathbb{Q}) \\ &\qquad=H^1(C,j_\ast R^{2d-n-1}\pi'_\ast\mathbb{Q})\,\smile\,H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})\hookrightarrow H^1(C,j_\ast R^{2d-1}\pi'_\ast\mathbb{Q})=0. \end{aligned}
\end{equation*}
\notag
$$
Hence by (2.14), (2.22) and Lemma 2.8, the algebraic correspondence
$$
\begin{equation*}
\begin{aligned} \, &u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-2,0}}+h_{n^2-2n+1}+u_{H^1(C,R^{n-1}\pi_\ast \mathbb{Q}),\Sigma_{n-4,2}} \\ &\qquad +h_{n^2-2n+2}+\dots +u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{0,n-2}}+h_{n^2-n-2} \\ &\qquad +u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}+h_{n^2-n-1} \\ &\qquad +u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),H^0(C',R^n\pi'_\ast\mathbb{Q})}+h_{n^2-n} \end{aligned}
\end{equation*}
\notag
$$
yields the algebraic isomorphism $H^1(C,R^{2d-n-1}\pi_\ast\mathbb{Q})\,\widetilde{\to}\, H^1(C,R^{n-1}\pi_\ast\mathbb{Q})$. In addition, this correspondence annihilates direct summands
$$
\begin{equation*}
\begin{gathered} \, \Sigma_{n-2,0}\,\smile\,H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})^{\smile\,d-n},\ \Sigma_{n-4,2} \,\smile\,H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})^{\smile\,d-n},\ \dots, \\ H^0(C,j_\ast R^{2d-n}\pi'_\ast\mathbb{Q}) \end{gathered}
\end{equation*}
\notag
$$
of decomposition (2.26) of the $\mathbb Q$-space $H^{2d-n}(X,\mathbb Q)$, because by (1.7), (2.11) and (2.31) we have
$$
\begin{equation*}
\begin{aligned} \, &(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n} \,\smile\,H^1(C,R^{n-1}\pi_\ast\mathbb{Q})=0, \\ &H^0(C,j_\ast R^{2d-n}\pi'_\ast\mathbb{Q})\,\smile\,H^1(C,R^{n-1}\pi_\ast\mathbb{Q}) \\ &\qquad=H^0(C,j_\ast R^{2d-n}\pi'_\ast\mathbb{Q})\,\smile\,H^1(C,j_\ast R^{n-1}\pi'_\ast\mathbb{Q}) \hookrightarrow H^1(C,j_\ast R^{2d-1}\pi'_\ast\mathbb{Q})=0. \end{aligned}
\end{equation*}
\notag
$$
As a result, we have the equality
$$
\begin{equation}
\begin{aligned} \, &[[(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\, \smile\,\operatorname{cl}_X(H)^{\smile\,d-n}]\oplus H^0(C,j_\ast R^{2d-n}\pi'_\ast\mathbb{Q})] \nonumber \\ &\qquad\smile\,H^1(C,R^{n-1}\pi_\ast\mathbb{Q})=0. \end{aligned}
\end{equation}
\tag{3.1}
$$
From these facts, (2.22), from the non-degeneracy of the bilinear form
$$
\begin{equation*}
\Phi\colon H^n(X,\mathbb{Q})\times H^n(X,\mathbb{Q})\xrightarrow{x\times y\mapsto \langle x\,\smile\,y\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n}\rangle} \mathbb{Q}
\end{equation*}
\notag
$$
(see [2], § 1.2.A), and since the restriction
$$
\begin{equation*}
\begin{aligned} \, \Psi_1 &\stackrel{\mathrm{def}}{=}\Phi|_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}\colon H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\times H^1(C,R^{n-1}\pi_\ast\mathbb{Q}) \\ &\xrightarrow{x\times y\mapsto \langle x\,\smile\,y\,\smile\,\operatorname{cl}_X(H)^{\smile\,d-n}\rangle} \mathbb{Q} \end{aligned}
\end{equation*}
\notag
$$
is non-degenerate (see [42], Proposition (10.5)), it follows that the restriction $\Psi_2$ of the form $\Phi$ to the $\mathbb Q$-subspace
$$
\begin{equation*}
(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})
\end{equation*}
\notag
$$
is also non-degenerate, where by (2.35) the $\mathbb{Q}$-space $H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})$ is $1$-dimensional and is generated by an algebraic class by surjectivity of the canonical map $H^{2q}(X,\mathbb{Q})\to H^0(C',R^{2q}\pi'_\ast\mathbb{Q})$ of rational Hodge structures (see [43], Theorem 4.1.1, the proof of Corollary 4.1.2). By (2.14) we have the decomposition
$$
\begin{equation}
\begin{aligned} \, &(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q}) \nonumber \\ &\qquad =\Sigma_{2q-2,0}\oplus \Sigma_{2q-4,2}\oplus \dots\oplus \Sigma_{0,2q-2}\oplus H^0(C,j_\ast R^{2q}\pi'_\ast\mathbb{Q}) \nonumber \\ &\qquad=\sum_{\substack{r\in\{0,\dots,q-1\}, \, \delta\in\Delta\\ i_\delta\in\{1,\dots,m_\delta\}}} H^{2q-2-2r}(A_\delta,\mathbb{Q})\otimes_\mathbb{Q} H^{2r}(Z_{\delta i_\delta},\mathbb{Q}) \,\smile\, \operatorname{cl}_X(X_{\delta i_\delta}) \nonumber \\ &\qquad\qquad\oplus \,H^0(C,j_\ast R^{2q}\pi'_\ast\mathbb{Q}). \end{aligned}
\end{equation}
\tag{3.2}
$$
Since $X_{\delta i_\delta}$ is an irreducible component of the divisor $\pi^{-1}(\Delta)$, it follows from (1.1) that the cohomology class $\operatorname{cl}_X(X_{\delta i_\delta})\in H^2(X,\mathbb Q)$ vanishes on $X'=X\setminus \pi^{-1}(\Delta)$ in the sense of the theory of coniveau (arithmetic) filtration (see [47], § 1.B); in addition, the elements
$$
\begin{equation*}
\begin{aligned} \, \operatorname{cl}_X(X_{\delta i_\delta}) &\in \operatorname{Ker}[H^2(X,\mathbb{Q})\xrightarrow{\varphi_2} H^2(X',\mathbb{Q})] \\ &\qquad=(i_\Delta f)_\ast H^0(Z,\mathbb{Q}),\qquad i_\delta\in\{1,\dots,m_\delta\}, \end{aligned}
\end{equation*}
\notag
$$
generate a $\mathbb Q$-subspace of dimension at least $2$ in $(i_\Delta f)_\ast H^0(Z,\mathbb{Q})\subseteq H^2(X,\mathbb{Q})$, because cohomology classes of irreducible components of a singular fibre $X_\delta$ and the class $\operatorname{cl}_X(H)$ of a hyperplane section generate a subgroup of $\operatorname{NS}(X)$ of rank at least $3$ (see [44], § 2.14). By the strong Lefschetz theorem and (1.1), the elements
$$
\begin{equation*}
\begin{aligned} \, \operatorname{cl}_X(X_{\delta i_\delta})\,\smile\,\operatorname{cl}_X(H)^{q-1}&\in \operatorname{Ker}[H^{2q}(X,\mathbb{Q})\xrightarrow{\varphi_{2q}} H^{2q}(X',\mathbb{Q})] \\ &\qquad=(i_\Delta f)_\ast H^{2q-2}(Z,\mathbb{Q}), \qquad i_\delta\in\{1,\dots,m_\delta\}, \end{aligned}
\end{equation*}
\notag
$$
generate a $\mathbb Q$-subspace of dimension at least $2$. Therefore, by (2.35) we obtain the inequality
$$
\begin{equation*}
\dim_\mathbb{Q}[(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})]\geqslant 3.
\end{equation*}
\notag
$$
Consequently, by results of § 1.2 in [26], there is a unique (up to a non-zero scalar multiple) Poincaré class
$$
\begin{equation*}
\wp((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})),
\end{equation*}
\notag
$$
generating the $1$-dimensional space of invariants of the diagonal action of the group
$$
\begin{equation*}
\operatorname{SO}(((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})),\Psi_2)
\end{equation*}
\notag
$$
on the tensor square of the $\mathbb Q$-space $(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})$. Recall that the $\mathbb Q$-space $H^{2r}(Z_{\delta i_\delta},\mathbb Q)$ is generated by classes of algebraic cycles on the smooth projective toric variety $Z_{\delta i_\delta}$ (see [34], Theorem 10.8) and the $1$-dimensional $\mathbb Q$-space $H^0(C,j_\ast R^{2q}\pi'_\ast\mathbb Q)$ is generated by the image of an algebraic class on the variety $X$ with respect to the canonical surjective morphism of Hodge $\mathbb Q$-structures $H^{2q}(X,\mathbb{Q})\to H^0(C,j_\ast R^{2q}\pi'_\ast\mathbb{Q})=H^0(C',R^{2q}\pi'_\ast\mathbb{Q})$ (see [43], Theorem (4.1.1)). By Condition (B) of the theorem, the $\mathbb Q$-space of Hodge cycles on the Abelian variety $A_\delta \times A_{\delta'}$, $\delta,\delta'\in\Delta$ is generated by classes of algebraic cycles. Now decomposition (3.2) shows that the Poincaré class $\wp((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q}))$ is algebraic, because this class is a Hodge cycle (see [26], § 1.2). On the other hand, decomposition (2.22) and the non-degeneracy of symmetric bilinear forms $\Psi_1$, $\Psi_2$ yield the canonical embedding of algebraic groups
$$
\begin{equation*}
\begin{aligned} \, &\operatorname{SO}\bigl(H^1(C,R^{n-1}\pi_\ast\mathbb{Q}), \Psi_1\bigr) \times \operatorname{SO}\bigl((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q}),\Psi_2\bigr) \\ &\qquad\hookrightarrow \operatorname{SO}(H^n(X,\mathbb{Q}),\Phi), \end{aligned}
\end{equation*}
\notag
$$
which in turn gives the inclusion
$$
\begin{equation}
\begin{aligned} \, \mathbb{Q}\cdot\wp(H^n(X,\mathbb{Q})) &\hookrightarrow H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\otimes_\mathbb{Q} H^1(C,R^{n-1}\pi_\ast\mathbb{Q}) \nonumber \\ &\qquad+\mathbb{Q}\cdot\wp((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})). \end{aligned}
\end{equation}
\tag{3.3}
$$
The correspondence $\wp(H^n(X,\mathbb Q))$ defines the isomorphism
$$
\begin{equation*}
H^{2d-n}(X,\mathbb{Q})\xrightarrow[\widetilde{\qquad}]{x\mapsto \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast(x)\,\smile\,\wp(H^n(X,\mathbb{Q})))} H^n(X,\mathbb{Q})
\end{equation*}
\notag
$$
(see [26], § 1.2), and so from (2.14), (2.26), (3.1), (3.3) it follows that the algebraic Poincaré class
$$
\begin{equation*}
\wp((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q}))
\end{equation*}
\notag
$$
defines the algebraic isomorphism
$$
\begin{equation*}
\begin{aligned} \, &(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\,\smile\,\operatorname{cl}_X(H)^{d-n}\oplus H^0(C,j_\ast R^{2d-n}\pi'_\ast\mathbb{Q}) \\ &\qquad\widetilde{\to}\, (i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q}). \end{aligned}
\end{equation*}
\notag
$$
Therefore, the algebraic correspondence
$$
\begin{equation*}
\begin{gathered} \, u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-2,0}}+h_{n^2-2n+1}+u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-4,2}} +h_{n^2-2n+2}+\cdots \\ {}+u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{0,n-2}}+h_{n^2-n-2} \\ {}+u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}+h_{n^2-n-1}+u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),H^0(C',R^n\pi'_\ast\mathbb{Q})} \\ {}+h_{n^2-n} +\wp((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\oplus H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})) \end{gathered}
\end{equation*}
\notag
$$
yields an algebraic isomorphism $H^{2d-n}(X,\mathbb{Q})\,\widetilde{\to}\,H^n(X,\mathbb{Q})$. 3.2.We finally assume that the number $n=2q+1$ is odd. Let
$$
\begin{equation*}
\begin{gathered} \, u_{\Sigma_{n-2,0},\Sigma_{n-2,0}},\ u_{\Sigma_{n-2,0},\Sigma_{n-4,2}},\ \dots,\ u_{\Sigma_{n-2,0},\Sigma_{1,n-3}},\ u_{\Sigma_{n-2,0},H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}, \\ u_{\Sigma_{n-4,2},\Sigma_{n-2,0}},\ u_{\Sigma_{n-4,2},\Sigma_{n-4,2}},\ \dots,\ u_{\Sigma_{n-4,2},\Sigma_{1,n-3}},\ u_{\Sigma_{n-4,2},H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}, \\ \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \\ u_{\Sigma_{1,n-3},\Sigma_{n-2,0}},\ u_{\Sigma_{1,n-3},\Sigma_{n-4,2}},\ \dots,\ u_{\Sigma_{1,n-3},\Sigma_{1,n-3}},\ u_{\Sigma_{1,n-3},H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}, \\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-2,0}},\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-4,2}},\ \dots,\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{1,n-3}}, \\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),H^1(C,R^{n-1}\pi_\ast\mathbb{Q})},\ h \end{gathered}
\end{equation*}
\notag
$$
be the components of the algebraic correspondence $u=(\iota\sigma)_\ast[[\operatorname{cl}_Y(D^{(1)})]^{\smile\,n-1}]$ in direct summands
$$
\begin{equation*}
\begin{gathered} \, \Sigma_{n-2,0}\otimes_\mathbb{Q} \Sigma_{n-2,0}; \ \dots; \ H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\otimes_\mathbb{Q}H^1(C,R^{n-1}\pi_\ast\mathbb{Q}); \\ H_\mathbb{Q} \stackrel{\mathrm{def}}{=}\bigoplus_{p+q=2n,\, p\neq n} H^p(X,\mathbb{Q})\otimes_\mathbb{Q} H^q(X,\mathbb{Q}), \end{gathered}
\end{equation*}
\notag
$$
defined by decomposition (2.23) and by the Künneth decomposition of the $\mathbb Q$-space $H^{2n}(X\times X,\mathbb Q)$. Taking into account a description of the action of the operator
$$
\begin{equation*}
[p_{X/C}^{m!}]^\ast\colon H^n(X,\mathbb{Q})\to H^n(X,\mathbb{Q})
\end{equation*}
\notag
$$
on direct summands of decompositions (2.23) and acting by the operators
$$
\begin{equation*}
[p_{X/C}^{m!}]^\ast\otimes_\mathbb{Q}[1_{X/C}]^\ast,\qquad [1_{X/C}]^\ast\otimes_\mathbb{Q}[p_{X/C}^{m!}]^\ast
\end{equation*}
\notag
$$
on the algebraic class $u$ by the modified Lieberman method (see [2], Theorem 2A11), it is easy to see that, for some elements $h_j\in H_\mathbb Q$, the classes
$$
\begin{equation*}
\begin{gathered} \, u_{\Sigma_{n-2,0},\Sigma_{n-2,0}}+h_1,\ u_{\Sigma_{n-2,0},\Sigma_{n-4,2}}+h_2,\ \dots,\ u_{\Sigma_{n-2,0},\Sigma_{1,n-3}}+h_{n-3}, \\ u_{\Sigma_{n-2,0},H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}+h_{n-2},\ u_{\Sigma_{n-4,2},\Sigma_{n-2,0}}+h_{n-1},\ u_{\Sigma_{n-4,2},\Sigma_{n-4,2}}+h_n,\ \dots, \\ u_{\Sigma_{n-4,2},\Sigma_{1,n-3}}+h_{2n-5},\ u_{\Sigma_{n-4,2},H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}+h_{2n-4}, \\ \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-2,0}}+h_{(n-2)^2-n+2},\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-4,2}}+h_{(n-2)^2-n+3},\ \dots, \\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{0,n-2}}+h_{(n-2)^2-1},\ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}+h_{(n-2)^2} \end{gathered}
\end{equation*}
\notag
$$
are algebraic. From (2.26) and (2.31) we have
$$
\begin{equation*}
H^1(C,R^{2d-n-1}\pi_\ast\mathbb{Q})\,\smile\,(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})=0.
\end{equation*}
\notag
$$
Hence by (2.15), (2.23) and by Lemma 2.8, the algebraic correspondence
$$
\begin{equation*}
\begin{aligned} \, &u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-2,0}}+h_{(n-2)^2-n+2} +u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-4,2}}+h_{(n-2)^2-n+3} \\ &\qquad+\dots+u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{0,n-2}}+h_{(n-2)^2-1} \\ &\qquad+u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}+h_{(n-2)^2} \end{aligned}
\end{equation*}
\notag
$$
yields the algebraic isomorphism $H^1(C,R^{2d-n-1}\pi_\ast\mathbb{Q})\,\widetilde{\to}\,H^1(C,R^{n-1}\pi_\ast\mathbb{Q})$. Since the number $n$ is odd, it follows in the case under consideration that (2.35) and (2.36) imply the equality (3.1). From the results of § 1.2 in [26] it follows that there is the correctly defined Poincaré class which generates the $1$-dimensional space of invariants of the diagonal action of the symplectic group
$$
\begin{equation*}
\operatorname{Sp}((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q}),\Psi_2)
\end{equation*}
\notag
$$
on the tensor square of the $\mathbb Q$-space
$$
\begin{equation*}
\begin{aligned} \, &(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})=\Sigma_{2q-1,0}\oplus \Sigma_{2q-3,2}\oplus \dots\oplus \Sigma_{1,2q-2} \\ &\qquad=\sum_{\substack{r\in\{0,\dots,q-1\}, \, \delta\in\Delta\\ i_\delta\in\{1,\dots,m_\delta\}}} H^{2q-1-2r}(A_\delta,\mathbb{Q})\otimes_\mathbb{Q} H^{2r}(Z_{\delta i_\delta},\mathbb{Q})\,\smile\,\operatorname{cl}_X(X_{\delta i_\delta}). \end{aligned}
\end{equation*}
\notag
$$
Proceeding as in § 3.1, we see that the Poincaré class $\wp((i_\Delta f)_\ast H^{n-2}(Z,\mathbb Q))$ is algebraic by Condition (B) of the theorem. On the other hand, decomposition (2.23) and the non-degeneracy of skew-symmetric bilinear forms $\Psi_1$, $\Psi_2$ yield the canonical embedding of algebraic groups
$$
\begin{equation*}
\operatorname{Sp}\bigl(H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Psi_1\bigr) \times \operatorname{Sp}\bigl((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q}),\Psi_2\bigr) \hookrightarrow \operatorname{Sp}(H^n(X,\mathbb{Q}),\Phi),
\end{equation*}
\notag
$$
which, in turn, yields the inclusion
$$
\begin{equation}
\begin{aligned} \, \mathbb{Q}\cdot\wp(H^n(X,\mathbb{Q})) &\hookrightarrow H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\otimes_\mathbb{Q} H^1(C,R^{n-1}\pi_\ast\mathbb{Q}) \nonumber \\ &\qquad+\mathbb{Q}\cdot\wp((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})). \end{aligned}
\end{equation}
\tag{3.4}
$$
The correspondence $\wp(H^n(X,\mathbb Q))$ defines the isomorphism
$$
\begin{equation*}
H^{2d-n}(X,\mathbb{Q})\xrightarrow[\widetilde{\qquad}]{x\mapsto \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast(x) \,\smile\,\wp(H^n(X,\mathbb{Q})))} H^n(X,\mathbb{Q})
\end{equation*}
\notag
$$
(see [26], § 1.2) and hence from (2.15), (2.26), (3.1), (3.4) it follows that the algebraic Poincaré class defines the algebraic isomorphism
$$
\begin{equation*}
(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})\,\smile\,\operatorname{cl}_X(H)^{d-n} \,\widetilde{\to}\, (i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q}).
\end{equation*}
\notag
$$
Therefore, the algebraic correspondence
$$
\begin{equation*}
\begin{aligned} \, &u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-2,0}}+h_{(n-2)^2-n+2} +u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{n-4,2}}+h_{(n-2)^2-n+3} \\ &\qquad+\dots+u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),\Sigma_{0,n-2}}+h_{(n-2)^2-1} \\ &\qquad+ u_{H^1(C,R^{n-1}\pi_\ast\mathbb{Q}),H^1(C,R^{n-1}\pi_\ast\mathbb{Q})}+h_{(n-2)^2} +\wp((i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q})) \end{aligned}
\end{equation*}
\notag
$$
yields an algebraic isomorphism $H^{2d-n}(X,\mathbb Q)\,\widetilde{\to}\,H^n(X,\mathbb Q)$. The theorem is proved.
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