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On the standard conjecture for a fourfold with S. G. Tankeev Vladimir State University
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2024, Volume 88, Issue 2, DOI: https://doi.org/10.4213/im9481 
Bibliographic databases:
    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 Lefschetz type conjecture $B(X)$ asserts that there exists an algebraic $\mathbb{Q}$-cycle $Z$ on the Cartesian product $X\times X$ which determines the inverse algebraic isomorphism
$$
\begin{equation*}
H^{2d-i}(X,\mathbb{Q})\underset{\widetilde{\qquad}} {\xrightarrow{x\,{\mapsto} \operatorname{pr}_{2\ast} (\operatorname{pr}_1^\ast x\smile\operatorname{cl}_{X\times X}(Z))}} H^i(X,\mathbb{Q}).
\end{equation*}
\notag
$$
It is known that the Lefschetz theorem on $(1, 1)$-classes 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 operators $\ast$ and $\Lambda$ of Hodge theory (see Proposition 2.3 in [2]), to the coincidence of the numerical and homological equivalences of algebraic cycles on the Cartesian product $X\times X$ (see formula (1.11) in [3]), to the semi-simplicity (see Proposition 1.7 in [4]) of the $\mathbb{Q}$-algebra $\mathcal A(X)=\operatorname{cl}_{X\times X}(\operatorname{CH}^\ast (X\times X))\otimes_\mathbb{Z}\mathbb{Q}$ of algebraic self-correspondences on the variety $X$ with the bilinear composition law (see § 1.3.1 in [2]), $g\circ f=\operatorname{pr}_{13\ast}(\operatorname{pr}_{12}^\ast(f) \smile\operatorname{pr}_{23}^\ast(g))$. On the other hand, $B(X)\Rightarrow C(X)$, where the standard Künneth type conjecture $C(X)$ asserts the algebraicity of Künneth components of the class of the diagonal $\Delta_X\hookrightarrow X\times X$ (see Lemma 2.4 in [2]). Finally, the conjecture $B(X)$ is compatible with monoidal transformations along smooth centres (see Theorem 4.3 in [5]). It is known that the standard conjecture $B(X$) is true for all smooth complex projective curves, surfaces, Abelian varieties [6], and threefolds of Kodaira dimension $\varkappa(X) < 3$ [7] (in particular, it holds for all complex elliptic threefolds and for compactifications of Néron minimal models of Abelian surfaces over fields of algebraic functions of one variable with the field of constants $\mathbb{C}$). In addition, $B(X$) holds for Hilbert schemes of points on surfaces (see Corollary 7.5 in [8]) and for hyperkähler varieties deformation equivalent to Hilbert schemes of points of $K3$ surfaces [9]. In this article, we prove the following main result. Theorem. The Grothendieck standard conjecture $B(X)$ of Lefschetz type holds for a smooth complex projective $4$-dimensional variety $X$ provided that there exists a morphism of $X$ onto a smooth projective curve whose generic scheme fibre is an Abelian variety with bad semi-stable reduction at some place of the curve. The author is grateful to a referee for essential improvements of the original text. § 1. Reduction of the proof to a construction of certain algebraic isomorphism1.1.Let $\mathcal M\to C$ be the Néron minimal model of an Abelian variety $\mathcal M_\eta$ over the field $\kappa(\eta)$ of rational functions of a smooth projective curve $C$ with the generic scheme point $\eta$ (by the definition, the Néron model represents the functor $S\mapsto\operatorname{Hom}(S_\eta,\mathcal M_\eta)$ on the category of smooth $C$-schemes $S\to C$ (see formula (1.1.1) in [10]). According to Künnemann (see [11], § 5.8, and [12], §§ 1.9, 4.1, 4.2, 4.4, 4.5, Theorem 4.6), after the base change determined by an appropriate ramified covering $\widetilde{C}\to C$, we may assume that, for the Néron minimal model $\mathcal M\to C$, there exists (not necessary unique) smooth compactification $X$ of the variety $\mathcal M$ which is flat and projective over the curve $C$ such that the following conditions hold: (i) the model $X/C$ has strictly semi-stable reductions (in particular, all fibres of the structure morphism $\pi\colon X\to C$ are unions of smooth irreducible components of multiplicity 1 with normal crossings); (ii) the variety $X$ contains the variety $\mathcal M$ as an open dense subscheme; (iii) the restriction $\pi|_{\mathcal M}\colon \mathcal M\to C$ coincides with the structure morphism of the Néron model; (iv) the connected component $\mathcal M^0_s$ of the neutral element of any fibre $\mathcal M_s$, $s \in C$ is an extension of an Abelian variety by a linear torus of dimension $r_s$; (v) the $C$-group law $\mathcal M^0\times_C\mathcal M^0\to\mathcal M^0$ can be expanded to a group $C$-action $\mathcal M^0\times_C X\to X$. Such compactifications of the Néron model will be called Künnemann compactifications. 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 has no non-trivial constant Abelian subscheme.It is known that if $\dim_{\kappa(\eta)}\mathcal M_\eta=3$ and the endomorphism ring
$$
\begin{equation*}
\operatorname{End}_{\overline{\kappa(\eta)}}(\mathcal M_\eta \otimes_{\kappa(\eta)}\overline{\kappa(\eta)})
\end{equation*}
\notag
$$
of the generic geometric fibre is not the order of an imaginary quadratic field, then the standard conjecture $B(X)$ is true [13]. 1.2.From now on, we assume that $\dim_{\kappa(\eta)}\mathcal M_\eta=3$ and $\operatorname{End}_{\overline{\kappa(\eta)}}(\mathcal M_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)}\,)$ is an order of an imaginary quadratic field. Since the standard conjecture holds for smooth projective surfaces and it is compatible with monoidal transformations along smooth centres, it follows from Hironaka results that, for any rational dominant map $X- \to Y$ of smooth projective varieties of dimensions $4$, we have $B(X)\Rightarrow B(Y)$ (see Lemma 1.1 in [7]). Therefore, we may assume that $X$ is a Künnemann compactification of the Néron minimal model $\mathcal M$ and Indeed, for the base change determined by 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 a point $\widetilde{s}\in\widetilde{C}$ lying over the point $s\in C$ is isomorphic to the connected component of the neutral element $\mathcal M_s^0$ of the Néron model $\mathcal M\to C$ (see [10], Corollary 3.3, Corollary 3.9); in particular, the toric rank is preserved under a base change. As a result, the condition of the theorem on the existence of bad semi-stable reduction of the generic scheme fibre is preserved under a base change; it remains to note that in the case under consideration $\operatorname{rank} \operatorname{NS}(X)\geqslant 3$ (see [14], formula (2.24)), because cohomology classes of irreducible components of a singular fibre and the class $\operatorname{cl}_X(H)$ of a hyperplane section generate a subgroup of $\operatorname{NS}(X)$ of rank at least $3$ (see [14], § 2.14). Let $\pi'\colon X'\to C'$ be the smooth part of the structure morphism $\pi\colon X\to C$, $\Delta=C\setminus C'$, and let $C'\stackrel{j}{\hookrightarrow} C$ the canonical embedding. One may assume that the closure $G$ of the image of the global monodromy in the Zariski topology of the group $\operatorname{GL}(H^1(X_s, \mathbb{Q})$) is a connected $\mathbb{Q}$-group, and, by the theorem on monodromy (see Theorem (6.1) in [15], local monodromies (Picard–Lefschetz transformations) are unipotent. In addition, we may assume that $\operatorname{End}_{\kappa(\eta)}(X_\eta)= \operatorname{End}_{\overline{\kappa(\eta)}}(X_{\overline\eta})$, and the Abelian variety $X_\eta$ over the field $\kappa(\eta)$ of rational functions of the curve $C$ is an absolutely simple Abelian variety with principal polarization, the ring $\operatorname{End}_{\kappa(\eta)}(X_\eta)$ is an order of an imaginary quadratic field, the trace of the Abelian scheme $\pi'\colon X'\to C'$ is trivial (otherwise the variety $X$ is rationally dominated by a product of a three-dimensional Abelian variety and a curve, so that, for every factor the standard conjecture holds, therefore, $B(X)$ is true by the well-known compatibility of the standard conjecture with Cartesian products (see Corollary 2.5 in [2]).Consider the canonical diagrams of fibre products Let $\iota\colon X\times_CX \hookrightarrow X\times X$ be the canonical embedding, $\sigma\colon Y \to X\times_CX$ be a resolution of singularities of the variety $X\times_CX$. One may assume that $\sigma$ induces an isomorphism over $C'$. In particular, $Y$ may be considered as a smooth projective compactification of the fibre product $X'\times_{C'}X'$. In addition, after the base change determined by certain ramified covering $\widetilde{C}\to C$, we may assume in view of Hironaka’s results and by the theorem on the existence of a Künnemann model of the generic scheme fibre of the Abelian scheme $X'\times_{C'}X'\to C'$ that, for all points $s\in C$, the fibre $Y_s$ is a union of smooth irreducible components of multiplicity $1$ with normal crossings. 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 $f\colon Z\to\pi^{-1}(\Delta)$ is a resolution of singularities of the closed subscheme $i_\Delta\colon \pi^{-1}(\Delta)\hookrightarrow X$, we have the equality of mixed Hodge $\mathbb{Q}$-structures (see Corollary (8.2.8) in [16])
$$
\begin{equation}
(i_\Delta f)_\ast H^{n-2}(Z,\mathbb{Q}) = \operatorname{Ker}[H^n(X,\mathbb{Q}) \xrightarrow{\varphi_n}H^n(X',\mathbb{Q})],
\end{equation}
\tag{1.1}
$$
where $(i_\Delta f)_\ast$ is a morphism of bidegree $(1,1)$ of pure Hodge structures and $\varphi_n$ is the restriction map. 1.3.We claim that
$$
\begin{equation}
H^0(C',R^1\pi'_\ast\mathbb{Q})=H^0(C',R^5\pi'_\ast\mathbb{Q})=0,
\end{equation}
\tag{1.2}
$$
Indeed, since $R^1\pi'_\ast\mathbb{Q}$ is a polarizable family of Hodge $\mathbb{Q}$-structures of weight 1, there exists an isomorphism of families of Hodge $\mathbb{Q}$-structures (see [17], § 4.2.3)
$$
\begin{equation*}
R_1\pi'_\ast\mathbb{Q} \stackrel{\mathrm{def}}{=} [R^1\pi'_\ast\mathbb{Q}]^\vee\,\widetilde{\to}\,R^1\pi'_\ast\mathbb{Q}(1).
\end{equation*}
\notag
$$
It is evident that $H^0(C',R^1\pi'_\ast\mathbb{Q})=0$, because otherwise by the results of Deligne, Grothendieck and Katz (see [17], § 4.4.3, Corollary 4.1.2, (4.1.3.2), (4.1.3.3)), there exists a non-trivial constant Hodge substructure
$$
\begin{equation*}
\mathcal H_\mathbb{Z}\hookrightarrow R_1\pi'_\ast\mathbb{Z} \stackrel{\mathrm{def}}{=}[R^1\pi'_\ast\mathbb{Z}]^\vee
\end{equation*}
\notag
$$
of type $(-1,0)+(0,-1)$ on $C'$ corresponding to a non-trivial constant Abelian subscheme of the Abelian scheme $\pi'\colon X'\to C'$ (see § 4.4.3 in [17]), contradicting the assumption on the triviality of the trace. The Poincaré duality on fibres of the smooth morphisms $\pi'\colon X'\to C'$ yields an isomorphism of local systems $R^5\pi'_\ast\mathbb{Q}\,\widetilde{\to}\,R^1\pi'_\ast\mathbb{Q}$, therefore,
$$
\begin{equation*}
H^0(C',R^5\pi'_\ast\mathbb{Q})\,\widetilde{\to}\, H^0(C',R^1\pi'_\ast\mathbb{Q})=0.
\end{equation*}
\notag
$$
On the other hand, natural $\smile$-products (together with polarization of the local system $R^n\pi'_\ast\mathbb{Q}$) determine a non-degenerate pairing (see Proposition (10.5) in [18])
$$
\begin{equation*}
H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})\times H^2(C,j_\ast R^n\pi'_\ast\mathbb{Q})\to H^2(C,\mathbb{Q})\,\widetilde{\to}\,\mathbb{Q}.
\end{equation*}
\notag
$$
By the theorem on local invariant cycles, the canonical map $R^n\pi_\ast\mathbb{Q}\to j_\ast R^n\pi'_\ast\mathbb{Q}$ is surjective with the kernel concentrated on the finite set $\Delta$ (see Proposition (15.12) in [18] and § 3 in [19]). Therefore,
$$
\begin{equation}
H^2(C,R^n\pi_\ast\mathbb{Q})=H^2(C,j_\ast R^n\pi'_\ast\mathbb{Q})\, \widetilde{\to}\, H^0(C,j_\ast R^n\pi'_\ast\mathbb{Q})^\vee=H^0(C',R^n\pi'_\ast\mathbb{Q})^\vee.
\end{equation}
\tag{1.4}
$$
Now (1.3) follows from (1.2). 1.4.We claim that Indeed, there exists a countable subset $\Delta_{\mathrm{countable}}\subset C'$ such that, for any point $s\in C'\setminus \Delta_{\mathrm{countable}}$, the closure $G$ of the image of the monodromy representation $\pi_1(C',s)\to\operatorname{GL}(H^1(X_s,\mathbb{Q}))$ in the Zariski topology of the group $\operatorname{GL}(H^1(X_s,\mathbb{Q}))$ is a connected semi-simple (see Corollary 4.2.9 in [17]) normal (see Theorem 7.3 in [20]) subgroup of the Hodge group (see Definition B.51 in [21]) $\operatorname{Hg}(X_s)\stackrel{\mathrm{def}}{=} \operatorname{Hg}(H^1(X_s,\mathbb{Q}))$ of the Abelian variety $X_s$. Since the trace of the Abelian variety $X_\eta$ is trivial, the equality yields the canonical isomorphism (see Corollary 4.4.13 in [17])
$$
\begin{equation}
\operatorname{End}_{C'}(X')\,\widetilde{\to} \operatorname{End}_{C'}(R^1\pi'_\ast\mathbb{Z}),
\end{equation}
\tag{1.6}
$$
which, by the existence of determined by a choice of a point $s\in C'\setminus\Delta_{\mathrm{countable}}$ canonical inclusions $\operatorname{Im}[\pi_1(C',s)\to \operatorname{GL}(H^1(X_s,\mathbb{Q}))] \hookrightarrow G\hookrightarrow\operatorname{Hg}(X_s)$ and the well-defined equality $\operatorname{End}_{\operatorname{Hg}(X_s)}H^1(X_s,\mathbb{Q}) = \operatorname{End}_\mathbb{C}(X_s)\otimes_\mathbb{Z}\mathbb{Q}$ (see Lemma B.60 in [21]), defines canonical maps
$$
\begin{equation}
\begin{aligned} \, &\operatorname{End}_{\kappa(\eta)}(X_\eta)\otimes_\mathbb{Z}\mathbb{Q}\, \widetilde{\to} \operatorname{End}_{\pi_1(C',s)}H^1(X_s,\mathbb{Q}) \nonumber \\ &\qquad\hookleftarrow \operatorname{End}_{\operatorname{Hg}(X_s)}H^1(X_s,\mathbb{Q}) = \operatorname{End}_\mathbb{C}(X_s)\otimes_\mathbb{Z}\mathbb{Q}. \end{aligned}
\end{equation}
\tag{1.7}
$$
The specialization map $\operatorname{End}_{\kappa(\eta)}(X_\eta) \otimes_\mathbb{Z}\mathbb{Q} \to\operatorname{End}_\mathbb{C}(X_s) \otimes_\mathbb{Z}\mathbb{Q}$ is injective. Therefore it follows from (1.7) that
$$
\begin{equation}
\operatorname{End}_{\kappa(\eta)}(X_\eta)\otimes_\mathbb{Z}\mathbb{Q}\, \widetilde{\to} \operatorname{End}_\mathbb{C}(X_s)\otimes_\mathbb{Z}\mathbb{Q} \quad \forall\, s\in C'\setminus\Delta_{\mathrm{countable}}.
\end{equation}
\tag{1.8}
$$
Consequently, the Abelian variety $X_s$ is simple because, by construction, the ring $\operatorname{End}_{\kappa(\eta)}(X_\eta)\otimes_\mathbb{Z}\mathbb{Q}$ is an imaginary quadratic field coinciding with the field $F\stackrel{\mathrm{def}}{=}\operatorname{End}_\mathbb{C}(X_s) \otimes_\mathbb{Z}\mathbb{Q}$. Therefore, the Hodge group $\operatorname{Hg}(X_s)$ is obtained by the Weil restriction of the field of scalars from $F$ to $\mathbb{Q}$ from the unitary group over the field $F$ associated to certain non-degenerate $F$-Hermitian form $\psi\colon H_1(X_s,\mathbb{Q}) \times H_1(X_s,\mathbb{Q})\to F$ (see § (2.3), Type IV(1,1) in [22]). It easily follows from the Schur lemma, from the existence of the isomorphism $\operatorname{NS}(X_s)\,\widetilde{\to}\,\mathbb{Z}$ (see § 21 in [23]) and from (1.6) that, in N. Bourbaki’s notation (see Ch. VIII, § 13, § 1 in [24]), the pair
$$
\begin{equation*}
(\text{type }\operatorname{Lie} G\otimes_\mathbb{Q}\overline{\mathbb{Q}},\, H^1(X_s,\mathbb{Q})\otimes_\mathbb{Q}\overline{\mathbb{Q}})
\end{equation*}
\notag
$$
takes the value $(A_2,E(\omega_1)+E(\omega_1)^\vee)$. Therefore, by the Schur lemma,
$$
\begin{equation*}
\begin{aligned} \, &H^0(C',R^3\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\, [\wedge^3(E(\omega_1)+E(\omega_1)^\vee)]^{\operatorname{Lie} G \otimes_\mathbb{Q}\overline{\mathbb{Q}}} \\ &\qquad=[E(0)^{\oplus 2} +E(\omega_1)^\vee \otimes_{\overline{\mathbb{Q}}}\, E(\omega_1)^\vee +E(\omega_1)\otimes_{\overline{\mathbb{Q}}} E(\omega_1)]^{\operatorname{Lie} G\otimes_\mathbb{Q}\overline{\mathbb{Q}}} \\ &\qquad=E(0)^{\oplus 2}+ \operatorname{Hom}_{\operatorname{Lie} G \otimes_\mathbb{Q}\overline{\mathbb{Q}}}(E(\omega_1),E(\omega_1)^\vee)+ \operatorname{Hom}_{\operatorname{Lie} G\otimes_\mathbb{Q} \overline{\mathbb{Q}}}(E(\omega_1)^\vee,E(\omega_1)) \\ &\qquad=E(0)^{\oplus 2}. \end{aligned}
\end{equation*}
\notag
$$
This proves (1.5). 1.5.By construction, 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 determined 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 determined (uniquely, up to an isomorphism) by the following properties (see § 2.5 in [25]): (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 principal polarization, we have the isomorphism $X_s\,\widetilde{\to}\operatorname{Pic}^0(X_s)=\overset\vee{X}_s$, which will be regarded in future as an identification. It follows easily from the above 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 Lemma 14.1.9 in [25]), so that
$$
\begin{equation*}
\begin{aligned} \, \operatorname{c}_1(\mathcal P'_s)&\in [H^1(X_s,\mathbb{Q}) \otimes_\mathbb{Q} H^1(X_s,\mathbb{Q})] \cap H^{1,1} (X_s\times X_s,\mathbb{C}) \\ &\qquad=[H^1(X_s,\mathbb{Q})\otimes_\mathbb{Q} H^1(X_s,\mathbb{Q})]^{\operatorname{Hg}(X_s)}. \end{aligned}
\end{equation*}
\notag
$$
Finally, the correspondence $\operatorname{c}_1(\mathcal P'_s)$ induces the algebraic isomorphism (see § 2A1(ii), Theorem 2A9 in [2] and § 16.4 in [25])
$$
\begin{equation*}
H^5(X_s,\mathbb{Q}) \underset{\widetilde{\qquad}}{\xrightarrow{x\,{\mapsto} \operatorname{pr}_{2s\ast}(\operatorname{pr}^\ast_{1s}(x)\, \smile\operatorname{c}_1(\mathcal P'_s))}}H^1(\operatorname{Pic}^0(X_s), \mathbb{Q})=H^1(X_s,\mathbb{Q}).
\end{equation*}
\notag
$$
In addition, for any point $s\in C'$ outside some countable subset $\Delta_{\mathrm{countable}}$, the group $G$ is a normal $\mathbb{Q}$-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 [20]). We fix such a point $s$. It follows from the existence of an inclusion $G\hookrightarrow\operatorname{Hg}(X_s)$ that the correspondence $\operatorname{c}_1(\mathcal P'_s)$ determines 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 Theorem 4.1.1, the proof of Corollary 4.1.2 in [17]). Since $\Lambda'_{1,1}\in H^0(C',R^1\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^1\pi'_\ast\mathbb{Q})\subset H^0(C',R^2\tau'_\ast\mathbb{Q})$ 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 $\Lambda'_{1,1}$. We will use the $\mathbb{Q}$-divisor $D^{(1)}$ in §§ 1.9 and 2.2. 1.6.Let $X$ be a smooth projective $d$-dimensional variety over the field $\mathbb{C}$. It is well known that the Hodge decomposition
$$
\begin{equation*}
V_\mathbb{Q}\otimes_\mathbb{Q}\mathbb{C}=\bigoplus_{p+q=n}V^{p,q}_\mathbb{C}
\end{equation*}
\notag
$$
of the Hodge $\mathbb{Q}$-substructure $V_\mathbb{Q}\hookrightarrow H^n(X,\mathbb{Q})$ yields the action $h_1\colon U^1\to\operatorname{GL}(V_\mathbb{R})$ of the group $U^1\stackrel{\mathrm{def}}{=}\{e^{i\theta}|\theta\in\mathbb{R}\}$ on the real space $V_\mathbb{R}\stackrel{\mathrm{def}}{=} V_\mathbb{Q}\otimes_\mathbb{Q}\mathbb{R}$ such that $h_1(e^{i\theta})(v^{p,q})= e^{-i\theta(p-q)}\cdot v^{p,q}$ for any element $v^{p,q}\in V_\mathbb{C}^{p,q}$. By definition, the Hodge group $\operatorname{Hg}(V_\mathbb{Q})$ is the smallest algebraic $\mathbb{Q}$-subgroup of $\operatorname{GL}(V_\mathbb{Q})$ whose group of $\mathbb{R}$-points contains the group $h_1(U^1)$ (see Definition B.51 in [21]). It is known that the group $\operatorname{Hg}(V_\mathbb{Q})$ is a connected reductive group, and in the case $(r-l)n=2p$ the space of invariants $[V_\mathbb{Q}^{\otimes\,r} \otimes_\mathbb{Q}( V_\mathbb{Q}^{\vee})^{\otimes\,l}]^{\operatorname{Hg}(V_\mathbb{Q})}$ coincides with the space of Hodge cycles $[V_\mathbb{Q}^{\otimes\,r} \otimes_\mathbb{Q}(V_\mathbb{Q}^{\vee})^{\otimes\,l}]\cap [V_\mathbb{Q}^{\otimes\,r}\otimes_\mathbb{Q} (V_\mathbb{Q}^{\vee})^{\otimes\,l}]_\mathbb{C}^{p,p}$ (see Corollary B.55 in [21]). 1.7.For a natural number $n\leqslant d$, by the strong Lefschetz theorem and by the Poincaré duality, the bilinear form
$$
\begin{equation*}
\Phi\colon H^n(X,\mathbb{Q})\times H^n(X,\mathbb{Q}) \xrightarrow{x\times y\mapsto\langle x\,\smile\,y\, \smile\operatorname{cl}_X(H)^{\smile \,d-n}\rangle}\mathbb{Q}
\end{equation*}
\notag
$$
is non-degenerate (see § 1.2A in [2]), where $\langle\ \rangle\colon H^\ast(X,\mathbb{Q})\to \mathbb{Q}$ is the degree map defined as zero on $H^n(X,\mathbb{Q})$ for $n<2d$ and as the orientation isomorphism $\langle\ \rangle\colon H^{2d}(X,\mathbb{Q})\,\widetilde{\to}\,\mathbb{Q}$ on $H^{2d}(X,\mathbb{Q})$. Since the group $U^1$ acts trivially on $H^{1,1}(X,\mathbb{C})$ and $\operatorname{NS}(X)\otimes_\mathbb{Z}\mathbb{Q}$ is naturally included into $H^{1,1}(X,\mathbb{C})$ by the Lefschetz theorem on divisors, we have (with trivial action of $U^1$ on $\mathbb{R}$)
$$
\begin{equation*}
\begin{aligned} \, \forall\, \sigma\in U^1\quad \Phi_\mathbb{R}(x,y) &= [\Phi_\mathbb{R}(x,y)]^\sigma= \langle x\smile\operatorname{cl}_X(H)^{d-n}y\rangle^{\sigma} \\ &=\langle x^\sigma \smile\operatorname{cl}_X(H)^{d-n} \smile y^\sigma\rangle= \Phi_\mathbb{R}(x^\sigma,y^\sigma). \end{aligned}
\end{equation*}
\notag
$$
Therefore, the form $\Phi_\mathbb{R}$ is $U^1$-invariant, so there exists a canonical embedding
$$
\begin{equation*}
\operatorname{Hg}(H^n(X,\mathbb{Q}))\hookrightarrow \operatorname{Aut}(\Phi)^0= \begin{cases} \operatorname{Sp}(\Phi) &\text{for odd } n, \\ \operatorname{SO}(\Phi) &\text{for even } n, \end{cases}
\end{equation*}
\notag
$$
and $\Phi$ is a $\operatorname{Hg}(H^n(X,\mathbb{Q}))$-invariant form. If $H^n(X,\mathbb{Q}) \neq 0$, it is well-known that $\operatorname{Lie}\operatorname{Aut}(\Phi)^0$ is a semisimple Lie algebra over $\mathbb{Q}$ and the $\mathbb{Q}$-space $H^n(X,\mathbb{Q})$ is an absolutely irreducible $\operatorname{Aut}(\Phi)^0$-module with exception in the case where $n$ is even and $\dim_\mathbb{Q} H^n(X,\mathbb{Q})\,{=}\,2$ (see Ch. I, §§ 6, 7, Proposition 9 in [24]). If the number $n$ is odd or $n$ is even and $\dim_\mathbb{Q} H^n(X,\mathbb{Q})\neq 2$, then it follows from the existence of the canonical inclusion $\operatorname{Hg}(H^n(X,\mathbb{Q}))\hookrightarrow \operatorname{Aut}(\Phi)^0$ and from the Schur lemma that the $1$-dimensional $\mathbb{Q}$-space $[H^n(X,\mathbb{Q})\otimes_\mathbb{Q} H^n(X,\mathbb{Q})]^{\operatorname{Aut}(\Phi)^0}$ of invariants of the diagonal action $\sigma(x\otimes y)=\sigma(x)\otimes\sigma(y)$ of the group $\operatorname{Aut}(\Phi)^0$ is generated by a Hodge cycle $\wp(H^n(X,\mathbb{Q}))$, which is called the Poincaré class (it is defined uniquely up to a non-zero scalar multiple). It is clear that this class determines an isomorphism of $[\operatorname{Aut}(\Phi)]^0$-modules $H^n(X,\mathbb{Q})^\vee\,\widetilde{\to}\,H^n(X,\mathbb{Q})$, which is the composite (see § 1.3 in [2])
$$
\begin{equation}
\begin{aligned} \, H^{2d-n}(X,\mathbb{Q}) &\xrightarrow{\operatorname{pr}_1^\ast} H^{2d-n}(X,\mathbb{Q}) \otimes_\mathbb{Q}H^0(X,\mathbb{Q}) \nonumber \\ &\xrightarrow{\smile\,\wp(H^n(X,\mathbb{Q}))}H^{2d}(X,\mathbb{Q}) \otimes_\mathbb{Q}H^n(X,\mathbb{Q}) \xrightarrow{\operatorname{pr}_{2\ast}} H^n(X,\mathbb{Q}). \end{aligned}
\end{equation}
\tag{1.9}
$$
A restriction of the form $\Phi$ to a non-trivial rational Hodge substructure $V_\mathbb{Q}\hookrightarrow H^n(X,\mathbb{Q})$ may be degenerate (for example, if $X_s$ is a smooth fibre of a morphism $\pi\colon X\to C$ of the variety $X$ of dimension $d\geqslant 2$ onto a smooth projective curve $C$, then, by the equality $\operatorname{cl}_X(X_s) \smile \operatorname{cl}_X(X_s)=0$, the restriction of the form $\Phi\colon H^2(X,\mathbb{Q})\times H^2(X,\mathbb{Q})\to\mathbb{Q}$ to the non-trivial rational Hodge substructure $\mathbb{Q}\cdot \operatorname{cl}_X(X_s)\hookrightarrow H^2(X,\mathbb{Q})$ is trivial). Nevertheless, if a restriction $\Phi|_{V_\mathbb{Q}}$ is non-degenerate, then there is a decomposition of Hodge $\mathbb{Q}$-structures (see Ch. IX, § 4, § 1, Corollary of Proposition 1 in [26])
$$
\begin{equation*}
H^n(X,\mathbb{Q})=V_\mathbb{Q}\oplus V^\perp_\mathbb{Q},
\end{equation*}
\notag
$$
where $V^\perp_\mathbb{Q}$ is the orthogonal complement of the $\mathbb{Q}$-space $V_\mathbb{Q}$ with respect to the form $\Phi$ and the Poincaré classes (which are also Hodge cycles)
$$
\begin{equation*}
\begin{aligned} \, \wp(V_\mathbb{Q}) &\in [V_\mathbb{Q}\otimes_\mathbb{Q} V_\mathbb{Q}]^{[\operatorname{Aut}(\Phi|_{V_\mathbb{Q}})]^0}, \\ \wp(V^\perp_\mathbb{Q}) &\in [V^\perp_\mathbb{Q}\otimes_\mathbb{Q} V^\perp_\mathbb{Q}]^{[\operatorname{Aut} (\Phi|_{V^\perp_\mathbb{Q}})]^0} \end{aligned}
\end{equation*}
\notag
$$
are well defined under the assumption that the number $n$ is odd or $n$ is even and $\dim_\mathbb{Q}V_\mathbb{Q}\neq 2$, $\dim_\mathbb{Q}V^\perp_\mathbb{Q}\neq 2$. 1.8.Theorem. Let $X$ be a smooth $d$-dimensional complex projective variety, $\pi\colon X\to C$ be a surjective morphism onto a smooth curve $C$, every geometric fibre $X_s$ be a union of smooth varieties of multiplicity $1$ with normal crossings, and $\pi'\colon X'\to C'$ be a smooth part of the morphism $\pi$. Assume that the space $H^0(C',R^2\pi'_\ast\mathbb{Q})$ of invariant cycles is a rational Hodge structure of type $(1,1)$ and that, for the generic geometric fibre $X_{\overline\eta}$, the standard conjecture $B(X_{\overline\eta})$ of Lefschetz type holds. Then there exists an algebraic isomorphism $H^{2d-2}(X,\mathbb{Q})\,\widetilde{\to}\,H^2(X,\mathbb{Q})$. A proof of this theorem is a verbatim repetition of that of Theorem 1.2 in [27], where the case $d=3$ is explored. 1.9.Now we return to a Künnemann model $\pi\colon X\to C$. Recall that, by a construction, $\operatorname{rank}\operatorname{NS}(X)\geqslant 3$. Taking into account the arguments of § 1.4, it is easy to see by the Schur lemma that
$$
\begin{equation*}
\begin{aligned} \, &H^0(C',R^2\pi'_\ast\overline{\mathbb{Q}})\,\widetilde{\to}\, [\wedge^2(E(\omega_1)+E(\omega_1)^\vee)]^{\operatorname{Lie} G \otimes_\mathbb{Q}\overline{\mathbb{Q}}} \\ &\qquad=[\wedge^2 E(\omega_1)+E(\omega_1)\otimes_{\overline{\mathbb{Q}}} E(\omega_1)^\vee+\wedge^2E(\omega_1)^\vee]^{\operatorname{Lie} G \otimes_\mathbb{Q}\overline{\mathbb{Q}}} \\ &\qquad=[E(\omega_2)+E(\omega_1)\otimes_{\overline{\mathbb{Q}}} E(\omega_1)^\vee+E(\omega_2)^\vee]^{\operatorname{Lie} G \otimes_\mathbb{Q}\overline{\mathbb{Q}}} \\ &\qquad=\operatorname{Hom}_{\operatorname{Lie} G\otimes_\mathbb{Q} \overline{\mathbb{Q}}}(E(\omega_1)^\vee,E(\omega_1)^\vee)=E(0). \end{aligned}
\end{equation*}
\notag
$$
Thus the $1$-dimensional Hodge $\mathbb{Q}$-structure $H^0(C',R^2\pi'_\ast\mathbb{Q})$ has type $(1,1)$, therefore, the standard algorithms (see § 1.2 in [27] and § 2 in [28]) show that in the case under consideration an algebraic isomorphism mentioned in Theorem 1.8 may be defined by the formula
$$
\begin{equation*}
H^6(X,\mathbb{Q})\underset{\widetilde{\qquad}}{\xrightarrow{x\,\mapsto\, \operatorname{pr}_{2\ast}(\operatorname{pr}^\ast_1(x)\,\smile\, ((\iota\sigma)_\ast\operatorname{cl}_Y(D^{(1)})-n_{2,2} + \wp(\operatorname{NS}_\mathbb{Q}(X))))}}H^2(X,\mathbb{Q}),
\end{equation*}
\notag
$$
where: $\bullet$ $\iota\sigma\colon Y\to X\times X$ is the composite of a resolution of singularities of the variety $X\,{\times_C}\,X$ and the canonical embedding $\iota\colon X\times_CX\hookrightarrow X\times X$; $\bullet$ 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 $\Lambda'_{1,1}$, inducing the correspondence $\operatorname{c}_1(\mathcal P'_t)$ for any point $t\in C'$ and for the Poincaré bundle $\mathcal P'_t$ on $X_t\times \overset{\vee}{X}_t$; $\bullet$ $\wp(\operatorname{NS}_\mathbb{Q}(X))$ is the Poincaré class of Hodge $\mathbb{Q}$-structure $\operatorname{NS}_\mathbb{Q}(X)= \operatorname{NS}(X)\otimes_\mathbb{Z}\mathbb{Q}$ (which is isomorphic by the Lefschetz theorem on divisors to the algebraic part of Hodge $\mathbb{Q}$-structure $H^2(X,\mathbb{Q})=\operatorname{NS}_\mathbb{Q}(X)\oplus T^2_\mathbb{Q}(X)$ (where we identify $\operatorname{NS}_\mathbb{Q}(X)$ with the sum of all $1$-dimensional Hodge $\mathbb{Q}$-substructures and $T^2_\mathbb{Q}(X)$ (the transcendental part) is the sum of irreducible Hodge $\mathbb{Q}$-substructures of dimension greater than $1$); $\bullet$ $n_{2,2}$ is the $\operatorname{NS}_\mathbb{Q}(X) \otimes_\mathbb{Q}\operatorname{NS}_\mathbb{Q}(X)$-component in the canonical decomposition of Künneth’s $H^2(X,\mathbb{Q})\otimes_\mathbb{Q} H^2(X,\mathbb{Q})$-component of the algebraic class
$$
\begin{equation*}
(\iota\sigma)_\ast\operatorname{cl}_Y(D^{(1)})\in H^4(X\times X,\mathbb{Q}).
\end{equation*}
\notag
$$
Using the arguments of § 1.7, (1.9) and the Lefschetz theorem on divisors, we see that the algebraic cycle
$$
\begin{equation*}
\wp(H^1(X,\mathbb{Q}))\in [H^1(X,\mathbb{Q})\otimes_\mathbb{Q} H^1(X,\mathbb{Q})]\cap H^{1,1}(X\times X,\mathbb{C})= \operatorname{NS}_\mathbb{Q}(X\times X)
\end{equation*}
\notag
$$
determines an algebraic isomorphism $H^7(X,\mathbb{Q})\,\widetilde{\to}\,H^1(X,\mathbb{Q})$. Since the coefficients of the characteristic polynomial of any endomorphism of the $\mathbb{Q}$-space $H^i(X,\mathbb{Q})$ are rational numbers, by Theorem 2.9 in [2] it suffices to construct an algebraic isomorphism $H^5(X,\mathbb{Q})\,\widetilde{\to}\,H^3(X,\mathbb{Q})$.
§ 2. Some isomorphisms and decompositions of rational Hodge structures2.1.From now on we denote by
$$
\begin{equation*}
K_{nX}\stackrel{\mathrm{def}}{=}\operatorname{Ker}[H^n(X,\mathbb{Q})\to H^0(C,R^n\pi_\ast\mathbb{Q})]
\end{equation*}
\notag
$$
the kernel of the edge map of the Leray spectral sequence $E_2^{p,q}(\pi)$ of the structure morphism $\pi\colon X\to C$. Also set
$$
\begin{equation*}
K_{nY}\stackrel{\mathrm{def}}{=}\operatorname{Ker}[H^n(Y,\mathbb{Q})\to H^0(C,R^n(\tau\sigma)_\ast\mathbb{Q})].
\end{equation*}
\notag
$$
The Leray spectral sequences
$$
\begin{equation*}
E_2^{p,q}(\pi)=H^p(C,R^q\pi_\ast\mathbb{Q})\quad \text{and}\quad E_2^{p,q}(\tau\sigma)=H^p(C,R^q(\tau\sigma)_\ast\mathbb{Q})
\end{equation*}
\notag
$$
are degenerate: $E_2^{p,q}=E_\infty^{p,q}$ (see Corollary (15.15) in [18]), so that, for any natural number $n$, there are exact sequences of Hodge $\mathbb{Q}$-structures (see formula (2.4) in [14])
$$
\begin{equation}
0\to H^2(C,R^{n-2}\pi_\ast\mathbb{Q})\to K_{nX}\xrightarrow{\alpha_{nX}} H^1(C,R^{n-1}\pi_\ast\mathbb{Q})\to 0,
\end{equation}
\tag{2.1}
$$
$$
\begin{equation}
0\to H^2(C,R^{n-2}(\tau\sigma)_\ast\mathbb{Q})\to K_{nY} \xrightarrow{\alpha_{nY}} H^1(C,R^{n-1}(\tau\sigma)_\ast\mathbb{Q})\to 0,
\end{equation}
\tag{2.2}
$$
and, by (1.3), the sequence (2.1) yields the identification In addition, there is the canonical embedding (see formula (1.21) in [29]) 2.2.For any point $s\in C'$, the correspondence $\operatorname{c}_1(\mathcal P'_s)^{\smile\,2}$ yields the algebraic isomorphism $H^4(X_s,\mathbb{Q})\,\widetilde{\to}\, H^2(X_s,\mathbb{Q})$ (see Lemma 2A12, Remark 2A13 in [2] and § 16.4 in [25])). Therefore, the section ${\Lambda'_{1,1}}^{\smile\,2}$ yields the isomorphism of local systems $R^4\pi'_\ast\mathbb{Q}\,\widetilde{\to}\, R^2\pi'_\ast\mathbb{Q}$, determined by the composite of maps
$$
\begin{equation}
R^4\pi'_\ast\mathbb{Q} \xrightarrow{(p'_1)^\ast} R^4\pi'_\ast\mathbb{Q} \otimes{\pi'_\ast\mathbb{Q}} \xrightarrow{\smile{\Lambda'_{1,1}}^{\smile \,2}} R^6\pi'_\ast\mathbb{Q} \otimes{R^2\pi'_\ast\mathbb{Q}} \xrightarrow{(p'_2)_\ast} R^2\pi'_\ast\mathbb{Q}.
\end{equation}
\tag{2.5}
$$
From the compatibility of the $\smile$-products with the Leray spectral sequences $E_2^{p,q}(\tau\sigma)=H^p(C,R^q(\tau\sigma)_\ast\mathbb{Q})$ (see [30], Vol II, Ch. 4, § 4.2.1, formula (4.5), Lemma 4.13), and the standard algorithm (see [31], § 2.3, Construction of formula (2.10)) it follows that we cam expand (2.5) to a sequence of maps
$$
\begin{equation*}
R^4\pi_\ast\mathbb{Q}\xrightarrow{(p_1\sigma)^\ast}R^4(\tau\sigma)_\ast \mathbb{Q}\xrightarrow{\,\smile\operatorname{cl}_Y(D^{(1)})^{\smile\,2}} R^8(\tau\sigma)_\ast\mathbb{Q}\xrightarrow{(p_2\sigma)_\ast} R^2\pi_\ast\mathbb{Q},
\end{equation*}
\notag
$$
so that their composite is an isomorphism outside the finite set $\Delta$; in turn, these maps yield a sequence of canonical maps of cohomology
$$
\begin{equation}
\begin{aligned} \, &H^1(C,R^4\pi_\ast\mathbb{Q})\xrightarrow{[(p_1\sigma)^\ast]_1} H^1(C,R^4(\tau\sigma)_\ast\mathbb{Q}) \nonumber \\ &\qquad\xrightarrow{[\smile\operatorname{cl}_Y(D^{(1)})^{\smile\,2}]_1} H^1(C,R^8(\tau\sigma)_\ast\mathbb{Q})\xrightarrow{[(p_2\sigma)_\ast]_1} H^1(C,R^2\pi_\ast\mathbb{Q}). \end{aligned}
\end{equation}
\tag{2.6}
$$
By the property of the functoriality (see Ch. II, Theorem 3.11 in [32]), the composite of these maps coincides with the canonical map
$$
\begin{equation}
H^1(C,R^4\pi_\ast\mathbb{Q})\xrightarrow{[x\,\mapsto\, (p_2\sigma)_\ast((p_1\sigma)^\ast x\,\smile\operatorname{cl}_Y(D^{(1)})^{\smile\,2})]_1} H^1(C,R^2\pi_\ast\mathbb{Q}),
\end{equation}
\tag{2.7}
$$
corresponding to the morphism of sheaves
$$
\begin{equation}
R^4\pi_\ast\mathbb{Q}\xrightarrow{x\,\mapsto\, (p_2\sigma)_\ast((p_1\sigma)^\ast x\,\smile\operatorname{cl}_Y (D^{(1)})^{\smile\,2})}R^2\pi_\ast\mathbb{Q}.
\end{equation}
\tag{2.8}
$$
Since the kernel and the cokernel of the map (2.8) are concentrated on $\Delta$, their higher cohomology vanish, therefore, the map (2.7) is surjective. On the other hand, in the notation of § 1.1, the strong Lefschetz theorem on fibres of a smooth morphism $\pi'$ determines the isomorphism of sheaves
$$
\begin{equation}
j_\ast R^2\pi'_\ast \mathbb{Q}\underset{\widetilde{\qquad}} {\xrightarrow{\smile\,\operatorname{cl}_X(H)}}j_\ast R^4\pi'_\ast\mathbb{Q}.
\end{equation}
\tag{2.9}
$$
Finally, by the theorem on local invariant cycles (see § 3 in [19] and Proposition (15.12) in [18]), the canonical map $R^p\pi_\ast\mathbb{Q}\to j_\ast R^p\pi'_\ast\mathbb{Q}$ is surjective, with kernel concentrated on the finite set $\Delta$. Hence there is a canonical isomorphism $H^1(C,R^p\pi_\ast\mathbb{Q})\,\widetilde{\to}\, H^1(C,j_\ast R^p\pi'_\ast\mathbb{Q})$. Therefore, by (2.7) and (2.9), the surjective map (2.7) is an isomorphism of bidegree $(-1,-1)$ of rational Hodge structures. 2.3.For any point $s\in C$, we denote by $\iota_{X_s/X}\colon X_s\hookrightarrow X$ the canonical embedding. The morphism $\pi$ is proper, therefore, the fibre of the sheaf $R^n\pi_\ast\mathbb{Q}$ over a point $s\in C$ coincides with the space $H^n(X_s,\mathbb{Q})$ (see Ch. II, § 4, Remark 4.17.1 in [33], and Ch. VI, § 2, Corollary 2.5 in [34]). Consequently, the restriction map $\iota_{X_s/X}^\ast$ coincides with the composite (see Vol. II, Ch. 4, § 4.3.1 in [30])
$$
\begin{equation*}
H^n(X,\mathbb{Q})\to E_\infty^{0,n}(\pi)\to E_2^{0,n}(\pi)= H^0(C,R^n\pi_\ast\mathbb{Q})\to H^n(X_s,\mathbb{Q}).
\end{equation*}
\notag
$$
Thus the map $\iota_{X_s/X}^\ast$ is the composite of canonical maps
$$
\begin{equation*}
H^n(X,\mathbb{Q})\to H^0(C,R^n\pi_\ast\mathbb{Q})\hookrightarrow \prod_{s\in C}H^n(X_s,\mathbb{Q})\to H^n(X_s,\mathbb{Q}),
\end{equation*}
\notag
$$
where the $\mathbb{Q}$-space $\prod_{s\in C}H^n(X_s,\mathbb{Q})$ is identified with the $\mathbb{Q}$-space of discontinuous global sections of the sheaf $R^n\pi_\ast\mathbb{Q}$ (see Ch. II, § 3, § 3.1 in [33]). It is clear that
$$
\begin{equation*}
\omega\in K_{nX}\quad \Longleftrightarrow\quad (\forall\, s\in C)\quad \iota_{X_s/X}^\ast(\omega)=0.
\end{equation*}
\notag
$$
Therefore, for all $\omega\in K_{nX}$, according to [35], Ch. 2, § 8, formula (5),
$$
\begin{equation*}
\iota_{X_s/X}^\ast(\operatorname{cl}_X(H)\smile\omega) = \iota_{X_s/X}^\ast(\operatorname{cl}_X(H))\smile \iota_{X_s/X}^\ast(\omega)=0;
\end{equation*}
\notag
$$
in particular, we obtain the inclusion
$$
\begin{equation}
\operatorname{cl}_X(H)\smile K_{3X}\hookrightarrow K_{5X}.
\end{equation}
\tag{2.10}
$$
According to § 1.9, the Hodge $\mathbb{Q}$-structure $H^0(C',R^2\pi'_\ast\mathbb{Q})=H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})$ is $1$-dimensional and has type $(1,1)$. Taking into account identification (2.3) and the fact that the isomorphism (2.9) is determined by the $\smile$-multiplication by the image $\omega$ of the class $\operatorname{cl}_X(H)$ in the $1$-dimensional $\mathbb{Q}$-space $H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})$ and employing the to arguments of § 4.2.2 in Vol. II, Ch. 4 of [30], we see that the map
$$
\begin{equation*}
K_{3X}\xrightarrow{x\,\mapsto\,\operatorname{cl}_X(H)\,\smile\,x} \operatorname{cl}_X(H)\smile K_{3X}
\end{equation*}
\notag
$$
is an isomorphism of $\mathbb{Q}$-spaces by the strong Lefschetz theorem, and the $\mathbb{Q}$-subspace
$$
\begin{equation*}
\operatorname{cl}_X(H)\smile H^1(C,R^2\pi_\ast\mathbb{Q})= \operatorname{cl}_X(H) \smile K_{3X} \hookrightarrow H^5(X,\mathbb{Q})
\end{equation*}
\notag
$$
does not depend on the choice an ample divisor $H$, because, by of the theorem on local invariant cycles and the compatibility of the Leray spectral sequence $E_2^{p,q}(\pi)$ with the $\smile$-product (see Vol. II, Ch. 4, Lemma 4.13, formula (4.8) in [30] and Ch. 4, §§ 6, 6.5 in [35]), it coincides with the $\mathbb{Q}$-subspace
$$
\begin{equation*}
\begin{aligned} \, &H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q})\,\smile\, H^1(C,j_\ast R^2\pi'_\ast\mathbb{Q}) \\ &\qquad=\operatorname{Im}\bigl[H^1(C,j_\ast R^2\pi'_\ast\mathbb{Q}) \xrightarrow{[\smile\,\omega]_1} H^1(C,j_\ast R^4\pi'_\ast\mathbb{Q})\bigr] \\ &\qquad=H^1(C,j_\ast R^4\pi'_\ast\mathbb{Q})=H^1(C,R^4\pi_\ast\mathbb{Q}) \hookrightarrow H^5(X,\mathbb{Q}). \end{aligned}
\end{equation*}
\notag
$$
2.4.The restriction of the non-degenerate (see § 1.2A in [2]) bilinear form
$$
\begin{equation*}
\Phi\colon H^3(X,\mathbb{Q})\times H^3(X,\mathbb{Q})\xrightarrow{x\times y\, \mapsto \langle x\,\smile\,y\,\smile\operatorname{cl}_X(H)\rangle}\mathbb{Q}
\end{equation*}
\notag
$$
to the subspace $K_{3X}\hookrightarrow H^3(X,\mathbb{Q})$ is non-degenerate in view of identification (2.3), the theorem on local invariant cycles (see § 3 in [19] and Proposition (15.12) in [18]) (allowing us to identify the $\mathbb{Q}$-subspaces $H^1(C',j_\ast R^n\pi'_\ast\mathbb{Q})$ and $H^1(C,R^n\pi_\ast\mathbb{Q})$) and the non-degeneracy (see Proposition (10.5) in [18]) of the canonical pairing
$$
\begin{equation*}
H^1(C,R^2\pi_\ast\mathbb{Q})\times H^1(C,R^2\pi_\ast\mathbb{Q}) \xrightarrow{x\times y\,\mapsto\,x\,\smile\,y\, \smile \operatorname{cl}_X(H)} H^2(C,R^6\pi_\ast\mathbb{Q})=H^8(X,\mathbb{Q}).
\end{equation*}
\notag
$$
Therefore, by the arguments of § 1.7 there exists the decomposition of Hodge $\mathbb{Q}$-structures where, according to § 2.3, the Hodge $\mathbb{Q}$-structure
$$
\begin{equation*}
K_{3X}^\perp=\{x\in H^3(X,\mathbb{Q})\mid x\smile \operatorname{cl}_X(H)\smile K_{3X}=0\}
\end{equation*}
\notag
$$
does not depend on a choice of an ample divisor $H$ (so that decomposition (2.11) is canonical), and the restriction $\Phi|_{K_{3X}^\perp}$ is a non-degenerate form. 2.5.In accordance with (2.11), the strong Lefschetz theorem, and the theorem on local invariant cycles, there is the canonical (independent of a choice of a divisor $H$) decomposition of rational Hodge structures
$$
\begin{equation}
H^5(X,\mathbb{Q}) =\operatorname{cl}_X(H) \smile K_{3X}\oplus \operatorname{cl}_X(H) \smile K_{3X}^\perp
\end{equation}
\tag{2.12}
$$
with canonical identifications
$$
\begin{equation}
\operatorname{cl}_X(H)\smile K_{3X}=H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q}) \smile H^1(C,j_\ast R^2\pi'_\ast\mathbb{Q})=H^1(C,R^4\pi_\ast\mathbb{Q}),
\end{equation}
\tag{2.13}
$$
$$
\begin{equation}
\operatorname{cl}_X(H)\smile K_{3X}^\perp=\operatorname{cl}_X(H) \smile \{x\in H^3(X,\mathbb{Q})\mid x \smile y \smile \operatorname{cl}_X(H)= 0\ \forall\, y\in K_{3X}\} \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad=\{x\in H^5(X,\mathbb{Q})\mid x \smile y=0\ \forall\, y\in K_{3X}\}= \{x\in H^5(X,\mathbb{Q})\mid x\smile K_{3X}=0\}.
\end{equation}
\tag{2.14}
$$
2.6.We set
$$
\begin{equation*}
m_\delta\stackrel{\mathrm{def}}{=} \operatorname{Card}(\mathcal M_\delta/\mathcal M_\delta^0), \quad \delta\in\Delta,\qquad m\stackrel{\mathrm{def}}{=} \prod_{\delta\in\Delta}m_\delta.
\end{equation*}
\notag
$$
We fix a prime number $p$ not dividing the number $m$. Let $p^{m!}_{X/C}\colon X- \to X$ be the rational map coinciding 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 $p^{m!}$. Let
$$
\begin{equation*}
[p^{m!}_{X/C}]^\ast\colon H^\ast(X,\mathbb{Q}) \xrightarrow{x\,\mapsto\,\sigma_\ast\nu^\ast(x)}H^\ast(X,\mathbb{Q})
\end{equation*}
\notag
$$
be the linear operator determined by the commutative diagram of a resolution of indeterminacies of the rational map $p^{m!}_{X/C}$. By Hironaka’s results and by the existence of the canonical isomorphism (see formula (1.1.2) in [10])
$$
\begin{equation*}
\operatorname{End}_C(\mathcal M)\,\,\widetilde{\to} \operatorname{End}_{\kappa(\eta)}(X_\eta)
\end{equation*}
\notag
$$
we may assume that the morphism $\sigma$ is the composite of monoidal transformations along smooth centres and $\sigma|_{\sigma^{-1}(\mathcal M)}\colon \sigma^{-1}(\mathcal M)\to \mathcal M$ is the identity morphism. 2.7.Lemma. There is a canonical decomposition of $[p^{m!}_{X/C}]^\ast$-modules
$$
\begin{equation*}
H^3(X,\mathbb{Q})=(i_\Delta f)_\ast H^1(Z,\mathbb{Q})\oplus H^1(C,R^2\pi_\ast\mathbb{Q})\oplus H^0(C',R^3\pi'_\ast\mathbb{Q}),
\end{equation*}
\notag
$$
where the operator $[p^{m!}_{X/C}]^\ast$ acts on the summands as the multiplication by the numbers $p^{m!}$, $p^{2\cdot m!}$, $p^{3\cdot m!}$, respectively. Proof. The theorem on local invariant cycles and the Leray spectral sequence for the embedding $j\colon C'\hookrightarrow C$ yield the embedding of mixed Hodge $\mathbb{Q}$-structures $H^1(C,R^2\pi_\ast\mathbb{Q}) \hookrightarrow H^1(C',R^2\pi'_\ast\mathbb{Q})$ (see Corollary (13.10), Remark (14.5) in [18]). In addition, by Deligne’s theorem, the canonical maps
$$
\begin{equation*}
H^3(X,\mathbb{Q})\to H^0(C',R^3\pi'_\ast\mathbb{Q}), \qquad H^3(X',\mathbb{Q})\to H^0(C',R^3\pi'_\ast\mathbb{Q})
\end{equation*}
\notag
$$
are surjective (see § 4.1.1 in [17]). Finally, the canonical map is surjective (see Corollary (15.14) in [18]). By (1.3), $H^2(C,R^1\pi_\ast\mathbb{Q})=0$. Since $\Delta\neq\varnothing$, we also have $H^2(C',R^1\pi'_\ast\mathbb{Q})=0$, because the cohomological dimension of the affine curve $C'$ equals $1$ (see Ch. VI, § 7, Theorem 7.2 in [34]).
Taking into account (2.3), we obtain the equality
$$
\begin{equation*}
H^1(C,R^2\pi_\ast\mathbb{Q})= \operatorname{Ker}[H^3(X,\mathbb{Q})\to H^0(C,R^3\pi_\ast\mathbb{Q})]
\end{equation*}
\notag
$$
and the exact sequence of rational Hodge structures
$$
\begin{equation*}
0\to H^1(C,R^2\pi_\ast\mathbb{Q})\to H^3(X,\mathbb{Q})\to H^0(C,R^3\pi_\ast\mathbb{Q})\to 0.
\end{equation*}
\notag
$$
Similarly, the equality $H^2(C',R^1\pi'_\ast\mathbb{Q})=0$ and the degeneracy of the Leray spectral sequence $E_2^{p,q}(\pi')=H^p(C',R^q\pi'_\ast\mathbb{Q})$ (see Theorem 4.1.1 in [17]) yield the exact sequence of mixed rational Hodge structures
$$
\begin{equation*}
0\to H^1(C',R^2\pi'_\ast\mathbb{Q})\to H^3(X',\mathbb{Q})\to H^0(C',R^3\pi'_\ast\mathbb{Q})\to 0.
\end{equation*}
\notag
$$
By functoriality of the Leray spectral sequence, the commutative diagram of morphisms yields the homomorphisms $E_2^{p,q}(\pi)\to E_2^{p,q}(\pi')$, which are compatible with differentials and filtrations (see § 2.4 in [36]). Consequently, taking into account the diagram (15.1) in [18], we obtain the commutative diagram of morphisms of mixed Hodge $\mathbb{Q}$-structures with exact rows and with surjective morphism $\psi_3$Clearly, the restriction of the map $p^{m!}_{X/C}$ to the Abelian scheme $\pi'\colon X'\to C'$ is a $C'$-isogeny $p^{m!}_{X'/C'}$. Therefore, there is the linear operator $[p^{m!}_{X'/C'}]^\ast$: $H^3(X',\mathbb{Q})\to H^3(X',\mathbb{Q})$ acting on the subspace $H^1(C',R^2\pi'_\ast\mathbb{Q})\hookrightarrow H^3(X',\mathbb{Q})$ and on the factor- space $H^0(C',R^3\pi'_\ast\mathbb{Q})$ as multiplications by the numbers $p^{2\cdot m!}$ and $p^{3\cdot m!}$, respectively, since the isogeny of the multiplication by the number $p^{m!}$ on fibres of a smooth morphism $\pi'$ induces the multiplication by the number $p^{m!}$ in the sheaf $R^1\pi'_\ast\mathbb{Q}$ (see Lemma 2A3, § 2A11 in [2]) and $R^n\pi'_\ast\mathbb{Q}=\wedge^n R^1\pi'_\ast\mathbb{Q}$. These properties of the operator $[p^{m!}_{X'/C'}]^\ast$ and Lieberman’s trick (see [37], § 3, the proof of Theorem B) imply the existence of the canonical decomposition of mixed Hodge $\mathbb{Q}$-structures
$$
\begin{equation*}
H^3(X',\mathbb{Q})=H^1(C',R^2\pi'_\ast\mathbb{Q})\oplus H^0(C',R^3\pi'_\ast\mathbb{Q}),
\end{equation*}
\notag
$$
which by the commutativity of diagram (2.15) and the surjectivity of the map $\psi_3$ yields the canonical splitting
$$
\begin{equation}
\operatorname{Im}(\varphi_3)=H^1(C,R^2\pi_\ast\mathbb{Q})\oplus H^0(C',R^3\pi'_\ast\mathbb{Q})
\end{equation}
\tag{2.16}
$$
of Hodge $\mathbb{Q}$-structures by restriction (see formula (3.5) in [37]). Consequently, we have the commutative diagram where $[p^{m!}_{X'/C'}]^\ast|_{\operatorname{Im}(\varphi_3)}$ acts on summands of decomposition (2.16) as multiplications by the numbers $p^{2\cdot m!}$ and $p^{3\cdot m!}$, respectively (because, by the arguments of the beginning of § 3.8 in [13], the element $[p^{m!}_{X/C}]^\ast$ acts on $H^1(C,j_\ast R^2\pi'_\ast\mathbb{Q})$ as the multiplication by the number $p^{2\cdot m!}$, and, by similar arguments. it acts on $H^0(C',R^3\pi'_\ast\mathbb{Q})$ as the multiplication by the number $p^{3\cdot m!}$).
The irreducible components of the smooth variety $Z$ are naturally identified with those $X_{\delta i}$ of the divisor $\pi^{-1}(\Delta)=\sum_{\delta\in\Delta}X_{\delta}$. Denote by $\iota_{X_{\delta i}/X}\colon X_{\delta i}\hookrightarrow X$, $\iota_{X_{\delta i}/Z}\colon X_{\delta i}\hookrightarrow Z$ the canonical embeddings. By commutativity of the diagram of canonical morphisms, we obtain the equality
$$
\begin{equation}
(i_\Delta f)_\ast (\iota_{X_{\delta i}/Z})_\ast|_{H^1(X_{\delta i},\mathbb{Q})}= \iota_{X_{\delta i}/X\ast}|_{H^1(X_{\delta i},\mathbb{Q})}.
\end{equation}
\tag{2.18}
$$
Since the prime number $p$ does not divide $m=\prod_{\delta\in\Delta}\operatorname{Card} (\mathcal M_\delta/\mathcal M_\delta^0)$, it follows that the multiplication by the invertible in the ring $\mathbb{Z}/m_\delta\mathbb{Z}$ element $p\ \operatorname{mod} m_\delta$ 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 $p^{m!}\ \operatorname{mod} m_\delta$ is the identity bijection of the set $\mathcal M_\delta/\mathcal M^0_\delta$. The restriction of the rational map ${p^{m!}_{X/C}}$ to $\mathcal M$ is a regular map ${p^{m!}_{X/C}}_{|_{\mathcal M}}\mathcal M\to \mathcal M$ in view of the existence of the canonical isomorphism
$$
\begin{equation*}
\operatorname{End}_C(\mathcal M)\,\widetilde{\to} \operatorname{End}_{\kappa(\eta)}(X_\eta)
\end{equation*}
\notag
$$
(see [10], formula (1.1.2)), we have by formula (3.32) in [13] that
$$
\begin{equation*}
(\forall\, i)\quad {p^{m!}_{X/C}}(\mathcal M_{\delta i})= \mathcal M_{\delta i},
\end{equation*}
\notag
$$
where $\mathcal M_{\delta i}$ is an irreducible component of the fibre $\mathcal M_\delta$, for which $X_{\delta i}=\overline{\mathcal M_{\delta i}}$ (the Zariski closure).
It is known that the morphism ${p^{m!}_{X/C}}_{|_{\mathcal M}}$ is étale, and, consequently, there are the equalities of smooth divisors (see [13], formula (3.39))
$$
\begin{equation*}
\bigl[{p^{m!}_{X/C}}_{|_{\mathcal M}}\bigr]^{-1}(\mathcal M_{\delta i}) = \bigl[{p^{m!}_{X/C}}_{|_{\mathcal M}}\bigr]^\ast(\mathcal M_{\delta i})= \mathcal M_{\delta i}.
\end{equation*}
\notag
$$
Moreover, there is the decomposition (see [13], formula (3.40)) of groups of divisors
$$
\begin{equation*}
\operatorname{Div}(\widetilde{X})=\sigma^\ast(\operatorname{Div}(X)) \oplus\operatorname{Ker}(\sigma_\ast),
\end{equation*}
\notag
$$
which in turn yields the equalities (see the proof of formula (3.41) in [13])
$$
\begin{equation*}
[p^{m!}_{X/C}]^\ast|_{H^2(X,\mathbb{Q})}(\operatorname{cl}_X(X_{\delta i}))= \operatorname{cl}_X(X_{\delta i}).
\end{equation*}
\notag
$$
On the other hand, we have the commutative diagram (see diagram (3.33) in [13]) of a resolution of indeterminacies of the rational map $p^{m!}_{X/C}|_{X_{\delta i}}$, where the morphism $\sigma_{\delta i}$ is the composite of monoidal transformations with non-singular centres lying over the variety $X_{\delta i}\setminus\mathcal M_\delta$. It is extendable to the commutative diagram of rational maps (where $A_\delta=\operatorname{Alb}(X_{\delta i})$ is the Albanese variety (see formula (3.25) in [13]), $\mathcal M_{\delta 1}=\mathcal M^0_\delta$, $\mathcal M_{\delta i}=a_{\delta i}\mathcal M^0_\delta$ for certain $a_{\delta i}\in\mathcal M_{\delta i}$ and $b_{\delta 1}\in\mathcal M_{\delta 1}$): which in turn yields the commutative diagram (see [13], diagram (3.34)) of isomorphisms of rational Hodge structures Recall that the Gysin map $\iota_{X_{\delta i}/X\ast}$ is defined by formula (3.37) in [13]
$$
\begin{equation*}
\alpha\mapsto\alpha\smile\operatorname{cl}_X(X_{\delta i}).
\end{equation*}
\notag
$$
Hence, for any element $\alpha\in H^1(X_{\delta i},\mathbb{Q})$, using (3.41) in [13], we have
$$
\begin{equation*}
\begin{aligned} \, [p^{m!}_{X/C}]^\ast(\iota_{X_{\delta i}/X\ast}(\alpha)) &= [p^{m!}_{X/C}]^\ast|_{H^3(X,\mathbb{Q})} (\alpha \smile \operatorname{cl}_X(X_{\delta i})) \\ &=[p^{m!}_{X/C}|_{X_{\delta i}}]^\ast|_{H^1(X_{\delta i},\mathbb{Q})}(\alpha) \smile [{p^{m!}_{X/C}}]^\ast|_{H^2(X,\mathbb{Q})} (\operatorname{cl}_X(X_{\delta i})) \\ &=p^{m!} \alpha \smile \operatorname{cl}_X(X_{\delta i})= p^{m!}\iota_{X_{\delta i}/X\ast}(\alpha). \end{aligned}
\end{equation*}
\notag
$$
Therefore, by (2.18) the operator $[p^{m!}_{X/C}]^\ast$ acts on the $\mathbb{Q}$-subspace
$$
\begin{equation*}
(i_\Delta f)_\ast H^1(Z,\mathbb{Q})=\operatorname{Ker}(\varphi_3)
\end{equation*}
\notag
$$
as the multiplication by the number $p^{m!}$.
Since the operator $[p^{m!}_{X/C}]^\ast$ acts by pairwise different eigenvalues on the $\mathbb{Q}$-spaces $(i_\Delta f)_\ast H^1(Z,\mathbb{Q})$, $H^1(C,j_\ast R^2\pi'_\ast\mathbb{Q})$, $H^0(C',R^3\pi'_\ast\mathbb{Q})$ in the exact sequence
$$
\begin{equation*}
0\to (i_\Delta f)_\ast H^1(Z,\mathbb{Q})\to H^3(X,\mathbb{Q})\to H^1(C,j_\ast R^2\pi'_\ast\mathbb{Q}) \oplus H^0(C',R^3\pi'_\ast\mathbb{Q})\to 0,
\end{equation*}
\notag
$$
arising from diagram (2.17), we see that the sequence under consideration admits a canonical splitting by eigenspaces. This proves the lemma. 2.8.Lemma. There is a canonical decomposition of $[p^{m!}_{X/C}]^\ast$-modules
$$
\begin{equation*}
K^\perp_{3X}=(i_\Delta f)_\ast H^1(Z,\mathbb{Q})\oplus H^0(C',R^3\pi'_\ast\mathbb{Q}).
\end{equation*}
\notag
$$
Proof. By the definition (see Vol. II, Ch. 4, § 4.2.1 in [30]), 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 follows from the evident equality that
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 (see formula (3.37) in [13]) $\alpha \mapsto\alpha\smile\operatorname{cl}_X(X_{\delta i_\delta})$. On the other hand, the strong Lefschetz theorem for the variety $X_{\delta i_\delta}$ yields the existence of the embedding
$$
\begin{equation*}
H^1(X_{\delta i_\delta},\mathbb{Q}) \smile \iota^\ast_{X_{\delta i_\delta}/X} \operatorname{cl}_X(H)\hookrightarrow H^3(X_{\delta i_\delta},\mathbb{Q}).
\end{equation*}
\notag
$$
Therefore, the projection formula (see § 1.2.A in [2]) yields the inclusion
$$
\begin{equation}
\begin{aligned} \, &\iota_{X_{\delta i_\delta}/X\ast}H^1(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X(H)\hookrightarrow \iota_{X_{\delta i_\delta}/X\ast}H^3(X_{\delta i_\delta},\mathbb{Q}) \nonumber \\ &\qquad=H^3(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X(X_{\delta i_\delta}). \end{aligned}
\end{equation}
\tag{2.20}
$$
Finally, the $\smile$-product by the class $\operatorname{cl}_X(X_{\delta i_\delta})$ induces the endomorphism of degree $2$ of the Leray spectral sequence which at $E_2$ coincides with the morphism induced in cohomology by the map of sheaves $R^q\pi_\ast\mathbb{Q} \xrightarrow{\smile\operatorname{cl}_X(X_{\delta i_\delta})} R^{q+2}\pi_\ast\mathbb{Q}$ (see Vol. II, Ch. 4, § 4.2.1, Lemma 4.13 in [30]). Consequently, using (2.18), (2.3), and the theorem on local invariant cycles, we have
$$
\begin{equation*}
\begin{aligned} \, &(i_\Delta f)_\ast H^1(Z,\mathbb{Q}) \smile \operatorname{cl}_X(H) \smile K_{3X} \\ &\qquad\hookrightarrow \sum_{\delta\in\Delta,\,i_\delta\in\{1,\dots,m_\delta\}} H^3(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X(X_{\delta i_\delta}) \smile H^1(C,j_\ast R^2\pi'_\ast\mathbb{Q})=0, \end{aligned}
\end{equation*}
\notag
$$
because the canonical map of cohomology
$$
\begin{equation*}
H^1(C,j_\ast R^2\pi'_\ast\mathbb{Q})\xrightarrow{[\smile \operatorname{cl}_X (X_{\delta i_\delta})]_1} H^1(C,j_\ast R^4\pi'_\ast\mathbb{Q})
\end{equation*}
\notag
$$
is induced by the zero map of sheaves $j_\ast R^2\pi'_\ast\mathbb{Q}\xrightarrow{\smile \operatorname{cl}_X (X_{\delta i_\delta})}j_\ast R^4\pi'_\ast\mathbb{Q}$ in view of (2.19). Therefore,
$$
\begin{equation*}
(i_\Delta f)_\ast H^1(Z,\mathbb{Q})\hookrightarrow K^\perp_{3X}.
\end{equation*}
\notag
$$
On the other hand, it follows from the equality $R^7\pi'_\ast\mathbb{Q}=0$, (2.3) and (2.13) that
$$
\begin{equation*}
\begin{aligned} \, &H^0(C',R^3\pi'_\ast\mathbb{Q})\smile\operatorname{cl}_X(H)\smile K_{3X} \\ &\qquad=H^0(C,j_\ast R^3\pi'_\ast\mathbb{Q}) \smile H^0(C,j_\ast R^2\pi'_\ast\mathbb{Q}) \smile H^1(C,j_\ast R^2\pi'_\ast\mathbb{Q}) \\ &\qquad\qquad\hookrightarrow H^1(C,j_\ast R^7\pi'_\ast\mathbb{Q})=0. \end{aligned}
\end{equation*}
\notag
$$
Hence $H^0(C',R^3\pi'_\ast\mathbb{Q})\hookrightarrow K^\perp_{3X}$, and now Lemma 2.8 follows from Lemma 2.7. 2.9.Taking into account (2.1) for $n=5$ and the canonical identification (2.13), we obtain the canonical exact sequence of Hodge $\mathbb{Q}$-structures
$$
\begin{equation*}
0\to H^2(C,R^3\pi_\ast\mathbb{Q})\to K_{5X} \xrightarrow{\alpha_{5X}}K_{3X} \smile \operatorname{cl}_X(H)\to 0,
\end{equation*}
\notag
$$
which, by (2.10) and (2.13), yields the canonical splittings of rational Hodge structures
$$
\begin{equation}
K_{5X}=K_{3X}\smile\operatorname{cl}_X(H)\oplus H^2(C,R^3\pi_\ast\mathbb{Q}) = H^1(C,R^4\pi_\ast\mathbb{Q})\,{\oplus}\, H^2(C,R^3\pi_\ast\mathbb{Q}).
\end{equation}
\tag{2.21}
$$
On the other hand, (2.1), (2.2), (2.4) and the functoriality of constructions under consideration yield the commutative diagram of canonical maps of rational Hodge structures Finally, proceeding as in § 2.3, it is easy to check that there is the commutative diagram 2.10.Lemma. There is a canonical embedding Proof. Since
$$
\begin{equation*}
K_{3X}=\{x\in H^3(X,\mathbb{Q})\mid x \smile K_{3X}^\perp \smile \operatorname{cl}_X(H)=0\},
\end{equation*}
\notag
$$
it suffices to check the equality
$$
\begin{equation*}
(p_2\sigma)_\ast(K_{9Y}) \smile K^\perp_{3X} \smile \operatorname{cl}_X(H)=0,
\end{equation*}
\notag
$$
which in turn is equivalent to the equality
$$
\begin{equation}
K_{9Y} \smile (p_2\sigma)^\ast(K^\perp_{3X} \smile \operatorname{cl}_X(H))=0,
\end{equation}
\tag{2.24}
$$
because according to § 1.2.A in [2] and Ch. VI, § 11, Remark 11.6 in [34],
It follows from (2.18) and (2.20) that there is the inclusion
$$
\begin{equation}
(i_\Delta f)_\ast H^1(Z,\mathbb{Q})\smile \operatorname{cl}_X(H) \hookrightarrow \sum_{\delta\in\Delta,\,i_\delta\in\{1,\dots,m_\delta\}} H^3(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X(X_{\delta i_\delta}).
\end{equation}
\tag{2.25}
$$
We claim that there exists a decomposition of rational Hodge structures
$$
\begin{equation}
K_{3X}^\perp \smile \operatorname{cl}_X(H)= (i_\Delta f)_\ast H^1(Z,\mathbb{Q}) \smile \operatorname{cl}_X(H) \oplus H^2(C,R^3\pi_\ast\mathbb{Q}).
\end{equation}
\tag{2.26}
$$
Indeed, the curve $C$ has cohomological dimension $2$ (see [34], Ch. VI, Theorem 1.1) and the $\smile$-product is compatible with the Leray spectral sequence $E_2^{p,q}(\pi)$ (see [35], Ch. 4, § 6, § 6.5), therefore, taking into account that $H^2(C,R^3\pi_\ast\mathbb{Q})\hookrightarrow H^5(X,\mathbb{Q})$ by (2.21), we have
$$
\begin{equation*}
K_{3X} \smile H^2(C,R^3\pi_\ast\mathbb{Q}) =H^1(C,R^2\pi_\ast\mathbb{Q}) \smile H^2(C,R^3\pi_\ast\mathbb{Q})\hookrightarrow H^3(C,R^5\pi_\ast\mathbb{Q})=0,
\end{equation*}
\notag
$$
so that by (2.14) and Lemma 2.8 we obtain the inclusion
$$
\begin{equation}
\begin{aligned} \, &H^2(C,R^3\pi_\ast\mathbb{Q})\hookrightarrow K_{3X}^\perp \smile \operatorname{cl}_X(H) \nonumber \\ &\qquad= (i_\Delta f)_\ast H^1(Z,\mathbb{Q})\smile \operatorname{cl}_X(H) \oplus H^0(C',R^3\pi'_\ast\mathbb{Q}) \smile \operatorname{cl}_X(H). \end{aligned}
\end{equation}
\tag{2.27}
$$
The non-degeneracy of the canonical pairing (see Proposition (10.5) in [18])
$$
\begin{equation*}
H^2(C,j_\ast R^3\pi'_\ast\mathbb{Q})\times H^0(C,j_\ast R^3\pi'_\ast\mathbb{Q}) \xrightarrow{x\times x'\mapsto\,x\,\smile\,x'} H^2(C,j_\ast R^6\pi'_\ast\mathbb{Q}) \,\widetilde{\to}\, \mathbb{Q}
\end{equation*}
\notag
$$
and the theorem on local invariant cycles show that, by (1.5), the 2-dimensional Hodge $\mathbb{Q}$-structure of odd weight
$$
\begin{equation*}
H^2(C,R^3\pi_\ast\mathbb{Q})=H^0(C',R^3\pi'_\ast\mathbb{Q})^\vee
\end{equation*}
\notag
$$
is irreducible and
$$
\begin{equation}
H^2(C,R^3\pi_\ast\mathbb{Q}) \smile H^0(C,j_\ast R^3\pi'_\ast\mathbb{Q}) = H^2(C,R^6\pi_\ast\mathbb{Q}) \, \widetilde{\to}\,\mathbb{Q}.
\end{equation}
\tag{2.28}
$$
So by (2.27) it suffices to check the equality
$$
\begin{equation*}
H^2(C,R^3\pi_\ast\mathbb{Q})\cap [(i_\Delta f)_\ast H^1(Z,\mathbb{Q}) \smile \operatorname{cl}_X(H)]=0.
\end{equation*}
\notag
$$
Assume on the contrary that
$$
\begin{equation*}
H^2(C,R^3\pi_\ast\mathbb{Q})\cap [(i_\Delta f)_\ast H^1(Z,\mathbb{Q}) \smile \operatorname{cl}_X(H)]\neq 0.
\end{equation*}
\notag
$$
Hence the irreducibility of the Hodge $\mathbb{Q}$-structure $H^2(C,R^3\pi_\ast\mathbb{Q})$ and (2.25), (2.19) yield the inclusions
$$
\begin{equation*}
\begin{gathered} \, H^2(C,R^3\pi_\ast\mathbb{Q})\hookrightarrow (i_\Delta f)_\ast H^1(Z,\mathbb{Q})\smile\,\operatorname{cl}_X(H), \\ \begin{aligned} \, &H^2(C,R^3\pi_\ast\mathbb{Q}) \smile H^0(C,j_\ast R^3\pi'_\ast\mathbb{Q}) \\ &\qquad\hookrightarrow (i_\Delta f)_\ast H^1(Z,\mathbb{Q})\smile \operatorname{cl}_X(H) \smile H^0(C,j_\ast R^3\pi'_\ast\mathbb{Q}) \\ &\qquad\hookrightarrow\sum_{\delta\in\Delta,\,i_\delta\in\{1,\dots,m_\delta\}} H^3(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X(X_{\delta i_\delta}) \smile H^0(C,j_\ast R^3\pi'_\ast\mathbb{Q})=0, \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
contradicting (2.28).
In particular, we have the equality of Hodge $\mathbb{Q}$-substructures in $H^5(X,\mathbb{Q})$
$$
\begin{equation*}
H^2(C,R^3\pi_\ast\mathbb{Q})=H^0(C',R^3\pi'_\ast\mathbb{Q}) \smile\operatorname{cl}_X(H),
\end{equation*}
\notag
$$
where $\smile$ is the usual $\smile$-multiplication in the $\mathbb{Q}$-algebra $H^\ast(X,\mathbb{Q})$ and it yields the $\smile$-multiplication by the class $\operatorname{cl}_X(H)$, which is different from the $\smile$-multiplication induced by the map of sheaves $j_\ast R^q\pi'_\ast\mathbb{Q}\xrightarrow{\smile\, \operatorname{cl}_X(H)} j_\ast R^{q+2}\pi'_\ast\mathbb{Q}$ (see [30], Vol. II, Ch. 4, § 4.2.1, Lemma 4.13).
In the case $n=9$, the exact sequence (2.2) takes the form
$$
\begin{equation}
0\to H^2(C,R^7(\tau\sigma)_\ast\mathbb{Q})\to K_{9Y} \xrightarrow{\alpha_{9Y}} H^1(C,R^8(\tau\sigma)_\ast\mathbb{Q})\to 0.
\end{equation}
\tag{2.29}
$$
On the other hand, it follows from (2.26) that
$$
\begin{equation}
(p_2\sigma)^\ast(K^\perp_{3X} \smile \operatorname{cl}_X(H)) \hookleftarrow(p_2\sigma)^\ast(H^2(C,R^3\pi_\ast\mathbb{Q}));
\end{equation}
\tag{2.30}
$$
besides, the surjective $C$-morphism $p_2\sigma\colon Y\to X$ determines the canonical injection $(p_2\sigma)^\ast\colon H^5(X,\mathbb{Q})\hookrightarrow H^5(Y,\mathbb{Q})$ (see Proposition 1.2.4 in [2]), whose restriction to the subspace $H^2(C,R^3\pi_\ast\mathbb{Q})\hookrightarrow H^5(X,\mathbb{Q})$ coincides with the composite of injective canonical maps
$$
\begin{equation}
H^2(C,R^3\pi_\ast\mathbb{Q})\xrightarrow{(p_2\sigma)^\ast} H^2(C,R^3(\tau\sigma)_\ast\mathbb{Q}) =H^2(C,R^3(\pi p_2\sigma)_\ast \mathbb{Q})\hookrightarrow H^5(Y,\mathbb{Q}),
\end{equation}
\tag{2.31}
$$
this follows from the commutativity of the diagram
It is well-known that the $\smile$-product in the Leray spectral sequence $E_2^{p,q}(\tau\sigma)$ is given by the composite (see [35], Ch. 4, § 6, § 6.5)
$$
\begin{equation*}
\begin{aligned} \, &H^p(C,R^q(\tau\sigma)_\ast\mathbb{Q})\otimes_\mathbb{Q} H^m(C,R^n(\tau\sigma)_\ast\mathbb{Q}) \\ &\qquad\xrightarrow{(-1)^{qm}\,\smile} H^{p+m}(C,R^q(\tau\sigma)_\ast \mathbb{Q} \otimes_\mathbb{Q}R^n(\tau\sigma)_\ast\mathbb{Q}) \to H^{p+m}(C,R^{q+n}(\tau\sigma)_\ast\mathbb{Q}). \end{aligned}
\end{equation*}
\notag
$$
Therefore, from (2.26), (2.29)–(2.31) and from the fact that the curve $C$ has cohomological dimension $2$ (see Ch. VI, § 1, Theorem 1.1 in [34]), it easily follows that
$$
\begin{equation*}
\begin{gathered} \, H^2(C,R^7(\tau\sigma)_\ast\mathbb{Q}) \smile H^2(C,R^3(\tau\sigma)_\ast\mathbb{Q})\hookrightarrow H^4(C,R^{10}(\tau\sigma)_\ast\mathbb{Q})=0, \\ H^1(C,R^8(\tau\sigma)_\ast\mathbb{Q}) \smile H^2(C,R^3(\tau\sigma)_\ast\mathbb{Q})\hookrightarrow H^3(C,R^{11}(\tau\sigma)_\ast\mathbb{Q})=0, \\ K_{9Y}\smile H^2(C,R^3(\tau\sigma)_\ast\mathbb{Q})=0, \\ K_{9Y} \smile (p_2\sigma)^\ast(H^2(C,R^3\pi_\ast\mathbb{Q}))=0. \end{gathered}
\end{equation*}
\notag
$$
Hence by (2.26), § 1.2.A in [2], and Ch. VI, § 11, Remark 11.6 in [34] we see that (2.24) is equivalent to the equalities
$$
\begin{equation}
\begin{aligned} \, &\langle K_{9Y} \smile (p_2\sigma)^\ast((i_\Delta f)_\ast H^1(Z,\mathbb{Q}) \smile \operatorname{cl}_X(H)) \rangle \nonumber \\ &\qquad=\langle (p_2\sigma)_\ast K_{9Y} \smile (i_\Delta f)_\ast H^1(Z,\mathbb{Q}) \smile \operatorname{cl}_X(H)\rangle =0. \end{aligned}
\end{equation}
\tag{2.32}
$$
By the theorem on local invariant cycles (see § 3 in [19] and Proposition (15.12) in [18]) and by Künneth’s formula on fibres of the smooth morphism there are canonical decompositions
$$
\begin{equation*}
\begin{gathered} \, H^2(C,R^7(\tau\sigma)_\ast\mathbb{Q})=\bigoplus_{p+q=7} H^2(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q})), \\ H^1(C,R^8(\tau\sigma)_\ast\mathbb{Q})=\bigoplus_{p+q=8} H^1(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q})). \end{gathered}
\end{equation*}
\notag
$$
On the other hand, there exists a section $e\colon C\to X$ of the structure morphism $\pi\colon X\to C$ determined 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 $p_2\colon X\times_CX\to X$, identifying $X$ with a subvariety of the fibre product $X\times_CX$ and identifying 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 the coniveau (arithmetic) filtration (see § 1.B in [37]). On the other hand, according to [30], Vol. II, Ch. 4, § 4.2.1, for any point $s\in C'=C\setminus \Delta$, the $\smile$-multiplication by the class
$$
\begin{equation*}
\omega\in\sigma^\ast p_2^\ast(H^3(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X(X_{\delta i_\delta}))\hookrightarrow H^5(Y,\mathbb{Q})
\end{equation*}
\notag
$$
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 $\smile$-multiplication by the class
$$
\begin{equation*}
\iota^\ast_{X_s\times X_s/Y}(\omega)\in\iota^\ast_{X_s\times X_s/Y} \bigl(\sigma^\ast p_2^\ast(H^3(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X(X_{\delta i_\delta}))\bigr),
\end{equation*}
\notag
$$
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
$$
\begin{equation*}
\iota^\ast_{X_s\times X_s/Y}\bigl(\sigma^\ast p_2^\ast (H^3(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X(X_{\delta i_\delta}))\bigr)=0,
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q}) \smile \sigma^\ast p_2^\ast (H^3(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X(X_{\delta i_\delta}))=0.
\end{equation*}
\notag
$$
In particular, for any $\omega\in\sigma^\ast p_2^\ast (H^3(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X (X_{\delta i_\delta}))$, 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 which induces (by the functoriality) the zero map of cohomology (see Vol. II, Ch. 4, § 4.2.1, Lemma 4.13 in [30]). Now from (2.25), (2.29) we have
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, &H^2(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q})) \smile (p_2\sigma)^\ast((i_\Delta f)_\ast H^1(Z,\mathbb{Q})\smile\,\operatorname{cl}_X(H)) \\ &\hookrightarrow H^2(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q})) \\ &\qquad \smile (p_2\sigma)^\ast\biggl(\sum_{\delta\in\Delta,\,i_\delta\in \{1,\dots,m_\delta\}}H^3(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X(X_{\delta i_\delta})\biggr) \\ &=H^2(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q})) \\ &\qquad\smile \sigma^\ast p_2^\ast\biggl(\sum_{\delta\in\Delta,\,i_\delta \in\{1,\dots,m_\delta\}}H^3(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X(X_{\delta i_\delta})\biggr)=0, \end{aligned} \\ \begin{aligned} \, &H^1(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q})) \smile (p_2\sigma)^\ast((i_\Delta f)_\ast H^1(Z,\mathbb{Q})\smile\,\operatorname{cl}_X(H)) \\ &\hookrightarrow H^1(C,j_\ast (R^p\pi'_\ast\mathbb{Q}\otimes_\mathbb{Q} R^q\pi'_\ast\mathbb{Q})) \\ &\qquad\smile \sigma^\ast p_2^\ast \biggl(\sum_{\delta\in\Delta,\,i_\delta\in \{1,\dots,m_\delta\}} H^3(X_{\delta i_\delta},\mathbb{Q})\,\smile\, \operatorname{cl}_X(X_{\delta i_\delta})\biggr)=0, \end{aligned} \\ K_{9Y} \smile (p_2\sigma)^\ast((i_\Delta f)_\ast H^1(Z,\mathbb{Q})\smile \operatorname{cl}_X(H))=0. \end{gathered}
\end{equation*}
\notag
$$
Therefore, (2.32) is true and the lemma is proved. 2.11.Lemma. The algebraic class
$$
\begin{equation*}
u\stackrel{\mathrm{def}}{=} (\iota\sigma)_\ast \bigl[[\operatorname{cl}_Y(D^{(1)})]^{\smile\,2}\bigr]\in H^6(X\times X,\mathbb{Q})
\end{equation*}
\notag
$$
determines the algebraic isomorphism $K_{3X} \smile \operatorname{cl}_X(H) \underset{\widetilde{\qquad}} {\xrightarrow{x\, \mapsto \operatorname{pr}_{2\ast} (\operatorname{pr}_1^\ast x\,\smile\,u)}} K_{3X}$. Proof. By (2.21), the canonical map
$$
\begin{equation*}
\widetilde{\alpha_{5X}}\stackrel{\mathrm{def}}=\alpha_{5X}|_{K_{3X} \smile \operatorname{cl}_X(H)}\colon K_{3X} \smile \operatorname{cl}_X(H)\to H^1(C,R^4\pi_\ast\mathbb{Q})
\end{equation*}
\notag
$$
is an isomorphism. Setting $\widetilde{K_{5X}}=K_{3X}\,\smile\,\operatorname{cl}_X(H)$, using Lemma 2.10, and also diagrams (2.22), (2.23), we obtain the commutative diagram which is glued from the commutative diagrams and
On the other hand, denote by $\operatorname{pr}_i\colon X\times X\to X$ the canonical projection of the Cartesian square of the variety $X$. It is clear that $p_i=\operatorname{pr}_i\iota$, therefore, the projection formula (see § 1.2.A in [2]) yields, for $x\in H^5(X,\mathbb{Q})$,
$$
\begin{equation*}
\begin{aligned} \, (p_2\sigma)_\ast\bigl((p_1\sigma)^\ast x \smile [\operatorname{cl}_Y(D^{(1)})]^{\smile\,2}\bigr) &= [\operatorname{pr}_2\iota\sigma]_\ast \bigl([\operatorname{pr}_1\iota\sigma]^\ast x \smile [\operatorname{cl}_Y(D^{(1)})]^{\smile\,2}\bigr) \\ &=\operatorname{pr}_{2\ast}(\iota\sigma)_\ast \bigl((\iota\sigma)^\ast\operatorname{pr}_1^\ast x \smile [\operatorname{cl}_Y(D^{(1)})]^{\smile\,2}\bigr) \\ &=\operatorname{pr}_{2\ast}\bigl(\operatorname{pr}_1^\ast x \smile (\iota\sigma)_\ast \bigl[[\operatorname{cl}_Y(D^{(1)})]^{\,\smile\,2}\bigr]\bigr). \end{aligned}
\end{equation*}
\notag
$$
Since the composite of maps in the last row of diagram (2.33) is the isomorphism (2.7), Lemma is proved. 2.12.Using the arguments of § 1.7 and those of § 3.5 in [28], we may assume that
$$
\begin{equation*}
\wp(H^3(X,\mathbb{Q}))=\wp(K_{3X})+\wp(K_{3X}^\perp);
\end{equation*}
\notag
$$
in addition, according to § 1.7, the class $\wp(H^3(X,\mathbb{Q}))$ yields the isomorphism
$$
\begin{equation*}
H^5(X,\mathbb{Q}) \underset{\widetilde{\qquad}}{\xrightarrow{x\,\mapsto \operatorname{pr}_{2\ast} (\operatorname{pr}_1^\ast(x) \smile \wp(H^3(X,\mathbb{Q})))}}H^3(X,\mathbb{Q}).
\end{equation*}
\notag
$$
Now from (2.12) and the evident equality
$$
\begin{equation*}
K_{3X} \smile K_{3X}^\perp \smile\operatorname{cl}_X(H)=0
\end{equation*}
\notag
$$
it follows that the Poncaré class $\wp(K_{3X}^\perp)$ yields the isomorphism
$$
\begin{equation}
K_{3X}^\perp \smile \operatorname{cl}_X(H) \underset{\widetilde{\qquad}}{\xrightarrow{x\,\mapsto \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast(x) \smile \wp(K_{3X}^\perp))}} K_{3X}^\perp.
\end{equation}
\tag{2.34}
$$
By (2.25) and (2.19) we obtain
$$
\begin{equation*}
\begin{aligned} \, &(i_\Delta f)_\ast H^1(Z,\mathbb{Q})\smile \operatorname{cl}_X(H) \smile H^0(C,j_\ast R^3\pi'_\ast\mathbb{Q}) \\ &\qquad\hookrightarrow\sum_{\delta\in\Delta,\,i_\delta\in\{1,\dots,m_\delta\}} H^3(X_{\delta i_\delta},\mathbb{Q}) \smile \operatorname{cl}_X(X_{\delta i_\delta}) \smile H^0(C,j_\ast R^3\pi'_\ast\mathbb{Q})=0. \end{aligned}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\Phi\bigl((i_\Delta f)_\ast H^1(Z,\mathbb{Q}), H^0(C',R^3\pi'_\ast\mathbb{Q})\bigr)=0,
\end{equation*}
\notag
$$
where $\Phi$ is the bilinear form
$$
\begin{equation*}
H^3(X,\mathbb{Q})\times H^3(X,\mathbb{Q})\xrightarrow{x\times y\,\mapsto\, \langle x\,\smile\,y\,\smile \operatorname{cl}_X(H)\rangle}\mathbb{Q}.
\end{equation*}
\notag
$$
Now from the arguments of § 1.7, Lemma 2.8 and the non-degeneracy of the form $\Phi|_{K_{3X}^\perp}$ (see § 2.4) we obtain the non-degeneracy of bilinear forms $\Phi|_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q})}$, $\Phi|_{H^0(C',R^3\pi'_\ast\mathbb{Q})}$ and the existence of the Poincaré classes
$$
\begin{equation*}
\wp\bigl((i_\Delta f)_\ast H^1(Z,\mathbb{Q})\bigr),\qquad \wp\bigl(H^0(C',R^3\pi'_\ast\mathbb{Q})\bigr).
\end{equation*}
\notag
$$
The Lefschetz theorem on divisors implies the algebraicity of the Poincaré class $\wp((i_\Delta f)_\ast H^1(Z,\mathbb{Q}))$ (see § 3.6 in [13]). On the other hand, following the arguments of § 3.5 in [28] and Lemma 2.8, we may assume that
$$
\begin{equation*}
\wp(K_{3X}^\perp)=\wp\bigl((i_\Delta f)_\ast H^1(Z,\mathbb{Q})\bigr) +\wp\bigl(H^0(C',R^3\pi'_\ast\mathbb{Q})\bigr).
\end{equation*}
\notag
$$
Consequently, the Poincaré class $\wp(K_{3X}^\perp)$ is algebraic if and only if so is the Poincaré class $\wp(H^0(C',R^3\pi'_\ast\mathbb{Q}))$. 2.13.Let $A$ be a complex Abelian variety. By definition (see Definition B.69 in [21]), the Lefschetz group
$$
\begin{equation*}
\operatorname{Lf}(A)=\{g\in\operatorname{Sp}(H^1(A,\mathbb{Q}),E)\mid g\circ\varphi=\varphi\circ g\ \forall\, \varphi\in \operatorname{End}^0_\mathbb{C}(A)\}^0
\end{equation*}
\notag
$$
is the connected component of unity of the centralizer of the ring
$$
\begin{equation*}
\operatorname{End}^0_\mathbb{C}(A)\stackrel{\mathrm{def}}{=} \operatorname{End}_\mathbb{C}(A) \otimes_\mathbb{Z}\mathbb{Q}
\end{equation*}
\notag
$$
in the group $\operatorname{Sp}(H^1(A,\mathbb{Q}),E)$, where $E$ is the Riemann form of a polarization of the Abelian variety $A$. It is known (see § B.69 in [21]) that the Lefschetz group $\operatorname{Lf}(A)$ does not depend on a choice of a polarization, and there is the canonical embedding
$$
\begin{equation*}
\operatorname{Hg}(A)\hookrightarrow\operatorname{Lf}(A).
\end{equation*}
\notag
$$
If $A$ is a simple Abelian variety of prime dimension or an elliptic curve, then this result follows from Theorems 0–3 in [38], and, in particular, from the equality $\dim_\mathbb{Q}\operatorname{Hg}(A)=\dim_\mathbb{C} A$ for a simple Abelian variety of CM-type (in the case of prime dimension or of dimension $1$) (see Corollary 2 in [39]). Since the Abelian variety $A$ does not belong to type III by Albert’s classification (because there do not exist simple Abelian surfaces of type III (see § (2.2) in [22])) it follows from (2.35) that, for any natural number $n$, taking into account Theorem 3.1 in [40], Theorem (2.7) in [41], and Theorem B.114 in [21], the $\mathbb{Q}$-space of Hodge cycles on the Abelian variety $A^n$ is generated by classes of intersections of divisors. In particular, the $\mathbb{Q}$-space of Hodge cycles on the Abelian variety
$$
\begin{equation*}
[X_\eta\otimes_{\kappa(\eta)}\mathbb{C}]\times [X_\eta\otimes_{\kappa(\eta)}\mathbb{C}]
\end{equation*}
\notag
$$
is generated by classes of algebraic cycles. 2.14.Lemma. The Poincaré class $\wp(H^0(C',R^3\pi'_\ast\mathbb{Q}))$ is algebraic. Proof. According to § 2.13, the $\mathbb{Q}$-space of Hodge cycles is spanned by algebraic cohomology classes, so that, by the Künneth formula, the $\mathbb{Q}$-space also is spanned by algebraic classes.
By definition, the Poincaré class $\wp(H^0(C',R^3\pi'_\ast\mathbb{Q}))$ is a generator of the $1$-dimensional space of invariants
$$
\begin{equation*}
[H^0(C',R^3\pi'_\ast\mathbb{Q})\otimes_\mathbb{Q}H^0(C',R^3\pi'_\ast \mathbb{Q})]^{\operatorname{Sp} (H^0(C',R^3\pi'_\ast\mathbb{Q}), \Phi|_{H^0(C',R^3\pi'_\ast\mathbb{Q})})}
\end{equation*}
\notag
$$
of the diagonal action of the group $\operatorname{Sp}(H^0(C',R^3\pi'_\ast \mathbb{Q}),\Phi|_{H^0(C',R^3\pi'_\ast\mathbb{Q})})$ on the tensor product $H^0(C',R^3\pi'_\ast\mathbb{Q}) \otimes_\mathbb{Q}H^0(C',R^3\pi'_\ast\mathbb{Q})$. On the other hand, there exist the canonical identifications
$$
\begin{equation*}
H^0(C',R^3\pi'_\ast\mathbb{Q}) =H^3(X_\eta\otimes_{\kappa(\eta)} \mathbb{C},\mathbb{Q})^{\pi_1(C',\overline\eta)} =H^3(X_\eta \otimes_{\kappa(\eta)}\mathbb{C},\mathbb{Q})^G,
\end{equation*}
\notag
$$
where $\overline\eta$ is the generic geometric point of the curve $C'$, so that the Poincaré class under consideration may be identified with an element of the $\mathbb{Q}$-subspace
$$
\begin{equation*}
H^3(X_\eta\otimes_{\kappa(\eta)}\mathbb{C},\mathbb{Q}) \otimes_\mathbb{Q}H^3(X_\eta\otimes_{\kappa(\eta)}\mathbb{C},\mathbb{Q}).
\end{equation*}
\notag
$$
It remains to note that the Poincaré class is a Hodge cycle according to arguments of § 1.7, and, consequently, it belongs to the $\mathbb{Q}$-space which is spanned by algebraic cohomology classes. This proves the lemma. § 3. A proof of the theorem3.1.Let
$$
\begin{equation*}
\begin{gathered} \, u_{K_{3X},K_{3X}},\quad u_{K_{3X},(i_\Delta f)_\ast H^1(Z,\mathbb{Q})},\quad u_{K_{3X},H^0(C',R^3\pi'_\ast\mathbb{Q})}, \\ u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),K_{3X}},\quad u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),(i_\Delta f)_\ast H^1(Z,\mathbb{Q})},\quad u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),H^0(C',R^3\pi'_\ast\mathbb{Q})}, \\ u_{H^0(C',R^3\pi'_\ast\mathbb{Q}),K_{3X}},\quad u_{H^0(C',R^3\pi'_\ast\mathbb{Q}),(i_\Delta f)_\ast H^1(Z,\mathbb{Q})},\quad u_{H^0(C',R^3\pi'_\ast\mathbb{Q}),H^0(C',R^3\pi'_\ast\mathbb{Q})}, \quad h \end{gathered}
\end{equation*}
\notag
$$
be components of the algebraic correspondence $u$ in direct summands
$$
\begin{equation*}
\begin{gathered} \, K_{3X}\otimes_\mathbb{Q} K_{3X},\quad \dots, \quad H^0(C',R^3\pi'_\ast\mathbb{Q})\otimes_\mathbb{Q} H^0(C',R^3\pi'_\ast\mathbb{Q}), \\ H_\mathbb{Q}\stackrel{\mathrm{def}}{=}\bigoplus_{p+q=6,\,p\neq 3} H^p(X,\mathbb{Q})\otimes_\mathbb{Q} H^q(X,\mathbb{Q}), \end{gathered}
\end{equation*}
\notag
$$
determined by Lemma 2.7, identification (2.3), and the Künneth decomposition of the $\mathbb{Q}$-space $H^6(X\times X,\mathbb{Q})$. It is clear that the operators
$$
\begin{equation*}
\begin{gathered} \, [p_{X/C}^{m!}]^\ast\otimes_\mathbb{Q}[1_{X/C}]^\ast= [\sigma_\ast\nu^\ast]\otimes_\mathbb{Q}[1_{X/C}]^\ast, \\ [1_{X/C}]^\ast\otimes_\mathbb{Q}[p_{X/C}^{m!}]^\ast =[1_{X/C}]^\ast \otimes_\mathbb{Q}[\sigma_\ast\nu^\ast] \end{gathered}
\end{equation*}
\notag
$$
transform the $\mathbb{Q}$-subspace $H_\mathbb{Q}\subset H^6(X\times X,\mathbb{Q})$ into the space $H_\mathbb{Q}$ and they transform algebraic cohomology classes into algebraic classes (see Proposition 1.3.7 in [2]). Taking into account Lemma 2.7, we see that the class
$$
\begin{equation*}
\begin{aligned} \, &[p_{X/C}^{m!}]^\ast\otimes_\mathbb{Q}[1_{X/C}]^\ast(u) = p^{2m!}u_{K_{3X},K_{3X}}+p^{2m!}u_{K_{3X},(i_\Delta f)_\ast H^1(Z,\mathbb{Q})} \\ &\qquad+p^{2m!}u_{K_{3X},H^0(C',R^3\pi'_\ast\mathbb{Q})} +p^{m!}u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),K_{3X}} \\ &\qquad +p^{m!}u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),(i_\Delta f)_\ast H^1(Z,\mathbb{Q})}+ p^{m!}u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),H^0(C',R^3\pi'_\ast\mathbb{Q})} \\ &\qquad +p^{3m!}u_{H^0(C',R^3\pi'_\ast\mathbb{Q}),K_{3X}} +p^{3m!}u_{H^0(C',R^3\pi'_\ast\mathbb{Q}),(i_\Delta f)_\ast H^1(Z,\mathbb{Q})} \\ &\qquad+p^{3m!}u_{H^0(C',R^3\pi'_\ast\mathbb{Q}), H^0(C',R^3\pi'_\ast\mathbb{Q})}+h_1 \end{aligned}
\end{equation*}
\notag
$$
is algebraic and $h_1\stackrel{\mathrm{def}}{=}[p_{X/C}^{m!}]^\ast \otimes_\mathbb{Q}[1_{X/C}]^\ast(h)\in H_\mathbb{Q}$. Subtracting this class from the algebraic class $p^{3m!}u$, we obtain the algebraic class
$$
\begin{equation}
\begin{aligned} \, &(p^{3m!}-p^{2m!})u_{K_{3X},K_{3X}}+ (p^{3m!}-p^{2m!})u_{K_{3X},(i_\Delta f)_\ast H^1(Z,\mathbb{Q})} \nonumber \\ &\qquad+(p^{3m!}-p^{2m!})u_{K_{3X},H^0(C',R^3\pi'_\ast\mathbb{Q})} +(p^{3m!}-p^{m!})u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),K_{3X}} \nonumber \\ &\qquad +(p^{3m!}-p^{m!})u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),(i_\Delta f)_\ast H^1(Z,\mathbb{Q})} \nonumber \\ &\qquad+(p^{3m!}-p^{m!})u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}), H^0(C',R^3\pi'_\ast\mathbb{Q})}+p^{3m!}h-h_1. \end{aligned}
\end{equation}
\tag{3.1}
$$
Acting on class (3.1) by the operator $[p_{X/C}^{m!}]^\ast\otimes_\mathbb{Q}[1_{X/C}]^\ast$, we obtain the algebraic class Subtracting from this class the preceding one multiplied by the number $p^{2m!}$, we obtain the algebraic class
$$
\begin{equation*}
\begin{aligned} \, &(p^{3m!}-p^{m!})(p^{m!}-p^{2m!})u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),K_{3X}} \\ &\qquad+(p^{3m!}-p^{m!})(p^{m!}-p^{2m!})u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),(i_\Delta f)_\ast H^1(Z,\mathbb{Q})} \\ &\qquad+(p^{3m!}-p^{m!})(p^{m!}-p^{2m!})u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),H^0(C',R^3\pi'_\ast\mathbb{Q})} \\ &\qquad+[p_{X/C}^{m!}]^\ast\otimes_\mathbb{Q}[1_{X/C}]^\ast(p^{3m!}h-h_1)- p^{2m!}(p^{3m!}h-h_1). \end{aligned}
\end{equation*}
\notag
$$
Thus, for some element $h_2\in H_\mathbb{Q}$, the class
$$
\begin{equation*}
u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),K_{3X}} +u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),(i_\Delta f)_\ast H^1(Z,\mathbb{Q})} + u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),H^0(C',R^3\pi'_\ast\mathbb{Q})}+h_2
\end{equation*}
\notag
$$
is algebraic. Therefore, it follows from the algebraicity of class (3.1) that, for some element $h_3\in H_\mathbb{Q}$, the class
$$
\begin{equation}
u_{K_{3X},K_{3X}}+u_{K_{3X},(i_\Delta f)_\ast H^1(Z,\mathbb{Q})}+ u_{K_{3X},H^0(C',R^3\pi'_\ast\mathbb{Q})}+h_3
\end{equation}
\tag{3.2}
$$
is algebraic. 3.2.Since an 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 the formula $v^\ast(\gamma)=\langle \gamma\smile\alpha\rangle\beta$ (see § 1.3 in [2]), it is evident that
$$
\begin{equation}
\operatorname{pr}_{2\ast}(H^i(X,\mathbb{Q})\otimes_\mathbb{Q} H^\ast(X,\mathbb{Q}))=0\quad{\text{for all }} i\neq 8.
\end{equation}
\tag{3.3}
$$
It follows from Lemma 2.8 that the correspondence
$$
\begin{equation*}
\begin{aligned} \, &u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),K_{3X}} +u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),(i_\Delta f)_\ast H^1(Z,\mathbb{Q})} +u_{(i_\Delta f)_\ast H^1(Z,\mathbb{Q}),H^0(C',R^3\pi'_\ast\mathbb{Q})} \\ &\quad+u_{H^0(C',R^3\pi'_\ast\mathbb{Q}),K_{3X}}+ u_{H^0(C',R^3\pi'_\ast\mathbb{Q}),(i_\Delta f)_\ast H^1(Z,\mathbb{Q})}+ u_{H^0(C',R^3\pi'_\ast\mathbb{Q}),H^0(C',R^3\pi'_\ast\mathbb{Q})} \end{aligned}
\end{equation*}
\notag
$$
annihilates the space $K_{3X}\smile\operatorname{cl}_X(H)$. In addition, the elements of $H_\mathbb{Q}$ annihilate this space in accordance with (3.3). Therefore, it follows from the algebraicity of the class (3.2) that the algebraic isomorphism
$$
\begin{equation*}
K_{3X} \smile \operatorname{cl}_X(H)\underset{\widetilde{\qquad}} {\xrightarrow{x\, \mapsto \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast x\,\smile\,u)}} K_{3X}
\end{equation*}
\notag
$$
of Lemma 2.11 in fact coincides with the algebraic isomorphism
$$
\begin{equation*}
\begin{aligned} \, &K_{3X}\smile \operatorname{cl}_X(H) \\ &\qquad\underset{\widetilde{\qquad}}{\xrightarrow{x\, \mapsto \operatorname{pr}_{2\ast}(\operatorname{pr}_1^\ast x\,\smile\, [u_{K_{3X},K_{3X}}+ u_{K_{3X},(i_\Delta f)_\ast H^1(Z,\mathbb{Q})} +u_{K_{3X},H^0(C',R^3\pi'_\ast\mathbb{Q})}+h_3])}} K_{3X}. \end{aligned}
\end{equation*}
\notag
$$
On the other hand, following the arguments of § 2.12, (2.34), and Lemma 2.14, it can be shown that the Poincaré class $\wp(K_{3X}^\perp)$ determines the algebraic isomorphism (2.34) and it annihilates the $\mathbb{Q}$-space $K_{3X}\smile\operatorname{cl}_X(H)$. Taking into account canonical decompositions (2.11), (2.12), we see that the algebraic correspondence
$$
\begin{equation*}
u_{K_{3X},K_{3X}}+u_{K_{3X},(i_\Delta f)_\ast H^1(Z,\mathbb{Q})} +u_{K_{3X},H^0(C',R^3\pi'_\ast\mathbb{Q})}+h_3+\wp(K_{3X}^\perp)
\end{equation*}
\notag
$$
determines the algebraic isomorphism
$$
\begin{equation*}
H^5(X,\mathbb{Q})\,\widetilde{\to}\,H^3(X,\mathbb{Q}).
\end{equation*}
\notag
$$
This proves the theorem.
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