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\define\ab{\operatorname{ab}}
\define\ad{\operatorname{ad}}
\define\Alb{\operatorname{Alb}}
\define\alg{\operatorname{alg}}
\define\an{\operatorname{an}}
\define\Ann{\operatorname{Ann}}
\define\anti{\operatorname{anti}}
\define\arith{\operatorname{arith}}
\define\analytic{\operatorname{analytic}}
\define\Aut{\operatorname{Aut}}
\define\bb{\operatorname{b}}
\define\cc{\operatorname{c}}
\define\Card{\operatorname{Card}}
\define\Cent{\operatorname{Cent}}
\define\Centralizer{\operatorname{Centralizer}}
\define\cd{\operatorname{cd}}
\define\ch{\operatorname{ch}}
\define\CH{\operatorname{CH}}
\define\charr{\operatorname{char}}
\define\cl{\operatorname{cl}}
\define\Coker{\operatorname{Coker}}
\define\codim{\operatorname{codim}}
\define\corr{\operatorname{corr}}
\define\cont{\operatorname{cont}}
\define\const{\operatorname{const}}
\define\DC{\operatorname{DC}}
\define\ddef{\operatorname{def}}
\define\diag{\operatorname{diag}}
\define\DR{\operatorname{DR}}
\define\End{\operatorname{End}}
\define\et{\operatorname{et}}
\define\Ex{\operatorname{Ex}}
\define\GG{\operatorname{G}}
\define\Gal{\operatorname{Gal}}
\define\geom{\operatorname{geom}}
\define\GHC{\operatorname{GHC}}
\define\GHodge{\operatorname{GHodge}}
\define\GL{\operatorname{GL}}
\define\gl{\operatorname{gl}}
\define\Gr{\operatorname{Gr}}
\define\Hg{\operatorname{Hg}}
\define\Hodge{\operatorname{Hodge}}
\define\Hom{\operatorname{Hom}}
\define\Hilb{\operatorname{Hilb}}
\define\id{\operatorname{id}}
\define\Imm{\operatorname{Im}}
\define\Int{\operatorname{Int}}
\define\isog{\operatorname{isog}}
\define\Jac{\operatorname{Jac}}
\define\sK{\Cal K}
\define\Km{\operatorname{Km}}
\define\Ker{\operatorname{Ker}}
\define\Lef{\operatorname{Lef}}
\define\level{\operatorname{level}}
\define\Lf{\operatorname{Lf}}
\define\length{\operatorname{length}}
\define\Lie{\operatorname{Lie}}
\define\Norm{\operatorname{Norm}}
\define\num{\operatorname{num}}
%\define\max{\operatorname{max}}
%\define\mod{\operatorname{mod}}
\define\M{\operatorname{M}}
\define\MT{\operatorname{MT}}
\define\mult{\operatorname{mult}}
\define\MumfordTate{\operatorname{Mumford-Tate}}
\define\NS{\operatorname{NS}}
\define\Pic{\operatorname{Pic}}
\define\pr{\operatorname{pr}}
\define\Proj{\operatorname{Proj}}
\define\Rad{\operatorname{Rad}}
\define\rank{\operatorname{rank}}
\define\red{\operatorname{red}}
\define\reg{\operatorname{reg}}
\define\Res{\operatorname{Res}}
\define\sep{\operatorname{s}}
\define\SO{\operatorname{SO}}
\define\SSp{\operatorname{Sp}}
\define\SS{\operatorname{S}}
\define\ttt{\operatorname{t}}
\define\Td{\operatorname{Td}}
\define\ttd{\operatorname{td}}
\define\Tr{\operatorname{Tr}}
\define\Sect{\operatorname{Sect}}
\define\sH{\Cal H}
\define\ssh{\operatorname{sh}}
\define\Sim{\operatorname{Sim}}
\define\Sing{\operatorname{Sing}}
\define\SL{\operatorname{SL}}
\define\ssl{\operatorname{sl}}
\define\smooth{\operatorname{smooth}}
\define\sso{\operatorname{so}}
\define\ssp{\operatorname{sp}}
\define\Ssp{\operatorname{Sp}}
\define\Spec{\operatorname{Spec}}
\define\ssss{\operatorname{ss}}
\define\Sym{\operatorname{Sym}}
\define\Tate{\operatorname{Tate}}
\define\tors{\operatorname{tors}}
\define\ttop{\operatorname{top}}
\define\tr{\operatorname{tr}}
\define\U{\operatorname{U}}
\define\un{\operatorname{un}}
\define\Zar{\operatorname{Zar}}
\define\Zariski{\operatorname{Zariski}}

% Bourbaki letters
\define\C{\Bbb C} % complex numbers
\define\K{\Bbb K} % quaternions
\define\N{\Bbb N} % natural numbers
\define\Q{\Bbb Q} % rational numbers
\define\R{\Bbb R} % real numbers
\define\Z{\Bbb Z} % integers

\topmatter
\title
On the standard conjecture
\endtitle
\title
On the standard conjecture of Lefschetz type for
a compactification of a complex Abelian
scheme over an affine curve
\endtitle
\footnotetext"\ "{This work was carried out with the financial support of
the Russian Foundation for
Basic Research
(grant no. 06-01-00181).}
\author  S.G.Tankeev
\endauthor
\author  S.G.Tankeev
\endauthor
\address  Vladimir State University
\endaddress
\email
tankeev\@vpti.vladimir.ru
\endemail
\abstract
For a compactification $\pi:X\to C$ (with semi-stable fibres)
of a principally
polarized Abelian scheme $\pi':X'\to C'$
of relative dimension $n$ over an affine curve $C'\hookrightarrow C$,
we construct
an algebraic isomorphism
$H^1(C,R^{2n-i}\pi_\ast\Q)\,\,
\widetilde{\rightarrow}\,\,$
\newline
$H^1(C,R^i\pi_\ast\Q)$.
If \,\, $\pi':X'\to C'$ has a trivial trace, we also prove
the Grothendieck standard conjecture $B^2(X)$ of Lefschetz type
on algebraicity
of the operator $\Lambda^{n-1}:H^{2n}(X,\Q)\to H^2(X,\Q)$
under the assumptions that, for the generic scheme fibre
$X_\eta$ of the Abelian scheme, $\End(X_{\overline\eta})=\End(X_\eta)$,
$\dim H^0(C,R^2\pi_\ast\Q)\neq 2,\quad\dim H^2(X,\Q)\neq 2$ and
$\End(X_\eta)$
is contained in an imaginary quadratic field;
moreover, $B^2(X)$ holds for a compactification
of the Jacobian scheme of a smooth $1$-parametric family (with a section)
of curves of
genus $n$ over
an affine curve with bad reduction of purely multiplicative type at infinity.
If $n=3$,
all fibres of the N\'eron minimal model
of the Jacobian scheme over points at infinity are tori
and $\End(X_{\overline\eta})=\Z$,
then $B(X)$ holds.

\endabstract
\endtopmatter

\document

\head
{\bf Introduction }
\endhead

For a $d$-dimensional complex
smooth projective variety $X$ and for an integer
\newline
$p\geq 0$ we denote by
$H^{2p}_{\alg}(X,\Q)$ the $\Q$-subspace of
$H^{2p}(X,\Q)$ generated by cohomology classes
$\cl_X(Z)\in
H^{2p}(X,\Q)$
of algebraic cycles $Z$ of codimension $p$ on $X$.

Let $H$ be an ample divisor on $X$. For $i\leq d$, the canonical map
$$
L^{d-i}:H^i(X,\Q)\,\,@>{x\mapsto x
\cup\,\cl_X(H)^{\cup\,d-i}}>>\,\,H^{2d-i}(X,\Q)
$$
is an isomorphism by the strong Lefschetz theorem.
Let $\Lambda^{d-i}:H^{2d-i}(X,\Q)\,\,\widetilde{\rightarrow}\,\,$
\newline
$H^i(X,\Q)$ be the isomorphism inverse
to $L^{d-i}$.

{\bf The Grothendieck standard conjecture of Lefschetz type }
$B^i(X)$ \cite{1}.
{\it Given a natural
number $i\leq d$,
the isomorphism $\Lambda^{d-i}:H^{2d-i}(X,\Q)\,\,\widetilde{\rightarrow}
\,\,H^i(X,\Q)$
is algebraic}.

If $B^i(X)$ is true for any natural number $i$, then we say that
the standard conjecture $B(X)$ of Lefschetz type holds for $X$.

In terms of the Lefschetz primitive decomposition
$$
x=\sum_{k\geq{\max}(0,j-d)}L^k\,x_{j-2k},
$$
where $x_{j-2k}\in P^{j-2k}(X)$,
\quad $P^i(X)=H^i(X,\Q)\,\cap\,\Ker\,L^{d-i+1}
\quad(i\leq d)$ (\cite{2}, \S \, 3, (3.3)),
the abstract operators $\Lambda$ and $\ast$ are defined by the
following formulae:
$$
\gather
\Lambda\,x=\sum_{k\geq{\max}(1,j-d)}L^{k-1}\,x_{j-2k};\\
\ast x=\sum_{k\geq{\max}
(0,j-d)}(-1)^{(j-2k)(j-2k+1)/2}L^{d-j+k}x_{j-2k}.
\endgather
$$

It is well known that $B(X)$
is equivalent to algebraicity of $\Lambda$ and $\ast$.
It holds for Grassmann varieties,
curves, surfaces and Abelian varieties \cite{3},
for a complex projective threefold $X$ with a holomorphic
1-parametric family $\pi:X\to C$ whose generic geometric fibre is
a smooth surface of Kodaira dimension $\varkappa\leq 0$ \cite{4}.

In this article,
for a compactification $\pi:X\to C$ (with semi-stable fibres)
of a principally
polarized Abelian scheme $\pi':X'\to C'$
of relative dimension $n$ over an affine curve $C'\hookrightarrow C$,
we construct
an algebraic isomorphism
$H^1(C,R^{2n-i}\pi_\ast\Q)\,\,
\widetilde{\rightarrow}\,\,$
\newline
$H^1(C,R^i\pi_\ast\Q)$ (Theorem 1.5).
If \,\, $\pi':X'\to C'$ has a trivial trace, we also prove
the Grothendieck standard conjecture $B^2(X)$ of Lefschetz type
on algebraicity
of the operator $\Lambda^{n-1}:H^{2n}(X,\Q)\to H^2(X,\Q)$
under the assumptions that, for the generic scheme fibre
$X_\eta$ of the Abelian scheme, $\End(X_{\overline\eta})=\End(X_\eta)$,
$\dim H^0(C,R^2\pi_\ast\Q)\neq 2,\quad\dim H^2(X,\Q)\neq 2$ and
$\End(X_\eta)$
is contained in an imaginary quadratic field;
moreover, $B^2(X)$ holds for a compactification
of the Jacobian scheme of a smooth $1$-parametric family (with a section)
of curves of genus $n$ over
an affine curve with bad reduction of purely multiplicative type at infinity
(Theorem 2.11).
If $n=3$,
all fibres of the N\'eron minimal model
of the Jacobian scheme over points at infinity are tori
and $\End(X_{\overline\eta})=\Z$,
then $B(X)$ holds (Theorem 3.4).

\head
{\bf \S \, 1.
A construction of an algebraic isomorphism
$\pmb{H^1(C,R^{2n-i}\pi_\ast\Q)\,\,
\widetilde{\rightarrow}\,\,H^1(C,R^i\pi_\ast\Q)}$ for a compactification
of an Abelian scheme of relative dimension $\pmb{n}$ over an affine curve}
\endhead

{\bf 1.1.} Let $\pi':X'\to C'$ be a principally polarized
Abelian scheme of relative dimension $n$ over
a complex smooth algebraic curve $C'$, \quad
$\overset\vee\to\pi':\overset\vee\to X'\to C'$ the dual Abelian scheme,
$\tau':X'\times_{C'}X'\to C'$ the structure morphism,
$\Cal P'$ the normalized Poincar\'e bundle on
$X'\times_{C'}\overset\vee\to X'\,\,\widetilde{\rightarrow}\,\,
X'\times_{C'}X'$ \,\,\,\cite{5}.  There is a section of $\tau'$, so
we may consider
$\Cal P'$ as an element of the relative Picard group
$H^0\left(C',R^1\tau'_\ast
\Cal O^\times_{X'\times_{C'}X'}\right)=\Pic(X'\times_{C'}X')/\Pic(C')$.
The canonical exact exponential
sequence
$$
0\to\Z\to\Cal O_{X'\times_{C'}X'}@>f\mapsto e^{2\pi if}>>
\Cal O^\times_{X'\times_{C'}X'}\to 1
$$
yields a canonical map
$H^0\left(C',R^1\tau'_\ast
\Cal O^\times_{X'\times_{C'}X'}
\right)@>\Cal L\mapsto\cc_1(\Cal L)>>H^0(C',R^2\tau'_\ast\Z)$.
Hence an element $\cc_1(\Cal P')\in H^0(C',
R^2\tau'_\ast\Q)$ is well defined.

We claim that
$$
\cc_1(\Cal P')\in H^0(C',R^1\pi'_\ast\Q\otimes
R^1\pi'_\ast\Q).
\tag{1.1}
$$

Indeed, since
$\tau':X'\times_{C'}\overset\vee\to X'\to C'$ is an Abelian scheme
over a curve $C'$, the Leray spectral sequence
$E_2^{p,q}=H^p(C',R^q\tau'_\ast\Q)\,\,\Rightarrow\,\,H^\ast(X'\times_{C'}X',
\Q)$ degenerates
\cite{6}, so we have the
decompositions
$$
\gather
H^2(X'\times_{C'}\overset\vee\to X',\Q)=
H^0(C',R^2\tau'_\ast\Q)\oplus
H^1(C',R^1\tau'_\ast\Q)\oplus
H^2(C',\Q)=\\
\left[\bigoplus_{a+b=2}H^0(C',R^a\pi'_\ast\Q\otimes
R^b\overset\vee\to\pi'_\ast\Q)\right]\oplus\\
H^1(C',R^1\pi'_\ast\Q)\oplus H^1(C',R^1\overset\vee\to\pi'_\ast\Q)\oplus
H^2(C',\Q)
\endgather
$$
(\cite{7}, (3.5)). By definition, $\Cal P'$ is normalized
along $e'\times_{C'}\overset\vee\to X'$, where $e':C'\to X'$ is
the zero section (in other words,
$\Cal O_{C'\times_{C'}\overset\vee\to X'}
\,\,\widetilde{\rightarrow}\,\,(e'\times_{C'}\overset\vee\to X')
^\ast(\Cal P')$).
Therefore
$$
0=\cc_1\left(\Cal O_{C'\times_{C'}\overset\vee\to X'}\right)=
\cc_1((e'\times_{C'}\overset\vee\to X')
^\ast(\Cal P'))=(e'\times_{C'}\overset\vee\to X')
^\ast(\cc_1(\Cal P'))
$$
is the
$[H^0(C',R^2\overset\vee\to\pi'_\ast\Q)\oplus
H^1(C',R^1\overset\vee\to\pi'_\ast\Q)\oplus H^2(C',\Q)]$-component
of the Chern class $\cc_1(\Cal P')$. Since $\Cal P'$ is normalized
along
$X'\times_{C'}\overset\vee\to e'$, the $[H^0(C',R^2\pi'_\ast\Q)\oplus
H^1(C',R^1\pi'_\ast\Q)$
\newline
$\oplus H^2(C',\Q)]$-component
of Chern class $\cc_1(\Cal P')$ is trivial by the same reason.
Therefore
the claim \thetag{1.1} is proved.

Let $s\in C'$,\quad $\Cal P_s=\Cal P'\vert_{X_s\times\overset\vee\to X_s}$
the Poincar\'e bundle on $X_s\times\overset\vee\to X_s=X_s\times X_s$ and
$\mu_s:X_s\times X_s\to X_s$ the group law for
$X_s$ (the sum morphism). We recall some results of Kleiman and
Lieberman.

{\bf 1.2. Lemma} (\cite{3}, Lemma 2A12). {\it
Let $X_s$ be an Abelian variety, $Y_s$ an arbitrary smooth
projective variety and
$u\in H^2(X_s\times Y_s,\Q)$ an element of K\" unneth type $(1,1)$.
Then the correspondence
$\exp(u)=\sum u^i/i!: H^\ast(X_s,\Q)\to H^\ast(Y_s,\Q)$
takes Pontrjagin product
$\alpha\vee\beta=\mu_{s\ast}\left(\alpha\otimes\beta\right)$
to the $\cup$-product.}

{\bf 1.3. Lemma} (\cite{3}, Remark 2A13). {\it
Let $X_s$ be an Abelian variety, $\overset\vee\to X_s$
the dual Abelian variety and
$u\in H^2(X_s\times\overset\vee\to X_s,\Q)$ the $(1,1)$-K\" unneth
component of a Poincar\' e divisor.
Then the correspondence
$\exp(u)=\sum u^i/i!: H^\ast(X_s,\Q)\to H^\ast(\overset\vee\to X_s,\Q)$
is an algebraic isomorphism of the
Pontrjagin algebra $H^\ast(X_s,\Q)$ onto the $\cup$-product algebra }
$H^\ast(\overset\vee\to X_s,\Q)$.

{\bf 1.4.} In the situation under consideration,  $X_s=
\overset\vee\to X_s$ and ${\cl}_{X_s\times X_s}(\Cal P_s)$
has K\" unneth type $(1,1)$. In virtue of Lemmas 1.2 -- 1.3,
for each natural number $i\leq n$, the map
$H^{2n-i}(X_s,\Q)\to H^i(X_s,\Q)$
defined by the class
$[{\cl}_{X_s\times X_s}(\Cal P_s)]^{\cup\,i}=
{\cl}_{X_s\times X_s}(\Cal P_s)\cup\dots\cup
{\cl}_{X_s\times X_s}(\Cal P_s)\in H^{2i}(X_s\times X_s,\Q)$
is an algebraic isomorphism. This isomorphism
is the composite of the following maps:
$$
\gather
H^{2n-i}(X_s,\Q)\underset{\widetilde{\quad}}\to{@>{\pr}_{1s}^{\ast}>>}
H^{2n-i}(X_s,\Q)\otimes H^0(X_s,\Q)
\underset{\widetilde{\quad}}\to{@>
x\mapsto x\cup\,[{\cl}_{X_s\times X_s}(\Cal P_s)]^{\cup\,i}>>}\\
H^{2n}(X_s,\Q)\otimes H^i(X_s,\Q)
\underset{\widetilde{\quad}}\to{@>
{\pr}_{2s\ast}>>}H^i(X_s,\Q),
\tag{1.2}
\endgather
$$
where ${\pr}_{ks}:X_s\times X_s\to X_s$
is the canonical projection.

{\bf 1.5. Theorem.}
{\it Let $X$ be a smooth complex projective $(n+1)$-dimensional variety,
$\pi:X\to C$
a surjective morphism onto a smooth curve $C$. Assume that
each geometric fibre of the morphism $\pi$ is a union of smooth
irreducible components of multiplicity $1$ with normal crossings.

If a generic scheme fibre $X_\eta$ of the morphism $\pi$
is a principally polarized
Abelian variety,
then there exist a resolution of singularities $\sigma:Y\to
X\times_CX$ of a subscheme
$\iota:X\times_CX\hookrightarrow X\times X$
and an algebraic class $\Cal D_{\Cal P'}\in H^2_{\alg}(Y,\Q)$ such
that, for each natural number $i\leq n$, the algebraic class
$(\iota\sigma)_\ast\Cal D_{\Cal P'}^{\cup i}\in H^{2i+2}_{\alg}(X\times X,\Q)$
defines an algebraic isomorphism}
$$
H^1(C,R^{2n-i}\pi_\ast\Q)\,\,
\widetilde{\rightarrow}\,\,H^1(C,R^i\pi_\ast\Q).
$$

{\it Proof.}
We may assume that $\pi$ is smooth over $C\setminus\Delta$,
where $\Delta\hookrightarrow C$ is a finite closed subset.
Let $C'=C\setminus\Delta$, \quad $X'=X\setminus\pi^{-1}(\Delta)$,
\quad $\pi'=\pi\vert_{X'}:X'\to C'$,
\quad $\tau':X'\times_{C'}X'\to C'$ a structure
morphism of fibre product.

Let
$$
\matrix
X\times_CX&@>p_1>>&X\\
\downarrow^{p_2}&\searrow^{\tau}&\downarrow^\pi\\
X&@>\pi>>&C
\endmatrix
$$
be the canonical diagram of fibre product, and
$\sigma:Y\to X\times_CX$ a resolution of singularities of $X\times_CX$.
We may assume that $\sigma$ induces an isomorphism over $C'$. In particular,
$Y$ may be considered as a smooth projective compactification
of $X'\times_{C'}X'$. Moreover, we may assume that
$(\tau\sigma)^{-1}(\Delta)$
is a union (with some positive multiplicities) of smooth divisors with normal crossings.
By Deligne's theorem,
the canonical morphism
$H^2(Y,\Q)\to H^0(C',R^2\tau'_\ast\Q)$
is a surjective morphism of Hodge $\Q$-structures
(\cite{8}, Theorem 4.1.1, a proof of Corollary 4.1.2). Moreover,
the canonical morphism
$H^2(Y,\Q)\to H^0(C,R^2(\tau\sigma)_\ast\Q)$ is
a surjective morphism of Hodge $\Q$-structures
(\cite{9}, Corollary (15.14)).

In virtue of \thetag{1.1} we have:
$\cc_1(\Cal P')\in H^0(C',R^2\tau'_\ast\Q)$. On the other hand,
$\cc_1(\Cal P')$ is an element of Hodge type $(1,1)$. By
the Lefschetz theorem for divisors on $Y$, there exists an algebraic
class $\cc_1(\Cal O_Y(D))\in
H^2_{\alg}(Y,\Q)$ such that its image under the canonical surjective
morphism
$H^2(Y,\Q)\to H^0(C',R^2\tau'_\ast\Q)$ coincides with
$\cc_1(\Cal P')$.

Consider the canonical diagram of fibre product
$$
\matrix
X'\times_{C'}X'&@>p'_1>>&X'\\
\downarrow^{p'_2}&\searrow^{\tau'}&\downarrow^{\pi'}\\
X'&@>\pi'>>&C'.
\endmatrix
$$

For any open subset $U\hookrightarrow (C')^{\an}$,
a proper morphism
$p'_k:(\tau')^{-1}(U)\to
(\pi')^{-1}(U)$ defines a canonical map
${p'_k}^\ast:H^q((\pi')^{-1}(U),\Q)
\to H^q((\tau')^{-1}(U),{p'_k}^\ast\Q)$
(\cite{10}, Ch. II, 4.16). Since
${p'_k}^\ast\Q=\Q$, it takes the form
${p'_k}^\ast:H^q((\pi')^{-1}(U),\Q)
\to H^q((\tau')^{-1}(U),\Q)$. Consequently,
the K\"unneth decompositions on fibres of $\tau'$ yield
the canonical injective map
$$
{p_k'}^\ast:R^q\pi'_\ast\Q\,\,
\hookrightarrow\,\,
R^q\tau'_\ast\Q=\bigoplus_{a+b=q}
R^a\pi'_\ast\Q\otimes
R^b\pi'_\ast\Q.
$$

On the other hand, a proper morphism
$p'_k:(\tau')^{-1}(U)\to
(\pi')^{-1}(U)$ defines a canonical map
${p'_k}^\ast:H^q_{\cc}((\pi')^{-1}(U),\Q)
\to H^q_{\cc}((\tau')^{-1}(U),\Q)$
of cohomology with compact supports (\cite{10}, Ch. II, 4.16).

By the Poincar\'e duality
theorem for a complex manifold $V$ of complex dimension $d$,
there is a canonical isomorphism (\cite{11}, \S \, 2, Theorem (2.8))
$H^{2d}_{\cc}(V,\Q)\,\,\widetilde{\rightarrow}\,\,\Q$,
inducing isomorphism
$$
H^m_{\cc}(V,\Q)\,\,\widetilde{\rightarrow}\,\, H^{\cc}_{2d-m}(V,\Q)
$$
(\cite{12}, Ch. I, \S \, 4, 4.3;
\cite{13}, 7.8;
\cite{3},
p. 359; \cite{14}, Ch. VI, \S \, 11, Corollary 11.2).
Consequently, one can identify $H^{2d-m}(V,\Q)$
with the dual of $H^m_{\cc}(V,\Q)$.

In virtue of the Poincar\'e duality theorem, the dual map to
$$
{p'_2}^\ast:H^{2(n+1)-i}_{\cc}((\pi')^{-1}(U),\Q)
\to H^{2(n+1)-i}_{\cc}((\tau')^{-1}(U),\Q)
$$
takes the form
$p'_{2\ast}:
H^{2n+i}((\tau')^{-1}(U),\Q)\to H^i((\pi')^{-1}(U),\Q)$.
It yields the canonical map
$p'_{2\ast}:R^{2n+i}\tau'_\ast\Q\to
R^i\pi'_\ast\Q$.
A restriction of $p'_{2\ast}$ to a subsheaf
$R^{2n}\pi'_\ast\Q\otimes R^i\pi'_\ast\Q\hookrightarrow
R^{2n+i}\tau'_\ast\Q$ induces a canonical
isomorphism
$p'_{2\ast}:
R^{2n}\pi'_\ast\Q\otimes
R^i\pi'_\ast\Q
\,\,{\widetilde{\rightarrow}}\,\,
R^i\pi'_\ast\Q$.
In particular, the globalization of \thetag{1.2} is the composite
$$
\gather
R^{2n-i}\pi'_\ast\Q\underset{\widetilde{\quad}}\to{@>{p'_1}^{\ast}>>}
R^{2n-i}\pi'_\ast\Q\otimes R^0\pi'_\ast\Q
\underset{\widetilde{\quad}}\to{@>
\cup\,[\cc_1(\Cal P')]^{\cup\,i}>>}
R^{2n}\pi'_\ast\Q\otimes R^i\pi'_\ast\Q
\underset{\widetilde{\quad}}\to{@>
{p'_2}_\ast>>}R^i\pi'_\ast\Q.
\tag{1.3}
\endgather
$$

Similarly, for any open subset $U\hookrightarrow C^{\an}$,
a commutative diagram
$$
\matrix
Y&@>\sigma>>&X\times_CX&@>p_1>>&X\\
\ &\ &\downarrow^{p_2}&\searrow^{\tau}&\downarrow^\pi\\
\ &\ &X&@>\pi>>&C
\endmatrix
$$
yields a canonical proper morphism
$p_1\sigma:(\tau\sigma)^{-1}(U)\to
\pi^{-1}(U)$ and a canonical map
$(p_1\sigma)^\ast:H^q(\pi^{-1}(U),\Q)\to
H^q((\tau\sigma)^{-1}(U),\Q)$
(\cite{10}, Ch. II, 4.16). Consequently, there is a canonical
map
$(p_1\sigma)^\ast:R^q\pi_\ast\Q\to
R^q(\tau\sigma)_\ast\Q$, and
$$
(p_1\sigma)^\ast\vert_{C'}={p'_1}^\ast.
\tag{1.4}
$$

There is a commutative diagram of canonical maps
$$
\matrix
H^2(Y,\Q)\\
\downarrow&\searrow\\
H^0(C,R^2(\tau\sigma)_\ast\Q)&\to&H^0(C',R^2\tau'_\ast\Q).
\endmatrix
$$
Recall that $\cc_1(\Cal P')$ is the image of $\cc_1(\Cal O_Y(D))$ under
the canonical surjective map $H^2(Y,\Q)
\to H^0(C',R^2\tau'_\ast\Q)$.
We denote by
$\Cal D_{\Cal P'}$ the image of an algebraic class
$\cc_1(\Cal O_Y(D))$ under
the canonical edge morphism $H^2(Y,\Q)\to H^0(C,R^2(\tau\sigma)_\ast\Q)$.
Since $\tau\sigma:Y\to C$ is a $1$-parametric family with smooth
generic fibre,
the Leray spectral sequence for $\tau\sigma$ de\-ge\-ne\-rates
(\cite{9}, Corollary (15.15)): $E_2^{p,q}=E_\infty^{p,q}$. In particular,
$$
H^2(Y,\Q)=H^2(C,\Q)\oplus H^1(C,R^1(\tau\sigma)_\ast\Q)\oplus
H^0(C,R^2(\tau\sigma)_\ast\Q).
\tag{1.5}
$$
Moreover, the canonical map
$H^2(Y,\Q)\to
H^0(C,R^2(\tau\sigma)_\ast\Q)$
is a surjective morphism of Hodge $\Q$-stuctures
(\cite{9}, Corollary (15.14)).
On the other hand,
$(\tau\sigma)^\ast:
H^2(C,\Q)\hookrightarrow H^2(Y,\Q)$
is a morphism of Hodge $\Q$-structures.
Consequently,
\newline
$H^1(C,R^1(\tau\sigma)_\ast\Q)$
is a Hodge $\Q$-substructure of $H^2(Y,\Q)$, and
\thetag{1.5} is a decomposition
of Hodge $\Q$-structures.
In particular, the component
$\Cal D_{\Cal P'}\in
H^0(C,R^2(\tau\sigma)_\ast\Q)$
\newline
$\hookrightarrow H^2(Y,\Q)$
of an algebraic class $\cc_1(\Cal O_Y(D))\in H^2_{\alg}(Y,\Q)$
is a class of Hodge type $(1,1)$. Hence it is algebraic by
the Lefschetz theorem.

For any open subset $U\hookrightarrow C^{\an}$ one has a canonical
$\cup$-product map (\cite{10}, Ch. II, 6.6)
$$
H^p((\tau\sigma)^{-1}(U),\Q)\otimes_\Q H^q((\tau\sigma)^{-1}(U),\Q)\to
H^{p+q}((\tau\sigma)^{-1}(U),\Q),
$$
which in turn defines the canonical maps
$$
\gather
R^p(\tau\sigma)_\ast\Q\otimes_\Q R^q(\tau\sigma)_\ast\Q\to
R^{p+q}(\tau\sigma)_\ast\Q;\\
R^p(\tau\sigma)_\ast\Q\otimes_\Q H^0(C,R^q(\tau\sigma)_\ast\Q)@>x\otimes y
\mapsto x\cup y>>
R^{p+q}(\tau\sigma)_\ast\Q.
\endgather
$$
The second map reflects the compatibility
of the $\cup$-products with the Leray spectral
sequence $E_2^{p,q}=H^p(C,R^q(\tau\sigma)_\ast\Q)$
(\cite{15}, p. 143), and the composite
$$
R^{2n-i}\pi_\ast\Q@>
(p_1\sigma)^\ast>>R^{2n-i}(\tau\sigma)_\ast\Q@>\cup\Cal D_{\Cal P'}^{\cup\,i}>>
R^{2n-i}(\tau\sigma)_\ast\Q\cup\Cal D_{\Cal P'}^{\cup\,i}\hookrightarrow
R^{2n+i}(\tau\sigma)_\ast\Q
\tag{1.6}
$$
is well defined.

For any open subset $U\hookrightarrow C^{\an}$,
a canonical proper morphism
$p_2\sigma:(\tau\sigma)^{-1}(U)\to
\pi^{-1}(U)$ defines a canonical map
$(p_2\sigma)^\ast:H^{2(n+1)-i}_{\cc}(\pi^{-1}(U),\Q)
\to$
\newline
$ H^{2(n+1)-i}_{\cc}((\tau\sigma)^{-1}(U),\Q)$
of cohomology with compact supports
(\cite{10}, Ch. II, 4.16).
By the Poincar\'e duality
theorem,
the dual map to $(p_2\sigma)^\ast$ takes the form
$(p_2\sigma)_\ast:
H^{2n+i}((\tau\sigma)^{-1}(U),\Q)\to H^i(\pi^{-1}(U),\Q)$.
Consequently, there is a canonical map
$(p_2\sigma)_\ast:
R^{2n+i}(\tau\sigma)_\ast\Q\to R^i\pi_\ast\Q$,
and one can extend \thetag{1.6} to the composite
$$
\gather
R^{2n-i}\pi_\ast\Q@>
(p_1\sigma)^\ast>>R^{2n-i}(\tau\sigma)_\ast\Q@>\cup\Cal D_{\Cal P'}^{\cup\,i}>>
R^{2n-i}(\tau\sigma)_\ast\Q\cup\Cal D_{\Cal P'}^{\cup\,i}\hookrightarrow\\
R^{2n+i}(\tau\sigma)_\ast\Q@>
(p_2\sigma)_\ast>>R^i\pi_\ast\Q.
\tag{1.7}
\endgather
$$
Since
$(p_2\sigma)_\ast\vert_{C'}=
p'_{2\ast}$, it is easy to see that
\thetag{1.3}, \thetag{1.4} and \thetag{1.7} yield
an exact sequence of sheaves on $C^{\an}$
$$
0\to\Cal E\to R^{2n-i}\pi_\ast\Q@>x\mapsto(p_2\sigma)_\ast[(p_1\sigma)^\ast x
\cup\Cal D_{\Cal P'}^{\cup\,i}]>>R^i\pi_\ast\Q\to\Cal G\to 0,
\tag{1.8}
$$
where $\Cal E,\,\,\,\Cal G$ are supported on $\Delta$.

Let
$\Cal F=\Imm[
R^{2n-i}\pi_\ast\Q@>x\mapsto(p_2\sigma)_\ast[(p_1\sigma)^\ast x
\cup\Cal D_{\Cal P'}^{\cup\,i}]>>R^i\pi_\ast\Q]$.
Then \thetag{1.8} yields the exact sequences
$$
\gather
0\to\Cal E\to R^{2n-i}\pi_\ast\Q\to\Cal F\to 0;
\tag{1.9}\\
0\to\Cal F\to
R^i\pi_\ast\Q\to\Cal G\to 0.
\tag{1.10}
\endgather
$$

Since $H^1(C,\Cal E)=H^2(C,\Cal E)=0$, it follows from \thetag{1.9} that
$$
H^1(C,R^{2n-i}\pi_\ast\Q)
\,\,\widetilde{\rightarrow}\,\,H^1(C,\Cal F).
\tag{1.11}
$$
On the other hand, $H^1(C,\Cal G)=0$, so \thetag{1.10} -- \thetag{1.11}
yield an exact sequence
$$
H^1(C,R^{2n-i}\pi_\ast\Q)@>x\mapsto
(p_2\sigma)_\ast[(p_1\sigma)^\ast x\cup\Cal D_{\Cal P'}^{\cup\,i}]>>
H^1(C,R^i\pi_\ast\Q)\to 0.
\tag{1.12}
$$

It is well known that the
$\cup$-products are compatible with the Leray spectral
sequence $E_2^{p,q}=H^p(C,R^q\pi_\ast\Q)$
(\cite{15}, p. 143). In particular,
$$
H^i(C,R^p\pi_\ast\Q)\cup
H^j(C,R^q\pi_\ast\Q)\hookrightarrow
H^{i+j}(C,R^{p+q}\pi_\ast\Q).
\tag{1.13}
$$

Indeed, for any open subset $U\hookrightarrow C^{\an}$ one has a canonical
$\cup$-product map (\cite{10}, Ch. II,  6.6)
$H^p(\pi^{-1}(U),\Q)\otimes_\Q H^q(\pi^{-1}(U),\Q)\to
H^{p+q}(\pi^{-1}(U),\Q)$,
which in turn defines a canonical map
$R^p\pi_\ast\Q\otimes_\Q R^q\pi_\ast\Q\to R^{p+q}\pi_\ast\Q$.
Therefore \thetag{1.13} is defined by the composite of canonical maps
(\cite{10}, Ch. II, 6.6)
$$
\gather
H^i(C,R^p\pi_\ast\Q)\otimes_\Q H^j(C,R^q\pi_\ast\Q)\to
H^{i+j}(C,R^p\pi_\ast\Q\otimes_\Q R^q\pi_\ast\Q)\to\\
H^{i+j}(C,R^{p+q}\pi_\ast\Q).
\endgather
$$

The canonical pairing
$H^{2n+1-i}(X,\Q)\times H^{i+1}(X,\Q)@>x\times y\mapsto x\cup y>>
H^{2n+2}(X,\Q)$
is non-degenerate by the Poincar\'e duality theorem.
Since $H^{2n+1}(X_s,\Q)=0$ for all $s\in C\setminus\Delta$, the sheaf
$R^{2n+1}\pi_\ast\Q$ is supported on $\Delta$.
On the other hand,
the cohomological dimension of $C$ equals $2$
(\cite{10}, Ch. II, \S \, 5, Theorem 5.13.1). Consequently,
in virtue of \thetag{1.13}, one has
$$
\gather
H^1(C,R^{2n-i}\pi_\ast\Q)
\cup H^0(C,R^{i+1}\pi_\ast\Q)\hookrightarrow
H^1(C,R^{2n+1}\pi_\ast\Q)=0,
\tag{1.14}\\
H^1(C,R^{2n-i}\pi_\ast\Q)
\cup H^2(C,R^{i-1}\pi_\ast\Q)\hookrightarrow
H^3(C,R^{2n-1}\pi_\ast\Q)=0.
\tag{1.15}
\endgather
$$
It is well known that the Leray spectral sequence
$E_2^{p,q}=H^p(C,R^q\pi_\ast\Q)\Rightarrow $
\newline
$H^\ast(X,\Q)$ degenerates
(\cite{9}, Corollary (15.15)):
$E_\infty^{p,q}=E_2^{p,q}$.
Hence
$$
H^k(X,\Q)=H^0(C,R^k\pi_\ast\Q)\oplus
H^1(C,R^{k-1}\pi_\ast\Q)\oplus
H^2(C,R^{k-2}\pi_\ast\Q);
\tag{1.16}
$$
moreover, \thetag{1.16} is a decomposition of Hodge $\Q$-structures
because the canonical map
$H^2(C,R^{k-2}\pi_\ast\Q)\to H^k(X,\Q)$
is a morphism of Hodge $\Q$-structures
(\cite{9}, Theorem
(15.11), Proposition (15.12)), the canonical map
$H^k(X,\Q)\to H^0(C,R^k\pi_\ast\Q)$ is a surjective
morphism of Hodge $\Q$-structures
(\cite{9}, Corollary
(15.14)), and the kernel of a morphism of Hodge $\Q$-structures is
a Hodge $\Q$-substructure.
Therefore \thetag{1.14} - \thetag{1.16} yield the non-degeneracy
of the canonical pairing
$$
\gather
H^1(C,R^{2n-i}\pi_\ast\Q)\times H^1(C,R^i\pi_\ast\Q)@>x\times y\mapsto x\cup y>>
H^2(C,R^{2n}\pi_\ast\Q)=\\
H^{2n+2}(X,\Q)
\,\,\widetilde{\rightarrow}\,\,\Q.
\endgather
$$
In particular,
$$
\dim H^1(C,R^{2n-i}\pi_\ast\Q)=\dim H^1(C,R^i\pi_\ast\Q).
\tag{1.17}
$$
It follows from \thetag{1.12} and \thetag{1.17} that
there is a canonical isomorphism
$$
H^1(C,R^{2n-i}\pi_\ast\Q)\underset{\widetilde{\quad}}\to{@>x\mapsto
(p_2\sigma)_\ast[(p_1\sigma)^\ast x\cup\Cal D_{\Cal P'}^{\cup\,i}]>>}
H^1(C,R^i\pi_\ast\Q).
\tag{1.18}
$$

Let $\pr_k:X\times X\to X$ be the canonical projection.
Then
$\pr_k\iota\sigma=p_k\sigma$.
Since $\pr_k$ and $\iota\sigma$ are morphisms of smooth projective
varieties, we have by the projection formula:
$$
\gather
(p_2\sigma)_\ast[(p_1\sigma)^\ast x\cup\Cal D_{\Cal P'}^{\cup\,i}]=
[\pr_2\iota\sigma]_\ast
\left([\pr_1\iota\sigma]^\ast x\cup\Cal D_{\Cal P'}^{\cup\,i}\right)=\\
\pr_{2\ast}(\iota\sigma)_\ast
\left((\iota\sigma)^\ast\pr_1^\ast x\cup\Cal D_{\Cal P'}^{\cup\,i}\right)=
\pr_{2\ast}\left(\pr_1^\ast x\cup(\iota\sigma)_\ast\Cal D_{\Cal P'}^{\cup\,i}\right).
\tag{1.19}
\endgather
$$
It follows from \thetag{1.18} - \thetag{1.19} that
an algebraic class
$(\iota\sigma)_\ast\Cal D_{\Cal P'}^{\cup\,i}\in H^{2i+2}_{\alg}(X\times X,\Q)$
defines an algebraic isomorphism
$H^1(C,R^{2n-i}\pi_\ast\Q)\,\,\widetilde{\rightarrow}\,\,
H^1(C,R^i\pi_\ast\Q)$.
The theorem is proved.

\head
{\bf \S \, 2.
Algebraicity of invariant cycles and $\pmb{B^2(X)}$
for a compactification of an Abelian scheme with semi-stable degenerations}
\endhead

{\bf 2.1.}
In the notations and assumptions of Theorem 1.5,
the K\"unneth formula and \thetag{1.16} yield a decomposition
$$
(\iota\sigma)_\ast\Cal D_{\Cal P'}=\sum_{p+q+l+m=4}
u_{pqlm},
\tag{2.1}
$$
where
$u_{pqlm}\in
H^p(C,R^q\pi_\ast\Q)\otimes
H^l(C,R^m\pi_\ast\Q)$.

Fix a point $s\in C'$.
By Deligne's theorem (\cite{8}, 4.1.1 - 4.1.2), the canonical
restriction map
$H^2(X,\Q)\to H^2(X_s,\Q)$
of Hodge $\Q$-structures is the composite of the surjective morphism
$H^2(X,\Q)\to H^0(C',R^2\pi'_\ast\Q)$
and of
canonical morphisms of Hodge $\Q$-structures
$$
H^0(C',R^2\pi'_\ast\Q)\,\,
\widetilde{\rightarrow}\,\,
H^2(X_s,\Q)^
{\pi_1(C',s)}\hookrightarrow H^2(X_s,\Q).
$$
If \,\,
$H^0(C',R^2\pi'_\ast\Q)$ is a Hodge $\Q$-structure of type $(1,1)$
then, by the Lefschetz theorem,
the $\Q$-space $H^2(X_s,\Q)^{\pi_1(C',s)}$ of
invariant cycles is generated by classes of algebraic cycles.

The canonical surjective morphism of Hodge $\Q$-structures
$H^2(X,\Q)\to$
\newline
$H^0(C',R^2\pi'_\ast\Q)$ (\cite{8}, Theorem 4.1.1)
is a part of the commutative diagram (with exact lower row)
of morphisms of mixed Hodge
$\Q$-structures (\cite{8}, Corollary 3.2.18, a proof of Theorem 4.1.1)
$$
\matrix
H^2(X,\Q)&\to&H^2(X',\Q)\\
\downarrow&\searrow&\downarrow\\
H^0(C,R^2\pi_\ast\Q)&\to&H^0(C',R^2\pi'_\ast\Q)&\to &0.
\endmatrix
$$
Therefore
$$
\dim H^0(C,R^2\pi_\ast\Q)\geq\dim H^0(C',R^2\pi'_\ast\Q)\geq 1.
$$

{\bf 2.2. Lemma} (\cite{4}, Lemma 3.2).
{\it Assume that $H^0(C',R^2\pi'_\ast\Q)$
is a Hodge $\Q$-structure of type $(1,1)$. Then there is a canonical
embedding
$H^0(C,R^2\pi_\ast\Q)\hookrightarrow\NS_\Q(X)\,\,\overset\ddef\to=\,\,$
$\NS(X)\otimes\Q$.}

{\bf 2.3. Lemma.} {\it Assume that $H^0(C',R^2\pi'_\ast\Q)$
is a Hodge $\Q$-structure of type $(1,1)$. Then}
$$
u_{1111}+\sum_{p+q+l+m=4\atop p+q\neq 2}u_{pqlm}\in H^4_{\alg}(X\times
X,\Q).
$$

{\it Proof.}
First of all, in virtue of \thetag{1.16} we have:
$u_{pqlm}\in H^4(X\times X,\Q)\cap H^{2,2}(X\times X,\C)$. On
the other hand, $H^2(C,\Q)=\Q\cdot\cl_X(X_s)\hookrightarrow
\NS_\Q(X)$.
The Lefschetz theorem and Lemma 2.2 yield
$$
\gather
u_{02lm}\in[H^0(C,R^2\pi_\ast\Q)\otimes H^2(X,\Q)]\cap H^{2,2}(X\times X,\C)
=\\
H^0(C,R^2\pi_\ast\Q)\otimes[H^2(X,\Q)\cap H^{1,1}(X,\C)]
\hookrightarrow\\
\NS_\Q(X)\otimes\NS_\Q(X)
\hookrightarrow H^4_{\alg}(X\times X,\Q);\\
u_{1102}\in[H^1(C,R^1\pi_\ast\Q)\otimes H^0(C,R^2\pi_\ast\Q)]
\cap H^{2,2}(X\times X,\C)
=\\
[H^1(C,R^1\pi_\ast\Q)\cap H^{1,1}(X,\C)]\otimes H^0(C,R^2\pi_\ast\Q)
\hookrightarrow\\
\NS_\Q(X)\otimes\NS_\Q(X)
\hookrightarrow H^4_{\alg}(X\times X,\Q);\\
u_{1120}\in[H^1(C,R^1\pi_\ast\Q)\otimes H^2(C,\Q)]
\cap H^{2,2}(X\times X,\C)
=\\
[H^1(C,R^1\pi_\ast\Q)\cap H^{1,1}(X,\C)]\otimes H^2(C,\Q)
\hookrightarrow\\
\NS_\Q(X)\otimes\NS_\Q(X)
\hookrightarrow H^4_{\alg}(X\times X,\Q);\\
u_{20lm}\in[H^2(C,\Q)\otimes H^2(X,\Q)]\cap H^{2,2}(X\times X,\C)=\\
H^2(C,\Q)\otimes[H^2(X,\Q)\cap H^{1,1}(X,\C)]
\hookrightarrow\NS_\Q(X)\otimes\NS_\Q(X)
\hookrightarrow H^4_{\alg}(X\times X,\Q).
\endgather
$$
It remains to note that
$(\iota\sigma)_\ast\Cal D_{\Cal P'}=\sum_{p+q+l+m=4}
u_{pqlm}\in H^4_{\alg}(X\times X,\Q)$.
The lemma is proved.

{\bf 2.4. Lemma.} {\it If $p+q\neq 2$, then}
$\pr_{2\ast}\left(\pr_1^\ast H^{2n}(X,\Q)\cup u_{pqlm}\right)=0$.

{\it Proof.}
It is evident that $u_{pqlm}\in H^{p+q}(X,\Q)\otimes
H^{4-p-q}(X,\Q)$ and
$$
\gather
\pr_{2\ast}\left(\pr_1^\ast H^{2n}(X,\Q)\cup u_{pqlm}\right)=
\pr_{2\ast}\left([H^{2n}(X,\Q)\otimes H^0(X,\Q)]
\cup u_{pqlm}\right)\hookrightarrow\\
\pr_{2\ast}\left(H^{2n+p+q}(X,\Q)\otimes H^{4-p-q}(X,\Q)\right).
\endgather
$$
On the other hand,
choose an element $\sqrt{-1}\in\C$ and consider
the degree map $<>:H^\ast(X,\Q)\to\Q$ defined
as the orientation isomorphism $H^{2n+2}(X,\Q)\,\,
\widetilde{\rightarrow}\,\,\Q$ on $H^{2n+2}(X,\Q)$ and as zero on $H^i(X,\Q)$
for $i<2n+2$. The
Poincar\'e duality states that the map
$x\mapsto<\cdot\,\,\cup\,\, x>$
induces isomorphisms $H^{2n+2-i}(X,\Q)\,\,\widetilde{\rightarrow}\,\,
H_i(X,\Q)$, which will be viewed as identifications (\cite{3}, 1.2).
Then for all $\alpha\in H^{2n+p+q}(X,\Q)$, \,\,\, $\beta\in H^{4-p-q}(X,\Q)$,
\,\,\, $\gamma\in H^\ast(X,\Q)$, one can apply the following standard
calculations:
$$
\gather
<\alpha>=0;
\tag{2.2}\\
<\pr_{2\ast}(\alpha\otimes\beta)\cup\gamma>=
<(\alpha\otimes\beta)\cup\pr_2^\ast\gamma>=\\
<(\alpha\otimes\beta)\cup(\cl_X(X)\otimes\gamma)>=
<\alpha\otimes(\beta\cup\gamma)>=
<\alpha>\cdot<\beta\cup\gamma>=0;
\tag{2.3}\\
\pr_{2\ast}(\alpha\otimes\beta)=0.
\tag{2.4}
\endgather
$$
The lemma is proved.

{\bf 2.5. Lemma.} {\it Assume that $H^0(C',R^2\pi'_\ast\Q)$
is a Hodge $\Q$-structure of type $(1,1)$.
Then an algebraic class
$$
u_{1111}+\sum_{p+q+l+m=4\atop p+q\neq 2}u_{pqlm}
$$
defines an algebraic
isomorphism $H^1(C,R^{2n-1}\pi_\ast\Q)\,\,\widetilde{\rightarrow}\,\,
H^1(C,R^1\pi_\ast\Q)$. }

{\it Proof.}
By \thetag{1.13} we have:
$$
[H^1(C,R^{2n-1}\pi_\ast\Q)\otimes H^0(C,\Q)]\cup
u_{pqlm}\hookrightarrow
H^{p+1}(C,R^{q+2n-1}\pi_\ast\Q)\otimes
H^l(C,R^m\pi_\ast\Q).
$$
If $(p,q)\neq(1,1)$, then it is evident that the degree map $<>$
is trivial on the space
$H^{p+1}(C,R^{q+{2n-1}}\pi_\ast\Q)$ because
$H^{p+1}(C,R^{q+2n-1}\pi_\ast\Q)\neq H^2(C,R^{2n}\pi_\ast\Q)=H^{2n+2}(X,\Q)$.
For all $\alpha\in
H^{p+1}(C,R^{q+2n-1}\pi_\ast\Q)$,
\quad $\beta\in
H^l(C,R^m\pi_\ast\Q)$, \quad $\gamma\in H^\ast(X,\Q)$
one can apply the standard calculations \thetag{2.2} - \thetag{2.4}
in order to get
$$
\pr_{2\ast}\left([H^1(C,R^{2n-1}\pi_\ast\Q)\otimes H^0(C,\Q)]\cup
u_{pqlm}\right)=0\quad{\text{for }}\quad(p,q)\neq(1,1).
$$
On the other hand,
$$
\gather
[H^1(C,R^{2n-1}\pi_\ast\Q)\otimes H^0(C,\Q)]\cup
u_{11lm}\hookrightarrow
H^2(C,R^{2n}\pi_\ast\Q)\otimes H^l(C,R^m\pi_\ast\Q)=\\
H^{2n+2}(X,\Q)\otimes H^l(C,R^m\pi_\ast\Q);\\
\left[\pr_{2\ast}\left([H^1(C,R^{2n-1}\pi_\ast\Q)\otimes H^0(C,\Q)]\cup
u_{11lm}\right)\hookrightarrow H^1(C,R^1\pi_\ast\Q)
\right]\,\,\Leftrightarrow\\
\,\,(l,m)=(1,1).
\endgather
$$
Therefore Lemmas 2.3, 2.4  and
Theorem 1.5 show that an algebraic class
$$
u_{1111}+\sum_{p+q+l+m=4\atop p+q\neq 2}u_{pqlm}
$$
defines an algebraic isomorphism
$H^1(C,R^{2n-1}\pi_\ast\Q)\,\,\widetilde{\rightarrow}\,\,
H^1(C,R^1\pi_\ast\Q)$. The lemma is proved.

{\bf 2.6.} We recall a definition of the Poincar\'e class
of a Hodge
$\Q$-structure of geometric origin
(\cite{4},  1.5).

Let $W$ be a smooth $d$-dimensional projective variety,
$V_\Q\hookrightarrow H^m(W,\Q)$ a non-zero
Hodge $\Q$-substructure and $\dim
V_\Q\neq 2$ for $m$ even.
The degree map
$<>:H^{2d}(W,\Q)\,\widetilde{\rightarrow}\,\Q$ defines
a non-degenerate pairing
$$
\gather
\Phi:H^m(W,\Q)\times H^m(W,\Q)\to\Q\\
x\times y\mapsto\,\Phi(x,y)=<x\cup L^{d-m}y>
\endgather
$$
with non-degenerate restriction $\Phi\vert_{V_\Q\times V_\Q}\,\,
\overset \ddef\to=\,\,\Phi_{V_\Q}$. It
follows from the Schur lemma that
$\left[V_\Q\otimes V_\Q\right]^{\Aut\left(\Phi_{V_\Q}\right)^0}$
is $1$-dimensional subspace generated by some Hodge cycle
$\goth p(V_\Q)$.
We call
$\goth p(V_\Q)$ the {\it Poincar\'e
class} of the Hodge substructure $V_\Q$.

{\bf 2.7. Lemma.} {\it Assume that $H^0(C',R^2\pi'_\ast\Q)$
is a Hodge $\Q$-structure of type $(1,1)$, \,\,\,
$\dim H^1(C,R^1\pi_\ast\Q)\leq 1$ and $\dim H^2(X,\Q)\neq 2$.
Then $B^2(X)$ holds.}

Indeed, in the case under consideration,
$H^1(C,R^1\pi_\ast\Q)=0$ or else
$H^1(C,R^1\pi_\ast\Q)$ is a Hodge $\Q$-structure of type $(1,1)$.
By Lemma 2.2, $H^0(C,R^2\pi_\ast\Q)$
is generated by algebraic classes.
Hence, by the Lefschetz theorem,
$H^2(X,\Q)=H^0(C,R^2\pi_\ast\Q)\oplus H^1(C,R^1\pi_\ast\Q)\oplus H^2(C,\Q)$
is generated by algebraic classes. We see that
\newline
$\goth p(H^2(X,\Q))\in H^2(X,\Q)\otimes H^2(X,\Q)$ is algebraic.
Since $\dim H^2(X,\Q)\neq 2$, an algebraic class
$\goth p(H^2(X,\Q))$
defines an algebraic isomorphism
$H^{2n}(X,\Q)\,\,\widetilde{\rightarrow}\,\, H^2(X,\Q)$
by Lemma 1.7 in \cite{4}. Hence $B^2(X)$ holds (\cite{4}, 1.8).
The lemma is proved.

{\bf 2.8. Lemma.} {\it The Poincar\'e class
$\goth p(H^2(C,\Q))$ defines an algebraic isomorphism}
$$
H^0(C,R^{2n}\pi_\ast\Q)\,\,
\widetilde{\rightarrow}\,\,
H^2(C,\Q).
$$

The case $n=2$ is treated in (\cite{4}, Lemma 3.7).
A proof carries over
verbatim to the case $n\geq 3$.

{\bf 2.9. Lemma.} {\it Assume that $H^0(C',R^2\pi'_\ast\Q)$
is a Hodge $\Q$-structure of type $(1,1)$, $\dim H^1(C,R^1\pi_\ast\Q)\geq 2$
and \,\,\, $\dim H^0(C,R^2\pi_\ast\Q)\neq 2$. Then the Poincar\'e class
\newline
$\goth p(H^0(C,R^2\pi_\ast\Q))$ defines an algebraic
isomorphism $H^2(C,R^{2n-2}\pi_\ast\Q)\,\,\widetilde{\rightarrow}\,\,$
\newline
$H^0(C,R^2\pi_\ast\Q)$.}

The case $n=2$ is treated in (\cite{4}, Lemma 3.8).
A proof carries over
verbatim to the case $n\geq 3$.

{\bf 2.10. Lemma.} {\it Assume that $H^0(C',R^2\pi'_\ast\Q)$
is a Hodge $\Q$-structure of type $(1,1)$, $\dim H^1(C,R^1\pi_\ast\Q)\geq 2$
and \,\,\, $\dim H^0(C,R^2\pi_\ast\Q)\neq 2$. Then an algebraic
class
$$
w=u_{1111}+\sum_{p+q+l+m=4\atop p+q\neq 2}u_{pqlm}+
\goth p(H^0(C,R^2\pi_\ast\Q))+\goth p(H^2(C,\Q))
$$
defines an algebraic
isomorphism $H^{2n}(X,\Q)\,\,\widetilde{\rightarrow}\,\,
H^2(X,\Q)$, and $B^2(X)$ holds.}

{\it Proof.}
First of all,
$u_{1111}$
and $\goth p(H^0(C,R^2\pi_\ast\Q))$ annihilate $H^0(C,R^{2n}\pi_\ast\Q)
\otimes H^0(C,\Q)$. By Lemmas 2.3, 2.4 and 2.8, an algebraic class $w$
defines an algebraic isomorphism
$H^0(C,R^{2n}\pi_\ast\Q)\,\,\widetilde{\rightarrow}\,\,H^2(C,\Q)$.

On the other hand,
$\goth p(H^0(C,R^2\pi_\ast\Q))$ and $\goth p(H^2(C,\Q))$ annihilate
\newline
$H^1(C,R^{2n-1}\pi_\ast\Q)\otimes H^0(C,\Q)$.
It follows from Lemma 2.5 that $w$ defines an algebraic isomorphism
$H^1(C,R^{2n-1}\pi_\ast\Q)\,\,\widetilde{\rightarrow}\,\,
H^1(C,R^1\pi_\ast\Q)$.

Finally,
$u_{1111}$
and $\goth p(H^2(C,\Q))$ annihilate $H^2(C,R^{2n-2}\pi_\ast\Q)\otimes
H^0(C,\Q)$;
in vir\-tue of Lemmas 2.3, 2.4 and
2.9 an element $w$ defines
an algebraic isomorphism
$H^2(C,R^{2n-2}\pi_\ast\Q)\,\,
\widetilde{\rightarrow}\,\,H^0(C,R^2\pi_\ast\Q)$.
It remains to use the decompositions \thetag{1.16} for $k\in\{2,2n\}$
in order to show that $w$ defines
an algebraic isomorphism
$H^{2n}(X,\Q)$
\newline
$\,\,\widetilde{\rightarrow}\,\,
H^2(X,\Q)$.
Hence $B^2(X)$ holds (\cite{4},  1.8).
The lemma is proved.

{\bf 2.11. Theorem.}
{\it Let $X$ be a smooth complex projective $(n+1)$-dimensional variety,
$\pi:X\to C$
a surjective morphism onto a smooth curve $C$,
every geometric fibre of the morphism $\pi$ is a union of smooth
irreducible components of multiplicity $1$ with normal crossings.

Assume that the generic scheme fibre $X_\eta$ of the morphism $\pi$
is a principally polarized
Abelian variety with a trivial trace
and at least one of the following conditions
holds}:

(i) {\it a generic geometric fibre $X_{\overline\eta}$
has no nontrivial endomorphisms $($in other words,
$\End(X_{\overline\eta})\,\,\widetilde{\rightarrow}\,\,\Z)$};

(ii) {\it $\End(X_\eta)\otimes\Q=
\End(X_{\overline\eta})\otimes\Q$ and $\End(X_{\overline\eta})\otimes\Q$
is an imaginary quadratic field};

(iii) {\it $\pi:X\to C$ is a compactification of the Jacobian scheme
$\pi':\Pic^0(Y'/C')\to C'$ associated to a smooth family $\tau':Y'\to C'$
$($with a section$)$ of curves of genus $n$ over an affine
curve $C'\hookrightarrow C$, and the generic scheme
fibre of the morphism \,\,
$\pi'$ has bad reduction of purely
multiplicative type at some place $\delta\in\Delta=C\setminus C'$}.

{\it If \,\, $\dim H^0(C,R^2\pi_\ast\Q)\neq 2$ \, and \,
$\dim H^2(X,\Q)\neq 2$,
then the operator $\Lambda^{n-1}:$
\newline
$H^{2n}(X,\Q)\to H^2(X,\Q)$
is algebraic.}

{\it Proof.} Assume first that (i) or (ii) holds.
Then
$\End_{C'}(X')\otimes\Q$
is contained in an imaginary quadratic field.
Fix a point $s\in C'$.
The representation of monodromy $\rho:\pi_1(C',s)\to\GL(H^1(X_s,\Q))$
is semisimple
(\cite{8}, 4.2.6), and hence the $\Q$-algebra
$H=\End(R^1\pi'_\ast\Q)$ is isomorphic to a direct sum of matrix
algebras of the form
$M_{n_i}(D_i)$, where $D_i$ is a certain skew field. On the other hand,
the centre $Z$ of the algebra $H$ has the Hodge type $(0,0)$, and hence it is
contained in the subalgebra $\End_{C'}(X')\otimes\Q$
(\cite{8}, 4.4.7).
Therefore $H=M_n(D)$,
where $D$ is some skew field, $Z=\Cent(D)$,\quad$[Z:\Q]\leq 2$.
If $Z$ is an imaginary quadratic extension of $\Q$, then
$\End_{C'}(X')\otimes\Q=\End(R^1\pi'_\ast\Q)$
(\cite{8}, Proposition 4.4.11).
If $Z=\Q$, then $D$ is a central division algebra over $\Q$ and hence
$D\otimes_{\Q}\R$ is a matrix algebra over $\R$ or a matrix algebra over the
skew field $\K$ of classical quaternions. Consequenly,
$\End_{C'}(X')\otimes\Q=\End(R^1\pi'_\ast\Q)$
(\cite{8}, Proposition 4.4.11).
In either of these cases we have:
$$
\End_{C'}(X')\otimes\Q=\End(R^1\pi'_\ast\Q).
\tag{2.5}
$$
Moreover,
$\End_{C'}(X')\otimes\Q$ is a
Hodge $\Q$-structure of type $(0,0)$ (\cite{8}, 4.4.6). On the other hand,
a polarization of the Abelian variety $X_s$ ($s\in C'$)
defines an isomorphism
of Hodge $\Q$-structures
$H^1(X_s,\Q)^\vee\,\,\widetilde{\rightarrow}\,\,H^1(X_s,\Q)(1)$
(\cite{8}, 4.2.3).
Therefore
$$
\gather
\End(R^1\pi'_\ast\Q)
\,\,\widetilde{\rightarrow}\,\,
\left[H^1(X_s,\Q)^\vee\otimes H^1(X_s,\Q)\right]^{\pi_1(C', s)}
\,\,\widetilde{\rightarrow}\,\,\\
\left[H^1(X_s,\Q)\otimes H^1(X_s,\Q)\right]^{\pi_1(C', s)}(1)
\endgather
$$
is a Hodge $\Q$-structure of type $(0,0)$. Since there are canonical
morphisms of Hodge $\Q$-structures
(\cite{16}, Ch. VIII, \S \, 7, Exercise 11 (a))
$$
H^2(X_s,\Q)^{\pi_1(C',s)}\,\,\widetilde{\rightarrow}\,\,
\left[\wedge^2H^1(X_s,\Q)\right]^{\pi_1(C', s)}
\hookrightarrow
\left[H^1(X_s,\Q)\otimes H^1(X_s,\Q)\right]^{\pi_1(C', s)},
$$
we see that
$H^0(C',R^2\pi'_\ast\Q)=H^2(X_s,\Q)^{\pi_1(C',s)}$ is a Hodge $\Q$-structure
of type $(1,1)$.

Finally, assume that (iii) holds. Let $X'=\Pic^0(Y'/C')$, \quad
$$
\matrix
Y'&\hookrightarrow&Y\\
\downarrow^{\tau'}&\ &\downarrow^\tau\\
C'&\hookrightarrow&C
\endmatrix
$$
a diagram of fibre product defined by a compactification,
\quad $Y$ a smooth projective surface, \quad $\Cal J\to C$
the N\'eron minimal model of the generic scheme fibre of the
canonical morphism $\pi':X'\to C'$.

Using the well known results of Deligne and Mumford on degenerations
of curves and Jacobians (\cite{17}, \S \, 2),
it is easy to show that the following conditions are equi\-valent:

(a) the geometric fibre of $\tau:Y\to C$ over $\delta$ is a union of
rational curves of multiplicity 1 with normal crossings;

(b) the geometric fibre of $\Cal J\to C$ over $\delta$ is an extension
of a linear torus by a finite group;

(c) the generic scheme fibre of the morphism $\pi': X'\to C'$ has bad
reduction of purely multiplicative type at $\delta$.

Using (if necessary) the
monoidal transformation $\widetilde{Y}\to Y$ of the surface $Y$
along the set of all singular points of irreducible components of the
fibre $Y_\delta$, we may assume that $Y_\delta$
is a union of {\it smooth} rational curves of multiplicity 1 with normal
crossings.

If $t$ is a local holomorphic coordinate on a neigbourhood
of a point $\delta\in C$ (such that $\delta$ is defined by the
equation $t=0$), then we may assume that $\tau:Y\to C$ is given locally by
$t=z_i$, or else $t=z_1z_2$
where $z_1,z_2$ are local holomorphic coordinates
on $Y$.

Fix a singular point $y_0\in\Sing(Y\times_CY)$. Then
$y_0=y_{12}\times y_{34}$, where $y_{12},y_{34}\in Y_\delta$,
\quad $y_{12}\in F_1\cap F_2$ for appropriate smooth rational
curves $F_i\hookrightarrow Y_\delta,\quad F_1\neq F_2$,
\quad
$F_i$ is defined locally by $z_i=0$ on the first copy of $Y$,
\quad
$y_{34}\in F_3\cap F_4$ for appropriate smooth rational
curves $F_j\hookrightarrow Y_\delta,\quad F_3\neq F_4$,
\quad $F_j$ is defined locally by $z_j=0$ on the second copy of $Y$.
A restriction of the
canonical embedding $Y\times_CY\hookrightarrow Y\times Y$
to a small neighbourhood of a point $y_0$ is analytically equivalent to the
embedding of the affine cone $W=\{z_1z_2=z_3z_4\}\hookrightarrow
\C^4$.

It is easy to see that a monoidal transformation of $\C^4$ along the
plane
\newline
$\{z_1=z_3=0\}$ yields a resolution of singularity
$W_{\text{strict}}\to W$ of the cone $W$.
Consequently, a monoidal transformation of the embedded subvariety
$Y\times_CY\hookrightarrow Y\times Y$ along $F_1\times F_3$ yields
a resolution of singularity of $Y\times_CY$ at a point $y_0$.
Therefore
a sequence of monoidal transformations of
the embedded subvariety
$Y\times_CY\hookrightarrow Y\times Y$ along appropriate strict images of
irreducible components of $Y_\delta\times Y_\delta$
yields a resolution of singularities of $Y\times_CY$ over some
neighbourhood of $\delta\in C$. Let $\sigma:V\to
Y\times_CY$ be the resolution of singularities under consideration.
By construction, the geometric fibre $V_\delta$
is a union of smooth rational surfaces of multiplicity $1$ with normal
crossings. We may assume that all fibres of
the canonical morphism $V\to C$ are divisors with normal crossings.

Let $\tau'\times_{C'}\tau':Y'\times_{C'}Y'\to C'$ be the structure morphism.
Since $V$ is a smooth compactification of $Y'\times_{C'}Y'$,
it follows from Deligne's results that the canonical map
$$
H^2(V,\Q)\to H^0(C',R^2(\tau'\times_{C'}\tau')_\ast\Q)=
H^2(Y'_s\times Y'_s,\Q)^{\pi_1(C',s)}
$$
is a surjective morphism of Hodge $\Q$-structures
(\cite{8}, Corollary 4.1.2). On the other hand,
$H^2(V,\Cal O_V)$
coincides with the component of type $(0,2)$ of the Hodge decomposition
of $H^2(V,\Q)\otimes\C$.

We claim that
$$
\forall\alpha\in H^2(V,\Cal O_V)\qquad\forall s\in C'
\qquad\alpha_s\,\,\overset\ddef\to=\,\,\alpha\vert_{V_s}=0.
\tag{2.6}
$$

Indeed, in virtue of
the classical Hodge theory, $\alpha=\overline\beta$
for a holomorphic $2$-form $\beta\in
H^0(V,\Omega^2_V)$.
If $z_1,z_2$ are local holomorphic coordinates on $V_s$, then we have:
$$
\gather
\beta_s
\,\,\overset\ddef\to=\,\,
\beta\vert_{V_s}=b(z_1,z_2)\,dz_1\wedge dz_2,\\
\beta_s\wedge\overline{\beta_s}=\vert b(z_1,z_2)\vert^2 dz_1\wedge
dz_2\wedge d\overline{z_1}\wedge d\overline{z_2}.
\endgather
$$
Therefore
$$
\int_{V_s}\beta_s\wedge\overline{\beta_s}=0\quad\Leftrightarrow\quad\beta_s=0.
\tag{2.7}
$$
Since
$$
\int_{V_s}\beta_s\wedge\overline{\beta_s}=\int_{V_s}\beta\wedge\overline
\beta,
$$
\thetag{2.7} takes the form
$$
\int_{V_s}\beta\wedge\overline\beta=0\quad\Leftrightarrow\quad\beta_s=0.
\tag{2.8}
$$
Every irreducible component of the geometric
fibre $V_\delta=\bigcup_{i=1}^nE_i$ is a smooth rational surface. Hence
$H^0(E_i,\Omega^2_{E_i})=0$
and $\beta\vert_{E_i}=0$ for all $i$,
$$
\int_{V_\delta}\beta\wedge\overline\beta=\sum_{i=1}^n\int_{E_i}\beta\wedge
\overline\beta=
\sum_{i=1}^n\int_{E_i}\beta\vert_{E_i}\wedge
\overline{\beta\vert_{E_i}}=0.
\tag{2.9}
$$
Since $V_\delta$ is homologous to $V_s$ on $V$,
and $\beta\wedge\overline\beta$ is a $d$-closed form, we obtain
by the Stokes theorem:
$$
\int_{V_\delta}\beta\wedge\overline\beta=
\int_{V_s}\beta\wedge\overline\beta.
$$
Therefore \thetag{2.8} - \thetag{2.9} yield $\alpha_s=
\overline\beta_s=0$.
The claim \thetag{2.6} is proved.

It follows from \thetag{2.6} that
$H^2(Y'_s\times Y'_s,\Q)^{\pi_1(C',s)}$
is a Hodge $\Q$-structure of type $(1,1)$. In virtue of the K\"unneth
formula we have:
$$
\gather
[H^1(Y'_s,\Q)\otimes H^1(Y'_s,\Q)]^{\pi_1(C',s)}=
[H^1(\Pic^0(Y'_s),\Q)\otimes H^1(\Pic^0(Y'_s),\Q)]^{\pi_1(C',s)}\\
{\text{is a Hodge }}\Q-{\text{structure of type }}(1,1).
\tag{2.10}
\endgather
$$
Since there are canonical
morphisms of Hodge $\Q$-structures
(\cite{16}, Ch. VIII, \S \, 7, Exercise 11 (a))
$$
\gather
H^2(X_s,\Q)^{\pi_1(C',s)}\,\,\widetilde{\rightarrow}\,\,
\left[\wedge^2H^1(\Pic(Y'_s),\Q)\right]^{\pi_1(C', s)}
\hookrightarrow\\
\left[H^1(\Pic^0(Y'_s),\Q)\otimes H^1(\Pic^0(Y'_s),\Q)\right]^{\pi_1(C', s)},
\endgather
$$
we see that
$H^0(C',R^2\pi'_\ast\Q)=H^2(X_s,\Q)^{\pi_1(C',s)}$ is a Hodge $\Q$-structure
of type $(1,1)$.

Consequently, if at least one of the conditions (i) -- (iii) holds, then
\newline
$H^0(C',R^2\pi'_\ast\Q)$ is a Hodge $\Q$-structure
of type $(1,1)$.

If \,\, $\dim H^0(C,R^2\pi_\ast\Q)\neq 2$ \, and \,
$\dim H^2(X,\Q)\neq 2$, then
the operator $\Lambda^{n-1}:H^{2n}(X,\Q)$
$\to H^2(X,\Q)$
is algebraic in virtue of Lemmas 2.7 and 2.10.
The theorem is proved.

\head
{\bf \S \, 3.
On $\pmb{B(X)}$ for a compactification of a Jacobian scheme
of relative dimension 3
over an affine curve with bad reductions of purely multiplicative
type}
\endhead

{\bf 3.1. Lemma.} {\it Let
$\tau:Y\to C$ be a morphism of a smooth
projective surface $Y$ onto a smooth curve $C$. Assume that all singular
fibres are unions of smooth rational curves of multiplicity $1$ with normal
crossings. Then there exists a desingularization
$\sigma:[Y\times_CY\times_CY]^{\smooth}\to Y\times_CY\times_CY$
such that all singular fibres of the composite
$$[Y\times_CY\times_CY]^{\smooth}@>\sigma>>Y\times_CY\times_CY\to C$$
are unions of smooth irreducible components $E'_i$ of
multiplicity $1$ with normal crossings, and $E'_i$ is obtained from
$\Bbb P^1\times\Bbb P^1\times\Bbb P^1$ by a composite of
blowings up of points and smooth rational curves. In particular,
$H^3(E'_i,\Q)=0$. }

{\it Proof.} If $t$ is a local holomorphic coordinate on a neigbourhood
of a point $\delta\in C$ (such that $\delta$ is defined by the
equation $t=0$), then we may assume that $\tau:Y\to C$ is given locally by
$t=z_i$, or else $t=z_1z_2$
where $z_1,z_2$ are local holomorphic coordinates
on $Y$.

Let $z_1,\dots,z_6$ be local holomorphic coordinates on
$X\,\,\overset\ddef\to=\,\,Y\times Y\times Y$. We may assume that
a subvariety $V\,\,\overset\ddef\to=\,\,
Y\times_CY\times_CY\hookrightarrow Y\times Y\times Y=X$
is defined locally by one of the following systems of equations:
$$
\gather
z_1z_2=z_3z_4=z_5z_6;
\tag{3.1}\\
z_1=z_3z_4=z_5z_6;
\tag{3.2}\\
z_1=z_3=z_5z_6;
\tag{3.3}\\
z_1=z_3=z_5.
\tag{3.4}
\endgather
$$

Let
$D\hookrightarrow\C^6$ be a polydisk with coordinates $z_1,\dots,z_6$,
\quad $W=V\cap D$
an open subset of $V^{\an}$.

We claim that
$$
\dim\Sing(W)=
\cases
\,\,\,\,1,\quad{\text{ if }}\,\,\, W=
\{z_1z_2=z_3z_4=z_5z_6\};\\
\,\,\,\,1,\quad{\text{ if }}\,\,\, W=
\{z_1=z_3z_4=z_5z_6\};\\
-1,\quad{\text{ if }}\,\,\, W=
\{z_1=z_3=z_5z_6\};\\
-1,\quad{\text{ if }}\,\,\, W=
\{z_1=z_3=z_5\}.
\endcases
\tag{3.5}
$$

Indeed, assume first that
$W=\{z_1z_2=z_3z_4=z_5z_6\}$. Since $W$ is the set of zeroes of holomorphic
functions $f_1=z_1z_2-z_3z_4,\quad f_2=z_1z_2-z_5z_6$, we have by (\cite{18},
Ch. 0, \, \S \, 2):
$$
\Sing(W)=\{(z_1,\dots,z_6)\in W\,\,\vert\,\,
\rank\Cal J(f_1,f_2)<2\},
$$
where $\Cal J=\Cal J(f_1,f_2)$ is the Jacobi matrix
$$
\gather
\Cal J=
\left(
\matrix
\partial f_1\over\partial z_1&\partial f_1\over\partial z_2&
\partial f_1\over\partial z_3&\partial f_1\over\partial z_4&
\partial f_1\over\partial z_5&\partial f_1\over\partial z_6\\
\partial f_2\over\partial z_1&\partial f_2\over\partial z_2&
\partial f_2\over\partial z_3&\partial f_2\over\partial z_4&
\partial f_2\over\partial z_5&\partial f_2\over\partial z_6
\endmatrix
\right)=
\left(
\matrix
z_2&z_1&-z_4&-z_3&0&0\\
z_2&z_1&0&0&-z_6&-z_5
\endmatrix
\right).
\endgather
$$
Therefore
$$
\gather
\Sing(W)=
\{(z_1,\dots,z_6)\in W\,\,\vert\,\,
z_1z_3=z_1z_4=z_1z_5=z_1z_6=\\
z_2z_3=z_2z_4=z_2z_5=z_2z_6=
z_3z_5=z_3z_6=
z_4z_5=z_4z_6=0\};\\
\Sing(W)=
\{(z_1,\dots,z_6)\in D\,\,\vert\,\,z_2=z_3=z_4=z_5=z_6=0\}\,\,\cup\\
\{(z_1,\dots,z_6)\in D\,\,\vert\,\,z_1=z_3=z_4=z_5=z_6=0\}\,\,\cup\dots\cup\\
\{(z_1,\dots,z_6)\in D\,\,\vert\,\,z_1=z_2=z_3=z_4=z_5=0\};
\tag{3.6}
\\
\dim\Sing(W)=1.
\endgather
$$

Similarly, for $W=\{z_1=z_3z_4=z_5z_6\}$ we have:
$$
\gather
\Cal J=
\left(
\matrix
1&0&-z_4&-z_3&0&0\\
1&0&0&0&-z_6&-z_5
\endmatrix
\right),\\
\Sing(W)=\{(z_1,\dots,z_6)\in W\,\,\vert\,\,
z_3=z_4=z_5=z_6=0\}=\\
\{(z_1,\dots,z_6)\in D\,\,\vert\,\,
z_1=z_3=z_4=z_5=z_6=0\};
\tag{3.7}
\\
\dim\Sing(W)=1.
\endgather
$$

Finally, for $W=\{z_1=z_3=z_5z_6\}$ we have:
$$
\Cal J=
\left(
\matrix
1&0&-1&0&0&0\\
1&0&0&0&-z_6&-z_5
\endmatrix
\right), \qquad \rank\Cal J=2,
$$
therefore  $W$ is non-singular. The same holds for
$W=\{z_1=z_3=z_5\}$. The claim \thetag{3.5} is proved.

It follows from \thetag{3.5} -- \thetag{3.7} that
$$
\Sing(Y\times_CY\times_CY)=\bigcup_{\delta\in\Delta,
\quad F\hookrightarrow Y_\delta
\atop{a,b\in\Sing(Y_\delta)}}
[F\times a\times b]\cup
[a\times F\times b]\cup
[a\times b\times F]
\tag{3.8}
$$
is a union of smooth rational curves,
where $F$ runs through
irreducible components of
$Y_\delta$ (which are smooth rational curves),
and $a,b$ run through singular points of $Y_\delta$.

Consider a monoidal transformation $f:X'\to X$ along
a smooth rational component $E_i\hookrightarrow V_\delta$.
It is evident that $E_i\,\,\widetilde{\rightarrow}\,\,\Bbb P^1\times
\Bbb P^1\times\Bbb P^1$. By the K\"unneth formula, we have:
$$
H^3(E_i,\Q)=0.
\tag{3.9}
$$

The canonical morphism $\varphi:V_{\text{strict}}\to V$
of the strict image of the subvariety $V\hookrightarrow X$
under the monoidal transformation $f$ is the monoidal
transformation of the variety $V$ along $E_i$
(\cite{19}, Ch. 0, \S \, 2). By the same reason,
the canonical morphism $\varphi_{E_j}:
(E_j)_{\text{strict}}\to E_j$ is the monoidal
transformation of $E_j$ along $E_i\cap E_j\quad(j\neq i)$.
Since $E_i,E_j$ are products of smooth rational curves, it
is easy to see that
there are
the following cases:
$$
E_i\cap E_j=
\cases
{\text{ a finite set of points}};\\
{\text{ a disjoint union of smooth rational curves}};\\
{\text{ a disjoint
union of products of smooth
rational curves}}
\endcases
\tag{3.10}
$$
(in the last case, $\varphi_{E_j}$ is an isomorphism).

In all cases, any irreducible component
$E'\hookrightarrow (V_{\text{strict}})_\delta$ is smooth. Moreover, we have:
$$
H^3(E',\Q)=0.
\tag{3.11}
$$

Indeed, let
$$
\matrix
B'&\overset j\to\hookrightarrow&E'\\
\downarrow^g&\ &\downarrow^{\varphi_E}\\
B&\overset i\to\hookrightarrow&E
\endmatrix
$$
be a canonical diagram of the monoidal
transformation $\varphi_E:E'\to E$ of a smooth rational threefold $E$
along a closed
smooth rational
centre $B$ of codimension $r\geq 2$,
\quad $N$ the conormal sheaf on $B$ (\cite{20}, 13.3).
Then there is a canonical decomposition of Hodge $\Q$-structures
$$
\gather
H^3(E',\Q)=\varphi_E^\ast H^3(E,\Q)\bigoplus
\left[\bigoplus_{p=0}^{r-2}
j_\ast[g^\ast H^{3-2p-2}(B,\Q)\cup\cc_1(\Cal O_{\Bbb P(N)}(1))^{\cup p}]
\right]\qquad
\tag{3.12}
\endgather
$$
(\cite{21}, Proposition 13.1). Since $H^1(B,\Q)=0$, \quad \thetag{3.12}
takes the form
$$
H^3(E',\Q)=\varphi_E^\ast H^3(E,\Q),
$$
therefore \thetag{3.11} follows from \thetag{3.9}.

We may assume that an irreducible
component $E_i$ of a singular fibre $V_\delta$
is defined locally by $\{z_1=z_3=z_5=0\}$.
It is well known that a monoidal transformation of a
polydisk $D$ with coordinates $z_1,\dots,z_6$
along the threefold $\{z_1=z_3=z_5=0\}$
is a canonical projection
of a subset
$$
\gather
\widetilde{D}=\{(z_1,z_2,z_3,z_4,z_5,z_6)\times(w_0:w_1:w_2)\,\,
\vert\,\,\\
z_1w_1=z_3w_0;\quad z_1w_2=z_5w_0;\quad z_3w_2=z_5w_1\}
\hookrightarrow D\times\Bbb P^2
\endgather
$$
onto $D$. Let $D_+(w_i)=\{(w_0:w_1:w_2)\in\Bbb P^2\,\,
\vert\,\,w_i\neq 0\}$,\quad
$u_1=w_1/w_0,\quad u_2=w_2/w_0$
the local coordinates
on $D_+(w_0)$, \quad
$u_3=w_0/w_1,\quad u_4=w_2/w_1$
the local coordinates
on $D_+(w_1)$, \quad
$u_5=w_0/w_2,\quad u_6=w_1/w_2$
the local coordinates
on $D_+(w_2)$.

Assume first that
$W$
is defined by the system of equations
$z_1=z_3z_4=z_5z_6$ of type
\thetag{3.2}.
Since $\widetilde{D}$ is
given in
an open subset
$U_0=D\times D_+(w_0)$ by
$\widetilde{D}\cap U_0=$
\newline
$\{z_3=z_1u_1;\quad
z_5=z_1u_2\}$,
the strict image
$W_{\text{strict}}$ of the variety $W$ is defined in
local coordinates
$z_1,z_2,z_4,z_6,u_1,u_2$ on
$\widetilde{D}\cap U_0$ by
$1=u_1z_4=u_2z_6$.
The rank of the corresponding Jacobi matrix
$$
\Cal J=
\left(
\matrix
0&0&-u_1&0&-z_4&0\\
0&0&0&-u_2&0&-z_6
\endmatrix
\right)
$$
equals $2$ because $u_1u_2\neq 0$ on $W_{\text{strict}}$. Hence
$W_{\text{strict}}$ has no singular points in
$\widetilde{D}\cap U_0$.
On the other hand,
$\widetilde{D}$ is
given in
an open subset
$U_1=D\times D_+(w_1)$ by
$\widetilde{D}\cap U_1=
\{z_1=z_3u_3;\quad z_5=z_3u_4\}$,
hence $W_{\text{strict}}$ is defined in
local coordinates
$z_2,z_3,z_4,z_6,u_3,u_4$ on
$\widetilde{D}\cap U_1$ by
$u_3=z_4=u_4z_6$,
$$
\rank\Cal J=
\rank\left(
\matrix
0&0&-1&0&1&0\\
0&0&0&-u_4&1&-z_6
\endmatrix
\right)=2,
$$
and $W_{\text{strict}}$ has no singular points in
$\widetilde{D}\cap U_1$.
Finally,
$\widetilde{D}$ is
given in
an open subset
$U_2=D\times D_+(w_2)$ by
$\widetilde{D}\cap U_2=
\{z_1=z_5u_5;\quad z_3=z_5u_6\}$,
hence $W_{\text{strict}}$ is defined in
local coordinates
$z_2,z_4,z_5,z_6,u_5,u_6$ on
$\widetilde{D}\cap U_2$ by
$u_5=u_6z_4=z_6$,
$$
\rank\Cal J=
\rank\left(
\matrix
0&-u_6&0&0&1&-z_4\\
0&0&0&-1&1&0
\endmatrix
\right)=2,
$$
and $W_{\text{strict}}$ has no singular points in
$\widetilde{D}\cap U_2$.

Now assume that
$W$
is defined by the system of equations
$z_1z_2=z_3z_4=z_5z_6$ of type
\thetag{3.1}.
Since $\widetilde{D}$ is
given in
an open subset
$U_0=D\times D_+(w_0)$ by
$\widetilde{D}\cap U_0=\{z_3=z_1u_1;\quad
z_5=z_1u_2\}$,
the strict image
$W_{\text{strict}}$ of the variety $W$ is defined in
local coordinates
$z_1,z_2,z_4,z_6,u_1,u_2$ on
$\widetilde{D}\cap U_0$ by the system of equations
$z_2=u_1z_4=u_2z_6$ of type \thetag{3.2}.
It follows from the calculations above
that the monoidal transformation of
$W_{\text{strict}}\cap
[\widetilde{D}\cap U_0]$ along $\{z_2=z_4=z_6=0\}$ yields a resolution
of singularities of
$W_{\text{strict}}\cap
[\widetilde{D}\cap U_0]$.
On the other hand,
$\widetilde{D}$ is
given in
an open subset
$U_1=D\times D_+(w_1)$ by
$\widetilde{D}\cap U_1=
\{z_1=z_3u_3;\quad z_5=z_3u_4\}$,
hence $W_{\text{strict}}$ is defined in
local coordinates
$z_2,z_3,z_4,z_6,u_3,u_4$ on
$\widetilde{D}\cap U_1$ by the system of equations
$u_3z_2=z_4=u_4z_6$ of type \thetag{3.2}. Hence
the monoidal transformation of
$W_{\text{strict}}\cap
[\widetilde{D}\cap U_1]$ along $\{z_2=z_4=z_6=0\}$ yields a resolution
of singularities of
$W_{\text{strict}}\cap
[\widetilde{D}\cap U_1]$.
Finally,
$\widetilde{D}$ is
given in
an open subset
$U_2=D\times D_+(w_2)$ by
$\widetilde{D}\cap U_2=
\{z_1=z_5u_5;\quad z_3=z_5u_6\}$,
hence $W_{\text{strict}}$ is defined in
local coordinates
$z_2,z_4,z_5,z_6,u_5,u_6$ on
$\widetilde{D}\cap U_2$ by the system of equations
$u_5z_2=u_6z_4=z_6$ of type \thetag{3.2},
and
the monoidal transformation of
$W_{\text{strict}}\cap
[\widetilde{D}\cap U_2]$ along $\{z_2=z_4=z_6=0\}$ yields a resolution
of singularities of
$W_{\text{strict}}\cap
[\widetilde{D}\cap U_2]$.

Let $E_1,E_2,\dots, E_m$ be all irreducible components of
singular fibres of the cano\-ni\-cal morphism $V=Y\times_CY\times_CY\to C$.
It is easy to see from calculations above and \thetag{3.11} that
a sequence of monoidal transformations
of $X=Y\times Y\times Y$ along strict images of smooth
rational threefolds $E_i\,\,\widetilde{\rightarrow}\,\,\Bbb P^1\times
\Bbb P^1\times\Bbb P^1$ allows us to construct a resolution of singularities
of the embedded subvariety $Y\times_CY\times_CY\hookrightarrow
Y\times Y\times Y$ satisfying the conclusions
of Lemma 3.1.

{\bf 3.2. Lemma.} {\it Let $\tau:Y\to C$ satisfies the assumptions of Lemma
$3.1$,
and $\mu:V\,\,\overset\ddef\to=\,\,
[Y\times_CY\times_CY]^{\smooth}\to C$ the canonical structure morphism.
If there is at least one singular fibre of $\tau$, \quad
$\End(\Pic^0(Y_{\overline\eta}))=\Z$
for the generic geometric fibre $Y_{\overline\eta}$ of the morphism $\tau$,
and the genus of
\,\, $Y_{\overline\eta}$ \,\, equals $3$,
then}
$$
\gather
H^5(V,\Q)=H^1(C,R^4\mu_\ast\Q);\\
H^3(V,\Q)=H^1(C,R^2\mu_\ast\Q).
\endgather
$$

{\it Proof.} We may assume that $\tau$ is smooth over
a non-empty open affine subset $C'\,\,\overset\ddef\to=\,\,
C\setminus\Delta\hookrightarrow C$.
Consider a diagram of fibre product
$$
\matrix
Y'&\hookrightarrow&Y\\
\downarrow^{\tau'}&\ &\downarrow^\tau\\
C'&\hookrightarrow&C.
\endmatrix
$$
Let $V'=\mu^{-1}(C')$.
Then $\mu'=\mu\vert_{\mu^{-1}(C')}:V'\to C'$ is smooth. By Lemma 3.1,
we may assume that $V_\delta\quad(\delta\in\Delta)$
is a union of smooth rational threefolds $E'_i$ of multiplicity 1
with normal crossings, and $H^3(E'_i,\Q)=0$.

Using arguments of the proof of \thetag{1.16}, we get
a canonical decomposition of Hodge $\Q$-structures
$$
H^m(V,\Q)=H^0(C,R^m\mu_\ast\Q)\oplus
H^1(C,R^{m-1}\mu_\ast\Q)\oplus
H^2(C,R^{m-2}\mu_\ast\Q).
\tag{3.13}
$$

Let
$$
f:Z\to\mu^{-1}(\Delta)
$$
be the normalization of the divisor
$\mu^{-1}(\Delta)$. Then $Z$ is a disjoint
union of smooth irreducible components of $\mu^{-1}(\Delta)$.
Since $f$ is a resolution of singularities
of a closed subscheme $i_\Delta:\mu^{-1}(\Delta)\hookrightarrow V$,
there is an exact sequence
of mixed Hodge $\Q$-structures
(\cite{22}, Corollary (8.2.8)):
$$
H^3(Z,\Q(-1))@>(i_\Delta f)_\ast>>
H^5(V,\Q)\to H^5(V',\Q).
$$
Therefore
$$
(i_\Delta f)_\ast
H^3(Z,\Q(-1))=
\Ker\left[H^5(V,\Q)\to H^5(V',\Q)\right].
\tag{3.14}
$$
On the other hand,
each irreducible component $E\hookrightarrow Z$ is a smooth
rational threefold with $H^3(E,\Q)=0$.
Then it follows from \thetag{3.14}
that there is a canonical embedding
$H^5(V,\Q)\hookrightarrow H^5(V',\Q)$.
Note that the canonical surjective morphism of Hodge $\Q$-structures
$H^5(V,\Q)\to H^0(C',R^5\mu'_\ast\Q)$ (\cite{8}, Theorem 4.1.1)
is a part of the commutative diagram (with exact columns and rows)
of morphisms of mixed Hodge
$\Q$-structures (\cite{8}, Corollary 3.2.18, a proof of Theorem 4.1.1)
$$
\matrix
H^5(V,\Q)&\hookrightarrow&H^5(V',\Q)\\
\downarrow&\searrow&\downarrow\\
H^0(C,R^5\mu_\ast\Q)&\to&H^0(C',R^5\mu'_\ast\Q)&\to &0\\
\downarrow&\ &\downarrow\\
0&\ &0.
\endmatrix
$$
A polarization allows us to identify $H^0(C',R^5\mu'_\ast\Q)$
with a Hodge $\Q$-substructure of
$H^0(C,R^5\mu_\ast\Q)\hookrightarrow H^5(V,\Q)$.
Hence a canonical decomposition \thetag{3.13} for $m=5$ yields
the identification
$H^0(C,R^5\mu_\ast\Q)=H^0(C',R^5\mu'_\ast\Q)$. On the other hand,
$H^0(C',R^5\mu'_\ast\Q)=H^5(V'_s,\Q)^{\pi_1(C',s)}$ for a point $s\in C'$.
By the Poincar\'e duality theorem,
$H^5(V'_s,\Q)$ is dual to $H^1(V'_s,\Q)$. Hence
$H^0(C,R^5\mu_\ast\Q)$ is dual to
$$
H^1(V'_s,\Q)^{\pi_1(C',s)}=H^1(Y'_s\times Y'_s\times Y'_s,\Q)^{\pi_1(C',s)}.
\tag{3.15}
$$

It is evident that the trace of the Abelian scheme $\Pic^0(Y'/C')\to C'$
is trivial (because the special fibre of the N\'eron
minimal model of
$\Pic^0(Y'/C')\to C'$ over an infinite point $\delta\in\Delta$ is an
extension of a finite group by a torus, and
this property is well known to be stable
under base change),
so
$$
H^1(Y'_s,\Q)^{\pi_1(C',s)}=0.
\tag{3.16}
$$
Indeed, this equality
has been proved in (\cite{7}, Lemma 2.6 and Remark 2.7) for Abelian schemes
over complete curves, and the proof carries over verbatim to the case
under consideration.
Therefore \thetag{3.15} --
\thetag{3.16} and the K\"unneth formula yield
$$
H^0(C,R^5\mu_\ast\Q)=0.
\tag{3.17}
$$

We claim that
$$
H^2(C,R^3\mu_\ast\Q)=0.
\tag{3.18}
$$

Indeed, there is the following analog of \thetag{1.13}:
$$
H^i(C,R^p\mu_\ast\Q)\cup
H^j(C,R^q\mu_\ast\Q)\hookrightarrow
H^{i+j}(C,R^{p+q}\mu_\ast\Q).
\tag{3.19}
$$
The canonical pairing
$H^5(V,\Q)\times H^3(V,\Q)@>x\times y\mapsto x\cup y>> H^8(V,\Q)$
is non-degenerate by the Poincar\'e duality theorem and,
in virtue of \thetag{3.19}, one has:
$$
\gather
H^2(C,R^3\mu_\ast\Q)
\cup H^1(C,R^2\mu_\ast\Q)\hookrightarrow
H^3(C,R^5\pi_\ast\Q)=0,\\
H^2(C,R^3\mu_\ast\Q)
\cup H^2(C,R^1\mu_\ast\Q)\hookrightarrow
H^4(C,R^4\mu_\ast\Q)=0.
\endgather
$$
Therefore the canonical pairing
$$
H^2(C,R^3\mu_\ast\Q)\times H^0(C,R^3\mu_\ast\Q)@>x\times y\mapsto x\cup y>>
H^2(C,R^6\mu_\ast\Q)=H^8(V,\Q)
\,\,\widetilde{\rightarrow}\,\,\Q
$$
is non-degenerate.
In particular,
$$
\dim H^2(C,R^3\mu_\ast\Q)=\dim H^0(C,R^3\mu_\ast\Q).
\tag{3.20}
$$
Using arguments of the proof of \thetag{3.14}, we get:
$$
(i_\Delta f)_\ast
H^1(Z,\Q(-1))=
\Ker\left[H^3(V,\Q)\to H^3(V',\Q)\right].
\tag{3.21}
$$
On the other hand,
each irreducible component $E\hookrightarrow Z$ is a smooth
rational threefold. Hence $H^1(E,\Q)=0$, because $2\dim_\C\Alb(E)=
\dim H^1(E,\Q)$ and every morphism of a rational variety $E$ to its Albanese
variety is constant. It follows from \thetag{3.21}
that there is a canonical embedding
$H^3(V,\Q)\hookrightarrow H^3(V',\Q)$, inducing the identifications
$$
H^0(C,R^3\mu_\ast\Q)=H^0(C',R^3\mu'_\ast\Q)=
H^3(V'_s,\Q)^{\pi_1(C',s)}=
H^3(Y'_s\times Y'_s\times Y'_s,\Q)^{\pi_1(C',s)}
$$
for a point $s\in C'$.
In virtue of \thetag{3.16} one has:
$$
[H^2(Y'_s,\Q)\otimes H^1(Y'_s,\Q)]^{\pi_1(C',s)}\,\,
\widetilde{\rightarrow}\,\,H^1(Y'_s,\Q)^{\pi_1(C',s)}=0.
$$
In order to prove \thetag{3.18} it remains to show that
$$
[H^1(Y'_s,\Q)\otimes H^1(Y'_s,\Q)\otimes H^1(Y'_s,\Q)]^{\pi_1(C',s)}=0.
$$

Note that
$$
\gather
[H^1(Y'_s,\Q)\otimes H^1(Y'_s,\Q)\otimes H^1(Y'_s,\Q)]^{\pi_1(C',s)}=\\
\Hom_{\pi_1(C',s)}(H^1(Y'_s,\Q),H^1(Y'_s,\Q)\otimes H^1(Y'_s,\Q)).
\tag{3.22}
\endgather
$$

Since $\End(\Pic^0(Y_{\overline\eta}))=\Z$ by the assumption of the lemma,
it follows that, for
a generic point $s\in C'$ (in the sense of A.Weil),
$\Hg(H^1(Y'_s,\Q))=$
\newline
$\Hg(H^1(\Pic^0(Y'_s),\Q))$ is a simple algebraic
$\Q$-group of type $C_3$ \,\, \cite{23}.

For a closed point $u\in C'$, we denote by
$\rho_u:\pi_1(C',u)\to\GL(H^1(Y'_u,\Q))$
the canonical representation of monodromy.
By Deligne's theorem, there exists a countable subset
$\Delta'_\infty\hookrightarrow C'$ such that, for all
$v\in C'\setminus\Delta'_\infty$, the closure
of
$\rho_v(\pi_1(C',v))$ in the Zariski topology of
$\GL(H^1(Y'_v,\Q))$ is a semisimple algebraic $\Q$-group $G$, and
the connected identity component $G^0$ of $G$ is a normal subgroup of
$\Hg(H^1(Y'_v,\Q))$
(\cite{24}, Theorem 7.3).  In particular, we have
$G^0=1$ or else $G^0$ is a simple algebraic $\Q$-group
of type $C_3$.

If $G^0=1$, then $\Pic^0(Y'/C')\to C'$ is an isotrivial Abelian scheme
(\cite{8}, (4.1.3.3), 4.4.3),
contradicting the fact that
the trace of this Abelian scheme
is trivial.
Therefore
$G^0$ is a simple algebraic $\Q$-group
of type $C_3$.

In virtue of \thetag{3.22}, it suffices to show that
$$
\Hom_{G^0}(H^1(Y'_s,\Q),H^1(Y'_s,\Q)\otimes H^1(Y'_s,\Q))=0.
$$
Since $H^1(Y'_s,\C)=E(\omega_1)$ is a standard representation
of a complex simple Lie algebra $g$ of type $C_3$ with the highest
weight $\omega_1$, it remains
to show that
$$
\Hom_g(E(\omega_1),E(\omega_1)\otimes E(\omega_1))=0.
\tag{3.23}
$$
Note that
$E(\omega_1)\otimes E(\omega_1)=\Sym^2E(\omega_1)\oplus\wedge^2E(\omega_1)$
is a representation of degree 36 containing an irreducible
subrepresentation $E(2\omega_1)$ (\cite{16}, Ch. VIII, \S \, 7, Exercise
17) of degree 21 by the H.Weyl formula (\cite{25}, 2.7).
Recall that $\wedge^2E(\omega_1)=E(\omega_2)\oplus E(0)$
(\cite{16}, Ch. VIII,
13.3, Lemma 2), \,\, $\dim E(\omega_2)=14$. Therefore we have
a canonical decomposition of $g$-modules
$E(\omega_1)\otimes E(\omega_1)=E(2\omega_1)\oplus E(\omega_2)\oplus E(0)$.
Hence
the Schur lemma yields \thetag{3.23}. We see that \thetag{3.18} holds.

It follows from \thetag{3.18} and \thetag{3.20} that
$$
H^0(C,R^3\mu_\ast\Q)=0.
\tag{3.24}
$$
On the other hand, \thetag{3.13} for $m=5$ and
\thetag{3.17} -- \thetag{3.18} yield
$$
H^5(V,\Q)=H^1(C,R^4\mu_\ast\Q).
\tag{3.25}
$$

The canonical pairing
$H^5(V,\Q)\times H^3(V,\Q)@>x\times y\mapsto x\cup y>> H^8(V,\Q)$
is non-de\-ge\-ne\-rate by the Poincar\'e duality theorem and,
in virtue of \thetag{3.19}, one has
$H^1(C,R^4\mu_\ast\Q)$
\newline
$\cup\,H^2(C,R^1\mu_\ast\Q)$
$\hookrightarrow
H^3(C,R^5\mu_\ast\Q)=0$.
Therefore \thetag{3.25} yields $H^2(C,R^1\mu_\ast\Q)=0$
and, in virtue of \thetag{3.13} for $m=3$ and
\thetag{3.24}, we have:
$H^3(V,\Q)=H^1(C,R^2\mu_\ast\Q)$.
The lemma is proved.

{\bf 3.3. Lemma.} {\it Let $Y_s$ be a smooth projective curve of genus
$3$, and $V_s=Y_s\times Y_s\times Y_s$. Then there
is a canonically defined algebraic class $\omega_s\in H^1(V_s,\Q)\otimes
H^1(V_s,\Q)$ such that an algebraic correspondence
$\omega_s^{\cup\,2}$ yields an algebraic
isomorphism $H^4(V_s,\Q)\,\,\widetilde{\rightarrow}
\,\,H^2(V_s,\Q)$.}

{\it Proof.} The canonical pairing
$$
\Phi:H^1(Y_s,\Q)\times H^1(Y_s,\Q)@>x\times y\mapsto x\cup y>>H^2(Y_s,\Q)\,\,
\widetilde{\rightarrow}\,\,\Q
$$
is skew and non-degenerate.
Recall that the matrix of $\Phi$ relative to the
Witt basis
$e_1,e_2,e_3,e_{-3},e_{-2},e_{-1}$ of $H^1(Y_s,\Q)$ is equal to
$\left(
\matrix
0&s\\
-s&0
\endmatrix
\right)$,
where
$s=\left(
\matrix
0&0&1\\
0&1&0\\
1&0&0
\endmatrix
\right)$
(\cite{16}, Ch. VIII, \S \, 13, n$^0$ 3).
Fix a point $p\in Y_s$. Then
$$
e_i\cup e_j=
\cases
\cl_{Y_s}(p)\qquad{\text{for }}i=-j=1,2,3;\\
-\cl_{Y_s}(p)\quad{\text{for }}i=-j=-1,-2,-3;\\
0\,\,\qquad\qquad{\text{for }}i\neq-j.
\endcases
\tag{3.26}
$$
One can identify the Lie algebra $\ssp(\Phi)$ with the Lie algebra of
matrices of the form
$a=\left(
\matrix
A&B\\
C&-s\,\,^tA\,\,s
\endmatrix
\right)$,
where $A,B,C$ are matrices of type $3\times 3$, and $B$, $C$ are
symmetric matrices relative to the secondary diagonal
(\cite{16}, Ch. VIII, \S \, 13, n$^0$ 3).
Then it is easy to show by direct computations that
the $1$-dimensional space
$[H^1(Y_s,\Q)\otimes H^1(Y_s,\Q)]^{{\Ssp}(\Phi)}$
is spanned by the Poincar\'e class
$$
\goth p(H^1(Y_s,\Q))=
e_1\otimes e_{-1}+e_2\otimes e_{-2}+e_3\otimes e_{-3}
-e_{-3}\otimes e_3-e_{-2}\otimes e_2-e_{-1}\otimes e_1.
\tag{3.27}
$$
Since
$\goth p(H^1(Y_s,\Q))$ is a Hodge cycle (\cite{4}, 1.5),
it is algebraic by the Lefschetz theorem.

Consider an algebraic class
$$
\gather
\omega_s=\goth p(H^1(Y^{(1)}_s,\Q))+
\goth p(H^1(Y^{(2)}_s,\Q))+
\goth p(H^1(Y^{(3)}_s,\Q))=\\
e_1^{(1)}\otimes e_{-1}^{(1)}+e_2^{(1)}\otimes e_{-2}^{(1)}+
e_3^{(1)}\otimes e_{-3}^{(1)}
-e_{-3}^{(1)}\otimes e_3^{(1)}-e_{-2}^{(1)}\otimes e_2^{(1)}-e_{-1}^{(1)}
\otimes e_1^{(1)}+\\
e_1^{(2)}\otimes e_{-1}^{(2)}+e_2^{(2)}\otimes e_{-2}^{(2)}+
e_3^{(2)}\otimes e_{-3}^{(2)}
-e_{-3}^{(2)}\otimes e_3^{(2)}-e_{-2}^{(2)}\otimes e_2^{(2)}-e_{-1}^{(2)}
\otimes e_1^{(2)}+\\
e_1^{(3)}\otimes e_{-1}^{(3)}+e_2^{(3)}\otimes e_{-2}^{(3)}+
e_3^{(3)}\otimes e_{-3}^{(3)}
-e_{-3}^{(3)}\otimes e_3^{(3)}-e_{-2}^{(3)}\otimes e_2^{(3)}-e_{-1}^{(3)}
\otimes e_1^{(3)},
\endgather
$$
where
$e_1^{(k)},e_2^{(k)},e_3^{(k)},e_{-3}^{(k)},e_{-2}^{(k)},e_{-1}^{(k)}$
is the Witt basis of the $k$-th copy $H^1(Y^{(k)}_s,\Q)$ of
\newline
$H^1(Y_s,\Q)$ in the K\"unneth
decomposition
$$
H^1(V_s,\Q)=H^1(Y_s\times Y_s\times Y_s,\Q)=
H^1(Y^{(1)}_s,\Q)\oplus
H^1(Y^{(2)}_s,\Q)\oplus
H^1(Y^{(3)}_s,\Q).
$$

It is well known that an algebraic correspondence
$\goth p(H^1(Y^{(k)}_s,\Q))$ on $Y^{(k)}_s\times Y^{(k)}_s$
defines a canonical isomorphism
$$
H^1(Y^{(k)}_s,\Q)\underset{\widetilde{\quad}}\to
{@>[\goth p(H^1(Y^{(k)}_s,\Q))]_\ast
>>}H^1(Y^{(k)}_s,\Q),
$$
where
$$
\gather
[\goth p(H^1(Y^{(k)}_s,\Q))]_\ast(\alpha)=\pr^{(k)}_{2s\ast}([\alpha
\otimes\cl_{Y^{(k)}_s}(Y^{(k)}_s)]\cup
\goth p(H^1(Y^{(k)}_s,\Q)))
\endgather
$$
for $\alpha\in H^1(Y^{(k)}_s,\Q)$, and $\pr^{(k)}_{2s}:Y^{(k)}_s\times
Y^{(k)}_s\to Y^{(k)}_s$ is
the canonical projection (\cite{4}, Lemma 1.7).
On the other hand, it follows from
\thetag{3.26} - \thetag{3.27} that
$$
\gather
[\goth p(H^1(Y^{(k)}_s,\Q))]_\ast(e^{(k)}_j)=
\pr^{(k)}_{2s\ast}(
[e^{(k)}_j\otimes\cl_{Y^{(k)}_s}(Y^{(k)}_s)]\,\,\cup\\
[e^{(k)}_1\otimes e^{(k)}_{-1}+e^{(k)}_2\otimes
e^{(k)}_{-2}+e^{(k)}_3\otimes e^{(k)}_{-3}
-e^{(k)}_{-3}\otimes e^{(k)}_3-e^{(k)}_{-2}\otimes e^{(k)}_2-e^{(k)}_{-1}
\otimes e^{(k)}_1])
=\\
\pr^{(k)}_{2s\ast}(
[-\cl_{Y^{(k)}_s}(p)\otimes e^{(k)}_j])=-e^{(k)}_j;
\tag{3.28}\\
[\goth p(H^1(Y^{(k)}_s,\Q))]^{\cup\,2}=
-6\cl_{Y^{(k)}_s}(p)\otimes\cl_{Y^{(k)}_s}(p).
\tag{3.29}
\endgather
$$

First of all, an algebraic correspondence
$$
\gather
\omega_s^{\cup\,2}=
-6\cl_{Y^{(1)}_s}(p)\otimes\cl_{Y^{(1)}_s}(p)
-6\cl_{Y^{(2)}_s}(p)\otimes\cl_{Y^{(2)}_s}(p)
-6\cl_{Y^{(3)}_s}(p)\otimes\cl_{Y^{(3)}_s}(p)\\
-2e_1^{(1)}\cup e_1^{(2)}\otimes e_{-1}^{(1)}\cup e_{-1}^{(2)}-
\dots
+2e_1^{(1)}\cup e_{-3}^{(2)}\otimes e_{-1}^{(1)}\cup e_3^{(2)}+
\dots\\
-2e_{-1}^{(2)}\cup e_{-1}^{(3)}\otimes e_{1}^{(2)}\cup e_1^{(3)}
\endgather
$$
induces a canonical isomorphism
$$
\gather
H^2(Y^{(1)}_s,\Q)\otimes
H^2(Y^{(2)}_s,\Q)
\,\,\widetilde{\rightarrow}\,\,
H^2(Y^{(3)}_s,\Q).
\tag{3.30}
\endgather
$$

Indeed, in virtue of \thetag{3.26} - \thetag{3.29},
the $\Q$-space $H^2(Y^{(1)}_s,\Q)\otimes
H^2(Y^{(2)}_s,\Q)=\Q\,\,
e_{1}^{(1)}\cup e_{-1}^{(1)}\otimes
e_{1}^{(2)}\cup e_{-1}^{(2)}$ is annihilated (relative to the $\cup$-product)
by all summands of $\omega_s^{\cup\,2}$ except
$-6\cl_{Y^{(3)}_s}(p)\otimes\cl_{Y^{(3)}_s}(p)$; on the other hand,
denoting by $\pr_{2s}:V_s\times V_s\to V_s$ the canonical projection,
we see that
$\cl_{Y^{(3)}_s}(p)\otimes\cl_{Y^{(3)}_s}(p)$ defines an isomorphism
\thetag{3.30} because
$$
\gather
\pr_{2s\ast}([H^2(Y^{(1)}_s,\Q)\otimes
H^2(Y^{(2)}_s,\Q)\otimes\cl_{V_s}(V_s)]\cup\omega_s^{\cup\,2})=\\
\pr_{2s\ast}(\Q\,\,
[\cl_{Y^{(1)}_s}(p)\cup\cl_{Y^{(2)}_s}(p)\cup
\cl_{Y^{(3)}_s}(p)]\otimes\cl_{Y^{(3)}_s}(p))=\\
\pr_{2s\ast}(\Q\,\,
\cl_{V_s}(p\times p\times p)\otimes
\cl_{Y^{(3)}_s}(p))=
\Q\,\,
\cl_{Y^{(3)}_s}(p)=
H^2(Y^{(3)}_s,\Q).
\endgather
$$

Finally, let
$$
\gather
[\omega_s^{\cup 2}]_{\overset\wedge\to 3}=\\
[\goth p(H^1(Y^{(1)}_s,\Q))]^{\cup\,2}+
2\,\,\goth p(H^1(Y^{(1)}_s,\Q))\,\cup\,
\goth p(H^1(Y^{(2)}_s,\Q))+
[\goth p(H^1(Y^{(2)}_s,\Q))]^{\cup\,2}
\endgather
$$
be the sum of all summands of $\omega_s^{\cup\,2}$
containing
no factors of type $e_j^{(3)}$. Since $e_j^{(3)}$ annihilates
$H^2(Y^{(3)}_s,\Q)$, one can deduce from \thetag{3.28} that
$[\omega_s^{\cup\,2}]_{\overset\wedge\to 3}$ and $\omega_s^{\cup\,2}$
induce an algebraic isomorphism
$$
H^1(Y^{(1)}_s,\Q)\otimes
H^1(Y^{(2)}_s,\Q)\otimes H^2(Y^{(3)}_s,\Q)\,\,\widetilde{\rightarrow}\,\,
H^1(Y^{(1)}_s,\Q)\otimes
H^1(Y^{(2)}_s,\Q).
$$
It remains to apply the K\"unneth formula.
The lemma is proved.

{\bf 3.4. Theorem.} {\it Let
$\tau:Y\to C$ be a morphism $($with a section$)$ of a smooth
projective surface $Y$ onto a smooth curve $C$. Assume that the generic fibre
is a smooth curve of genus $3$ and all singular
fibres are unions of rational curves of multiplicity $1$ with normal
crossings.

If there is at least one singular fibre of $\tau$ and
$\End(\Pic^0(Y_{\overline\eta}))=\Z$
for the generic geometric fibre $Y_{\overline\eta}$ of the morphism $\tau$,
then the standard conjecture $B(X)$ of Lefschetz type holds for
a compactification $X$ of a semi-Abelian scheme $\Pic^0(Y/C)\to C$.}

{\it Proof.}
Using (if necessary) the
monoidal transformation $\widetilde{Y}\to Y$ of the surface $Y$
along the set of all singular points of irreducible components of
singular fibres $Y_\delta\quad(\delta\in\Delta)$,
we may assume that all singular
fibres $Y_\delta$
are unions of {\it smooth} rational curves of multiplicity 1 with normal
crossings. Indeed,
since $B(S)$ holds for smooth projective curves and surfaces,
we see that, for any smooth projective fourfold $\widetilde{X}$ birationally
isomorphic to $X$, the conjectures $B(\widetilde{X})$ and $B(X)$ are equivalent in virtue of Hironaka's results
(\cite{19}, \cite{26}),
Theorem 1.6 in \cite{7} and the compatibility of the standard
conjecture of Lefschetz type with monoidal transformations
(\cite{27}, Theorem 4.3).

It is evident that the cohomology class
$\cl_{V\times V}(\Delta_V)\in H^8(V\times V,\Q)$ of the diagonal
defines an algebraic
isomorphism $H^4(V,\Q)\,\,\widetilde{\rightarrow}\,\, H^4(V,\Q)$.
In particular, $B^4(V)$ holds
(\cite{4}, 1.8).

By Lemma 3.3,
for any point $s\in C'$, there are canonical
isomorphisms
$$
\gather
H^4(V_s,\Q)
\underset{\widetilde{\quad}}\to{@>\pr^\ast_{1s}>>}
H^4(V_s,\Q)\otimes H^0(V_s,\Q)
\underset{\widetilde{\quad}}\to
{@>x\mapsto\,x\cup\,
\omega_s^{\cup\,2}>>}\\
H^6(V_s,\Q)\otimes H^2(V_s,\Q)
\underset{\widetilde{\quad}}\to
{@>\pr_{2s\ast}>>}
H^2(V_s,\Q).
\endgather
$$
A family $(\omega_s)_{s\in C'}$ yields a canonically defined
element $\omega'\in H^0(C',R^1\mu'_\ast\Q\otimes R^1\mu'_\ast\Q)
\hookrightarrow H^0(C',R^2(\mu'\times_{C'}\mu')_\ast\Q)$
of Hodge type $(1,1)$, which in turn defines
the composite of isomorphisms
$$
R^4\mu'_\ast\Q
\underset{\widetilde{\quad}}\to{@>{p'_1}^\ast>>}
R^4\mu'_\ast\Q\otimes R^0\mu'_\ast\Q
\underset{\widetilde{\quad}}\to{@>x\mapsto x\cup\,
[\omega']^2>>}
R^6\mu'_\ast\Q\otimes R^2\mu'_\ast\Q
\underset{\widetilde{\quad}}\to{@>p'_{2\ast}>>}
R^2\mu'_\ast\Q,
\tag{3.31}
$$
where $p'_i:V'\times_{C'}V'\to V'$ is the canonical projection.

There is a resolution $\sigma:[V\times_CV]^{\smooth}\to V\times_CV$
of singularities of the subvariety
$\iota:V\times_CV\hookrightarrow V\times V$ such that every geometric fibre
$[V\times_CV]^{\smooth}_\delta\quad(\delta\in\Delta)$
is a union (with some positive multiplicities) of smooth irreducible
components with normal crossings and $\sigma$ is an isomorphism over
$C'$. By Deligne's theorem,
the canonical map
$H^2([V\times_CV]^{\smooth},\Q)\to H^0(C',R^2(\mu'\times_{C'}\mu')_\ast\Q)$
is a surjective morphism of Hodge $\Q$-structures
(\cite{8}, Theorem 4.1.1, a proof of Corollary 4.1.2).
In particular, there is a divisor $\Cal D_{\omega'}$ on
$[V\times_CV]^{\smooth}$ (with coefficients in $\Q$) inducing $\omega'$.

Using an algorithm of the proof of Theorem 1.5,
it is easy to deduce from
\thetag{3.31} that an algebraic class
$(\iota\sigma)_\ast\cl_{[V\times_CV]^{\smooth}}(\Cal D_{\omega'})^{\cup\,2}
\in H^6_{\alg}(V\times V,\Q)$
defines an algebraic isomorphism
$$
H^1(C,R^4\mu_\ast\Q)\,\,
\widetilde{\rightarrow}\,\,H^1(C,R^2\mu_\ast\Q).
$$
By Lemma 3.2, the algebraic class
$(\iota\sigma)_\ast\cl_{[V\times_CV]^{\smooth}}(\Cal D_{\omega'})^{\cup\,2}$
defines an algebraic
isomorphism
$H^5(V,\Q)\,\,\widetilde{\rightarrow}\,\,
H^3(V,\Q)$.
Hence $B^3(V)$ holds (\cite{4}, 1.8).

It is easy to deduce from \thetag{3.26}, \thetag{3.28} -
\thetag{3.29} and the K\"unneth formula that,
for any point $s\in C'$, there are canonical
isomorphisms
$$
\gather
H^5(V_s,\Q)
\underset{\widetilde{\quad}}\to{@>\pr^\ast_{1s}>>}
H^5(V_s,\Q)\otimes H^0(V_s,\Q)
\underset{\widetilde{\quad}}\to
{@>x\mapsto x\cup\,
\omega_s>>}\\
H^6(V_s,\Q)\otimes H^1(V_s,\Q)
\underset{\widetilde{\quad}}\to
{@>\pr_{2s\ast}>>}
H^1(V_s,\Q).
\endgather
$$
Hence there are isomorphisms
$$
R^5\mu'_\ast\Q
\underset{\widetilde{\quad}}\to{@>{p'_1}^\ast>>}
R^5\mu'_\ast\Q\otimes R^0\mu'_\ast\Q
\underset{\widetilde{\quad}}\to{@>x\mapsto x\cup\,
\omega'>>}
R^6\mu'_\ast\Q\otimes R^1\mu'_\ast\Q
\underset{\widetilde{\quad}}\to{@>p'_{2\ast}>>}
R^1\mu'_\ast\Q.
\tag{3.32}
$$
Using an algorithm of the proof of Theorem 1.5,
it is easy to deduce from
\thetag{3.32} that an algebraic class
$(\iota\sigma)_\ast\cl_{[V\times_CV]^{\smooth}}(\Cal D_{\omega'})
\in H^4_{\alg}(V\times V,\Q)$
defines an algebraic isomorphism
$$
H^1(C,R^5\mu_\ast\Q)\,\,
\widetilde{\rightarrow}\,\,H^1(C,R^1\mu_\ast\Q).
\tag{3.33}
$$

There is an exact sequence of mixed Hodge $\Q$-structures
(\cite{22}, Corollary (8.2.8))
$$
H^0(Z,\Q(-1))@>(i_\Delta f)_\ast>>
H^2(V,\Q)\to H^2(V',\Q).
$$
Therefore
$$
\Ker\left[H^2(V,\Q)\to H^2(V',\Q)\right]
\quad{\text{is a Hodge }}\Q-{\text{structure of type }}(1,1).
\tag{3.34}
$$
The canonical surjective morphism of Hodge $\Q$-structures
$H^2(V,\Q)\to$
\newline
$H^0(C',R^2\mu'_\ast\Q)$ (\cite{8}, Theorem 4.1.1)
is a part of the commutative diagram (with exact columns and rows)
of morphisms of mixed Hodge
$\Q$-structures (\cite{8}, Corollary 3.2.18, a proof of Theorem 4.1.1)
$$
\matrix
H^2(V,\Q)&\to&H^2(V',\Q)\\
\downarrow&\searrow&\downarrow\\
H^0(C,R^2\mu_\ast\Q)&\to&H^0(C',R^2\mu'_\ast\Q)&\to &0\\
\downarrow&\ &\downarrow\\
0&\ &0.
\endmatrix
$$
A polarization allows us to identify $H^0(C',R^2\mu'_\ast\Q)$
with a Hodge $\Q$-substructure of
$H^0(C,R^2\mu_\ast\Q)\hookrightarrow H^2(V,\Q)$.
Hence there exists a
commutative diagram (with exact rows)
of canonical morphisms of mixed Hodge $\Q$-structures
$$
\matrix
\Ker\left[H^2(V,\Q)\to H^2(V',\Q)\right]&\to&
H^2(V,\Q)&\to&H^2(V',\Q)\quad\\
\bigcup&\ &\bigcup&\ &\bigcup\quad\\
K
&\to &H^0(C,R^2\mu_\ast\Q)&{\to\atop\hookleftarrow}&H^0(C',R^2\mu'_\ast\Q)\to
0,
\endmatrix
\tag{3.35}
$$
where
$K=\Ker\left[H^0(C,R^2\mu_\ast\Q)\to H^0(C',R^2\mu'_\ast\Q)\right]$.


We claim that
$$
H^0(C',R^2\mu'_\ast\Q)\quad{\text{is a Hodge }}\Q-{\text{structure
of type }}(1,1).
\tag{3.36}
$$

Indeed,
$H^0(C',R^2\mu'_\ast\Q)=H^2(V'_s,\Q)^{\pi_1(C',s)}$ for a point $s\in C'$.
Since
$$
H^2(V'_s,\Q)^{\pi_1(C',s)}=H^2(Y'_s\times Y'_s\times Y'_s,\Q)^{\pi_1(C',s)},
$$
it remains to apply the K\"unneth formula and \thetag{2.10}.

On the other hand, the K\"unneth formula yields
$$
\dim H^0(C',R^2\mu'_\ast\Q)\geq 3.
\tag{3.37}
$$

It follows from \thetag{3.34} -- \thetag{3.37} that
$$
\gather
H^0(C,R^2\mu_\ast\Q)\quad{\text{is a Hodge }}\Q-{\text{structure of type }}
(1,1);
\tag{3.38}\\
\dim H^0(C,R^2\mu_\ast\Q)\geq 3.
\tag{3.39}
\endgather
$$
Hence \thetag{3.33}, \thetag{3.37} - \thetag{3.39},
and an algorithm described in subsections 2.1 -- 2.10 allow us to construct
an algebraic isomorphism
$H^6(V,\Q)\,\,\widetilde{\rightarrow}\,\,H^2(V,\Q)$.
In particular, $B^2(V)$ holds.
Since $B^0(V)$ trivially holds and $B^1(V)$ is well known
(\cite{4}, Corollary 1.10), we see that $B(V)$ holds by Proposition 2.3 in
\cite{3}.

Let $e:C\to Y$ be a section of $\tau:Y\to C$.
It is well known that there exists a dominant rational map $V-\to X$
inducing a surjective morphism
$$
\gather
Y_s\times Y_s\times Y_s\to\Pic^0(Y_s)=X_s\\
y^{(1)}_s\times
y^{(2)}_s\times
y^{(3)}_s\mapsto\Cal O_{Y_s}(y^{(1)}_s+y^{(2)}_s+y^{(3)}_s-3e(s))
\endgather
$$
for every $s\in C'$.
Since $B(S)$ holds for smooth projective curves and surfaces,
we see that $B(X)$ follows from $B(V)$, the existence of
Hironaka's resolution of indeterminacies of a rational map $V-\to X$
(\cite{19}, \cite{26}),
Theorem 1.6 in \cite{7} and the compatibility of the standard
conjecture of Lefschetz type with monoidal transformations
(\cite{27}, Theorem 4.3). The theorem is proved.

\newpage

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\enddocument
