ON THE STANDARD CONJECTURE FOR COMPLEX ABELIAN

SCHEMES OVER SMOOTH PROJECTIVE CURVES*

S.G.Tankeev

Abstract. We reduce the Hodge conjecture for Abelian varieties to the question about the existence of an algebraic isomorphism H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{0}(C, R^{i}π_{∗}ℚ) for all i ≥ 2 and for all complex principally polarized Abelian schemes π : X → C of relative dimension d over smooth projective curves. If canonically defined Hodge cycles α_{i}(X/C) ∈ H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ) are algebraic for all natural numbers i ≥ 2, then the Grothendieck standard conjecture B(X) on the algebraicity of the operators Λ and ∗ holds for X. We prove B(X) for an Abelian scheme under the assumption that End(X_{s}) = ℤ for some geometric fiber X_{s} of non-exceptional dimension.

Introduction

Let X be a smooth projective d-dimensional variety over the field ℂ of complex numbers. The choice of \sqrt{−1} ∈ ℂ determines the isomorphism ℚ(1) ≃ ℚ. We denote by cl_{X}(Z) ∈ H^{2i}(X, ℚ) the class of an algebraic cycle Z of codimension i on X.

The Hodge conjecture Hodge^{i}(X) [13]. Given a natural number i, the ℚ-space H^{2i}(X, ℚ) ∩ H^{i,i}(X, ℂ) is generated by classes of algebraic cycles of codimension i on X.

If Hodge^{i}(X) is true for any natural number i, then we say that the Hodge conjecture Hodge(X) holds for X. By the Lefschetz theorem, Hodge^{1}(X) always holds.

The surveys of T.Shioda [27] and B.Gordon [10] give a good introduction to the Hodge conjecture for Fermat varieties and for Abelian varieties.

Recall that a linear map u^{∗} : H^{i}(X, ℚ) → H^{j}(X, ℚ) is called algebraic if it is the restriction of a map H^{∗}(X, ℚ) → H^{∗}(X, ℚ) induced by an element of the ℚvector space generated by the algebraic classes ([14], 1.3.6). In other words, u^{∗} is algebraic, if there exists an algebraic cycle Z with rational coefficients such that u = cl_{X×X}(Z) and

u^{∗} : H^{∗}(X, ℚ) ⟶^{pr_{1}^{∗}} H^{∗}(X × X, ℚ) ⟶^{∪ u} H^{∗}(X × X, ℚ) ⟶^{pr_{2∗}} H^{∗}(X, ℚ),

where pr_{1}, pr_{2} are canonical projections.

Typeset by AMS-TEX

*The article was completed with the financial support of the Russian Foundation for Fundamental Research (grant no. 00-01-00058). Partially supported by the Institut des Hautes Études Scientifiques (Bures-sur-Yvette, December, 2000) and by the Max-Planck-Institut für Mathematik (Bonn, February – May, 2001). I would like to thank both Institutions for their warm hospitality and excellent working conditions.

Let H be a smooth hyperplane section of X. We denote by L the Lefschetz operator on H^{∗}(X, ℚ) defined by the formula Lx = cl_{X}(H) ∪ x. By the strong Lefschetz theorem, the map

L^{d−i} : H^{i}(X, ℚ) → H^{2d−i}(X, ℚ)

is an isomorphism for any i ≤ d. Let Λ^{d−i} : H^{2d−i}(X, ℚ) ˜→ H^{i}(X, ℚ) be the isomorphism inverse to L^{d−i}.

The Grothendieck standard conjecture B^{i}(X) [12]. Given a natural number i ≤ d, the isomorphism Λ^{d−i} : H^{2d−i}(X, ℚ) → 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) holds for X.

The conjecture B(X) implies the algebraicity of the classical operators ^{c}∗ and cΛ of the Hodge theory for X, the coincidence of the numerical and homological equivalences of algebraic cycles on X, the algebraicity of the Künneth components of the class of an algebraic cycle on X × X.

The property B(X) is independent of the polarization L and is stable under product, smooth hyperplane section and specialization. It is satisfied by projective space and the other usual rational varieties having cellular decompositions, by curves, by surfaces and by Abelian varieties. In § 11, we add to this list some complex Abelian schemes over smooth projective curves.

Now let k be a finitely generated extension of a prime field, and let X be a smooth projective variety over k. We denote by k^{sep} the maximal separable algebraic extension of k.

The Tate conjecture Tate^{i}(X) [32]. Given a natural number i, there exists a finite Galois extension k′ of k such that the ℚ_{l}-space

[H_{et}^{2i}(X ⊗_{k} k^{sep}, ℚ_{l}(i))]^{Gal(k^{sep}/k′)}

is generated by classes of k′-rational algebraic cycles of codimension i on X ⊗_{k} k′.

The Tate conjecture and the corresponding conjecture about the poles of zeta functions are discussed in [32], [33].

The significance of the conjecture B(X) is based on the following important remarks:

i) By the Y.André theorem ([1], Theorem 0.6.2), the algebraicity of

Λ^{d+1−i} : H^{2d+2−i}(X, ℚ) → H^{i}(X, ℚ)

for all complex Abelian schemes π : X → C of relative dimension d over smooth projective curves C and all i implies the Hodge conjecture for all Abelian varieties.

ii) By the Pohlmann theorem ([25], Theorem 4), the Hodge conjecture for a complex Abelian variety X of CM-type is equivalent to the Tate conjectures [32] for algebraic cycles on a model X_{0} of X over a number field and for poles of zeta function of X_{0} ([25], Theorem 5).

iii) By the Milne theorem ([22], Theorem 7.1), the Hodge conjecture for all Abelian varieties of CM-type over ℂ implies the Tate conjectures [32] for algebraic cycles and poles of zeta functions for all Abelian varieties over finite fields.

Let π : X → C be a principally polarized complex Abelian scheme over a smooth projective curve, j : X ×_{C} X ↪ X × X the canonical embedding, τ : X ×_{C} X → C the structure morphism, Δ : X ↪ X × X the diagonal embedding.

We denote by P the Poincaré bundle on X ×_{C} X^{∧} ˜→ X ×_{C} X, and let v be the image of cl_{X×_{C}X}(P) ∈ H^{2}(X ×_{C} X, ℚ) under the canonical edge morphism H^{2}(X ×_{C} X, ℚ) → H^{0}(C, R^{2}τ_{∗}ℚ) of the Leray spectral sequence E_{2}^{p,q} = H^{p}(C, R^{q}τ_{∗}ℚ) ⇒ H^{∗}(X ×_{C} X, ℚ). Then j_{∗}v^{i} = j_{∗}(v^{∪ i}) is a canonically defined algebraic class, and it determines an algebraic isomorphism

[j_{∗}v^{i}]^{∗} : H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{2}(C, R^{i}π_{∗}ℚ).

We develop the Y.André approach and reduce the Hodge conjecture for Abelian varieties to the question about the existence of an algebraic isomorphism

H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{0}(C, R^{i}π_{∗}ℚ)

for all i ≥ 2 and for all complex principally polarized Abelian schemes π : X → C of relative dimension d over smooth projective curves.

In this paper, we prove the following technical result:

Let π : X → C be a principally polarized complex Abelian scheme of relative dimension d over a smooth projective curve C. Then for any natural number i ≥ 2, there exists the unique Hodge cycle

α_{i} = α_{i}(X/C) ∈ [H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)] ∩ H^{i,i}(X × X, ℂ),

characterized by the following condition: α_{i} ∪ cl_{X×X}(X × X_{s}) determines an isomorphism

H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{2}(C, R^{i}π_{∗}ℚ),

which coincides with an algebraic isomorphism

[j_{∗}v^{i}]^{∗} : H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{2}(C, R^{i}π_{∗}ℚ).

If α_{i} is algebraic, then the algebraic class

[Δ_{∗}cl_{X}(X_{s})] ∘ [j_{∗}v^{i−2}] + j_{∗}v^{i−1} + α_{i}

induces an algebraic isomorphism H^{2d+2−i}(X, ℚ) ˜→ H^{i}(X, ℚ), and the conjecture B^{i}(X) holds (Theorem 10.1).

If canonically defined Hodge cycles α_{i}(X/C) ∈ H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ) are algebraic for all natural numbers i ≥ 2, then the Grothendieck standard conjecture B(X) on the algebraicity of the operators Λ and ∗ holds for X.

We also prove B(X) for an Abelian scheme under the assumption that the Hodge group of the generic geometric fiber X_{0\overline{η}} is a simple algebraic group of type C_{d} (Theorem 11.2). This assumption holds for an Abelian scheme π : X → C, if End(X_{s}) = ℤ for some geometric fiber X_{s} and d is a non-exceptional dimension (for example, if d is an odd integer) (Corollaries 11.4, 11.5, 11.3).

Acknowledgement. I am grateful to Yu.I.Manin for interesting discussions of the problem.

§ 1. On some conjectures concerning algebraic cycles

1.1. Let X be a smooth d-dimensional projective variety over the field ℂ of complex numbers. For j ≥ 0, we have the Lefschetz primitive decomposition

H^{j}(X, ℚ) = ⨁_{k≥max(0,j−d)} L^{k}P^{j−2k}(X),

where P^{i}(X) = H^{i}(X, ℚ) ∩ Ker L^{d−i+1} (i ≤ d) is a primitive part of H^{i}(X, ℚ) ([4], § 3, (3.4)).

In particular, for any j, any x ∈ H^{j}(X, ℚ) can be written uniquely in the form

x = ∑_{k≥max(0,j−d)} L^{k}x_{j−2k},

where x_{j−2k} ∈ P^{j−2k}(X) ([4], § 3, (3.3)).

Let ^{c}Λ be the dual operator for L of the classical Hodge theory, and let ^{c}∗ be the classical operator star ([11], Chapter 0, § 6; [4], § 3). For an element φ ∈ P^{p}(X)⊗_{ℚ}ℂ,

cΛL^{r}φ = r(d − p − r + 1)L^{r−1}φ

([4], § 3.1). In particular,

cΛ x = ∑_{k≥max(0,j−d)}^{c}ΛL^{k}x_{j−2k} =

∑_{k≥max(0,j−d)} k(d − (j − 2k) − k + 1)L^{k−1}x_{j−2k}.

Hence

cΛ x = ∑_{k≥max(1,j−d)} k(d − j + k + 1)L^{k−1}x_{j−2k}.  (1.1)

On the space of forms of type (a, b), we define the Weil operator C by the following formula: Cφ = (\sqrt{−1})^{a−b}φ. If φ ∈ P^{p}(X) ⊗_{ℚ} ℂ, then

c ∗ L^{r}φ = \{ \begin{matrix}(−1)^{p(p+1)/2} \frac{r!}{(d−p−r)!}L^{d−p−r}Cφ, for r ≤ d − p, \\ 0, for r > d − p \end{matrix}

([4], § 3, (3.6)). In particular,

c ∗ x = ∑_{k≥max(0,j−d)}^{c} ∗ L^{k}x_{j−2k} =

∑_{k≥max(0,j−d)}(−1)^{(j−2k)(j−2k+1)/2} \frac{k!}{(d − (j − 2k) − k)!}L^{d−(j−2k)−k}Cx_{j−2k}.

Hence

c ∗ x = ∑_{k≥max(0,j−d)}(−1)^{(j−2k)(j−2k+1)/2} \frac{k!}{(d − j + k)!}L^{d−j+k}Cx_{j−2k}.  (1.2)

The decompositions (1.1) – (1.2) permit to define the following abstract operators:

Λx = ∑_{k≥max(1,j−d)} L^{k−1}x_{j−2k},  (1.3)
∗x = ∑_{k≥max(0,j−d)}(−1)^{(j−2k)(j−2k+1)/2}L^{d−j+k}x_{j−2k},  (1.4)
p^{k}x = δ_{jk}x_{m} where m = max(0, j − d) for k = 0, …, 2d.  (1.5)

The formula for p^{k} corrects ([14], 1.4.2.4). All these formulas are discussed in ([14], § 1.4; [15], § 4). It is easy to check that ∗^{2} = 1 and Λ = ∗L∗ ([14], Proposition 1.4.3).

For i ≤ d, Λ^{d−i} : H^{2d−i}(X, ℚ) → H^{i}(X, ℚ) is known to be inverse to L^{d−i} and ^{c}Λ^{d−i} : L^{d−i}P^{i}(X) → P^{i}(X) is inverse to a multiple of L^{d−i} ([14], Proposition 1.4.3), so one can define the Lefschetz involution on H^{∗}(X, ℚ) ([1], § 1.1):

∗_{L} x = \{ \begin{matrix}L^{d−j}x, for x ∈ H^{j}(X, ℚ), j ≤ d, \\ Λ^{j−d}x, for x ∈ H^{j}(X, ℚ), j > d. \end{matrix}  (1.6)

1.2. It is well known that the Grothendieck standard conjecture

B(X) : ”the operator Λ is algebraic” is equivalent to each of the following conjectures:

cB(X): ”the operator ^{c}Λ is algebraic”;

∗(X): ”the operator ∗ is algebraic” ([15], Theorem 4 -1);

ν(X): ”for any i ≤ dim X − 1, there exists an isomorphism

ν^{i} : H^{2·dim X−i}(X, ℚ) ≃ H^{i}(X, ℚ)

which is algebraic” ([14], p. 360; Theorem 2.9) (for the reader convenience, we reproduce the proof of the equivalence

B^{i}(X) ⇔ ν^{i}(X) :

consider the algebraic isomorphism

u = ν^{i}L^{d−i} : H^{i}(X, ℚ) → H^{i}(X, ℚ);

its characteristic polynomial P (t) has rational coefficients; by the Cayley – Hamilton theorem, P (u) = 0; it follows that u^{−1} is a linear combination of 1, u, u^{2}, … with rational coefficients; so, u^{−1} is algebraic; consequently, the isomorphism

Λ^{d−i} : H^{2d−i}(X, ℚ) ˜→ H^{i}(X, ℚ)

inverse to L^{d−i} : H^{i}(X, ℚ) ˜→ H^{2d−i}(X, ℚ), being equal to u^{−1}ν^{i}, is algebraic).

Let p ≤ d = dim X, and let H_{alg}^{2p}(X, ℚ) ↪ H^{2p}(X, ℚ) be the ℚ-vector space of classes of algebraic cycles of codimension p on X. The standard conjecture B(X) has the following weaker form:

1.3. Conjecture A(X, L). For any natural number p ≤ d/2, the injection

L^{d−2p} : H_{alg}^{2p}(X, ℚ) → H_{alg}^{2d−2p}(X, ℚ)

is an isomorphism.

The conjecture A(X, L) is equivalent to each of the following conjectures: ”H_{alg}^{∗}(X, ℚ) is stable under the operator ∗”; ”H_{alg}^{∗}(X, ℚ) is stable under the operator Λ”; ”H_{alg}^{∗}(X, ℚ) is stable under the operator ^{c}Λ”

([14], Proposition 2.1). By the classical result of D.Lieberman it is also equivalent to the following conjectures:

”H_{alg}^{∗}(X, ℚ) is stable under the operator ^{c}∗”; ”if x ∈ H_{alg}^{2p}(X, ℚ) and

x = ∑_{k≥max(0,2p−d)} L^{k}x_{2p−2k},

where x_{2p−2k} ∈ P^{2p−2k}(X), then x_{2p−2k} ∈ H_{alg}^{2p−2k}(X, ℚ)”;

D(X): ”numerical equivalence of algebraic cycles on X coincides with homological equivalence” ([19], Theorem 1).

The latter conjecture is independent of the polarization L, therefore the conjecture A(X, L) is independent of the polarization, and we denote it simply by A(X).

We claim that

A(X) ⇒ (H_{alg}^{∗}(X, ℚ) is stable under the Lefschetz involution ∗_{L}) .  (1.7)

Indeed, assume that A(X) holds. If x ∈ H_{alg}^{2p}(X, ℚ) and

x = ∑_{k≥max(0,2p−d)} L^{k}x_{2p−2k},

where x_{2p−2k} ∈ P^{2p−2k}(X), then x_{2p−2k} ∈ H_{alg}^{2p−2k}(X, ℚ). Since H_{alg}^{∗}(X, ℚ) is stable under the operators Λ and L, we see that ∗_{L} x ∈ H_{alg}^{∗}(X, ℚ).

Recall that

B(X) ⇒ A(X)  (1.8)

([14], Corollary 2.2), and

A(X × X) ⇒ B(X)  (1.9)

([15], Theorem 4 -1),

B(X) & B(Y) ⇒ B(X × Y)  (1.10)

([14], Corollary 2.5).

Moreover,

B(X) ⇔ A(X × X).  (1.11)

Indeed, (1.10) gives the relation B(X) ⇒ B(X × X); it remains to note that, in virtue of (1.8) – (1.9), B(X × X) ⇒ A(X × X) and A(X × X) ⇒ B(X).

1.4. Recall that

B(X) ⇒ C(X),

where

C(X): ”the Künneth components π^{i} ⊗ H^{2·dim X−i}(X, ℚ) ∈ H^{i}(X, ℚ) of the diagonal class Δ, which correspond to the projection operators

π^{i} : H^{∗}(X, ℚ) → H^{i}(X, ℚ),

are algebraic” ([14], Lemma 2.4). On the other hand, C(X) is equivalent to the following assertion:

”H_{alg}^{∗}(X × X, ℚ) is stable under Künneth decomposition” ([14], Proposition 2.6).

1.5. Theorem. Let X, Y be smooth projective varieties over the field of complex numbers. If there exists a surjective morphism f : X → Y, then A(X) ⇒ A(Y).

Proof. Recall that the conjecture A(X) is equivalent to the following statement: numerical and homological equivalences coincide on the set of algebraic cycles on X ([19], Theorem 1).

Assume that dim X > dim Y. Let X′ be a smooth hyperplane section of X. Then A(X) ⇒ A(X′). Indeed, let i : X′ ↪ X be the inclusion morphism of a smooth hyperplane section; then Λ′ = i^{∗}Λ^{2}i_{∗} ([14], Proposition 1.4.7(vii)), and therefore

Λ′ (H_{alg}^{∗}(X′, ℚ)) = i^{∗}Λ^{2}i_{∗} (H_{alg}^{∗}(X′, ℚ)) ↪
i^{∗}Λ^{2} (H_{alg}^{∗}(X, ℚ)) ↪ i^{∗} (H_{alg}^{∗}(X, ℚ)) ↪ H_{alg}^{∗}(X′, ℚ)

because i_{∗}, Λ, i^{∗} transform algebraic classes into algebraic classes. Since Λ′ transforms algebraic classes into algebraic classes, the conjecture A(X′) holds.

On the other hand, f_{|_{X′}} : X′ → Y is a surjective morphism. Therefore one can use an induction and reduce the problem to the case dim X = dim Y. In the latter case, let Z_{Y} be an algebraic cycle of codimension r on Y, which is numerically equivalent to zero on Y. Then for any algebraic cycle Z_{X} of codimension d − r on X, the projection formula gives the following relation:

deg(f_{∗}(f^{∗}(cl_{Y} (Z_{Y})) ∪ cl_{X}(Z_{X}))) = deg(cl_{Y} (Z_{Y}) ∪ f_{∗}(cl_{X}(Z_{X}))) = 0; (1.12)

on the other hand, f^{∗}(cl_{Y} (Z_{Y})) ∪ cl_{X}(Z_{X}) ∈ H^{2d}(X, ℚ) ≃ ℚ, and so f_{∗} : H^{2d}(X, ℚ) → H^{2d}(Y, ℚ) is an isomorphism. Therefore (1.12) gives an equality f^{∗}(cl_{Y} (Z_{Y})) ∪ cl_{X}(Z_{X}) = 0 for all algebraic cycles Z_{X} of codimension d − r on X. In particular, f^{∗}(cl_{Y} (Z_{Y})) is numerically equivalent to zero. Since A(X) holds, we get an equality f^{∗}(cl_{Y} (Z_{Y})) = 0. On the other hand, the projection formula shows that

f_{∗}f^{∗}(cl_{Y} (Z_{Y})) = deg f · cl_{Y} (Z_{Y})

([9], § 2.3). Consequently, cl_{Y} (Z_{Y}) = 0 and Z_{Y} is homologically equivalent to zero. Another proof: since f^{∗} is injective ([14], Proposition 1.2.4), we have the relation

f^{∗}(cl_{Y} (Z_{Y})) = 0 ⇒ cl_{Y} (Z_{Y}) = 0.

Therefore the numerical and homological equivalences of algebraic cycles coincide on Y, and so A(Y) holds. Theorem 1.5 is proved.

1.6. Theorem. Let X, Y be smooth projective varieties over the field of complex numbers. If there exists a surjective morphism f : X → Y, then B(X) ⇒ B(Y).

Proof. Since f × f : X × X → Y × Y is a surjective morphism, Theorem 1.5 gives the relation A(X × X) ⇒ A(Y × Y), and the statement follows from (1.11). Theorem 1.6 is proved.

1.7. Corollary. Let Sym^{n}(C) be a symmetric power of a smooth projective curve C over the field of complex numbers. Then the conjecture B(Sym^{n}(C)) holds.

Indeed, since B(X) trivially holds for curves, it remains to apply (1.10) and Theorem 1.6 to the natural surjective morphism C^{n} = C × ⋯ × C → Sym^{n}(C).

1.8. Corollary (D.Lieberman’s theorem) ([19], Theorem 3; [14], Theorem 2A11). The conjecture B(X) holds for all Abelian varieties.

Proof. For any smooth projective curve C of genus g, there exists a morphism Sym^{g}(C) → Pic^{0}(C) of finite degree, which is defined by the following formula:

f(c_{1} + c_{2} + ⋯ + c_{g}) = O_{C}(c_{1} + c_{2} + ⋯ + c_{g} − g · e)) ∈ Pic^{0}(C),

where e is the fixed point of C ([11], Chapter II, § 7; [26], Chapter V, § 1, Proposition 2, § 3.11). By Corollary 1.7 and Theorem 1.6, B(X) holds for all Jacobian varieties. On the other hand, any Abelian variety is isomorphic to a factor of a product of Jacobian varieties ([26], Chapter VII, § 2.13, Corollary). Since B(X) is stable under product (property (1.10)), it remains to apply Theorem 1.6.

1.9. Let X be a smooth projective d-dimensional variety over the field ℂ of complex numbers. Recall that there exists a canonical orientation isomorphism

tr : H^{2d}(X, ℚ) ˜→ ℚ,

and, by the Poincaré duality theorem ([14], 1.2.A), the bilinear form

tr(x ∪ y) : H^{i}(X, ℚ) × H^{2d−i}(X, ℚ) → ℚ

is a perfect duality (which identifies H^{i}(X, ℚ) with the dual of H^{2d−i}(X, ℚ)).

§ 2. On the Y.André theorem

2.1. Y.André’s theorem ([1], Theorem 0.6.2). Each Hodge cycle ξ on an Abelian variety J is motivated ( motivé ). More precisely, ξ is a sum of cycles of the form pr_{J∗}(α ∪ ∗_{L} β), where α and β are algebraic cycles ( with coefficients in ℚ ) on J × I × Y_{1} × ⋯ × Y_{n}, I is an Abelian variety, Y_{i} is a total space of a compact pencil of Abelian varieties, and pr_{J} is the canonical projection onto J.

Recall that the Lefschetz involution ∗_{L} is defined by the formula (1.6). If the conjecture A(J × I × Y_{1} × ⋯ × Y_{n}) holds, then in virtue of (1.7), ∗_{L} transforms algebraic classes into algebraic classes, and (in virtue of the Y.André theorem), the Hodge conjecture holds for an abelian variety J. On the other hand, by the relation (1.8),

B(J × I × Y_{1} × ⋯ × Y_{n}) ⇒ A(J × I × Y_{1} × ⋯ × Y_{n}).

2.2. By the property (1.10), B(X) is stable under product; in other words, B(X), B(Y) imply B(X × Y). Consequently, the Hodge conjecture holds for an Abelian variety J if the standard conjecture B(X) holds for all Abelian varieties and for all total spaces of compact pencils of Abelian varieties. By the classical D.Lieberman theorem (Corollary 1.8), B(X) holds for all Abelian varieties. It remains to prove B(X) for a total space of a compact pencil π : X → C of Abelian varieties.

Note that for some ramified covering C′ → C, the morphism X ×_{C} C′ → C′ has a section (in other words, X ×_{C} C′ → C′ is an Abelian scheme over C′). The canonical morphism X ×_{C} C′ → X is finite, and in virtue of Theorem 1.6, one may assume that π : X → C is an Abelian scheme.

Consequently, we arrive to the following result ([1], p. 33, Remark 2):

2.3. Corollary. If the Grothendieck standard conjecture B(X) holds for all Abelian schemes π : X → C over smooth projective curves C, then the Hodge conjecture holds for all Abelian varieties.

2.4. Let K be a function field with the field of constants k, and let A be an Abelian variety over K. By definition, the K/k-trace of A is the pair (B, τ) consisting of an Abelian variety B over k and a K-homomorphism τ : B → A which has the following universal property: for any Abelian variety J over k and for any K-homomorphism φ : J → A, there exists the unique k-homomorphism φ_{∗} : J → B such that the following diagram is commutative:

J ⟶^{φ} A
↘^{φ_{∗}} ↑^{τ}
B.

It is well known that K/k-trace exists ([16], Chapter VIII, § 3, Theorem 8), and τ is injective. We recall the Lang – Néron theorem:

2.5. Theorem ([17], Chapter 6, Theorem 2). Let K be a function field with the field of constants k, A an Abelian variety over K and the pair (B, τ) is a K/k-trace of A. Then the group A(K)/τB(k) is finitely generated.

2.6. Lemma. Let π : X → C be an Abelian scheme over a smooth projective curve C, J × C ⟶^{pr_{2}} C be a maximal constant Abelian subscheme of π : X → C, and let π′ : Y → C be the corresponding Abelian scheme factor. Then

B(X) ⇔ B(Y)

and

H^{0}(C, R^{1}π′_{∗}ℚ) = 0.

Proof. By Theorem 1.6, B(X) ⇒ B(Y) because there exists a canonical surjective morphism X → Y. On the other hand, the Poincaré reducibility theorem shows that there exists an isogeny of Abelian C-schemes

Y × J = Y ×_{C} (J × C) → X.

By Theorem 1.6, B(Y × J) ⇒ B(X). By the property (1.10), B(Y)&B(J) ⇒ B(Y ×J). Since B(J) holds for all Abelian varieties (by the D.Lieberman theorem), we get a relation B(Y) ⇒ B(X).

Let K = ℂ(C) be the field of rational functions on a curve C. In the situation under consideration, the generic fiber Y_{η} of the morphism π′ : Y → C has a trivial K/ℂ-trace. By the Lang – Néron theorem 2.5, Y_{η}(K) is a finitely generated group.

The canonical exponential map defines an exact sequence of sheaves of sections

0 → T_{ℤ}(Y) → [e^{∗}Ω_{Y/C}^{1}]^{∧} → Y → 0,

where e : C → Y is a zero section of π′ : Y → C and

T_{ℤ}(Y) = [R^{1}π′_{∗}ℤ]^{∧}

is a local system dual to R^{1}π′_{∗}ℤ. We arrive to an exact cohomology sequence

0 → H^{0}(C, T_{ℤ}(Y)) → H^{0}(C, [e^{∗}Ω_{Y/C}^{1}]^{∧}) → H^{0}(C, Y)  (2.1)

In virtue of Lemma 1 in Supplement to § 2 of [2], the set of all extensions of e : C → Y to C-morphisms C[ε]/(ε^{2}) → Y is isomorphic to Hom_{O_{C}} (e^{∗}Ω_{Y/C}^{1}, O_{C}). Therefore

Hom_{O_{C}} (e^{∗}Ω_{Y/C}^{1}, O_{C}) = H^{0}(C, [e^{∗}Ω_{Y/C}^{1}]^{∧})

is the tangent space at the origin of the ℂ-scheme Sect_{Y/C} of sections of Y over C, and Sect_{Y/C}(ℂ) = Y_{η}(K). In virtue of the Lang – Néron theorem 2.5, dim H^{0}(C, [e^{∗}Ω_{Y/C}^{1}]^{∧}) is equal to the dimension of K/ℂ-trace of Y_{η}. Therefore

H^{0}(C, [e^{∗}Ω_{Y/C}^{1}]^{∧}) = 0,

and an exact sequence (2.1) shows that

H^{0}(C, T_{ℤ}(Y)) = 0.  (2.2)

Since T_{ℤ}(Y) ⊗ ℚ is a polarizable system of Hodge ℚ-structures of weight −1, there exists an isomorphism of systems of Hodge ℚ-structures

R^{1}π′_{∗}ℚ = [T_{ℤ}(Y) ⊗ ℚ]^{∧} ≃ T_{ℤ}(Y) ⊗ ℚ(−1)

([6], § 4.2.3). Therefore (2.2) gives the relations

H^{0}(C, R^{1}π′_{∗}ℚ) = H^{0}(C, T_{ℤ}(Y) ⊗ ℚ(−1)) = H^{0}(C, T_{ℤ}(Y)) ⊗ ℚ(−1) = 0.

Lemma 2.6 is proved.

2.7. Remark. We may assume that an Abelian scheme π : X → C satisfies the following additional assumption: for any (ramified) covering C′ → C an Abelian scheme X ×_{C} C′ → C′ contains no nontrivial constant Abelian subschemes; in particular,

H^{0}(C, R^{1}π_{∗}ℚ) = 0.  (2.3)

It is well known that for an Abelian scheme π : X → C, there exists an isogeny X′ → X over C such that an Abelian scheme X′ → C has a principal polarization. By Theorem 1.6, B(X′) ⇒ B(X). Therefore we may also assume that π : X → C has a principal polarization.

§ 3. On cohomology of complex Abelian schemes over smooth projective curves

3.1. Let C be a smooth projective curve over the field of complex numbers and let π : X → C be an Abelian scheme of relative dimension d. Since π is a smooth morphism of compact Kähler varieties, the Leray spectral sequence E_{2}^{p,q} = H^{p}(C, R^{q}π_{∗}ℚ) ⇒ H^{∗}(X, ℚ) degenerates at the E_{2}-term, so

H^{i}(X, ℚ) = ⨁_{a+b=i} H^{a}(C, R^{b}π_{∗}ℚ)  (3.1)

([5]; [11], § 3.5).

Let Θ_{n} be the endomorphism of X induced by multiplication by n in the fibers of π : X → C, for any integer n > 1. Since [Θ_{n}]^{∗} commute with the d_{2} differentials of the Leray spectral sequence E_{2}^{p,q} = H^{p}(C, R^{q}π_{∗}ℚ) ⇒ H^{∗}(X, ℚ), we see that H^{a}(C, R^{b}π_{∗}ℚ) coincides with the n^{b}-eigenspace H^{<a,b>}(X, ℚ) of [Θ_{n}]^{∗} in H^{r}(X, ℚ) ([14], Lemma 2A3).

So we get the well known decomposition of Hodge ℚ-structures

H^{i}(X, ℚ) = ⨁_{a+b=i} H^{<a,b>}(X, ℚ) = ⨁_{a+b=i} H^{a}(C, R^{b}π_{∗}ℚ);  (3.2)

moreover, by the same reason, the ∪-product in H^{∗}(X, ℚ) has the following property:

H^{a}(C, R^{b}π_{∗}ℚ) ∪ H^{p}(C, R^{q}π_{∗}ℚ) ↪ H^{a+p}(C, R^{b+q}π_{∗}ℚ).  (3.3)

It is easy to see that H^{a}(C, R^{b}π_{∗}ℚ) = (0) for all a ≥ 3. Indeed, the lcohomological dimension cd_{l}(C_{et}) ≤ 2 · dim C = 2 ([21], Chapter VI, § 1, Theorem 1.1); therefore H_{et}^{a}(C, R^{b}π_{∗}ℤ/l^{n}ℤ) = (0) for a ≥ 3, and it remains to use the comparison theorem for the étale cohomology and classical cohomology ([21], Chapter III, § 3, Theorem 3.12). Consequently, we get the following variant of the decomposition (3.2):

H^{i}(X, ℚ) = H^{0}(C, R^{i}π_{∗}ℚ) ⨁ H^{1}(C, R^{i−1}π_{∗}ℚ) ⨁ H^{2}(C, R^{i−2}π_{∗}ℚ). (3.4)

3.2. Recall that R^{p}π_{∗}ℚ is completely determined by the monodromy representation π_{1}(C, s) → GL(H^{p}(X_{s}, ℚ)) ([11], Chapter 3, § 5).

Let τ : X ×_{C} X → C be the structure morphism. Then R^{p}τ_{∗}ℚ is completely determined by the corresponding monodromy representation

π_{1}(C, s) → GL(H^{p}(X_{s} × X_{s}, ℚ)) = GL ( ⨁_{a+b=p} H^{a}(X_{s}, ℚ) ⊗ H^{b}(X_{s}ℚ)) ,

where π_{1}(C, s) acts on H^{a}(X_{s}, ℚ) ⊗ H^{b}(X_{s}ℚ) via the diagonal embedding π_{1}(C, s) ↪ π_{1}(C, s) × π_{1}(C, s) = π_{1}(C × C, s × s) ([11], Chapter 3, § 5), and the Künneth decomposition

H^{p}(X_{s} × X_{s}, ℚ) = ⨁_{a+b=p} H^{a}(X_{s}, ℚ) ⊗ H^{b}(X_{s}, ℚ)

is compatible with monodromy representations. Therefore we have the Künneth decomposition

R^{p}τ_{∗}ℚ = ⨁_{a+b=p} R^{a}π_{∗}ℚ ⊗ R^{b}π_{∗}ℚ.

Since τ : X ×_{C} X → C is an Abelian scheme over a curve C, we have the decompositions

H^{p}(X ×_{C} X, ℚ) =
H^{0}(C, R^{p}τ_{∗}ℚ) ⨁ H^{1}(C, R^{p−1}τ_{∗}ℚ) ⨁ H^{2}(C, R^{p−2}τ_{∗}ℚ) =

[ ⨁_{a+b=p} H^{0}(C, R^{a}π_{∗}ℚ ⊗ R^{b}π_{∗}ℚ)] ⨁ [ ⨁_{a+b=p−1} H^{1}(C, R^{a}π_{∗}ℚ ⊗ R^{b}π_{∗}ℚ)] ⨁

[ ⨁_{a+b=p−2} H^{2}(C, R^{a}π_{∗}ℚ ⊗ R^{b}π_{∗}ℚ)] .  (3.5)

Similarly,

H^{p}(X × X, ℚ) =

⨁_{a+b=p} [H^{a}(X, ℚ) ⨂ H^{b}(X, ℚ)] =

⨁_{a+b=p} [H^{0}(C, R^{a}π_{∗}ℚ) ⨁ H^{1}(C, R^{a−1}π_{∗}ℚ) ⨁ H^{2}(C, R^{a−2}π_{∗}ℚ)] ⨂
[H^{0}(C, R^{b}π_{∗}ℚ) ⨁ H^{1}(C, R^{b−1}π_{∗}ℚ) ⨁ H^{2}(C, R^{b−2}π_{∗}ℚ)] .  (3.6)

We denote by u_{pqrt} ∈ H^{p}(C, R^{q}π_{∗}ℚ) ⨂ H^{r}(C, R^{t}π_{∗}ℚ) the corresponding component of u ∈ H^{2i}(X × X, ℚ) in the decomposition (3.6). In particular,

u_{0i0i} ∈ H^{0}(C, R^{i}π_{∗}ℚ) ⨂ H^{0}(C, R^{i}π_{∗}ℚ).

3.3. Lemma. If u ∈ H_{alg}^{2i}(X × X, ℚ), then u_{0i0i} ∈ H_{alg}^{2i}(X × X, ℚ). Proof. The following is a modification of the D.Lieberman method ([14], the

proof of Theorem 2A11). In virtue of (3.2), we have

[Θ_{n} × id_{X}]^{∗}u_{pqrt} = n^{q} · u_{pqrt},
[id_{X} ×Θ_{n}]^{∗}u_{pqrt} = n^{t} · u_{pqrt}.  (3.7)

For the sake of simplicity assume that i = 2. Then

u = u_{0004} + u_{0013} + u_{0022} + u_{0103} + u_{0112} + u_{0121}+

u_{0202} + u_{0211} + u_{0220} + u_{1102} + u_{1111} + u_{1120} + u_{2002} + u_{2011} + u_{2020}+ u_{0301} + u_{0310} + u_{1201} + u_{1210} + u_{2101} + u_{2110} + u_{0400} + u_{1300} + u_{2200}.

Applying [Θ_{n} × id_{X}]^{∗} − [id_{X} × id_{X}]^{∗} to this algebraic class, we get the relation

([Θ_{n} × id_{X}]^{∗} − [id_{X} × id_{X}]^{∗})(u) ∈ H_{alg}^{2i}(X × X, ℚ).

ON THE STANDARD CONJECTURE 13

In virtue of (3.7), the latter relation has the form (n − 1) · u_{0103} + (n − 1) · u_{0112} + (n − 1) · u_{0121} + (n^{2} − 1) · u_{0202}+

(n^{2} − 1) · u_{0211} + (n^{2} − 1) · u_{0220} + (n − 1) · u_{1102} + (n − 1) · u_{1111}+
(n − 1) · u_{1120} + (n^{3} − 1) · u_{0301} + (n^{3} − 1) · u_{0310} + (n^{2} − 1) · u_{1201}+
(n^{2} − 1) · u_{1210} + (n − 1) · u_{2101} + (n − 1) · u_{2110} + (n^{4} − 1) · u_{0400}+
(n^{3} − 1) · u_{1300} + (n^{2} − 1) · u_{2200} ∈ H_{alg}^{2i}(X × X, ℚ).

Consequently, u_{0103} + u_{0112} + u_{0121} + (n + 1) · u_{0202} + (n + 1) · u_{0211} + (n + 1) · u_{0220} + u_{1102}+

u_{1111} + u_{1120} + (n^{2} + n + 1) · u_{0301} + (n^{2} + n + 1) · u_{0310} + (n + 1) · u_{1201}+

(n + 1) · u_{1210} + u_{2101} + u_{2110} + (n^{3} + n^{2} + n + 1) · u_{0400}+
(n^{2} + n + 1) · u_{1300} + (n + 1) · u_{2200} ∈ H_{alg}^{2i}(X × X, ℚ).

Applying [id_{X} ×Θ_{n}]^{∗} − [id_{X} × id_{X}]^{∗} to the latter algebraic class, we get the relation

(n^{3} − 1) · u_{0103} + (n^{2} − 1) · u_{0112} + (n − 1) · u_{0121} + (n + 1)(n^{2} − 1) · u_{0202}+

(n + 1)(n − 1) · u_{0211} + (n^{2} − 1) · u_{1102} + (n − 1) · u_{1111}+

(n^{2} + n + 1)(n − 1) · u_{0301} + (n + 1)(n − 1) · u_{1201}+
(n − 1) · u_{2101} ∈ H_{alg}^{2i}(X × X, ℚ).

Hence,

(n^{2} + n + 1) · u_{0103} + (n + 1) · u_{0112} + u_{0121}+
(n + 1)^{2} · u_{0202} + (n + 1) · u_{0211} + (n + 1) · u_{1102} + u_{1111}+
(n^{2} + n + 1) · u_{0301} + (n + 1) · u_{1201} + u_{2101} ∈ H_{alg}^{2i}(X × X, ℚ).  (3.8)

Applying [Θ_{n} × id_{X}]^{∗} − [id_{X} × id_{X}]^{∗} to this algebraic class, we get the relation

(n^{2} + n + 1)(n − 1) · u_{0103} + (n + 1)(n − 1) · u_{0112} + (n − 1) · u_{0121}+

(n + 1)^{2}(n^{2} − 1) · u_{0202} + (n + 1)(n^{2} − 1) · u_{0211}+
(n + 1)(n − 1) · u_{1102} + (n − 1) · u_{1111} + (n^{2} + n + 1)(n^{3} − 1) · u_{0301}
+(n + 1)(n^{2} − 1) · u_{1201} + (n − 1) · u_{2101} ∈ H_{alg}^{2i}(X × X, ℚ).

Therefore

(n^{2} + n + 1) · u_{0103} + (n + 1) · u_{0112} + u_{0121}+
(n + 1)^{3} · u_{0202} + (n + 1)^{2} · u_{0211} + (n + 1) · u_{1102} + u_{1111}+
(n^{2} + n + 1)^{2} · u_{0301} + (n + 1)^{2} · u_{1201} + u_{2101} ∈ H_{alg}^{2i}(X × X, ℚ). (3.9)

Subtracting (3.8) from (3.9), we get the following relation:

(n + 1)^{2}n · u_{0202} + (n + 1)n · u_{0211}+
(n^{2} + n + 1)n(n + 1) · u_{0301} + (n + 1)n · u_{1201} ∈ H_{alg}^{2i}(X × X, ℚ).

Consequently,

(n + 1) · u_{0202} + u_{0211} + (n^{2} + n + 1) · u_{0301} + u_{1201} ∈ H_{alg}^{2i}(X × X, ℚ).(3.10)

Applying [id_{X} ×Θ_{n}]^{∗} − [id_{X} × id_{X}]^{∗} to the latter algebraic class, we get the relation

(n + 1)(n^{2} − 1) · u_{0202} + (n − 1) · u_{0211}+
(n^{2} + n + 1)(n − 1) · u_{0301} + (n − 1) · u_{1201} ∈ H_{alg}^{2i}(X × X, ℚ).

Therefore

(n + 1)^{2} · u_{0202} + u_{0211} + (n^{2} + n + 1) · u_{0301} + u_{1201} ∈ H_{alg}^{2i}(X × X, ℚ).(3.11)

Subtracting (3.10) from (3.11), we arrive to the relation (n + 1)n · u_{0202} ∈ H_{alg}^{2i}(X × X, ℚ). In particular, u_{0i0i} ∈ H_{alg}^{2i}(X × X, ℚ). Lemma 3.3 is proved.

§ 4. Poincaré bundles

4.1. Let π : X → C be an Abelian scheme of relative dimension d over a smooth projective curve C, and let s ∈ C. Recall that there exists a Poincaré bundle P_{s}

on X_{s} × X^{∧}_{s}, uniquely determined up to isomorphisms by the following properties:

i) P_{s}|_{X_{s}×{L_{s}}} ˜→ L_{s} for every L_{s} ∈ X^{∧}_{s},

ii) P_{s}|_{{0}×X^{∧}_{s}} is trivial ([18], Chapter 2, § 5).

We claim that cl_{X_{s}×X^{∧}_{s}}(P_{s}) has Künneth type (1,1). Indeed, by the Künneth formula we have

cl_{X_{s}×X_{s}}(P_{s}) ∈ H^{2}(X_{s} × X_{s}, ℚ) =
[H^{2}(X_{s}, ℚ) ⊗ H^{0}(X_{s}, ℚ)] ⨁[H^{0}(X_{s}, ℚ) ⊗ H^{2}(X_{s}, ℚ)] ⨁
[H^{1}(X_{s}, ℚ) ⊗ H^{1}(X_{s}, ℚ)].

On the other hand, by the properties i), ii) above,

P_{s}|X_{s} × {0} ˜→ O_{X_{s}}, P_{s}|{0} × X_{s} ˜→ O_{X_{s}}.

Therefore the (2,0) and (0,2) components of the Künneth decomposition of cl_{X_{s}×X_{s}}(P_{s}) are trivial.

4.2. Let e : C → X be the zero section, and let π^{∧} : X^{∧} = Pic^{0}(X/C) → C be the dual Abelian scheme with a zero section e^{∧}. For any invertible sheaf L on X, a rigidification of L is an isomorphism O_{C} ˜→ e^{∗}L. By definition of Pic^{0}(X/C), for any C-scheme T we have:

Hom_{C}(T, Pic^{0}(X/C)) =

{ isomorphism classes of invertible sheaves L on X ×_{C} T with rigidification

along e_{T} = e ×_{C} T such that ∀t ∈ T L|_{X×_{C}t} is algebraically equivalent

to zero on X ×_{C} t}

([8], Chapter I, § 1, Definition 1.4).

The tautological invertible sheaf P on X ×_{C} X^{∧} corresponding to the identity

morphism id_{X^{∧}} : X^{∧} → X^{∧} = Pic^{0}(X/C) is called the Poincaré bundle. By definition,

P is rigidified along e ×_{C} X^{∧}, and ∀t ∈ X^{∧} P|_{X×_{C}t} represents the element of

Pic^{0}(X ×_{C} t) given by t under the identification Pic^{0}(X ×_{C} t) = X^{∧}_{π^{∧}(t)}.

It is easy to see that P induces the Poincaré bundle P_{s} on X_{s} × X^{∧}_{s} via the

canonical embedding X_{s}×X^{∧}_{s} ↪ X×_{C} X^{∧}. Indeed, we have the canonical embedding of C-schemes

X^{∧}_{s} ↪ X^{∧}.

It is determined by an invertible sheaf L = P|_{X×_{C} X^{∧}_{s}} on X ×_{C} X^{∧}_{s} = X_{s} × X^{∧}_{s}, and

for any t ∈ X^{∧}_{s},

L|_{X_{s}×_{C}t} = L|_{X_{s}×t} = P|_{X×_{C}t}

represents the element of Pic^{0}(X_{s}×t) = Pic^{0}(X_{s}) given by t under the identification

Pic^{0}(X_{s}) = X^{∧}_{s}. Therefore L = P_{s}.

4.3. By Remark 2.7, we may assume that there exists a principal polarization

λ : X ˜→ X^{∧},

so we may consider λ as an identification X = X^{∧}.

§ 5. Correspondences and algebraic isomorphisms of rational Hodge structures

5.1. Recall some basic properties of correspondences ([14], § 1.3; [20]).

Let X, Y be smooth projective varieties over the field of complex numbers. There are canonical ℚ-linear isomorphisms

H^{∗}(X × Y, ℚ) ˜→ H^{∗}(X, ℚ) ⊗_{ℚ} H^{∗}(Y, ℚ) (Künneth formula)
˜→ H_{∗}(X, ℚ) ⊗_{ℚ} H^{∗}(Y, ℚ) (Poincaré duality)
˜→ Hom_{ℚ}(H^{∗}(X, ℚ), H^{∗}(Y, ℚ)).

Specifically, an element u = a ⊗ b ∈ H^{∗}(X × Y, ℚ) corresponds to the ℚ-linear map u^{∗} : H^{∗}(X, ℚ) → H^{∗}(Y, ℚ) defined by u^{∗}(c) = deg(c ∪ a) · b, where deg : H^{∗}(X) → ℚ is defined by the following formula:

deg = \{ \begin{matrix}the orientation isomorphism tr : H^{2·dimX}(X, ℚ) ˜→ ℚ, \\ 0, on H^{i}(X, ℚ) for i < 2 · dimX. \end{matrix}

Again, an arbitrary element u ∈ H^{∗}(X × Y, ℚ) corresponds to the composition

u^{∗} : H^{∗}(X, ℚ) ⟶^{pr_{1}^{∗}} H^{∗}(X × Y, ℚ) − − − ⟶^{w→w∪u} H^{∗}(X × Y, ℚ) ⟶^{pr_{2∗}} H^{∗}(Y, ℚ).

5.2. A linear map u^{∗} : H^{i}(X, ℚ) → H^{j}(Y, ℚ) is called algebraic if it is the restriction of a map u^{∗} : H^{∗}(X, ℚ) → H^{∗}(Y, ℚ) induced by an algebraic class u ∈ H_{alg}^{∗}(X × Y, ℚ).

Let V ↪ H^{i}(X, ℚ) and W ↪ H^{j}(Y, ℚ) be Hodge ℚ-substructures. We call a linear map V → W algebraic if it is the restriction of a map u^{∗} : H^{i}(X, ℚ) → H^{j}(Y, ℚ) induced by an algebraic class u ∈ H_{alg}^{∗}(X × Y, ℚ).

5.3. Composition of linear maps defines a composition of correspondences: if u ∈ H^{∗}(X × Y, ℚ) and v ∈ H^{∗}(Y × Z, ℚ), then

v ∘ u = pr_{13∗}(u ⊗ 1_{Z} ∪ 1_{X} ⊗ v) ∈ H^{∗}(X × Z, ℚ),

where pr_{13} : X × Y × Z → X × Z is the projection. If X = Y = Z, then the composition v ∘u of algebraic correspondences is again an algebraic correspondence ([14], Proposition 1.3.7). It is defined by an algebraic class

v ∘ u = pr_{13∗}(u ⊗ 1_{X} ∪ 1_{X} ⊗ v) ∈ H^{∗}(X × X, ℚ).

§ 6. Basic construction

6.1. Let μ_{s} : X_{s} × X_{s} → X_{s} be the group law for X_{s} (the sum morphism). Recall some results of S.L.Kleiman and D.Lieberman.

6.2. Lemma (S.L.Kleiman) ([14], Lemma 2A12). Let X_{s} be an Abelian variety, Y_{s} an arbitrary smooth projective variety and u ∈ H^{2}(X_{s} × Y_{s}, ℚ) an element of Künneth type (1, 1). Then the correspondence

exp(u) = ∑ u^{i}/i! : H^{∗}(X_{s}, ℚ) → H^{∗}(Y_{s}, ℚ)

takes Pontrjagin product

α ⋁ β = μ_{s∗} (α ⨂ β)

into cup product.

6.3. Lemma (D.Lieberman) ([14], Remark 2A13). Let X_{s} be an Abelian

variety, X^{∧}_{s} the dual Abelian variety and u ∈ H^{2}(X_{s} × X^{∧}_{s}, ℚ) the (1, 1)-Künneth component of a Poincaré divisor. Then the correspondence

exp(u) = ∑ u^{i}/i! : H^{∗}(X_{s}, ℚ) → H^{∗}(X^{∧}_{s}, ℚ)

is an algebraic isomorphism of the Pontrjagin algebra H^{∗}(X_{s}, ℚ) with the cup-

product algebra H^{∗}(X^{∧}_{s}, ℚ).

6.4. In the situation under consideration, X_{s} = X^{∧}_{s} and cl_{X_{s}×X_{s}}(P_{s}) has Künneth type (1, 1), so (by Lemma 6.3) for any natural number i, the map

H^{2d−i}(X_{s}, ℚ) → H^{i}(X_{s}, ℚ)

defined by

[cl_{X_{s}×X_{s}}(P_{s})]^{∪ i} = cl_{X_{s}×X_{s}}(P_{s}) ∪ ⋯ ∪ cl_{X_{s}×X_{s}}(P_{s}) ∈ H^{2i}(X_{s} × X_{s}, ℚ),

is an algebraic isomorphism. More precisely, this isomorphism is a composition of the following isomorphisms:

H^{2d−i}(X_{s}, ℚ) ˜⟶^{pr_{1s}^{∗}} H^{2d−i}(X_{s}, ℚ) ⊗ H^{0}(X_{s}, ℚ) − − − − − − −˜→^{∪ [cl_{Xs×Xs} (P_{s})]^{∪ i}}

H^{2d}(X_{s}, ℚ) ⊗ H^{i}(X_{s}, ℚ) ˜⟶^{pr_{2s∗}} H^{i}(X_{s}, ℚ),  (6.1)

where pr_{ks} : X_{s} × X_{s} → X_{s} is the canonical projection.

6.5. Let i_{s} : X_{s} ↪ X be the canonical embedding. Then we have canonical embeddings

X_{s} × X_{s} ↪^{i_{s}×i_{s}} X ×_{C} X ↪^{j} X × X.

By the Deligne theorem ([6], 4.1.2), the canonical restriction map

H^{2}(X ×_{C} X, ℚ) → H^{2}(X_{s} × X_{s}, ℚ)

is a composition of the canonical surjective edge morphism (of rational Hodge structures)

H^{2}(X ×_{C} X, ℚ) → H^{0}(C, R^{2}τ_{∗}ℚ)

associated to the Leray spectral sequence E_{2}^{p,q} = H^{p}(C, R^{q}τ_{∗}ℚ) ⇒ H^{∗}(X ×_{C} X, ℚ) and the canonical maps

H^{0}(C, R^{2}τ_{∗}ℚ) ˜→ H^{2}(X_{s} × X_{s}, ℚ)^{π_{1}(C,s)} ↪ H^{2}(X_{s} × X_{s}, ℚ).

It is evident that cl_{X×_{C}X}(P) ∈ H^{2}(X ×_{C} X, ℚ) maps to an element v_{s} = cl_{X_{s}×X_{s}}(P_{s}) ∈ H^{2}(X_{s} × X_{s}, ℚ)^{π_{1}(C,s)} of Künneth type (1,1) because

[i_{s} × i_{s}]^{∗}cl_{X×_{C}X}(P) = cl_{X_{s}×X_{s}}([i_{s} × i_{s}]^{∗}(P))

([14], 1.2.C) and [i_{s} × i_{s}]^{∗}(P) = P_{s}. Therefore

cl_{X×_{C}X}(P) ∈

H^{0}(C, R^{1}π_{∗}ℚ ⊗ R^{1}π_{∗}ℚ) ⨁ [ ⨁_{a+b=1} H^{1}(C, R^{a}π_{∗}ℚ ⊗ R^{b}π_{∗}ℚ)] ⨁ H^{2}(C, ℚ).

In virtue of the Lefschetz theorem, the H^{0}(C, R^{2}τ_{∗}ℚ)-component v of an algebraic class cl_{X×_{C}X}(P) is again algebraic. On the other hand, v ∈ H^{0}(C, R^{1}π_{∗}ℚ⊗R^{1}π_{∗}ℚ).

6.6. We claim that the algebraic class

v^{i} = v ∪ v ∪ ⋯ ∪ v ∈ H^{0}(C, R^{i}π_{∗}ℚ ⊗ R^{i}π_{∗}ℚ) ↪ H^{2i}(X ×_{C} X, ℚ)

determines a canonical isomorphism of local systems

ν^{i} : R^{2d−i}π_{∗}ℚ ˜→ R^{i}π_{∗}ℚ

over C. Indeed, let ν^{i} be the composition

R^{2d−i}π_{∗}ℚ ˜⟶^{p_{1}^{∗}} R^{2d−i}π_{∗}ℚ ⊗ R^{0}π_{∗}ℚ ˜⟶^{∪ v^{i}} R^{2d}π_{∗}ℚ ⊗ R^{i}π_{∗}ℚ ˜⟶^{p_{2∗}} R^{i}π_{∗}ℚ,

where p_{1}, p_{2} are canonical projections X ×_{C} X → X; in terms of monodromy representations of π_{1}(C, s) we have:

v^{i} ∈ H^{0}(C, R^{i}π_{∗}ℚ ⊗ R^{i}π_{∗}ℚ) ˜→ [H^{i}(X_{s}, ℚ) ⊗ H^{i}(X_{s}, ℚ)]^{π_{1}(C,s)} ,

and ν^{i} is the composition of isomorphisms of π_{1}(C, s)-modules

H^{2d−i}(X_{s}, ℚ) ˜⟶^{pr_{1s}^{∗}} H^{2d−i}(X_{s}, ℚ) ⊗ H^{0}(X_{s}, ℚ) ˜⟶^{∪ v_{s}^{i}}

H^{2d}(X_{s}, ℚ) ⊗ H^{i}(X_{s}, ℚ) ˜⟶^{pr_{2s∗}} H^{i}(X_{s}, ℚ).

These sequences induce the following sequence of isomorphisms:

H^{k}(C, R^{2d−i}π_{∗}ℚ) ˜⟶^{p_{1}^{∗}} H^{k}(C, R^{2d−i}π_{∗}ℚ ⊗ R^{0}π_{∗}ℚ) ˜⟶^{∪ v^{i}}

H^{k}(C, R^{2d}π_{∗}ℚ ⊗ R^{i}π_{∗}ℚ) ˜⟶^{p_{2∗}} H^{k}(C, R^{i}π_{∗}ℚ).

It is evident that p_{k} = pr_{k} ∘ j, therefore the latter sequence may be rewritten as

H^{k}(C, R^{2d−i}π_{∗}ℚ) ˜⟶^{j^{∗}pr_{1}^{∗}} H^{k}(C, R^{2d−i}π_{∗}ℚ ⊗ R^{0}π_{∗}ℚ) ˜⟶^{∪ v^{i}}

H^{k}(C, R^{2d}π_{∗}ℚ ⊗ R^{i}π_{∗}ℚ) ˜⟶^{pr_{2∗}j_{∗}} H^{k}(C, R^{i}π_{∗}ℚ).

Let x ∈ H^{k}(C, R^{2d−i}π_{∗}ℚ). Then the projection formula gives the relation

pr_{2∗}j_{∗}(j^{∗}pr_{1}^{∗}(x) ∪ v^{i}) = pr_{2∗}(pr_{1}^{∗}(x) ∪ j_{∗}v^{i}).

In particular, an algebraic class j_{∗}v^{i} ∈ H^{2i+2}(X ×X, ℚ) determines an algebraic isomorphism

H^{k}(C, R^{2d−i}π_{∗}ℚ) − − −˜→^{[j_{∗}v^{i}]^{∗}} H^{k}(C, R^{i}π_{∗}ℚ).  (6.2)

6.7. Consider the canonical maps

α : H^{1}(Alb(X), ℚ) → H^{1}(X, ℚ),
β : H^{2d+1}(X, ℚ) → H^{1}(Pic^{0}(X), ℚ)

from ([14], 2A1), where β is induced by a Poincare divisor D on X × Pic^{0}(X). It is known that α is an injection and β is a surjection ([14], Theorem 2A9(1)). Since we are in characteristic zero, both maps are isomorphisms. Moreover, there exists a divisor E on X × X, which induces the multiple of the map Λ^{d} : H^{2d+1}(X, ℚ) → H^{1}(X, ℚ) inverse to L^{d} ([14], Theorem 2A9(5)). In particular, Λ^{d} is algebraic, so the conjecture B^{1}(X) holds.

Another proof of B^{1}(X): in virtue of Lemma 2.6, we may assume that H^{0}(C, R^{1}π_{∗}ℚ) = 0; in this case,

H^{1}(X, ℚ) = H^{1}(C, ℚ),
H^{2d+1}(X, ℚ) = H^{1}(C, R^{2d}π_{∗}ℚ) ⨁ H^{2}(C, R^{2d−1}π_{∗}ℚ) = H^{1}(C, R^{2d}π_{∗}ℚ)

because R^{2d}π_{∗}ℚ ˜→ ℚ; in virtue of (6.2), an algebraic class j_{∗}1 ∈ H^{2}(X × X, ℚ) determines an algebraic isomorphism H^{1}(C, R^{2d}π_{∗}ℚ) ˜→ H^{1}(C, ℚ), and the conjecture B^{1}(X) holds.

It is evident that for a generator γ of the 1-dimensional space H^{0}(X, ℚ) = H_{alg}^{0}(X, ℚ), the canonical sequence

H^{2d+2}(X, ℚ) ⟶^{pr_{1}^{∗}} H^{2d+2}(X, ℚ) ⊗ H^{0}(X, ℚ) ⟶^{∪ γ}

H^{2d+2}(X, ℚ) ⊗ H^{0}(X, ℚ) ⟶^{pr_{2∗}} H^{0}(X, ℚ)

determines an algebraic isomorphism

H^{2d+2}(X, ℚ) ˜→ H^{0}(X, ℚ).

In particular, B^{0}(X) holds.

So we may assume that i ≥ 2.

In this case, algebraic isomorphisms (6.2) determine the following diagram:

\begin{matrix}H^{0}(C, R^{2d+2−i}π_{∗}ℚ) & ⨁ & H^{1}(C, R^{2d+1−i}π_{∗}ℚ) & ⨁ & H^{2}(C, R^{2d−i}π_{∗}ℚ) \\ ^{≃} ↓^{[j_{∗}v^{i−2}]^{∗}} & & ^{≃} ↓^{[j_{∗}v^{i−1}]^{∗}} & & ^{≃} ↓^{[j_{∗}v^{i}]^{∗}} \\ H^{0}(C, R^{i−2}π_{∗}ℚ) & ⨁ & H^{1}(C, R^{i−1}π_{∗}ℚ) & ⨁ & H^{2}(C, R^{i}π_{∗}ℚ), \end{matrix} (6.3)

where all vertical arrows are algebraic isomorphisms.

Compairing the rows of the diagram with the decompositions

H^{2d+2−i}(X, ℚ) =
H^{0}(C, R^{2d+2−i}π_{∗}ℚ) ⨁ H^{1}(C, R^{2d+1−i}π_{∗}ℚ) ⨁ H^{2}(C, R^{2d−i}π_{∗}ℚ),
H^{i}(X, ℚ) = H^{2}(C, R^{i−2}π_{∗}ℚ) ⨁ H^{1}(C, R^{i−1}π_{∗}ℚ) ⨁ H^{0}(C, R^{i}π_{∗}ℚ),

we see that the middle vertical arrow is good, but the left and the right vertical arrows of the diagram have to be corrected in order to obtain an algebraic isomorphism H^{2d+2−i}(X, ℚ) ˜→ H^{i}(X, ℚ).

§ 7. Correction of the left vertical arrow

7.1. Let H be a hyperplane section of X, and let h ∈ H^{0}(C, R^{2}π_{∗}ℚ) be the component of

cl_{X}(H) ∈ H^{2}(X, ℚ) = H^{0}(C, R^{2}π_{∗}ℚ) ⨁ H^{1}(C, R^{1}π_{∗}ℚ) ⨁ H^{2}(C, ℚ).

The Lefschetz theorem shows that h is an algebraic class, inducing the polarization on each fiber X_{s}. By the strong Lefschetz theorem, the canonical map

H^{i}(X_{s}, ℚ) − − −→^{∪h^{d−i}} H^{2d−i}(X_{s}, ℚ) is an isomorphism of π_{1}(C, s)-modules (i ≤ d). Therefore h^{d−i} determines an isomorphism

H^{0}(C, R^{i}π_{∗}ℚ) = H^{i}(X_{s}, ℚ)^{π_{1}(C,s)} − − −˜→^{∪h^{d−i}}

H^{2d−i}(X_{s}, ℚ)^{π_{1}(C,s)} = H^{0}(C, R^{2d−i}π_{∗}ℚ),

and this isomorphism is induced by an algebraic class Δ_{∗}h^{d−i}, where Δ : X ↪ X ×X is the diagonal embedding ([14], Proposition 1.3.5). We also have an equality

dim H^{0}(C, R^{i}π_{∗}ℚ) = dim H^{0}(C, R^{2d−i}π_{∗}ℚ).  (7.1)

7.2. We have a canonical nondegenerate pairing of π_{1}(C, s)-modules

H^{2d−i}(X_{s}, ℚ) × H^{i}(X_{s}, ℚ) → H^{2d}(X_{s}, ℚ) ˜→^{tr_{Xs}} ℚ.

It gives an identification of π_{1}(C, s)-modules

H^{i}(X_{s}, ℚ)^{∧} = H^{2d−i}(X_{s}, ℚ),

so we have an identification

[H^{i}(X_{s}, ℚ)^{∧}]^{π_{1}(C,s)} = H^{2d−i}(X_{s}, ℚ)^{π_{1}(C,s)}.

In particular,

H^{0}(C, R^{i}π_{∗}ℚ)^{∧} = [H^{i}(X_{s}, ℚ)^{π_{1}(C,s)}]^{∧} =
[H^{i}(X_{s}, ℚ)^{∧}]^{π_{1}(C,s)} = H^{2d−i}(X_{s}, ℚ)^{π_{1}(C,s)} = H^{0}(C, R^{2d−i}π_{∗}ℚ),

and the canonical pairing

H^{0}(C, R^{2d−i}π_{∗}ℚ) × H^{0}(C, R^{i}π_{∗}ℚ) → H^{0}(C, R^{2d}π_{∗}ℚ)

is nondegenerate: for any element 0 ≠ y ∈ H^{0}(C, R^{i}π_{∗}ℚ) there exists an element x ∈ H^{0}(C, R^{2d−i}π_{∗}ℚ) such that

0 ≠ x ∪ y ∈ H^{0}(C, R^{2d}π_{∗}ℚ);

let c be the image of cl_{X}(e(C)) ∈ H^{2d}(X, ℚ) under the canonical morphism

H^{2d}(X, ℚ) → H^{0}(C, R^{2d}π_{∗}ℚ);

we have

cl_{X}(e(C)) = c + c_{1} + c_{2},

where c_{j} ∈ H^{j}(C, R^{2d−j}π_{∗}ℚ) (j = 1, 2); since cl_{X}(X_{s}) ∈ H^{2}(C, ℚ) is a generator of 1-dimensional space H^{2}(C, ℚ), we see that

cl_{X}(X_{s}) ∪ c_{j} = 0 (j = 1, 2),

and

cl_{X}(X_{s}) ∪ c = cl_{X}(X_{s}) ∪ cl_{X}(e(C)) = cl_{X}(X_{s} ∩ e(C)) = cl_{X}(e(s)) ≠ 0;

in particular,

0 ≠ cl_{X}(X_{s}) ∪ H^{0}(C, R^{2d}π_{∗}ℚ);

since the latter space is contained in 1-dimensional space H^{2}(C, R^{2d}π_{∗}ℚ), we get the equalities

H^{2}(C, R^{2d}π_{∗}ℚ) = cl_{X}(X_{s}) ∪ H^{0}(C, R^{2d}π_{∗}ℚ) = H^{2}(C, ℚ) ∪ H^{0}(C, R^{2d}π_{∗}ℚ);

consequently, the canonical morphism

H^{0}(C, R^{2d}π_{∗}ℚ) − − − →^{∪ cl_{X}(X_{s})} H^{2}(C, R^{2d}π_{∗}ℚ)

is bijective, and

0 ≠ cl_{X}(X_{s}) ∪ x ∪ y ∈ H^{2}(C, R^{2d}π_{∗}ℚ);

in particular,

0 ≠ cl_{X}(X_{s}) ∪ y ∈ H^{2}(C, R^{i}π_{∗}ℚ)

for any 0 ≠ y ∈ H^{0}(C, R^{i}π_{∗}ℚ); so the canonical morphism

H^{0}(C, R^{i}π_{∗}ℚ) − − − →^{∪ cl_{X}(X_{s})} H^{2}(C, R^{i}π_{∗}ℚ)  (7.2)

is injective.

Note that elements of H^{2}(C, R^{i}π_{∗}ℚ) annihilate (by the ∪-product) the components H^{1}(C, R^{2d−i−1}π_{∗}ℚ) and H^{2}(C, R^{2d−i−2}π_{∗}ℚ) of the space

H^{2d−i}(X, ℚ) =
H^{0}(C, R^{2d−i}π_{∗}ℚ) ⨁ H^{1}(C, R^{2d−i−1}π_{∗}ℚ) ⨁ H^{2}(C, R^{2d−i−2}π_{∗}ℚ),

so the nondegeneracy of the canonical pairing

H^{i+2}(X, ℚ) × H^{2d−i}(X, ℚ) → H^{2d+2}(X, ℚ) ˜→^{tr} ℚ

(the Poincaré duality theorem) implies the nondegeneracy of the canonical pairing

H^{2}(C, R^{i}π_{∗}ℚ) × H^{0}(C, R^{2d−i}π_{∗}ℚ) → H^{2}(C, R^{2d}π_{∗}ℚ) ˜→^{tr} ℚ.

In particular,

H^{2}(C, R^{i}π_{∗}ℚ)^{∧} ˜→ H^{0}(C, R^{2d−i}π_{∗}ℚ).

Therefore, by (7.1),

dim H^{2}(C, R^{i}π_{∗}ℚ) = dim H^{2}(C, R^{i}π_{∗}ℚ)^{∧} =
dim H^{0}(C, R^{2d−i}π_{∗}ℚ) = dim H^{0}(C, R^{i}π_{∗}ℚ),

and an injective morphism (7.2) is bijective:

H^{0}(C, R^{i}π_{∗}ℚ) − − −˜→^{∪ cl_{X}(X_{s})} H^{2}(C, R^{i}π_{∗}ℚ).  (7.3)

7.3. In the situation under consideration,

H^{2}(C, R^{i−2}π_{∗}ℚ) = cl_{X}(X_{s}) ∪ H^{0}(C, R^{i−2}π_{∗}ℚ) = H^{2}(C, ℚ) ∪ H^{0}(C, R^{i−2}π_{∗}ℚ),

and the isomorphism

H^{0}(C, R^{i−2}π_{∗}ℚ) − − −˜→^{∪ cl_{X}(X_{s})} H^{2}(C, R^{i−2}π_{∗}ℚ)

is induced by an algebraic class Δ_{∗}cl_{X}(X_{s}), where Δ : X ↪ X × X is the diagonal embedding ([14], Proposition 1.3.5).

The composition of the algebraic isomorphisms

φ_{i} : H^{0}(C, R^{2d+2−i}π_{∗}ℚ) − − − ⟶^{[j_{∗}v^{i−2}]^{∗}} H^{0}(C, R^{i−2}π_{∗}ℚ) − − − ⟶^{[Δ_{∗}cl_{X}(X_{s})]^{∗}}

H^{2}(C, R^{i−2}π_{∗}ℚ)

is an algebraic isomorphism, defined by an algebraic class

[Δ_{∗}cl_{X}(X_{s})] ∘ [j_{∗}v^{i−2}] =
pr_{13∗}([j_{∗}v^{i−2}] ⊗ 1_{X} ∪ 1_{X} ⊗ [Δ_{∗}cl_{X}(X_{s})]) ∈ H^{2i}(X × X, ℚ).

Indeed, it is evident that

j_{∗}v^{i−2} ∈ H^{2i−2}(X × X, ℚ),
Δ_{∗}cl_{X}(X_{s}) ∈ H^{2+2·dim X}(X × X, ℚ),
[j_{∗}v^{i−2}] ⊗ 1_{X} ∪ 1_{X} ⊗ [Δ_{∗}cl_{X}(X_{s})] ∈ H^{2i+2·dim X}(X × X × X, ℚ),
pr_{13∗}([j_{∗}v^{i−2}] ⊗ 1_{X} ∪ 1_{X} ⊗ [Δ_{∗}cl_{X}(X_{s})]) ∈ H^{2i}(X × X, ℚ).

We transform the diagram (6.3) into the new diagram

\begin{matrix}H^{0}(C, R^{2d+2−i}π_{∗}ℚ) & ⨁ & H^{1}(C, R^{2d+1−i}π_{∗}ℚ) & ⨁ & H^{2}(C, R^{2d−i}π_{∗}ℚ) \\ ^{≃} ↓^{φ_{i}} & & ^{≃} ↓^{[j_{∗}v^{i−1}]^{∗}} & & ^{≃} ↓^{[j_{∗}v^{i}]^{∗}} \\ H^{2}(C, R^{i−2}π_{∗}ℚ) & ⨁ & H^{1}(C, R^{i−1}π_{∗}ℚ) & ⨁ & H^{2}(C, R^{i}π_{∗}ℚ), \end{matrix} (7.4)

where all vertical arrows are algebraic isomorphisms.

It remains to correct the right vertical arrow in (7.4).

§ 8. On the components of j_{∗}v^{i}

8.1. Lemma. We have

j_{∗}v^{i} ∈ [H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{2}(C, R^{i}π_{∗}ℚ)] ⨁[H^{2}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)]

⨁[H^{1}(C, R^{i}π_{∗}ℚ) ⊗ H^{1}(C, R^{i}π_{∗}ℚ)].

Proof. Let Θ_{n} be the endomorphism of X induced by multiplication by n in the fibers of π : X → C, for any integer n > 1. It is well known that H^{a}(C, R^{b}π_{∗}ℚ) coincides with the n^{b}-eigenspace H^{<a,b>}(X, ℚ) of Θ_{n}^{∗} in H^{r}(X, ℚ) ([14], Lemma 2A3).

First of all note that the map Θ_{n} : X → X is a morphism of schemes over C, and so it determines the C-morphism Θ_{n} ×_{C} id_{X} : X ×_{C} X → X ×_{C} X and the C × C-morphism Θ_{n} × id_{X} : X × X → X × X. Moreover,

j ∘ [Θ_{n} ×_{C} id_{X}] = [Θ_{n} × id_{X}] ∘ j.

Therefore

j_{∗}[Θ_{n} ×_{C} id_{X}]_{∗}v^{i} = [Θ_{n} × id_{X}]_{∗}j_{∗}v^{i}.

Since v^{i} ∈ H^{0}(C, R^{i}π_{∗}ℚ ⊗ R^{i}π_{∗}ℚ), we have

[Θ_{n} ×_{C} id_{X}]^{∗}v^{i} = n^{i} · v^{i}.

On the other hand, deg [Θ_{n} ×_{C} id_{X}] = n^{2d}. Therefore the projection formula

[Θ_{n} ×_{C} id_{X}]_{∗}[Θ_{n} ×_{C} id_{X}]^{∗} = deg [Θ_{n} ×_{C} id_{X}]

gives the relations

n^{2d} · v^{i} = [Θ_{n} ×_{C} id_{X}]_{∗}[Θ_{n} ×_{C} id_{X}]^{∗}v^{i} =
[Θ_{n} ×_{C} id_{X}]_{∗}(n^{i} · v^{i}) = n^{i} · [Θ_{n} ×_{C} id_{X}]_{∗}v^{i}.

We see that

[Θ_{n} ×_{C} id_{X}]_{∗}v^{i} = n^{2d−i} · v^{i}.

Consequently,

n^{2d−i} · j_{∗}v^{i} = j_{∗}(n^{2d−i} · v^{i}) = j_{∗}[Θ_{n} ×_{C} id_{X}]_{∗}v^{i} = [Θ_{n} × id_{X}]_{∗}j_{∗}v^{i}.

Let u = j_{∗}v^{i}. We denote by u_{pqrt} ∈ H^{p}(C, R^{q}π_{∗}ℚ) ⊗ H^{r}(C, R^{t}π_{∗}ℚ) the corresponding component of u. In virtue of (3.7),

[Θ_{n} × id_{X}]^{∗}u_{pqrt} = n^{q} · u_{pqrt}.

On the other hand, deg [Θ_{n} × id_{X}] = n^{2d}, and the projection formula

[Θ_{n} × id_{X}]_{∗}[Θ_{n} × id_{X}]^{∗} = deg [Θ_{n} × id_{X}]

gives the relations

n^{2d} · u_{pqrt} = [Θ_{n} × id_{X}]_{∗}[Θ_{n} × id_{X}]^{∗}u_{pqrt} =
[Θ_{n} × id_{X}]_{∗}(n^{q} · u_{pqrt}) = n^{q} · [Θ_{n} × id_{X}]_{∗}u_{pqrt}.

Therefore

[Θ_{n} × id_{X}]_{∗}u_{pqrt} = n^{2d−q} · u_{pqrt}.

Since

∑ n^{2d−q} · u_{pqrt} = [Θ_{n} × id_{X}]_{∗} ∑ u_{pqrt} =
[Θ_{n} × id_{X}]_{∗}j_{∗}v^{i} = n^{2d−i} · j_{∗}v^{i} = n^{2d−i} · ∑ u_{pqrt},

one has the equality q = i.

Similarly, using id_{X} × Θ_{n} instead of Θ_{n} × id_{X}, we arrive to the relation t = i. We see that j_{∗}v^{i} ∈ H^{2i+2}(X × X, ℚ) belongs to

[H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{2}(C, R^{i}π_{∗}ℚ)] ⨁[H^{2}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)]
⨁[H^{1}(C, R^{i}π_{∗}ℚ) ⊗ H^{1}(C, R^{i}π_{∗}ℚ)].

Lemma 8.1 is proved.

8.2. Corollary. For i = 0 we have:

j_{∗}1 = cl_{X×X}(X ×_{C} X) ∈

[H^{0}(C, ℚ) ⊗ H^{2}(C, ℚ)] ⨁[H^{2}(C, ℚ) ⊗ H^{0}(C, ℚ)] ⨁[H^{1}(C, ℚ) ⊗ H^{1}(C, ℚ)].

8.3. In virtue of Lemma 8.1, j_{∗}v^{i−1} ∈ H^{2i}(X × X, ℚ) belongs to

[H^{0}(C, R^{i−1}π_{∗}ℚ) ⊗ H^{2}(C, R^{i−1}π_{∗}ℚ)] ⨁[H^{2}(C, R^{i−1}π_{∗}ℚ) ⊗ H^{0}(C, R^{i−1}π_{∗}ℚ)]

⨁[H^{1}(C, R^{i−1}π_{∗}ℚ) ⊗ H^{1}(C, R^{i−1}π_{∗}ℚ)].

The latter algebraic class defines an algebraic isomorphism

H^{1}(C, R^{2d+1−i}π_{∗}ℚ) − − −˜→^{[j_{∗}v^{i−1}]^{∗}} H^{1}(C, R^{i−1}π_{∗}ℚ),

which is the composition

H^{1}(C, R^{2d+1−i}π_{∗}ℚ) ˜→^{pr_{1}^{∗}} H^{1}(C, R^{2d+1−i}π_{∗}ℚ) ⊗ H^{0}(C, ℚ) − − − →^{∪ j_{∗}v^{i−1}}

[H^{1}(C, R^{2d}π_{∗}ℚ) ⊗ H^{2}(C, R^{i−1}π_{∗}ℚ)] ⨁[H^{2}(C, R^{2d}π_{∗}ℚ) ⊗ H^{1}(C, R^{i−1}π_{∗}ℚ)]

→^{pr_{2∗}} H^{1}(C, R^{i−1}π_{∗}ℚ).

Finally, j_{∗}v^{i−2} ∈ H^{2i−2}(X × X, ℚ) belongs to

[H^{0}(C, R^{i−2}π_{∗}ℚ) ⊗ H^{2}(C, R^{i−2}π_{∗}ℚ)] ⨁[H^{2}(C, R^{i−2}π_{∗}ℚ) ⊗ H^{0}(C, R^{i−2}π_{∗}ℚ)]

⨁[H^{1}(C, R^{i−2}π_{∗}ℚ) ⊗ H^{1}(C, R^{i−2}π_{∗}ℚ)].

This algebraic class defines an algebraic isomorphism

H^{0}(C, R^{2d+2−i}π_{∗}ℚ) − − −˜→^{[j_{∗}v^{i−2}]^{∗}} H^{0}(C, R^{i−2}π_{∗}ℚ),

which is the composition

H^{0}(C, R^{2d+2−i}π_{∗}ℚ) ˜→^{pr_{1}^{∗}} H^{0}(C, R^{2d+2−i}π_{∗}ℚ) ⊗ H^{0}(C, ℚ) − − − →^{∪ j_{∗}v^{i−2}}

[H^{0}(C, R^{2d}π_{∗}ℚ) ⊗ H^{2}(C, R^{i−2}π_{∗}ℚ)] ⨁[H^{2}(C, R^{2d}π_{∗}ℚ) ⊗ H^{0}(C, R^{i−2}π_{∗}ℚ)]

⨁[H^{1}(C, R^{2d}π_{∗}ℚ) ⊗ H^{1}(C, R^{i−2}π_{∗}ℚ)] →^{pr_{2∗}} H^{0}(C, R^{i−2}π_{∗}ℚ).

§ 9. Restriction of line bundles

9.1. Lemma. If π : X → C has no nontrivial constant Abelian subschemes, then the restriction morphism

j^{∗} : Pic^{0}(X × X) → Pic^{0}(X ×_{C} X)

of the Picard varieties is surjective.

Proof. Recall that H^{1}(X × X, O_{X×X}) is a tangent space at the origin of the Picard variety Pic^{0}(X × X), and H^{1}(X ×_{C} X, O_{X×_{C}X}) is a tangent space at the origin of the Picard variety Pic^{0}(X ×_{C} X). By the Hodge theory,

H^{1}(X × X, ℚ) ⊗ ℂ = H^{1}(X × X, O_{X×X}) ⨁ \overline{H^{1}(X × X, O_{X×X})};
H^{1}(X ×_{C} X, ℚ) ⊗ ℂ = H^{1}(X ×_{C} X, O_{X×_{C}X}) ⨁ \overline{H^{1}(X ×_{C} X, O_{X×_{C}X})}.

Consequently, it remains to check the surjectivity of the restriction map

j^{∗} : H^{1}(X × X, ℚ) → H^{1}(X ×_{C} X, ℚ).

The Leray spectral sequence

E_{2}^{p,q} = H^{p}(C × C, R^{q}(π × π)_{∗}ℚ) ⇒ H^{∗}(X × X, ℚ)

induces a canonical exact sequence 0 → E_{2}^{1,0} → E^{1} → E_{2}^{0,1} ([21], Appendix B). It has the form

0 → H^{1}(C × C, ℚ) → H^{1}(X × X, ℚ) → H^{0}(C × C, R^{1}(π × π)_{∗}ℚ).

Similarly we arrive to the exact sequence

0 → H^{1}(C, ℚ) → H^{1}(X ×_{C} X, ℚ) → H^{0}(C, R^{1}τ_{∗}ℚ),

where τ : X ×_{C} X → C is the structure morphism.

In virtue of Lemma 2.6, we have the following relations:

H^{0}(C × C, R^{1}(π × π)_{∗}ℚ) = H^{0}(C, R^{1}π_{∗}ℚ) ⨁ H^{0}(C, R^{1}π_{∗}ℚ) = (0);
H^{0}(C, R^{1}τ_{∗}ℚ) = H^{0}(C, R^{1}π_{∗}ℚ ⊗ π_{∗}ℚ) ⨁ H^{0}(C, π_{∗}ℚ ⊗ R^{1}π_{∗}ℚ) =
H^{0}(C, R^{1}π_{∗}ℚ) ⨁ H^{0}(C, R^{1}π_{∗}ℚ) = (0).

Let Δ_{C} : C → C × C be the diagonal embedding. The commutative diagram with exact rows

0 → H^{1}(C × C, ℚ) → H^{1}(X × X, ℚ) → 0
↓^{[Δ_{C}]^{∗}}  ↓^{j^{∗}}
0 →  H^{1}(C, ℚ)  → H^{1}(X ×_{C} X, ℚ) → 0

shows that it remains to prove the surjectivity of the canonical morphism

[Δ_{C}]^{∗} : H^{1}(C × C, ℚ) → H^{1}(C, ℚ).

Since

H^{1}(C × C, ℚ) = [H^{1}(C, ℚ) ⊗ H^{0}(C, ℚ)] ⨁[H^{0}(C, ℚ) ⊗ H^{1}(C, ℚ)],

and (by definition of the cup product), for any a, b ∈ H^{∗}(C, ℚ),

[Δ_{C}]^{∗}(a ⊗ b) = a ∪ b,

we have a surjective morphism [Δ_{C}]^{∗} : H^{1}(C, ℚ) ⊗ H^{0}(C, ℚ) → H^{1}(C, ℚ). Lemma 9.1 is proved.

9.2. Lemma. If π : X → C has no nontrivial constant Abelian subschemes, then the restriction morphism

j^{∗} : H^{1}(X × X, O_{X×X}^{×}) → H^{1}(X ×_{C} X, O_{X×_{C}X}^{×})

of the Picard groups is not surjective; moreover,

P ∉ j^{∗} (H^{1}(X × X, O_{X×X}^{×})) .

Proof. Assume, for a moment, that there exists a line bundle L on X × X such that j^{∗}(L) = P. Let i_{s} : X_{s} ↪ X be the canonical embedding. Then we have canonical embeddings

X_{s} × X_{s} ↪^{i_{s}×i_{s}} X ×_{C} X ↪^{j} X × X.

By the Deligne theorem ([6], 4.1.2), the canonical restriction map

H^{2}(X × X, ℚ) → H^{2}(X_{s} × X_{s}, ℚ)

is a composition of the canonical surjective edge morphism

H^{2}(X × X, ℚ) → H^{0}(C × C, R^{2}(π × π)_{∗}ℚ))

of the Leray spectral sequence E_{2}^{p,q} = H^{p}(C × C, R^{q}(π × π)_{∗}ℚ) ⇒ H^{∗}(X × X, ℚ) and the canonical maps

H^{0}(C × C, R^{2}(π × π)_{∗}ℚ) ˜→ H^{2}(X_{s} × X_{s}, ℚ)^{π_{1}(C×C,s×s)} ↪ H^{2}(X_{s} × X_{s}, ℚ).

It is evident that cl_{X×X}(L) ∈ H^{2}(X × X, ℚ) maps to an element

v_{s} = cl_{X_{s}×X_{s}}(P_{s}) ∈ H^{2}(X_{s} × X_{s}, ℚ)^{π_{1}(C×C,s×s)}

of Künneth type (1,1) because j^{∗}(cl_{X×X}(L)) = cl_{X×_{C}X}(j^{∗}(L)) = cl_{X×_{C}X}(P) ([14], 1.2.C), [i_{s} × i_{s}]^{∗}cl_{X×_{C}X}(P) = cl_{X_{s}×X_{s}}([i_{s} × i_{s}]^{∗}(P)) ([14], 1.2.C) and [i_{s} × i_{s}]^{∗}(P) = P_{s}. Since v_{s} = cl_{X_{s}×X_{s}}(P_{s}) is of Künneth type (1,1), we see that the image of cl_{X×X}(L) under the canonical edge morphism

H^{2}(X × X, ℚ) → H^{0}(C × C, R^{2}(π × π)_{∗}ℚ)

belongs to

H^{0}(C, R^{1}π_{∗}ℚ) ⨂ H^{0}(C, R^{1}π_{∗}ℚ) ˜→ [H^{1}(X_{s}, ℚ) ⊗ H^{1}(X_{s}, ℚ)]^{π_{1}(C×C,s×s)} =

[H^{1}(X_{s}, ℚ)]^{π_{1}(C,s)} ⊗ [H^{1}(X_{s}, ℚ)]^{π_{1}(C,s)} .

In virtue of Lemma 2.6, H^{0}(C, R^{1}π_{∗}ℚ) = 0. Therefore v_{s} = cl_{X_{s}×X_{s}}(P_{s}) = 0 contradicting to (6.1). Lemma 9.2 is proved.

9.3. Lemma. If π : X → C has no nontrivial constant Abelian subschemes, then the restriction morphism

j^{∗} : H_{alg}^{2}(X × X, ℚ) → H_{alg}^{2}(X ×_{C} X, ℚ)

is not surjective; moreover,

v ∉ j^{∗} (H_{alg}^{2}(X × X, ℚ)) .

Proof. Assume, for a moment, that there exists a line bundle L on X × X such that j^{∗}(cl_{X×X}(L)) = v.

It is evident that the restriction of cl_{X×X}(L) ∈ H^{2}(X × X, ℚ) equals to an element

v_{s} = cl_{X_{s}×X_{s}}(P_{s}) ∈ H^{2}(X_{s} × X_{s}, ℚ)^{π_{1}(C×C,s×s)}

because j^{∗}(cl_{X×X}(L)) = v. Since v_{s} = cl_{X_{s}×X_{s}}(P_{s}) is of Künneth type (1,1), we see that the image of cl_{X×X}(L) under the canonical edge morphism

H^{2}(X × X, ℚ) → H^{0}(C × C, R^{2}(π × π)_{∗}ℚ)

belongs to

H^{0}(C, R^{1}π_{∗}ℚ) ⨂ H^{0}(C, R^{1}π_{∗}ℚ) ˜→ [H^{1}(X_{s}, ℚ) ⊗ H^{1}(X_{s}, ℚ)]^{π_{1}(C×C,s×s)} =

[H^{1}(X_{s}, ℚ)]^{π_{1}(C,s)} ⊗ [H^{1}(X_{s}, ℚ)]^{π_{1}(C,s)} .

In virtue of Lemma 2.6, H^{0}(C, R^{1}π_{∗}ℚ) = 0. Therefore v_{s} = cl_{X_{s}×X_{s}}(P_{s}) = 0 contradicting to (6.1). Lemma 9.3 is proved.

§ 10. Hodge cycles and a conditional correction of the right vertical arrow

10.1. Theorem. Let π : X → C be a principally polarized complex Abelian scheme of relative dimension d over a smooth projective curve C. Then for any natural number i ≥ 2, there exists the unique Hodge cycle

α_{i} = α_{i}(X/C) ∈ [H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)] ∩ H^{i,i}(X × X, ℂ),

characterized by the following condition: α_{i} ∪ cl_{X×X}(X × X_{s}) determines an isomorphism

H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{2}(C, R^{i}π_{∗}ℚ),

which coincides with an algebraic isomorphism

[j_{∗}v^{i}]^{∗} : H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{2}(C, R^{i}π_{∗}ℚ).

If α_{i} is algebraic, then the algebraic class

[Δ_{∗}cl_{X}(X_{s})] ∘ [j_{∗}v^{i−2}] + j_{∗}v^{i−1} + α_{i}

induces an algebraic isomorphism H^{2d+2−i}(X, ℚ) ˜→ H^{i}(X, ℚ), and the conjecture B^{i}(X) holds.

Proof. By Lemma 8.1,

j_{∗}v^{i} ∈ [H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{2}(C, R^{i}π_{∗}ℚ)] ⨁[H^{2}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)]

⨁[H^{1}(C, R^{i}π_{∗}ℚ) ⊗ H^{1}(C, R^{i}π_{∗}ℚ)].

In virtue of (7.3),

H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{2}(C, R^{i}π_{∗}ℚ) =
[H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)] ∪ cl_{X×X}(X × X_{s});
[H^{2}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)] =
[H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)] ∪ cl_{X×X}(X_{s} × X).

Consequently, there exist uniquely determined elements

α_{i}, β_{i} ∈ H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ),
δ_{i} ∈ H^{1}(C, R^{i}π_{∗}ℚ) ⊗ H^{1}(C, R^{i}π_{∗}ℚ)

such that

j_{∗}v^{i} = α_{i} ∪ cl_{X×X}(X × X_{s}) + β_{i} ∪ cl_{X×X}(X_{s} × X) + δ_{i}.

Let V_{ℚ} be a rational Hodge structure. From the general theory of groups of multiplicative type it follows that to define a Hodge decomposition

V_{ℚ} ⊗ ℂ = ⨁ V_{ℂ}^{p,q}

is the same as to define a morphism of real algebraic groups h : Res_{ℂ/ℝ}(G_{m}) → GL(V_{ℚ} ⊗ ℝ), where Res_{ℂ/ℝ}(G_{m}) is the real algebraic group obtained from the multiplicative group G_{m} by restricting the field of scalars from ℂ to ℝ ([10], Definition B.42); an element z ∈ ℂ^{×} = Res_{ℂ/ℝ}(G_{m})(ℝ) acts on V_{ℂ}^{p,q} as the multiplication by z^{p}\overline{z}^{q}. Let U^{1} = \{z ∈ ℂ^{×} = Res_{ℂ/ℝ}(G_{m})(ℝ) | |z| = 1\}. The smallest ℚ-subgroup in GL(V_{ℚ}) whose group of ℝ-points contains h(U^{1}) is called the Hodge group of a Hodge structure V_{ℚ} and is denoted by Hg(V_{ℚ}). The smallest ℚ-subgroup in GL(V_{ℚ}) whose group of ℝ-points contains h(Res_{ℂ/ℝ}(G_{m})) is called the Mumford – Tate group of a Hodge structure V_{ℚ} and is denoted by MT(V_{ℚ}) ([10], Definition B.51).

It is well known that there exists an isogeny G_{m} × Hg(V_{ℚ}) → MT(V_{ℚ}) ([10], Lemma B.53); if V_{ℚ} is a polarizable rational Hodge structure, then Hg(V_{ℚ}) and MT(V_{ℚ}) are reductive ℚ-groups acting in a natural way in the space V_{ℚ} ([10], Proposition B.57).

The vital role of Hodge groups in analizing Hodge structures comes from the fact, that the rational Hg(V_{ℚ})-subrepresentations in V^{⊗r} ⊗ (V^{∧})^{⊗s} are precisely the rational sub-Hodge structures of V^{⊗r} ⊗ (V^{∧})^{⊗s} ([10], Proposition B54).

Assume that V_{ℚ} is a rational Hodge structure of even weight 2r. Then

V_{ℚ}^{Hg(V_{ℚ})} = V_{ℚ} ∩ V_{ℂ}^{r,r},  (10.1)

where V_{ℚ} ⊗ ℂ = ⨁ V_{ℂ}^{p,q} is the Hodge decomposition ([10], Corollary B.55). By definition, elements of V_{ℚ}^{Hg(V_{ℚ})} = V_{ℚ} ∩ V_{ℂ}^{r,r} are called Hodge cycles.

In the situation under consideration, the components

H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{2}(C, R^{i}π_{∗}ℚ) =
[H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)] ∪ cl_{X×X}(X × X_{s}),
[H^{2}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)] =
[H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)] ∪ cl_{X×X}(X_{s} × X),
H^{1}(C, R^{i}π_{∗}ℚ) ⊗ H^{1}(C, R^{i}π_{∗}ℚ)

are rational Hodge substructures of H^{2i+2}(X × X, ℚ) (recall that H^{0}(C, R^{i}π_{∗}ℚ) is a rational Hodge substructure of H^{i}(X, ℚ) which is canonically identified (by (3.2)) with the n^{i}-eigenspace of the operator [Θ_{n}]^{∗} on H^{i}(X, ℚ); similarly, H^{1}(C, R^{i}π_{∗}ℚ) is canonically identified with the n^{i}-eigenspace of the operator [Θ_{n}]^{∗} on H^{i+1}(X, ℚ)). Consequently, they are stable under the canonical action of Hg(H^{2i+2}(X × X, ℚ)).

By the Lefschetz theorem,

NS(X × X) ⊗ ℚ ˜→ H^{2}(X × X, ℚ) ∩ H^{1,1}(X × X, ℂ).

In virtue of (10.1),

NS(X × X) ⊗ ℚ ˜→ H^{2}(X × X, ℚ)^{Hg(H^{2}(X×X,ℚ))}.

Therefore cl_{X×X}(X ×X_{s}) and cl_{X×X}(X_{s} ×X) are fixed under the canonical action of U^{1} on H^{2}(X × X, ℂ). Similarly, the algebraic class j_{∗}v^{i} belongs to

H_{alg}^{2i+2}(X × X, ℚ) ↪ H^{2i+2}(X × X, ℚ) ∩ H^{i+1,i+1}(X × X, ℂ) =
H^{2i+2}(X × X, ℚ)^{Hg(H^{2i+2}(X×X,ℚ))}

and is fixed under the action of U^{1} on H^{2i+2}(X × X, ℂ). Since this action is compatible with the cup product, the elements

α_{i}, β_{i} ∈ H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)

are fixed under the action of U^{1} on H^{2i}(X × X, ℂ). Consequently, α_{i} and β_{i} are fixed under the action of the Hodge group Hg(H^{2i}(X × X, ℚ)). In other words, α_{i} and β_{i} are Hodge cycles,

α_{i}, β_{i} ∈ [H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)] ∩ H^{i,i}(X × X, ℂ).

We claim that the Hodge cycle α_{i} ∪ cl_{X×X}(X × X_{s}) induces an algebraic isomorphism

[j_{∗}v^{i}]^{∗} : H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{2}(C, R^{i}π_{∗}ℚ)

and the Hodge cycle α_{i} induces an isomorphism

H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{0}(C, R^{i}π_{∗}ℚ),

which is not assumed to be algebraic.

Indeed, in virtue of (6.2), an algebraic class j_{∗}v^{i} induces an algebraic isomorphism

H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{2}(C, R^{i}π_{∗}ℚ),  (10.2)

determined by the composition

H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→^{pr_{1}^{∗}} H^{2}(C, R^{2d−i}π_{∗}ℚ) ⨂ H^{0}(C, ℚ) − − − →^{∪ j_{∗}v^{i}}

H^{2d+4+i}(X × X, ℚ) ˜→^{pr_{2∗}}H^{i+2}(X, ℚ).

Since an element β_{i}∪cl_{X×X}(X_{s}×X)+δ_{i} annihilates H^{2}(C, R^{2d−i}π_{∗}ℚ) ⨂ H^{0}(C, ℚ), an algebraic isomorphism (10.2) coincides with the composition

H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→^{pr_{1}^{∗}} H^{2}(C, R^{2d−i}π_{∗}ℚ) ⨂ H^{0}(C, ℚ) − − − − − →^{∪ α_{i}∪cl_{X×X}(X×X_{s})}

H^{2}(C, R^{2d}π_{∗}ℚ) ⊗ H^{2}(C, R^{i}π_{∗}ℚ) ˜→^{pr_{2∗}}H^{2}(C, R^{i}π_{∗}ℚ).

In particular, the map

H^{2}(C, R^{2d−i}π_{∗}ℚ) ⨂ H^{0}(C, ℚ) →^{∪ α_{i}} H^{2}(C, R^{2d}π_{∗}ℚ) ⨂ H^{0}(C, R^{i}π_{∗}ℚ)(10.3)

is injective.

On the other hand, (7.3) and (10.2) give the relations

dim H^{2}(C, R^{2d−i}π_{∗}ℚ) = dim H^{2}(C, R^{i}π_{∗}ℚ) = dim H^{0}(C, R^{i}π_{∗}ℚ).

Consequently, the map (10.3) is bijective, and the claim is proved.

If α_{i} is algebraic, then it induces an algebraic isomorphism

H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{0}(C, R^{i}π_{∗}ℚ),

therefore there exists an algebraic correction

H^{0}(C, R^{2d+2−i}π_{∗}ℚ) ⨁ H^{1}(C, R^{2d+1−i}π_{∗}ℚ) ⨁ H^{2}(C, R^{2d−i}π_{∗}ℚ)
≃ ↓^{φ_{i}} ^{≃} ↓^{[j_{∗}v^{i−1}]^{∗}} ^{≃} ↓^{[α_{i}]^{∗}}
H^{2}(C, R^{i−2}π_{∗}ℚ) ⨁ H^{1}(C, R^{i−1}π_{∗}ℚ) ⨁ H^{0}(C, R^{i}π_{∗}ℚ),

of the right vertical arrow of the diagram (7.4), and the conjecture B^{i}(X) holds. Theorem 10.1 is proved.

10.2. Corollary. If the Hodge cycles α_{i}(X/C) are algebraic for all natural numbers i ≥ 2 and for all complex principally polarized Abelian schemes π : X → C without nontrivial constant Abelian subschemes over smooth projective curves, then the Hodge conjecture holds for all Abelian varieties.

Proof. In virtue of Theorem 10.1, the conjecture B^{i}(X) holds for all i ≥ 2. Since B^{0}(X) and B^{1}(X) hold by the results of § 6.7, the conjecture B^{i}(X) holds for all i ≥ 0. It remains to apply the Y.André theorem 2.1, Corollary 2.3 and Remark 2.7.

§ 11. The proof of B^{i}(X) for some Abelian schemes

11.1. Let π : X → C be a complex Abelian scheme of relative dimension d over a smooth projective curve. There exists a finitely generated extension k of ℚ such that X, C, π are defined over k ↪ ℂ:

X = X_{0} ⊗_{k} ℂ, C = C_{0} ⊗_{k} ℂ, π = π_{0} ⊗_{k} ℂ.

Let X_{0η} be a generic fiber of an Abelian scheme π_{0} : X_{0} → C_{0}, and let X_{0\overline{η}} = X_{0η}⊗_{k(η)}ℂ for some embedding k(η) ↪ ℂ which is a prolongation of the embedding k ↪ ℂ.

We call X_{0\overline{η}} the generic geometric fiber of an Abelian scheme π_{0} : X_{0} → C_{0}. For any complex Abelian variety X_{s}, we denote by Hg(X_{s}) =^{def} Hg(H^{1}(X_{s}, ℚ))

the Hodge group of X_{s}. It is a reductive connected algebraic ℚ-group with a canonical action on H^{∗}(X_{s}, ℚ) = ⋀^{∗} H^{1}(X_{s}, ℚ). In virtue of (10.1),

H^{2i}(X_{s}, ℚ)^{Hg(X_{s})} = H^{2i}(X_{s}, ℚ) ∩ H^{i,i}(X_{s}, ℂ).

11.2. Theorem. Assume that for a complex Abelian scheme π : X → C of relative dimension d over a smooth projective curve C, Lie Hg(X_{0\overline{η}}) is a simple Lie algebra of type C_{d}.

Then the conjecture B(X) holds. In particular, the operators ^{c}∗, ∗, ^{c}Λ, Λ on H^{∗}(X, ℚ) are algebraic, the numerical equivalence of algebraic cycles on X coincides with the homological equivalence, the Künneth components of an arbitrary algebraic cycle on X × X are algebraic.

Proof. Let

ρ : π_{1}(C, s) → GL(H^{1}(X_{s}, ℚ))

be the canonical monodromy representation. We denote by G the Zariski closure of ρ(π_{1}(C, s)) in GL(H^{1}(X_{s}, ℚ)). By definition,

H^{i}(X_{s}, ℚ)^{G} = H^{i}(X_{s}, ℚ)^{π_{1}(C,s)}

([34], § 7.0).

In virtue of Theorem 1.6, we may consider a ramified covering C′ → C and an Abelian scheme X′ = X ×_{C} C′ → C′ instead of X → C. So we may assume that G is a connected ℚ-group. It is a semisimple group ([6], Corollary 4.2.9).

By the Deligne theorem, for all s outside a countable subset of C, the group G is a normal subgroup of the Hodge group Hg(X_{s}) ([7], Proposition 7.5; [34], Theorem 7.3). In particular, there exists an embedding G ↪ Hg(X_{0\overline{η}}).

If G = 1, then an Abelian scheme π : X → C is constant ([6], 4.1.3.3; [7], § 5.2): there exists an Abelian variety J such that X ˜→ J × C. Since B(C) is true and B(J) holds by the D.Lieberman theorem (Corollary 1.8), the conjecture B(X) holds by the property (1.10).

Consequently, we may assume that G is a nontrivial normal algebraic subgroup of a simple algebraic group Hg(X_{0\overline{η}}), and therefore

Lie G = Lie Hg(X_{0\overline{η}})

is a simple Lie algebra of type C_{d}.

We have:

H^{0}(C, R^{i}π_{∗}ℚ) ˜→ H^{i}(X_{s}, ℚ)^{π_{1}(C,s)} = H^{i}(X_{s}, ℚ)^{G} =

H^{i}(X_{s}, ℚ)^{Lie G} = H^{i}(X_{s}, ℚ)^{sp(H^{1}(X_{s},ℚ))} = [⋀^{i} H^{1}(X_{s}, ℚ)]^{sp(H^{1}(X_{s},ℚ))} .

On the other hand, let Γ be a generator of the 1-dimensional space

[⋀^{2} H^{1}(X_{s}, ℚ)]^{sp(H^{1}(X_{s},ℚ))} . Then

[⋀^{i} H^{1}(X_{s}, ℚ)]^{sp(H^{1}(X_{s},ℚ))} = \{ \begin{matrix}0, for i odd, \\ ℚ · Γ^{∧ i/2} for i even \end{matrix}

([28], Lemma 4.8.4).

Consequently,

H^{0}(C, R^{i}π_{∗}ℚ) = \{ \begin{matrix}0, for i odd, \\ ℚ · h^{∪ i/2} for i even, \end{matrix}

where h is an image of the class cl_{X}(H) of a hyperplane section under the canonical surjective edge morphism H^{2}(X, ℚ) → H^{0}(C, R^{2}π_{∗}ℚ) of the Leray spectral sequence E_{2}^{p,q} = H^{p}(C, R^{q}π_{∗}ℚ) ⇒ H^{∗}(X, ℚ).

We see that H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ) is trivial or else is generated by an algebraic class h^{∪ i/2} ⊗ h^{∪ i/2}, so

α_{i}(X/C) ∈ H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ)

is algebraic, and Theorem 11.2 follows from Theorem 10.1.

11.3. Corollary. Assume that End(X_{s}) = ℤ for some geometric fiber of a complex Abelian scheme π : X → C of odd relative dimension d over a smooth projective curve C.

Then the conjecture B(X) holds.

Indeed,

End(X_{0\overline{η}}) ↪ End(X_{s}) = ℤ.

Therefore End(X_{0\overline{η}}) = ℤ. Since d is odd, Hg(X_{0\overline{η}}) is a simple algebraic ℚ-group of type C_{d} ([28], Theorem 4.7; [29], Theorem 6.2).

11.4. Corollary. Let

Ex(1) = {4, 10, 16, 32, 64, 108, 126, 256, 500, 512, 864, 1024, … }

be the set of exceptional numbers in the sense of [31] for simple Abelian varieties of type I by Albert’s classification. If End(X_{s}) = ℤ for some geometric fiber of a complex Abelian scheme π : X → C of relative dimension d ∉ Ex(1) over a smooth projective curve C, then the conjecture B(X) holds.

Indeed, it is known that in the case under consideration, Hg(X_{0\overline{η}}) is a simple algebraic ℚ-group of type C_{d} ([31], Theorem 1.1).

11.5. Corollary. If End(X_{s}) = ℤ for some geometric fiber of a complex Abelian scheme π : X → C of relative dimension d ∈ {4, 10} over a smooth projective curve C, then the conjecture B(X) holds.

Indeed, it is known that in the case under consideration, Hg(X_{0\overline{η}}) is a simple algebraic ℚ-group of type C_{d} or else of type A_{1}×A_{1}×A_{1} (d = 4) (over \overline{ℚ}), A_{5} (d = 10). Therefore G = 1 or else G is a simple algebraic ℚ-group of the same type as Hg(X_{0\overline{η}}), and

H^{0}(C, R^{i}π_{∗}ℚ) = \{ \begin{matrix}(0), for i odd, \\ ℚ · h^{∪ i/2} for i even \end{matrix}

([31], Lemma 2.6), H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ) is trivial or else is generated by an algebraic class h^{∪ i/2} ⊗ h^{∪ i/2}. It remains to apply on Theorem 10.1.

11.6. Theorem. Assume that for a complex Abelian scheme π : X → C of relative dimension d over a smooth projective curve C, End(X_{0\overline{η}}) ⊗ ℚ is an imaginary quadratic extension of ℚ and the semisimple part [Lie Hg(X_{0\overline{η}})]^{ss}of a reductive Lie algebra Lie Hg(X_{0\overline{η}}) is a simple Lie algebra of type A_{d−1}.

Then for any i ≠ d, the isomorphism

Λ^{d+1−i} : H^{2d+2−i}(X, ℚ) ˜→ H^{i}(X, ℚ)

is algebraic.

Proof. Since G is a normal semisimple subgroup of Hg(X_{0\overline{η}}), it is a normal subgroup of [Hg(X_{0\overline{η}})]^{ss}. Therefore G = 1 (in this situation an Abelian scheme π : X → C is constant and B(X) holds), or else G is a simple ℚ-group of type A_{d−1}. In the latter case,

H^{1}(X_{s}, ℚ) ⊗ \overline{ℚ} = E(ω_{1}) ⨁ E(ω_{1})^{∗} = E(ω_{1}) ⨁ E(ω_{d−1})

is the sum of irreducible representations V = E(ω_{1}) and V^{∗} = E(ω_{d−1}) with highest weights ω_{1} and ω_{d−1} of a simple Lie algebra g = [Lie G] ⊗ \overline{ℚ} of type A_{d−1} ([31], § 2.9).

If i ≤ d − 1, then

[H^{i}(X_{s}, ℚ) ⊗ \overline{ℚ}]^{g} = ∧^{i}(V ⊕ V^{∗})^{g} = ∧^{i}(V ⊕ V^{∗})^{sl(V)} =

⨁_{m=0}^{i}[∧^{m}V ⊗ ∧^{i−m}V^{∗}]^{sl(V)} = ⨁_{m=0}^{i} Hom_{sl(V)}(∧^{i−m}V, ∧^{m}V).

Recall that

∧^{m}V = ∧^{m}E(ω_{1}) = E(ω_{m})

is an irreducible representation in the case d − 1 ≥ m ≥ 1, and

∧^{i−m}V = ∧^{i−m}E(ω_{1}) = E(ω_{i−m})

is an irreducible representation in the case d−1 ≥ i−m ≥ 1 ([3], Chapter 8, § 13.1); hence by Schur’s lemma,

Hom_{sl(V)}(∧^{i−m}V, ∧^{m}V) ≠ (0) ⇔ i − m = m (for d − 1 ≥ i ≥ i − m ≥ 1)

with

dim [∧^{m}V ⊗ ∧^{m}V^{∗}]^{sl(V)} = 1.

If d − 1 ≥ i ≥ i − m = 0, then

dim Hom_{sl(V)}(∧^{i−m}V, ∧^{m}V) = dim Hom_{sl(V)}(∧^{0}V, ∧^{i}V) = \{ \begin{matrix}0, for i ≥ 1, \\ 1, for i = 0. \end{matrix}

In particular, for all i ≤ d − 1,

dim [H^{i}(X_{s}, ℚ)]^{G} = dim [H^{i}(X_{s}, ℚ)]^{π_{1}(C,s)} = \{ \begin{matrix}0, for i odd, \\ 1, for i even. \end{matrix}

The case i > d reduces to the previous case with the aid of the strong Lefschetz theorem.

We see that for i ≠ d,

H^{0}(C, R^{i}π_{∗}ℚ) = \{ \begin{matrix}(0), for i odd, \\ ℚ · h^{∪ i/2} for i even, \end{matrix}

H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ) is trivial or else is generated by an algebraic class h^{∪ i/2} ⊗ h^{∪ i/2}. It remains to apply on Theorem 10.1.

Finally, for i = d we have:

[H^{d}(X_{s}, ℚ) ⊗ \overline{ℚ}]^{g} = ∧^{d}(V ⊕ V^{∗})^{g} = ∧^{d}(V ⊕ V^{∗})^{sl(V)} =

⨁_{m=1}^{d−1}[∧^{m}V ⊗ ∧^{d−m}V^{∗}]^{sl(V)} ⨁[∧^{0}V ⊗ ∧^{d}V^{∗}] ⨁[∧^{d}V ⊗ ∧^{0}V^{∗}] =

⨁_{m=1}^{d−1} Hom_{sl(V)}(∧^{d−m}V, ∧^{m}V) ⨁[∧^{0}V ⊗ ∧^{d}V^{∗}] ⨁[∧^{d}V ⊗ ∧^{0}V^{∗}],

and

dim [H^{d}(X_{s}, ℚ)]^{G} = dim [H^{d}(X_{s}, ℚ)]^{π_{1}(C,s)} = \{ \begin{matrix}2, for d odd, \\ 3, for d even. \end{matrix}

Theorem 11.6 is proved.

11.7. Corollary. Let

Ex(4) = {9, 10, 15, 16, 21, 25, 27, 28, 35, 36, 45, 49,
55, 56, 64, 66, 78, 81, 84, 91, 100, … }

be the set of exceptional numbers in the sense of [31] for simple Abelian varieties of type IV by Albert’s classification. If End_{C}(X)⊗ℚ and End(X_{s})⊗ℚ are imaginary quadratic extensions of ℚ, where X_{s} is some geometric fiber of a complex Abelian scheme π : X → C of relative dimension d ∉ Ex(4) over a smooth projective curve C, then for any i ≠ d, the isomorphism

Λ^{d+1−i} : H^{2d+2−i}(X, ℚ) ˜→ H^{i}(X, ℚ)

is algebraic.

Indeed, it is known that in the case under consideration, Hg(X_{0\overline{η}}) is a reductive algebraic ℚ-group with a semisimple part [Hg(X_{0\overline{η}})]^{ss} of type A_{d−1} ([31], Theorem 1.1).

11.8. Theorem. Let π : X → C be an Abelian scheme over a smooth projective curve C. Assume that some geometric fiber X_{s} is a simple Abelian variety of prime dimension d.

Then for any i ≠ d, the isomorphism

Λ^{d+1−i} : H^{2d+2−i}(X, ℚ) ˜→ H^{i}(X, ℚ)

is algebraic.

Proof. In virtue of Remark 2.7, we may assume that an Abelian scheme π : X → C contains no nontrivial constant Abelian subschemes. Since X_{s} is a simple Abelian variety, the fiber X_{0\overline{η}} is a simple Abelian variety of prime dimension d.

It is well known that End(X_{0\overline{η}}) ⊗ ℚ is isomorphic to a totally real extension K_{0} of ℚ of degree [K_{0} : ℚ] ∈ {1, d} or else to an imaginary quadratic extension K of a totally real field K_{0} of degree [K : ℚ] ∈ {2, 2d} ([29], 6.1).

Assume that End(X_{0\overline{η}}) ⊗ ℚ = K_{0}. If K_{0} = ℚ, then we have a situation of Corollaries 11.3 – 11.4, and B(X) holds.

If K_{0} ≠ ℚ, then K_{0} is a totally real field of degree [K_{0} : ℚ] = d. Since the monodromy representation

ρ : π_{1}(C, s) → GL(H^{1}(X_{s}, ℚ))

is semisimple ([6], 4.2.6), the ℚ-algebra H = End(R^{1}π_{∗}ℚ) is isomorphic to a direct sum of algebras of the form M_{n_{α}}(D_{α}), where D_{α} is a division algebra over ℚ. On the other hand, the center Z of the algebra H is of Hodge type (0, 0) and hence is contained in the center of the division algebra End_{C}(X)⊗ℚ ↪ End(X_{0\overline{η}})⊗ℚ = K_{0} ([6], 4.4.7). Therefore H ˜→ M_{n}(D), where D is a central division algebra over a totally real field Z ↪ K_{0}. Let e = dim_{ℚ} Z and f^{2} = dim_{Z} D. The space H^{1}(X_{s}, ℚ) is decomposed into a direct sum of irreducible M_{n}(D)-modules each of which is a linear space over ℚ of dimension enf^{2} (this is a well-known fact from the theory of simple central algebras). Hence

enf^{2}|2d.  (11.1)

We claim that

End_{C}(X) ⊗ ℚ ˜→ End(R^{1}π_{∗}ℚ).  (11.2)

The case d ≤ 3 is considered in ([6], 4.4.13). Hence we may assume that d ≥ 5. From the relation (11.1) it follows that f = 1. Consequently, D = Z, and for any embedding φ : Z ↪ ℝ, the algebra H ⊗_{Z} ℝ_{(φ)} is a matrix algebra over ℝ. Therefore (11.2) follows from ([6], 4.4.11).

Replacing X → C by X ×_{C} C′ → C′ for some finite covering C′ → C, we may assume that

K_{0} = End_{C}(X) ⊗ ℚ ˜→ End(R^{1}π_{∗}ℚ).

In this situation,

K_{0} = End(R^{1}π_{∗}ℚ) = End_{π_{1}(C,s)}(H^{1}(X_{s}, ℚ)) = End_{G}(H^{1}(X_{s}, ℚ)),

and by the Schur lemma,

H^{1}(X_{s}, ℚ) ⊗ \overline{ℚ} ˜→ V_{1} ⊕ ⋯ ⊕ V_{d}

is a direct sum of mutually nonisomorphic [Lie G] ⊗ \overline{ℚ}-modules of degree 2 over \overline{ℚ}. Therefore [Lie G] ⊗ \overline{ℚ} is a semisimple Lie algebra of type A_{1} × ⋯ × A_{1} = A_{1}^{d}.

On the other hand, it is known that Hg(X_{0\overline{η}}) is a semisimple ℚ-group of type A_{1} × ⋯ × A_{1} = A_{1}^{d}, so the geometric analog of the Mumford – Tate conjecture [23] holds. In particular, the Hodge conjecture for X_{0\overline{η}} is equivalent to the geometric analog of the Tate conjecture for X_{η}, where X_{η} is the generic fiber of π. Since the Hodge conjecture holds for all simple Abelian varieties of prime dimension (moreover,

H^{2i}(X_{0\overline{η}}, ℚ) ∩ H^{i,i}(X_{0\overline{η}}, ℂ)

is spanned by classes of intersections of divisors [30]), it is easy to verify that H^{0}(C, R^{i}π_{∗}ℚ) is trivial for odd i; for i even, it is generated by cup products of elements of H^{0}(C, R^{2}π_{∗}ℚ). Moreover, elements of H^{0}(C, R^{2}π_{∗}ℚ) are algebraic (they come from classes of divisors on the generic fiber). Therefore H^{0}(C, R^{i}π_{∗}ℚ)⊗ H^{0}(C, R^{i}π_{∗}ℚ) is generated by algebraic classes, and the conjecture B(X) holds.

If K is an imaginary quadratic extension of ℚ, then we have the situation of Corollary 11.7, and for any i ≠ d, the isomorphism

Λ^{d+1−i} : H^{2d+2−i}(X, ℚ) ˜→ H^{i}(X, ℚ)

is algebraic.

If [K : ℚ] = 2d, then X_{0\overline{η}} is an Abelian variety of CM-type, the Hodge group Hg(X_{0\overline{η}}) is a torus [24], so the relation G ↪ Hg(X_{0\overline{η}}) and semisimplicity of G imply the relation G = 1. It follows that an Abelian scheme π : X → C is potentially constant, so B(X) holds. Theorem 11.8 is proved.

§ 12. Variations

12.1. Theorem. Assume that for some natural number i ≥ 2, there exists an algebraic cycle Z_{i} with rational coefficients of codimension i on X × X such that

j_{∗}v^{i} = cl_{X×X}(Z_{i}) ∪ j_{∗}1.

Then the algebraic class [cl_{X×X}(Z_{i})]_{0i0i} induces an algebraic isomorphism

H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{0}(C, R^{i}π_{∗}ℚ).

Proof. For u = cl_{X×X}(Z_{i}), we have the decomposition u = ∑ u_{pqrt}, where u_{pqrt} ∈ H^{p}(C, R^{q}π_{∗}ℚ)⊗H^{r}(C, R^{t}π_{∗}ℚ). In virtue of Lemma 3.3, u_{0i0i} is an algebraic class.

We claim that u_{0i0i} induces an algebraic isomorphism

H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{0}(C, R^{i}π_{∗}ℚ).

Indeed, consider the composition

H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→^{pr_{1}^{∗}} H^{2}(C, R^{2d−i}π_{∗}ℚ) ⨂ H^{0}(C, ℚ) − − − →^{∪ u_{0i0i}}

H^{2}(C, R^{2d}π_{∗}ℚ) ⨂ H^{0}(C, R^{i}π_{∗}ℚ) = H^{2d+2}(X, ℚ) ⨂ H^{0}(C, R^{i}π_{∗}ℚ) ˜→^{pr_{2∗}}

H^{0}(C, R^{i}π_{∗}ℚ).

In virtue of (6.2), an algebraic class j_{∗}v^{i} induces an algebraic isomorphism

H^{2}(C, R^{2d−i}π_{∗}ℚ) ˜→ H^{2}(C, R^{i}π_{∗}ℚ).  (12.1)

Therefore (7.3) gives the relations

dim H^{2}(C, R^{2d−i}π_{∗}ℚ) = dim H^{2}(C, R^{i}π_{∗}ℚ) = dim H^{0}(C, R^{i}π_{∗}ℚ),

and it remains to prove the injectivity of the map

H^{2}(C, R^{2d−i}π_{∗}ℚ) ⨂ H^{0}(C, ℚ) − − − →^{∪ u_{0i0i}}

H^{2}(C, R^{2d}π_{∗}ℚ) ⨂ H^{0}(C, R^{i}π_{∗}ℚ).  (12.2)

We have

cl_{X×X}(Z_{i}) = u = u_{0i0i} + ∑_{p≠0} u_{pqrt} + ∑_{r≠0} u_{0qrt} + ∑_{q≠i} u_{0q0t}.

Since ∑_{p≠0} u_{pqrt} annihilates H^{2}(C, R^{2d−i}π_{∗}ℚ) ⨂ H^{0}(C, ℚ), an isomorphism (12.1) is determined by the correspondence

[u_{0i0i} + ∑_{r≠0} u_{0qrt} + ∑_{q≠i} u_{0q0t}] ∪ j_{∗}1.

By Corollary 8.2,

j_{∗}1 = cl_{X×X}(X ×_{C} X) ∈

[H^{0}(C, ℚ) ⊗ H^{2}(C, ℚ)] ⨁[H^{2}(C, ℚ) ⊗ H^{0}(C, ℚ)] ⨁[H^{1}(C, ℚ) ⊗ H^{1}(C, ℚ)].

Therefore ∑_{r≠0} u_{0qrt} ∪ j_{∗}1 annihilates H^{2}(C, R^{2d−i}π_{∗}ℚ) ⨂ H^{0}(C, ℚ), and an isomorphism (12.1) is determined by the correspondence

[u_{0i0i} + ∑_{q≠i} u_{0q0t}] ∪ j_{∗}1.

Finally,

pr_{2∗} ([H^{2}(C, R^{2d−i}π_{∗}ℚ) ⨂ H^{0}(C, ℚ)] ∪ [∑_{q≠i} u_{0q0t} ∪ j_{∗}1]) = (0),

therefore an isomorphism (12.1) is determined by the algebraic correspondence u_{0i0i} ∪ j_{∗}1. In particular, the map

H^{2}(C, R^{2d−i}π_{∗}ℚ) ⨂ H^{0}(C, ℚ) − − − →^{∪ u_{0i0i}∪ j_{∗}1}

H^{2}(C, R^{2d}π_{∗}ℚ) ⨂ H^{2}(C, R^{i}π_{∗}ℚ)

is injective. Therefore the map (12.2) is injective. Theorem 12.1 is proved.

12.2. Under the assumption of Theorem 12.1, the diagram (7.4) can be transformed into a new diagram

\begin{matrix}H^{0}(C, R^{2d+2−i}π_{∗}ℚ) & ⨁ & H^{1}(C, R^{2d+1−i}π_{∗}ℚ) & ⨁ & H^{2}(C, R^{2d−i}π_{∗}ℚ) \\ ^{≃} ↓^{φ_{i}} & & ^{≃} ↓^{[j_{∗}v^{i−1}]^{∗}} & & ^{≃} ↓^{[u_{0i0i}]^{∗}} \\ H^{2}(C, R^{i−2}π_{∗}ℚ) & ⨁ & H^{1}(C, R^{i−1}π_{∗}ℚ) & ⨁ & H^{0}(C, R^{i}π_{∗}ℚ), \end{matrix} (12.3)

where u = cl_{X×X}(Z_{i}), and all vertical arrows are algebraic isomorphisms. We arrive to the following result:

12.3. Theorem. Let π : X → C be a principally polarized complex Abelian scheme of relative dimension d over a smooth projective curve C. Assume that for some natural number i ≥ 2, there exists an algebraic cycle Z_{i} with rational coefficients of codimension i on X × X such that

j_{∗}v^{i} = cl_{X×X}(Z_{i}) ∪ j_{∗}1.

Let u = cl_{X×X}(Z_{i}) and let u_{0i0i} ∈ H^{0}(C, R^{i}π_{∗}ℚ) ⊗ H^{0}(C, R^{i}π_{∗}ℚ) be the corresponding component of u.

Then the algebraic class

[Δ_{∗}cl_{X}(X_{s})] ∘ [j_{∗}v^{i−2}] + j_{∗}v^{i−1} + u_{0i0i}

determines an algebraic isomorphism

H^{2d+2−i}(X, ℚ) ˜→ H^{i}(X, ℚ).

In particular, the conjecture B^{i}(X) holds.

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Vladimir State University, ul. Gorkogo, 87, Vladimir, 600000, Russia

E-mail address: tankeev-m2@vpti.vladimir.su