The middle of the diagonal of a surface with p_{g} = 0 and q = 1

V. Guletskiǐ

Institute of Mathematics, Surganova str. 11, Minsk 220072, Belarus e-mail: guletskii@im.bas-net.by

April 17, 2000

Abstract We prove that if X is a smooth projective complex surface with the invariants p_{g} = 0 and q = 1 then the middle Murre projector π_{2} (see [Mu90] or [Sch94] for the definition of π_{2}) can be generated by two natural divisors on X whose cohomology classes form a basis for the second cohomology group H^{2}(X, ℚ). As a consequence, this provides a second, in fact, Chow-motivic, proof of the triviality of the Albanese kernel for surfaces with p_{g} = 0 and q = 1 (the first proof was made in [BKL76]).

1 Introduction

All varieties will be smooth projective and defined over the field of complex numbers ℂ. If Y is a variety then let A^{t}(Y) be the Chow-group of codimension t cycles on Y with coefficients in the field of rational numbers ℚ and let A(Y) = ⊕_{t}A^{t}(Y) be the Chow-ring of Y.

For any smooth projective surface X by A_{0}^{2}(X) denote the ℚ-subspace in A^{2}(X) consisting of zero-cycles of degree zero on X. Also let A(X) be the Albanese variety for the surface X. By definition, the Albanese kernel T (X) is the kernel of the surjective homomorphism

A_{0}^{2}(X) ⟶ A(X) ⊗ ℚ

induced by the Albanese mapping X → A(X). The Bloch conjecture predicts that if p_{g} = 0, then T (X) = 0. Here p_{g} = dim Γ(X, Ω_{X}^{2}) is the dimension of the space of global holomorphic 2-forms on X.

In [BKL76] Bloch, Kas and Lieberman proved that if p_{g} = 0 and q = 1, where q is the irregularity of X, then T (X) = 0. The purpose of this paper is to prove that the middle Murre projector π_{2} for a such surface X (see [Mu90] or [Sch94] for the definition of π_{2}) is rationally equivalent to a sum of external products of two natural divisors on X whose cohomology classes form a basis for the cohomology group H^{2}(X, ℚ). As a consequence this provides a second, in fact, Chow-motovic, proof of the triviality of T (X) for surfaces with p_{g} = 0 and q = 1.

Recall the definition of the category ℳ of Chow motives over ℂ with coefficients in ℚ. Objects in ℳ are pairs (X, p) where X is variety and p is a projector of X, that is a class in A(X × X), such that p ∘ p = p in the sense of compositions of correspondences. If M = (X, p) and N = (Y, q) are two motives then a morphism f : M → N is a class f ∈ A(X × Y), such that q ∘ f = f ∘ p. If Δ_{X} is a diagonal of a variety X then Δ_{X} is a projector of X. The motive h(X) = (X, Δ_{X}) is called the motive of the variety X. At the same time Δ_{X} can be viewed as an identity morphism 1_{M} : M → M for any motive M = (X, p).

Let X be a smooth projective surface over ℂ. According to the Murre’s results (see [Mu90] and [Mu93]), we have that there exist the projectors π_{0}, π_{1}, π_{2}, π_{3} and π_{4} of the surface X, such that

Δ_{X} = π_{0} + π_{1} + π_{2} + π_{3} + π_{4}

in A^{2}(X × X) and the cohomology class of the correspondence π_{j} coincides with the (4 − j, j) Künneth component of the cohomology class of the diagonal Δ_{X} in the group H^{4}(X × X, ℚ). The projectors π_{j} are pairwise orthogonal, whence

h(X) = h^{0}(X) ⊕ h^{1}(X) ⊕ h^{2}(X) ⊕ h^{3}(X) ⊕ h^{4}(X)

in the additive category ℳ, where h^{j}(X) = (X, π_{j}).

This decomposition of Δ_{X} allows us to study zero cycles on X following the first Bloch’s lecture in [Bloch80]. Indeed, correspondences act on Chow-groups: for any two varieties X and Y, if ν ∈ A^{d}(X × Y), where d = dim(X), and if ξ ∈ A^{p}(X), then one can define

ν(ξ) = p_{Y} _{∗}(ν · p_{X}^{∗}(ξ)) .

Here p_{Y} and p_{X} are projections, p_{Y} _{∗} and p_{X}^{∗} are direct and inverse images respectively. In particular we have the homomorphisms

π_{j} : A^{2}(X) ⟶ A^{2}(X)

on zero cycles induced by the projectors π_{j}. Murre proved (see [Mu90], [Mu93] or [J94]) that π_{0} and π_{1} operate as zero on the whole Chow-group A^{2}(X), the kernel ker(π_{4}) coincides with the group A_{0}^{2}(X) and ker(π_{3} |_{A_{0}^{2}(X)}) coincides with the Albanese kernel T (X). Consequently the action of the middle Murre projector π_{2} on T (X) is the same as the identical action of the diagonal Δ_{X} on T (X). Therefore, if π_{2} acts as zero on T (X), then T (X) = 0.

Now let X be a smooth projective surface with p_{g} = 0 and q = 1. Since q = 1, it follows that the Albanese variety A(X) is an elliptic curve A over ℂ. Let F be a general fiber of the Albanese mapping

α : X ⟶ A .

According to the classification of surfaces (see [Sh65], chapter IV) we have that, since p_{g} = 0 and q = 1, it follows that X ≅ (E × F)/G where E is a curve and G is a finite abelian group acting on E and on F. The action of G on E × F is diagonal free. Let

η : E × F ⟶ X

be the corresponding quotient map. The quotient curve E/G coincides with the elliptic curve A and the regular map X → E/G = A induced by the projection p_{E} : E × F → E coincides with the Albanese mapping α.

There are two different cases: either genus of the curve F is greater than one or F is a curve of genus one. In the first case E is an elliptic curve, i.e. a curve of genus one with fixed zero-point e ∈ E. The group G is a finite subgroup in E acting on E by translations. Therefore the quotient map η_{E} : E ⟶ A is unramified finite covering. We take o = η_{E}(e) as a zero point on A. Then η_{E} is in fact an isogeny of elliptic curves. We fix any point f ∈ F and take x_{0} = η(e × f) as a fixed point on X. If genus of the general fiber F is one then the quotient morphism η_{E} can be ramified. In this case we choose a zero-point o ∈ A in such a way that any point e ∈ η_{E}^{−1}(o) is unramified over o. Let e be a fixed point on E lying in η_{E}^{−1}(o) and let f be a fixed point on F. Then let x_{0} = η(e, f) be a fixed point on the surface X. So in both cases e is unramified point of the quotient map η_{E}. Also note that p_{g} = 0 and q = 1 imply that genus of the curve F/G is zero (see [Sh65], ch. IV, § 8), so that F/G = ℙ^{1}.

Since p_{q} = 0 and q = 1, it can be shown that the second Betti number of X is 2. Let n be the order of the group G. Consider the divisors

D_{1} = η_{∗}[e × F]  and  D_{2} = \frac{1}{n} η_{∗}[E × f]

on X. Its cohomology classes form a basis for the ℚ-vector space H^{2}(X, ℚ). The Poincaré dual basis is formed by the cohomology classes of divisors D_{2} and D_{1} (the same divisors but in inverse order).

Theorem 1.1 Let X be a smooth projective surface with p_{g} = 0 and q = 1. Let D_{1} and D_{2} be the above divisors on X. Then the middle Murre projector π_{2} of the surface X is rationally equivalent (with coefficients in ℚ) to the correspondence D_{1} × D_{2} + D_{2} × D_{1}.

Now one can imitate the proof of Proposition 1.11 in [Bloch80] and deduce the triviality of T (X) in the Chow-motivic way. Namely, we know that the middle projector π_{2} acts on T (X) identically. At the same time, one can move the divisors D_{1} and D_{2} from points on X and obtain that π_{2} acts as zero on T (X). Hence T (X) = 0.

For any cycle Z on X of codimension t its class in A^{t}(X) will be denoted by [Z]. Let f : Y → X be a morphism of varieties and d = dim(X). Then the graph Γ_{f} is either a closed subvariety {(f(y), y) ∈ X × Y | y ∈ Y } or its class in A^{d}(X × Y) (depending from a context). The diagonal Δ_{X} of a variety X can be viewed as a graph of the identical map 1_{X} : X → X. All computations with algebraic cycles are based on the book [F84].

Acknowledgements. The author would like to thank Ivan Panin and Claudio Pedrini for useful discussions and comments. The author gratefully acknowledge the support of TMR ERB FMRX CT-97-0107 and the hospitality of the University of Bielefeld.

2 Linear section

Let Ỹ be a variety and let G be a finite group of order n acting on Ỹ. Let Y = Ỹ /G be the quotient variety of Ỹ by G and let τ : Ỹ → Y be the corresponding finite covering. It is well known that the composition τ_{∗}τ^{∗} : A^{t}(Y) → A^{t}(Y) coincides with the multiplication-by-n homomorphism for any codimesion t. By

N_{Ỹ /G} : A^{t}(Ỹ) → A^{t}(Ỹ)

denote the norm-homomorphism

w ↦ ∑_{g∈G} g(w) .

Lemma 2.1 The composition τ^{∗}τ_{∗} : A^{t}(Ỹ) → A^{t}(Ỹ) coincides with the norm homomorphism N_{Ỹ /G} : A^{t}(Ỹ) → A^{t}(Ỹ).

Proof. This lemma is a well known fact and it can be easily deduced, for example, from the considerations in [B62]. Recall the proof for the convenience of the reader. Let V be a prime cycle on Ỹ and W = τ(V). Then τ_{∗}(V) = dW where d = [ℂ(V) : ℂ(W)] is the degree of V over W. Let V_{1}, …, V_{s} be the distinct irreducible components of τ^{−1}(W). If l is a number of elements in the inertia group {g ∈ G | g(y) = y for all y ∈ V }, then τ^{∗}(W) = l ∑_{j} V_{j}. Consequently, τ^{∗}τ_{∗}(V) = dl ∑_{j} V_{j}. But sdl = n, where n is the order of G, and dl = \frac{n}{s} is the order of the splitting group \{g ∈ G| g(V) = V \} (loc. cit.). Thus we obtain that τ^{∗}τ_{∗}(V) = \frac{n}{s} ∑_{j} V_{j} = N_{Ỹ /G}(V).

Let X be a surface with p_{g} = 0 and q = 1 and let X = (E × F)/G be the presentation of X described in Introduction. Since a point on an algebraic curve is an ample divisor, it follows that for any g ∈ G the divisor

[g(e) × F] + [E × g(f)]

is ample on E × F. Then

η^{∗}η_{∗}([e × F] + [E × f]) = N_{(E×F)/G}([e × F] + [E × f]) = ∑_{g∈G}([g(e) × F] + [E × g(f)])

is an ample divisor on E × F. Since η is a finite surjective morphism of varieties, it follows that the divisor

η_{∗}([e × F] + [E × f])

is ample on X (see [H70], p. 25). Hence there exists a positive integer m such that

mη_{∗}([e × F] + [E × f])

is very ample.

Fix an embedding X ↪ ℙ^{N} induced by this very ample divisor. Using Bertini theorem one can choose a hyperplane section C of X, such that the following properties hold: (i) C is a smooth projective curve; (ii) x_{0} ∈ C; (iii) the curve C has a positive genus. Let i : C ↪ X be the corresponding closed embedding of the curve C into X.

For any variety Y denote by P(Y) and A(Y) the Picard and Albanese varieties of Y respectively. In fact we have the functors P and A on the category of connected varieties with fixed points. To be precise let 𝒞𝒱_{•} be the category whose objects are pairs (Y, y_{0}) where Y is a connected variety and y_{0} ∈ Y is a fixed point on Y. Morphisms in 𝒞𝒱_{•} are morphisms of varieties preserving fixed points. For short let Hom_{•}(Y, Z) = Hom_{𝒞𝒱 _{•}}((Y, y_{0}), (Z, z_{0})). If Y is an abelian variety, then, automatically, y_{0} is a zero point on Y. Now A is a functor from 𝒞𝒱_{•} into the category of abelian varieties and P is a functor from the opposite category 𝒞𝒱_{•}^{opp} into the category of abelian varieties. In particular, for the closed embedding i : C → X there exist the morphisms P(i) : P(X) → P(C) and A(i) : A(C) → A(X).

The Picard variety is an object presenting the Picard functor. Hence P(X) is defined up to an isomorphism. Since A is an elliptic curve, it follows that it isomorphic to its dual. Therefore we can take P(X) = A = A(X). Identifying P(C) with A(C), J = P(C) = A(C), we get an isogeny

A = P(X) ⟶^{P(i)} J = P(C) = A(C) ⟶^{A(i)} A(X) = A

(see [Sch94]).

Lemma 2.2 Let n be the order of the group G and let m be the above natural number. Then the isogeny A(i) ∘ P(i) coincides with the multiplication-by-mn homomorphism of the abelian variety A.

Proof. Let a be a point on A and let F_{a} be a set-theoretic fiber of the morphism α over the point a. Then for any point t ∈ η_{E}^{−1}(a) we have that F_{a} = η(t × F). Suppose that η_{E} : E → A is unramified at a ∈ A. Then the restriction of the morphism η on the subvariety t × F ↪ E × F is a closed embedding. Hence F_{a} = η(t × F) ≅ F for any t ∈ η_{E}^{−1}(a). Moreover,

α^{∗}[a] = \frac{1}{n} nα^{∗}[a] = \frac{1}{n} η_{∗}η^{∗}α^{∗}[a] = \frac{1}{n} η_{∗}p_{E}^{∗}η_{E}^{∗}[a] =

= \frac{1}{n} η_{∗} ( ∑_{t∈η_{E}^{−1}(a)}[t × F]) = \frac{1}{n} ∑_{t∈η_{E}^{−1}(a)} η_{∗}[t × F] = \frac{1}{n} ∑_{t∈η_{E}^{−1}(a)}[F_{a}] =

= \frac{1}{n} n[F_{a}] = [F_{a}] = η_{∗}[t × F] .

So we see that the fiber F_{a} is not multiply and α^{∗}[a] = η_{∗}[t × F].

By a construction,

[C] = mη_{∗}([E × f + e × F]) .

Then

[C] · [F_{a}] = [C] · η_{∗}[t × F] = (mη_{∗}[E × f] + mη_{∗}[e × F]) · η_{∗}[t × F] =
= mη_{∗}[E × f] · η_{∗}[t × F] + mη_{∗}[e × F] · η_{∗}[t × F] = mη_{∗}[E × f] · η_{∗}[t × F] + 0 =
= mη_{∗}(η^{∗}η_{∗}[E × f] · [t × F]) = mη_{∗}(N_{E×F/G}[E × f] · [t × F]) =

= mη_{∗} (∑_{g∈G}[E × g(f)] · [t × F]) = mn .

For any connected variety Y by Pic^{0}(Y) denote the group of Weil divisors on Y algebraically equivalent to zero. Consider the commutative diagram

\xymatrix{ & Pic^{0}(X) \ar[r]^{i^{∗}} & Pic^{0}(C) & C \ar[dl] \ar[r]^{i} & X \ar[dl]_{α} \\ Pic^{0}(A) \ar[r]^{s}_{≅} \ar[ur]^{α^{∗}} & A \ar[u]^{≅} \ar[r]^{P(i)} & J \ar[u]^{≅} \ar[r]^{A(i)} \ar[u]^{≅} & A}

Here s is an isomorphism defined by the formula s^{−1}(a) = [a − o] for any point a ∈ A, C → J is the Albanese mapping for the curve C. The vertical isomorphisms are isomorphisms of functors presentations.

Let a be a point on A such that η_{E} is unramified at a. The fibre F_{a} is a prime cycle on X and [F_{a}] · [F_{a}] = 0. Hence, by Nakai-Moishezon criterion, we have that C is not a fiber of α because [C]·[C] = 2nm^{2} > 0. It follows that the restriction α|_{C}: C → A is a surjective map. Let t ∈ η_{E}^{−1}(a). Note that e and t are unramified over o and a respectively. Therefore the multiplicity of fibers F_{a} and F_{o} are equal to 1. Then

i^{∗}(α^{∗}(s^{−1}(a))) = i^{∗}α^{∗}([a − o]) = i^{∗}(α^{∗}[a]) − i^{∗}(α^{∗}[o]) =
= i^{∗}[F_{a}] − i^{∗}[F_{o}] = [C] · [F_{a}] − [C] · [F_{o}] .

The first summand is

[C] · [F_{a}] = [C · η_{∗}(t × F)] = d_{1}v_{1} + … + d_{z}v_{z} ,

where v_{1}, …, v_{z} are the points of intersection of C with F_{a} = η(t × F) and d_{j} is the multiplicity of this intersection at the point v_{j}. The second summand is

[C] · [F_{o}] = [C · η_{∗}(e × F)] = t_{1}w_{1} + … + t_{h}w_{h} ,

where w_{1}, …, w_{h} are the points of intersection of C with F_{o} = η(e × F) and t_{l} is its multiplicity at the point w_{l}. Then we have

(A(i) ∘ P(i))(a) = ∑_{j=1}^{z} d_{j}(α ∘ i)(v_{j}) + ∑_{l=1}^{h} t_{l}(α ∘ i)(w_{l}) .

Since any point i(w_{l}) lies in the fiber F_{o}, the point α(i(w_{l})) is equal to zero. Therefore,

(A(i) ∘ P(i))(a) = ∑_{j=1}^{z} d_{j}(α ∘ i)(v_{j}) .

Since i(v_{j}) ∈ F_{a}, it follows that α(i(v_{j})) = a for any j. Then

(A(i) ∘ P(i))(q) = ∑_{j=1}^{z} d_{j}a .

But the number ∑_{j=1}^{z} d_{j} coincides with the intersection number [C] · [F_{a}] = mn. Consequently we obtain that

(A(i) ∘ P(i))(a) = mna .

3 The Picard and Albanese projectors

Recall (see [Mu90]) that for any smooth projective surface Y the projector π_{1} is closely connected with the Picard variety P(Y) and so it is called the Picard projector of the surface Y. By definition, π_{3} is a transpose of π_{1}. The projector π_{3} is connected with the Albanese variety A(Y) and it is called the Albanese projector of Y. In this section we compute π_{1} and π_{3} for our surface X in terms of quotient structure on X.

Lemma 2.2 shows that the identical morphism 1_{A} : A → A can be viewed as an inverse isogeny for the isogeny A ⟶^{P(i)} J ⟶^{A(i)} A. Then, according to the general method of constructing of π_{1} (see [Mu90], [Sch94], [G99]), we can get the Picard projector in the form

π_{1} = \frac{1}{mn} [C × X] · ε ,

where ε is a divisor on X × X that corresponds to the identical isogeny 1_{A} : A → A. Observe that since [C] = mη_{∗}([E × f + e × F]), we have that

\frac{1}{mn} [C × X] = \frac{1}{mn} [C] × [X] = \frac{1}{n} (η_{∗}[E × f] + η_{∗}[e × F]) × [X] .

Hence one can write

π_{1} = \frac{1}{n} (η_{∗}[E × f] × [X] + η_{∗}[e × F] × [X]) · ε .  (1)

So to compute π_{1} in terms of quotient structure on the surface X we have to compute ε.

For any variety Y with fixed point y_{0} ∈ Y let P_{Y} be the Poincaré divisor on Y × P(Y). Also let α_{Y} : Y ⟶ A(Y) be the Albanese map for Y sending y_{0} into the zero point on A(Y). Then

P_{Y} = (α_{Y} × 1_{P(Y)})^{∗}P_{P(Y)}^{t} ,

where P_{P(Y)}^{t} is a transpose of P_{P(Y)} (see [G99], Section 4.2, p. 20). In particular,

P_{X} = (α × 1_{A})^{∗}P_{A}^{t} .

Hence, the transpose P_{X}^{t} of P_{X} is

P_{X}^{t} = (1_{A} × α)^{∗}P_{A} .

At the same time ε = (α × 1_{X})^{∗}(P_{X}^{t}). To show this we need some more notation. For any connected variety Y ∈ Ob(𝒞𝒱_{•}) one can consider two following functors, both from 𝒞𝒱_{•}^{opp} into the category of ℚ-vector spaces. The first one is a functor Z ↦ Hom_{•}(Z, P(Y))⊗ℚ. The second functor sends any object Z = (Z, z_{0}) ∈ Ob(𝒞𝒱_{•}) into the ℚ-space {c ∈ A^{1}(Z × Y)| c ∘ z_{0}_{∗} = 0 = y_{0}^{∗} ∘ c}. Here z_{0}_{∗} = [z_{0} × z_{0}] and y_{0}^{∗} = [y_{0} × y_{0}] are correspondences induced by the fixed points and compositions are compositions of correspondences. There exists an isomorphism of functors

ω_{Z}^{Y} : Hom_{•}(Z, P(Y)) ⊗ ℚ ⟶^{≅} {c ∈ A^{1}(Z × Y)| c ∘ z_{0}_{∗} = 0 = y_{0}^{∗} ∘ c}

(see [Sch94] or [G99], Section 3.2). For any Z ∈ Ob(𝒞𝒱_{•}), if α_{Z} : Z → A(Z) is the Albanese mapping for Z, then let Ω_{Z}^{Y} be the composition ω_{Z}^{Y} ∘ (Hom_{•}(α_{Z}, P(X)) ⊗ ℚ). In fact

Ω_{Z}^{Y} : Hom_{•}(A(Z), P(Y)) ⊗ ℚ ⟶^{≅} {c ∈ A^{1}(Z × Y)| c ∘ z_{0}_{∗} = 0 = y_{0}^{∗} ∘ c}

is an usual one-to-one correspondence between isogenies and divisors normalised by fixed points. Now applying the general method of a construction of the divisor ε in our case we have that

ε = Ω_{X}^{X}(1_{A})

(loc. cit.). Then the commutative diagram

\xymatrix{Hom_{•}(A, A) ⊗ ℚ \ar[r]^{Ω_{X}^{X}} \ar[d]_{Hom_{•}(α,A)⊗ℚ} & \{c ∈ A^{1}(X × X)| c ∘ x_{0}_{∗} = 0 = x_{0}^{∗} ∘ c\} \\ Hom_{•}(X, A) ⊗ ℚ \ar[ur]^{ω_{X}^{X}} & \{c ∈ A^{1}(A × X)| c ∘ 0_{∗} = 0 = x_{0}^{∗} ∘ c\} \ar[u]^{(α×1_{X})^{∗}} \\ Hom_{•}(A, A) ⊗ ℚ \ar[ur]^{ω_{A}^{X}} \ar[u]^{Hom_{•}(α,A)⊗ℚ}}

[figure: 1_{A}; ε; α; P_{X}^{t}; 1_{A}]

shows that

ε = (α × 1_{X})^{∗}(P_{X}^{t}) .

Therefore, we can write

ε = (α × 1_{X})^{∗}(1_{A} × α)^{∗}(P_{A}) = (α × α)^{∗}(P_{A}) .

But it is very well known that the Poincaré divisor for the curve A is

P_{A} = Δ_{A} − o × A − A × o .

Hence,

ε = (α × α)^{∗}(P_{A}) = (α × α)^{∗}(Δ_{A}) − η_{∗}[o × F] × [X] − [X] × η_{∗}[o × F] .

Now we will calculate the divisor (α × α)^{∗}(P_{A}). Let

l : E × E × F × F → E × F × E × F
(e_{1}, e_{2}, f_{1}, f_{2}) ↦ (e_{1}, f_{1}, e_{2}, f_{2})

be the isomorphism transposing points in the second and third factors. Also for any g ∈ G let Γ_{g} be the graph of the morphism E → E, q ↦ g(q). Consider the commutative square

[figure: E × F × E × F; η×η; l^{−1}; X × X; α×α; E × E × F × F; p_{E×E}; A × A; E × E; η_{E}×η_{E}; E/G × E/G]

Since

(η × η)_{∗}(η × η)^{∗}(α × α)^{∗}(Δ_{A}) = n^{2}(α × α)^{∗}(Δ_{A}) ,

one can write

(α × α)^{∗}(Δ_{A}) = \frac{1}{n^{2}} (η × η)_{∗}(η × η)^{∗}(α × α)^{∗}(Δ_{A}) = \frac{1}{n^{2}} (η × η)_{∗}(l^{−1})^{∗}p_{E×E}^{∗}(η_{E} × η_{E})^{∗}(Δ_{A}) =

= \frac{1}{n^{2}} (η × η)_{∗}(l^{−1})^{∗}p_{E×E}^{∗} [∑_{g∈G} Γ_{g}] = \frac{1}{n^{2}} (η × η)_{∗}l_{∗} ∑_{g∈G}[Γ_{g} × F × F] .

But for any g ∈ G we have that

(η × η)_{∗}l_{∗}[Γ_{g} × F × F] = (η × η)_{∗}l_{∗}[Δ_{E} × F × F] .

Therefore,

(α × α)^{∗}(Δ_{A}) = \frac{1}{n^{2}} (η × η)_{∗}l_{∗} ∑_{g∈G}[Γ_{g} × F × F] = \frac{1}{n} (η × η)_{∗}l_{∗}[Δ_{E} × F × F] .

So we have that

ε = \frac{1}{n} (η × η)_{∗}l_{∗}[Δ_{E} × F × F] − η_{∗}[e × F] × [X] − [X] × η_{∗}[e × F] .

Now one can substitute this formula for ε into the formula (1) and obtain that the projector π_{1} is the intersection of the correspondences

\frac{1}{n} (η_{∗}[E × f] × [X] + η_{∗}[e × F] × [X])

and

\frac{1}{n} (η × η)_{∗}l_{∗}(Δ_{E} × [F × F]) − η_{∗}[e × F] × [X] − [X] × η_{∗}[e × F] .

Intersecting summands we have:

π_{1} = \frac{1}{n} (η_{∗}[E × f] × [X]) · \frac{1}{n} (η × η)_{∗}l_{∗}[Δ_{E} × F × F]

−\frac{1}{n} (η_{∗}[E × f] × [X]) · (η_{∗}[e × F] × [X])

−\frac{1}{n} (η_{∗}[E × f] × [X]) · ([X] × η_{∗}[e × F])

+\frac{1}{n} (η_{∗}[e × F] × [X]) · \frac{1}{n} (η × η)_{∗}l_{∗}[Δ_{E} × F × F]

−\frac{1}{n} (η_{∗}[e × F] × [X]) · (η_{∗}[e × F] × [X])

−\frac{1}{n} (η_{∗}[e × F] × [X]) · ([X] × η_{∗}[e × F]) =

= \frac{1}{n^{2}} (η × η)_{∗}[E × f × E × F]) · \frac{1}{n} (η × η)_{∗}l_{∗}[Δ_{E} × F × F]

−\frac{1}{n^{2}} (η × η)_{∗}[E × f × E × F]) · \frac{1}{n} (η × η)_{∗}[e × F × E × F])

−\frac{1}{n^{2}} (η × η)_{∗}[E × f × E × F]) · \frac{1}{n} (η × η)_{∗}[E × F × e × F])

+\frac{1}{n^{2}} (η × η)_{∗}[e × F × E × F]) · \frac{1}{n} (η × η)_{∗}l_{∗}[Δ_{E} × F × F]

−\frac{1}{n^{2}} (η × η)_{∗}[e × F × E × F]) · \frac{1}{n} (η × η)_{∗}[e × F × E × F])

−\frac{1}{n^{2}} (η × η)_{∗}[e × F × E × F]) · \frac{1}{n} (η × η)_{∗}[E × F × e × F]) =

= \frac{1}{n^{3}} (η × η)_{∗}( N_{(E×F ×E×F)/(G×G)}[E × f × E × F] · l_{∗}[Δ_{E} × F × F]

− N_{(E×F ×E×F)/(G×G)}[E × f × E × F] · [e × F × E × F]

− N_{(E×F ×E×F)/(G×G)}[E × f × E × F] · [E × F × e × F]

+ N_{(E×F ×E×F)/(G×G)}[e × F × E × F] · l_{∗}[Δ_{E} × F × F]

− N_{(E×F ×E×F)/(G×G)}[e × F × E × F] · [e × F × E × F]

− N_{(E×F ×E×F)/(G×G)}[e × F × E × F] · [E × F × e × F] ) =

= \frac{1}{n^{3}} (η×η)_{∗} (n ∑_{g∈G}[E × g(f) × E × F] · l_{∗}[Δ_{E} × F × F] − n ∑_{g∈G}[E × g(f) × E × F] · [e × F × E × F]−

−n ∑_{g∈G}[E × g(f) × E × F] · [E × F × e × F] + n ∑_{g∈G}[g(e) × F × E × F] · l_{∗}[Δ_{E} × F × F]−

−n ∑_{g∈G}[g(e) × F × E × F] · [e × F × E × F] − n ∑_{g∈G}[g(e) × F × E × F] · [E × F × e × F]) =

= \frac{1}{n^{2}} (η×η)_{∗}l_{∗} (∑_{g∈G}[E × E × g(f) × F] · [Δ_{E} × F × F] − ∑_{g∈G}[E × E × g(f) × F] · [e × E × F × F]−

− ∑_{g∈G}[E × E × g(f) × F] · [E × e × F × F] + ∑_{g∈G}[g(e) × E × F × F] · [Δ_{E} × F × F]−

− ∑_{g∈G}[g(e) × E × F × F] · [e × E × F × F] − ∑_{g∈G}[g(e) × E × F × F] · [E × e × F × F]) .

Now one can simplify every summand in the obtained expression for the correspondence π_{1}. For any g ∈ G we have that

(η × η)_{∗}l_{∗}[E × E × g(f) × F] = (η × η)_{∗}l_{∗}[E × E × f × F] ,

whence

(η ×η)_{∗}l_{∗} (∑_{g∈G}[E × E × g(f) × F] · [Δ_{E} × F × F]) = n(η ×η)_{∗}l_{∗}([E ×E ×f ×F]·[Δ_{E} ×F ×F]) .

The other summands can be simplified similarly. Therefore we proceed the computation of the correspondence π_{1}:

π_{1} = \frac{1}{n} (η × η)_{∗}l_{∗}([E × E × f × F] · [Δ_{E} × F × F] − [E × E × f × F] · [e × E × F × F]−

−[E × E × f × F] · [E × e × F × F] + [e × E × F × F] · [Δ_{E} × F × F]−

−[e × E × F × F] · [e × E × F × F] − [e × E × F × F] · [E × e × F × F]) =

= \frac{1}{n} (η × η)_{∗}l_{∗}([Δ_{E} × f × F] − [e × E × f × F] − [E × e × f × F]+

+[e × e × F × F] − 0 − [e × e × F × F]) =

= \frac{1}{n} (η × η)_{∗}l_{∗}([Δ_{E} × f × F] − [e × E × f × F] − [E × e × f × F]) .

So we have that the Picard projector of the surface X can be expressed as

π_{1} = \frac{1}{n} (η × η)_{∗}l_{∗}([Δ_{E} × f × F] − [e × E × f × F] − [E × e × f × F]) .  (2)

Then

π_{3} = π_{1}^{t} = \frac{1}{n} (η × η)_{∗}l_{∗}([Δ_{E} × F × f] − [E × e × F × f] − [e × E × F × f])  (3)

is the Albanese projector of X. Also let

π_{0} = [x_{0} × X] and π_{4} = [X × x_{0}] .

As had shown in [Sch94], in the general situation the relations π_{i} ∘ π_{j} = δ_{ij} · π_{i} hold for any i, j ∈ {0, 1, 3, 4} except for, maybe, the relation π_{1} ∘ π_{3} = 0. This is the reason for the replacement π_{1} by π_{1} − \frac{1}{2}π_{1} ∘ π_{3} in loc. cit. But in our case the last relation holds as well:

Lemma 3.1 π_{1} ∘ π_{3} = 0 Proof. Let Ỹ be a variety, let G be a finite abelian group acting on Ỹ and let τ : Ỹ → Y = Ỹ /G be the quotient map. If S is a set then by S^{t} denote the product S^{t} = S × … × S with t factors. Let μ and ν be correspondences on Ỹ × Ỹ. It is easy to show that

(τ ×τ)_{∗}(μ)∘(τ ×τ)_{∗}(ν) = (τ ×τ)_{∗}q_{13}_{∗}(N_{Ỹ^{3}/G^{3}} q_{23}^{∗}(μ)·q_{12}^{∗}(ν)) = (τ ×τ)_{∗}q_{13}_{∗}(q_{23}^{∗}(μ)·N_{Ỹ^{3}/G^{3}} q_{12}^{∗}(ν))

where q_{12}, q_{23} and q_{13} are the projections

\xymatrix{ & Ỹ × Ỹ \\  & Ỹ × Ỹ × Ỹ \ar[dl]_{q_{12}} \ar[u]_{q_{13}} \ar[dr]^{q_{23}} \\ Ỹ × Ỹ &  & Ỹ × Ỹ}

In particular, we can write

π_{1} ∘ π_{3} = \frac{1}{n} (η × η)_{∗}l_{∗}[Δ_{E} × f × F − e × E × f × F − E × e × f × F] ∘

∘ \frac{1}{n} (η × η)_{∗}l_{∗}[Δ_{E} × F × f − E × e × F × f − e × E × F × f] =

= \frac{1}{n^{2}} (η × η)_{∗}q_{13}_{∗}(N_{(E×F)^{3}/G^{3}} q_{23}^{∗}l_{∗}[Δ_{E} × f × F − e × E × f × F − E × e × f × F] ·

· q_{12}^{∗}l_{∗}[Δ_{E} × F × f − E × e × F × f − e × E × F × f]) =

= \frac{1}{n^{2}} (η × η)_{∗}q_{13}_{∗} (n[E × F] × l_{∗} [∑_{g∈G} Γ_{g} × ∑_{g∈G}(g(f) × F) − n ∑_{g∈G}(g(e) × E × g(f) × F)−

− ∑_{g∈G}(E × g(e)) × ∑_{g∈G}(g(f) × F)] · l_{∗}[Δ_{E} × F × f − E × e × F × f − e × E × F × f] × [E × F]) =

= \frac{1}{n^{2}} (η × η)_{∗}q_{13}_{∗} (n[E × F] × l_{∗} [∑_{g∈G} Γ_{g} × ∑_{g∈G} g(f) × F − n ∑_{g∈G}(g(e) × E × g(f) × F)−

−E × ∑_{g∈G} g(e) × ∑_{g∈G} g(f) × F ] · l_{∗}[Δ_{E} × F × f − E × e × F × f − e × E × F × f] × [E × F]) =

= \frac{1}{n^{2}} (η × η)_{∗}q_{13}_{∗} ∑_{g∈G} ∑_{g∈G}(n[E × F] × l_{∗}[Γ_{g} × g(f) × F − g(e) × E × g(f) × F −

−E × g(e) × g(f) × F] · l_{∗}[Δ_{E} × F × f − E × e × F × f − e × E × F × f] × [E × F]) .

Now observe that we can move the divisor [f] ∈ A^{1}(F) from the finite number of points {g(f)| g ∈ G}. In other words, there exist points {f_{1}, …, f_{s}} on the curve F, such that

[f] = [∑_{j=1}^{s} r_{j}f_{j}] ,

where r_{j} are multiplicities, and f_{j} ∉ {g(f) | g ∈ G} for any j. Then the intersection of any summand in

[E × F] × l_{∗}[Γ_{g} × g(f) × F] − [E × F] × [g(e) × g(f) × E × F] − [E × F] × [E × g(f) × g(e) × F] with any summand in

l_{∗}[Δ_{E} × F × f] × [E × F] − [E × F × e × f] × [E × F] − [e × F × E × f] × [E × F]

is zero. Hence π_{1} ∘ π_{3} = 0.

So we see that π_{0}, π_{1}, π_{3} and π_{4} are pairwise orthogonal idempotences in the ring of correspondences A^{2}(X × X).

4 The middle projector

By definition,

π_{2} = Δ_{X} − π_{0} − π_{1} − π_{3} − π_{4}

(see [Mu90]). Hence, tautologically, we have the decomposition Δ_{X} = π_{0} + π_{1} + π_{2} + π_{3} + π_{4}. For any variety Y let

cl_{Y}^{t} : A^{t}(Y) ⟶ H^{2t}(Y, ℚ)

be the cycle map for codimension t cycles. As it was mentioned in Introduction,

cl_{X×X}^{2}(π_{j}) = Δ(4 − j, j)

is the (4 − j, j) Künneth component of the cohomology class of the diagonal Δ_{X}. Since the classes cl_{X}^{1}([D_{1}]), cl_{X}^{1}([D_{2}]) form a basis for H^{2}(X, ℚ) and cl_{X}^{1}([D_{2}]), cl_{X}^{1}([D_{1}]) form a Poincaré dual basis (see Introduction), it follows that

Δ(2, 2) = cl_{X×X}^{2}([D_{1} × D_{2}] + [D_{2} × D_{1}]) .

Therefore the correspondences π_{2} and [D_{1} × D_{2}] + [D_{2} × D_{1}] are the same modulo cohomological equivalence. Theorem 1.1 asserts, that, in fact, these correspondences are the same modulo rational equivalence. Hence to prove Theorem 1.1 we have to prove that the difference of correspondences

Ξ = [D_{1} × D_{2}] + [D_{2} × D_{1}] − π_{2}

is equal to zero in A^{2}(X × X).

Let us compute the cycle Ξ. Observe that

π_{0} = [x_{0} × X] = [x_{0}] × [X] = η_{∗}[e × f] × \frac{1}{n} η_{∗}[E × F]

= \frac{1}{n} (η × η)_{∗}[e × f × E × F] = \frac{1}{n} (η × η)_{∗}l_{∗}[e × E × f × F] .

Respectively,

π_{4} = \frac{1}{n} (η × η)_{∗}l_{∗}[E × e × F × f] .

Then it is easy to compute:

Ξ = [D_{1} × D_{2}] + [D_{2} × D_{1}] + π_{0} + π_{1} + π_{3} + π_{4} − Δ_{X} =

= \frac{1}{n} (η × η)_{∗}l_{∗}([e × E × F × f] + [E × e × f × F] + [e × E × f × F] + [Δ_{E} × f × F]−

−[e × E × f × F] − [E × e × f × F] + [Δ_{E} × F × f] − [E × e × F × f]−
−[e × E × F × f] + [E × e × F × f] − [Δ_{E} × Δ_{F}]) =

= \frac{1}{n} (η × η)_{∗}l_{∗}([Δ_{E} × f × F] + [Δ_{E} × F × f] − [Δ_{E}] × [Δ_{F}]) .

So we obtain that

Ξ = \frac{1}{n} (η × η)_{∗}l_{∗}([Δ_{E} × f × F] + [Δ_{E} × F × f] − [Δ_{E}] × [Δ_{F}]) .

Now, again, let Y = Ỹ /G be the quotient variety of a variety Ỹ by a finite abelian group G. Let τ : Ỹ → Y be the corresponding quotient map. Let θ ∈ A^{t}(Ỹ) be any cycle of codimension t on Ỹ. Then τ_{∗}(θ) is zero if and only if N_{Ỹ /G}(θ) is zero. Indeed, if τ_{∗}(θ) = 0 then N_{Ỹ /G}(θ) = τ^{∗}τ_{∗}(θ) = 0. Conversely, if N_{Ỹ /G}(θ) = 0 then τ_{∗} N_{Ỹ /G}(θ) = τ_{∗}τ^{∗}τ_{∗}(θ) = nτ_{∗}(θ) = 0 where n is order of the group G. Dividing by n we obtain that τ_{∗}(θ) = 0.

In particular Ξ = 0 if and only if the cycle

Υ = N_{(E×F ×E×F)/(G×G)}(l_{∗}([Δ_{E} × f × F] + [Δ_{E} × F × f] − [Δ_{E}] × [Δ_{F}]))

is zero. We omit \frac{1}{n} here because we can multiply and divide by n. For any g ∈ G let Θ_{g} be the graph of the regular map F → F, q ↦ g(q). Then we compute:

Υ = l_{∗} N_{E×E×F ×F/G×G}[Δ_{E} × f × F + Δ_{E} × F × f − Δ_{E} × Δ_{F}] =

= l_{∗} [ ∑_{g_{1},g_{2}∈G} Γ_{g_{1}g_{2}^{−1}} × g_{1}(f) × F + ∑_{g_{1},g_{2}∈G} Γ_{g_{1}g_{2}^{−1}} × F × g_{2}(f) − ∑_{g_{1},g_{2}∈G} Γ_{g_{1}g_{2}^{−1}} × Θ_{g_{1}g_{2}^{−1}} ] =

= l_{∗} [∑_{g∈G} Γ_{g} × ∑_{g∈G} g(f) × F + ∑_{g∈G} Γ_{g} × F × ∑_{g∈G} g(f) − n ∑_{g∈G} Γ_{g} × Θ_{g}] =

= l_{∗} (∑_{g∈G}[Γ_{g}] × ∑_{g∈G}[g(f) × F + F × g(f)] − n ∑_{g∈G}[Γ_{g} × Θ_{g}]) .

Let

λ = (η_{E}, η_{E}, η_{F} , η_{F}) : E × E × F × F ⟶ A × A × ℙ^{1} × ℙ^{1}

where η_{F} : F → ℙ^{1} is a quotient map. Then we have that

λ^{∗}λ_{∗}l^{∗}Υ = N_{(E×E×F ×F)/(G×G×G×G)} l^{∗}Υ =

= N_{(E×E×F ×F)/(G×G×G×G)} (∑_{g∈G}[Γ_{g}] × ∑_{g∈G}[g(f) × F + F × g(f)] − n ∑_{g∈G}[Γ_{g} × Θ_{g}]) =

= ∑_{g_{1},g_{2}∈G} ∑_{g∈G}[Γ_{g}] × ∑_{g_{3},g_{4}∈G} ∑_{g∈G}[g(f) × F + F × g(f)] − n ∑_{g_{1},g_{2},g_{3},g_{4}∈G} ∑_{g∈G}[Γ_{g} × Θ_{g}] =

= n^{2} ∑_{g∈G}[Γ_{g}] × n^{2} ∑_{g∈G}[g(f) × F + F × g(f)] − n^{5} ∑_{g∈G}[Γ_{g} × Θ_{g}] =

= n^{4}l^{∗}Υ .

Hence, if the cycle λ_{∗}l^{∗}Υ is trivial, then the cycle l^{∗}Υ is trivial too. We have that

λ_{∗}l^{∗}Υ = n^{4}([Δ_{A}] × [pt × ℙ^{1} + ℙ^{1} × pt] − [Δ_{A}] × [Δ_{ℙ^{1}}]) =
= n^{4}[Δ_{A}] × [pt × ℙ^{1} + ℙ^{1} × pt − Δ_{ℙ^{1}}] =
= n^{4}[Δ_{A}] × 0 = 0 .

So we obtain that

λ_{∗}l^{∗}Υ = 0 ⇒ λ^{∗}λ_{∗}l^{∗}Υ = 0 ⇒ l^{∗}Υ = 0 ⇒ Υ = 0 ⇒ Ξ = 0 .

Theorem 1.1 is proved.

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