Algebraic equivalence of real divisors

W. Kucharz^{∗}

1 Introduction and results

Let X be a smooth projective variety over ℝ of dimension n (that is, an integral smooth projective scheme over ℝ of dimension n). We endow the set X(ℝ) of ℝ-rational points of X, regarded here as a subset of X, with the topology induced by the usual metric topology on ℝ and assume that X(ℝ) is nonempty. Thus X(ℝ) is a closed n-dimensional manifold of class C^{∞}. The number of connected components of X(ℝ) will be denoted by c(X). Let, as usual, Div(X) denote the group of Weil divisors on X. There is a unique group homomorphism

cl_{ℝ} : Div(X) → H_{n−1}(X(ℝ), ℤ/2)

such that for any closed subvariety V of X of dimension n − 1 the homology class cl_{ℝ}(V) is determined by V (ℝ) if dim V (ℝ) = n − 1 (cf. [5] or [3, Section 11.3] for the definition of this homology class) and cl_{ℝ}(V) = 0 if dim V (ℝ) ≤ n − 2. Of considerable interest in real algebraic geometry is the image

H_{n−1}^{alg}(X(ℝ), ℤ/2) = cl_{ℝ}(Div(X))

of Div(X) via cl_{ℝ}, cf. [4] for a survey of results and references to the literature. This note is concerned with the subgroup

D(X) = cl_{ℝ}(Div_{a}(X))

of H_{n−1}^{alg}(X(ℝ), ℤ/2), where Div_{a}(X) is the subgroup of Div(X) consisting of all divisors algebraically equivalent to 0, cf. [7, Chapter 10] for the theory of algebraic equivalence. It has been noticed first by R. Silhol [14] that D(X) plays a nontrivial role in real algebraic geometry. More recent results are contained in [1, 12]. We prove here that the dimension

∗The author was partially supported by NSF Grant DMS-9503138. The paper was completed at the Max-Planck-Institut für Mathematik in Bonn, whose support and hospitality is gratefully acknowledged.

d(X) = dim_{ℤ/2} D(X)

of the ℤ/2-vector space D(X) is a birational invariant (it is a trivial observation that dim_{ℤ/2} H_{n−1}^{alg}(X(ℝ), ℤ/2) is not a birational invariant for n ≥ 2).

Theorem 1.1 Let X and Y be smooth projective varieties over ℝ with X(ℝ) and Y (ℝ) nonempty. If X and Y are birationally equivalent over ℝ, then d(X) = d(Y).

The behavior of d(−) will now be illustrated by several examples.

Example 1.2. Suppose n = 1, that is, X is a curve. Then

D(X) = H̃_{0}(X(ℝ), ℤ/2) and d(X) = c(X) − 1,

where H̃_{0}(X(ℝ), ℤ/2) is the reduced 0th homology group (in other words, H̃_{0}(X(ℝ), ℤ/2) consists precisely of these homology classes in H_{0}(X(ℝ), ℤ/2) that are represented by an even number of points). One argues as follows.

If p_{0} and p_{1} are distinct points of X(ℝ), then the divisor p_{1} − p_{0} is in Div_{a}(X) and hence cl_{ℝ}(p_{1} − p_{0}) is in D(X). Since cl_{ℝ}(p_{1} − p_{0}) is represented by {p_{0}, p_{1}}, it follows that any homology class in H_{0}(X(ℝ), ℤ/2) represented by an even number of points belongs to D(X). Hence H̃_{0}(X(ℝ), ℤ/2) ⊆ Div(X).

On the other hand, if D is in Div_{a}(X), then deg D = 0 [7, p. 180, Proposition 10.2]. Writing D in the form D = D_{1} + D_{2}, where D_{1} (resp. D_{2}) is a formal linear combination of ℝ-rational (resp. ℂ-rational) points, we obtain deg D_{1} = 0. Hence cl_{ℝ}(D_{1}) is represented by an even number of points of X(ℝ). It follows that D(X) ⊆ H̃_{0}(X(ℝ), ℤ/2).

We therefore have D(X) = H̃_{0}(X(ℝ), ℤ/2). The equality d(X) = c(X) − 1 is obvious since dim_{ℤ/2} H̃_{0}(X(ℝ), ℤ/2) = c(X) − 1.

Example 1.3. If X is an Abelian variety over ℝ, then 2^{d(X)} = c(X) [12, Theorem 2.3]. If X^{∨} is the dual Abelian variety of X, then c(X^{∨}) = c(X) [10, Proposition 1.2] and hence d(X^{∨}) = d(X). Recall that given an integer s satisfying 0 ≤ s ≤ n, there exists an Abelian variety Y over ℝ of dimension n and with c(Y) = 2^{s} [15, p. 90].

Example 1.4. If dim X = 1 and J is the Jacobian variety of X, then c(J) = 2^{c(X)−1} [10, Proposition 3.2], and hence d(X) = d(J) by virtue of Examples 1.2 and 1.3.

The invariant d(−) is subtler than the examples above might suggest. For instance, c(X) = 1 (resp. c(X) ≥ 2) does not imply d(X) = 0 (resp. d(X) ≥ 1) and Example 1.4 has no obvious generalization in dimensions greater than 1.

Example 1.5. Fix a positive integer r. Given nonnegative integers k and ℓ satisfying k + ℓ ≥ 1 and k ≤ r, there exists a smooth projective surface X^{k,ℓ} over ℝ such that

c(X^{k,ℓ}) = k + ℓ,

X^{k,ℓ}(ℝ) has k (resp. ℓ) connected components diffeomorphic to the real torus ℝ^{2}/ℤ^{2} (resp. the real projective plane ℙ_{ℝ}^{2}(ℝ)), and

d(X^{k,ℓ}) = k, dim_{ℤ/2} H_{1}^{alg}(X^{k,ℓ}(ℝ), ℤ/2) = k + ℓ

(note dim_{ℤ/2} H_{1}(X^{k,ℓ}(ℝ), ℤ/2) = 2k + ℓ). Furthermore, for all k and ℓ, the surfaces X^{k,ℓ} share the same Albanese variety and hence also the same Picard variety; if A denotes the Albanese variety and P denotes the Picard variety, then A is isomorphic to P over ℝ and

d(A) = d(P) = r.

In addition, the surfaces X_{ℂ}^{k,ℓ} = X^{k,ℓ} ×_{ℝ} ℂ are pairwise birationally equivalent over ℂ. It is explained in Section 3, Example 3.3 how to construct X^{k,ℓ} with such properties.

If X is as at the beginning of the section and dim X = 2, the intersection product

H_{1}(X(ℝ), ℤ/2) × H_{1}(X(ℝ), ℤ/2) ⟶ ℤ/2

(α, β) ⟶ α · β

makes of H_{1}(X(ℝ), ℤ/2) an inner product space, cf. [13]. For any ℤ/2-vector subspace N of H_{1}(X(ℝ), ℤ/2), let N^{⊥} denote the orthogonal complement of N:

N^{⊥} = {α ∈ H_{1}(X(ℝ), ℤ/2) | α · β = 0 for all β in N}.

Proposition 1.6 Let X be a smooth projective surface over ℝ with X(ℝ) nonempty. If X_{ℂ} = X ×_{ℝ} ℂ is a ruled surface defined over ℂ, then

H_{1}^{alg}(X(ℝ), ℤ/2) = D(X)^{⊥}.

We use here the definition of ruled surfaces (defined over ℂ) given in [2]; sometimes such surfaces are called birationally ruled. In Proposition 1.6 the groups D(X) and H_{1}^{alg}(X(ℝ), ℤ/2) determine each other since, obviously,

D(X) = H_{1}^{alg}(X(ℝ), ℤ/2)^{⊥}.

Let us mention that the surface X^{k,ℓ} in Example 1.5 can be chosen in such a way that X_{ℂ}^{k,ℓ} is ruled, cf. Example 3.3.

Theorem 1.1 and Proposition 1.6 are proved in Section 2 and Section 3, respectively. A method for computing the groups that appear in Proposition 1.6 is also described in Section 3.

2 Proof of Theorem 1.1

Let X be as at the beginning of Section 1. It will be more convenient to work with

H_{alg}^{1}(X(ℝ), ℤ/2) = P^{−1}(H_{n−1}^{alg}(X(ℝ), ℤ/2)),
D^{1}(X) = P^{−1}(D(X)),

where P : H^{1}(X(ℝ), ℤ/2) → H_{n−1}(X(ℝ), ℤ/2) is the Poincaré duality isomorphism. Clearly, d(X) = dim_{ℤ/2} D^{1}(X).

Given an invertible sheaf L on X, we denote by L(ℝ) the topological real line bundle on X(ℝ) determined by L in the usual way. The correspondence which assigns to L the first Stiefel-Whitney class w_{1}(L(ℝ)) of L(ℝ) gives rise to a canonical homomorphism

w_{1} : Pic(X) → H^{1}(X(ℝ), ℤ/2)

defined on the Picard group of isomorphism classes of invertible sheaves on X. If O(D) is the invertible sheaf associated with a divisor D on X, then P (w_{1}(O(D)(ℝ))) = cl_{ℝ}(D) [5, p. 498]. Since every invertible sheaf on X is isomorphic to O(D) for some D, we have

H_{alg}^{1}(X(ℝ), ℤ/2) = w_{1}(Pic(X)),
D^{1}(X) = w_{1}(Pic^{0}(X)),

where Pic^{0}(X) is the subgroup of Pic(X) consisting of the isomorphism classes of invertible sheaves of the form O(D) for D in Div_{a}(X).

Given a morphism f : X → Y over ℝ into a smooth projective variety Y over ℝ, we denote by f(ℝ) : X(ℝ) → Y (ℝ) the restriction of f. Thus f(ℝ) is a C^{∞} map. One readily checks that the diagram

Pic(Y)  ⟶^{f^{∗}}  Pic(X)
w_{1}↓  ↓^{w_{1}}
H^{1}(Y (ℝ), ℤ/2) ⟶^{f(ℝ)^{∗}} H^{1}(X(ℝ), ℤ/2),

where f^{∗} and f(ℝ)^{∗} are the induced homomorphisms, is commutative. Hence

f(ℝ)^{∗}(H_{alg}^{1}(Y (ℝ), ℤ/2)) ⊆ H_{alg}^{1}(X(ℝ), ℤ/2).

Since f^{∗}(Pic^{0}(Y)) ⊆ Pic^{0}(X), we also obtain

f(ℝ)^{∗}(D^{1}(Y)) ⊆ D^{1}(X).

Let us recall certain facts, which are special cases of a much more general theory [9] (note that X(ℝ) nonempty is assumed). The homomorphism p^{∗} : Pic(X) → Pic(X_{ℂ}), induced by the canonical projection p : X_{ℂ} = X ×_{ℝ} ℂ → X, is injective and p^{∗}(Pic(X)) = Pic(X_{ℂ})^{G}, where Pic(X_{ℂ})^{G} is the subgroup of Pic(X_{ℂ}) of all elements fixed under the usual action of the Galois group G = Gal(ℂ|ℝ) on Pic(X_{ℂ}). Furthermore, the action of G on Pic(X_{ℂ}) restricts to the action on Pic^{0}(X_{ℂ}) and p^{∗}(Pic^{0}(X)) = Pic^{0}(X_{ℂ})^{G}. The group Pic(X) (resp. Pic^{0}(X)) is often identified via p^{∗} with Pic(X_{ℂ})^{G} (resp. Pic^{0}(X_{ℂ})^{G}).

Lemma 2.1 Let A be the Albanese variety of X and let α : X → A be the Albanese morphism corresponding to some point in X(ℝ). Then

α(ℝ)^{∗}(D^{1}(A)) = D^{1}(X).

In particular, d(X) ≤ d(A).

Proof. It is well-known that the induced homomorphism α^{∗} : Pic^{0}(A) → Pic^{0}(X) is an isomorphism. (The reader accustomed to working with varieties over ℂ may first extend the base field to ℂ : α_{ℂ} : X_{ℂ} → A_{ℂ}. Then α_{ℂ}^{∗} : Pic^{0}(A_{ℂ}) → Pic^{0}(X_{ℂ}) is an isomorphism [8, p. 332, p. 462], which is obviously G-invariant. Hence α_{ℂ}^{∗} : Pic^{0}(A_{ℂ})^{G} → Pic^{0}(X_{ℂ})^{G} is an isomorphism corresponding to α^{∗} : Pic^{0}(A) → Pic^{0}(X) via the identification described immediately before the proof.) In the commutative diagram

Pic^{0}(A) ⟶^{α^{∗}} Pic^{0}(X)
w_{1}↓ ↓^{w_{1}}
D^{1}(A) ⟶^{α(ℝ)^{∗}} D^{1}(X)

α^{∗} is an isomorphism and the vertical arrows are surjective. Hence α(ℝ)^{∗} : D^{1}(A) → D^{1}(X) is also surjective, as required. The inequality d(X) ≤ d(A) is now obvious. □

If dim X = 1 in Lemma 2.1, then α(ℝ)^{∗} : D^{1}(A) → D^{1}(X) is an isomorphism by virtue of Example 1.4. This, however, is no longer true in higher dimensions, cf. Examples 1.5 and 3.3.

Proof of Theorem 1.1. Since X and Y are birationally equivalent over ℝ, it follows from Hironaka’s theorem on resolution of points of indeterminacy [11] that there exist a smooth projective variety V over ℝ and two birational morphisms f : V → X and g : V → Y over ℝ.

We claim that f(ℝ)^{∗}(D^{1}(X)) = D^{1}(V). Indeed, let A be the Albanese variety of X and let α : X → A be the Albanese morphism corresponding to f(p), where p is a point in V (ℝ). Then A is also the Albanese variety of V and α ∘ f : V → A is the Albanese morphism corresponding to p. By Lemma 2.1, we have α(ℝ)^{∗}(D^{1}(A)) = D^{1}(X) and ((α ∘ f)(ℝ))^{∗}(D^{1}(A)) = D^{1}(V). Hence in view of ((α ∘ f)(ℝ))^{∗} = (α(ℝ) ∘ f(ℝ))^{∗} = f(ℝ)^{∗} ∘ α(ℝ)^{∗}, we obtain

f(ℝ)^{∗}(D^{1}(X)) = f(ℝ)^{∗}(α(ℝ)^{∗}(D^{1}(A))) = ((α ∘ f)(ℝ))^{∗}(D^{1}(A)) = D^{1}(V),

as claimed.

Note that the homomorphism

f(ℝ)^{∗} : H^{1}(X(ℝ), ℝ/2) → H^{1}(V (ℝ), ℤ/2)

is injective, the map f(ℝ) : V (ℝ) → X(ℝ) having topological degree modulo 2 equal to 1. Hence the claim implies d(X) = d(V).

By summetry, we also get d(Y) = d(V). Thus d(X) = d(Y) and the proof is complete.□

3 Conic fibrations

Let X be a smooth projective surface over ℝ with X(ℝ) nonempty. Recall that

H^{1}(X(ℝ), ℤ/2) × H^{1}(X(ℝ), ℤ/2) → ℤ/2
(u, v) →< u ∪ v, [X(ℝ)] >

is a dual pairing [6, p. 300, Proposition 8.13] (standard notation is used: ∪ and < −, − > denote the cup product and scalar product, respectively, and [X(ℝ)] stands for the fundamental class of X(ℝ) in H_{2}(X(ℝ), ℤ/2)). Of course, this dual pairing corresponds via Poincaré duality to the intersection product considered in Section 1. More precisely, the Poincaré duality H^{1}(X(ℝ), ℤ/2) → H_{1}(X(ℝ), ℤ/2) is an isomorphism of inner product spaces. For any ℤ/2-vector subspace N of H^{1}(X(ℝ), ℤ/2), let N^{⊥} denote its orthogonal complement:

N^{⊥} = {u ∈ H^{1}(X(ℝ), ℤ/2) | < u ∪ v, [X(ℝ)] >= 0 for all v in N}.

By [12, Theorem 2.1], we have

H_{alg}^{1}(X(ℝ), ℤ/2) ⊆ D^{1}(X)^{⊥}.  (3.1)

It is interesting to distinguish a class of surfaces for which (3.1) is an equality.

Let π : X → B be a conic fibration defined over ℝ. This means that π is a morphism over ℝ from a smooth projective surface X over ℝ into a smooth projective curve B over ℝ such that the generic fiber of π is isomorphic to a plane conic over the function field of B (equivalently, the generic fiber of π is an integral smooth curve of genus 0). If X(ℝ) is nonempty (hence B(ℝ) is also nonempty), then a detailed topological description of the map π(ℝ) : X(ℝ) → B(ℝ) is available [15, Chapter V]. In particular, one fully understands the induced homomorphism

π(ℝ)^{∗} : H^{1}(B(ℝ), ℤ/2) → H^{1}(X(ℝ), ℤ/2),

which, as we shall see below, completely determines D^{1}(X) and H_{alg}^{1}(X(ℝ), ℤ/2).

Denote by H_{0}^{1}(B(ℝ), ℤ/2) the subgroup of H^{1}(B(ℝ), ℤ/2) that consists of all cohomology classes z such that the homology class Poincaré dual to z belongs to H̃_{0}(B(ℝ), ℤ/2). By Example 1.2, D^{1}(B) = H_{0}^{1}(B(ℝ), ℤ/2).

Proposition 3.2 Let π : X → B be a conic fibration defined over ℝ, with X(ℝ) nonempty. Let A be the Albanese variety of X. Then

(i) D^{1}(X) = π(ℝ)^{∗}(D^{1}(B)) = π(ℝ)^{∗}(H_{0}^{1}(B(ℝ), ℤ/2)),

(ii) d(X) ≤ d(A) = d(B) = c(B) − 1,

(iii) H_{alg}^{1}(X(ℝ), ℤ/2) = D^{1}(X)^{⊥} = (π(ℝ)^{∗}(H_{0}^{1}(B(ℝ), ℤ/2)))^{⊥},

(iv) D^{1}(X) ≠ H_{alg}^{1}(X(ℝ), ℤ/2) if the map π(ℝ) : X(ℝ) → B(ℝ) is surjective.

Proof. The morphism π_{ℂ} : X_{ℂ} → B_{ℂ} over ℂ, obtained by extending the base field to ℂ, is a trivial ℙ_{ℂ}^{1}-bundle over a nonempty Zariski open subset of B_{ℂ} [2, Theorem III.4]. It therefore readily follows that the Albanese variety A of X is also the Jacobian variety of B. Hence we have

d(A) = d(B) = c(B) − 1

in view of Examples 1.2 and 1.4. Furthermore, if x_{0} is a point in X(ℝ) and β : B → A is the Abel-Jacobi (that is, Albanese) morphism corresponding to π(x_{0}), then β ∘ π : X → A is the Albanese morphism corresponding to x_{0}. By Lemma 2.1, β(ℝ)^{∗}(D^{1}(A)) = D^{1}(B) and ((β ∘ π)(ℝ))^{∗}(D^{1}(A)) = D^{1}(X). Since ((β ∘ π)(ℝ))^{∗} = (β(ℝ) ∘ π(ℝ))^{∗} = π(ℝ)^{∗} ∘ β(ℝ)^{∗}, we obtain

D^{1}(X) = π(ℝ)^{∗}(D^{1}(B)).

By virtue of Example 1.2,

D^{1}(B) = H_{0}^{1}(B(ℝ), ℤ/2).

Hence (i) and (ii) are proved.

In order to prove (iii) and (iv) we argue as follows. Suppose that there is a commutative diagram of morphisms over ℝ:

(∗) \xymatrix{X \ar[rr]^{p} \ar[dr]_{π} &  & Y \ar[dl]^{ρ} \\  & B}

where p : X → Y is the blowup at a closed point y of Y. Then ρ : Y → B is also a conic fibration defined over ℝ and Y (ℝ) is nonempty. We shall consider two cases, depending on whether the point y is ℝ-rational or ℂ-rational.

If y is an ℝ-rational point, then H^{1}(X(ℝ), ℤ/2) = p(ℝ)^{∗}(H^{1}(Y (ℝ), ℤ/2)) ⊕ (ℤ/2)e,

H_{alg}^{1}(X(ℝ), ℤ/2) = p(ℝ)^{∗}(H_{alg}^{1}(Y (ℝ), ℤ/2)) ⊕ (ℤ/2)e,

where e is the cohomology class in H^{1}(X(ℝ), ℤ/2) Poincaré dual to the homology class in H_{1}(X(ℝ), ℤ/2) represented by p(ℝ)^{−1}(y). Note that e is in (π(ℝ)^{∗}(H^{1}(B(ℝ), ℤ/2)))^{⊥} and therefore also in (π(ℝ)^{∗}(H_{0}^{1}(B(ℝ), ℤ/2)))^{⊥}. Keeping in mind p(ℝ)_{∗}([X(ℝ)]) = [Y (ℝ)], one readily checks that the homomorphism

p(ℝ)^{∗} : H^{1}(Y (ℝ), ℤ/2) → H^{1}(X(ℝ), ℤ/2)

preserves the inner product. Since π(ℝ)^{∗} = (ρ(ℝ) ∘ p(ℝ))^{∗} = p(ℝ)^{∗} ∘ p(ℝ)^{∗}, we get (π(ℝ)^{∗}(H_{0}^{1}(B(ℝ), ℤ/2)))^{⊥} = p(ℝ)^{∗}((ρ(ℝ)^{∗}(H_{0}^{1}(B(ℝ), ℤ/2)))^{⊥}) ⊕ (ℤ/2)e.

If y is a ℂ-rational point, then p is an isomorphism (over ℝ) of a neighborhood of X(ℝ) onto a neighborhood of Y (ℝ), and hence

p(ℝ)^{∗} : H^{1}(Y (ℝ), ℤ/2) → H^{1}(X(ℝ), ℤ/2)

is an isomorphism of inner product spaces satisfying

H_{alg}^{1}(X(ℝ), ℤ/2) = p(ℝ)^{∗}(H_{alg}^{1}(Y (ℝ), ℤ/2))
(π(ℝ)^{∗}(H_{0}^{1}(B(ℝ), ℤ/2)))^{⊥} = p(ℝ)^{∗}((ρ(ℝ)^{∗}(H_{0}^{1}(B(ℝ), ℤ/2)))^{⊥}).

In conclusion, in either case, if

H_{alg}^{1}(Y (ℝ), ℤ/2) = (ρ(ℝ)^{∗}(H_{0}^{1}(B(ℝ), ℤ/2)))^{⊥},

then

H_{alg}^{1}(X(ℝ), ℤ/2) = (π(ℝ)^{∗}(H_{0}^{1}(B(ℝ), ℤ/2)))^{⊥}.

Thus, since (i) is already proved, we get that (iii) and (iv) hold for π : X → B whenever they hold for ρ : Y → B.

It is well-known that a diagram like (∗) exists if and only if some fiber of π over a closed point of B contains an exceptional curve of the first kind (according to a suitable definition for varieties over ℝ [15, p. 51, Definition 6.3]), and then p : X → Y is obtained by contracting this exceptional curve [15, p. 50, Proposition 6.2]. Thus for the proof of (iii) and (iv) we may assume that π : X → B is B-minimal, that is, no diagram of the form (∗) exists (or equivalently, no fiber of π over a closed point of B contains an exceptional curve of the first kind).

Assuming that π : X → B is B-minimal, the map π(ℝ) : X(ℝ) → B(ℝ) has the following properties (cf. [15, p. 105, Theorem 3.4 and its proof]): Each connected component M of X(ℝ) is homeomorphic either to a 2-sphere or a 2-torus or a Klein bottle. Denote by B(ℝ)_{M} the connected component of B(ℝ) containing π(ℝ)(M). If M is homeomorphic to a 2sphere, then π(ℝ)(π(ℝ)^{−1}(B(ℝ)_{M})) ≠ B(ℝ)_{M}. If M is not homeomorphic to a 2-sphere, then M = π(ℝ)^{−1}(B(ℝ)_{M}) and π(ℝ) : M → B(ℝ)_{M} is a C^{∞} circle bundle over B(ℝ)_{M}. In particular, the map π(ℝ) : X(ℝ) → B(ℝ) is surjective if and only if it is a C^{∞} circle bundle over B(ℝ), in which case there is a C^{∞} section s : B(ℝ) → X(ℝ) of the bundle.

It follows from these properties that

(π(ℝ)^{∗}(H_{0}^{1}(B(ℝ), ℤ/2)))^{⊥} = π(ℝ)^{∗}(H_{0}^{1}(B(ℝ), ℤ/2)) ⊕ (ℤ/2)u,

where u = 0 if π(ℝ) : X(ℝ) → B(ℝ) is not surjective, while otherwise u is the cohomology class in H^{1}(X(ℝ), ℤ/2) Poincaré dual to the homology class in H_{1}(X(ℝ), ℤ/2) represented by s(B(ℝ)). Since H_{alg}^{1}(B(ℝ), ℤ/2) = H^{1}(B(ℝ), ℤ/2), we have

π(ℝ)^{∗}(H^{1}(B(ℝ), ℤ/2)) ⊆ H_{alg}^{1}(X(ℝ), ℤ/2).

Supposing for a moment that u is in H_{alg}^{1}(X(ℝ), ℤ/2), we obtain

(π(ℝ)^{∗}(H_{0}^{1}(B(ℝ), ℤ/2)))^{⊥} ⊆ H_{alg}^{1}(X(ℝ), ℤ/2).

In view of (3.1) and (already proved)(i), we conclude that (iii) and (iv) hold.

Thus in order to complete the proof it sufficies to show that u belongs to H_{alg}^{1}(X(ℝ), ℤ/2). This is obvious if π(ℝ) : X(ℝ) → B(ℝ) is not surjective since then u = 0.

Suppose π(ℝ) : X(ℝ) → B(ℝ) is surjective. We just have to demonstrate that π : X → B has a section (which is a morphism over ℝ). By definition, the generic fiber of π is isomorphic to a plane conic C over the function field ℝ(B) of B. It is known that in suitably chosen coordinates C has an affine equation of the form

x_{1}^{2} + x_{2}^{2} = h,

where h is in ℝ(B) and has no poles on B(ℝ) [15, p. 102, Corollary 2.7]. Let h(ℝ) : B(ℝ) → ℝ be the function determined by h. Since π(ℝ) : X(ℝ) → B(ℝ) is surjective, it follows that h(ℝ)(b) ≥ 0 for all b in B(ℝ). Hence h is a sum of 2 squares in ℝ(B) [3, Theorem 6.4.18]. In other words, C has an ℝ(B)-rational point. Consequently, π : X → B has a rational section (defined over ℝ), which automatically is a morphism. The proof is complete. □

Example 3.3. We shall use Proposition 3.2 and its proof. Let r be a positive integer and let B be any smooth projective curve over ℝ with c(B) = r + 1. Let k and ℓ be nonnegative integers satisfying k+ℓ ≥ 1 and k ≤ r. By [15, p. 105, Lemma 3.2], there exists a B-minimal conic fibration ρ^{k,ℓ} : Y^{k,ℓ} → B, defined over ℝ, such that c(Y^{k,ℓ}) = k + ℓ and Y^{k,ℓ}(ℝ) has k (resp. ℓ connected comoponents diffeomorphic to a 2-torus (resp. 2-sphere). Let p : X^{k,ℓ} → Y^{k,ℓ} be the blowup at ℓ points y_{1}, …, y_{ℓ} such that the intersection of {y_{1}, …, y_{ℓ}} with each connect component of Y^{k,ℓ}(ℝ) diffeomorphic to a 2-sphere consists precisely of one point. Then c(X^{k,ℓ}) = k+ℓ and X^{k,ℓ}(ℝ) has k (resp. ℓ) connected components diffeomorphic to a 2-torus (resp. ℙ_{ℝ}^{2}(ℝ)). It follows from the proof of Proposition 3.2 that

d(X^{k,ℓ}) = k, dim_{ℤ/2} H_{alg}^{1}(X^{k,ℓ}(ℝ), ℤ/2) = k + ℓ.

Furthermore, the Albanese variety A of X^{k,ℓ} coincides with the Jacobian variety of B, and hence the Picard variety P of X^{k,ℓ} is isomorphic to A over ℝ. We have

d(A) = d(P) = d(B) = c(B) − 1 = r.

In addition, X_{ℂ}^{k,ℓ} is birationally equivalent over ℂ to B_{ℂ} ×_{ℂ} ℙ_{ℂ}^{1}.

Proof of Proposition 1.6. If the surface X_{ℂ} over ℂ is rational, then D(X) = 0 since Pic^{0}(X) = 0. On the other hand, H_{alg}^{1}(X(ℝ), ℤ/2) = H_{1}(X(ℝ), ℤ/2) [15, p. 67, Theorem 3.4]. Hence the conclusion holds in this case.

If X_{ℂ} is not rational, then there exist a smooth projective curve B over ℝ and a conic fibration π : X → B defined over ℝ [15, p. 98, Proposition 1.8]. Now the conclusion follows from Proposition 3.2. □

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Max-Planck-Institut für Mathematik Vivatsgasse 7 53111 Bonn Germany

Department of Mathematics and Statistics University of New Mexico Albuquerque, New Mexico 87131-1141 U.S.A. e-mail: kucharz@math.unm.edu