Rational and homological equivalence of real algebraic cycles

W. Kucharz^{∗}

1 Introduction and main results

Throughout this paper X denotes a smooth n-dimensional quasiprojective variety over ℝ (that is an integral smooth n-dimensional quasiprojective scheme over ℝ). We endow the set X(ℝ) of ℝ-rational points of X with the topology induced by the usual metric topology on ℝ, and assume that X(ℝ) is nonempty and compact. Thus X(ℝ) is a closed n-dimensional C^{∞} manifold. Since there is an affine neighborhood of X(ℝ) in X, we can regard X(ℝ) as an algebraic subset of ℝ^{N} for some N. Given a nonnegative integer k, we let Z_{k}(X) denote the group of algebraic k-cycles on X (that is, the free Abelian group on the set of closed k-dimensional subvarieties of X). There is a unique group homomorphism

cℓ_{k}^{ℝ} : Z_{k}(X) → H_{k}(X(ℝ), ℤ/2)

such that for every closed k-dimensional subvariety Y of X, the homology class cℓ_{k}^{ℝ}(Y) is represented by Y (ℝ) assuming dim Y (ℝ) = k, and cℓ_{k}^{ℝ}(Y) = 0 otherwise [3] (dim Y (ℝ) refers to the dimension of Y (ℝ) regarded as an algebraic subset of X(ℝ); of course, X(ℝ) = X(ℝ)∩Y and dim Y (ℝ) ≤ k, with dim Y (ℝ) = k if and only if Y (ℝ) is dense in Y). A k-cycle z on X is said to be homologous to zero (or homologically equivalent to zero) if cℓ_{k}^{ℝ}(z) = 0. Below we recall a beautiful characterization of such cycles due to Ischebeck and Schülting [11]. Following [11], we say that a k-cycle z on X is thin, provided

∗The paper was written at the Max-Planck-Institut für Mathematik in Bonn, whose support and hospitality are gratefully acknowledged.

dim(X(ℝ) ∩ supp z) < k

(supp z stands for the support of z : if z = n_{1}Y_{1} + ⋯ + n_{r}Y_{r}, where n_{1}, …, n_{r} are nonzero integers and Y_{i} ≠ Y_{j} for i ≠ j, then supp z = Y_{1} ∪ … ∪ Y_{r}, while supp 0 = ∅). We also say that z supports real points (resp. supports no real points) if X(ℝ) ∩ supp z is nonempty (resp. empty). Clearly, z is thin if it supports no real points. By definition of cℓ_{k}^{ℝ}, every thin cycle is homologous to zero. Moreover, it is well known that every cycle on X rationally equivalent to zero is also homologous to zero [3].

We can now state the main result of [11].

Theorem A. A given k-cycle on X, where 0 ≤ k ≤ n − 1, is homologous to zero if and only if it is rationally equivalent to a thin cycle.

In [11] X is assumed to be a complete variety. However, Theorem A is true as stated since, by virtue of [9], X has a smooth projectivization with the set of ℝ-rational points equal to X(ℝ). Some special cases of Theorem A had been known before [11] was published. If k = 0 and n = 1, the result follows from [16], while for k = 0 and k = 1 it is contained in [6] and [10], respectively. Note that thin cycles in Z_{0}(X) support no real points. This remark corresponds to an earlier result for k = n − 1, due to Bröcker [4], asserting that every homologous to zero (n − 1)-cycle on X is rationally equivalent to a cycle supporting no real points (cf. also [7] for a related theorem formulated in terms of vector bundles). As we demonstrate below, the cases k = 0 and k = n − 1 are somewhat misleading since for many values of k, 1 ≤ k ≤ n − 2, a k-cycle on X that is homologous to zero need not be rationally equivalent to any cycle supporting no real points. We actually prove much more (cf. also [15]).

Theorem 1.1 Let n and ℓ be positive integers. If 4ℓ ≤ n, then there exist a smooth projective variety X_{nℓ} over ℝ and a cycle z_{nℓ} in Z_{n−2ℓ}(X_{nℓ}) such that dim X_{nℓ} = n, X_{nℓ}(ℝ) is nonempty, z_{nℓ} is homologous to zero, and every cycle rationally equivalent to rz_{nℓ}, where r is an arbitrary nonzero integer, supports real points. Furthermore, X_{nℓ} and z_{nℓ} can be described by explicitly given equations.

In the light of Theorem 1.1, the following problem becomes natural.

Problem B. Assuming 1 ≤ k ≤ n − 2, identify cases in which one of the following two conditions is satisfied:

(i) Every homologous to zero k-cycle on X is rationally equivalent to a cycle supporting no real points.

(ii) For every k-cycle z on X, there exists a nonzero integer r such that rz is rationally equivalent to a cycle supporting no real points (note that 2z is always homologous to zero).

The only contribution we can make to Problem B(i) is the following.

Theorem 1.2 If n = 3, then every homologous to zero 1-cycle on X is rationally equivalent to a cycle supporting no real points.

Our next result deals with Problem B(ii) and covers a wider range of cases.

Theorem 1.3 There is a positive integer r, depending only on X, such that for every kcycle z on X, where 2k + 1 ≤ n, the multiple rz is rationally equivalent to a cycle supporting no real points.

If n − k is even, then in view of Theorem 1.1, the assumption 2k + 1 ≤ n in Theorem 1.3 is optimal and cannot be changed without imposing some extra conditions on X.

We precede our last result with a definition of a technical nature. Regard X(ℝ) as a subset of the set X(ℂ) of ℂ-rational points of X. Let e : X(ℝ) ↪ X(ℂ) be the inclusion map and

e^{∗} : H^{∗}(X(ℂ), ℤ) → H^{∗}(X(ℝ), ℤ)

the induced homomorphism. The variety X is said to have property (D), provided that the image e^{∗}(H^{n}(X(ℂ), ℤ)) is a torsion group. Note that H^{n}(X(ℝ), ℤ) ≅ ℤ^{p} ⊕ (ℤ/2)^{q} for some nonnegative integers p and q. If each connected component of X(ℝ) is nonorientable, then p = 0, and hence X has property (D). Another example of X having property (D) is the product of n smooth projective curves C_{1}, …, C_{n}, where C_{i}(ℝ) is connected and C_{i}(ℂ)\C_{i}(ℝ) is disconnected for each i.

Theorem 1.4 Assume that X has property (D). If 2k = n and k ≥ 2, then there is a positive integer r, depending only on X, such that for every k-cycle z on X, the multiple rz is rationally equivalent to a cycle supporting no real points. Furthermore, if n = 4 and a 2-cycle z on X is homologous to zero, then 2z is rationally equivalent to a cycle supporting no real points.

Theorem 1.1 implies that one cannot drop the assumption about X having property (D) if k is even.

2 Proofs

We need some facts from algebraic geometry over ℂ. Let V be a smooth n-dimensional quasiprojective variety over ℂ. The set V (ℂ) of ℂ-rational points of V is an n-dimensional complex manifold. There is a unique homomorphism

cℓ_{k}^{ℂ} : Z_{k}(V) → H_{2k}^{BM}(V (ℂ), ℤ)

which assigns to each closed k-dimensional subvariety W of V the homology class represented by W (ℂ) [3, 8] (H_{2k}^{BM}(V (ℂ), ℤ) stands for the Borel-Moore homology, H_{2k}^{BM}(V (ℂ), ℤ) = H_{2k}(V (ℂ), ℤ) if V (ℂ) is compact). This homomorphism respects rational equivalence, that is, cℓ_{k}^{ℂ}(u) = 0 assuming u is rationally equivalent to zero [3, 8].

For any variety Y over ℝ, we set Y_{ℂ} = Y ×_{ℝ} ℂ. In order to ease the notation, we will always identify Y_{ℂ}(ℂ) with Y (ℂ). Thus Y (ℝ) ⊂ Y (ℂ) = Y_{ℂ}(ℂ). The canonical projection π : Y_{ℂ} → Y gives rise to the pullback homomorphism

π^{∗} : Z_{k}(Y) → Z_{k}(Y_{ℂ}),
π^{∗}(n_{1}Y_{1} + ⋯ + n_{r}Y_{r}) = n_{1}Y_{1ℂ} + ⋯ + n_{r}Y_{rℂ}.

If a k-cycle z on Y is rationally equivalent to zero, then so is the cycle π^{∗}(z) on Y_{ℂ}. The following example will play a crucial role in the proof of Theorem 1.1.

Example 2.1. For any positive integer n denote by X_{n} the subvariety of ℙ^{n+1} (projective (n + 1)-space over ℝ) defined by the equation

x_{1}^{2} + ⋯ + x_{n+1}^{2} = x_{n+2}^{2}.

Clearly, X_{n} is smooth of dimension n and X_{n}(ℝ) is the unit n-sphere in ℝ^{n+1} ⊂ ℙ^{n+1}(ℝ). In particular, H_{i}(X_{n}(ℝ), ℤ/2) = 0 and hence every cycle in Z_{i}(X_{n}) is homologous to zero for 1 ≤ i ≤ n − 1.

Assertion. Assume n = 2k, where k is even and positive. Let Y_{k} be the reduced subscheme of ℙ^{n+1} defined by the system of equations

x_{1}^{2} + x_{2}^{2} = 0, …, x_{n−1}^{2} + x_{n}^{2} = 0, x_{n+1} = x_{n+2}.

Then Y_{k} is a closed k-dimensional subvariety of X_{n} and the k-cycle z_{k} = Y_{k} on X_{n} (we introduce z_{k} here just to emphasize that Y_{k} is regarded as a cycle) has the following property: given a nonzero integer r, every cycle rationally equivalent to rz_{k} supports real points.

We prove the assertion as follows. Note that

Y_{k}(ℂ) = L ∪ σ(L) and L ∩ σ(L) = {p},

where L is the linear subspace of ℙ^{n+1}(ℂ) defined by the system of equations

x_{1} − \sqrt{−1}x_{2} = 0, …, x_{n−1} − \sqrt{−1}x_{n} = 0, x_{n+1} = x_{n+2},

σ : ℙ^{n+1}(ℂ) → ℙ^{n+1}(ℂ) is the complex conjugation, and p = (0 : …: 0 : 1 : 1). It follows that Y_{k} is irreducible of dimension k. Obviously, Y_{k} is contained in X_{n}. In other words, Y_{k} is a closed k-dimensional subvariety of X_{n} and z_{k} = Y_{k} is a cycle in Z_{k}(X_{n}).

Observe that

X_{n}(ℝ) ∩ L = {p} and X_{n}(ℝ) ∩ σ(L) = {p}.

Moreover, X_{n}(ℝ) and L (resp. X_{n}(ℝ) and σ(L)) are transverse C^{∞} submanifolds of X_{n}(ℂ). The manifolds L, σ(L), and X_{n}(ℂ) come with the canonical orientations determined by their complex structure. We fix an orientation on X_{n}(ℝ) so that the intersection number of X_{n}(ℝ) and L in X_{n}(ℂ) is equal to 1. One readily shows that the intersection number of X_{n}(ℝ) and σ(L) in X_{n}(ℂ) is also 1 (this is the only place, where k even is used). It follows that if ξ, η, and α are the homology classes in H_{n}(X_{n}(ℂ), ℤ) represented by L, σ(L), and X_{n}(ℝ), respectively, then the intersection number α • (ξ + η) of α and ξ + η is equal to 2.

Let π : X_{nℂ} = X_{n} ×_{ℝ} ℂ → X_{n} be the canonical projection. By definition of cℓ_{k}^{ℂ}, we have cℓ_{k}^{ℂ}(π^{∗}(z_{k})) = ξ + η. Hence for any integer r,

(∗) α • cℓ_{k}^{ℂ}(π^{∗}(rz_{k})) = 2r.

Suppose now that rz_{k} is rationally equivalent to a cycle w which supports no real points. Since π^{∗}(rz_{k}) and π^{∗}(w) are also rationally equivalent, we obtain cℓ_{k}^{ℂ}(π^{∗}(rz_{k})) = cℓ_{k}^{ℂ}(π^{∗}(w)). The fact that w supports no real points means X_{n}(ℝ) ∩ supp π^{∗}(w) = ∅, and therefore α • cℓ_{k}^{ℂ}(π^{∗}(w)) = 0. Thus α • cℓ_{k}^{ℂ}(π^{∗}(rz_{k})) = 0, which in view of (∗) implies r = 0. The assertion is proved. □

Henceforth X is as in Section 1. Denote by A_{k}(X) the kth Chow group of X (Z_{k}(X) modulo rational equivalence). As usual, we set A^{i}(X) = A_{n−i}(X). For any kcycle z on X, we let [z] denote its class in A_{k}(X). The following simple observation will be useful.

Lemma 2.2 A given k-cycle z on X is rationally equivalent to a cycle supporting no real points if and only if there exists an open neighborhood U of X(ℝ) in X such that the restriction homomorphism A_{k}(X) → A_{k}(U) sends [z] to 0.

Proof. It is obvious that such a neighborhood U exists if z is rationally equivalent to a cycle supporting no real points. The converse holds since for any open neighborhood U of X(ℝ) in X, the sequence

A_{k}(X\U) → A_{k}(X) → A_{k}(U) → 0,

where the first homomorphism is induced by the inclusion X\U ↪ X, is exact [8, Proposition 1.8]. □

Proof of Theorem 1.1. We will refer to X_{n}, Y_{k}, and z_{k} constructed in Example 2.1. If n = 4ℓ, it suffices to take X_{nℓ} = X_{n} and z_{nℓ} = z_{2ℓ}. Suppose n ≥ 4ℓ + 1. We set

X_{nℓ} = X_{4ℓ} ×_{ℝ} X_{n−4ℓ} and z_{nℓ} = Y_{2ℓ} ×_{ℝ} X_{n−4ℓ}.

Since the 2ℓ-cycle z_{2ℓ} = Y_{2ℓ} on X_{4ℓ} is homologous to zero, it follows that the (n − 2ℓ)-cycle z_{nℓ} on X_{nℓ} is also homologous to zero. Suppose that for some integer r the cycle rz_{nℓ} on X_{nℓ} is rationally equivalent to a cycle supporting no real points. In order to complete the proof it suffices to show r = 0. We argue as follows. Choose any point p in X_{n−4ℓ}(ℝ). Let i : X_{4ℓ} → X_{nℓ} be the obvious embedding identifying X_{4ℓ} with X_{4ℓ} ×_{ℝ} {p}. We have α = i^{∗}(β), where α = [z_{2ℓ}] in A_{2ℓ}(X_{4ℓ}) = A^{2ℓ}(X_{4ℓ}), β = [z_{nℓ}] in A_{n−2ℓ}(X_{nℓ}) = A^{2ℓ}(X_{nℓ}), and i^{∗} : A^{2ℓ}(X_{nℓ}) → A^{2ℓ}(X_{n}) is the pullback homomorphism. Since rz_{nℓ} is rationally equivalent to a cycle supporting no real points, one can find an open neighborhood U of X_{nℓ}(ℝ) in X_{nℓ} such that rβ restricted to U is 0 in A^{2ℓ}(U). Then rα = i^{∗}(rβ) restricted to i^{−1}(U) is 0 in A^{2ℓ}(i^{−1}(U)). By Lemma 2.2, the cycle rz_{2ℓ} on X_{4ℓ} is rationally equivalent to a cycle supporting no real points. This implies r = 0 in view of Example 2.1. The proof is complete. □

In the proofs of other theorems from Section 1 we will need some facts concerning vector bundles. Any vector bundle E on X determines in a canonical way a real vector bundle on X(ℝ), which we denote by E(ℝ).

Lemma 2.3 If the vector bundle E(ℝ) on X(ℝ) is topologically stably trivial, then there exists an open neighborhood U of X(ℝ) in X such that the restriction E|U of E to U is algebraically stably trivial.

Proof. Replacing X by an affine neighborhood of X(ℝ), we may assume that X itself is affine. Thus the lemma follows from either of [7] or [2, Theorem 12.3.3] (recall that X(ℝ) is compact). □

By abuse of notation, we let

cℓ_{k}^{ℝ} : A_{k}(X) → H_{k}(X(ℝ), ℤ/2)

denote the homomorphism induced by cℓ_{k}^{ℝ} defined in Section 1. Let

cℓ_{ℝ}^{i} : A^{i}(X) → H^{i}(X(ℝ), ℤ/2)

be the composite homomorphism D^{−1} ∘ cℓ_{n−i}^{ℝ}, where

D : H^{i}(X(ℝ), ℤ/2) → H_{n−i}(X(ℝ), ℤ/2)

is the Poincaré duality and

cℓ_{n−i}^{ℝ} : A^{i}(X) = A_{n−i}(X) → H_{n−i}(X(ℝ), ℤ/2).

Given a vector bundle E on X, denote by c_{i}(E) its ith Chern class in A^{i}(X) [3, 8]. According to [3],

w_{i}(E(ℝ)) = cℓ_{ℝ}^{i}(c_{i}(E)) for all i ≥ 0, (2.4)

where w_{i}(−) is the ith Stiefel-Whitney class.

Proof of Theorem 1.2. Replacing X by an affine neighborhood of X(ℝ), we may assume that X is affine. Let z be a 1-cycle on X that is homologous to zero, that is, satisfies cℓ_{1}^{ℝ}(z) = 0. Then the class α = [z] is in A_{1}(X) = A^{2}(X) and

(1) cℓ_{ℝ}^{2}(α) = 0.

By Grothendieck’s formula [8, Example 15.3.6], there exists a vector bundle E on X with

(2) c_{1}(E) = 0 and c_{2}(E) = α.

It follows from (1), (2), and (2.4) that

(3) w_{1}(E(ℝ)) = 0 and w_{2}(E(ℝ)) = 0.

Since E is of constant rank (X being irreducible) and n = 3, condition (3) implies that the vector bundle E(ℝ) on X(ℝ) is topologically stably trivial, cf. for example [5, Lemma 1.2]. Thus, by Lemma 2.3, there exists an open neighborhood U of X(ℝ) in X such that the restriction E|U is algebraically stably trivial. As a consequence, c_{2}(E|U) = 0 in A^{2}(U). This means, in view of the second equality in (2), that the restriction homomorphism A^{2}(X) → A^{2}(U) sends α to 0. Therefore, according to Lemma 2.2, z is rationally equivalent to a cycle supporting no real points. □

We will again consider varieties over ℂ. Let V be a smooth n-dimensional quasiprojective variety over ℂ. Denote by

cℓ_{k}^{ℂ} : A_{k}(V) → H_{2k}^{BM}(V (ℂ), ℤ)

the homomorphism induced by the previously defined

cℓ_{k}^{ℂ} : Z_{k}(V) → H_{2k}^{BM}(V (ℂ), ℤ).

Let

cℓ_{ℂ}^{i} : A^{i}(V) → H^{2i}(V (ℂ), ℤ)

be the composite homomorphism D^{−1} ∘ cℓ_{n−i}^{ℂ}, where

D : H^{2i}(V (ℂ), ℤ) → H_{2n−2i}^{BM}(V (ℂ), ℤ)

is the Poincaré duality and

cℓ_{n−i}^{ℂ} : A^{i}(V) = A_{n−i}(V) → H_{2n−2i}^{BM}(V (ℂ), ℤ).

Any vector bundle F on V determines in a canonical way a complex vector bundle F (ℂ) on V (ℂ). By [3, 8],

c_{i}^{top}(F (ℂ)) = cℓ_{ℂ}^{i}(c_{i}(F)) for all i ≥ 0,

where c_{i}(F) is the ith Chern class of F in A^{i}(V) and c_{i}^{top}(F (ℂ)) is the ith topological Chern class of F (ℂ) in H^{2i}(V (ℂ), ℤ).

Any vector bundle E on X determines in a canonical way a complex vector bundle on X(ℂ), denoted E(ℂ). The restriction E(ℂ)|X(ℝ), regarded as a real vector bundle, is isomorphic to E(ℝ) ⊕ E(ℝ). If π : X_{ℂ} → X is the canonical projection, then π^{∗}E is a vector bundle on X_{ℂ}. We identify (π^{∗}E)(ℂ) and E(ℂ) as vector bundles on X(ℂ) = X_{ℂ}(ℂ). Since c_{i}(π^{∗}E) = π^{∗}(c_{i}(E)), we have

(2.5) c_{i}^{top}(E(ℂ)) = cℓ_{ℂ}^{i}(π^{∗}(c_{i}(E))) for all i ≥ 0.

We also need the following topological result.

Lemma 2.6 Let M be a compact C^{∞} manifold. There exists a positive integer s = s(M) such that for every complex vector bundle ξ on M, which is of constant rank and satisfies c_{i}^{top}(ξ) = 0 for all i ≥ 1, the direct sum ξ ⊕ ⋯ ⊕ ξ of s copies of ξ is topologically stably trivial.

Proof. The Grothendieck group K(M) of complex vector bundles on M is a finitely generated Abelian group, cf. for example [1, p. 110]. Thus there is a positive integer s = s(M) such that sa = 0 for every torsion element a of K(M). If ξ is a complex vector bundle of constant rank on M and with c_{i}(ξ) = 0 for all i ≥ 1, then the Chern character of ξ is equal to 0, and hence the class of ξ in K(M) is a torsion element [12, p. 283, Theorem 3.2.5]. This implies that the direct sum of s copies of ξ is stably trivial. □

Proofs of Theorems 1.3 and 1.4. Let us set r = 2s(n − k − 1)!, where s = s(X(ℝ)) is as in Lemma 2.6 with M = X(ℝ). We claim that r has the required property if the hypothesis of either Theorem 1.3 or Theorem 1.4, in the latter case with n ≥ 6, are satisfied. The special case of Theorem 1.4 with n = 4 will be handled at the end of this proof.

The claim is proved as follows. We may assume that X is affine. Let z be a k-cycle on X and let α = [z] be its class in A_{k}(X) = A^{n−k}(X). By Grothendieck’s formula [8, Example 15.3.6], there exists a vector bundle E on X satisfying

(1) c_{i}(E) = 0 for 1 ≤ i ≤ n − k − 1,

c_{n−k}(E) = (n − k − 1)! α.

Making use of (1) and (2.5), we get

c_{i}^{top}(E(ℂ)) = 0 for 1 ≤ i ≤ n − k − 1,
c_{n−k}^{top}(E(ℂ)) = (n − k − 1)! cℓ_{ℂ}^{n−k}(π^{∗}(α)),

where π : X_{ℂ} → X is the canonical projection. For the restriction E(ℂ)|X(ℝ) this yields

(2) c_{i}^{top}(E(ℂ)|X(ℝ)) = 0 for 1 ≤ i ≤ n − k − 1,

c_{n−k}^{top}(E(ℂ)|X(ℝ)) = (n − k − 1)! e^{∗}(cℓ_{ℂ}^{n−k}(π^{∗}(α))),

where e : X(ℝ) ↪ X(ℂ) is the inclusion map and

e^{∗} : H^{2n−2k}(X(ℂ), ℤ) → H^{2n−2k}(X(ℝ), ℤ)

is the induced homomorphism.

If 2k+1 ≤ n, then 2n−2k ≥ n+1 and hence H^{2n−2k}(X(ℝ), ℤ) = 0. If 2k = n, k ≥ 3, and X has property (D), then (n−k−1)! is divisible by 2, 2n−2k = n, and e^{∗}(H^{2n−2k}(X(ℂ), ℤ)) is a torsion subgroup of H^{2n−2k}(X(ℝ), ℤ) ≅ ℤ^{p} ⊕ (ℤ/2)^{q}. Therefore in either of these two cases

(n − k − 1)! e^{∗}(cℓ_{ℂ}^{n−k}(π^{∗}(α))) = 0.

In view of (2), c_{j}^{top}(E(ℂ)|X(ℝ)) = 0 for 1 ≤ j ≤ n − k. This actually yields

(3)

c_{j}^{top}(E(ℂ)|X(ℝ)) = 0 for all j ≥ 0.

Indeed, in the cases under consideration, 2k ≤ n. Hence for j ≥ n − k + 1 we always have 2j ≥ n + 1, and therefore c_{j}^{top}(E(ℂ)|X(ℝ)) = 0.

By Lemma 2.6 and (3), if F = E ⊕ ⋯ ⊕ E is the direct sum of s = s(X(ℝ)) copies of E, then the complex vector bundle F (ℂ)|X(ℝ) on X(ℝ) is topologically stably trivial (note that F is of constant rank, X being irreducible). Since F (ℂ)|X(ℝ) and F (ℝ) ⊕ F (ℝ) are isomorphic as real vector bundles, it follows that F (ℝ) ⊕ F (ℝ) is topologically stably trivial on X(ℝ). The vector bundle G = F ⊕ F on X is the direct sum of 2s copies of E, and G(ℝ) is topologically stably trivial on X(ℝ). By Lemma 2.3, there exists an open neighborhood U of X(ℝ) in X such that the restriction G|U is algebraically stably trivial on U. In particular,

(4) c_{n−k}(G|U) = 0 in A^{n−k}(U).

On the other hand, (1) yields

(5) c_{n−k}(G) = 2s(n − k − 1)! α = rα in A^{n−k}(X).

By (4) and (5), the restriction of rα to U is 0 in A^{n−k}(U). Hence, according to Lemma 2.2, rz is rationally equivalent to a cycle supporting no real points. The claim is proved.

The proof of Theorems 1.3 and 1.4 is now complete, except the case n = 4 in Theorem 1.4.

Suppose then that n = 4, X has property (D), and z is a 2-cycle on X homologous to zero. We will show that 2z is rationally equivalent to a cycle supporting no real points.

Let α = [z] be the class of z in A_{2}(X) = A^{2}(X). By assumption, cℓ_{ℝ}^{2}(α) = 0. One still can find a vector bundle E on X for which (1) and (2) hold with n = 4 and k = 2. In other words,

(6) c_{1}(E) = 0, c_{2}(E) = α

(7) c_{1}^{top}(E(ℂ)|X(ℝ)) = 0, c_{2}^{top}(E(ℂ)|X(ℝ)) = e^{∗}(cℓ_{ℂ}^{2}(π^{∗}(α))).

We assert that c_{2}^{top}(E(ℂ)|X(ℝ)) = 0. Indeed, in view of (6) and (2.4), we obtain

w_{1}(E(ℝ)) = 0 and w_{2}(E(ℝ)) = cℓ_{ℝ}^{2}(α) = 0,

which in turn imply w_{4}(E(ℝ)⊕E(ℝ)) = 0. Since E(ℝ)⊕E(ℝ) and E(ℂ)|X(ℝ) are isomorphic as real vector bundles, the last equality yields

w_{4}(E(ℂ)|X(ℝ)) = 0.

If ρ : H^{4}(X(ℝ), ℤ) → H^{4}(X(ℝ), ℤ/2) is the homomorphism induced by the canonical projection ℤ → ℤ/2, then ρ(c_{2}(E(ℂ)|X(ℝ))) = w_{4}(E(ℂ)|X(ℝ)) [13, p. 171, Problem 14-B] and hence

(8) ρ(c_{2}(E(ℂ)|X(ℝ))) = 0.

Since X(ℝ) is a closed 4-dimensional manifold, ρ is injective on the torsion part of H^{4}(X(ℝ), ℤ). It follows from the second equality in (7) and the fact that X has property (D) that c_{2}(E(ℂ)|X(ℝ)) is a torsion element of H^{4}(X(ℝ), ℤ). Hence (8) implies c_{2}(E(ℂ)|X(ℝ)) = 0 and the assertion is proved.

One readily sees (but it is also a special case of [14]) that the conditions n = 4, E is of constant rank, and c_{i}(E(ℂ)|X(ℝ)) = 0 for i = 1, 2 imply that the complex vector bundle E(ℂ)|X(ℝ) is topologically stably trivial. Thus the real vector bundle E(ℝ) ⊕ E(ℝ) is also topologically stably trivial on X(ℝ). By Lemma 2.3, there exists an open neighborhood U of X(ℝ) in X for which the restriction (E ⊕ E)|U is algebraically stably trivial on U. In particular, c_{2}((E ⊕E)|U) = 0 in A^{2}(U). On the other hand, using (6), we get c_{2}(E ⊕E) = 2α in A^{2}(X). The last two equalities imply that the restriction of 2α to U is 0 in A^{2}(U) and hence, according to Lemma 2.2, 2z is rationally equivalent to a cycle supporting no real points.

If now z is an arbitrary 2-cycle on X, then 2z is homologous to zero and therefore, according to what we just proved, 4z is rationally equivalent to a cycle supporting no real points. Thus one can take r = 4 for n = 4 in Theorem 1.4. The proof is complete in all

cases. □

References

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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)