ALGEBRAIC CYCLES ON REAL VARIETIES AND Z/2-EQUIVARIANT ...

ALGEBRAIC CYCLES ON REAL VARIETIES AND Z/2-EQUIVARIANT HOMOTOPY THEORY PEDRO F. DOS SANTOS

Abstract. In this paper the spaces of algebraic cycles on a real projective variety X are studied as Z/2-spaces under the action of the Galois group Gal(C/R). In particular, the equivariant homotopy type of the group of algebraic p-cycles Zp (Pn C ) is computed. A version of Lawson homology for real varieties is proposed. The real Lawson homology groups are computed for a class of real varieties.

1. Introduction In the past ten years there has been renewed interest in the study of the topological groups of algebraic cycles on complex projective varieties, using homotopy theoretic techniques. The idea behind these results is that, for a complex projective variety X, the homotopy invariants of the groups of p-dimensional algebraic cycles (denoted here by Zp (X), with p ≤ dim X) carry information about the algebraic structure of X. In this context, the case X = PnC was particularly important. The computation of the homotopy type of Zp (PnC ) in [16] was the starting point for the definition, due to Friedlander, of a homology theory for projective varieties called Lawson homology [7]. The Lawson homology groups of a projective variety X are a set of bigraded invariants, Lp Hk (X), defined by Lp Hk (X) = πk−2p Zp (X), for all k ≥ 2p. This theory was later extended to arbitrary complex varieties by Lima-Filho [22]. The importance of Lawson homology comes from the fact that it has a rather accessible definition and still carries a lot of information about the algebraic structure of the variety X. For example, the groups of algebraic equivalence classes of p-cycles on X are the Lawson homology groups Lp H2p (X) (see [7]). Another reason for the interest in Lawson homology comes from the fact that it interpolates between motivic homology and 2000 Mathematics Subject Classification. 55P91; Secondary 14C05, 19L47, 55N91. 1

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PEDRO F. DOS SANTOS

singular homology (see [8]). This can be seen as a generalization of the fact that algebraic equivalence interpolates between rational and homological equivalence of algebraic cycles. Lawson homology groups also have independent interest as they are as homotopy groups of colimits of Chow varieties. Therefore they provide information about the topological properties of these varieties which are classical objects in algebraic geometry. The definition of Lawson homology for complex varieties relies heavily on the fact that, when X is a complex variety, the analytic topology can be used to endow Zp (X) with a Hausdorff topology. For varieties over more general fields no such topology is available and the definition of similar invariants requires the much more sophisticated machinery of etale homotopy theory [7]. In this paper we address the of problem defining a version of Lawson homology for varieties defined over R, namely, one that reflects the real structure. For this, we will adopt the point of view that a real variety X is (by Galois descent) a complex variety with the extra structure provided by the action of the Galois group of Gal(C/R) on its set of complex points. This Z/2-action naturally extends to the groups of cycles Zp (X) and is continuous w.r.t. the analytic topology. Thus, it is natural to define a version Lawson homology for real varieties in terms of equivariant homotopy invariants of its groups of algebraic cycles, endowed with the analytic topology. To introduce such a theory we must at least understand the equivariant homotopy type of Zp (PnC ). The first steps in this direction were taken by Lam. In his thesis [15] he proved that Lawson’s suspension Theorem holds equivariantly. More recently, the homotopy type of the spaces of real cycles (i.e. invariant under the action of the Galois group) was computed in [18]. Non-equivariantly, Zp (PnC ) is a product of classifying spaces for singular cohomology with integer coefficients in the even dimensions 0, 2, . . . , 2(n − p). Our first result is an equivariant version of this. It parallels the non-equivariant result very closely with the appropriate changes. Singular cohomology is replaced with RO(Z/2)-graded equivariant cohomology; Z is replaced with the constant Mackey functor Z; the dimensions 2, . . . , 2(n − p) above are replaced with the non-integral dimensions R1,1 , . . . , Rn−p,n−p — we use Rr,s to denote Rr+s with the Z/2-action of multiplication by −1 in the last s coordinates. Equivariant cohomology with Z coefficients in dimension Rr,s is denoted by H r,s (−; Z).

ALGEBRAIC CYCLES ON REAL VARIETIES

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Theorem 3.7. The space Zp (PnC ) is Z/2-homotopy equivalent to the following product of equivariant Eilenberg-Mac Lane spaces

(3.1)

'

Zp (PnC ) − →

n−p Y

K(Z, Rk,k ).

k=0

Furthermore, under these equivalences, Zp (Pn−1 ) includes in Zp (PnC ) as a factor C in the product (3.1) with constant last coordinate. Hence, the space of stabilized cycles, Z = colimn,p→∞ Zp (PnC ), is Z/2-homotopy equivalent to a weak product of equivariant Eilenberg-Mac Lane spaces (3.2)

'

Z− →

∞ Y

K(Z, Rk,k ).

k=0

Before considering cycles on other real varieties we look more closely at the space of stabilized cycles Z of (3.2). In [2] it was shown that Z has an infinite loop space structure induced by the operation of joining algebraic cycles. The inclusion of linear spaces into all cycles induces an infinite loop space map P : BU → Z that classifies the total Chern class. The existence an infinite loop structure with these properties proved a longstanding conjecture of Segal in homotopy theory. This result has an equivariant counterpart. Indeed, the pairing on the space of stabilized cycles, µ : Z × Z → Z induced by the join of algebraic cycles, and the map P : BU → Z are both equivariant (here BU is given the Z/2-action induced by complex conjugation). In Proposition 3.10 we show that µ classifies the cup product in H ∗,∗ (−; Z). The equivariant map P : BU → Z leads to an interesting connection with Atiyah’s KR-theory. Recall that a real bundle in the sense of Atiyah [1] is a complex bundle over a Z/2-space with an anti-linear involution which covers the involution on the base. Since BU with the Z/2-action mentioned above is the classifying space for real bundles, it is natural to expect that P should be closely related to characteristic classes for real bundles. A simple computation shows that H ∗,∗ (BU; Z) is a polynomial ring on certain classes ce1 , . . . , cen , . . . in the dimensions R1,1 , . . . , Rn,n , . . . which appear is the decomposition (3.2). In Proposition 3.12 we show that, under the isomorphism (3.2), P classifies the total Chern class 1 + e c1 + e c2 · · · . Having obtained a complete description of the Z/2-homotopy type of Zp (PnC ) and of the effect of the join map we propose a definition of real Lawson homology for real quasi-projective varieties. For a projective real variety X, we define the

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PEDRO F. DOS SANTOS

real Lawson homology groups by (Definition 4.4)

def

Lp Rn,m (X) = [S n−p,m−p , Zp (X)]Z/2

for n, m ≥ p and p ≤ dim X;

where S n−p,m−p is the one-point compactification of the representation Rn−p,m−p and [−, −]Z/2 is the set of equivariant homotopy classes. Given that Lp H2p (−) is the group of p-cycles modulo algebraic equivalence, a relation between Lp Rp,p (−) and algebraic equivalence is to be expected. In Proposition 4.6, we show that, for a real variety X, Lp Rp,p (X) is the quotient of the group of real p-cycles (i.e. defined over R) on X modulo an equivalence relation closely related to algebraic equivalence. This equivalence relation is a version in the level of algebraic cycles of Friedlander and Walker’s real topological equivalence for real algebraic bundles [11, Prop.1.6]. Using the techniques developed by Friedlander, Gabber [8] and Lima-Filho [23], we establish the existence of exact sequences, excision and a cycle map which takes values in the Z/2-equivariant homology of X with Z coefficients. In Proposition 4.20 we relate real Lawson homology to Lawson homology for complex varieties by showing that, considering a complex variety U as a real variety UR (see Example 4.19), we recover the Lawson homology of the complex variety U , i.e. we have Lp Rr,s (UR ) ∼ = Lp Hr+s (U ). Furthermore, we prove the following result.

Proposition 4.21. Let U be a real quasi-projective variety. Then there is a transfer map π ∗ : Lp Rr,s (U ) → Lp Hr+s (U ) and a restriction map π∗ : Lp Hr+s (U ) → Lp Rr,s (U ) such that the composition π∗ ◦ π ∗ is multiplication by 2. In particular, for any finitely generated module M over Z[ 21 ], Lp Rr,s (U ) ⊗ M is isomorphic to a submodule of Lp Hr+s (U ) ⊗ M . In Section 5 we compute compute real Lawson homology in several examples. In particular, the real Lawson homology groups of affine space An = PnC −Pn−1 , with its C standard real structure, and the effect of the cycle map are computed. Moreover, excision and the cycle map are used to establish a general result for a class of examples: real varieties with a real cell decomposition. A real variety X has a real cell decomposition if there is a filtration X = Xn ⊃ Xn−1 ⊃ · · · X0 ⊃ X−1 = ∅ by real subvarieties, such that Xi −Xi−1 is a union of affine spaces Anij (Definition 5.3).


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Theorem 5.4. Let X be a real quasi-projective variety with a real cell decomposition, then the map sp : Zp (X) −→ Ωp,p Z0 (X) is an equivariant homotopy equivalence. In particular, the cycle map induces an isomorphism Lp Rn,m (X) ∼ = Hn,m (X; Z), so that Lp Rn,m (X) is independent of p in this case. This class includes, for example, the Grassmannians Gp (Cn ) (with the real structure induced by sending a complex plane to its complex conjugate plane) and n m PnC × Pm C with the product of the standard real structures on PC and PC .

By considering different real structures one can produce many of different examples of real varieties for which the non-equivariant Lawson homology groups are known. Examples of this are PnC ×PnC with the action τ ·(x, y) = (¯ y, x ¯), real quadrics and the real Severi-Brauer varieties, PC (Hn ) (whose real structure is induced by multiplication of the homogeneous coordinates by the imaginary quaternion j). The real Lawson homology groups of some of these examples are computed in Section 5 and we expect that they should all be possible to compute. In the case of the Severi-Brauer varieties PC (Hn ) the homotopy groups of the real cycles have been computed in [17]. The complete equivariant homotopy type of their spaces of cycles is computed in [5] where a beautiful connection between the spaces Zp (PC (Hn )), Atiyah’s KR-theory and Dupont’s symplectic K-theory [6] is established. The paper is organized as follows. We summarize in §2 the results, definitions and notation from equivariant homotopy theory we will need. In §3 we compute the equivariant homotopy type of Zp (PnC ) and Z and study the natural map BU → Z. In §4 we propose a definition of real Lawson homology and prove its basic properties: exactness, homotopy invariance and existence of cycle map with values in equivariant homology with Z coefficients. In §5 we compute examples and show that the cycle map is an isomorphism for the class of real varieties with a real cell decomposition. The details of the proof of proposition 4.9 are given in §6. Acknowledgement. The author wishes to thank H. Blaine Lawson, Jr., for introducing him to this subject, suggesting the problem and for his guidance and support during the elaboration of this work; Daniel Dugger and Gustavo Granja for many fruitful discussions; Paulo Lima-Filho for suggestions and corrections.

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2. Definitions and results from equivariant homotopy theory In this section we summarize the definitions, results and notation from equivariant homotopy theory needed to state the results concerning the Z/2-equivariant homotopy type of spaces of algebraic cycles on real varieties. Our general reference for equivariant homotopy theory is [27]. 2.1. G-CW-complexes and G-homotopies. Let Dn , S n−1 be the unit disk and unit sphere of Rn , respectively, considered as a G-spaces with the trivial action. The unit interval I = [0, 1] is also equipped with the trivial action. Definition 2.1. A G-CW complex X is the union of G-spaces X0 ⊂ X1 ⊂ · · · Xn ⊂ · · · such that X0 is a disjoint union of orbits G/H and Xn is obtained from Xn−1 by attaching G-cells Dn × G/H along maps S n−1 × G/H → Xn−1 . We say that Xn is the n-skeleton of X. Given G-maps f, g : X → Y , a G-homotopy from f to g is a homotopy from f to g which is equivariant for the diagonal action of G on X × I. A G-map f : X → Y is a G-homotopy equivalence (or equivariant homotopy equivalence) if there exist a G-map h : Y → X such that f ◦ h and h ◦ f are G-homotopic to the identity. We will use the symbol ' to denote G-homotopy equivalence. 2.2. Coefficient systems. Let G be a finite group and let FG be category of finite G-sets and G-maps. The coefficients for ordinary equivariant (co)homology are (contravariant) covariant functors from FG to the category Ab of abelian groups which sends disjoint unions to direct sums. Notation 2.2. If V is a representation of a finite group G, S V denotes the one point compactification of V and, for a based G-space X, ΩV X denotes the space of based maps F (S V , X). The space F (S V , X) is equipped with the its standard G-space structure. The set of equivariant homotopy classes [S V , X]G is denoted by πV (X). Example 2.3. For each G-space X and G-representation V , there is a contravariant coefficient system π V (X) whose value on G/H is [S V ∧ G/H+ , X]G . The value of π V (X) on a G-map f : G/H → G/K is induced by id ∧f . Actually, without extra assumptions on X, we must assume V contains two copies of the trivial


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representation to ensure that π V (X) takes values in Ab. If X is, for example, a topological abelian G-group this extra hypothesis is not necessary. If V is the trivial representation Rn , the coefficient system π V is just denoted by π n . The functors π n are also called Bredon homotopy groups. The following result will be used throughout: a G-map f : X → Y between GCW complexes is a G-homotopy equivalence if and only if it induces an isomorphism on the Bredon homotopy groups. A Mackey functor M is a pair (M∗ , M ∗ ) of additive functors M∗ : FG → Ab and op M ∗ : FG → Ab with the same value on objects and which transform each pull-back

diagram f

A −−−−→   gy

B   yh

k

in FG

C −−−−→ D into a commutative diagram in Ab M∗ (f )

M (A) −−−−→ M (B) x x  M ∗ (h) M ∗ (g)  M∗ (k)

M (C) −−−−→ M (D) In this paper we will be interested in the case where M = Z is the Mackey functor constant at Z. This Mackey functor is uniquely determined by the conditions: (i) Z(G/H) = Z, for H ≤ G; (ii) If K ≤ H, the value of the contravariant functor Z∗ on the projection ρ : G/K → G/H is the identity. 2.3. RO(G)-graded homology and cohomology. Given a Mackey functor M ∗ 0 there exists an RO(G)-graded cohomology theory HG (−; M ) such that HG (pt; M ) =

M . For each real orthogonal representation V there is a classifying space K(M, V ) ∗ which classifies HG (−; M ) in dimension V . It is important to note that the spaces

K(M, V ) can have homotopy in several integral dimensions. This is precisely the case of the K(M, V )-spaces considered below (see 3.6). The spaces K(M, V ) fit together to give an equivariant Eilenberg-Mac Lane spectrum HM. In particular, this means that, given G-representations V , W , there is a G-homotopy equivalence K(M, V ) ' ΩW K(M, V + W ). In this paper we will be interested in the case where G = Z/2 and M = Z is the Mackey functor constant at Z. Recall that RO(Z/2) ∼ = Z ⊕ Z and let 1 and ρ denote the one-dimensional trivial representation, and the one-dimensional

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PEDRO F. DOS SANTOS

non-trivial representation, respectively. Then, for p, q ∈ N, Rp,q ∼ = p·1+q·ρ where Rp,q denotes Rp+q with the Z/2-action of multiplication by −1 in the last q coordinates. For simplicity, we will use the notation H p,q (−; Z) for Z/2-equivariant cohomology with Z coefficients in dimension Rp,q . In order to avoid confusion, ∗ singular cohomology groups with Z coefficients will be denoted by Hsing (−; Z).

Similar conventions will be used for homology groups. Notation 2.4. In the case G = Z/2, we will use the notation S p,q instead of S R πp,q (−) instead of πRp,q , Ωp,q instead of ΩR

p,q

p,q

,

and π p,q instead of π Rp,q .

∗ Given a Mackey functor M the computation of HG (pt; M ) is a non-trivial prob-

lem. In the case G = Z/p and M = Z these groups were originally computed by Stong and appear in [3]. In the case p = 2 they are as follows

(2.1)

H n,m (pt; Z) =

   Z/2       Z    Z/2      0

n even, 0 ≤ −n < m n even, −n = m n odd, 1 < n ≤ −m otherwise 0

0

0

0

There is also a cup product ∪ : H n,m (−; Z) ⊗ H n ,m (−; Z) → H n+n ,m+m (−; Z) which, in particular, gives a ring structure to H ∗ (pt; Z). We will denote this ring be R. The product on R appears also in [3] but we will not give here as it is not so easy to describe and we will not need it. It is important to note that for any Z/2-space X, H∗ (X; Z) is a module over R. 2.4. G-fibrations. Since restricting to the case of Z/2 is not more elucidating we will again give the definitions for a finite group G. A G-fibration is an equivariant map π : E → B with the G-homotopy lifting property with respect to for G-CW-complexes. A G-fibration π : E → B gives rise to exact sequences in homotopy groups. In the particular case of G = Z/2, for each q, there is an exact sequence · · · → πp,q E → πp,q B → πp−1,q F → · · · ending at · · · → π0,q F → π0,q E → π0,q B.


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2.5. Dold-Thom theorem. In the non-equivariant case, the computation of the homotopy type of the spaces of algebraic cycles Zp (PnC ) is a direct consequence of the suspension theorem and classical Dold-Thom Theorem. To compute the equivariant homotopy type of Zp (PnC ) we will need a generalization of the this classical theorem. Definition 2.5. Let X be a G-space. The topological group of zero cycles on X is P denoted by Z0 (X). Its elements are formal sums i ni xi , with ni ∈ Z and xi ∈ X. There is an augmentation homomorphism deg : Z0 (X) → Z. Its kernel is denoted by Ze0 (X). Note that, denoting by X+ the union of X with a disjoint point fixed by the G-action, Z0 (X) is isomorphic to Ze0 (X+ ). Theorem 2.6. [4] Let G be a finite group, let X be a based G-CW-complex and let V be a finite dimensional G-representation, then there is a natural equivalence e VG (X; Z). πV Ze0 (X) ∼ =H In particular, Ze0 S V

is a K(Z, V ) space.

Given a G-space X there is a G-spectrum HZ ∧ X+ such that π? (HZ ∧ X+ ) = H? (X; Z) (where ? refers to RO(G)-grading). The theorem above shows that Z0 (X) is G-homotopy equivalent to the zero-th space of the G-spectrum HZ ∧ X+ .

3. The equivariant homotopy type of Z and real vector bundles In this section we study the space of algebraic p-cycles in PnC as a Z/2-space under the action of Gal(C/R). We begin by reviewing some basic definitions concerning algebraic cycles. An effective algebraic p-cycle in PnC is a finite formal c =

P

i

ni Vi where each ni

is a positive integer and the Vi ’s are irreducible subvarieties of dimension p in PnC . P The degree of c is defined as deg(c) = i ni deg(Vi ), where deg(Vi ) is the degree of S as an irreducible subvariety of PnC . The support of c is the algebraic set i Vi and is denoted by |c|. The set of effective algebraic p-cycles of degree d in PnC can be given the structure of an algebraic set, denoted by Cp,d (PnC ) (see [28]). If X is an algebraic subset of PnC , the subset Cp,d (X) ⊂ Cp,d (PnC ) consisting of those cycles whose support is contained in X, is an algebraic subset of Cp,d (PnC ). The algebraic structure of Cp,d (PnC ) depends on the embedding of X in PnC .

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Definition 3.1. Let X ⊂ PnC be a projective subvariety. The Chow monoid of X is the set ( def

Cp (X) =

a

Cp,d (X) = {0}

a

d≥0

) a

Cp,d (X) .

d>0

Here we consider Cp,d (X) with its analytic topology. The Grothendieck group (or naive group completion) of Cp (X) is denoted Zp (X) and will be called the group of p-cycles on X. The group Zp (X) is endowed with the quotient topology induced by the map π : Cp (X) × Cp (X) → Zp (X) such that π(c, c0 ) = c − c0 . Remark 3.2. It is important to note that although the algebraic structure of Cp,d (X) depends on the embedding, the homeomorphism type of Cp (X) does not, cf. [7]. Definition 3.3. Let X be a subvariety of PnC . The algebraic suspension of X is the subset of Pn+1 consisting of all points of the lines joining points of X to the C point (0 : · · · : 0 : 1). It is a subvariety of Pn+1 defined by the same equations as C X but now considered as equations in n + 2 variables. The operation Σ / increases dimension by one and keeps the codimension fixed. Note that, as a topological space, Σ / X is the Thom space of the line bundle O(1)|X. More generally, given varieties X ⊂ PnC and Y ⊂ Pm C the set of points of the lines joining points of X to points of Y — which we denote by X#Y — is a subvariety of Pn+m+1 = PnC #PnC . Observe that dim X#Y = dim X + dim Y + 1 and C Σ / X = X#P0C . The importance of the operation Σ / for the computation of the Z/2-homotopy type of Zp (PnC ) is given by the following result of Lam. This result is an equivariant version of Lawson’s suspension theorem [16]. Theorem 3.4. [15] The suspension map Σ / : Zp (PnC ) −→ Zp+1 (Pn+1 ) C is a Z/2-homotopy equivalence. Proof. This is proved in [18, Prop.9.1]. Definition 3.5. The space Z of stabilized cycles is Z = lim Zp (PnC ) n,p→∞


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where the limit is defined w.r.t. the suspension map and the natural inclusions Zp (PnC ) ⊂ Zp (Pn+1 ). C Observe that, by Theorem 3.4, Z is Z/2-homotopy equivalent to Z0 (P∞ C ). The join extends to a map Z ∧ Z → Z which is equivariant. Definition 3.6. Let σ : Cn+1 × Cn+1 → C2n+2 be the “shuffle” isomorphism defined by σ(z, w) = (z0 , w0 , . . . , zn , wn ). Consider the composition #

σ

∗ 2n+1 Zp (PnC ) ∧ Zp (PnC ) −→ Z2p+1 (PC ) −→ Z2p+1 (P2n+1 ). C

The compositions above are compatible with the inclusions Zp (PnC ) → Zp (Pn+1 ) C and the suspension map. Thus they define a pairing µ : Z ∧ Z → Z. It is clear that µ is equivariant. Using the equivariant version of the Dold-Thom Theorem we can now compute the equivariant homotopy type of Zp (PnC ) and Z. Theorem 3.7. The space Zp (PnC ) is Z/2-homotopy equivalent to the following of equivariant Eilenberg-Mac Lane spaces (3.1)

n−p Y

'

Zp (PnC ) − →

K(Z, Rk,k ).

k=0

Furthermore, under these equivalences, Zp (Pn−1 ) includes in Zp (PnC ) as a factor C in the product (3.1) with constant last coordinate. Hence, the space of stabilized cycles, Z = colimn,p→∞ Zp (PnC ), is Z/2-homotopy equivalent to a weak product of equivariant Eilenberg-Mac Lane spaces (3.2)

'

Z− →

∞ Y

K(Z, Rn,n ).

k=0

Proof. By Theorem 3.4, Zp (PnC ) ' Z0 (Pn−p ) hence we need to show C Z0 (PnC ) '

n Y

K(Z, Rk,k ).

k=0

This equivalence is a consequence of Theorem 2.6 and the fact that PnC has an equivariant cell decomposition with one cell of dimension Rk,k for k = 0, . . . , n; the cell decomposition is given by the filtration P0C ⊂ P1C ⊂ · · · ⊂ PnC of PnC (note that ∼ PnC /Pn−1 = S n,n ). Using this decomposition, it can be proved that (see Lemma 5.5) C

(3.3)

H∗,∗ (PnC ; Z) ∼ =

n M k=0

e ∗,∗ (S k,k ; Z); H

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PEDRO F. DOS SANTOS

It follows that H∗,∗ (PnC ; Z) is a free module over the cohomology ring of a point, R, with one generator xk in dimension (k, k), for k = 0, . . . , n. By the equivariant version of Dold-Thom (2.6), the same is true of π∗,∗ Z0 (PnC ). We proceed to make an explicit choice of generators xk . Let x1 be the equivariant map S 1,1 3 x 7→ x − ∞ ∈ Z0 (P1C ). For k > 1 we observe that S k,k ∼ = S 1,1 ∧ . . . ∧ S 1,1 (k times) and set xk equal to the composition Σ / −k+1

x1 #···#x1

S k,k −−−−−−−→ Zk−1 (P2k−1 ) −−−−→ Z0 (PkC ). C We claim that xk is a generator of πk,k Z0 (PnC ), for n ≥ k. To see this we observe that its image under the functor F that forgets the Z/2-action is a generator of π2k Z0 (PnC ) because it coincides with the choice of generator for π2k Z0 (PnC ) made in [19]. Now, (3.3) and theorem 2.6 imply that π k,k Z0 (PnC ) is the Mackey functor Z (see the proof of Proposition 3.10 for a more detailed explanation). Since the map Z(pt) → Z(Z/2) induced by Z/2 → pt is the identity we conclude that xk is a generator of πk,k Z0 (PnC ). Define F :

n Y

Ze0 S k,k → Z0 (PnC )

k=0

as the group homomorphism extension of x1 + · · · + xn . Since F is a group homoWn morphism it extends to a morphism of spectra HZ ∧ k=0 S k,k → HZ ∧ PnC . The map π(F ) induced by F on the homotopy groups is a map between homology of Wn k,k and of PnC which are R-modules. Moreover, since F is the zero-th map of k=0 S a morphism of spectra, the induced map on homotopy groups π(F ) is a morphism of R-modules. But the homologies of these two spaces are isomorphic free R-modules, and F is defined so that π(F ) is a bijection on the generators. It follows that π(F ) is an isomorphism hence F is an equivariant homotopy equivalence. By construction the equivalence F sends the inclusion Z0 (Pn−1 ) ⊂ Z0 (PnC ) to C the inclusion n−1 Y

K(Z, Rk,k ) × {∗} ⊂

k=0

n Y

K(Z, Rk,k );

k=0

hence the statement about the inclusion of Zp (Pn−1 ) in Zp (PnC ) is a consequence of C the commutative diagram Zp (Pn−1 ) C x  Σ /

−−−−→

Zp (PnC ) x Σ /

Z0 (Pn−p−1 ) −−−−→ Z0 (Pn−p ) C C


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The equivalence (3.2) now follows from the fact that Z is a colimit of the spaces Zp (PnC ) w.r.t. the maps Zp (Pn−1 ) ⊂ Zp (PnC ) and the equivalences Σ / : Zp (PnC ) → C Zp+1 (Pn+1 ). C

Remark 3.8. The Z/2-homotopy equivalence F :

Q

k≥0

Ze0 S k,k → Z is com-

pletely determined by a choice of generators xk of πk,k Z and the fact that it is an abelian group homomorphism. Henceforth when we refer to the equivalence (3.2) we mean the group homomorphism F determined by the choice generators xk made in the proof above. Q

This equivalence agrees with the equivalence

k≥0

K(Z, 2k) ' Z of [19] when

we forget the Z/2-action. It is also possible to define an equivalence G : Z → Q k,k e using the Z/2-homeomorphisms PkC ∼ = SP k (P1C ) ( see [18]). k≥0 Z0 S We now analyze the equivariant pairing µ.

In [19] it is proved that, non-

equivariantly, µ classifies the cup product in singular cohomology with integer coefficients. As mentioned before, there is a notion of cup product in H ∗,∗ (−; Z). In Proposition 3.10 we show that µ classifies the cup product in equivariant cohomology with Z coefficients. Note 3.9. We will need to following fact: let G be a finite group and let {Xα } be W a family of G-spaces. Then Ze0 ( α Xα ) is G-homeomorphic to the weak product Q e W α Z0 (Xα ). If iα0 denotes the inclusion Xα0 ⊂ α Xα the homeomorphism is given by ⊕α iα∗ ; see [26] for a proof. Proposition 3.10. Under the equivalence of Theorem 3.7 the map µ:Z ∧Z →Z is Z/2-homotopic to the map ν:

∞ Y

! Ze0 S k,k

k=0

∧

∞ Y

! Ze0 S k,k

→

k=0

∞ Y

Ze0 S k,k ,

k=0 0

0

defined as the biadditive extension of the smash product of spheres, S k,k ∧ S k ,k → 0

0

S k+k ,k+k . In particular, µ classifies the cup product in Z/2-equivariant cohomology with Z coefficients. 0

0

Proof. Consider the inclusion of ik,k0 : S k,k ∧ S k ,k → Z ∧ Z given by 0 0 0 0 S k,k ∧ S k ,k 3 x ∧ y 7→ (x − ∞) ∧ (y − ∞) ∈ Ze0 S k,k ∧ Ze0 S k ,k ⊂ Z ∧ Z.

14

PEDRO F. DOS SANTOS

We start by showing that, for every k, k 0 , the compositions µ ◦ ik,k0 and ν ◦ ik,k0 are Z/2-homotopic. In [19] it is proved that these compositions are non-equivariantly homotopic. We will see that the forgetful map F : [S n,n , Z]Z/2 −→ [S n,n , Z]

(3.4)

is an isomorphism. Thus, the fact that µ ◦ ik,k0 and ν ◦ ik,k0 are homotopic implies that they are also Z/2-homotopic. Note that F is the map induced in equivariant cohomology by the projection p : S n,n ∧Z/2+ → S n,n and the cofiber of p is S n,n+1 . Since, for any Z/2-space X [X, Z]Z/2 ∼ =

∞ M

e k,k (X; Z), H

k=0

F fits into an exact sequence ∞ M

∞ M

e k,k (S n,n+1 ; Z) → [S n,n , Z]Z/2 → [S n,n , Z] → H

k=0

e k+1,k (S n,n+1 ; Z). H

k=0

From (2.1) it follows that the first and last groups on this sequence are both trivial and hence F is an isomorphism. We now use the Z/2-homeomorphism ∞ Y

Ze0 S

k,k

∞ _

∼ = Ze0

k=0

! S

k,k

k=0

mentioned before. From what was said above we see that the restrictions of µ and W∞ W∞ 0 0 ν to k=0 S k,k ∧ k=0 S k ,k are Z/2-homotopic. Let H:

∞ _

S k,k ∧

k=0

∞ _

0

0

S k ,k ∧ I+ → Z

k=0

be an equivariant homotopy from the restriction of ν to the restriction of µ. Extend H to an equivariant homotopy through biadditive maps ! ! ∞ ∞ _ _ 0 0 F : Ze0 S k,k × Ze0 S k ,k × I → Z. k=0

k=0

Since ∧ is biadditive we have F (−, −; 0) = ∧. Now, µ is biadditive up to Z/2homotopy, so F (−, −; 1) is Z/2-homotopic to µ. Since F (−, −; t) is biadditive, F descends to Ze0

∞ _ k=0

! S

k,k

∧ Ze0

∞ _

! S

k0 ,k0

∧ I+ .

k=0

This completes the proof that ν and µ are Z/2-homotopic. The statement regarding the cup product is a consequence of fact that the pairing ∧ on the Z/2-Ω-prespectrum Rp,q 7→ Ze0 (S p,q ) induces the cup product in H ∗,∗ (−; Z). Indeed, the parings HZ ∧ HZ → HZ are in bijective correspondence


15

with the pairings Z Z → Z — where denotes the box product of Mackey functors defined by Lewis (see [21] and [27]). This correspondence sends a pairing HZ ∧ HZ → HZ to the map induced on π 0 , using the fact that, given two Mackey functors M and M 0 , π 0 (HM ∧ HM 0 ) = M M 0 . But Z Z is just Z so, up to equivariant homotopy, there is only one pairing HZ ∧ HZ → HZ which agrees with the usual pairing on the non-equivariant Eilenberg-Mac Lane spectrum HZ, when we forget the Z/2-action. Our product does this hence it classifies the cup product in H ∗,∗ (−; Z).

One of the interesting features of the space Z is that the classifying space BU maps naturally into it, as follows. We have BU = lim Gn (C2n ). n→∞

2n−1 Linear spaces in C2n are degree one cycles on PC , thus BU maps to the compo-

nent of degree one, Z(1), of space of stabilized cycles and this map is equivariant. Definition 3.11. Let P : BU → Z(1) be the equivariant map induced at the finite level by the inclusions Gn (C2n ) ⊂ Zn (P2n−1 ). The compatibility of these inclusions C with the maps used to define the limits BU and Z guarantees that P is well-defined. The space BU with the action induced by complex conjugation is the classifying space for the (connective) KR-theory of Atiyah [1]. Its Z/2-equivariant cohomology can be easily computed using the equivariant cell decomposition coming from the Schubert cells. Denoting by R the cohomology ring of a point H ∗,∗ (pt; Z), as before, we get H ∗,∗ (BU; Z) ∼ c1 , . . . , e cn , . . .], = R[e where the e cn ’s are classes of dimension (n, n), n ∈ Z+ , whose images under the forgetful functor to singular cohomology are the Chern classes, cn ; see Lemma 5.5 for a similar computation. The classes e cn are universal characteristic classes for real vector bundles. We call them equivariant Chern classes for real vector bundles. We show that, as in the non-equivariant case, the map P : BU → Z classifies the total equivariant Chern class, i.e., it classifies 1+e c1 + e c2 + · · · + e cn + · · · Proposition 3.12. Let ιen denote the universal (n, n)-dimensional class in the cohomology of K(Z, Rn,n ). Using the isomorphism of Equation (3.2), we consider

16

PEDRO F. DOS SANTOS

ιen has an element in the cohomology of Z. Then P ∗ (ιen ) = e cn . Proof. The proof goes exactly as in the non-equivariant case [19]. One observes that BU(n) = limk→∞ Gk (Cn+k ) maps to lim Zk (Pn+k ) ' Z0 (PnC ) ' C

k→∞

n Y

Ze0 S k,k ,

k=0

where the limit is defined using the map Σ / : Zk (Pn+k ) → Zk+1 (Pn+k+1 ). Recall C C that under the isomorphism of Equation (3.2) the inclusion Z0 (Pn−1 ) ⊂ Z0 (PnC ) C is the inclusion as a factor in the product. Thus P ∗ (ιen )| BU(n − 1) = 0 and so P ∗ (ιen ) = λe cn , for some λ ∈ C. Let F denote the forgetful functor from equivariant cohomology to singular cohomology. Since F ιen = ιn (the generator of 2n Hsing (Ze0 S 2n ; Z)) and by [19] P ∗ (ιn ) = cn we conclude that λ = 1. def

The equivariant product µ restricts to a product on the fixed point set ZR =

Z Z/2 giving a ring structure to the Z-graded homotopy groups of ZR . The computation of this ring is one of the main results of [18]. In view of Proposition 3.10 this ring can be interpreted as a subring of the equivariant cohomology of a point: Lemma 3.13. The ring (ZR , µ) is isomorphic to a subring of H ∗,∗ (pt; Z). Proof. For k > 0, Theorem 3.7 gives,

(3.5)

πk (ZR ) ∼ =

M

[S k , Ze0 (S n,n )]Z/2 ∼ =

n≥0

M

e n,n (S k,0 ; Z) ∼ H =

n≥0

M

H n−k,n (pt; Z).

n≥0

So we see from (3.5) that π∗ (ZR ) is isomorphic as a group to the subring consisting of the H p,q (pt; Z) such that q ≥ 0. The fact that the product structure is the same follows from Proposition 3.10.

From (2.1) we can also conclude that the homotopy type of K(Z, Rn,n )Z/2 ' Z/2 Ze0 (S n,n ) is

(3.6)

  K(Z, 2n) × K(Z/2, 2n − 2) × K(Z/2, 2n − 4) × · · · × K(Z/2, n) n even,  K(Z/2, 2n − 1) × K(Z/2, 2n − 3) × · · · × K(Z/2, n)

n odd.


17

From this decomposition it follows that, for a space X with trivial Z/2 action there is a natural equivalence

H n,n (X; Z) ∼ =

 n/2  M   2n−2k 2n  H (X; Z) ⊕ Hsing (X; Z/2) n even,   sing k=1

(n−1)/2  M   2n−2k−1  Hsing (X; Z/2)  

n odd.

k=0

4. A version of Lawson homology for real varieties In this section we propose a definition of Lawson homology for real algebraic varieties. This definition is a natural equivariant generalization of Lawson homology for projective varieties, and we show that it still carries all the basic properties which make Lawson homology computable.

4.1. Lawson homology for complex varieties. We start by recalling the definition of the Lawson homology groups for complex projective varieties. These groups are a hybrid of algebraic geometry and algebraic topology: for a projective variety X, the Lawson homology groups of X, Lp Hk (X) are the following set of invariants def

Lp Hk (X) = πk−2p Zp (X)

for k ≥ 2p and p ≤ dim X.

sing In the case p = 0, it follows by the Dold-Thom theorem that L0 Hk (X) ∼ = Hk (X; Z).

For k = 2p, we get Lp H2p (X) = π0 Zp (X) and Friedlander has shown [7] that this is isomorphic to the group of p-cycles on X modulo algebraic equivalence. We recall that algebraic equivalence is generated by the following relation: c0 , c1 ∈ Cp (X) are equivalent if there exists a smooth curve C, a (p + 1)-cycle , Z, on X × C equidimensional over C (i.e. Z meets each fiber X × {t} properly, t ∈ C), and points t0 , t1 of C, such that ci = Z • (PnC × {ti }), i = 0, 1. Here we assume that X is embedded in PnC and • denotes intersection of cycles which is well defined as a cycle since these cycles intersect properly [13]. The functorial properties of Lawson homology are a consequence of the following basic facts (see [7]): let X, Y, W be projective varieties then (i) a morphism f : X → Y induces a continuous map f∗ : Zp (X) → Zp (Y ), for any 0 ≤ p ≤ dim X; (ii) a flat morphism g : W → X of relative dimension r ≥ 0 induces a continuous map g ∗ : Zp (X) → Zp+r (W ), for any 0 ≤ p ≤ dim X.

18

PEDRO F. DOS SANTOS

The maps f∗ and g ∗ are induced by push-forward and flat pull-back of cycles, respectively. They induce maps f∗ : Lp Hn (X) → Lp Hn (Y ) and g ∗ : Lp Hn (X) → Lp+r Rn+2r (Y ). 4.2. Real varieties. We will adopt the point of view that real varieties are complex varieties with extra structure. Definition 4.1. A real quasi-projective algebraic variety U is a quasi-projective variety with an anti-holomorphic involution τ : U → U . A morphism of real quasi-projective varieties (U 0 , τ 0 ), (U, τ ) is a morphism of quasi-projective varieties f : U 0 → U such that f ◦ τ 0 = τ ◦ f . The projective space PnC with τ (x0 : · · · : xn ) = (x0 : · · · : xn ) is an example of a real variety. Any real quasi-projective variety has a real embedding into a projective space, i.e., there is an embedding φ : U → PnC which is equivariant w.r.t the action induced by complex conjugation on PnC ( see [29], for example). If X is a real variety the anti-holomorphic involution τ induces a Z/2-action on X. The fixed points X Z/2 are the real points of X ( i.e. defined over R) and are denoted X(R). Example 4.2. Let H = C ⊕ Cj denote the quaternions, and let PC (Hn ) be the projective space of complex lines in Hn . The anti-homolomorphic involution on PC (Hn ) given by multiplication by j from the left on Hn gives it a real structure. As a complex variety, PC (Hn ) is isomorphic to P2n−1 , but is clear these are very C 2n−1 different as real varieties since PC (Hn )(R) = ∅ and PC (R) = P2n−1 . The real R

varieties PC (Hn ) are examples of Severi-Brauer varieties. 4.3. Lawson homology for real varieties. A version of Lawson homology for real varieties should make use of the extra structure provided by the anti-holomorphic involution in order to provide information about the real structure. In fact, if X is a real variety the Chow varieties Cp,d (X) are also real varieties [28, Chapter I.9]. The corresponding anti-holomorphic involution Cp,d (X) → Cp,d (X) is induced from the involution τ on X as follows: observe that, for a subvariety V , τ∗ V is a subvaP P riety of the same degree as V . Given a cycle c = i ni Vi set τ∗ (c) = i ni τ∗ (Vi ). The real points of Cp,d (X) are the real p-cycles of degree d on X, i.e., of the form P i ni Vi , where the Vi ’s are irreducible real subvarieties of X. The Chow monoid Cp (X) and the group of p-cycles Zp (X) are Z/2-spaces under the action of the continuous involution τ∗ .


19

Notation 4.3. The fixed points of Cp (X) and Zp (X) under the action of τ∗ will be denoted by Cp (X)(R) and Zp (X)(R), respectively. We will refer to these spaces as the monoid of real p-cycles and the group of real p-cycles on X, respectively. It is important to note that the Z/2-homeomorphism type of Zp (X) is an invariant of the real variety X. Indeed, suppose f : (X 0 , τ 0 ) → (X, τ ) is a real isomorphism. It follows that f induces a Z/2-equivariant homeomorphism f∗ : Zp (X 0 ) → P P Zp (X) defined by f∗ i ni Vi = i ni f∗ (Vi ); see [7]. Thus the equivariant homeomorphism type of Zp (X), equipped with the action τ∗ , is an invariant of the real structure on X. From here onwards the groups Zp (X) are always considered as Z/2-spaces with this action. Going back to the problem of defining a version of the Lawson homology groups for real varieties which are invariants of the real structure, it is natural to look at spheres with Z/2-actions and consider equivariant homotopy classes. We are, therefore, naturally led to the following definition. Definition 4.4. Let X be a real projective variety. The real Lawson homology groups of X, are the groups def

Lp Rn,m (X) = πn−p,m−p Zp (X)

for n, m ≥ p and p ≤ dim X.

The previous observations imply that the groups Lp Hn,m (X) are invariants of the real structure on X. Remark 4.5. It follows from Theorem 2.6 that, for cycles of dimension zero, the real Lawson homology groups of a real projective variety X are the Z/2-equivariant homology groups of X with coefficients in the Mackey functor Z, i.e. L0 Rn,m (X) ∼ = Hn,m (X; Z). In the case (n, m) = (p, p) we have Lp Rn,m (X) = π0 (Zp (X)(R)) and it is natural to expect that these groups should be related to the notion of algebraic equivalence. The Proposition below answers this question. The answer is formulated in terms of an equivalence relation for real cycles introduced by Friedlander and Walker in [11], which uses the analytic topology on the set of real points of a real curve. Proposition 4.6. Let X ⊂ PnC be a real variety. The real Lawson homology group Lp Rp,p (X) is the group of real p-cycles modulo the equivalence relation generated by: two cycles c0 , c1 ∈ Cp (X)(R) are equivalent if there exists a smooth real curve C, a real (p + 1)-cycle Z on X × C equidimensional over C ( i.e. Z meets each fiber

20

PEDRO F. DOS SANTOS

X × {t} properly, t ∈ C), and points t0 , t1 lying in the same connected component of C(R), such that ci = Z • (PnC × {ti }), i = 0, 1. Proof. Since π0 (Zp (X)(R)) is the Grothendieck group of the monoid π0 (Cp (X)(R)) it suffices to show that π0 (Cp (X)(R)) is the quotient of Cp (X)(R) by the equivalence relation ∼ defined above. Let C be a smooth real curve, by [7, Cor.1.5] there is 1-1 correspondence between morphisms C → Cp (X) of real varieties and real (p + 1)-cycles on X × C which are equidimensional over C. Under this correspondence, the morphism f corresponding to a cycle Z ∈ Cp+1 (X × C), as above, satisfies f (t) = Z • (PnC × {t}), t ∈ C. Thus if c0 , c1 are real p-cycles such that c0 ∼ c1 then c0 and c1 lie in the same connected component of Cp (X)(R). Suppose now that c0 , c1 are real p-cycles which lie in the same connected component of Cp (X)(R). Then c0 , c1 lie in the same component of Cp,d (X) for some d. We need to show that c0 ∼ c1 . We will use following fact shown in [11, Prop.1.6]: given a real quasi-projective variety T and points t0 , t1 lying in the same connected component of T (R) there exist smooth real curves C0 , . . . , Ck and points ai , bi , lying in the same connected component of Ci (R), for each i ∈ {0, . . . , k}, and morphisms of real varieties fi : Ci → T such that f0 (a0 ) = t0 , fi (bi ) = fi+1 (ai+1 ) and fk (bk ) = t1 . Applying this with T = Cp,d (X), t0 = c0 , t1 = c1 and the correspondence between real morphisms Ti → Cp,d (X) and equidimensional (p + 1)-cycles on X × C we conclude that c0 ∼ c1 .

Remark 4.7. The computation of the real Lawson homology groups of a real variety X completely determines the homotopy type of the spaces of real cycles Zp (X)(R). This is an immediate consequence of that fact that these spaces are topological abelian groups and hence products of Eilenberg-Mac Lane spaces and so their homotopy type is completely determined by the homotopy groups. The homotopy groups πk (Zp (X))(R) are the real Lawson homology groups Lp Rk+p,p (X). This computation is important in itself as the Chow varieties Cp,d (X) are classical objects in algebraic geometry about which not much is known. The functorial properties of Lawson homology mentioned above also hold in the real case: let X, Y and W be projective varieties, let f : X → Y be morphism of real varieties and let g : W → X be a real flat morphism of relative dimension r. The continuous maps f∗ : Zp (X) → Zp (Y ) and g ∗ : Zp (X) → Zp+r (W ) are equivariant


21

and thus induce maps f∗ : Lp Rn,m (X) → Lp Rn,m (Y ) and g ∗ : Lp Rn,m (X) → Lp+r Rn+r,m+r (Y ). Next we establish the basic properties of real Lawson homology such as the existence of relative groups and exact sequences for pairs. Following Lima-Filho’s definition in the non-equivariant case, we define Definition 4.8. Let (X, X 0 ) be a real pair, i.e. X 0 is a real subvariety of X. The group of relative p-cycles is the quotient def

Zp (X, X 0 ) =

Zp (X) , Zp (X 0 )

with the quotient topology. Note that Zp (X, X 0 ) is a Z/2-space with the action induced from Zp (X). The real Lawson homology groups of the pair (X, X 0 ) are def

Lp Rn,m (X, X 0 ) = πn−p,m−p Zp (X, X 0 ) where, as above, n, m ≥ p and 0 ≤ p ≤ dim X. The next result is the main step in showing the existence of long exact sequences in real Lawson homology. The proof is a simple generalization to the equivariant context of [23, Thm.3.1]. Proposition 4.9. The short exact sequence of topological groups (4.1)

0 −→ Zp (X 0 ) −→ Zp (X) −→ Zp (X, X 0 ) −→ 0

is an equivariant fibration sequence. Proof. See Section 6.

Proposition 4.10. Let (X, X 0 , X 00 ) be a real triple. Then the short exact sequence of topological groups 0 −→ Zp (X 0 , X 00 ) −→ Zp (X, X 00 ) −→ Zp (X, X 0 ) −→ 0 is an equivariant fibration sequence. As a consequence, there is a long exact sequence of real Lawson homology groups → Lp Rn,m (X 0 , X 00 ) → Lp Rn,m (X, X 00 ) → Lp Rn,m (X, X 0 ) → Lp Rn−1,m (X 0 , X 00 ) → Proof. As in Proposition 4.9 we only need to show that the exact sequence of topological groups (4.2)

0 −→ Zp (X 0 , X 00 )(R) −→ Zp (X, X 00 )(R) −→ Zp (X, X 0 )(R) −→ 0

22

PEDRO F. DOS SANTOS

is a fibration sequence. Just as in the non-equivariant case [23, Prop.3.1] , this follows from Proposition 4.9 in a standard fashion. Since (4.2) is a sequence of topological groups the result will follow if we can show that (4.2) has a local crosssection at zero (see [30]). By the proof of Proposition 4.9 there is a neighborhood U of zero in Zp (X, X 0 )(R) and a section s : U → Zp (X)(R) to the projection π1 : Zp (X)(R) → Zp (X, X 0 )(R). Composing s with the projection π2 : Zp (X)(R) → Zp (X, X 00 )(R) we get the desired section.

Finally, we recall a fundamental result of Lima-Filho that provides a definition of Lawson homology for quasi-projective varieties. This also yields a localization sequence which is an important computational tool. Theorem 4.11. [23, Thm.4.3] A relative isomorphism Ψ : (X, X 0 ) → (Y, Y 0 ) induces an isomorphism of topological groups: Ψ∗ : Zp (X, X 0 ) → Zp (Y, Y 0 ) for all p ≥ 0. Remark 4.12. Our observation here is that, if Ψ : (X, X 0 ) → (Y, Y 0 ) is a relative real isomorphism of real pairs then Ψ∗ : Zp (X, X 0 ) → Zp (Y, Y 0 ) is an equivariant homeomorphism. Following [23] we define: Definition 4.13. Let U be a real quasi-projective variety. The group of p-cycles on U is the topological group def

Zp (U ) = Zp (X, X 0 ) where (X, X 0 ) is a real pair such that X − X 0 is isomorphic to U as real varieties. Such a pair is called a real compactification of U . The group Zp (U ) is considered as a Z/2-space with the action induced from the action on Zp (X). Note that such compatification always exists: if U ⊂ PnC is a real quasi-projective variety then let X be the Zariski closure of U in PnC and set X 0 = X − U . The real Lawson homology groups of U are defined as the groups of the pair (X, X 0 ): def

Lp Rn,m (U ) = πn−p,m−p Zp (U ) where, as before, n, m ≥ p and 0 ≤ p ≤ dim X.


23

Remark 4.14 (Independence of compactification and functoriality). Theorem 4.11 shows that the definition of Lp Rn,m (U ) is independent of the compactification and is covariant w.r.t proper maps of real quasi-projective varieties. In fact, a proper map of real quasi-projective varieties, Ψ : U → V , induces a set-theoretic map Ψ∗ : Zp (U ) → Zp (V ). The question is whether Ψ∗ is continuous. We recall Lima-Filho’s argument to show continuity of Ψ∗ . Suppose U, V have real compactifications (X, X 0 ) and (Y, Y 0 ), respectively. Let Γ ⊂ X × Y be the closure of the graph Graph(Ψ), where X × Y is endowed with the product real structure. Set Γ0 = Γ − Graph(Ψ). Let π1 and π2 denote the projections on the first and second factors, respectively. Then π1 : (Γ, Γ0 ) → (X, X 0 ) is a relative real isomorphism and π2 : (Γ, Γ0 ) → (Y, Y 0 ) is a map of real pairs because Ψ is proper. From Theorem 4.11 it follows that π1∗ is an equivariant homeomorphism. The continuity of Ψ∗ follows because it coincides with the composition 0 0 π2∗ ◦ π1 −1 ∗ : Zp (X, X ) → Zp (Y, Y ).

If Ψ is an isomorphism of real quasi-projective varieties π2∗ is also equivariant homeomorphism and hence so is Ψ∗ . The long exact sequence for triple now gives the localization sequence for real Lawson homology: let V be a real closed subset of a real quasi-projective variety U and let n, m ≥ p. Then there is a long exact sequence of real Lawson homology groups (4.3)

· · · → Lp Rn,m (V ) → Lp Rn,m (U ) → Lp Rn,m (U − V ) → Lp Rn−1,m (V ) → · · ·

ending at · · · → Lp Rp,m (V ) → Lp Rp,m (U ) → Lp Rp,m (U − V ). As a consequence we can now prove the real version of the “homotopy property” for Lawson homology. π

Proposition 4.15. Let U be a real quasi-projective variety and let E − → U be a real algebraic vector bundle of rank k. Then the flat pull-back of cycles π ∗ : Zp (U ) −→ Zp+k (E) is an equivariant homotopy equivalence.

24

PEDRO F. DOS SANTOS

Proof. The proof goes exactly as in the non-equivariant context [8]: it suffices to show that the map induced by π ∗ on the Bredon homotopy groups is an isomorphism. Using localization and the 5-lemma we can reduce to the case where E is trivial. At this point one can use induction on k to reduce to the case of k = 1. Then one can further reduce to the case where U has a projective closure U such that E → U is the restriction to U of O(1)|U → U . The result now follows from the suspension Theorem.

4.4. Cycle map. An important feature of Lawson homology is the existence of a natural map sp : Lp Hk (X) → Hksing (X; Z) called cycle map (or cycle class map). In in the case k = 2p, we have Lp H2p (X) = π0 (Zp (X)) and sp is the map which sends the component of c ∈ Zp (X) to the 2p-homology class that c represents (see [25]). This is the motivation for calling it the cycle map. Our next goal is to define an appropriate version of this map for real Lawson homology. It will play an important role in the examples of Section 5. There are several ways to define the cycle map for Lawson homology (see [25], [12] and [8]). The definition we give here is a natural modification of that of Friedlander and Gabber [8]. The two main ingredients in this construction are intersection with divisors to get a map s : Zp (X) → Ω2 Zp−1 (X), and the Dold-Thom theorem to map to singular homology by composition with sp (p iterations of s). To apply this construction to real Lawson homology we need to check that intersecting with a real divisor is equivariant w.r.t the action of Gal(C/R). Once this is achieved we apply Theorem 2.6 to get a cycle map which takes values in equivariant homology with Z coefficients. Let U be a real quasi-projective variety and let D be a real Cartier divisor, whose inclusion of D in U is denoted by iD : D ,→ U . This means that D is defined by the vanishing of a real section, sD , of a real algebraic line bundle π : LD → U . Let P V be the complement of |D| in U (recall that if D = i ni V , the support of D is the algebraic set |D| = ∪i Vi ). Since i∗D LD is closed in LD and LD |V = LD − i∗D LD we have an exact sequence 0 −→ Zp (i∗D LD ) −→ Zp (LD ) −→ Zp (LD |V ) −→ 0, which by Proposition 4.9 is an equivariant fibration sequence. Consider the composition res ◦ sD ∗ : Zp (U ) → Zp (LD ) → Zp (LD |V ).


25

We claim that res ◦ sD ∗ is equivariantly homotopic to the zero map: in [8] a homotopy H is defined such that Ht is multiplication by t in the fibres of LD , for t ∈ [1, ∞[, and as the zero map for t = ∞. It is clear that H is equivariant. This equivariant homotopy determines an equivariant map σD : Zp (U ) → Zp (i∗D LD ), well defined up to equivariant homotopy. Definition 4.16. The intersection with a real divisor D is defined as the composition def

i!D = (π ∗ )−1 ◦ σD : Zp (U ) → Zp (i∗D LD ) → Zp−1 (|D|). We now explain how, using intersection with divisors, we can define a version of the map s of [8] in the context of real varieties. Let U be a real quasi-projective variety. Consider the composition (4.4)

ω i!U Zp (U ) ∧ Ze0 P1C − → Zp (U × P1C ) −→ Zp−1 (U )

where ω(V, t) = V × {t} and i!U denotes intersection with U × {∞} which is a real divisor in U × P1C . This map is clearly equivariant for the diagonal action on Zp (U ) × Ze0 P1C where P1C is equipped with the action induced by complex conjugation. Definition 4.17. For a real quasi-projective U variety let i!U and ω be as above. Consider P1C embedded in Ze0 P1C by t 7→ t − ∞. We define s : Zp (U ) → Ω1,1 Zp−1 (U ) as the adjoint of the restriction of i!U ◦ ω to Zp (U ) ∧ P1C . The map s induces a map in real Lawson homology groups s∗ : Lp Rn,m (U ) → Lp−1 Rn,m (U ). We define the cycle map for the real Lawson homology of U as the map sp∗ : Lp Rn,m (U ) → L0 Rn,m (U ). Frequently we abuse notation and denote s∗ and sp∗ by s and sp , respectively. The cycle map is the motivation for the indexing in real Lawson homology. If U is a projective variety, the group L0 Rn,m (U ) is isomorphic to Hn,m (U ; Z). We think of the elements of Lp Rn,m (U ) as having algebraic dimension p and homological dimension (n, m).

26

PEDRO F. DOS SANTOS

The map s also induces a map in usual Lawson homology, Lp Hn (U ) → Lp−1 Hn (U ), which we denote by s∗ as well. In [8] it is proved that , in usual Lawson homology, s∗ can be defined by a different construction. Consider P1C embedded in Ze0 P1C , as above, by mapping t ∈ P1C to t − ∞. The adjoint of the composition Σ / −2

#

Zp (U ) ∧ P1C −→ Zp+1 (U #P1C ) −−−→ Zp−1 (U ) is another a map s0 : Zp (U ) → Ω2 Zp−1 (U ), which is homotopic to s. Therefore the map s∗ can also be realized in the following way. The inclusion of P1C in Ze0 P1C is a generator x for π2 Ze0 P1C ∼ = Z. Joining with x gives a map Σ−2

πn−2p Zp (U ) → πn+2−2p Zp+1 (U #P1C ) −−−→ πn+2−2p Zp−1 (U ) that coincides with s∗ in usual Lawson homology. All this works equivariantly but now x is seen as the generator of the group π1,1 Ze0 P1C ∼ = Z, so joining with it gives a map Σ / −2

πn−p,m−p Zp (U ) → πn+1−p,m+1−p Zp+1 (U #P1C ) −−−→ πn+1−p,m+1−p Zp−1 (U ) that coincides with s∗ in real Lawson homology. Remark 4.18. The following observation is often useful. Suppose the map s : Zp (X) → Ω1,1 Zp−1 (X) is an equivariant homotopy equivalence. Assuming the diagram Zp (X)   Σ /y

s

−−−−→ Ω1,1 Zp−1 (X)   Ω1,1 Σ /y s

Zp+1 (Σ / X) −−−−→ Ω1,1 Zp (Σ / X) is commutative, it follows that s : Zp+1 (Σ / X) → Ω1,1 Zp (Σ / X) is also an equivariant homotopy equivalence. To see that the diagram above commutes we use the definition of the adjoint of s given by joining with a generator of π1,1 Ze0 P1C . The result follows from the commutativity of the diagram Zp (X) × Ze0 P1C   Σ / ×idy

Zp+1 (Σ / X) × Ze0 P1C up to Z/2-homotopy.

#

2

Σ / −2

#

3

Σ / −2

−−−−→ Zp+1 (Σ / X) −−−−→ Zp−1 (X)     Σ /y Σ /y

−−−−→ Zp+2 (Σ / X) −−−−→ Zp (Σ / X)


27

4.5. Relation to Lawson homology for complex varieties. To understand the relation between real Lawson homology and usual Lawson homology we start by observing that every complex variety gives rise to a real variety. Example 4.19. Let U ⊂ PnC be a quasi-projective variety. Then U q U is a real variety with the involution induced by complex conjugation (it can be embedded as a real subvariety in PnC × A1 , for example). We have Zp (U q U ) ∼ = Zp (U ) × Zp (U ) and, under this isomorphism, the involution is given by τ · (C1 , C2 ) = (C2 , C1 ). Recall that, for pointed Z/2-spaces S and X, F (S, X) denotes the space of based maps with the conjugation action: f 7→ τ ◦ f ◦ τ . It follows that, if we consider Zp (U ) equipped with the trivial Z/2-action, then (4.5)

Zp (U q U ) ∼ = F (Z/2+ , Zp (U )).

A more elucidating explanation of this example can be given using the language of schemes, as follows. The natural inclusion R ⊂ C determines a morphism Spec C → Spec R. Hence any scheme X over C can be seen as a scheme over R. We will denote this scheme by XR . For affine varieties this construction corresponds to thinking of an algebra over C as an algebra over R, by restriction of scalars. The set of complex ` points of XR is precisely X(C) X(C), and the corresponding anti-involution sends a point x in one component to x in the other component. Thus, if U is a complex variety, we can view it as a real variety. The following Proposition shows that real Lawson homology gives the usual Lawson homology when applied to a complex variety. Proposition 4.20. Let U be a quasi-projective complex variety. Then Lp Rr,s (UR ) ∼ = Lp Hr+s (U ). Proof. By Equation (4.5) we have Lp Rr,s (UR ) ∼ = [S r−p,s−p ∧ Z/2+ , Zp (UR )]Z/2 ∼ = [S r+s−2p , Zp (U )] = Lp Hr+s (U ). Suppose now that (U, τ ) is a real variety. Forgetting the real structure we can ` think of it as a complex variety U . The corresponding real variety UR is just U U with the involution which sends x of one component to the point τ (x) in the other

28

PEDRO F. DOS SANTOS

component. The folding map π : U

`

U → U is a proper flat map of real varieties

of relative dimension zero. Hence it induces maps π ∗ : Zp (U ) → Zp (UR ) π∗ : Zp (UR ) → Zp (U ). It is easy to check that π∗ ◦ π ∗ is multiplication by 2. As a consequence we have Proposition 4.21 (cf.[11, Cor.5.9]). Let U be a real quasi-projective variety. Then there is a transfer map π ∗ : Lp Rr,s (U ) → Lp Hr+s (U ) and a restriction map π∗ : Lp Hr+s (U ) → Lp Rr,s (U ) such that the composition π∗ ◦ π ∗ is multiplication by 2. In particular, for any finitely generated module M over Z[ 12 ], Lp Rr,s (U ) ⊗ M is isomorphic to a submodule of Lp Hr+s (U ) ⊗ M . Proof. Clear.

4.6. Relation to other theories. There is also a cohomological version of Lawson homology, called morphic cohomology, defined by Friedlander and Lawson as follows. For simplicity we only consider the case of normal varieties. Given normal projective varieties X and Y , we let Mor(X, Cr (Y )) denote the abelian monoid of algebraic morphisms from X to Cr (Y ), endowed with the compact open topology. The monoid of algebraic cocyles of codimension t on X is the topological quotient monoid def

Ct (X) = Mor(X, C0 (PtC ))/Mor(X, C0 (Pt−1 C )). Its associated Grothendieck group is called the topological group of codimension t algebraic cocycles on X; it is denoted Z t (X). The morphic cohomology groups of X, Lp H k (X), are the bigraded groups defined by (see [10]) def

Lt H k (X) = π2t−k Z t (X). If X is a real variety then Z t (X) becomes naturally a Z/2-space and it is natural to define a version of morphic cohomology for real varieties in terms equivariant homotopy groups of Z t (X). This has been done recently by Friedlander and Walker in [11]. There they introduce a set of invariants for real varieties called semitopological K-theory and denoted KRsemi (−), which interpolate between algebraic K-theory and Atiyah’s KR-theory. Moreover, they show real morphic cohomology interpolates between motivic cohomology and singular cohomology and use it to define higher Chern classes for KRsemi (−). Using techniques similar to those of [11] it would be possible to show that, for real varieties, real Lawson interpolates


29

between motivic homology and equivariant homology with Z coefficients. We will not pursue this here as it would take to far afield. A duality between Lawson homology and morphic cohomology has been established by Friedlander and Lawson in [9]. In a forthcoming paper with Lima-Filho we will show that for smooth real varieties real Lawson homology is dual to real morphic cohomology. With this result, the computations performed in Section 5 concerning spaces of cycles yield the equivariant homotopy type of the corresponding spaces of algebraic morphisms. 5. Examples and computations In this section we compute the real Lawson homology groups for some real varieties. The main tools in these computations are the localization sequence (4.3) and the cycle map sp . We start with the fundamental example of affine space An with its standard real structure. Example 5.1 (The real Lawson homology of affine space An ). Let An have the real structure given by complex conjugation of its coordinates. We call this real structure on An the standard real structure. By definition, Zp (An ) =

Zp (PnC ) . Zp (Pn−1 ) C

Since An → An−p is a real algebraic bundle, it follows from Proposition 4.15 Zp (An ) ' Z0 (An−p ) =

Z0 (Pn−p ) C Z0 (Pn−1−p ) C

' K(Z, Rn−p,n−p ).

Here we used the following important property of the equivalence in Theorem 3.7: under this equivalence the inclusion Zp (Pn−1 ) ⊂ Zp (PnC ) corresponds to the incluC sion n−p−1 Y k=0

n−p Y Ze0 S k,k → Ze0 S k,k k=0

as a factor. Thus, for 0 ≤ p ≤ n and r, s ≥ p, Lp Rr,s (An ) ∼ = πr−p,s−p K(Z, Rn−p,n−p ) ∼ = H n−r,n−s (pt; Z). Moreover we see that the cycle map sp gives an equivariant homotopy equivalence sp : Zp (An ) −→ Ωp,p Z0 (An ). Since sp will play a central role in the examples to follow we will explain this in detail.

30

PEDRO F. DOS SANTOS

Recall the description of s on Zp (PnC ) as the adjoint of the restriction of the composition (5.1)

# Σ / −2 Zp (PnC ) ∧ Ze0 P1C −→ Zp+1 (Pn+2 ) −−−→ Zp−1 (PnC ) C

to Zp (PnC ) ∧ P1C . Where we consider P1C embedded in Ze0 P1C by the map t 7→ t − ∞. Recall also that we have a complete description of the action of the join and suspension maps on the cycles of PnC : the suspension Theorem identifies Zp (PnC ), Zp−1 (PnC ), ), respectively. The commutativity of ) and Z0 (Pn+1−p canonically, with Z0 (Pn−p C C the diagram (up to equivariant homotopy) Zp (PnC ) × P1C   Σ / −p ×idy

#

Σ / −2

#

Σ / −1

−−−−→ Zp+1 (Pn+2 ) −−−−→ C   −p / yΣ

Zp−1 (PnC )   −(p−1) / yΣ

) ) −−−−→ Z0 (Pn+1−p Z0 (Pn−p ) × P1C −−−−→ Z1 (Pn+2−p C C C shows that the, under the above identifications, the map (5.1) is identified with the product µ (see Proposition 3.10) restricted to Z0 (Pn−p ) ∧ P1C . C By Theorem 3.7, Z0 (Pn−p ) C

'

n−p Y

Ze0 S k,k ,

k=0

and by Proposition 3.10, µ restricted to Z0 (Pn−p ) ∧ P1C is identified with the map C n−p Y

n−p+1 Y ∧ Ze0 S k,k ∧ P1C − → Ze0 S k,k

k=0

k=1

∧ induced by the smash map S k,k ∧ S 1,1 − → S k+1,k+1 (recall that P1C ∼ = S 1,1 ). It

follows that this map is one of the structural maps of the Ω-Z/2-prespectrum Rp,q → Ze0 (S p,q ) hence its adjoint is an equivariant homotopy equivalence (5.2)

n−p Y k=0

n−p+1 Y Ze0 S k,k ' Ω1,1 Ze0 S k,k k=1

This concludes the analysis of the map s : Zp (PnC ) → Ω1,1 Zp−1 (PnC ). To obtain the result for the affine space An we observe that s is natural, so s : Zp (An ) → Ω1,1 Zp−1 (An ) is obtained by passing to the quotient in (5.2) and we get that, for An , s is the adjoint of ∧ Zp (An ) ∧ S 1,1 ' Ze0 S n−p,n−p ∧ S 1,1 − → Ze0 S n+1−p,n+1−p ' Zp−1 (An ) which is an equivariant homotopy equivalence, as desired. The following summarizes our conclusions regarding the real Lawson homology of affine space An .


31

Lemma 5.2. The space Zp (An ) is an equivariant Eilenberg-Mac Lane space of type K(Z, Rn−p,n−p ), for every 0 ≤ p ≤ n. Moreover, cycle map sp : Zp (An ) −→ Ωp,p Z0 (An ) is an equivariant homotopy equivalence. We will now make use of the cycle map and the localization sequence to prove the following general result about the real Lawson homology of real varieties with a real cell decomposition. The next definition is an adaptation to the real case of [25, Definition 5.3] Definition 5.3. Let (X, Y ) be a pair of real projective varieties. We say that X is a real algebraic cellular extension of Y if there is a filtration X = Xn ⊃ Xn−1 ⊃ · · · X0 ⊃ X−1 = Y by real projective subvarieties Xi such that Xi − Xi−1 is a union of affine spaces Anij . If Y = ∅ we say that X has a real cell decomposition. Theorem 5.4. Let X be a real quasi-projective variety with a real cell decomposition, then the map sp : Zp (X) −→ Ωp,p Z0 (X) is an equivariant homotopy equivalence. In particular, the cycle map induces an isomorphism Lp Rn,m (X) ∼ = Hn,m (X; Z), so that Lp Rn,m (X) is independent of p in this case. Proof. The result is proved by induction using the localization sequence and the fact that, by Example 5.1, it holds for affine spaces. Assume that sp : Zp (Xi−1 ) −→ Ωp,p Z0 (Xi−1 ) is an equivariant homotopy equivalence. Applying the localization sequence and the cycle map sp we get a map of long exact sequences

Lp Rk+p,p (Xi )

sp

L0 Rk+p,p (Xi )

/ Lp Rk+p,p (Xi − Xi−1 ) sp

/ L0 Rk+p,p (Xi − Xi−1 )

/ Lp Rk+p−1,p (Xi−1 )

sp

/

/ L0 Rk+p−1,p (Xi−1 )

/

32

PEDRO F. DOS SANTOS

ending at Lp Rp,p (Xi−1 )

/ Lp Rp,p (Xi )

sp

L0 Rp,p (Xi−1 )

sp

/ L0 Rp,p (Xi )

/ Lp Rp,p (Xi − Xi−1 )

/ 0

sp

/ L0 Rp,p (Xi − Xi−1 )

/ 0.

Exactness at the last group of the bottom row follows from the fact that, since Xi−1 has a real cell decomposition, Hm−1+p,p (Xi−1 ; Z) = 0, for all m ∈ Z. This follows from Lemma 5.5 since, Hr,s (pt; Z) ∼ = R−r,−s = 0 for all r, s > 0. By the assumptions and the 5-Lemma it follows that sp∗ : Lp Rk+p,p (Xi ) −→ L0 Rk+p,p (Xi ) is an isomorphism for all k ≥ 0. Translating this into homotopy groups, it means that sp∗ : πk (Zp (Xi )(R)) −→ πk (Ωp,p Z0 (Xi ))

Z/2

is an isomorphism for all k ≥ 0. Since we already know that sp is a non-equivariant homotopy equivalence [25], this implies that sp : Zp (Xi ) → Ωp,p Z0 (Xi ) is an equivariant homotopy equivalence.

Lemma 5.5. Let X be a real variety with a real cell decomposition X = Xn ⊃ Xn−1 ⊃ · · · X0 ⊃ X−1 = ∅ such that Xi − Xi−1 is a union of affine spaces Anij . Let R denote the cohomology ring of a point, H ∗,∗ (pt; Z). Then H∗,∗ (X; Z) is an R-free module. Each cell Anij gives rise to a generator xi,j in dimension Rnij ,nij . Proof. Denote the unit disk of the representation Rnij ,nij by D(Rnij ,nij ). The real cell decomposition gives X an equivariant cell decomposition with cells of type D(Rnij ,nij ). The proof is by induction on the cells: by Definition 5.3, X0 is a disjoint union of points fixed by the action, so the result holds. Assume it also holds for Xi−1 and consider the cofibration sequence Xi−1 + −→ Xi+ −→

_

S nij ,nij .

j

There is a long exact sequence (5.3)

−→ Hr,s (Xi ; Z) −→

L e δ nij ,nij ; Z) −→ Hr−1,s (Xi−1 ; Z) −→ j Hr,s (S


33

Observe that this is an exact sequence of R-modules and, by assumption, the homology of Xi−1 is free on generators xk,j , k < i, of dimensions (nkj , nkj ). Also e ∗,∗ (S nij ,nij ; Z) ∼ H = H nij −∗,nij −∗ (pt; Z) = Rnij −∗,nij −∗ so, in particular, this R-module is free and generated by an element xi,j in dimension (nij , nij ) (xi,j is sent to the identity element in R by the isomorphism above). The connecting homomorphism δ in the sequence (5.3) is determined by the image of the generators xi,j . But the induction hypothesis implies that this image is zero because Rr−1,r = 0 for all r ∈ Z; see Equation (2.1). This completes the proof.

The following are examples of real varieties with a real cell decomposition. Example 5.6 (The Grassmannians Gq (Cn+1 )). The variety Gq (Cn+1 ) has a real structure given by the action induced by complex conjugation in Cn+1 . The Schubert cells give a real cell decomposition for Gq (Cn+1 ). Example 5.7 (Products of varieties with real cell decompositions). Real varieties with a real cell decomposition form a class which is closed under products. So, for example, PnC × Pm C has a real cell decomposition and we have that the group Lp Rα (PnC × Pm C ) is isomorphic to the α degree part of the RO(Z/2)-graded module R[x, y]/(xn , y m ) where R is the cohomology of a point and x, y have degree (1, 1). Example 5.8 (Quadrics with signature zero). Any real smooth quadric in Pn−1 is C equivalent to a quadric of the form def Qn,k = (x1 : · · · : xn ) ∈ Pn−1 |x21 + · · · + x2k − x2k+1 − · · · − x2n = 0 C where k ≤ n/2. We consider the case, n = 2k, i.e. the quadratic form defining the quadric has signature zero. We will show that the cycle map is an isomorphism. From now on we use homogeneous coordinates (X : Y ) = (x1 : . . . : xn : y1 : . . . : yn ) for the points of P2n−1 . In these coordinates the quadratic form is XX T − Y Y T . The C point p0 = (X0 : Y0 ) = (0 : . . . : 0 : 1 : 0 : . . . : 0 : 1) is a real point of Q2n,n and the tangent plane to Q2n,n through p0 is H = (X : Y ) ∈ P2n−1 |(X : Y ) · (X0 : −Y0 ) = 0 C

34

PEDRO F. DOS SANTOS

and Q2n,n ∩ H = (X : Y ) ∈ P2n−1 |xn − yn = 0 and XX T − Y Y T = 0 . C Using coordinates x1 , . . . , xn−1 , y1 , . . . , yn−1 and t = xn + yn for H we see that the 2 quadric Q2n,n ∩ H is given by the equation x21 + · · · + x2n−1 − y12 − · · · − yn−1 = 0. Let

H 0 be the real hyperplane given by the equation t = xn + yn = 0. The intersection Q2n,n ∩ H ∩ H 0 is a quadric Q2n−2,n−1 and we have Q2n,n ∩ H = Q2n−2,n−1 #p0 . Thus Q2n,n ∩ H ∼ / Q2n−2,n−1 . Assume that the cycle map is an isomorphism = Σ Lp Rα (Q2n−2,n−1 ) → Hα (Q2n−2,n−1 ; Z). From Remark 4.18 it follows that the same holds for Σ / Q2n−2,n−1 . It is also easy to see that, if π : P2n−1 −p0 → H 0 is the projection onto H 0 centered C at p0 , then π|Q2n,n − Q2n,n ∩ H is an isomorphism onto H 0 − H ∩ H 0 ∼ = A2n−2 . Since everything is real, this is a real isomorphism. It is easy to check that Q4,2 ∼ = P1C × P1C with the standard real structure, hence, by Example 5.7 the cycle map is an isomorphism in this case. Using induction on n, the sequence for the real pair (Q2n,n , Q2n,n ∩ H) and the five Lemma, it follows that the cycle map Lp Rr,s (Q2n,n ) −→ Hr,s (Q2n,n ; Z) is an isomorphism. This reduces the computation of real Lawson homology to the computation of the equivariant homology of Q2n,n . The next example is a very simple case — albeit somewhat artificial — in which the cycle map is an isomorphism but the variety doesn’t have a real cell decomposition. Example 5.9. Let U ⊂ PnC be a quasi-projective variety for which the nonequivariant s map Zp (U ) → Ω2 Zp−1 (U ) is a homotopy equivalence. Let UR be the real variety obtained from U by restriction of scalars. Recall from Example 4.19 ` that UR = U U with the anti-holomorphic involution x 7→ x. We will show that s : Zp (UR ) → Ω1,1 Zp−1 (UR ) is an equivariant homotopy equivalence. From (4.5) we have Zp (UR ) ∼ = F (Z/2+ , Zp (U )). Hence there is a commutative diagram Zp (UR )(R)   sy

∼ =

−−−−→

Zp (U )   sy

Z/2 ∼ = Ω1,1 Zp−1 (UR ) −−−−→ Ω2 Zp−1 (U )


35

where the left and right vertical arrows denote the equivariant and the non-equivariant s maps, respectively. It follows that the equivariant s map is a Z/2-homotopy equivalence. Example 5.10. Consider the variety PnC × PnC with the real structure given by (5.4)

τ · (X, Y ) = (Y , X)

(X, Y ) ∈ PnC × PnC .

We will show that the s map is an equivariant homotopy equivalence. In the case n = 0 there is nothing to prove. Assume the result holds for n − 1. We have (5.5)

PnC × PnC = An × An ∪ An × Pn−1 ∪ Pn−1 × An ∪ Pn−1 × Pn−1 . C C C C

Note that An × An (with the action of (5.4)) is a real subvariety and it is actually isomorphic to A2n with the standard real structure. The isomorphism is √ (X, Y ) 7→ (X + Y, −1(X − Y )). Also the second factor in the decomposition (5.5) can be written as the disjoint union U q τ · U where U = An × Pn−1 . By Example 5.9 we know that the s map C Zp (U q τ · U ) → Ω1,1 Zp−1 (U q τ · U ) is a Z/2-homotopy equivalence. Finally, by induction, the cycle map is also an equivalence in the case of the last factor, Pn−1 × C Pn−1 . By localization and the 5-lemma it follows that the map s : Zp (PnC × PnC ) → C Ω1,1 Zp−1 (PnC × PnC ) is an equivariant homotopy equivalence. Example 5.11 (The real Severi-Brauer curve). Let X be the Severi-Brauer curve PC (H) defined in Example 4.2. If one identifies PC (H) with the two sphere S 2 , the involution is the antipodal map. In particular, there are no fixed points so X cannot have a real cell decomposition. The cycle map is very simple in this case because there are no cycles above dimension 1 and Z1 (X) ∼ = Z. The s map, in this case, sends Z1 (X) to Ω1,1 Z0 (X) and we know from [23] it is a non-equivariant homotopy equivalence. A direct homology computation shows that the induced map s∗ : π0 Z1 (X) → π1,1 Z0 (X) is an isomorphism. Also π1+n,1 Z0 (X) = 0, for n > 0 — because PC (H) has no homology in dimensions (r, s) with r + s > 2 — hence s is an equivariant homotopy equivalence. Example 5.12 (Quadrics with signature 3). It is easy to check that PC (H) is isomorphic as a real variety to the plane quadric Q3,0 . From Example 5.8 it follows that the quadric of signature 3, Q2n−1,n−1 , is obtained from Q3,0 by adding real

36

PEDRO F. DOS SANTOS

cells Ak and taking suspensions. Using exact sequences, the 5-lemma and the results of the previous examples, it follows that the cycle map sp : Zp (Q2n−1,n−1 ) −→ Ωp,p Z0 (Q2n−1,n−1 ) is an equivariant homotopy equivalence. Example 5.13 (Quadrics with signature 2). From Example 5.8 it follows that the signature 2 quadric, Q2n+2,n , is obtained from Q4,1 by adding real cells Ak and taking suspensions. One can check that Q4,1 ∼ = P1C × P1C with the real structure of Example 5.10. It follows that the cycle map is also an equivariant homotopy equivalence in this case. Of course the cycle map is not an isomorphism in general, otherwise Lawson homology would not be very interesting. Products of elliptic curves (and abelian varieties in general) provide examples for which the non-equivariant cycle map is not an isomorphism. The reason is the following. The homology classes in the image sing of the cycle map for usual Lawson homology, sp : Lp H2p (−) → H2p (−; Z), have

Hodge type (p, p) and abelian varieties have homology classes which are not of this type, hence sp is not surjective (see [12] and [20] for details). The same argument can be adapted to produce examples of real varieties for which the (equivariant) cycle map is not an isomorphism. Example 5.14. Let X be the product of elliptic curves C/Λ × C/Λ where Λ is the √ lattice Z ⊕ Z · −1. Complex conjugation in C descends to an anti-holomorphic involution on C/Λ, giving it a real structure. As a topological space, X is Z/2homeomorphic to S 1,0 × S 0,1 × S 1,0 × S 0,1 . We will show that the cycle map is not an isomorphism from L1 R1,1 (X) to H1,1 (X; Z). Let α ∈ H1,1 (S 1,0 × S 0,1 ; Z) ∼ = Z, be the generator and set β = i∗ (α), where i embeds S 1,0 × S 0,1 in X as the subspace S 1,0 × {0} × {0} × S 0,1 . If β were in the image of the cycle map, then its image under the forgetful functor to singular homology, F(β), would be in the image of the cycle map for usual Lawson homology. But this is impossible s

because, as mentioned above, all classes in the image of the composite L1 R1,1 (X) − → F

H1,1 (X; Z) −→ H2sing (X; Z) are of Hodge type (1, 1) and F(β) is not of Hodge type (1, 1): let z, w be the complex coordinates for C2 . Then dz ∧ dw is a closed (2, 0)R form on X. We have F (β) dz ∧ dw 6= 0 hence F(β) is not a (1, 1)-cycle.


37

6. Proof of Proposition 4.9 The existence an of exact sequence associated to a pair of real varieties (X, X 0 ) is one of the basic properties of real Lawson homology. We could have defined the relative Lawson homology groups as relative homotopy groups of the pair (Zp (X), Zp (X 0 )) thus yielding a long exact sequence in real Lawson homology. Definition 4.8 has the advantage of being very geometric giving us a lot of control over Zp (X, X 0 ). Its disadvantage is that it is not obvious that it yields the desired long exact sequence. The purpose of this Section is to prove this fact. Proof of Proposition 4.9. The Proposition is a consequence of a result of Lima-Filho [24, Thm.5.2], which shows that under certain conditions, a pair of abelian topof0 → C e → C/ e C f0 , where logical monoids (C, C 0 ) gives rise to a fibration sequence C e and C f0 are the Grothendieck groups of C, C 0 , respectively, with the quotient C topology. We will check that this result applies. Let (X, X 0 ) be a pair of real algebraic varieties. Recall that def

a

Cp (X) =

Cp,d (X)

d≥0

is a monoid under addition of cycles. It is endowed with the disjoint union topology; the algebraic sets Cp,d (X) are equipped with their analytic topology. This monoid is filtered by def

Cp,≤d (X) =

a

Cp,k (X).

k≤d

The real structures on X and X 0 induce real structures on the Chow varieties Cp,d (X), Cp,d (X 0 ). Set C = {Cp (X)}(R) and C 0 = {Cp (X 0 )}(R). Note that the Grothendieck groups of C, C 0 are Zp (X)(R) and Zp (X 0 )(R), respectively. The monoid C is free and C 0 is freely generated by a subset of generators of C. In the language of [24, Def.5.1(b)], (C, C 0 ) is a free pair. The monoid C is filtered by Cd = {Cp,≤d (X)}(R). This is a filtration by compact subsets that satisfy Cd + Cd0 ⊂ Cd+d0 . In the language of [24, Def.2.4(ii)], C is cS filtered. We endow C ×C with the product filtration: (C ×C)d = n+m≤d Cn ×Cm . The last condition we need to check is that with id

def

(C × C)d = ((C × C)d−1 + ∆ + C 0 × C 0 ) ∩ (C × C)d ,

(where ∆ denotes the diagonal) the inclusion id (C ×C)d ⊂ (C ×C)d is a cofibration. This will show that the pair (C, C 0 ) is properly c-filtered [24, Def.5.(a)].

38

PEDRO F. DOS SANTOS

Observe that Cd and Cd0 = Cd ∩ C 0 are the real points of the Chow varieties Cp,≤d (X) and Cp,≤d (X 0 ) so, in particular, (Cd , Cd0 ) is a pair of algebraic sets. Since the sum operation +

Cp (X) × Cp (X) − → Cp (X) is algebraic [7] it follows that ((C × C)d ,id (C × C)d ) is also a pair of algebraic sets and hence can be triangulated [14]. We conclude that [24, Thm.5.2] apf0 → C e → C/ e C f0 is a principal fibration. We have C e = Zp (X)(R), plies and so C f0 = Zp (X 0 )(R) and it is easy to check that C/ e C f0 = Zp (X, X 0 )(R) (the natural C map Zp (X)(R)/Zp (X 0 )(R) → Zp (X, X 0 )(R) is a homeomorphism). Thus the exact sequence of topological groups 0 −→ Zp (X 0 )(R) −→ Zp (X)(R) −→ Zp (X, X 0 )(R) −→ 0 is a principal fibration. Since the same holds for the sequence Zp (X 0 ) → Zp (X) → Zp (X, X 0 ) [23], and the maps are all equivariant, it follows that this sequence is actually a Z/2-fibration sequence.

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Department of Mathematics, Texas A&M University, USA Current address: Department of Mathematics, Instituto Superior T´ ecnico, Portugal