arXiv:1110.1836v1 [math.AG] 9 Oct 2011

REMARKS ON FILTRATIONS OF THE HOMOLOGY OF REAL
VARIETIESJEREMIAH HELLER AND MIRCEA VOINEAGU
Abstract. We demonstrate that a conjecture of Teh which relates the niveau
ﬁltration on Borel-Moore homology of real varieties and the images of generalized cycle maps from reduced Lawson homology is false. We show that theniveau ﬁltration on reduced Lawson homology is trivial and construct an explicit class of examples for which Teh’s conjecture fails by generalizing a resultof Schülting. We compare various cycle maps and in particular we show thatthe Borel-Haeﬂinger cycle map naturally factors through the reduced Lawson
homology cycle map.

Contents
1. Introduction
2. Preliminaries
3. Birational invariants and examples
4. Coniveau spectral sequences
5. Filtrations in homology
6. Cycle Maps
References

1
3
5
7
10
11
17

1. Introduction
Let X be a quasi-projective real variety. In [Teh10] the reduced Lawson homology groups RLq Hn (X) are introduced as homotopy groups of certain spacesof “reduced” algebraic cycles. For q = 0 we have RL0 Hn (X) = Hn (X(R); Z/2),
where Hn (X(R); Z/2) is the Borel-Moore homology. At the other extreme we havethat RLn Hn (X) is a quotient of the Chow group CHn (X). There are generalized
cycle maps
cycq,n : RLq Hn (X) → Hn (X(R); Z/2)
and the images of these cycle maps form a filtration of the homology
Im(cycn,n ) ⊆ Im(cycn−1,n ) ⊆ · · · ⊆ Im(cyc0,n ) = Hn (X(R); Z/2).
The first step of this filtration is the image of the Borel-Haeflinger cycle map
Im(cycn,n ) = Hn (X(R); Z/2)alg .
The construction of the reduced Lawson homology is based on Friedlander’sconstruction of Lawson homology groups for complex varieties. Friedlander-Mazur[FM94] have conjectured a relationship between the filtration on singular homologyof the space of complex points given by images of the generalized cycle map and1

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JEREMIAH HELLER AND MIRCEA VOINEAGU

the niveau filtration. Teh makes an analogous conjecture for the reduced Lawsongroups.
Conjecture 1.1 ([Teh10, Conjecture 7.9]). Let X be a smooth projective real variety. Then Im(cycq,n ) = N2n−q Hn (X(R); Z/2) for any 0 ≤ q ≤ n.
Here Np Hn (X(R); Z/2) is the niveau filtration which is the sum over all images
Im (Hn (V (R); Z/2) → Hn (X(R); Z/2))
such that dim V ≤ p.
In the complex case, Friedlander-Mazur’s conjecture is a very difficult and interesting question. It is known to be true with arbitrary finite coefficients (in particularwith Z/2-coefficients), as a corollary of Beilinson-Lichtenbaum conjecture. With integral coefficients, it is known that Suslin’s conjecture for a smooth quasi-projective
variety X implies Friedlander-Mazur’s conjecture for X.
Surprisingly, the real case is totally different. Our main result says that theanalog of this conjecture over the field of real numbers is false.
Theorem 1.2. Conjecture 1.1 is false.
To see the failure of this conjecture we first observe that the niveau filtration on
reduced Lawson homology is uninteresting (the case of Borel-Moore homology is
q = 0). Specifically we have that(RLq Hn (X) j ≥ n
Nj RLq Hn (X) =0else.
This is a consequence of Corollary 4.4 which asserts that the coniveau spectral
sequence for the reduced morphic cohomology collapses. Conjecture 1.1 is therefore
equivalent to the surjectivity of the generalized cycle maps. It is known that ingeneral Hn (X(R); Z/2) 6= Hn (X(R); Z/2)alg although it is difficult to find explicit
examples. In Example 5.2 and Proposition 5.3 we give an explicit class of such
examples. Our examples are based on a decomposition given in Theorem 3.1 of
the reduced cycle spaces of a blow-up with smooth center. This decomposition is a
generalization to reduced cycle spaces of the main result of [Sch85].
As another application of thee decomposition we have the following theorem,
which is also different from the complex case where the group of divisors modulo
algebraic equivalence of an irreducible complex variety always injects into the corresponding homology group. Theorem 1.3 gives examples of thin divisors that are
non-trivial in the reduced Lawson group (see Remark 5.6).
birational

Theorem 1.3. There exists a smooth real variety X∼
PN
R such that the
1 1
1cycle map on divisors RL H (X) → H (X(R), Z/2) is not injective.
The collapse of the coniveau spectral sequence for reduced morphic cohomology
is a consequence of local vanishing of motivic cohomology in degrees larger than theweight together with the vanishing theorem proved in [HV09]. For the purposes ofseeing that Conjecture 1.1 is false one does not need the full strength of the collapsing, it suffices to use only the local vanishing of motivic cohomology. However,an interesting consequence of the collapse of this spectral sequence is that we cancompute reduced morphic cohomology as the sheaf cohomologynHZar(X; RLq H0 ) = RLq H n (X).

REMARKS ON FILTRATIONS OF THE HOMOLOGY OF REAL VARIETIES3

As a consequence, we identify a family of birational invariants given by0
HZar(X; RLq H0 ) = RLq H 0 (X),for any q ≥ 0. In case q = dim(X) we obtain that the number s of connected components of X(R) is a birational invariant of an algebraic nature (i.e. RLdim(X) H 0 (X) =(Z/2)s ). The purely algebraic nature of s forms part of the main result of [CTP90],
where they use étale cohomology. The relation between reduced morphic cohomology and étale cohomology is discussed in the final section. As an application ofthese birational invariants we compute reduced Lawson homology of a real rationalsurface in Corollary 3.10.
In Section 6 we discuss various cycle maps. We show that there is basically
one cycle map from the mod-2 motivic cohomology to mod-2 singular cohomologyof the space of real points. As a consequence we see in Theorem 6.8 that the
Borel-Haeflinger cycle map factors through this cycle map.
2. Preliminaries
Let Y be a projective complex variety and Cq (Y ) the Chow variety of effective q-cycles on Y . Write Zq (Y ) = (Cq (Y )(C))+ for the group completion of thismonoid. The group completion is done algebraically and Zq (Y ) is given the quotient topology. It turns out that this naive group completion is actually a homotopy
group completion [FG93], [LF92]. When U is quasi-projective with projectivization
U ⊆ U then define Zq (U ) = Zq (U )/Zq (U∞ ) where U∞ = U \U (and the quotientis a group quotient). This definition is independent of choice of projectivization[LF92], [FG93].
If X is a real variety then G = Z/2 acts on Zq (XC ) via complex conjugation.
The space of real cycles on X is defined to be the subgroup Zq (XC )G of cycles
invariant under conjugation. Write Zq (XC )av for the subgroup generated by cyclesof the form α + α. The space of reduced cycles on X is defined to be the quotientgroup
Zq (XC )G.
Rq (X) =Zq (XC )av
Definition 2.1 ([Teh10]). Let X be a quasi-projective real variety. The reduced
Lawson homology of X is defined by
RLq Hq+i (X) = πi Rq (X).
When q = 0 we have that R0 (X) = Z0 (X(R))/2 so by the Dold-Thom theorem
RL0 Hi (X) = Hi (X(R); Z/2) is the Borel-Moore homology of X(R). In general
RLq Hq+i (X) are all Z/2-vector spaces. It is not known whether these are finitelygenerated vector spaces or not however we do have the following vanishing theorem.
Theorem 2.2 ([HV09]). Let X be a quasi-projective real variety. ThenRLq Hq+i (X) = 0
if q + i > dim(X).
There is also a space Rq (X) = Z q (XC )G /Z q (XC )av of reduced algebraic cocycles
on X when X is normal and projective. We refer to [Teh10] for the details of its
construction. It is convenient to extend this definition to quasi-projective normal
varieties, which is done in [HV09] although not introduced formally as such an

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JEREMIAH HELLER AND MIRCEA VOINEAGU

extension. We avoid difficulties with point-set topology by giving the extension asa simplicial abelian group. Define the simplicial abelian group of reduced cocyles
on a quasi-projective normal real varietyqGe q (X) = Sing• Z/2 (XC ) .
R
Sing Z• /2q (XC )av≃
eq (X) −If X is a projective, normal real variety then R→ Sing• Rq (X) is a homotopy
equivalence [HV09, Lemma 6.7].

Definition 2.3 ([Teh10]). Let X be a normal quasi-projective variety. Define thereduced morphic cohomology of X bye q (X).
RLq H q−i (X) = πi R

The reduced Lawson homology and reduced morphic cohomology are related bya Poincare duality. This is proved in [Teh10, Theorem 6.2] for smooth projective
varieties. We give a quick proof below which applies to both the projective andquasi-projective case. If T is a topological abelian group we write Te = Sing• T for
the associated simplicial abelian group. If M is a G-module and σ is the nontrivial
element of G we write N = 1 + σ and define M av = Im(N ). If in addition M is
2-torsion then we have the two fundamental short exact sequences of abelian groupsN
0 → M G → M −→ M av → 0 and 0 → M av → M G → M G /M av → 0.

Theorem 2.4. Let X be a smooth quasi-projective real variety of dimension d.
The inclusione q (X) → R
e d (X × Aq )R
is a homotopy equivalence of simplicial abelian groups. Consequently there is anatural isomorphism RLq H n (X) = RLd−q Hd−q (X).
Proof. This follows immediately from consideration of the following comparison diagrams of homotopy fiber sequences of simplicial abelian groups, where the displayed
homotopy equivalences follow from [Fri98, Theorem 5.2] and [HV09, Corollary 4.20]
Zeq /2(XC )G
≃

and


e
Zd /2(XC ×C AqC ))G/ Zeq /2(XC )N/ Zed /2(XC ×C Aq ))

N

≃



C



/ Zeq /2(X )G

Zed /2(XC ×C AqC )av/ Zed /2(XC ×C Aq )G

Zeq /2(XC )av

C

≃



C

/ Zeq /2(XC )av/ Zed /2((XC ×C Aq )av
C/R
e q (X)

/R
e d (X × Aq ).
R

The inclusion of algebraic cocycles into topological cocycles gives a generalized
cycle map
cycq,n : RLq H n (X) → H n (X(R); Z/2).
If X is smooth and q ≥ dim X then the cycle map cycq,n is an isomorphism. For X
projective this follows from [Teh10, Corollary 6.5, Theorem 8.1] (the results there

REMARKS ON FILTRATIONS OF THE HOMOLOGY OF REAL VARIETIES

5

are stated under the assumption that X(R) is nonempty and connected but thisis unnecessary). The isomorphism for projective varieties implies the isomorphism
for quasi-projective varieties (for example by using cohomology with supports andan argument as in [HV10, Corollary 4.2]).
There are operations called the s-map in both reduced Lawson homology and reduced morphic cohomology s : RLq Hn (X) → RLq−1 Hn (X) and s : RLq H n (X) →RLq+1 H n (X). Iterated compositions of s-maps give rise to generalized cycle maps
in reduced Lawson homologyssscycq,n : RLq Hn (X) −→ RLq−1 Hn (X) −→ ··· −
→ RL0 Hn (X) = Hn (X(R); Z/2).
The s-map in morphic cohomology is compatible with the s-map in the sense that
cycq,n agrees with the compositions

cycq,ncycq,n : RLq H n (X) −→ RLq+1 H n (X) −−−−→ H n (X(R); Z/2).
Let X be a normal quasi-projective real variety and Z ⊆ X a closed subvariety.
The reduced morphic cohomology with supports is defined in the usual way withe q (X)Z where R
e q (X)Z = hofib(Re q (X) → R
e q (X − Z)).
RLq H q−i (X)Z = πi R
Theorem 2.5 (Cohomological purity for reduced morphic cohomology). Let X bea smooth, quasi-projective real variety of dimension d and Z ⊂ X a closed smoothe q (X)Z ≃ R
eq−p (Z).
subvariety of codimension p. There are homotopy equivalences R
These induce natural isomorphisms
RLq H n (X)Z = RLq−p H n−p (Z).

Proof. This follows from the localization sequence for reduced Lawson homology[Teh10, Corollary 3.14] together with Poincare duality between reduced Lawson
homology and reduced morphic cohomology (see [Teh10, Theorem 6.2] and Theorem
2.4).

Recall that a presheaf F (−) of cochain complexes satisfies Nisnevich descentprovided that for any smooth X, any étale map f : Y → X, and open embeddingi : U ⊆ X such that f : Y − f −1 (U ) → X − U is an isomorphism, we have a
Mayer-Vietoris exact triangle (in the derived category of abelian groups):
F (X) → F (Y ) ⊕ F (U ) → F (f −1 (V )) → F (X)[1].
e q (−) is homotopy invariant theory and satisfies
Corollary 2.6. The presheaf R
Nisnevich descent.

Proof. It is homotopy invariant by [Teh10, Theorem 5.13]. Nisnevich descent follows immediately from Theorem 2.5.

3. Birational invariants and examples
We use the following decomposition theorem for spaces of reduced cycles on
blow-ups with smooth center in order to obtain a decomposition of the cokernelsof the cycle map from reduced Lawson homology. Later we use this decomposition
to exhibit spaces whose cycle map has nontrivial cokernel i.e. it is not surjective.
Recall that R−q (X) = R0 (X × A−q ) for q ≥ 0.

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JEREMIAH HELLER AND MIRCEA VOINEAGU

Theorem 3.1. Let X be a smooth real projective variety and Z ⊂ X a smoothclosed irreducible subvariety of codimension d > 1. Let π : X ∗ → X be the blow upof X by the smooth center Z. Then, for any 0 ≤ q ≤ dim(X) we have a homotopy
equivalence of topological abelian groups
(3.2)

h.e.

Rq (X ∗ ) ≃ Rq (X) ⊕ Rq−d+1 (Z) ⊕ Rq−d+2 (Z) ⊕ · · · ⊕ Rq−1 (Z).

Moreover this decomposition is compatible with s-maps.
Proof. We follow the ideas used to prove [Voi, Theorem 2.5] and the main theoremof [Sch85]. We work in H−1 AbT op, the category of topological abelian groups witha CW-structure with inverted homotopy equivalences.
Recall that π −1 (Z) → Z is the projective bundle p : P(NZ X) → Z of dimensiond − 1. The decomposition in the statement of the theorem is given as follows. The
first component of the map is π∗ (notice that π∗ ◦ π ∗ = id). The other maps are
given via compositions−∩c1 (O(1))lp∗φl : Rq−d+1+l (Z) −→ Rq+l (X ∗ ) −−−−−−−−→ Rq (X ∗ ),i∗
Rq+d−1 (X ∗ ).
where p∗ : Rq (Z) → Rq+d−1 (P(NZ X)) →Using the Mayer-Vietoris sequence (see Corollary 2.6), we have thath.e.

Rk (X ∗ ) → Rk (X) ⊕ Ker(p∗ ),with p∗ : Rk (NZ (X)) → Rk (Z) and then identify in H−1 AbT op
(3.3)

h.e.

Ker(p∗ ) ≃ Rq−d+1 (Z) ⊕ Rq−d+2 (Z) ⊕ · · · ⊕ Rq−1 (Z).

Using the properties of Segre classes sl (NZ X) ∩ − : Rk (Z) → Rk+d−1−l (Z) i.e.
sl (NZ (X)) ∩ − = 0 if l < 0 and s0 (NZ (X)) ∩ − = id, one can prove the projective
bundle formula for the reduced cycle groups in the usual way, see [Teh10]. Observethat the projective bundle formula works also for negative indexes. We obtain
Rk (NZ (X)) = ⊕0≤l≤d−1 Rk−d+1+l (Z).
Now one can conclude the homotopy equivalence (3.3).
The s-maps are compatible with all of the maps involved in the decomposition
3.2 (see [Teh10]) and therefore the decomposition of the theorem is preserved by
the s-maps.

The generalized cycle maps cycq,n : RLq Hn (X) → Hn (X(R), Z/2), are defined
as a composite of s-maps together with the Dold-Thom isomorphism. Write
Tq,n := Coker(cycq,n : RLq Hn (X) → Hn (X(R), Z/2))
and
Kq,n (X) = ker(cycq,n : RLq Hn (X) → Hn (X(R), Z/2)).
Notice that Tq,n = 0 = Kq,n , for q ≤ 0.
Corollary 3.4. Let π : X ∗ → X and Z be as in the above theorem. Then
Tq,n (X ∗ ) = Tq,n (X) ⊕ Tq−1,n−1 (Z) ⊕ ... ⊕ Tq−d+1,n−d+1 (Z),
and
(3.5)

Kq,n (X ∗ ) = Kq,n (X) ⊕ Kq−1,n−1 (Z) ⊕ ... ⊕ Kq−d+1,n−d+1 (Z).

REMARKS ON FILTRATIONS OF THE HOMOLOGY OF REAL VARIETIES

7

Proof. In the case k = 0 the decomposition (3.2) givesh.e.

R0 (X ∗ (R)) ≃ R0 (X(R)) ⊕ R−d+1 (Z(R)) ⊕ R−d+2 (Z(R)) ⊕ · · · ⊕ R−1 (Z(R)),
therefore producing the decomposition of Borel-Moore homology
Hk (X ∗ (R), Z/2) = Hk (X(R), Z/2) ⊕ Hk−d−1 (Z(R), Z/2) ⊕ · · · ⊕ Hk−1 (Z(R), Z/2).
The s-maps respect the decomposition (3.2) and comparing the decomposition for
q = 0 and for q > 0 yields the result.

Corollary 3.6. If Z(R) = ∅, then Tq,n (X ∗ ) = Tq,n (X) for any 0 ≤ q ≤ n ≤dim(X).
Remark 3.7. An analog of Corollary 3.4 for Tq,q was originally proven by Schülting
in [Sch85]. There separate arguments are needed to give a decomposition algebraically and a decomposition topologically. An advantage that our uniform proof
has is that is entirely algebraic, the homology of real points being expressed interms of homotopy of the group of algebraic cycles R0 (X).
Remark 3.8. Using similar techniques one can prove a decomposition analogous to(3.2) for the spaces of real cycles defining dos Santos equivariant Lawson homology
groups.
Corollary 3.9. The groups T1,n (X) and K1,n (X) are birational invariants forsmooth projective real varieties.
Proof. By [AKMW02, Theorem 0.3.1] every birational map between smooth projective varieties factors as a composition of blow-ups and blow-downs with smoothcenters. The result then follows from Corollary 3.4.

We close the section with the following computation.
Corollary 3.10. Let X be a rational smooth projective surface i.e. X
Then the cycle map

birational

≃

P2R .

cycq,n : RLq Hn (X) → Hn (X(R), Z/2)
is an isomorphism for q ≤ n.
Proof. We have R0 (X) = R0 (X(R)). By Corollary 3.9 we have that T1,n (X) andK1,n (X) are birational invariants and we know that T1,n (P2R ) = 0 = K1,n (P2R ). By
Theorem 4.4 the group π0 R2 (X) is a birational invariant.

Remark 3.11. For X as in the previous corollary we have that RLq Hn (X) = 0 for
n < q and n > 2 and RL0 H0 (X) = RL0 H2 (X) = RL1 H2 (X) = RL2 H2 (X) = Z/2.
4. Coniveau spectral sequences
In this section we show that the coniveau spectral sequence for reduced morphic
cohomology collapses. We make use of [CTHK97] for the Bloch-Ogus theorem
identifying the E2 -term of this spectral sequence. Let X be a smooth, quasiprojective real variety and write X (p) for the set of points x ∈ X whose closure has codimension p. Let h∗ be a cohomology theory with supports. Define

8

JEREMIAH HELLER AND MIRCEA VOINEAGUi(U ) and hi (k(x)) = colimU⊆x hi (U ) (where in both colimhix (X) = colimU⊆X hx∩Uits, U ranges over nonempty opens). One may form the Gersten complexMMM
0→hnx (X) →hn+1(X) → · · · →hn+p(X) → · · · .
xxx∈X (0)x∈X (1)x∈X (p)This complex gives rise to the coniveau spectral sequenceMp+q
E1p,q =
hp+q
(X)
x (X) =⇒ hx∈X (p)

The associated filtration is N p hn (X) = ∪Z Im(hnZ (X) → h(X)), where the unionis over closed subvarieties Z ⊆ X of codimension p.
Proposition 4.1. [CTHK97, Corollary 5.1.11, Theorem 8.5.1] Let h∗ be a cohomology theory with supports on Sm/R which satisfies Nisnevich excision and ishomotopy invariant. Let Hq be the Zariski sheafification of the presheaf U 7→ hq (U ).
Then the Gersten complex E1•,q is a flasque resolution of Hq and the coniveau spectral sequence has the formp
E2p,q = HZar
(X; Hq ) =⇒ H p+q (X).

For every q, the group H 0 (X, Hq ) is a birational invariant for smooth proper varieties.
Corollary 4.2. Let X be a smooth quasi-projective real variety. For each k wehave spectral sequences
M
E1p,q (k) =RLk−p H q (k(x)) =⇒ RLk H p+q (X).
x∈X (p)p
0
The E2 -terms are E2p,q (k) = HZar
(X; RLk Hq ) and each HZar
(X; RLk Hq ) is a bi-

rational invariant for smooth projective real varieties. Moreover, the s-maps induce
maps of spectral sequences {Erp,q (k)} → {Erp,q (k + 1)}.
Proof. By Corollary 2.6 reduced morphic cohomology is homotopy invariant andsatisfies Nisnevich excision. By Theorem 2.5 we have for any x ∈ X (p) the isomorphism RLk H p+q (X)x = RLk−p H q (k(x)). Thus the E1 -page of the coniveau
spectral sequence can be rewritten in the displayed form. The s-maps are natural transformations and so induce maps of the exact couples defining the coniveau
spectral sequence.

The following result gives us the collapsing of the coniveau spectral sequence. It
is a consequence of the main vanishing theorem of [HV09] and the local vanishingof equivariant morphic cohomology and morphic cohomology.
Theorem 4.3. Let R = OT,t1 ,··· ,tn be the semi-local ring of a smooth, real variety
T at the closed points t1 , . . . , tn ∈ T . ThenRLk H q (Spec R) = 0
if q 6= 0 and any k > 0.
Proof. For convenience write Y = Spec R. Recall that by definition
RLk H q (Y ) = colim RLk H q (U ),where the colimit is over all open U ⊆ T such that all ti ∈ U . Recall also thatfiltered colimits commute with homotopy groups and preserve exact sequences.

REMARKS ON FILTRATIONS OF THE HOMOLOGY OF REAL VARIETIES

9

We need to see that RLk H k−s (Y ) = πs Rk (Y ) = 0 for s 6= k. The main vanishing
result in [HV09, Theorem 6.10] implies that πs Rk (Y ) = 0 for s > k.
Consider the homotopy fiber sequences of simplicial abelian groups

andZek /2(YC )G/ Zek /2(Y )C

Zek /2(YC )av/ Zek /2(Y )GN/ Zek /2(Y )av

C

C

/R
e k (Y ).

Because πs Zek /2(YC )G = 0 = πs Zek /2(YC ) if s ≤ k − 1 ([FHW04, Theorem 7.3] and[HV10, Lemma 3.22]) we see that πs Zek /2(YC )av = 0 if s ≤ k − 1. Using the second
e k (Y ) = 0 if s ≤ k − 1.
homotopy fiber sequence we conclude that πs RCorollary 4.4. Let X be a smooth quasi-projective real variety. For any k, theconiveau spectral sequence for reduced morphic cohomology satisfies
E1p,q (k) = 0p,0for q 6= 0. Consequently E2p,0 (k) = E∞(k) and so we have natural isomorphismsp
HZar(X; RLk H 0 ) = RLk H p (X).

In particular H 0 (X, RLq H 0 ) = RLq H 0 (X) = πq (Rq (X)) is a birational invariant
for smooth projective real varieties.
Remark 4.5. The case k = dim X tells us that HRi = 0 for any i > 0 where HRi
is the Zariski sheafification of the presheaf U 7→ H i (U (R); Z/2). This also follows
from [Sch94, Theorem 19.2].
Remark 4.6. Corollary 4.4 gives us birational invariants RLq H 0 (X) for 0 ≤ q ≤d = dim(X). If q = d then we have
RLd H 0 (X) = H 0 (X(R); Z/2) = (Z/2)⊕sand therefore s = s(X) = #(connected components of X(R)) is a birational invariant of an algebraic nature. This also follows from the main result of [CTP90] wherenthey show that H 0 (X, Het) = H 0 (X(R), Z/2) for any n ≥ dim(X) + 1. Here Hetn(U, µ⊗nis the sheaf associated to the presheaf U → Het
2 ).
At the other extreme if one takes q = 0,
RL0 H 0 (X) = (Z/2)⊕rwhere r = r(X) = #(geometrically irreducible components of X) and so r is also
a birational invariant.
Remark 4.7. By Corollary 4.4 and Corollary 4.2 the s-mapss

RLq H n (X) → RLq+1 H n (X)are obtained as the map induced on Zariski sheaf cohomology by the sheafified
s-maps s : RLq H0 → RLq+1 H0 . In particular we see that the generalized cycle
map
cycq,n : RLq H n (X) → H n (X(R); Z/2)
is obtained from the sheafified cycle map RLq H0 → HR0 . In the last section weshow that this cycle map is naturally related to the Borel-Haeflinger cycle map[BH61].

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JEREMIAH HELLER AND MIRCEA VOINEAGU

We finish by observing that Poincare duality gives the collapsing of the niveau
spectral sequence for reduced Lawson homology (of possibly singular varieties).
Proposition 4.8. Let X be a quasi-projective real variety. Write X(p) for the setof points x ∈ X whose closure has dimension p. The niveau spectral sequence1Ep,q(k) = ⊕x∈X(p) RLk Hp+q (k(x)) =⇒ RLk Hp+q (X)12∞
satisfies Ep,q(k) = 0 for any q 6= 0 and therefore Ep,q(k) = Ep,q(k).

Proof. The niveau spectral sequence is constructed as in [BO74]. Let x ∈ X(p) .
Using Corollary 4.4 we see that for any q 6= 0
RLk Hp+q (k(x)) = RLp−k H −q (k(x)) = 0.

5. Filtrations in homology
Let X be a quasi-projective real variety of dimension d. The generalized cycle
map φq,n : RLq Hn (X) → Hn (X(R), Z/2) is the composition of q iterations of thes-map together with the Dold-Thom isomorphism. Write
RTq Hn (X) = Im(φq,n : RLq Hn (X) → Hn (X(R); Z/2).
This gives us a decreasing filtration of the homology of the space of real points andis called the topological filtration.
Associated to the niveau spectral sequence is the niveau filtrationX
Np RLk Hn (X) =Im (RLk Hn (V ) → RLk Hn (X)) .
dim V ≤p

Notice that in the complex case, if Y is a complex variety of dimension d then Weak
Lefschetz theorem says thatNn Hn (Y (C); Z) = Nn+1 Hn (Y (C); Z) = · · · = Nd Hn (Y (C); Z)).

In particular, in the complex case there are only n + 1 steps in the filtration, andanother d − n are equal to the homology. In the real case, we don’t have thistheorem and so apriori all one has is a filtration.
N0 Hn (X(R); Z/2) ⊆ · · · ⊆ Nd Hn (X(R); Z/2) = Hn (X(R); Z/2).
Teh has formulated the following conjecture which is made in analogy with aconjecture of Friedlander-Mazur [FM94, Conjecture p.71] for complex varieties.
Conjecture 5.1 ([Teh10, Conjecture 7.9]). Let X be a smooth projective real variety. Then RTq Hn (X) ⊆ N2n−q Hn (X(R); Z/2) and moreover this containment is
an equality RTq Hn (X) = N2n−q Hn (X(R); Z/2) for any 0 ≤ q ≤ n.
From Proposition 4.8 we have the following equality:∞
En−q,q(k) = Nn−q RLk Hn (X)/Nn−q−1 RLk Hn (X) = 0

for any q 6= 0. This means that for any k we have
RLk Hn (X) = Nd RLk Hn (X) = · · · = Nn+1 RLk Hn (X) = Nn RLk Hn (X),and
0 = N−1 RLk Hn (X) = N0 RLk Hn (X) = · · · = Nn−1 RLk Hn (X).

REMARKS ON FILTRATIONS OF THE HOMOLOGY OF REAL VARIETIES

11

The first row of equalities contains the groups that appear in Conjecture 5.1.
Consequently the first part of the conjecture is obviously true because by the above
we have that Nj Hn (X(R); Z/2) = Hn (X(R); Z/2) for all j ≥ n.
The second part of the conjecture is false because the s-maps are not always be
surjective. Using the material from Section 3 we give an explicit example of this failure. Recall that we write Tq,n (X) = Coker(cycq,n : RLq Hn (X) → Hn (X(R); Z/2)).
Example 5.2. Let Z ⊆ P3R be the smooth irreducible elliptic curve given by theequation t2 x + ty 2 − x3 = 0, z = 0. Then Z(R) is well known to consist of 2
connected components (see for example [BCR98, Example 3.1.2]). Let X ∗ be theblow-up of P3 along Z. Then T2,2 (X ∗ ) = T2,2 (P3R ) ⊕ T1,1 (Z) by Corollary 3.4.
Because RL1 H1 (Z) = π0 R1 (Z) = Z/2 and Z(R) has two components we conclude
that T2,2 (X ∗ ) = Z/2.
More generally we have the following.
Proposition 5.3. For each N ≥ 3, there is a smooth projective real variety X
of dimension N (which is topologically connected) such that Tq,q (X) 6= 0 for all
2 ≤ q ≤ N − 1.
Proof. Let Z ⊆ PN be a smooth irreducible real curve such that Z(R) has at least 2
connected components. Let s denote the number of connected components of Z(R).
Since Z is irreducible we have that R1 (Z) = Z/2. Therefore T1,1 (Z) = (Z/2)s−1 .
Since Z is a curve Ti,n (Z) = 0 for all other values of i and n. We take X → PN to
be the blow up of PN along Z. Then Tq,q (X) = T1,1 (Z) = Z/2s−1 by Corollary 3.4
because 2 ≤ q ≤ N − 1 = codim(Z).
We also have a similar result for the kernel.
Proposition 5.4. For each N ≥ 3, there is a smooth projective real variety X,
birational to PN , such that Kq,q (X) 6= 0 for all 2 ≤ q ≤ N − 1.
Proof. Let Z ⊆ PN be a smooth irreducible real curve such that Z(R) = ∅. Then
K1,1 (Z) = Z/2 and Ki,n (Z) = 0 for all other values of i and n. Take X → PN to
be the blow up of PN along Z. We have Kq,q (X) = K1,1 (Z) = Z/2 by Corollary
3.4.

As an interesting particular case we have the following which is different thanthe complex analog.
birational

Corollary 5.5. There exists a smooth real variety X∼
PN
R such that the
1 1
1cycle map on divisors RL H (X) → H (X(R), Z/2) is not injective.
Remark 5.6. A k-cycle is said to be thin if it is a sum of closed subvarieties Z ⊆ X
with dim Z(R) < k. The kernel of the Borel-Haeflinger cycle map consists entirely ofthin cycles by [IS88] and the composite CHq (X) → RLq Hq (X) → Hq (X(R); Z/2)
agrees with the Borel-Haeflinger cycle map by Theorem 6.8. This means that the
proposition above gives examples of nonzero classes which are represented by thincycles in RLq Hq (X).
6. Cycle Maps
Let X be a smooth quasi-projective real variety. We discuss two natural cycle
maps from motivic cohomology to the singular cohomology of X(R). Based on this,12

JEREMIAH HELLER AND MIRCEA VOINEAGU

we show that Borel-Haeflinger map [BH61] factors through the reduced Lawson
homology cycle map. We end the section with a discussion of the maps involved in
the Suslin conjecture from the view of the methods in this section.
i(M ; Z/2) for the Borel
Recall that G = Z/2. If M is a G-space we write HG
cohomology with Z/2-coefficients. The reduced morphic cohomology of X comesequipped with a natural generalized cycle map to the singular cohomology of realpoints. Composing this with the canonical map from real morphic cohomology andits isomorphism with motivic cohomology (with Z/2-coefficients) gives us the cycle
map
(6.1)

cyc

p
HM
(X; Z/2(q)) → RLq H p−q (X) −−→ H p−q (X(R); Z/2).

On the other hand the real morphic cohomology maps naturally to the Borel cohon
mology of the space of complex points. In turn there is a map HG(X(C); Z/2) →n−i⊕H(X(R); Z/2) obtained by restricting to real points together with the decomposition
Mn
HG(X(R), Z/2) = H n (X(R) × RP ∞ ; Z/2) =H i (X(R), Z/2).
0≤i≤nComposing with the appropriate projection gives us
(6.2)

p
p
HM
(X; Z/2(q)) → HG
(X(C); Z/2) −→ H p−q (X(R); Z/2).

We show that the cycle maps (6.1) and (6.2) agree with each other. Basicallythese agree because they can be seen as induced by maps of presheaves of cochain
complexes and so Theorem 6.12 applies.
Write (T op)an for the category of topological spaces homeomorphic to a finite
dimensional CW-complex given the usual topology and φ : (T op)an → (Sm/R)Zar
for the map of sites induced by X 7→ X(R).
Theorem 6.3. Let X be a smooth quasi-projective real variety. The cycle maps
given by (6.1) and (6.2) agree. Moreover the intermediate maps in (6.2) can bechosen so that the following diagram commutesp(X, Z/2(q))
HMRLp H p−q (X)/ H p (X, µ⊗q )
et

cyc

2


/ H p−q (X(R), Z/2) o∼=/ H p (X(C), Z/2)
G


p
HG
(X(R), Z/2).

for any p, q ≥ 0.
Proof. Consider the following complexes of Zariski sheaves on Sm/R:
q/q−1

Z/2(q)(X) = (zequi (PR, 0)(X ×R ∆•R ) ⊗ Z/2)[−2q]Z/2(q)sst (X) = Sing• (Z q /2(XC )G )[−2q]Z/2(q)top (X) = Homcts (X(C) × ∆•top , Z/20 (AqC ))G [−2q]Z/2(q)Bor (X) = Homcts (X(C) × EG × ∆•top , Z/20(AqC ))G [−2q]q G•
Z/2(q)Bor
R (X) = Homcts (X(R) × EG × ∆top , Z/20 (AC )) [−2q],

R(q)(X) = (Sing• Z q /2(XC )G / Sing• Z q /2(XC )av )[−2q]These complexes all satsify Nisnevich descent for standard reasons. See e.g.
[HV09, Section 5] for the first three. Similarly the complex Z/2(q)Bor
R (X) because

REMARKS ON FILTRATIONS OF THE HOMOLOGY OF REAL VARIETIES

13

taking real points of a distinguished Nisnevich square of real varieties gives a homotopy pushout square of spaces. The last complex satisfies Nisnevich descent by
Proposition 2.6. As a consequence we havei
HiZar (Z; Z/2(q)) = HM
(X; Z/2(q))

HiZar (X; Z/2(q)sst ) = Lq HRi−q,q (X; Z/2)
HiZar (X; Z/2(q)top ) = H i−q,q (X(C); Z/2)i
HiZar (X; Z/2(q)Bor ) = HG
(X(C); Z/2)i
HiZar (X; Z/2(q)Bor
R ) = HG (X(R); Z/2)

HiZar (X; R(q)) = RLq H i−q (X),
where Lq HRi−q,q (X; Z/2) denotes Friedlander-Walker’s real morphic cohomology[FW02] and H i−q,q (X(C); Z/2) is Bredon cohomology.
The map (6.1) is induced by the map of complexes1)

2)

Z/2(q) −→ R(q) −→ φ∗ Z/2[−q].

(6.4)

The map 1) is given by the “usual” cycle map from motivic cohomology to reduced
morphic cohomology. It is defined as the composite Z/2(q) → Z/2(q)sst → R(q).
The map 2) is obtained by adjunction from the composite≃≃φ∗ (R(q)) −→ M apcts (X(R) × ∆∗top , R0 (AqR ))[−2q] −→ Z/2[−q]which arises because R0 (AqR ) ≃ K(Z/2, q) and any CW complex has an open cover
given by contractibles.
We show that the map (6.2) is induced by a composite of maps:
(6.5)

3)

4)

5)

Z/2(q) −→ tr≤2q Rǫ∗ Z/2 −→ Z/2(q)Bor −→ Z/2(q)Bor−→ φ∗ Z/2[−q].
Rof Zariski complexes of sheaves in D− (ShvZar (Sm/R)) which we now explain.
Write ǫ : Xet → XZar for the usual map of sites. The first unlabeled map isthe cycle map from motivic cohomology to etale cohomology. The map 3) is obtained in Proposition 6.6 using Cox’s theorem [Cox79]. The map 4) is obtained by
restriction to real points. The map 5) will be obtained from the adjoint of a map
φ∗ (Z/2(q)Bor
R ) → Z/2[−q] as follows. Every CW-complex is locally contractible
and soq G•φ∗ (Z/2(q)Bor
R ) ≃ Homcts (EG × ∆top , Z/20 (AC )) [−2q]is a quasi-isomorphism of complexes of Zariski sheaves where the right-hand side is−
the constant sheaf. In DZ/2
(Ab), any complex is quasi-isomorphic with the complex
given by its cohomology. We have that H p,q (EG; Z/2) = H p+q (BG; Z/2) andtherefore
H k Homcts (EG × ∆•top , Z/20 (AqC ))G [−2q] = H q−k,q (EG; Z/2) = Z/2
for 0 ≤ k ≤ 2q and is 0 otherwise. This gives us the map
φ∗ (Z/2(q)Bor
R ) ≃ ⊕0≤i≤2q Z/2[−i] → Z/2[−i],which induces the ith projection on cohomologyn
HG(X(R), Z/2) = ⊕0≤i≤n H n−i (X(R), Z/2) → H n−i (X(R), Z/2).

14

JEREMIAH HELLER AND MIRCEA VOINEAGU

In particular the adjoint of this map for i = q gives us the map 5)
Z/2(q)Bor→ φ∗ Z/2[−q].
R
Because both map 6.5 and map 6.4 induce non-trivial maps in cohomology they
have to coincide by Theorem 6.12.

In the above proof, we used the following proposition which relies on Cox’s
theorem identifying the etale cohomology of a real variety with Borel cohomology.
Proposition 6.6. There is a quasi-isomorphism ρ : tr≤2q Rǫ∗ Z/2 → Z/2(q)Bor ofcomplexes of Zariski sheaves.
Proof. We show that the canonical map Z/2(q)Bor → Rǫ∗ Z/2, constructed in[HV09, Proposition 5.5] induces a quasi-isomorphism Z/2(q)Bor → tr≤2q Rǫ∗ Z/2.
The map ρ is its inverse in the derived category. Its hypercohomology gives thenncycle map HG(X, Z/2) → Het(X, Z/2), for every n ≥ 0.
≃
→ Rǫ∗ µ⊗q+i
and a commutative diagram
There is a quasi-isomorphism Rǫ∗ µ⊗q
2
2 −
(6.7)

/ tr≤2q Rǫ∗ µ⊗q

Z/2(q)Bor

2

≃

≃


/ tr≤2q Rǫ∗ µ⊗q+i .
2tr≤2q Z/2(q + i)Bor

Take i = q. The result follows by showing the bottom map is a quasi-isomorphism.
In [HV09, Section 5] it is shown that the composite
Z/2(2q) → tr≤2q Z/2(2q)Bor → tr≤2q Rǫ∗ µ⊗2q
2is the usual cycle map Z/2(2q) → tr≤2q Rǫ∗ µ⊗2q
. By Voevodsky’s resolution of the2
Milnor conjecture [Voe03] this cycle map is a quasi-isomorphism. This implies thatnnthat HG(X, Z/2) → Het(X, Z/2) is a surjective map between finite-dimensional
spaces for n ≤ 2q. By [Cox79] both vector spaces have the same dimension and so
the map is an isomorphism. Therefore Z/2(q)Bor ≃ tr≤2q Rǫ∗ Z/2.

Let aZar denote Zariski sheafification and define the following sheavesnn(U (C)), Z/2(q)))HM(q) =aZar (U 7→ HMnHCn (G) =aZar (U 7→ HG(U (C)), Z/2))n(U (R), Z/2)),
HRn (G) =aZar (U 7→ HGnnHet(q) =aZar (U 7→ Het(U, µ⊗q2 )),HRn =aZar (U 7→ H n (U (R), Z/2)).
Sheafifying the diagram in Theorem 6.3 for p = q gives the commutative diagram
q
(q)
HM

∼
=


cycq,0RLq H0/ Hq (q)et/ H0 oR∼=/ Hq (G)C


HRq (G).

REMARKS ON FILTRATIONS OF THE HOMOLOGY OF REAL VARIETIES

15

By Corollary 4.4 and Corollary 4.2 the sheafified cycle map cycq,0 induces theusual cycle map between reduced morphic cohomology and singular cohomologynn
RLq H n (X) = HZar
(X; RLq H0 ) → HZar
(X; HR0 ) = H n (X(R); Z/2).
qq
The map HM(q) → Het(q) induces the Bloch-Ogus isomorphismqqq
CH (X) = H q (X; Hq (q)) ∼= H (X; H (q)).
Zar

Zar

M

et

q
The composite Het(q) → HCq (G) → HRq (G) → HR0 induces a mapqqq
CH q (X) = HZar
(X; Het(q)) → HZar
(X; HR0 ) = H q (X(R); Z/2)

which by [CTS96, Remark 2.3.5] is just the Borel-Haeflinger cycle map sending aclosed irreducible Z ⊆ X to the Poincare dual of the fundamental class of Z if
dim Z(R) = dim Z and zero otherwise. The above commutative diagram tells us
qthat this agrees with the composite Hq (q)et ∼= HM (q) → LRq H0 → HR0 and so we
immediately obtain the following.
Theorem 6.8. Let X be a smooth quasi-projective real variety. For any q ≥ 0, theBorel-Haeflinger cycle map factors as the composite
cycq,nCH q (X)/2 → RLq H q (X) −−−−→ H q (X(R), Z/2),
where the first map is the natural quotient.
Next we compare the s-map in reduced morphic cohomology with the operai(X; µ⊗qtion (−1) in étale cohomology. Recall that the operation (−1) : Het
2 ) →
⊗q+1
i+1
Het (X; µ2) is defined to be multiplication with the class (−1) which is the0
1
image of −1 under the boundary map Het(X; Gm ) → Het(X; µ2 ) in the Kummer
1(Spec(R); µ2 )sequence. By naturality this is equal to the pullback of (−1) ∈ Het
under the structure map X → Spec(R). Sheafifying gives the operation on Zariskiqq+1
sheaves (−1) : Het(q) → Het
(q + 1).
Proposition 6.9. Let X be a smooth quasi-projective real variety of dimension d.
The following square commutes for any q ≥ 0
qi
HZar(X; Het(q))
∪(−1)y−−−−→

RLq H i (X)

∪s
y

q+1
i
HZar
(X; Het(q + 1)) −−−−→ RLq+1 H i (X).
For any q ≥ d + 1 all maps are isomorphisms.

Proof. The s-operation is induced by multiplication with s ∈ RL1 H 0 (Spec(R)),
where s is the generator. Sheafifying the s-map gives a map of Zariski sheaves
and the composite RLq H0 −→ RLq+1 H0 → HR0 induces the usual s-map on sheaf
cohomology.
1
The class (−1) ∈ Het(Spec(R); µ2 ) and (−1) maps to s under the isomor1
phism Het
(Spec(R); µ2 ) ∼= RL1 H 0 (Spec(R)) because they both map to the generator of H 0 (pt; Z/2). Therefore the above square commutes. When q ≥ d thenqi(X; Hetthe vertical maps are isomorphisms, both HZar(q)) → H i (X(R); Z/2) andRLq H i (X) → H i (X(R); Z/2) are isomorphisms.
Corollary 6.10. Let X be a smooth quasi-projective real variety. We have theisomorphism of rings∗−1
Het(X, µ⊗∗] ≃ RL∗ H ∗ (X)[s−1 ] ≃ H ∗ (X(R), Z/2),
2 )[s

16

JEREMIAH HELLER AND MIRCEA VOINEAGU

where s = (−1) under the left-hand isomorphism.
q
Remark 6.11. The map Het(q) → RLq H 0 is not in general an isomorphism ofsheaves as we can see for the case of a smooth projective variety of dimension
dim(X) = q. In this case the first group surjects with non-trivial kernel in codimension 0, 1, 2 (under some mild conditions on X) into cohomology of real points
X(R) by [CTS96]. On the other hand the latter group is the cohomology of realpoints by [Teh10].

We now turn our attention to natural transformations from various cycle cohomology theories of interest to singular cohomology.
Theorem 6.12. Let k = R or C and write φ : (T op)an → (Sm/k)Zar the map ofsites that send X 7→ X(k) where (T op) is the category of spaces homeomorphic tofinite dimensional CW complexes equipped with the usual topology. Then we have(1) HomD− ((Sm/R)Zar ) (Z/2(q), φ∗ Z/2[−q]) = Z/2,
(2) HomD− ((Sm/R)Zar ) (R(q), φ∗ Z/2[−q]) = Z/2,
(3) HomD− ((Sm/C)Zar ) (Z(q)sst , Rφ∗ Z/n) = Z/n, for any n ≥ 1.
Proof. We have a quasi-isomorphism Z/2(q) ≃ Z/2(q)sst and that
Hom(Z/2(q)sst , φ∗ Z/2[−q]) = Hom(φ∗ (Z/2(q)sst ), Z/2[−q]).
Because every CW complex is locally contractible in D− (T op) we haveφ∗ (Z/2(q)sst )[2q] ≃ Homcts (− × ∆•top , Z/20 (AqC )G ) ≃ Sing• Z/20 (AqC )G .
Q2q
From [dS03, (3.6)] it follows that Z/20 (AqC )G ≃ i=q K(Z/2, i). This yields theresult because we then have
HomD− (T op) (φ∗ (Z/2(q)), Z/2[−q]) =2q
M

H q (K(Z/2, i); Z/2) = Z/2.

i=q

In the proof of Theorem 6.3 we observed that φ∗ R(q) ≃ Z/2[−q] and so
HomD− ((Sm/R)Zar ) (R(q), φ∗ Z/2[−q]) = HomD− ((Sm/R)Zar ) (φ∗ R(q), Z/2[−q])= HomD− ((Sm/R)Zar ) (Z/2[−q], Z/2[−q]) = Z/2.
The last item follows from the equivalence φ∗ Z(q)sst ≃ Z. We have
HomD− ((Sm/C)Zar ) (Zsst (q), Rφ∗ Z/n) = HomD− ((Sm/R)Zar ) (Z, Z/n) = Z/n,
for any n ≥ 1.

Because Zsst (q) = tr≤q Z(q)sst we have, according to Theorem 6.12, 3), that
HomD− (Sm/C) (Z(q)sst , tr≤n Rφ∗ Z) = Z.
Let α be a generator of this group. Recall that Suslin’s conjecture is the statementthat α is a quasi-isomorphism for all q ≥ 0. We end the section with the following
corollary of Voevodsky’s resolution of the Beilinson-Lichtenbaum conjectures.
Theorem 6.13. The map α⊗Z/n : Z/n(q)sst → tr≤q Rφ∗ Z/n is a quasi-isomorphism
for any n ≥ 2.

REMARKS ON FILTRATIONS OF THE HOMOLOGY OF REAL VARIETIES

17

Proof. Theorem 6.12 and the quasi-isomorphism Z(q)sst ⊗ Z/p ≃ Z/p(q)sst gives
HomD− (Z(q)sst , tr≤q Rφ∗ Z/p) = HomD− (Z/p(q)sst , tr≤q Rφ∗ Z/p) = Z/pf or any prime p > 1. Let ǫ : (Sm/C)et → (Sm/C)Zar be the usual map of sites.
The cycle map β : Z/p(q) → tr≤q Rǫ∗ Z/p can be seen as a generator of the group
HomD− (Sm/C) (Z/p(q)sst , tr≤n Rφ∗ Z/p) because we have Z/p(q)sst ≃ Z/p(q) andRφ∗ Z/p = Rǫ∗ Z/p by the classical comparison theorem between étale and singularcohomology. This means that there is 0 < N < p such that α ⊗ Z/p = N β. Since
N is prime to p it follows that α ⊗ Z/p is a quasi-isomorphism.
One easily obtains the result for powers of primes and then products of distinct
primes using 5-lemma.

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E-mail address: mircea.voineagu@ipmu.jp