arXiv:1008.3685v1 [math.AG] 22 Aug 2010

EQUIVARIANT SEMI-TOPOLOGICAL INVARIANTS, ATIYAH’S
KR-THEORY, AND REAL ALGEBRAIC CYCLESJEREMIAH HELLER AND MIRCEA VOINEAGU
Abstract. We establish an Atiyah-Hirzebruch type spectral sequence relating
real morphic cohomology and real semi-topological K-theory and prove it to becompatible with the Atiyah-Hirzebruch spectral sequence relating Bredon cohomology and Atiyah’s KR-theory constructed by Dugger. An equivariant anda real version of Suslin’s conjecture on morphic cohomology are formulated,proved to come from the complex version of Suslin conjecture and verified forcertain real varieties. In conjunction with the spectral sequences constructedhere this allows the computation of the real semi-topological K-theory of somereal varieties. As another application of this spectral sequence we give an alternate proof of the Lichtenbaum-Quillen conjecture over R, extending an earlier
proof of Karoubi and Weibel.

Contents
1. Introduction
2. Atiyah-Hirzebruch Spectral Sequences
3. Conjectures and Computations
4. Compatibility of Poincare Dualities
Appendix A. Γ-spaces
Appendix B. Globalization
References

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1. Introduction
A semi-topological Atiyah-Hirzebruch spectral sequence relating the morphic
cohomology and the semi-topological K-theory of smooth complex variety X isconstructed is constructed in [FHW04]. There is a comparison map from the motivic Atiyah-Hirzebruch spectral sequence to the semi-topological spectral sequence
and a map from the semi-topological spectral sequence to the topological AtiyahHirzebruch spectral sequence for connective complex K-theory. A first goal ofthis paper is to fill in this picture for real varieties. A second goal is to computeequivariant morphic cohomology and real semi-topological K-theory for some realvarieties.
In Section 2 we establish the semi-topological Atiyah-Hirzebruch spectral sequence and show that it receives a map from the motivic spectral sequence and2000 Mathematics Subject Classification. 19E15, 19E20, 14F43.
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JEREMIAH HELLER AND MIRCEA VOINEAGU

maps to the spectral sequence constructed by Dugger in [Dug05, Corollary 1.3],
E2p,q (alg) = Hp−q
M (X; A(−q)) =⇒ K−p−q (X; A)yE2p,q (sst) = L−q HRp,−q (X; A) =⇒ KRsst−p−q (X; A)yE2p,q (top) = H p,−q (X(C); A) =⇒ krp+q (X(C); A).

To construct the real semi-topological spectral sequence we apply FriedlanderHaesemeyer-Walker’s method [FHW04], with straightforward modifications for realvarieties. As their method makes clear, in order to produce the semi-topological
spectral sequence one may use as input any model for the motivic spectral sequence.
However, constructing the companion topological spectral sequence and identifying
the equivariant homotopy type of the spectra appearing in its construction requires
more care. Presumably this can be carried out with any of the available models,
however we find it convenient to make use of Grayson’s model (as described in[Wal96, Wal00]). Suslin [Sus03] has proved that the motivic cohomology appears
as the E2 -term of Grayson’s spectral sequence. Moreover it is widely believed thatFriedlander-Suslin’s spectral sequence [FS02], Levine’s spectral sequence [Lev08],
and Grayson’s spectral sequence are in fact all the same spectral sequence. The
advantage that this model has for us is that the spectra in this tower arise fromhomotopy group completions of simplicial sets of algebraic maps to certain quot
schemes. This geometric description makes it straightforward to prove Proposition
B.1, which is one of the main technical tools we use to identify the form of our
topological spectral sequence. The topological spectral sequence we construct here
has the same form as the spectral sequence constructed by Dugger in [Dug05,
Corollary 1.3] and indeed in Proposition 2.35 we prove that these two spectral
sequences are the same.
Suslin’s conjecture on morphic cohomology of a smooth complex variety predictsnthat the cycle map Lq H n (X) → Hsing(X(C); Z) is an isomorphism for n ≤ q
and an injection for n = q + 1. We formulate appropriate analogues for the realmorphic cohomology and the equivariant morphic cohomology of a smooth realvariety (see Section 3 for precise statements). The real morphic cohomology groupsof real varieties and the morphic cohomology of complex varieties are particularinstances of equivariant morphic cohomology, and so at first glance the equivariant
conjecture would appear to be stronger than the rest of these conjectures. However,as we show in Theorem 3.4 all three of these conjectures are equivalent. As aconsequence we verify the equivariant Suslin conjecture in codimension one for allsmooth real varieties and give a complete computation of real morphic cohomology
in codimension one (see Theorem 3.9), generalizing a result of Teh [Teh08]. For
certain real varieties, such as geometrically rationally connected threefolds andgeneric cubics of small dimension we verify the equivariant Suslin conjecture in all
codimensions.
Moreover in Section 3, we discuss the real algebraic equivalence relation on real
cycles (as defined by [FW02a]). We prove that it satisfies a Bloch-Ogus formula for
Bredon cohomology (and also Borel cohomology) as well as a number of other good
properties satisfied by the usual algebraic equivalence over complex algebraic cycles

EQUIVARIANT INVARIANTS OF REAL VARIETIES

3(see Theorem 3.24 and Theorem 3.9). Together with other techniques, these allow
us to prove a stronger result than is predicted in general by the real Suslin’s conjecture in some cases (e.g. geometrically rationally connected threefolds), extending
results from the complex case of Voineagu [Voi08]. In particular we obtain that allof the equivariant morphic cohomology groups are finitely generated in these cases.
Using the spectral sequences constructed in Section 2 in conjunction with our
computations of the real morphic cohomology, we compute the real semi-topological
K-theory of some real varieties. As an additional application of the spectral sequences constructed here, in conjunction with Voevodsky’s verification of the Milnor
conjecture, we give an alternate proof of the 2-adic Lichtenbaum-Quillen over R.
This was proved [KW03] (in a slightly weaker form) and in general (for any real
closed field) in [RØ05]. One would like to use Voevodsky’s solution of Milnor conjecture [Voe03] together with a comparison of Atiyah-Hirzebruch spectral sequencesin order to prove this conjecture (indeed this method works just fine over non-realclosed fields). The difficulty which has to be worked around in [KW03] and [RØ05]
is that if X has a real point then its etale 2-cohomological dimension is infinite andthe etale Atiyah-Hirzebruch spectral sequence will not converge. The proof we give
avoids this problem by using the Atiyah-Hirzebruch spectral sequence based on the
Bredon cohomology of the Z/2-space X(C) (which is convergent). We view thisapplication as an example of the fact that Bredon cohomology is a more naturaland useful target for Chow groups of real algebraic cycles than Borel cohomology.
Finally in Section 4 we verify the expected compatibility between the duality
proved in [HV09] relating equivariant morphic cohomology and dos Santos’ equivariant Lawson homology groups of a smooth projective variety X and the Poincare
duality relating the Bredon cohomology and homology of the space X(C). An important consequence of this compatibility is Corollary 4.2 which asserts that inweights larger than dim U , the cycle map from the equivariant morphic cohomology of U to the Bredon cohomology of U is an isomorphism for any smooth quasiprojective real variety U . This identification is an important ingredient in thepreviously mentioned applications of the spectral sequences. Another applicationof this compatibility was given in [HV09, Corollary 5.14].
The techniques and computations presented in this paper are a sequel to our previous work on real algebraic cycles [HV09] and to the paper [FHW04] of FriedlanderHaesemeyer-Walker which had an important influence on this work.
Acknowledgements We were told by E. Friedlander that a version of the spectral sequence constructed in section 2 of this paper was also constructed independently by E. Friedlander, M. Walker and C. Haesemeyer. We thank him for
informing us about their work, for his interest in our paper, and his encouragements. We also thank Pedro dos Santos and Jens Hornbostel for interest in thispaper and Mark Walker for useful discussions.
1.1. Notation. By a k-variety we mean a reduced, separated quasi-projective
scheme of finite type over a field k. We write Sch/k for the category of k-varieties
and Sm/k for the subcategory of smooth quasi-projective k-varieties. Unless otherwise specified, G denotes the group Gal(C/R) = Z/2.
If X is a real variety then the set HomR (C, X) is denoted as X(C). This set
agrees with the set of closed points XC (C) of the complexification and is given an
analytic topology via this identification. Note that when X is a complex variety
viewed as a real variety this notation is potentially ambiguous but it will always be

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JEREMIAH HELLER AND MIRCEA VOINEAGU

clear in such situations which is meant. Via the action of complex conjugation thespace X(C) is a G-space.
By a spectrum we mean a spectrum in the sense of Bousfield-Friedlander, which
consists of a sequence A = (A1 , A2 , . . .) of simplicial sets together with connecting
maps σi : S 1 ∧ Ai → Ai+1 . A naive G-spectrum is a G-object in the category ofspectra.
A map of G-spaces (or naive G-spectra) f : A → B is a G-weak equivalence
provided both A → B and AG → B G are weak equivalences.
Let V be a real representation of G, write S V for the one-point compactificationof V . The V th homotopy group of a based G-space X isπV X = [S V , X]G ,(where [−, −]G denotes maps in the based G-homotopy category). When V = Rp,q
we use the notation πp,q X = πRp,q X where Rp,q is Rp+q with trivial action on thefirst p-components and is multiplication by −1 on the last q-components. In particular πk,0 X = πk X G . We use a similar convention for the RO(G)-graded Bredon
cohomology with coefficients in a Mackey functor M and write H p,q (X; M ) =p,q
H R (X; M ). For more details on equivariant homotopy and cohomology see[May96].
2. Atiyah-Hirzebruch Spectral Sequences
In this section we construct the semi-topological Atiyah-Hirzebruch spectral sequence for real varieties following the construction in [FHW04] for complex varieties. We show that this spectral sequence receives a map from the motivic AtiyahHirzebruch spectral sequence and maps to the Atiyah-Hirzebruch spectral sequence
for Bredon cohomology and real K-theory constructed by Dugger.
2.1. Motivic spectral sequence. There are several available constructions of amotivic Atiyah-Hirzebruch spectral sequence. For our purposes we find it convenient to make use of Grayson’s construction [Gra95] (as reinterpreted by Walker[Wal96, Wal00]).
Let f be a polynomial (integer valued with rational coefficients), E a coherent
sheaf on a quasi-projective variety Y , and Y → S a morphism of quasi-projective
varieties. The quot scheme QuotfE/Y /S represents the functor which sends X to∗the collection of quotient objects [πXE ։ F ] (where πX : X ×S Y → Y is the
projection) which are flat over X and have support proper over X with Hilbert
polynomial f .
`Definition 2.1. For an integer n define GY (n) ⊆ deg(f )=0 QuotfOn /Y / Spec k to beYnthe subfunctor whose value on X is the collection of quotient objects [OX×Y։ F]
which satisfy the conditions(1) F is flat over X and Supp(F ) is finite over X,nn(2) The induced map OX→ π∗ OX×Y→ π∗ F is surjective.
By [Wal02, Lemma 2.2] this functor is represented by a variety which we also`ndenote GY (n). In fact GY (n) ⊆ deg(f )=0 QuotfOn /Y / Spec k is an open subvariety.
Y`
In [GW00] Grayson-Walker use KY (n) = deg(f )=0 QuotfOn /Y / Spec k rather thanY
GY (n). Using GY (n) rather than KY (n) does not change the resulting bivariant

EQUIVARIANT INVARIANTS OF REAL VARIETIES

5

theory [Wal02] and has several advantages such as making certain induced maps ofspectra functorial rather than merely functorial up to homotopy.
The projection On+1 → On onto the first n basis elements defines a map
GY (n) → GY (n + 1) and let GY denote the ind-scheme GY = colimn GY (n). The
elements of GY (X) = colimn Hom(X, GY (n)) are considered as quotient objects∞[p : OX×Y։ F ] such that all but finitely many of the standard basis elements arein the kernel of p.
∞∞
Definition 2.2. An m-tuple ([p1 : OX×Y
։ F1 ], . . . , [pm : OX×Y։ Fm ]) is said
to be in general position if the induced map
X∞pi : OX×Y→ F1 ⊕ · · · ⊕ Fmi(m)

is again a surjection. Write GY (X) ⊆ GY (X)×m for the subset of all m-tuples
which are in general position.
(m)

By [GW00] the functor X 7→ GY (X) on Sch/k is represented by an open(m)(0)
ind-scheme GY ⊆ G×m
Y . By convention GY = Spec k.
(m)
The collection {GY }m≥0 gives a Γ-object in presheaves of sets on Sch/k (seeAppendix A for a recollection of Γ-spaces). Given a map σ : m → n then σ∗ :(m)(n)
GY (X) → GY (X) is defined by∞∞σ∗ ([p1 : OX×Y
։ F1 ], · · · , [pm : OX×Y։ Fm ]) =∞∞=([q1 : OX×Y
։ E1 ], · · · , [qn : OX×Y։ En ]),∞
where Ek = ⊕i∈σ−1 (k) Fi and qk : OX×Y։ Ek is the sum over i ∈ σ −1 (k) of themaps pi . The condition of being in general-position implies that this definition iswell-defined. This is functorial in X and so defines a Γ-object in the category ofpresheaves on Sch/k. The argument in [GW00, Lemma 2.2] shows that the Γ-space(m)m 7→ GY (X × ∆• ) is special for any X.

Definition 2.3. For quasi-projective k-varieties X and Y define Kgeom (X, Y ) to(m)be the Ω-spectrum associated with the special Γ-space m 7→ GY (X × ∆• ).
By [Wal02] the Ω-spectrum Kgeom (X, Y ) defined here is equivalent to the spec(m)trum obtained from the special Γ-space m 7→ KY (X × ∆• ). Let K(X, Y ) denote
the K-theory spectrum of the exact category P(X, Y ) of coherent OX×Y -modules
which are flat over X and have finite support over X. By [GW00, Theorem 2.3] for
any X and Y there is a natural map of Ω-spectra Kgeom (X, Y ) → |K(X × ∆• , Y )|which is always a weak equivalence. In particular whenever X is regular thenπ∗ Kgeom (X, Y ) = K∗ (X, Y ).
Lemma 2.4. ([Wal02, Lemma 2.3]) A map f : Y → Y ′ of k-varieties induces
(m)
(m)
n
a natural map of Γ-spaces f∗ : GY (X × ∆• ) → GY (X × ∆• ), by f∗ [OX×Y։nM] = [OX×Y ′ → (id × f )∗ M]. In particular we obtain a natural transformation ofpresheaves of Ω-spectraf∗ : Kgeom (−, Y ) → Kgeom (−, Y ′ ).
nProof. Let [OX×Y։ M] be an object of GY (n)(X). The second condition onn
M in Definition 2.1 guarantees that the induced map [OX×Y′ → (id × f )∗ M]•
is again a surjection. Thus f∗ : GY (X × ∆ ) → GY ′ (X × ∆• ) is well-defined.

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JEREMIAH HELLER AND MIRCEA VOINEAGU∞
The second condition in Definition 2.1 also guarantees that if ([p1 : OX×Y։∞M1 ], . . . , [pm : OX×Y ։ Mm ]) is an m-tuple in general position then the induced
m-tuple ((id × f )∗ (p1 ), . . . , (id × f )∗ (pm )) is again in general position. Thus we(m)(m)
have a natural map of Γ-spaces f∗ : GY (X × ∆• ) → GY ′ (X × ∆• ).


Let Cube(n) denote the category whose objects are the subsets of {1, . . . , n} andmorphisms are subset inclusions. An n-cube in Sch/k is a functor from Cube(n) to1
Sch/k. For a subset I ⊆ {1, . . . , n} define P∧n
I = P1 ×· · ·×Pn where Pi = P if i ∈ I
and otherwise Pi = {∞}. By the above lemma we have natural transformations∧n(n)
Kgeom (−, P∧n(X) to be homotopy
I ) → Kgeom (−, PJ ) for I ⊆ J. Define W∧ncolimit of the cube I 7→ Kgeom (−, PI ), this is natural in X and so defines thepresheaf W (n) (−). When X is regular, W (0) (X) ∼ K(X).
There are functorial maps W (n+1) (X) → W (n) (X) [Wal00, Section 2], fromwhich we obtain a tower of presheaves· · · → W (n+1) (−) → W (n) (−) → · · · → W (0) (−).
For a simplicial abelian group A• write M(A• ) for the associated Ω-spectrum,
M(A• ) = (A• , BA• , B 2 A• , · · · ).
Write zequi (Y, 0)(−) for the presheaf of equidimensional cycles of relative dimension 0 [SV00]. Using the localization property for the complex of equidimensional cycles [FV00] we see that the complex zequi (P∧n , 0)(X × ∆• ) is equivalent
to the complex zequi (An , 0)(X × ∆• ) which computes motivic cohomology (see[FV00, MVW06]). As explained in [Wal00, Section 4] there is a natural sequenceof spectra
W (n+1) (X) → W (n) (X) → M(zequi (P∧n , 0)(X × ∆• ))which is homotopic to the constant map. By [Sus03] this sequence is a homotopy
fiber sequence when X = Spec OY,y is the local ring of a smooth variety Y at apoint y ∈ Y .
This tower is globalized as in Friedlander-Suslin [FS02] via Godement resolutions (for the Zariski topology) which are recalled in Appendix B. Write G
for the Godement resolution and define K(n) (−) = GW (n) (−) and M(n) (−) =GM(zequi (P∧n , 0)(−×∆• )). Observe that because the presheaf of simplicial abeliangroups zequi (P∧n , 0)(− × ∆• ) satisfies Zariski descent on smooth varieties, the natural map M(zequi (P∧n , 0)(X × ∆• )) → M(n) (X) is a weak equivalence for anysmooth k-variety X.
Because W (n+1) (X) → W (n) (X) → M(zequi (P∧n , 0)(X × ∆• )) is a homotopy
fiber sequence when X = Spec OY,y for a smooth variety Y and a point y ∈ Y , we
see thatK(n+1) (X) → K(n) (X) → M(n) (X)(2.5)

is a homotopy fiber sequence for any smooth X. In general when X is not smooththe composition is homotopic to the constant map.
This gives the tower of presheaves of spectra on Sch/k
(2.6)

···

/ K(q+1)/ K(q)/ K(q−1)M(q)M(q−1)/ ···/ K(0)M(0)We summarize the properties of this tower which we use.

/ K.

EQUIVARIANT INVARIANTS OF REAL VARIETIES

7

Theorem 2.7. [Gra95, Sus03] Let X be a smooth k-variety.
(1) Each of K(q) (X) and M(q) (X) are (−1)-connected Ω-spectra.
(2) The sequenceK(q+1) (X) → K(q) (X) → M(q) (X)is a homotopy fibration sequence of spectra (in general the sequence is homotopic to the constant map if X is not smooth).
(3) For any abelian group A,
2q−kπk (M(q) (X); A) = HM
(X; A(q))(4) The augmentation K(0) (X) → K(X) is a weak equivalences of spectra.
(5) If k < q − dim X then πk K(q) (X) = 0.
Proof. The first four items have already been discussed above. Let πei W (q) denote(q)
the Zariski sheafification of the presheaf U 7→ πi W (U ) on Sm/k. For Y =Spec OX,x , the localization of a smooth variety X a point x ∈ X, there is a naturalq(q)homotopy equivalence of spectra Kgeom (Y, G∧q(Y ) by [Wal96, Theoremm)→Ω W(q)(q)
7.11]. Therefore π
ei W = 0 for i < q. That πk K (X) = 0 for k < q − dim X now
follows from the descent spectral sequence [BG73]s
E2s,t = HZar
(X; πet W (q) ) =⇒ πt−s K(q) (X)since E2s,t 6= 0 means that s ≤ dim X and t ≥ q.

Corollary 2.8. [Gra95, Sus03] For any smooth k-variety X and any abelian group
A, there is a strongly convergent spectral sequencep−q
E2p,q = HM
(X; A(−q)) =⇒ K−p−q (X; A).

Proof. For any abelian group A, the homotopy fiber sequences (2.5) yield an exact
couple,D2p,q = π−p−q (K(−q) (X); A) and E2p,q = π−p−q (M(−q) (X); A).
By the previous theorem this exact couple is bounded below and thus the resulting
spectral sequence converges to colimp D2p,n−p = π−n (K(p−n) (X); A) = π−n K(X; A)(see for example [Wei94, Corollary 5.9.7]).

We are primarily interested in the cases k = R and k = C. Let L/k be a finite
extension of fields. In the following lemma, in order to (temporarily) distinguishbetween the presheaf of spectra on Sch/k obtained using GPnk and the presheaf
(n)

(n)

on Sch/L using GPnL we use the notation Kk (X) and KL (X) (and similarly forthe presheaves M(n) ). Because of the following lemma we will not make careful
distinction between the two in the rest of this paper.
Lemma 2.9. Let L/k be a finite field extension and let p : Sch/k → Sch/L be
the map of sites specified by X 7→ XL . Then for an L-variety Y we have that
(n)
(n)
(n)
(n)
p∗ Kk (Y ) = KL (Y ) and p∗ Mk (Y ) = ML (Y )Proof. For a k-variety Y we have the agreement of presheaves of sets on Sch/L,
(m)

(m)

p∗ Homk (−, GY ) = HomL (−, GYL ).

8JEREMIAH HELLER AND MIRCEA VOINEAGU
(m)

(m)

Thus for any L-variety X we have, p∗ GY (X×L ∆•L ) = GYL (X×L ∆•L ). If F (−) is apresheaf of Γ-spaces then p∗ BF = Bp∗ F and so p∗ Kgeom,k (−, Y ) = Kgeom,L (−, YL ).
This implies that(n)(n)
p∗ Wk (−) = WL (−).
Finally, because the Godement resolution commutes with p∗ we have
(n)

(n)

(n)

(n)

p∗ Kk (−) = p∗ GWk (−) = GWL (−) = KL (−).
A similar argument gives the result for the presheaves M(n) (−).



2.2. Semi-topological Spectral Sequence. We now construct a semi-topological
spectral sequence for real varieties, using the motivic spectral sequence above, andfollowing the method of [FHW04]. We begin by recalling a construction used byFriedlander-Walker.
Let F (−) be a presheaf of abelian groups, simplicial sets, or spectra on Sch/R.
If T is a topological space and X is a real variety then define F (T ×R X) by thefiltered colimit
F (T ×R X) = colim F (V ×R X).
T →V (R)The colimit is over continuous maps f : T → V (R)an , where V is a real variety,and a map from (f : T → V (R)) to (g : T → U (R)) is a map α : V → U of realvarieties such that g = αf . Similarly define F (T ×C X) by the colimit
F (T ×C X) = colim F ((VC ) ×R X).
T →V (R)Via the action of Gal(C/R) on each F ((VC ) ×R X) we obtain an action of Gal(C/R)
on F (T ×C X).
Remark 2.10. When H is a presheaf on Sch/C, Friedlander-Walker [FW02a, proof
of Lemma 2.4] show thatcolim H(UC ) ∼= colim H(W )T →U(R)

T →W (C)where the first colimit is over continuous maps T → U (R) with U a real variety
and in the second T → W (C) with W a complex variety. In particular writingp : Sch/R → Sch/C for the map X 7→ XC we see that for a presheaf F (−) on
Sch/R we have the agreement
F (T ×C R) = (p∗ p∗ F )(T ×R R) = (p∗ F )(T ×C C).
Let F be a presheaf of simplicial sets on Sch/R. We obtain a bisimplicial set by
the formula d 7→ F (∆dtop ×R X) and defineF (∆•top ×R X) = diag(d 7→ F (∆dtop ×R X)).
Similarly, if F is a presheaf of spectra on Sch/R then d 7→ F (∆dtop ×R X) is a
simplicial spectrum. Define the spectrum F (∆•top ×R X) by the formulaF (∆•top ×R X)i = diag(d 7→ F (∆dtop ×R X)i )and bonding maps given byS 1 ∧ diag(d 7→ F (∆dtop ×R X)i ) ∼= diag(d 7→ S 1 ∧ F (∆dtop ×R X)i ) →→ diag(d 7→ F (∆dtop ×R X)i+1 )EQUIVARIANT INVARIANTS OF REAL VARIETIES

9Definition 2.11. Let F be a presheaf of simplicial sets of spectra. Define a newpresheaf Fsst on Sch/R by
Fsst (X) = F (∆•top ×R X).
Remark 2.12. For emphasis and to avoid confusion we will often write below
F (∆•top ×R R) instead of F (∆•top ) when F is a presheaf on Sch/R and similarly
we will write G(∆•top ×C C) instead of G(∆•top ) when G is a presheaf on Sch/C.
In [FW03] the uad-topology on Sch/C is used to establish a recognition principle
identifying when a map F (−) → G(−) of presheaves on Sch/C induces a weakequivalence F (∆•top ×C C) → G(∆•top ×C C). In [HV09] it is observed that theirwork carries over to Sch/R, using a suitable real uad-topology. We briefly recall
this topology, which is essentially due to Deligne.
Definition 2.13.
(1) A continuous map of topological spaces f : S → T issaid to satisfy cohomological descent if for any sheaf A of abelian groups
on T the natural map
H ∗ (T, A) → H ∗ (NT (S), f ∗ A)
is an isomorphism. Here NT (S) → T is the Cech nerve of f , i.e. NT (S) is
the simplicial space which in degree n is the n + 1-fold fiber product of S
over T . A map f : S → T is said to be of universal cohomological descentprovided the pullback S ×T T ′ → T ′ along any continuous map T ′ → T isagain of cohomological descent.
(2) The uad-topology on Sch/C is the Grothendieck topology associatedto the`pretopology generated by collections {Ui → X} such that
Ui (C)an →X(C)an is a surjective map of universal cohomological descent.
(3) The uad-topology on Sch/R is the Grothendieck topology associatedto the`pretopology generated by collections {Ui → X} such that
Ui (R)an →X(R)an is a surjective map of universal cohomological descent.
A useful feature of the uad-topologies is that a complex variety is locally smoothin the uad-topology and a real variety is locally smooth in the real uad-topology.
The following is the recognition principle proved by Friedlander-Walker [FW03](for the real analogue see [HV09]).
Theorem 2.14 ([FW03, Theorem 2.2]). Let k = C or R. Suppose that F → G is anatural transformation of presheaves of abelian groups on Sch/k. If Fuad −→ Guadis an isomorphism of uad-sheaves, thenF (∆•top ×k k) → G(∆•top ×k k)
is a homotopy equivalence of simplicial abelian groups.
The following two propositions were proved in the complex case in [FHW04].
Proposition 2.15. (c.f. [FHW04, Theorem 2.6] Let k = C or R. Suppose thatf : F → G is a natural transformation of presheaves of spectra on Sch/k. Supposethat there is an integer N with the property that the maps (πq F )uad → (πq G)uad
are isomorphisms of uad-sheaves for q ≤ N and a surjection for q = N + 1. Then
πq F (∆•top ×k k) → πq G(∆•top ×k k)

10

JEREMIAH HELLER AND MIRCEA VOINEAGU

is an isomorphism for q ≤ N and a surjection for q = N + 1 . In particular if
F (X) → G(X) is a weak equivalence of spectra for all smooth X thenF (∆•top ×k k) → G(∆•top ×k k)
is also a weak equivalence of spectra.
Proof. Define the presheaf of spectra Q(−) by Q(X) = hocofib(F (X) → G(X)).
Then πq Quad = 0 for all q ≤ N + 1. By the recognition principle the simplicial
abelian group (d 7→ πq Q(∆dtop ×k k)) is contractible for any q ≤ N + 1.
From the strongly convergent spectral sequence [Jar97, Corollary 4.22]
2Es,t= πs (d 7→ πt Q(∆dtop ×k k)) =⇒ πs+t Q(∆•top ×k k)

we conclude that πq Q(∆•top ×k k) = 0 for q ≤ N + 1. Finally observe thatQ(∆dtop ×k k) ≃ hocofib(F (∆dtop ×k k) → G(∆dtop ×k k))because filtered colimits preserve homotopy cofiber sequences. Since the diagonal
converts degreewise homotopy cofiber sequences of simplicial spectra into homotopy
cofiber sequences of spectra, we have thatQ(∆•top ×k k) ≃ hocofib(F (∆•top ×k k) → G(∆•top ×k k)).
Finally from the long exact sequence for homotopy cofiber sequences of spectra weconclude that πk F (∆•top ×k k) → πk G(∆•top ×k k) is an isomorphism for k ≤ N anda surjection for k = N + 1.
Proposition 2.16. ([FHW04, Corollary 2.7]) Let k = C or R. Suppose that
F →G→His a sequence of presheaves of spectra on Sch/k such that the composition is homotopic to the constant map and the sequence
F (X) → G(X) → H(X)
is a homotopy fiber sequence for any smooth X. Then
(2.17)F (∆•top ×k X) → G(∆•top ×k X) → H(∆•top ×k X)

is a homotopy fiber sequence of spectra.
Proof. A sequence of spectra is a homotopy fiber sequence if and only if it is a
homotopy cofiber sequence. Write Q(X) = hocofib(F (X) → G(X)). Filteredcolimits preserve homotopy cofiber sequences of spectra. Therefore for each d thesequence
F (∆dtop ×k k) → G(∆dtop ×k k) → Q(∆dtop ×k k)
is a homotopy cofiber sequence. The diagonal converts degreewise homotopy cofiber
sequences of simplicial spectra into homotopy cofiber sequences of spectra and thusF (∆•top ×k k) → G(∆•top ×k k) → Q(∆•top ×k k)
is a homotopy cofiber sequence. The natural map Q → H is a weak equivalenceon smooth varieties which implies that (πq Q)uad → (πq H)uad is an isomorphism
for all q. Therefore Q(∆•top ×k k) → H(∆•top ×k k) is also a weak equivalence by
Proposition 2.15. We conclude that the sequence (2.17) is a homotopy cofiber (andthus a homotopy fiber) sequence of spectra.
EQUIVARIANT INVARIANTS OF REAL VARIETIES
(q)

11

(q)

Write KRsst (X) = K(q) (X ×R ∆•top ) and MRsst (X) = M(q) (X ×R ∆•top ). We
obtain from the motivic tower (2.6) the semi-topological tower of presheaves ofspectra,···/ KR(q+1)
sst/ KR(q)sst/ KR(q−1)


(q)
MRsst


(q−1)
MRsst

sst

/ KR(0)

/ ···

sst

/ KRsst .


(0)
MRsst

Theorem 2.18. Let X be a smooth real variety.
(q)

(q)

(1) The spectra KRsst (X) and MRsst (X) are (−1)-connected spectra. More(q)over πk KRsst (X) = 0 for k < q − dim X.
(2) The sequence(2.19)

(q+1)

KRsst

(q)

(q)

(X) → KRsst (X) → MRsst (X)is a homotopy fiber sequence of spectra.
(3) For any abelian group A,(q)πk (MRsst (X); A) = Lq HRq−k,q (X; A),
where Lq HRq−k,q (X; A) denotes the real morphic cohomology defined in[FW02a].
(4) The augmentation(0)

KRsst (X) → KRsst (X)is a weak equivalence of spectra, where KRsst (X) is the real semi-topological
K-theory spectrum defined in [FW02a, Definition 2.1] (denote there by
Kalg (∆•top ×R X).
Proof.
(1) This follows from Theorem 2.7 and Proposition 2.15.
(2) This follows from Theorem 2.7 and Proposition 2.16.
(3) The spectrum M(q) (X) is the spectrum associated to the simplicial abelian(q)•group zequi (P∧q
R , 0)(XR ×R ∆R ). Consequently we have that MRsst (X) is
naturally equivalent to the spectrum associated to the simplicial abeliangroup zequi (AqR , 0)(XR × ∆•R × ∆•top ). By [HV09] the homotopy groups ofthis simplicial abelian group compute real morphic cohomology.
(4) Follows from Theorem 2.14 and Theorem 2.7.
The following is the Atiyah-Hirzebruch type spectral sequence relating the realmorphic cohomology and the real semi-topological algebraic K-theory, of a smoothreal variety X.
Corollary 2.20. For any smooth k-variety X and any abelian group A, there is astrongly convergent spectral sequence
E2p,q = L−q HRp,−q (X; A) =⇒ KRsst−p−q (X; A).
Proof. For any abelian group A, the homotopy fiber sequences (2.19) yield an exact
couple,(−q)(−q)

D2p,q (sst) = π−p−q (KRsst (X); A) and E2p,q (sst) = π−p−q (MRsst (X); A).

12

JEREMIAH HELLER AND MIRCEA VOINEAGU

This exact couple is bounded below by the previous theorem and therefore theresulting spectral sequence converges to colimp D2p,n−p = π−n KRsst (X; A) (see forexample [Wei94, Corollary 5.9.7]).
Remark 2.21. More generally we could use the equivariant homotopy groups πs,trather than πs,0 in order to produce other spectral sequences relating equivaridef

ant morphic cohomology and equivariant semi-topological K-theory KRs,t (X) =πs,t K(XC ×C ∆•top ).

2.3. Topological spectral sequence. We continue to write G = Gal(C/R). Let
F be a presheaf of simplicial sets or spectra on Sch/R. We remind the reader
that for a topological space T we have a natural G-action on F (T ×C X) =colimT →U(R) F (UC ×R X) and that F (T ×C X) = p∗ F (T ×C XC ) where p : Sch/R →Sch/C is the map of sites given by X 7→ XC (see Remark 2.10).
The construction of [FHW04, Lemma 3.2] yields an equivariant natural transformation of presheaves of G-simplicial setsp∗ F (∆•top ×C XC ) → Hom( Sing• X(C), p∗ F (∆•top )),where Hom(−, −) denotes the simplicial function complex equipped with the usual
G-action. If F is a presheaf of spectra then this is an equivariant natural transformation of naive G-spectra.
The projections UC → U induce the natural mapG= p∗ F (T ×C XC )G .
F (T ×R X) = colim F (U ×R X) −→ colim F (UC ×R X)
T →U(R)

T →U(R)

Composing with the map from [FHW04, Lemma 3.2] yields the natural transformation
F (∆•top ×R X) → HomG ( Sing• X(C), p∗ F (∆•top )),where HomG (−, −) is the simplicial complex of equivariant maps.
Definition 2.22. If F is a presheaf of simplicial sets on Sch/R define a presheafof G-simplicial sets on Sch/R by
Ftop (X) = Hom( Sing• X(C), p∗ F (∆•top )).
If F is a presheaf of spectra on Sch/R define Ftop by the same formula. In thiscase Ftop is a presheaf of naive G-spectra.
The preceding discussion shows that we have the sequence of natural transformations
G
F → Fsst → Ftop
of presheaves on Sch/R.
Applying the functor (−)top to the motivic tower on Sch/R gives rise to thetower of presheaves of naive G-spectra on Sch/R···/ K(q+1)
top/ K(q)top/ K(q−1)


(q)
Mtop

(q−1)
Mtop

top



/ ···

/ K(0)

top/ Ktop .


(0)
Mtop

We have taken the Grayson spectral sequence as our model for the motivic spectral sequence because with this choice it is straightforward to see that K(n) (XC )G =

EQUIVARIANT INVARIANTS OF REAL VARIETIES

13

K(n) (X). This observation is crucial in the following proposition, which together
with the two propositions after it, form the basis of the companion topological
spectral sequence to the previously constructed semi-topological spectral sequence.
Proposition 2.23. Let X be a smooth real variety. Then
(q+1)

(q)

(q)

Ktop (X) → Ktop (X) → Mtop (X)(2.24)

is an equivariant homotopy fiber sequence.
Proof. It is enough to verify the claim for X = Spec R and to verify that thesequence is a homotopy fiber sequence upon forgetting the G-action and upontaking fixed points. Evaluating this sequence on Spec R and using the observation
at the end of Remark 2.10 and Lemma 2.9, the sequence becomesK(q+1) (∆•top ×C C) → K(q) (∆•top ×C C) → M(q) (∆•top ×C C).
By Proposition B.1, upon taking fixed points this sequence becomesK(q+1) (∆•top ×R R) → K(q) (∆•top ×R R) → M(q) (∆•top ×R R).
That both of these are homotopy fiber sequences of spectra now follows from Proposition 2.16 and Theorem 2.7.

Recall that Atiyah’s real K-theory KR(W ) of the space W equipped with involution [Ati66] is represented by the G-space Z × BU (where BU has G-action given
by complex conjugation). For a special Γ-space F we write SF for the associatedΩ-spectrum. If F has a G-action compatible with the Γ-space structure then SF isa naive G-spectrum (see Appendix A for a recollection of these matters). We define
the naive G-spectrum spectrum kr to be the the spectrum kr = SGrassC (C)an(where GrassC (C)an is given the structure of a Γ-space in a manner analogous tothe Γ-space structure on GY (− × ∆• ) in Section 2.1). For a G-space W we write
kri (W ) = kri,0 . Note that for n ≥ 0 we have that kr−n (W ) = πn MapG (W, kr0 )where MapG (−, −) denotes the space of equivariant maps. Furthermore we havethat Z × BU = kr0 and therefore kr−n (W ) = KR−n (W ) for n ≥ 0.
Proposition 2.25. The naive G-spectrum Ktop (R) is equivariantly equivalent to
kr. In particular Ktop (R)0 ≃ Z × BU .
Proof. The map(0)

Ktop (R) = SHomC (∆•top ×C ∆•C , GrassC ) ←− SHomC (∆•top , GrassC )is a homotopy equivalence by [FW01a, Lemma 1.2]. On fixed points(0)

Ktop (R)G = SHomR (∆•top ×R ∆•R , GrassR ) ←− SHomR (∆•top , GrassR )(0)is a homotopy equivalence by [FW02a]. Thus SHomC (∆•top , GrassC ) → Ktop (R)is an equivariant equivalence. In general if T is a compact Hausdorff topological space and Y is a real variety then HomC (T, YC ) ∼= Homcts (T, Y (C)) is anequivariant isomorphism (by an argument similar to [FW02b, Proposition 4.2]).
(n)(n)
Therefore HomC (∆•top , GrassC ) ∼= Sing• GrassC (C)an is an isomorphism of equi(0)variant Γ-spaces and so Ktop (R) is equivariantly equivalent to the the Ω-spectrum
anSGrassC (C) = kr.
14

JEREMIAH HELLER AND MIRCEA VOINEAGU(q)

Proposition 2.26. The 0th space of the Ω-spectrum Mtop (R) is naturally weakly
equivalent to Sing• Z0 (AqC ) for g ≥ 0. In particular for any real variety X,(q)πk Mtop (X) = H q−k,q (X(C); Z).
(q)

Proof. The spectrum M(R)top is equivalent to the spectrum associated to the simplicial G-module zequi (AqC , 0)(∆•top ). By [FW03, Corollary 3.5] and [HV09, Proposition 4.15] this is equivariantly homotopic to Sing• Z0 (AqC ). By [dS03b] the space
Z0 (AqC ) is the Eilenberg-MacLane space K(Z, Rq,q ).
That this spectral sequence associated to the fiber sequences (2.24) is strongly
convergent spectral sequence will follows from the following proposition.
Proposition 2.27. Let W be a finite-dimensional G-CW complex of dimension d.
There are natural isomorphisms, for any abelian group A
H n,m (W ; A) → H n,m+1 (W ; A)
for n + m > d and for n + m + 1 < 0. In particular(1) if m ≥ 0, then H n,m (X, A) = 0 for n > d and for n + 1 < −m,(2) if m ≤ 0 then H n,m (X, A) = 0 for n + m > d and for n < 0.
Proof. Consider the homotopy cofibration sequence
W+ ∧ Z/2+ → W+ → W+ ∧ S 0,1 .
This induces a long exact sequence in reduced Bredon cohomology· · · → H̃ n−1,m+1 (W+ ∧ Z/2+ ) → H̃ n,m+1 (W+ ∧ S 0,1 ) → H̃ n,m+1 (W+ ) → · · · ,which may be rewritten as(2.28)(n−1)+(m+1)
n+m+1(W ; A) → H n,m (W ; A) → H n,m+1 (W ; A) → Hsing→ Hsing(W ; A) → .
(n−1)+(m+1)(W ; A) =This implies the first statement because for n, m as stated Hsing
n+m+1
0 = Hsing(W ; A).
n
If m ≥ 0 and n > d or n + 1 < −m ≤ 0 then we have 0 = Hsing(W/G) =n,0n,1n,m
H (W ) = H (W ) = · · · = H(W ). Similarly if m ≤ 0 and d < n + m ≤ n
or n < 0 then we have H n,m (W ) = H n,m+1 (W ) = · · · H n,−1 (W ) = H n,0 (W ) =n
Hsing(W/G) = 0.

Theorem 2.29. If X is a smooth real variety then for any abelian group A, there
is a strongly convergent spectral sequence
E2p,q (top) =⇒ krp+q (X; A).

(2.30)

where E2p,q (top) = H p,−q (X; A) for q ≤ 0 and E2p,q (top) = 0 for q > 0.
Proof. The homotopy fiber sequences (2.24) give the exact couple,
(−q)

(−q)

D2p,q (top) = π−p−q,0 (Ktop (X); A) and E2p,q (top) = π−p−q,0 (Mtop (X); A),
which gives rise to a half-plane spectral sequence with exiting differentials in the(n)(n)
terminology of [Boa99, Section 6]. We have πs Ktop (R) = 0 = πs Ktop (R)G for s < n(n)by Theorem 2.7(5) together with Proposition 2.15. Therefore limn πs Ktop (R) = 0 =(n)limn πs Ktop (R)G for all s. These towers satisfy the trivial Mittag-Leffler condition
(n)

(n)

and so lim1n πs Ktop (R) = 0 = lim1n πs Ktop (R)G for all s (see for example [Wei94,

EQUIVARIANT INVARIANTS OF REAL VARIETIES

15

Section 3.5]). From the Milnor exact sequence [Tho85, Lemma 5.41] we conclude(n)(n)
that holim Ktop (R) and holim Ktop (R)G are both weakly contractible. The homo(n)topy limit holim Ktop (R) is computed by first replacing the maps of this tower with
G-fibrations between fibrant objects and then taking an inverse limit and so we
(n)
(n)
(n)
see that (holim Ktop (R))G ≃ holim Ktop (R)G . We conclude that holim Ktop (R) is(n)equivariantly weakly contractible, and therefore holim Hom( Sing• X(C), Ktop (R))
is weakly contractible as well. In turn this implies that limp D2p,n−p (top) = 0. Finally, because E2p,q (top) = 0 when p > 2 dim X by Proposition 2.27(1) we conclude
by [Boa99, Theorem 6.1] that the spectral sequence is strongly convergent andconverges to colimp D2p,n−p (top) = π−n,0 (Ktop (X); A).

To summarize, we have three strongly convergent spectral sequences and natural
maps between them for a smooth real variety X and abelian group A,(2.31)

E2p,q (alg) = Hp−q
M (X; A(−q)) =⇒ K−p−q (X; A)yE2p,q (sst) = L−q HRp,−q (X; A) =⇒ KRsst−p−q (X; A)yE2p,q (top) = H p,−q (X(C); A) =⇒ krp+q (X(C); A).

We finish by comparing our construction with Dugger’s [Dug05]. We warn thereader that the indexing conventions used there are different from ours.
top
For n ≥ 0, define K(n−1)(R) so that it fits into the homotopy fiber sequence of
naive G-spectra(n)top
Ktop (R) → Ktop (R) → K(n−1)(R).

(2.32)

There are homotopy fiber sequences of naive G-spectra
(n)

top
top
Mtop (R) → K(n)(R) → K(n−1)(R)(2.33)(n)top
and the towers {K(n−1)(R)} and {Ktop (R)} give rise to the same spectral sequence(see for example [GM95, Appendix B]).
Write Ω∞ E for the infinite loop space associated to a naive G-spectrum E. By(n)
Proposition 2.25 and Proposition 2.26 we have that Ω∞ Mtop (R) ≃ K(Z, Rn,n ) andΩ∞ Ktop (R) ≃ Z× BU . Taking associated infinite loop spaces of the homotopy fiber
sequences (2.33) gives the homotopy fiber sequences of G-spaces
top
top(R) → Ω∞ K(n−1)(R).
K(Z, Rn,n ) → Ω∞ K(n)(2.34)

We briefly recall Dugger’s construction, for complete details see [Dug05, Section
3]. For a based G-space X, define P2n X to be the nullification of X at the set ofdef

G-spaces ARn,n = {S W | Rn,n ( W } ∪ {S W ∧ G+ | Rn,n ( W } (where the W are
G-representations). Equivalently, this may be viewed as the Bousfield localization
at the set of maps {S W → ∗ | Rn,n ( W } ∪ {S W ∧ G+ → ∗ | Rn,n ( W }. Dugger
defines his spectral sequence [Dug05, Corollary 1.3] via the tower {P2n (Z × BU )}.
Proposition 2.35. There is a map of towers of G-spaces
top{P2n Ω∞ Ktop (R)} → {Ω∞ K(n)
(R)}

16

JEREMIAH HELLER AND MIRCEA VOINEAGU

topsuch that P2n Ω∞ Ktop (R) → Ω∞ K(n)(R) is an equivariant weak equivalence.

Proof. From the homotopy fiber sequences (2.34) and the fact that πa,b K(Z, Rk,k ) =H k−a,k−b (pt, Z) = 0 whenever a ≥ k, b ≥ k and a + b > 2k, we see that if W
top
top
is a G-representation with Rn,n ( W then πW Ω∞ K(n)(R) ⊆ πW Ω∞ K(−1)
(R) =
top
(R)]G =
0. Similarly if Rn,n ( W , then we also have that [S W ∧ G+ , Ω∞ K(k)topπ|W | Ω∞ K(k)(R) = 0. Therefore by the universal property of the functors P2n ,top
the map Ω∞ Ktop (R) → Ω∞ K(n)(R) factors through the localization Ω∞ Ktop (R) →∞ top
P2n Ω K (R). Thus we obtain unique up to homotopy maps P2n Ω∞ Ktop (R) →topΩ∞ K(n)(R) from Dugger’s Postnikov tower so thattop

/ Ω∞ K (R)Ω∞ Ktop (R)
(n)
PPP
O
PPP
PPP
PPP
(
P2n Ω∞ Ktop (R)/ Ω∞ Ktop (R)(n−1)
O/ P2n−2 P2n Ω∞ Ktop (R) = P2n−2 Ω∞ Ktop (R)commutes up to homotopy. Thus we obtain the homotopy commutative comparison
diagram of homotopy fiber sequences
K(Z, Rn,n )/ P2n Ω∞ Ktop (R)K(Z, Rn,n )/ Ω∞ Ktop (R)(n)/ P2n−2 Ω∞ Ktop (R)/ Ω∞ Ktop(n−1) (R).

To see that the map K(Z, Rn,n ) → K(Z, Rn,n ) is an equivariant weak equivalence
it is enough to verify that it is an isomorphism on πn,n and an isomorphism on π2nupon forgetting the equivariant structure [Lew92, Definition 1.4, Lemma 3.7]. It is
an isomorphism on π2n because the tower {K(n) (∆•top ×C C)} has the nonequivariant
homotopy type of the usual Postnikov tower [FHW04, Theorem 3.4]. The forgetfulmap πn,n K(Z, Rn,n ) → π2n K(Z, Rn,n ) is seen to be an isomorphism using thesequence (2.28) because H 0,−1 (pt; Z) = 0 = H 1,−1 (pt; Z). We conclude that all themaps K(Z, Rn,n ) → K(Z, Rn,n ) appearing in the comparison diagram above are
equivariant weak equivalences.
top
By induction we see that the induced maps πk P2n Ω∞ Ktop (R) → πk Ω∞ K(n)(R)
topand πk P2n Ω∞ Ktop (R)G → πk Ω∞ K(n)(R)G are isomorphisms for all k ≥ 1. Now
we address the case k = 0. By Theorem 2.7 together with Proposition 2.15 we see(n)(n)
that πs Ktop (R) = 0 = πs Ktop (R)G for s < n. This together with the homotopy
top
fiber sequence (2.32) of naive G-spectra shows that the maps πs Ω∞ K(n)
(R) →
top
top
top
πs Ω∞ K(n−1)(R) and πs Ω∞ K(n)(R)G → πs Ω∞ K(n−1)(R)G are isomorphisms fors ≤ n − 1. Together with the corresponding fact for the tower P2n Ω∞ Ktop (R)(which follows from [Dug05, Proposition 3.8]) allows us to conclude the case k = 0.
top
Thus the maps P2n Ω∞ Ktop (R) → Ω∞ K(n)(R) are equivariant weak equivalences
and we are done.
Remark 2.36. As a consequence the spectral sequence {E2p,q (top)} agrees with Dugger’s for p + q ≤ 0, with the caveat that his indexing conventions differ from ours.
(Note that Dugger’s spectral sequence is fringed and confined to p + q ≤ 0 while weEQUIVARIANT INVARIANTS OF REAL VARIETIES

17

can have non-zero E2p,q terms for p + q > 0, as long as q ≤ 0, because our spectral
sequence is constructed using naive G-spectra).
3. Conjectures and Computations
In this section we introduce an equivariant version of Suslin’s conjecture on
morphic cohomology and we show that the complex, real, and equivariant versionsof these conjectures are all equivalent. Based on this we compute real morphic
cohomology in weight one, generalizing a partial computation of [Teh08]. We also
prove a real version of the Bloch-Ogus formula [BO74, Corollary 7.4]. This allows
us to prove that real algebraic equivalence coincides with homological equivalencein codimension two for a large class of real varieties. Using these techniques, we
compute real and equivariant morphic cohomology of certain real varieties which
together with the spectral sequences constructed in the previous section allows us tomake new computations of real semi-topological K-theory in terms of Atiyah’s KRtheory. As another application of the spectral sequence we give a simple proof ofthe mod-2 Lichtenbaum-Quillen conjecture over R generalizing a proof of KaroubiWeibel [KW03].
Recall Suslin’s conjecture on morphic cohomology:
Conjecture 3.1 (Suslin’s Conjecture in weight q). Let X be a smooth, quasiprojective complex variety. The cycle mapp
Lq H p (X) → Hsing(X(C); Z)

is an isomorphism for p ≤ q and an injection for p ≤ q + 1.
We introduce the analogues of this conjecture for real morphic cohomology andfor equivariant morphic cohomology. Real morphic cohomology is introduced byFriedlander-Walker in [FW02a] (for equivariant morphic cohomology see [HV09]).
Conjecture 3.2 (Real Suslin’s Conjecture in weight q). Let X be a smooth, quasiprojective real variety. The cycle map
Lq HRa+q (X) = Lq HRa,q (X) → H a,q (X(C); Z)
is an isomorphism for a ≤ 0 and an injection for a ≤ 1.
Conjecture 3.3 (Equivariant Suslin’s Conjecture in weight q). Let X be a smooth,quasi-projective real variety. The cycle map
Lq HRa,b (X) → H a,b (X(C); Z)
is an isomorphism for a ≤ 0 (and a ≤ q, b ≤ q) and an injection for a = 1 (anda ≤ q, b ≤ q).
Theorem 3.4. The following are all equivalent statements:(1) Suslin’s conjecture is valid in weight q,(2) Real Suslin’s conjecture is valid in weight q,(3) Equivariant Suslin’s conjecture is valid in weight q.
Proof. Recall that if X is a complex variety considered as a real variety, thenZ q (XC ) = Z q (X) × Z q (X), where G acts by switching the factors. Consequently
Lq HRa,q (X) = Lq H a+q (X) when X is a complex variety viewed as a real variety.
Thus if the real Suslin’s conjecture is valid in weight q then Suslin’s conjecture isvalid as well. The real Suslin’s conjecture is a special case of the equivariant Suslin’s

18

JEREMIAH HELLER AND MIRCEA VOINEAGU

conjecture. We show that the validity of Suslin’s conjecture in weight q implies
the real Suslin’s conjecture in weight q and that the validity of the real Suslin’s
conjecture in weight q implies the validity of the equivariant Suslin’s conjecture inweight q.
Assume that Suslin’s conjecture is true in weight q. For any ℓ ≥ 1, by [HV09,
Corollary 5.11] the map Lq HRa,q (X; Z/2ℓ ) → H a,q (X(C); Z/2ℓ ) is an isomorphism
for a ≤ 0 and an injection for a = 1. Therefore it is enough to verify the real Suslin’s
conjecture with Z[1/2]-coefficients. For a smooth real variety X, let π : XC → X
be the projection and consider the commutative diagram(3.5)
Lk HRa,q (X; Z[1/2])
H a,q (X(C); Z[1/2])π∗π∗/ Lk H a+q (XC ; Z[1/2])
/ H a+q (X(C); Z[1/2])

π∗

π∗

sing

/ Lk HRa,q (X; Z[1/2])
/ H a,q (X(C); Z[1/2]).

The compositions π∗ π ∗ = 2 and thus the horizontal compositions are isomorphisms.
It follows that Suslin’s conjecture implies the real Suslin’s conjecture.
Now we assume the validity of the real Suslin’s conjecture and conclude that theqequivariant Suslin’s conjecture is valid. Let Fq = hofib(Z q (XC ) → Ztop
(X(C))) be
the homotopy fiber of the cycle map. The homotopy fiber construction is equivariant and so we obtain the equivariant homotopy fiber sequenceqFq → Z q (XC ) → Ztop
(X(C)).

The real Suslin’s conjecture implies that πk FqG = 0 for k ≥ q − 1 and also πk F = 0
for k ≥ q − 1 (because the real Suslin’s conjecture implies Suslin’s conjecture). We
see therefore that Ωq−1 Fq is equivariantly contractible for q ≥ 1 and for q = 0, F0 is
equivariantly contractible as well. It follows that the equivariant Suslin’s conjectureholds.

Remark 3.6. In general, from Diagram 3.5 we see that if Suslin’s conjecture is valid
in weight k then
Lk HRa,b (X; Z[1/2]) → H a,b (X(C); Z[1/2])
is an isomorphism for a + b ≤ k and an injection for a + b ≤ k + 1. A consequenceof this observation is the fact that the cycle map(3.7)

Lk HRa,b (X; Z) → H a,b (X(C); Z)

has 2-torsion kernel a + b ≤ k + 1 and 2-torsion cokernel for any a + b ≤ k, wheneverSuslin’s conjecture is valid in weight k. Moreover if the above isomorphism extends
for a + b ≤ k + 1 then the map 3.7 has a 2-torsion cokernel for any a + b ≤ k + 1.
This remark will be used below in Theorem 3.17.
Corollary 3.8. Let X be a smooth, quasi-projective real variety. Then the cyclemap
L1 HRa,b (X) → H a,b (X(C); Z)
is an isomorphism for a ≤ 0 and an monomorphism for a ≤ 1.
Proof. By [FL92a] Suslin’s conjecture holds in weight 1. According to Theorem
3.4, this implies that the equivariant Suslin’s conjecture holds in weight 1.
EQUIVARIANT INVARIANTS OF REAL VARIETIES

19Friedlander-Walker introduce in [FW02a] the equivalence relation of real algebraic equivalence for algebraic cycles on a real variety. Briefly two cycles α, β ona real variety X are real algebraically equivalent provided there is a smooth real
curve C, two real points c0 , c1 in the same connected component (in the analytic
topology) of C(R), and a cycle γ on X × C such that α = γ|c0 and β = γ|c1 . We
write Aqrae (X) for the group of codimension-q cycles modulo real algebraic equivalence. If X is smooth then Aqrae (X) = Lq HRq,q (X) by [dS03b, Proposition 4.6]and [HV09, Corollary 4.19].
We obtain an improvement of the computation of real Lawson homology of thespace of divisors given in [Teh08].
Corollary 3.9. Let X be a smooth, quasi-projective real variety. Then
∼ H a,1 (X(C); Z) =∼ H a+1 (X(C); Z(1))
L1 HRa,1 (X) =G

for a ≤ 0 and
2(X(C), Z(1)).
L1 HR1,1 (X) = A1rae (X) ֒→ H 1,1 (X(C); Z) ֒→ HG∗
Here HG(X(C); Z(q)) denotes Borel cohomology, and Z(q) is the sheaf Z where G
acts by (−1)q .
In particular, real algebraic equivalence coincides with homological equivalence
for divisors on real smooth varieties.

Proof. This follows by setting b = 1 in Corollary 3.8. The relation between Bredon
cohomology and Borel cohomology is a particular case of [HV09, Proposition 5.17].
The last statement is just a reformulation of the monomorphism L1 HR1,1 (X) =A1rae (X) ֒→ H 1,1 (X(C); Z) and the equality L1 HR1,1 (X) = A1rae (X) is proved in[dS03a] and in [HV09].
r
Remark 3.10. Notice that L1 HRa,1 (X) = 0 for any a ≤ −1 because HG(X, Z(1)) =0 for any r ≤ 0. In particular, the above computation gives exactly the vanishingrange obtained in [Teh08].

Conjecture 3.11. (2-adic Lichtenbaum-Quillen conjecture) Let X be a smooth realvariety. For any k ≥ 1 the map
πn (K(X); Z/2k ) → πn (K(X)hG ; Z/2k )is an isomorphism for n ≥ dim X − 1 and an injection for n ≥ dim X − 2.
By [FW02a, Proposition 4.7, Corollary 3.10] this is equivalent to the statementthat for any smooth real variety X and k ≥ 1, the mapk−n(X(C); Z/2k )KRsstn (X; Z/2 ) → kr

is an isomorphism for n ≥ dim X − 1 and an injection for n ≥ dim X − 2.
This was conjecture was verified in some cases by Friedlander-Walker [FW02a],by Karoubi-Weibel [KW03] (in slightly weaker form for varieties with real points),
and in general by Rosenschon-Ostvaer [RØ05]. We provide a simple proof of thisresult using Voevodsky’s verification of the Milnor conjecture [Voe03] in conjunction
with the comparison of spectral sequences (2.31).
Theorem 3.12. Let X be a smooth, quasi-projective real variety. Then for any
k ≥ 1,k−n
KRsst(X(C); Z/2k )n (X; Z/2 ) → kr
is an isomorphism for n ≥ dim X − 1 and a monomorphism for n = dim X − 2

20

JEREMIAH HELLER AND MIRCEA VOINEAGU

Proof. We write E(sst) (respectively E(top)) for the semi-topological (respectively
the topological) spectral sequence constructed in Section 2.2 and 2.3. We show thatthe map Erp,q (sst) → Erp,q (top) from (2.31) is an isomorphism for p + q ≤ 1 − dim X
and a monomorphism for p + q = 2 − dim X. Since these spectral sequences arestrongly convergent, this will yield the result.
First the case r = 2. Suppose that p + q ≤ 1 − dim X then either p ≤ 0 orq ≤ −d. If p ≤ 0 then by [HV09, Corollary 5.10]
E2p,q (sst) = L−q H p,−q (X; A) → H p,−q (X(C); A) = E2p,q (top)is an isomorphism (note that if both p, −q ≤ 0 then E2p,q (sst) = 0 = E2p,q (top)).
If q ≤ −d then by Corollary 4.2 then E2p,q (sst) → E2p,q (top) is an isomorphism.
Similarly if p + q ≤ 2 − dim X then either p ≤ 1 or q ≤ −d and we see by [HV09,
Corollary 5.10] and Corollary 4.2 that E2p,q (sst) → E2p,q (top) is a monomorphism.
Proceeding by induction as in [FHW04, Theorem 6.1] shows that Erp,q (sst) →Erp,q (top) is an isomorphism for all r and p + q ≤ 1 − dim X and a monomorphism
for all r and p + q ≤ 2 − dim X.
Remark 3.13. The above proof is along the same lines as in [KW03], a key difference
being that there they use a spectral sequence based on the Borel cohomology of
X(C). If X(R) 6= ∅ (i.e. the G-action on X(C) is not free) then this spectral
sequence will not converge and in this way the use of Bredon cohomology simplifies
the situation.
If we know that Suslin’s conjecture is valid for a particular variety then we candeduce an integral agreement of real semi-topological K-theory and kr-theory.
Theorem 3.14. Suppose that X is a smooth, quasi-projective real variety suchthat the cycle maps Lq HRa,q (X) → H a,q (X(C); Z) are isomorphisms for any q,
any a ≤ 0 and injections for a ≤ 1 (in other words X satisfies the real Suslin’s
conjecture). Then−n
KRsst
(X(C))n (X) → kr
is an isomorphism for n ≥ dim X − 1 and a monomorphism for n ≥ dim X − 2.
Proof. Follows from the comparison of spectral sequences by the same argument asin Theorem 3.12.
Corollary 3.15. Let X be a smooth, quasi-projective real surface. Then−n
(X(C); Z)
KRsstn (X; Z) → kr

is an isomorphism for n ≥ 1 and a monomorphism for n = 0.
Proof. This follows from Corollary 3.9 and Theorem 3.14.

The following corollary was proved in [FW02a, Corollary 6.9] using different
methods.
Corollary 3.16. Let X be a smooth, quasi-projective real curve. Then−n
KRsst
(X(C); Z)n (X; Z) → kr

is an isomorphism for n ≥ 0.
Proof. It follows again from Corollary 3.9 and Theorem 3.14.



EQUIVARIANT INVARIANTS OF REAL VARIETIES21

In certain special examples of real smooth projective varieties, some of the semitopological invariants can be computed. For example, suppose that X is a smooth,projective geometrical rationally connected threefold defined over real numbers orthat X is generic smooth, projective real cubic of dimension d where 3 ≤ d ≤ 8,
d 6= 7. Then:
Theorem 3.17. Let X be one of the varieties from the above list. Then
Lq HRa,b (X) → H a,b (X(C); Z)
is an isomorphism for a ≤ 1 (and b ≤ q).
Proof. By [Voi08] and [Voi09], for X is as in hypothesis, the complexification XC
satisfies
Lq H a+b (XC ) ∼= H a+b (XC )for a+b ≤ q+1. By [HV09, Corollary 5.11] Lq HRa,q (X; Z/2ℓ ) → H a,q (X(C); Z/2ℓ )is an isomorphism for a ≤ 0 and an injection for a = 1 and any ℓ > 0. In particular,
we can conclude that the kernel of the cycle map Lq HRa,q (X; Z) → H a,q (X(C); Z)
is 2-divisible for any a ≤ 1 and the cokernel is 2-torsion free for any a ≤ 1 (see[Voi08, Proposition 3.1]). According to the Remark 3.6, the kernel and cokernel ofthe cycle map for a ≤ 1 and b ≤ q are 2-torsion. This implies that the cycle map
Lq HRa,q (X; Z) → H a,q (X(C); Z) is an isomorphism for any a ≤ 1.
Recall that for a based G-space W and fixed value of q ≥ 0, there is a long exact
sequence relating the non-equivariant and equivariant homotopy groups(3.18)
φ(q)

δ (q)

δ (q)

φ(q)

· · · → πp+q+1 W −−→ πp,q+1 W → πp,q W −−→ πp+q W −−→ · · · → π0,q W −−→ πq W,(the map φ(q) is the forgetful map). This sequence arises by considering the homotopy cofiber sequence
Z/2+ → S 0 → S 0,1
and the above long-exact sequence is the one induced by the homotopy fiber sequence
F S 0,1 , W → F S 0 , W → F (Z/2+ , W ) .

q
Applying this homotopy fiber sequence to Z q (XC ) and Ztop
(X(C)) we obtain thefollowing diagram(3.19)/ Lq HRa,b−1/ Lq HRa,b/ Lq H a+b/ Lq H a−1+b
Lq HRa−1,b
H a−1,b/ H a−1+b/ H a,b−1/ H a,b/ H a+b

The theorem now follows from the above diagram and reverse induction on b ≤q.
Remark 3.20. Theorem 3.17 can be extended, using the same ideas, to real smooth
threefolds and fourfolds with zero cycles on the complexification supported on aproper subvariety of dimension less or equal than two. We also remark that equivariant Suslin’s conjecture holds for these varieties and those from Theorem 3.17.

22

JEREMIAH HELLER AND MIRCEA VOINEAGU

Our next results will involve the coniveau spectral sequences for real morphic cohomology and for Bredon cohomology. We make use of [CTHK97] for these coniveau
spectral sequences. Let Y ⊆ X be a closed subvariety, with open complement
U = X − Y . Let Z q (XC )YC = hofib(Z q (XC ) → Z q (UC )) and define the equivariant
morphic cohomology of X with support on Y by Lq HRq−a,q−b(X) = πa,b Z q (XC )YC .
Y
q−a,q−b
Similarly, HY (C) (X(C); Z) = H q−a,q−b (X(C), U (C); Z) denotes the Bredon cohomology with support on Y (C). The generalized cycle map extends to a map ons,tcohomology theories with support Φ : Lq HRs,t
Y (X) → HY (C) (X(C); Z). The following lemma makes explicit that both theories satisfy all the properties requiredin order to make use of the results of [CTHK97].
Lemma 3.21. Real morphic cohomology and Bredon cohomology are homotopy
invariant cohomologies that satisfy Nisnevich excision.
Proof. Both theories are homotopy invariant. That real morphic cohomology satisfies Nisnevich excision follows from the fact that motivic cohomology does. To be
explicit, let
/Y
V
 
U


/X

be a distinguished Nisnevich square. Let M(q) (−) be the presheaf of spectra on
Sch/R from section 2.1 which computes motivic cohomology (so that M(q) (− ×∆•top ) computes real morphic cohomology). The commutative square
M(q) (X × W )/ M(q) (Y × W )M(q) (U × W )/ M(q) (V × W )is homotopy cartesian whenever W is smooth. Let F (W ) and G(W ) denote thehomotopy fibers of the top and bottom horizontal arrows. By Proposition 2.15
F (∆•top ) → G(∆•top ) is a weak equivalence. By Proposition 2.16 we see that these
are the homotopy fibers of the horizontal arrows in the diagram
M(q) (X × ∆•top )/ M(q) (Y × ∆•top )M(q) (U × ∆•top )/ M(q) (V × ∆•top ).

It follows that real morphic cohomology satisfies Nisnevich descent.
That Bredon cohomology satisfies Nisnevich descent follows from the fact thatV (C)/ Y (C) 
U (C) 
/ X(C)

is an equivariant homotopy pushout diagram of G-spaces (see for example [DI04]).


EQUIVARIANT INVARIANTS OF REAL VARIETIES

23

We will use below the following lemma that proves the local vanishing for equivariant motivic cohomology.
Lemma 3.22. Let R = OX,x1 ,...,xn be a semi-local ring of a smooth real variety X
at the points x1 , . . . , xn ∈ X. Then Ls HRa,s (Spec R) = 0 if a > 0.
Proof. Let M(s) be a presheaf computing weight-s motivic cohomology (for example
the presheaf from Section 2.1). If R is as in the statement of the lemma, then2s−iπi M(s) (Spec R) = HM
(Spec R; Z(s)) = 0 for i < s. By Theorem 2.14 we see thatthe simplicial abelian group (d 7→ πi M(s) (Spec R ×R ∆dtop )) is contractible for alli < s. Applying the spectral sequence for a simplicial spectrum [Jar97, Corollary(s)4.22] shows that πi MRsst (Spec R) = Ls HRs−i,s (Spec R) = 0 for i < s.
We now prove a real analogue of the Bloch-Ogus formula [BO74, Corollary 7.4].
pNamely for a smooth quasi-projective real variety X we show that HZar
(X, H0,p ) ∼
=
p
a,b
a,b
Arae (X), where H is the Zariski sheafification of the presheaf U 7→ H (U (C); Z)
and Aprae (X) is the group of codimension-p cycles modulo real algebraic equivalence.
Observe that by [HV09, Proposition 5.17] and [dSLF04, Proposition 1.15], thesheaf H0,p agrees with the sheafification of the Borel cohomology presheaf U 7→p
HG
(U (C); Z(p)) (where Z(p) is the G-module Z with action given by multiplicationby (−1)p ). Thus the following theorem may be seen as an integral extension, forreal varieties, of the Bloch-Ogus formula for etale cohomology with finite coefficients[BO74, Theorem 7.7], because Borel cohomology with finite coefficients agrees withetale cohomology with finite coefficients for real varieties [Cox79].
Theorem 3.23. Let X be a smooth quasi-projective real variety. There is a natural
isomorphismp
HZar(X, H0,p ) ∼= Aprae (X).
Proof. Let X (p) denote the set of points x ∈ X whose closure has codimension p.
i
If H ∗ is a cohomology theory with supports, define Hxi (X) = colimU⊆X Hx∩U
(U )ii
and H (k(x)) = colimU⊆x H (U ) (where in both colimits, U ranges over nonempty
opens). By [CTHK97, Corollary 5.1.11] if H is homotopy invariant and satisfies
Nisnevich excision then for a smooth quasi-projective X, the coniveau spectral sep
quence E1p,q = ⊕x∈X (p) Hxp+q (X) =⇒ H p+q (X) has E2 -term given by HZar
(X; Hq ),qqwhere H is the sheafification of the presheaf U 7→ H (U ). The associated filtration
is F p H n (X) = ∪Z Im(HZn (X) → H(X)), where the union is over closed subvarieties
Z ⊆ X of codimension p.
Equivariant morphic cohomology and Bredon cohomology satisfy these properties by Lemma 3.21 and using Lemma 4.3, the coniveau spectral sequences can be
written as
M
E1p,q (k, b) =Lk−p HRq,b−p (k(x)) =⇒ Lk HRp+q,b (X)x∈X (p)ande p,q (b) =E
1

M

H q,b−p (k(x); Z) =⇒ H p+q,b (X(C); Z),x∈X (p)p
and the E2 terms are respectively given by E2p,q (k, b) = HZar
(X; Lk HRq,b ) and
p,q
p
q,b
p p
p,p
e
E2 (b) = HZar (X; H ). Observe that F L HR (X) = Lp HRp,p (X) (andp,0
F p+1 Lp HRp,p (X) = 0) and so E∞(p, p) = Lp HRp,p (X).

24

JEREMIAH HELLER AND MIRCEA VOINEAGU

In fact we show that we have E2p,0 (p, p) = Lp HRp,p (X). By Lemma 3.22
we see that E1p−r,r−1 (p, p) = ⊕x∈X (p−r) Lr HRr−1,r (k(x)) = 0 for r > 1. Alsop,0E1p+r,−r+1 (p, p) = 0 if r ≥ 1. We conclude that E2p,0 (p, p) = E∞(p, p), and therepp0,ppfore HZar (X, L HR ) = Arae (X).
Consider the commutative diagramLL
00,0 (k(x))
10,1 (k(x))/0/x∈X (p) L HR
x∈X (p−1) L HR
Lx∈X (p−1)
H 0,1 (k(x); Z)/Lx∈X (p)
H 0,0 (k(x))/ 0,pthe cohomology of the top row computes HZar
(X, Lp HR0,p ) and the bottom comp0,pputes HZar (X, HR ). The right vertical arrow is an isomorphism and by Corollary
3.9 the left vertical arrow is also an isomorphism. ThereforeppAprae (X) ∼= HZar (X; Lp HR0,p ) → HZar (X; H0,p )is an isomorphism.

We can now prove a real analogue of a result due to Bloch-Srinivas [BS83].
Theorem 3.24. Let X be a smooth projective real variety of dimension d. Supposethat there is a closed subvariety V ⊆ XC with dim V ≤ 2 such that CH0 (XC \V ) = 0.
Then
L2 HR2,2 (X) = A2rae (X) ֒→ H 2,2 (X(C); Z),
so that real algebraic equivalence on X in codimension 2 agrees with homologicalequivalence.
Proof. We use the same notations as in the previous theorem for the coniveau spec2,0
e∞tral sequence for Bredon cohomology. We know that E(2) = F 2 H 2,2 (X(C); Z)
p,q
3 2,2
e
because F H (X(C); Z) = 0. Because E1 (2) = 0 if q ≤ 1 and p > 2 or if p < 0
e 2,0 (2) = E
e 2,0 (2) = F 2 H 2,2 (X(C); Z) ⊆ H 2,2 (X(C); Z). We thus have
we see that E∞
3d2 e 2,00,1
e 2,0 (2) ⊆ H 2,2 (X(C); Z) which is thee (2) −→ E2 (2) → E
the exact sequence E
32exact sequence
α

0
2
HZar
(X, H1,2 ) → HZar
(X, H0,2 ) −→ H 2,2 (X(C), Z).

The cycle map Φ induces the comparison of coniveau spectral sequences which
together with the preceding theorem and its proof gives us the commutative diagram
2(X, L2 HR0,2 )E22,0 (2, 2) = HZar∼=

/ E 2,0 (2, 2) = L2 HR2,2 (X)
∞

∼
=



e 2,0 (2) = H 2 (X, H0,2 )
E
2
Zar

Φ



//E
e 2,0 (2) ⊆ H 2,2 (X(C); Z).
∞2
We see that under the identification HZar
(X, H0,2 ) = A2 (X)rae the map α isidentified with the cycle map and so the result will follow from showing that
0
k
k
HZar
(X, H1,2 ) = 0. Let Hcsing
denote the sheaf U 7→ Hsing(U (C); Z). WithZ[1/2]-coefficients the composition induced by the map π : XC → X
π∗

π

∗
0
0
0
3
(X, H1,2 [1/2]) −→ HZar
HZar
HZar
(X, H1,2 [1/2])(XC , (Hcsing)[1/2]) −→EQUIVARIANT INVARIANTS OF REAL VARIETIES25

03
is an isomorphism. Since HZar
(XC , (Hcsing)[1/2]) = 0 for a variety X satisfying
the condition from hypothesis (by the proof of [BS83, Theorem 1(ii)]) it is enough0
to show that HZar
(X, H1,2 ) is 2-torsion free. Write Lq HRa,bfor the sheafification
Z/2ka,bof U 7→ Lq HRa,b (U ; Z/2k ) and similarly for HZ/2
k . Consider the comparison ofshort exact sequences0/ L2 HR0,2 ⊗ Z/2k

/ L2 HR0,2 k/ T or(L2 HR1,2 , Z/2k )

/0

0


/ H0,2 ⊗ Z/2k


/ HR0,2 k/ T or(H1,2 , Z/2k )

/ 0.

Z/2

Z/2

0,2
0,2
The map L2 HRZ/2
k → HRZ/2k is an isomorphism by [HV09, Corollary 5.13] andL2 HR1,2 = 0 by Lemma 3.22. We conclude that T or(H1,2 , Z/2k ) = 0 and we aredone.


Because all the real varieties from Theorem 3.17 fulfill the condition of Theorem
3.24, we can use Theorem 3.24 to improve the range of indexes in Theorem 3.17.
We will restrict to the case of real geometrically rationally connected varieties ofdimension 3, although the same type of arguments and computations can be done
for more general varieties that fulfill the condition from Theorem 3.24.
Corollary 3.25. Let X be a smooth projective real geometrically rationally connected threefold. Then
Lq HRa,b (X) → H a,b (X(C); Z)
is an isomorphism for a < q (and b ≤ q) and a monomorphism if a = q (andb ≤ q). In particular, all equivariant morphic cohomology groups of X are finitely
generated.
Proof. Write Fq = hofib(Z q (XC ) → Map(XC (C), Z0 (AqC ))) for the homotopy fiberof the cycle map. The homotopy fiber construction is equivariant and yields anequivariant homotopy fiber sequence
Fq → Z q (XC ) → Map(XC (C), Z0 (AqC )).
Applying Theorem 3.24 and Theorem 3.17 we obtain that π∗ (FqG ) = 0. From [Voi08]
we have that π∗ (Fq ) = 0. We conclude that F is an equivariantly contractible
space. Applying the long exact sequences for the above equivariant homotopy fiber
sequence we obtain that π0,p (Z q (XC )) ֒→ π0,p (Map(XC (C), Z0 (AqC ))) for any p ≥ 0
and that πq,p (Z q (XC )) ≃ πq,p (Map(XC (C), Z0 (AqC ))) for any q > 0 and p ≥ 0. Thisconcludes the statement of the corollary.

The following corollary computes the real semi-topological K-theory of a realgeometrically connected variety.
Corollary 3.26. Let X be a smooth projective real geometrically rationally connected threefold. Then0KRsst
0 (X; Z) ֒→ kr (X; Z)
and
∼
=
KRsst
→ kr−p (X; Z)
p (X; Z) −

26

JEREMIAH HELLER AND MIRCEA VOINEAGU

for any p ≥ 1.
Proof. This follows from the previous corollary together with a comparison of spectral sequences constructed in the previous section (see 2.31).

4. Compatibility of Poincare Dualities
The main result of this section asserts that the Poincare duality proved in [HV09]
between the equivariant morphic cohomology and Dos Santos’s equivariant Lawson homology for a smooth, projective real variety is compatible with the Poincare
duality in Bredon homology and cohomology of a compact real manifold (see Theorem 4.1). The method used here is similar to that of [FL97] where FriedlanderLawson show that the duality between morphic cohomology and Lawson homology
for smooth projective complex varieties is compatible with Poincare duality between
singular cohomology and homology. An important consequence of this compatibility is Corollary 4.2 which is proved below and is needed for our applications in
section 3 of the spectral sequence constructed in this paper.
Let X be a smooth real variety of dimension d. There are generalized cycle
maps Φ : Lq HRa,b (X) → H a,b (X(C); Z) relating equivariant morphic cohomology
and Bredon cohomology (see [FW02a, HV09]). The equivariant Lawson homology
defined by dos Santos [dS03a] comes equipped with operations s : Lq HRa,b (X) →Lq−1 HRa,b (X). In particular, iterating the s-map together with the equivariant
Dold-Thom theorem [dS03b, LF97] defines the generalized cycle map Ψ = sq :Lq HRa,b (X) → L0 HRa,b (X) = Ha,b (X(C); Z).
A real n-bundle over a G-space W is a complex n-plane bundle π : E −→ W
together with an involution τ : E → E covering the involution σ : W → W . A
G-manifold is called a real n-manifold if the tangent bundle is a real n-bundle overW , with involution given by dσ. A real n-manifold W has a fundamental class[W ] ∈ Hn,n (W ; Z) and satisfies Poincare duality ([dSLF04],[LMSM86]),∼=→ Hn−p,n−q (W ; Z).
[W ] ∩ − : H p,q (W ; Z) −The following is the main theorem of this section, whose proof we postpone until
the end of the section.
Theorem 4.1. Let X be a smooth, projective real variety of dimension d. The
following square commutes for all k, l ≤ q, where the horizontal arrows are thePoincare duality isomorphisms
Lq HRk,l (X)D/ Ld−q HRd−k,d−l (X)Φ
H k,l (X(C), Z)ΨP/ Hd−k,d−l (X(C), Z).

We make use of cohomology with supports in the proof of Corollary 4.2 below.
The definitions of these are recalled prior to Theorem 3.23.
Corollary 4.2. Let U be a smooth, quasi-projective real variety of dimension d.
Then the cycle map
Φ : Lq HRa,b (U ) → H a,b (U (C); Z)
is an isomorphism for q ≥ d (and a ≤ q, b ≤ q).

EQUIVARIANT INVARIANTS OF REAL VARIETIES

27

Proof. If U is smooth and projective this follows immediately from Theorem 4.1
together with the fact that the map Ld−q HRd−a,d−b (U ) → Hd−a,d−b (U (C), Z) is
an isomorphism for any q ≥ d.
Reverse induction on b, together with the diagram (3.19) and the corresponding
result for morphic cohomology of complex varieties, shows that in order to prove
the general result we may assume that b = q. We proceed by induction on dim U
to show that Φ is an isomorphism whenever q ≥ dim U and U is smooth andquasi-projective.
The case dim U = 0 is done and so we may assume that the result is valid forall smooth quasi-projective U , with dim U < d. Let U be a quasi-projective realvariety with dim U = d and let U ⊆ X be a smooth projective closure with closed
complement Z ⊆ X (we may assume that U is connected so that dim Z < dim U ).
a,qFirst we argue that Lq HRa,q
W (X) → HW (C) (X(C); Z) is an isomorphism for q ≥ d
and a ≤ q whenever W ⊆ X is a proper closed subvariety. Using Lemma 4.3 below
we find that it is an isomorphism whenever W is smooth. In particular the casedim W = 0 is done and we proceed by induction on dim W . Let Ws ⊆ W be thesingular locus. Since dim Ws < dim W we have by induction that Lq HRa,qWs (X) →a,q
HW(X(C))
isan
isomorphism
for
q
≥
d
and
a
≤
q.
Considering
thecomparisons (C)of long exact sequence/ Lq HRa,q
W (X)/ Lq HRa,q (X)/ Lq HRa,q
W \W (X \ Ws )
/


a,q
/ HW
(X(C))
(C)


a,q
/ HW
(X(C))
(C)


a,q
/ H(W
((X
\ Ws )(C))\W )(C)

/,

W

s

s

s

s

together with Lemma 4.3 below and the inductive hypothesis, we see that for q ≥ da,qand a ≤ q the map Lq HRa,q
W \Ws (X \ Ws ) → H(W \Ws )(C) ((X \ Ws )(C)) is also ana,qisomorphism. We conclude that Lq HRa,q
W (X) → HW (C) (X(C)) is an isomorphism
for q ≥ d and a ≤ q and any closed, proper subset W ⊆ X (for the case a = qa,qobserve that Lq HRa,q
W \Ws (X \ Ws ) = 0 = H(W \Ws )(C) ((X \ Ws )(C)) because q ≥d > dim W \ Ws ) .
Now from the comparison of long exact sequences/ Lq HRa,q (X)Z/ Lq HRa,q (X)/ Lq HRa,q (U )a,q/ HZ(C)(X(C))

/ H a,q (X(C))

/ H a,q (U (C))//

we see that Lq HRa,q (U ) → H a,q (U (C); Z) is also an isomorphism for q ≥ d anda ≤ q (for the case a = q, note that Lq HRq,q (X) → Lq HRq,q (U ) is surjective and
q+1,q
HZ(C)
(X(C)) = 0 by Proposition 2.27 and the long exact sequence for cohomology
with supports, because 2q + 1 ≥ dim X(C)).

Lemma 4.3. Let X be a smooth, quasi-projective real variety and let Z ⊆ X
be a smooth closed subvariety of codimension c. There are natural isomorphisms
∼
∼
=
=
a,b
a−c,b−c→ HZ(C)(X(C); Z)
Lq−c HRa−c,b−c (Z) −→ Lq HRa,b(Z(C); Z) −Z (X) and H

28

JEREMIAH HELLER AND MIRCEA VOINEAGU

which fit into the commutative square
Lq−c HRa−c,b−c (Z)∼=/ Lq HRa,b (X)
Z

Φ

Φ


H a−c,b−c (Z(C); Z)/ H a,b (X(C); Z).
Z(C)

∼
=

Proof. Let N denote the normal bundle of Z ⊆ X. We consider Z as a closed
subvariety of N via the zero-section. The lemma will follow by seeing that we havethe commutative diagram
Lq−c HRa−c,b−c (Z)

∼
=

∼
=

Z

Φ


H a−c,b−c (Z(C); Z)/ Lq HRa,b (N )

Z

Φ
∼
=


/ H a,b (N (C); Z)
Z(C)/ Lq HRa,b (X)
Φ

∼
=


/ H a,b (X(C); Z).
Z(C)The first horizontal arrows are Thom isomorphisms and the second horizontal arrows arise from the deformation to the normal cone.
We apply the results of [Pan03]. To put ourselves in the context of this paper
we consider the total equivariant morphic cohomology and Bredon cohomology:a,b⊕q,a,b Lq HRa,b
Z (X) and ⊕a,b HZ(C) (X(C); Z). These are cohomology theories in thesense of [Pan03, Definition 2.1], which amounts to saying these are homotopy invariant, satisfy Nisnevich excision, and have localization exact sequences. Bothequivariant morphic cohomology and Bredon cohomology clearly satisfy the first
and third property, and the excision property is made explicit in Lemma 3.21. As
shown in [Pan03, Theorem 2.2] the deformation to the normal cone yields natu∼=
ral isomorphisms hZ (N ) −→ hZ (X) for any cohomology theory h(−). The maps
involved in this isomorphism preserve the gradings and therefore we obtain theright-hand commutative square in the diagram above.
Now we need to see that Φ is compatible with Thom isomorphisms (which Panincalls an orientation [Pan03, Definition 3.1]) in both theories. By [Pan03, Theorem
3.35, Theorem 3.5] in order for L∗ HR∗,∗ (−) to have Thom isomorphisms it isenough to see that it has a Chern structure [Pan03, Definition 3.2] which amounts
to having first Chern classes for line bundles which satisfy a naturality condition, aprojective bundle formula for X × P1 , and vanishing on trivial line bundles. Theseare easily deduced from the corresponding properties for motivic cohomology (overC and over R), Theorem 2.14, and the fact that equivariant morphic cohomology
may be computed using the complex zequi (AqC , 0)(XC ×C ∆•top ). Moreover the Thom
isomorphism for E is given by cupping with a Thom class th(E) [Pan03, Lemma
3.33] and keeping track of gradings we see that th(E) ∈ Ln HRn,n
X (E) where E is
an n-plane bundle.
In [dSLF04, Proposition 1.13] it is verified that Bredon cohomology has Thom
isomorphisms for real bundles (in the above sense). In particular Bredon cohomology viewed as a theory on Sm/R has Thom isomorphisms. To see that the naturaltransformation Φ is compatible with the Thom isomorphisms in suffices to see thatn,nΦ(µ) ∈ HX(C)(E(C)) is a Thom class whenever µ ∈ Ln HRn,n
X (E) is a Thom class.
By the description of the Thom class given in [dSLF04, Proposition 1.13], it suffices
to see that for any closed point i : x ֒→ X the restriction i∗ Φ(µ) is a generator of

EQUIVARIANT INVARIANTS OF REAL VARIETIES

29

Hx∗,∗ (i∗ E(C); Z). Note that i∗ E = AnR or AnC depending on whether k(x) = R, or∗,∗∗∗
C. In both cases Φ : L∗ HR∗,∗x (i E) → Hx(C) (i E(C); Z) is an isomorphism and weare done.

Let X be a quasi-projective real variety and Y a projective real variety. The
cap-product pairing ([FW01b, Proposition 1.6], [FL92b, Theorem 7.2])
Cs (XC ) × Mor(XC , Cr (YC )) → Cr+s (XC ×C YC )is defined by sending a pair (f, Z) to the graph of the composite function Z →XC → Cr (YC ). This pairing is G-equivariant and by [FW01b, Theorem 2.6] it
induces a continuous pairing
Cs (XC )an × Mor(XC , Cr (YC ))an → Cr+s (XC ×C YC )an ,which in turn induces the continuous equivariant pairing∩ : Zr (XC ) ∧ Mor(XC , Z0 (AqC )) → Zr (XC ×C AqC ),where for typographical convenience here and below we write
Mor(XC , Z0 (AqC )) =Mor(XC , C0 (PqC ))+q−1 +Mor(XC , C0 (PC)).

We also denote by ∩ the induced pairing,∩ : Lp HRs,t (X) ⊗ Lq HRk,l (X) → Lp−q HRs−k,t−l (X).
When X is irreducible of dimension d, the d-cycle XC ⊆ XC defines the fundamentalclass
ηX ∈ π0,0 Zd (XC ) = Ld HRd,d (XC ),and cap-product with the fundamental class gives a mapηX ∩ − : Lq HRk,l (X) → Ld−q HRd−k,d−l (X),
which is the Poincare duality isomorphism.
For a G-space W , define the equivariant pairing ⊓ to be the compositionµ→ Z0 (W ∧ S d,d )0− Z0 (W ) ∧ Z0 (S d,d)0 −⊓ : Z0 (W ) ∧ F W, Z0 (S d,d )0 →where the first arrow is defined by (x, f ) 7→ (x, f (x)) and µ is the map induced by(x, y) 7→ x ∧ y. Together with the suspension isomorphism in Bredon homology thisdefines a pairing⊓ : Hs,t (W ; Z) ⊗ H k,l (W ; Z) → Hs+d−k,t+d−l (W ∧ S d,d ; Z) ∼= Hs−k,t−k (W ; Z).
Remark 4.4. The pairing⊓

Z0 (XC )∧Mor(XC , Z0 (AqC )) → Z0 (XC )∧F (X(C), Z0 (S q,q )0 ) −→ Z0 (X(C)+ ∧S d,d )0
agrees with the pairings ∩ under the isomorphism Z0 (X(C)+ ∧ S d,d)0 ∼= Z0 (XC ×C
AdC ).
The following is merely a rephrasing of [dSLF04, Proposition 1.14].
Proposition 4.5. [dSLF04, Proposition 1.14] Let W be a smooth compact Gmanifold of dimension d and let [W ] ∈ Hd,d (W ; Z) denote its fundamental class.
The map [W ] ⊓ − defined above agrees with cap-product (in Bredon cohomology andhomology) with [W ][W ] ∩ − = [W ] ⊓ − : H k,l (W ; Z) → Hd−k,d−l (W ; Z).

30

JEREMIAH HELLER AND MIRCEA VOINEAGU

We finish our discussion of pairings by showing that the pairings ∩ and ⊓ arecompatible under the natural transformations Φ and Ψ.
Proposition 4.6. Let X be a smooth projective real variety. The diagram∩

π0,0 Zd (XC ) ⊗ πk,l Mor(XC , Z0 (AdC ))/ πk,l Zd (XC ×C Ad )
C

Ψ
 Ψ⊗Φ
⊓ /
d,d
d,d
d,d
π0,0 Ω Z0 (X(C)) ⊗ πk,l Map(X(C), Z0 (S )0 )πk,l Ω Z0 (X(C)+ ∧ S d,d )0
commutes.
Proof. Consider the following diagram
PdC /Pd−1∧ Zd (XC ) ∧ Mor(XC , Z0 (AdC ))C1∧ ∩/ Pd /Pd−1 ∧ Zd (XC ×C Ad )
C

C

C

u∧1

u∧1


Z0 (AdC ) ∧ Zd (XC ) ∧ Mor(XC , Z0 (AdC ))

1∧ ∩



/ Z0 (Ad ) ∧ Zd (XC ×C Ad )
C

C

µ∧1


Zd (AdC ×C XC ) ∧ Mor(XC , Z0 (AdC ))

µ

1∧π ∗


Zd (AdC ×C XC ) ∧ Mor(AdC ×C XC , Z0 (AdC ))O∩/ Zd (Ad ×C XC ×C Ad )CCO

π ∗ ∧π ∗Z0 (XC ) ∧ Mor(XC , Z0 (AdC ))π∗∩/ Z0 (XC ×C Ad )
C

∼
=

1∧Ψ


Z0 (X(C)) ∧ Map(X(C), Z0 (S d,d )0 )⊓



/ Z0 (X(C)+ ∧ S d,d)0 .

Here u is the map induced by sending x ∈ PdC to x − ∞ ∈ Zp (PdC )0 and µ is inducedby (x, V ) 7→ {x} × V . The composition of the left vertical maps induces (by taking
an adjoint) the map Ψ⊗Φ and the adjoint of the right vertical compositions induces
the map Ψ = sd . Thus to establish the proposition it is enough to establish thecommutativity of this diagram.
The first square of this diagram is evidently commutative. Denote the graph ofa morphism g : WC → C0 (PkC ) by Γ(g) ∈ Zd (WC ×C PkC ). Let a ∈ PdC and f : XC →C0 (PdC ). The equality of graphs {a} × Γ(f ) = Γ(f π|{a}×X ) ∈ Zd (AdC ×C XC ×C AdC )establishes the commutativity of the second square. Similarly for the third square.
The fourth square is commutative by remark 4.4.
The first step towards proving Theorem 4.1 will be to reduce to the case q = d.
For this we need to introduce the equivariant analogue of the s-map in morphic

EQUIVARIANT INVARIANTS OF REAL VARIETIES

31

cohomology. Consider the commutative diagram
Mor(XC , C0 (PqC ))+

Σ
\

q−1 +
Mor(XC , C0 (PC
))

q+1 +
/ Mor(XC , C1 (PC ))Mor(XC , C1 (PqC ))+
D

D



Σ
\

Zd (XC ×C Aq )

/ Zd+1 (XC ×C Aq+1 )C

where Σ\ denotes the algebraic suspension map. By [HV09, Corollary 4.20] the vertical maps are equivariant homotopy equivalences, by the real analogue of Lawson’s
suspension theorem proved by Lam [Lam90] (see [LLFM03]) the bottom horizontal map is an equivariant homotopy equivalence. We conclude that the the upperhorizontal arrow is a homotopy equivalence. Let Σ\−1 denote a choice of homotopy
inverse to this map and consider the following composition
Σ
\−1

#

q+1
Mor(XC , Z0 (AqC )) ∧ Z0 (P1C )0 −→ Mor(XC , Z1 (Aq+2
C )) −−→ Mor(XC , Z0 (AC )).

Define the s-map in equivariant morphic cohomologys : Lq HRa,b (X) → Lq+1 HRa,b (X)to be the map induced by this composition together with α ∈ π1,1 P1C where themap α : S 1,1 = P1C → Z0 (P1C )0 is defined by x 7→ {x} − {∞}. Observe that under
the duality morphism, the s-map in equivariant morphic cohomology is compatible
with the s-map in dos Santos’ equivariant Lawson homology in the sense that wehave a commutative diagrams

Lq HRa,b (X)(4.7)/ Lq+1 HRa,b (X)

D

D



s

Ld−q HRd−a,d−b (X)/ Ld−q−q HRd−a,d−b (X).

Proposition 4.8. The transformations Φ are compatible with the s-maps in thesense thats
/ Lq+1 H p,r (X)
Lq H p,r (X)
mm
m
mm
m
m
Φ
mm
vmmm Φ

H p,r (X(C); Z)
commutes.
Proof. The space Zk (AqC ) has the equivariant homotopy type of an equivariant
Eilenberg-MacLane space K(Z, Rq−k,q−k ). By [dS03a] the join pairing#Σ\Z0 (ApC ) ∧ Z0 (AqC ) −→ Z1 (Ap+q+1)←− Z0 (Ap+qC
C )represents the cup product pairing in Bredon cohomology.
Consider the homotopy commutative square
Mor(XC , C0 (PqC ))+∧ Z0 (P1C )0
Mor(XC , C0 (Pq−1
))+
C

Φ

1

Map(X(C), Z0 (A )) ∧ Z0 (P )0
q

#

q+2 +
q+1 +
\−1 Mor(XC , C0 (P))
C/ Mor(XC , C1 (PC )) Σ/q

Mor(XC , C1 (Pq+1
))+
C

#

Mor(XC , C0 (PC ))+

Φ
Φ
\−1
/ Map(X(C), Z0 (Aq+1 ))/ Map(X(C), Z1 (Aq+2 )) Σ

32

JEREMIAH HELLER AND MIRCEA VOINEAGU

Let α : S 1,1 = P1C → Z0 (P1C )0 be the map defined by x 7→ {x} − {∞}. The map
α represents the canonical generator of π1,1 Z0 (P1 )0 ∼= Z. The upper horizontal
composition restricted to α induces the s-map on equivariant morphic cohomology.
On the other hand the bottom horizontal composition restricted to α induces theidentity because it represents the product with α ∈ H 0,0 (pt; Z) = Z and α is the
canonical generator of this group.

The Poincare dualities D and P are both given by capping with a fundamental
class. We verify that the transformation Ψ takes fundamental classes in equivariant
Lawson homology to fundamental classes in Bredon homology. The cycle map
cyc : Zd (X) → Hd,d (X(C); Z) sends an irreducible closed subvariety i : V ⊆ X tocyc(V ) = i∗ [V (C)] ∈ Hd,d (X(C); Z).
Lemma 4.9. Let X be a projective real variety. The composition of s-maps together
with the Dold-Thom isomorphism
Ψ : Lk HRk,k (X) → Hk,k (X(C); Z),
coincides with the cycle-map V 7→ [V (C)].
Proof. The proof is the similar to [FM94, Proposition 6.4]. Let V ⊆ X be a closed
irreducible subvariety of dimension k and i : Ṽ → V ⊆ X be a resolution of
singularities. Then i∗ ηṼ = [VC ⊆ XC ] ∈ Lk HRk,k (X). By Noether normalization
there is a dominant map f : V → PkR of some degree d. Consider the commutative
diagram
Lk HRk,k (Pk ) of∗

Lk HRk,k (Ṽ )

∼
= Ψ

i∗

/ Lk HRk,k (X)Ψ
Hk,k (Pk ; Z) o

f∗



Ψ

Hk,k (Ṽ (C); Z)i∗/ Hk,k (X(C); Z).

The map Ψ : Lk HRk,k (Pk ) → Hk,k (Pk (C); Z) is an isomorphism [dS03a], Ψ(ηPk ) =[Pk (C)], and f∗ (ηV ) = d·ηPk ∈ Lk HRk,k (Pk ). Note that H 0,0 (Ṽ (C); Z) ∼= Z because
H 0,0 (Ṽ (C); Z) ∼= H 0 (Ṽ (C)/G; Z) ∼= Hom(H0 (Ṽ (C)/G; Z), Z) ∼=Z
sing

and so by equivariant Poincare duality we have Hk,k (Ṽ (C); Z) ∼= Z generated by
the fundamental class [Ṽ (C)].
If we see that f∗ : Hk,k (Ṽ (C); Z) → Hk,k (Pk (C); Z) is multiplication by d we aredone. Indeed if f∗ is multiplication by d then it follows that Ψ takes the naturalgenerator ηṼ of Lk Hk,k (Ṽ (C); Z) ∼= Z to the generator [Ṽ (C)] of Hk,k (Ṽ (C); Z) ∼=Z.
The fact that this map is multiplication by d follows from the commutativesquare/ H2k (Ṽ (C); Z)
Hk,k (Ṽ (C); Z)·d

Hk,k (Pk (C); Z)∼=/ H2k (Pk (C); Z),

where the horizontal maps are the forgetful maps and send the equivariant fundamental class to the non-equivariant fundamental class.
EQUIVARIANT INVARIANTS OF REAL VARIETIES33

We also need to know that the homotopy invariance in equivariant Lawson homology and the suspension isomorphism in Bredon homology are suitably related.
First recall that the suspension isomorphism is given by the following explicit mapof spaces.
Lemma 4.10. ([LF97, Remark 2.14]) The suspension isomorphism φ is inducedby the adjoint of the map
µ

u×1

τ

→ Z0 (W+ ∧ S d,d )0 ,S d,d ∧ Z0 (W ) −−−→ Z0 (S d,d )0 ∧ Z0 (W ) −→ Z0 (S d,d ∧ W+ )0 −where the first map is the map u(x) = {x} − {∞} .
Proposition 4.11. Let X be a smooth, projective real variety. Thenπ∗

Zd (XC ×C AdC ) o

Z0 (XC )

lll
lll
l
Ψ
l
ll φ
lv ll

Ωd,d Z0 (X(C)+ ∧ S d,d )0
is homotopy commutative.
Proof. By adjointness, the homotopy commutativity of the triangle follows fromthe homotopy commutativity of the following diagram:
PdC /Pd−1∧ Zd (XC ×C AdC ) o
C

1∧π ∗

u×1



Z0 (AdC ) ∧ Zd (XC ×C AdC ) o

u×1
1∧π

µ



PdC /Pd−1∧ Z0 (XC )C∗


Z0 (AdC ) ∧ Z0 (XC )µ
Zd (AdC ×C XC × AdC ) o
Z0 (AdC ×C XC )
O
iii
iiii
i
i
π2∗
i
ii τ ∗tiiii
π1∗

Z0 (XC ×C AdC )Here π1 : AdC ×C XC ×C AdC → AdC ×C XC and π2 : AdC ×C XC ×C AdC → XC ×C AdC
are the projections, µ is multiplication, and τ is the twist isomorphism. The left
vertical composition induces the s-map and the right vertical composition induces
the suspension isomorphism φ. The first two squares evidently commute. The twomaps τ π1 and π2 : AdC ×C XC ×C AdC → AdC ×C XC are homotopic via
F : AdC ×C XC ×C AdC ×C A1C → AdC ×C XC ,where F (a, x, b, t) = (ta+ (1 − t)b, x), and therefore π2∗ τ ∗ and π1∗ are homotopic. 
We now prove the compatibility(Theorem 4.1). Let X be a smooth, projective real variety of dimension d. The
following square commutes, where the horizontal arrows are the Poincare duality

34

JEREMIAH HELLER AND MIRCEA VOINEAGU

isomorphisms
Lq HRk,l (X)D/ Ld−q HRd−k,d−l (X)ΨΦ
H k,l (X(C), Z)P/ Hd−k,d−l (X(C), Z).

Proof. Proposition together with the commutativity of diagram (4.7) shows that inorder to prove the theorem it suffices to consider the case q = d and to prove thatthe following square commutes
Ld HRd−k,d−l (X) = πk,l Z d (XC )D/ πk,l Z0 (XC ) = L0 HRk,l (X)ΦΨ
H d−k,d−l (X(C), Z)/ Hk,l (X(C), Z).

P

In this case the map Ψ is simply the Dold-Thom isomorphism.
Consider the following diagram
πk,l Mor(XC , Z0 (AdC ))
Φ

ηX ∩−

/ πk,l Zd (XC ×C Ad ) o∼=πk,l Z0 (XC )

C

=

Ψ



[X(C)]⊓−/ πk,l Ωd,d Z0 (X(C)+ ∧ S d,d)0 oπk,l Map(X(C), Z0 (S d,d )0 )φπk,l Z0 (X(C)).

The upper-composition is the Poincare duality isomorphism between equivariant
morphic cohomology and equivariant Lawson homology and by Proposition 4.5 thecomposition along the bottom row is the Poincare duality isomorphism between
Bredon cohomology and homology.
The first square commutes by Proposition 4.6 together with Lemma 4.9 and thesecond square commutes by Proposition 4.11

Appendix A. Γ-spaces
We recall Segal’s notion of Γ-space, introduced by Segal in [Seg74], where heshowed that such objects give rise infinite loop spaces. Let n be the pointed set
n = {0, 1, 2, . . . , n}, pointed by 0. The category Γop is the category whose objects
are n for n ≥ 0 with pointed set maps (Γop is equivalent to the opposite of thecategory Γ considered by Segal). A Γ-object in a category C is a functor F : Γop →C. Generally C will be a pointed category in which case we additionally requirethat F (0) = ∗. When C is the category of spaces or of simplicial sets then we referto a Γ-object F as a Γ-space. A Γ-space is said said to be special if(p1∗ , . . . , pn∗ ) : F (n) → F (1)×n
is a weak equivalence, where pi : n → 1 is the function pi (i) = 1 and pi (j) = 0 if
i 6= j. If F is a presheaf of special Γ-spaces then we will say that F is special.
Given a special Γ-space F one can associate a spectrum, which we denote by SF .
First note that a Γ-space F is in particular a bisimplicial set (or simplicial space)n 7→ F (n). Write BF for the classifying Γ-space defined by
BF (m) = |n 7→ F (m ∧ n)|,

EQUIVARIANT INVARIANTS OF REAL VARIETIES

35

here m ∧ n is identified with mn via (i, j) 7→ n(i − 1) + j. For a Γ-space A there isa natural map S 1 ∧ A(1) → BA(1). The spectrum associated to F is obtained byevaluating iterated classifying spaces of F :SF = (F (1), BF (1), B 2 F (1), . . .).
By [Seg74] the fact that F is special implies that BF is special, that B k F (1) →ΩB k F (1) is a weak equivalence for k ≥ 1, and that F (1) → ΩBF (1) is a homotopy
group completion.
Now suppose that G = Z/2 acts on the Γ-space F and that both m 7→ F (m)and m 7→ F (m)G are special. Say that such a Γ-space is equivariantly special.
Realization of a simplicial space commutes with taking fixed points and therefore
we immediately conclude the following proposition.
Proposition A.1. Let F by an equivariantly special Γ-space and SF the associated Ω-spectrum. Then the naive fixed point spectrum (SF )G is the Ω-spectrum
associated to the fixed point Γ-space n 7→ F (n)G .
Appendix B. Globalization
We recall the Godement resolution which gives an explicit model for computing
the Zariski hypercohomology of X with coefficients in a presheaf of spectra [Tho85].
A particularly useful property of this resolution is that it behaves appropriately withrespect to the Galois action of Gal(C/R).
Suppose that F (−) is a presheaf of fibrant spectra on Sch/R. The Godement
complex F (−) → G• F (−) is defined as follows. LetYGF (U ) =F (Spec OU,u )u∈Unthen define G F = G ◦ · · · ◦ GF to be the (n + 1)-fold composition of G. The inclusions and projections of the various factors gives rise to the cosimplicial spectrum
n 7→ Gn F (U ) and define GF (U ) = Tot G• F (U ).
In general for a presheaf of spectra, F (−), define
GF (U ) = Tot G• QF (U )where Q is a fixed functorial fibrant replacement for spectra. Then QF (−) is apresheaf of fibrant spectra and the homotopy groups of GF (U ) define the Zariski
hypercohomology with coefficients in F ,πk GF (U ) = H−kZar (U ; F ).
Moreover πk GF (U ) = [S k ∧U+ , F ]Zar , where [−, −]Zar denotes maps in the Zariskilocal homotopy category of presheaves of spectra on Sch/R ([Jar87, Mit97]).
Now we consider the case when F ′ is a presheaf of naive Z/2-spectra on Sch/Robtained by F ′ = p∗ p∗ F for some presheaf F on Sch/R, where p : Sch/R → Sch/C
is the`map of sites specified by X 7→ XC ). Observe that if F is additive, i.e
F (U V ) = F (U ) × F (V ), then for a real variety U we haveYY
GF ′ (U ) =F ((Spec OU,u )C ) =F ((Spec OUC ,v ) = p∗ p∗ GF (U ).
u∈U

v∈UC

`
Indeed (Spec OU,u )C = Spec OUC ,π−1 {u} = u′ ∈π−1 {u} Spec OUC ,u′i , where π : UC →i
U is the projection. Take the fibrant replacement functor Q on spectra used above
to be a Z/2-equivariant fibrant replacement functor. Further suppose that canonical

36

JEREMIAH HELLER AND MIRCEA VOINEAGU

map F (U ) → F (UC )Z/2 is a weak equivalence. For a cosimplicial spectrum A• wehave that Tot(A• )n = Tot(A•n ) and from the description of Tot(A•n ) as the simplicial
function complex Tot(A•n ) = Hom(∆, A•n ) on cosimplicial simplicial sets, we seethat Tot(−) commutes with fixed points. Because (GF (UC ))Z/2 = G(F (UC )Z/2 ) =GF (U ) and [Tot G• F (U )]Z/2 = Tot(G• F (U )Z/2 ) we see that
GF (U ) → (GF (UC ))Z/2
is a weak equivalence as well.
The reason for using this specific fibrant replacement functor is the following
proposition which is needed during our discussion of the topological spectral sequence in Section 2.3. We remind the reader that the presheaf K(n) (−) is obtainedby applying the Godement resolution to the presheaf W (n) (−) = Kgeom (−, P∧n )and Kgeom (X, Y ) is defined to be the spectrum associated to a certain special Γspace (Definition 2.3). The presheaf M(n) is obtained by applying the Godement
resolution to the presheaf of spectra associated to the presheaf of simplical abeliangroups zequi (P∧n , 0)(− × ∆• ). See Section 2.1 for more details on these constructions and Section 2.2 for a recollection of the meaning of the notation F (∆•top ×R −)and F (∆•top ×C −) for a presheaf F (−) on Sch/R, used below.
Proposition B.1. Let X be a real variety. Then
K(n) (∆•top ×R X) → K(n) (∆•top ×C X)Z/2
and
M(n) (∆•top ×R X) → M(n) (∆•top ×C X)Z/2
are weak equivalences.
Proof. The spectrum Kgeom (X, Y ) is the spectrum associated to the special Γ(m)space m 7→ HomR (X, GY ). By Proposition A.1 we see that Kgeom (XC , Y )G =(m)(m)
Kgeom (X, Y ) because HomR (X, GY ) = HomR (XC , GY )G . In Section 2.1 thespectrum W (n) (X) is defined by a homotopy colimit of a certain diagram of spectra, W (n) (X) = hocolimI Kgeom (X, P∧nI ). Taking fixed points commutes withhomotopy colimits of naive G-spectra (see [DI04, Remark 5.6] for an argument
in terms of G-spaces, note that the same argument applies to G-simplicial sets
and homotopy colimits of spectra are computed degreewise) and so we have that
W (n) (X) ≃ W (n) (XC )G . Similarly we see that M(zequi (P∧n , 0)(X ×R ∆•R )) ≃M(zequi (P∧n , 0)(XC ×R ∆•R ))G . These are additive presheaves and so from the discussion above on Godement resolutions we conclude that K(n) (−) = GW (n) (−) andM(n) (−) = GM(zequi (P∧n , 0)(− ×R ∆•R )) also satisfy K(n) (X) ≃ K(n) (XC )Z/2 andM(n) (X) ≃ M(n) (XC )Z/2 .
For a presheaf F , the colimit F (∆ntop ×C X) = colimT →U(R) F (UC ×R X) is afiltered colimit and so fixed points commutes with this colimit,
F (∆ntop ×C X)Z/2 = colim F (UC ×R X)Z/2 .
T →U(R)

We conclude that K(n) (∆•top ×R X) ≃ K(n) (∆•top ×C X)Z/2 and M(n) (∆•top ×R X) ≃M(n) (∆•top ×C X)Z/2 .


EQUIVARIANT INVARIANTS OF REAL VARIETIES

37

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