Semi-topological cycle theory I

0.15 MB·English

by  J. H. Teh

Semi-topological cycle theory I
Jyh-Haur Teh
2 Abstract
1
0 We study algebraic varieties parametrizedby topologicalspaces and enlarge the domains of
2 Lawson homology and morphic cohomology to this category. We prove a Lawson suspension
theorem and splitting theorem. A version of Friedlander-Lawson moving is obtained to prove
n
a a duality theorem between Lawson homology and morphic for smooth semi-topological pro-
J jective varieties. K-groups for semi-topological projective varieties and Chern classes are also

3 constructed.

]
G
1 Introduction
A
. Algebraiccyclesarebasicingredientsinstudyinginvariantsofalgebraicvarieties. Thecollection
h
t of all r-dimensional algebraic cycles of a projective variety X forms a topologicalabelian group
a
Z (X). Lawson studied these groups from a homotopic viewpoint ([Law89]) and proved a
m r
suspensiontheoremwhichservesasacornerstoneforLawsonhomologyandmorphiccohomology
[
developed later by Lawson and Friedlander ([Fri91, FL92, FL98, FL97]). A continuous map

2 f : Sn → Zr(X) from the n-sphere to Zr(X) can be viewed as a family of algebraic cycles
v parametrizedbySn. Thisfamilycanalsobeconsideredas“an”algebraiccycleofX×Sn. This
5 motivates us to consider algebraic varieties parametrized by topological spaces and consider
5
algebraic cycles on them.
3
When the base space is an algebraic variety and the parametrization is algebraic, this is
2
. just the relative theory of algebraic varieties. The main point of our studying is that the base
1
space is a very general topological space, the ring of continuous complex-valued functions of
0
0 it is usually not Noetherian. It is well known in the algebraic case when we wish a family of
1 algebraic varieties behaves well we need to require the family to be flat over the base scheme.
: The flatnessofafamily ofvarietiesisequivalenttothe propertythatthe family isthe pullback
v
i of the universal family over a Hilbert scheme by an algebraic morphism to the Hilbert scheme.
X
So to obtain a nice theory, we define our “semi-topological variety” to be a continuous map
r from a topological space S to some Hilbert scheme with some additional technical assumption.
a
We are able to define semi-topological algebraic cycles on semi-topological projective varieties
and extend the definition of Lawson homology and morphic cohomology to them. This paper
is the first part of this theory. A Hodge theory and Riemann-Roch theorem will be given in a
forthcoming paper.
Letus give abrief overviewofthis paper. Insection2, wedefine semi-topologicalprojective
varietiesandalgebraiccycleonthem. Somebasictopologicalpropertiesofsemi-topologicalcycle
groupsarestudied. Insection3,weprovetheLawsonsuspensiontheoremandsplittingtheorem
for semi-topological projective varieties. In section 4, we give a version of Friedlander-Lawson
moving lemma for semi-topological projective varieties and use it to prove a duality theorem
between the Lawsonhomologyandmorphic cohomologyforsemi-topologicalsmoothprojective
varieties. Insection5,wecompute theLawsonhomologygroupofdivisorsinasemi-topological
smooth projective variety. In section 6, we construct K-groups and Chern classes.
1
Acknowledgements TheauthorthanksTaiwanNationalCenterforTheoreticalSciences(Hsinchu)
for proving a nice working environment, and thanks Yu-Wen Kao for her support.
2 Semi-topological varieties
Let us briefly recallthe constructionof cycle groupsofcomplex projectivevarieties. Fora com-
plex projectivevarietyX, we write C (X)for the collectionofeffective r-cycles ofdegreed on
r,d
X. According to Chow theorem, C (X) is a projective variety. Let C (X) = C (X)
be the Chow monoid and Z (X) r=,d [C (X)]+ the naive group complretion of`Cd≥(X0 ).r,dLet
r r r
K (X) = π(C (X)×C (X)) where π : C (X)×C (X) → Z (X) is the map
r,d `d1+d2≤d r,d1 r,d2 r r r
(a,b)7→a−b. We have a filtration
K (X)⊆K (X)⊆···=Z (X)
r,0 r,1 r
Each K (X) is compact and the topology of Z (X) is the weak topology induced from this
r,d r
filtration. With this topology, Z (X) is a topological abelian group. If Y is also a projective
r
variety, we write Z (Y)(X) for the group of algebraic r-cocycles on X with values in Y, i.e.,
r
c∈Z (Y)(X) if c∈Z (X×Y) where k is the dimension of X, the projection from c to X is
r r+k
surjective and fibres of c over X are r-cycles in Y.
Throughout this paper, S is a compactly generated topological space with based point s .
0
We write Pn for Pn×S.
S
Definition 2.1. A semi-topological projective variety over S is a continuous map X : S →
Hil (Pn) such that X is a normal projective variety for all s∈S where Hil (Pn) is the Hilbert
P s p
scheme of Pn associated to a Hilbert polynomial p. We write X ⊂ Pn in this case. We define
S
the dimension of X to be the dimension of X , and |X | for the algebraic variety corresponding
s0 s0
to X .
s0
Definition 2.2. Suppose X⊂Pn,Y⊂Pm are semi-topological projective varieties over S. Let
S S
Z (Y)(X):={α∈Map((S,s ),(Z (Pn×Pm),0))|α(s)∈Z (Y )(X )}
r 0 k r s s
where k =r+dimX. Let Z (Pn×Pm) be the image of C (Pn×Pm)×C (Pn×
Pm)inZ (Pn×Pm). Thetokp,≤oleogyofZ (Pn×Pm)isthewe`ake1t+oep2o≤loegyki,en1ducedfromthekfi,elt2ration
k k
Z (Pn×Pm)⊆Z (Pn×Pm)⊆···=Z (Pn×Pm)
k,≤0 k,≤1 k
Let Z (Y)(X):=Map((S,s ),(Z (Pn×Pm),0))∩Z (Y)(X). We give Z (Y)(X) the weak
r,≤e 0 k,≤e r r
topological induced from the filtration
Z (Y)(X)⊆Z (Y)(X)⊆···=Z (Y)(X)
r,≤0 r,≤1 r
where each Z (Y)(X) is endowed with the compact-open topology. Then Z (Y)(X) is a topo-
r,≤e r
logical abelian group.
Definition 2.3. If Y,Y′ are two semi-topological projective varieties, we say that Y′ is a sub-
variety of Y, denoted as Y′ ⊆Y if |Y′|⊆|Y | for all s∈S. A semi-topological Zariski open set
s s
if a set of the form |Y−Y′| where Y′ ⊆Y.
Definition 2.4. Suppose X⊂Pn, Y⊂Pm. Suppose for each s∈S, f :|X |→|Y | is a given
S S s s s
morphism of projective varieties. The assignment s 7→f f is said to be a morphism between X
s
and Y, denoted f : X →Y if s 7→grf ∈C (Pn×Pn) is continuous where grf is the graph of
s r s
f .
s
2
Definition 2.5. If f : X → Y is a morphism of semi-topological projective varieties, define
f :Z (X)(W)→Z (Y)(W) by
∗ r r
(f α)(s):=q ((|W |×grf )•p∗(α(s)))
∗ s∗ s s s
where p:W×X×Y→W×X,q :W×X×Y→W×Y are projections.
Proposition 2.6. f is continuous.
∗
Proof. Wenotethatitiswellknownthattheintersectionproductoncyclesintersectingproperly
is continuous [Ful98]. Suppose β : T → Z (X)(W) is continuous where T is some topological
r
space. Then (f (β(t)))(s) = q ((|W |×grf )•p∗(β(t)(s))) which is continuous in t. Thus f
∗ s∗ s s s ∗
is continuous.
Proposition 2.7. If Y′ is a subvariety of Y, then Z (Y′)(X) is closed in Z (Y)(X).
r r
Proof. Forα∈Z (Y)(X)−Z (Y′)(X),thereiss ∈S suchthatα(s )∈Z (Y′ )(X )−
Z (Y′ )(X ).r,≤Sience Z (Yr,′≤e)(X ) is closed in1Z (Y )(X ), 1see [Ter,h≤1e0, Ps1rops21.9],
thre,≤reeissV1 opse1n in Z (Prn,≤×ePms1) sucsh1 that α(s )∈V ∩r,≤Ze s(1Y )s(1X )⊂Z (Y )(X )−
k,≤e 1 r,≤e s1 s1 r,≤e s1 s1
Z (Y′ )(X ). Let
r,≤e s1 s1
W ={β ∈Map((S,s ),(Z (Pn×Pm),0))|β(s )∈V}
0 k,≤e 1
then W is openinMap((S,s ),(Z (Pn×Pm),0)) andα∈W∩Z (Y)(X)⊆Z (Y)(X)−
0 k,≤ r,≤e r,≤e
Z (Y′)(X). Therefore Z (Y)(X) − Z (Y′)(X) is open and Z (Y′)(X) is closed in
r,≤e r,≤e r,≤e r,≤e
Z (Y)(X). We then have Z (Y′)(X) is closed in Z (Y)(X).
r,≤e r r
Recall that there is a functor k (see [Ste67]) constructed by Steenrod from the category of
topological spaces to the category of compactly generated spaces which acts like a retraction.
Furthermore,for any topologicalspace X, X andk(X)have the same homologyandhomotopy
groups. Recall that by the construction in [Teh08], if H is a normal closed subgroup of G and
both are compactly generated, then the short exact sequence 0→H →G→G/H →0 gives a
fibration
//
B B
H G
(cid:15)(cid:15)
B
G/H
whereBGistheclassifyingspaceofG. Thuswehavealongexactsequenceofhomotopygroups
···→π (H)→π (G)→π (G/H)→π (H)→···
n n n n−1
CombinewiththeSteenrodfunctork,oncewehavesomecomplicatedtopologicalabeliangroups
that form the short exact sequence stated above, we get a long exact sequence of homotopy
groups. The following is an application of this result.
Definition 2.8. Let
Z (Y)(X)
Z (Y;Y′)(X):= r
r Z (Y′)(X)
r
and
Zt(X):=Z (Pt;Pt−1)(X)
r S S
where Y′ is a semi-topological subvariety of Y.
Corollary 2.9. We have a long exact sequence of homotopy groups
···→π Z (Y′)(X)→π Z (Y)(X)→π Z (Y;Y′)(X)→π Z (Y′)(X)→···
n r n r n r n−1 r
3
Definition 2.10. Let pt : S → Hil (Pn) be a constant map whose image is a point in Pn.
p
Then Z (Y)(pt) is isomorphic to Z (Y)(pt′) for any two such maps pt,pt′. We write Z (Y):=
r r r
Z (Y)(pt) without referring to which point we take. The map pt is called a point map.
r
Definition 2.11. Define
H (X):=π Z (X)
S,n n 0
and
Hn(X):=π Zm(X)
S 2m−n
where m is the dimension of X.
Example 2.12. When S = S0 the 0-dimensional sphere, X = X × S0 for some projective
variety X,
H (X)=π Z (X)∼=H (X)
S,n n 0 n
by the Dold-Thom theorem. If X is smooth, then
FL PD
Hn(X)=π Zm(X) ∼= π Z (X)∼=H (X) ∼= Hn(X)
S 2m−n 2m−n 0 2m−n
where FL is the Friedlander-Lawson duality isomorphism and PD is the Poincare duality iso-
morphism.
3 Suspension theorem and splitting theorem
Let us recall that if X ⊆ Pn and x ∈ Pn+1\Pn, the suspension Σ/X of X with respect to x
∞ ∞
is the join of X and x .
∞
Definition3.1. SupposethatY⊆Pm, ptapointmapwithimageinPm+1\Pm. Thesuspension
S
of Y with respect to pt is the semi-topological subvariety
(Σ/ Y)(s):=Σ/ Y
pt pt(s) s
SowehaveΣ/ Y⊆Pm+1. ThesuspensioninducesamapZ (Pn×Pm)−Σ/→pt Z (Pn×Pm+1)
pt S k k+1
by suspending Pm. Hence for each α ∈ Z (Y)(X), Σ/ induces a semi-topological cycle in
r pt
Z (Σ/Y)(X) in the following way
r+1
(Σ/ α)(s):=Σ/ α(s)
pt pt
Theorem 3.2. Let pt = [0 : ··· : 0 : 1] ∈ Pm+1,Y ⊆ Pm,X ⊆ Pn. Then Σ/ : Z (Y)(X) →
S S pt∗ r
Z (Σ/ Y)(X) is a weak homotopy equivalent.
r+1 pt
Proof. We write Σ/ for Σ/ . Let
pt
T (Σ/Y)(X):={α∈Z (Σ/Y)(X)|α(s) meets X ×Y properly in X ×Σ/Y , for all s∈S}
r+1 r+1 s s s s
Let Λ⊆Pm+1×P1×Pm+1 be the closedsubvariety constructed by Friedlander in [Fri91, Prop
3.2]whichisageometricdescriptionofLawson’sholomorphictaffy. LetΛ :=Λ•(Pm+1×{t}×
t
Pm+1),t∈C. Then for α∈T (Σ/Y)(X),
r+1
Φ (α):=q (p∗α•(Pn×Λ ))∈T (Σ/Y)(X)
t t∗ t t r+1
wherep :Pn×Pm+1×{t}×Pm+1 →Pn×Pm+1 andq :Pn×Pm+1×{t}×Pm+1 →Pn×Pm+1
t t
are the projections to the (1,2)-component and the (1,4)-component respectively. If t = 0,
Φ (α)∈Σ/Z (Y)(X). Not difficult to see that Φ is a strong deformation retract of T (Σ/Y)(X)
0 r r+1
to Σ/Z (Y)(X).
r
4
Let x = [0 : ··· : 0 : 1] ∈ Pm+2,x = [0 : ··· : 0 : 1 : 1] ∈ Pm+2. Recall that by [FL98,
1 2
Prop 2.3], for any d > 0, there is e(d) > 0 such that for any e > e(d), there is a line L in
e
C (Pm+2) containing ePm+1 such that we have a map
m+1,e
Ψ :Z (Pm+1)×L →Z (Pm+1)
e r+1,≤d e r+1,≤de
defined by
Ψ (Z,D):=p ((x #Z)•D)
e 2∗ 1
where p : Pm+2 −{x } → Pm+1 is the projection with center {x }. Furthermore, for D ∈
2 2 2
L −{ePm},Ψ (Z,D) ∈ T (Pm+1) and Ψ (Z,ePm) = eZ. When we restrict cycles having
e e r+1,de e
support in Σ/Y⊆Pm+1 by checking the definition of Ψ , we get a map
e
Ψ :Z (Σ/Y)×L →Z (Σ/Y)
e r+1,≤d e r+1,≤de
with the corresponding properties. For α∈Z (Σ/Y)(X), define
r+1,≤d
Ψ (Z,D)(s):=p ((x #α(s))•(D×|X |))
e 2∗ 1 s
This map is continuous in s and have the homotopy property as before. Note that if f : C →
Z (Σ/Y)(X) is a map from a compact topological space C, the image Imf is compact and
r+1
Imf ⊆Z (Pn×Pm),by[Teh10,Lemma2.8],Imf ⊆Z (Pn×Pm)forsomed>0. Therefore
k k,≤d
Imf ⊆Z (Σ/Y)(X).
r+1,≤d
We show that the map i : T (Σ/Y)(X) → Z (Σ/Y)(X) induced from the inclusion is a
∗ r+1 r+1
weakhomotopyequivalence. Let[f]∈π (Z (Σ/Y)(X))beabasedpointpreservingcontinuous
n r+1
map. Since Imf is compact, Imf ⊆ Z (Σ/Y)(X) for some d. Then by the result above,
r+1,≤d
there is a map Ψ :Z (Σ/Y)(X)×L →Z (Σ/Y)(X) such that Ψ (f(s),ePm)=f(s),
e r+1,≤d e r+1,≤de e
Ψ (f(s),D) ∈ T (Σ/Y)(X) for D ∈ L −{ePm}. Hence i [Ψ (f,D)] = [f] which implies that
e r+1 e ∗ e
i is surjective. For injectivity, if [g] ∈ π T (Σ/Y)(X) is mapped to 0 by i in Z (Σ/Y)(X),
∗ n r+1 ∗ r+1
then i g can be extended to a map g : Dn+1 → Z (Σ/Y)(X) where Dn+1 is the unit closed
∗ r+1
ball. Again, by choose some Ψ , we can show that g is homotopy to some Ψ (g,D) : Dn+1 →
e e e
T (Σ/Y)(X). Thus [g]=0. Combining with previous result, the proof is complete.
r+1 e e
Theorem 3.3. (The splitting theorem) If X is a semi-topological projective variety, then there
is
ξ :Z (Pt)(X)→Zt(X)×Zt−1(X)×···×Z0(X)
t 0
which is a weak homotopy equivalence.
Proof. Recall that there is an isomorphism Pn ∼= C (P1) for any positive integer n. The
0,n
projection map Pt ∼=C0,t(P1)→C0,(t)(Pk) defined by
k
x +···+x 7→ x
1 t X I
I⊂{1,...,t},|I|=k
wherex =x +···+x forI ={i ,...,i }inducesamapξ :Z (Pt)(X)→Z (Pk)(X)→Zk(X)
I i1 it 1 t t 0 S 0 S
for 0≤k≤t. We have a commutative diagram
Z (Pt−1)(X) ξt−1//Zt−1(X)×···×Z0(X)
0 S
(cid:15)(cid:15) (cid:15)(cid:15)
Z (Pt)(X) ξt //Zt(X)×Zt−1(X)×···×Z0(X)
0 S
q p
(cid:15)(cid:15) (cid:15)(cid:15)
Zt(X) = //Zt(X)
5
where q is the quotient map, p is the projection map. From the homotopy sequence associated
to the vertical columns, we get the result by induction on t.
4 Moving lemma
Let H = Hil (Pn) be the Hilbert scheme of Pn associated to the Hilbert polynomial p, and
p
π
H −→H be the universalfamily over H. Suppose each algebraic variety parametrized by H is
oefdimensionm. LetUHe(d)⊂P(Γ(OPnH(d)m+1))betheZariskiopensetofthoseF =(f0,...,fm)
such that
L :={(t,h)∈Pn|F (t)=0}
F H h
missesH whereF =(f ,...,f )isobtainedfrompullingbackF bythe inclusionz 7→(z,h)
h 0,h m,h
from Pneto Pn. Then F induces a finite morphism p :H →Pm by p (x):=p (x).
H F H F Fπ(x)
For Y ∈C (H)(H),Z ∈C (H)(H), let e
r,≤e ℓ,≤e
e e
Y ⋆ Z :={(y,z)∈Y × Z|y 6=z,p (y)6=p (z)}
F H F F
where Y × Z is the fibre product of Y and Z over H.
H
Follow similar approach as in [FL98, Prop 1.3], we get the following result
Proposition 4.1. Suppose that r+ℓ ≥ m, e ∈ N. There is a Zariski closed subset B(d) ⊂
e
UHe(d)withlimd→∞FcodimB(d)e =∞whereFcodimB(d)e =min{codimB(d)e,h}andB(d)e,h :=
h
{F |F ∈B(d) } such that for any Y ∈C (H)(H),Z ∈C (H)(H), |Y |⋆ |Z | has pure
h e,h r,≤e ℓ,≤e h Fh h
dimension r+ℓ−m whenever F ∈UHe(d)−B(ed)e. e
Now let X:S →H be a semi-topological projective variety. Then the pullback X∗Pn =Pn
H S
and for F ∈ P(Γ(OPn(d))m+1), X∗F(x,s) := F(x,X(s)),X∗F ∈ P(Γ(OPn(d)m+1)). Define
H S
pX∗F :X∗(H)→PmH by
e pX∗F(x,s):=pFX(s)(x)
where x∈H,s∈S. For α∈Z (X),β ∈Z (X),
r,≤e ℓ,≤e
e
α⋆X∗F β :={(a,b)∈|α|×S |β||a6=b,pX∗F(a,s)6=pX∗F(b,s)}
LetUHe(d)⊂P(Γ(OPnS(d)m+1))bethesemi-topologicalZariskiopensetofthoseF =(f0,...,fm)
such that
L :={(t,s)∈Pn|F (t)=0}
F S s
misses X∗H. Then by taking B(d) = X∗(B(d) ), from the above result, we also have enough
e e
good projeections for semi-topoleogicalprojective varieties when the degree is large enough.
Corollary 4.2. Let X ⊆ Pn be a semi-topological projective variety of dimension m. Suppose
S
that r+ℓ ≥ m, e ∈ N. There is a semi-topological Zariski closed subset B(d)e ⊆ UX(d) with
limd→∞FcodimB(d)e = ∞ such that for α ∈ Zr,≤e(X),β ∈ Zℓ,≤e(X), |αes|⋆Fs |βs| has pure
dimension r+ℓ−m where F ∈UX(d)−B(d)e.
e
Once we know how to find good projections for semi-topological projective varieties, follow
argument of Friedlander and Lawson in [FL98], we get a moving lemma for semi-topological
projective varieties.
Theorem 4.3. Let X ⊆ Pn be a semi-topological projective variety of dimension m. Let r,ℓ,e
S
be nonnegative integers with r +ℓ ≥ m. Then there exist an open set O of {0} in C and a
continuous map
Ψ:C (X)×O →C (X)2
ℓ ℓ
e
6
such that π◦Ψ induces by linearity a continuous map
e
Ψ:Z (X)×O →Z (X)
ℓ s
satisfying the following properties. Let ψp =Ψ|Zs(X)×{p} for p∈O.
1. ψ =Id.
0
2. For any p∈O, ψ is a continuous group homomorphism.
p
3. For any α ∈ Z (X), β ∈ Z (X), and any p 6= 0 in O, each component of excess
ℓ,≤e r,≤e
dimension of the intersection |α(s)|∩|ψ (Z)| is contained in the singular locus of |X |, for
p s
s∈S.
Let (X,Y) be a pair of semi-topological projective varieties in Pn where Y ⊆ X. We say
S
that a map f : (X,Y) → (X′,Y′) between two pairs of semi-topological varieties is a relative
isomorphism if f :X→X′ is a semi-topological morphism such that f :X−Y→X′−Y′ is an
isomorphism of semi-topological quasi-projective varieties. The following example is the most
important case to us.
Example 4.4. Define φ:(X× Pt,X× Pt−1)→(Σ/tX,Pt−1) by
S S S S S
φ(([x :···:x ],s),([a :···:a ],s)):=([a x :···:a x :a :···:a ],s)
0 n 0 t 0 0 0 n 1 t
where we identify the Pt−1 of the second pair to the hyperplane at infinity of Σ/tX. Then not
S
difficult to see that φ is a relative isomorphism.
The followinglemma is a specialcasein orderto define the cycle groupsforquasi-projective
variety (see [Li92]) but it is enough to prove the duality theorem.
Lemma 4.5. Suppose that f : (X,Y) → (X′,Y′) is a relative isomorphism where dimY′ < r,
then Zr(X) is weak homotopic equivalent to Zr(X′) and Zr(X′) is isomorphic to Z (X′).
Zr(Y) Zr(Y′) Zr(Y′) r
Proof. The morphism f : X → X′ induces group homomorphisms f : Z (X) → Z (X′) and
∗ r r
f : Z (Y) → Z (Y′). Since f restricts to X − Y is injective, this gives the injectivity of
∗ r r
f : Zr(X) → Zr(X′). But since r > dimY′, Z (Y′) = {0}. Hence f is surjective and Zr(X′) =
∗ Zr(Y) Zr(Y′) r ∗ Zr(Y′)
Z (X′).
r
Byusingthemovinglemma,wegotthefollowingdualitytheoremwhichisprovedbysimilar
arguments in [FL97].
Theorem 4.6. (Duality theorem) Suppose that X ⊆ Pn,Y ⊆ Pk where dimX = m, X,Y are
S S
smooth. Then there is a weak homotopic equivalence: i :Z (Y)(X)∼=Z (X×Y) where i is
∗ k m+k
the inclusion.
Corollary 4.7. Suppose that X ia a nonsingular semi-topological projective variety with dimen-
sion m. If 0≤t≤m, then Zt(X) is weak homotopic equivalent to Z (X).
m−t
Proof. Zt(X)= Z0(Pt)(X) ∼= Zm(X×PtS) ∼= Zm(Σ/tX) ∼=Z (Σ/tX)∼=Z (X)
Z0(Pt−1)(X) Zm(X×PSt−1) Zm(X×Pt−1) m m−t
5 Semi-topological divisors
Suppose that the dimension of X is greater than 0. Let
K[X]=C(S)[Z ,....,z ]/I(X)= K (X)
0 n M d
d≥0
whereC(S)istheringofcomplex-valuedcontinuousfunctionsonS,andK (X)isthecollection
d
of homogeneous polynomials of degree d in K[X].
7
Proposition 5.1. If f ∈C(S)[z ,...,z ] is homogenous of degree d which is not a zero polyno-
0 n
mial for any s∈S, then f defines an effective semi-topological divisor (f)∈C (Pn) by
n−1 S
(f)(s):=(f )
s
for s∈S where (f ) is the divisor on Pn defined by f .
s s
Proof. Since f is not a zero polynomial for any s ∈ S, (f ) is an effective divisor for any
s s
s ∈ S. From the definition of Chow form, we see that the coefficients of the Chow form
F of (f ) are continuous functions of the coefficients of f. This implies that the assignment
(fs) s
(f):S →C (Pn) is continuous.
n−1,d
Definition 5.2. Let C(S)[z0,...,zn]d,X be the collection of all f ∈ C(S)[z0,...,zn] of degree
d such that (f ) meets X properly in Pn for all s ∈ S. For f +I(X) ∈ K (X) where f ∈
s s d
C(S)[z0,...,zn]d,X, let
(f +I(X))(s):=(f )•X
s s
for s∈S Then (f +I(X)) is a semi-topological divisor on X. Let
Wd(X):={(f +I(X))|f ∈C(S)[z0,...,zn]d,X}
Let W(X)= W (X) and
`d≥0 d
Z (X)lin ={α−β| where α,β ∈W(X),α(s )=β(s )}
m−1 0 0
We say that a semi-topological divisor D ∈Z (X) is semi-topologically linearly equivalent to
m−1
zero if D ∈Z (X)lin.
m−1
Proposition 5.3. Let Td(X)={(f)|f ∈C(S)[z0,...,zn]d,X}, T(X)=`d≥0Td(X) and
T(X):={(f)−(g)|(f)(s )=(g)(s ),(f),(g)∈T(X)}⊆Z (X)
0 0 m−1
e
Then
1. T(X) is isomorphic as a topological group to Z (X)lin.
m−1
2. Te(X) is weak homotopy equivalent to Z (Pn) where X⊆Pn.
n−1 S S
e
Proof. The isomorphismbetweenT(X)andZ (X)lin isgivenbythenaturalmap(f)−(g)7→
m−1
(f+I(X))−(g+I(X)). By movineg lemma, for any e>0, there is an integer e(d) such that for
any k >e(d) there is a continuous function Θ :Z (Pn)×ℓ0 →Z (Pn) such that
k n−1,≤e n−1,ke
1. Θ (c,0)=kc,
k
2. Θ (c,t) meets X properly for t∈ℓ0\{0}.
k s
Then follow exactly the argument in proving the suspension theorem, we show that the
inclusion i :T(X)→Z (Pn) is a weak homotopy equivalence.
∗ n−1 S
e
Proposition 5.4. Suppose that X ⊆ Pn is a semi-topological variety of dimension m. Then
S
Z (X)lin is weak homotopy equivalent to Map((S,s ,(Z (P1),0)). In particular,
m−1 ) 0
H2(S), if ℓ=0

H1(S), if ℓ=1
πℓZm−1(X)lin = H0(S), if ℓ=2
0, otherwise

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Proof. We have a homeomorphism between C(S)[z0,...,zn]d and Map(S,C(n+dd)) by
f 7→ coefficients of f
This homeomorphism reduces to a homeomorphism
Cn−1,d(Pn)∼=Map(S,P(n+dd)−1)∼=Map(S,C0,(n+d)−1(P1))
d
ThusZ (Pn)∼=Map((S,s ),(Z (P1),0)). Fromtheresultabove,wehaveweakhomotopic
n−1 S 0 0
equivalent Z (X)lin ∼=T(X)∼=Z (Pn)∼=map((S,s ),(Z (P1),0).
m−1 n−1 S 0 0
e
6 Chern classes
Definition 6.1. Suppose that the dimension of a semi-topological variety X is k. Let
Cs,1(Pn)(X):={α∈Map(S,C (Pm×Pn))|α(s)∈Cs,1(Pn)(|X |)}
S k+n−s s
where Cs,1(Pn)(|X |):=C (|X |×Pn).
s n−s,1 s
By suspension, we have a sequence
···→Cs,1(Pn)(X)→Cs,1(Pn+1)(X)→Cs,1(Pn+2)(X)→···
Let
Cs,1(P∞)(X):= lim Cs,1(Pn)(X)
n→∞
and let
Vects(X):=[Cs,1(P∞)(X)]+
be the group completion. Note that we do not fix a based point for Vects(X). Let
Vects(X):={f −g ∈Vects(X)|f =g }
s0 s0
g
and let
Vects(X) :={f −g ∈Vects(X)|f,g ∈Cs,1(Pn)(X)}
n
g e
then we have the following sequences and maps
Vects(X) //Z (Pn)(X)
n n−s S
g
Σ/ Σ/
(cid:15)(cid:15) (cid:15)(cid:15)
Vects(X) //Z (Pn+1)(X)
n+1 n−s S
g
Σ/ (cid:15)(cid:15) (cid:15)(cid:15)Σ/
. .
. .
. .
and we get a map
Vects(X)→ lim Z (Pn)(X)
n−s S
n→∞
g
By taking π on both sides, we get a homomorphism
0
s
π Vects(X)−c→π ( lim Z (Pn)(X))∼=π Z (Pn)(X)∼= LiH2i(X)
0 0 n→∞ n−s S 0 o S M
g i=0
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Definition 6.2. For [α] ∈π Vects(X), c([α]) ∈ s LiH2i(X) is called the total Chern class
0 Li=0
of [α]. g
The inclusions
Cs,1(Pn)(X)֒→Cs+1,1(Pn+1)(X)֒→Cs+2,1(Pn+2)(X)֒→···
S S S
induce inclusions on
Cs,1(P∞)(X)֒→Cs+1,1(P∞)(X)֒→Cs+2,1(P∞)(X)֒→···
S S S
which induce again maps on
Vects(X)→Vects+1(X)→Vects+2(X)→···
g g g
Let
Vect(X):= lim Vects(X)
s→∞
g g
Definition 6.3. Suppose that X is a semi-topological projective variety. Let
K (X):=π Vect(X)
n n
g
This is called the n-th K-group of X.
This construction of Chern classes is a preparation for a proof of Grothendieck-Riemann-
Roch for semi-topologicalprojective varieties.
Example 6.4. When S =S0, X=X×S0 where X is a smooth projective variety. Then
K (X)=Ksemi(X)
n n
where Kn(X) is the semi-topological K-group of X constructed by Friedlander and Walker
[FW02, FW03].
References
[Fri91] E. Friedlander, Algebraic cycles, Chow varieties, and Lawson homology, Compositio
Math. 77 (1991), 55-93.
[FG93] E. Friedlander and O. Gabber, Cycles spaces and intersection theory, in Topological
Methods in Modern Mathematics (1993), 325-370.
[FL92] E. Friedlander and H.B. Lawson, A theory of algebraic cocycles, Annals of Math. 136
(1992), 361-428.
[FL98] E. Friedlander and H.B. Lawson, Moving algebraic cycles of bounded degree, Invent.
Math. 132 (1998), 91-119.
[FL97] E.FriedlanderandH.B.Lawson,Duality relating spaces of algebraic cocycles and cycles,
Topology 36 (1997), 533-565.
[FW02] E. Friedlander and M. Walker, Semi-topological K-theory using function complexes,
Topology 41(2002), 591-644.
[FW03] E. Friedlander and M. Walker, Rational isomorphisms between K-theories and coho-
mology theories, Invent. Math., 154 (2003), 1-61.
[Ful98] W. Fulton, Intersection Theory, Springer, 2nd, 1998.
[Law89] H.B.Lawson,Algebraic cycles and homotopy theory, AnnalsofMath.129(1989),253-
291.
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