HOLOMORPHIC K - THEORY, ALGEBRAIC CO-CYCLES, AND LOOP
GROUPS
RALPH L. COHEN AND PAULO LIMA-FILHO
Abstract.In this paper we study the "holomorphic K -theory" of a project*
*ive variety.
This K - theory is defined in terms of the homotopy type of spaces of ho*
*lomorphic maps
from the variety to Grassmannians and loop groups. This theory has been *
*introduced in
various places such as [12], [9], and a related theory was considered in*
* [11]. This theory
is built out of studying algebraic bundles over a variety up to "algebra*
*ic equivalence". In
this paper we will give calculations of this theory for "flag like varie*
*ties" which include
projective spaces, Grassmannians, flag manifolds, and more general homog*
*eneous spaces,
and also give a complete calculation for symmetric products of projectiv*
*e spaces. Using
the algebraic geometric definition of the Chern character studied by the*
* authors in
[6], we will show that there is a rational isomorphism of graded rings*
* between holomor-
phic K - theory and the appropriate "morphic cohomology" groups, defined*
* in [7] in terms
of algebraic co-cycles in the variety. In so doing we describe a geometr*
*ic model for ratio-
nal morphic cohomology groups in terms of the homotopy type of the space*
* of algebraic
maps from the variety to the "symmetrized loop group" U(n)=n where the s*
*ymmetric
group n acts on U(n) via conjugation. This is equivalent to studying alg*
*ebraic maps to
the quotient of the infinite Grassmannians BU(k) by a similar symmetric *
*group action.
We then use the Chern character isomorphism to prove a conjecture of Fri*
*edlander and
Walker stating that if one localizes holomorphic K - theory by inverting*
* the Bott class,
then rationally this is isomorphic to topological K - theory. Finally th*
*is will allows us
to produce explicit obstructions to periodicity in holomorphic K - theor*
*y, and show that
these obstructions vanish for generalized flag manifolds.
Introduction
The study of the topology of holomorphic mapping spaces Hol(X; Y ), where X a*
*nd Y
are complex manifolds has been of interest to topologists and geometers for man*
*y years.
In particular when Y is a Grassmannian or a loop group, the space of holomorphi*
*c maps
____________
Date: December 15, 1999.
The first author was partially supported by a grant from the NSF and a visiti*
*ng fellowship from St. Johns
College, Cambridge.
The second author was partially supported by a grant from the NSF.
1
2 R.L. COHEN AND P. LIMA-FILHO
yields parameter spaces for certain moduli spaces of holomorphic bundles (see [*
*21], [1], [4]).
In this paper we study the K -theoretic properties of such holomorphic mapping *
*spaces.
More specifically, let X be any projective variety, let Grn(CM ) denote the G*
*rassmannian
of n - planes in CM (with its usual structure as a smooth projective variety),*
* and let
U(n) denote the loop group of the unitary group U(n), with its structure as an *
*infinite
dimensional smooth algebraic variety ([21]). We let
Hol(X; Grn(CM )) and Hol(X; U(n))
denote the spaces of algebraic maps between these varieties (topologized as sub*
*spaces of the
corresponding spaces of continuous maps, with the compact open topologies). We *
*use this
notation because if X is smooth these spaces of algebraic maps are precisely th*
*e same as
holomorphic maps between the underlying complex manifolds. The holomorphic K -t*
*heory
space Khol(X) is defined to be the Quillen - Segal group completion of the unio*
*n of these
mapping spaces, which we write as
Khol(X) = Hol(X; Z x BU)+ = Hol(X; U)+ :
This group completion process will be described carefully below. The holomorph*
*ic K -
groups will be defined to be the homotopy groups
K-qhol(X) = ssq(Khol(X)):
A variant of this construction was first incidentally introduced in [12], and s*
*ubsequently
developed in [16] where one obtains various connective spectra associated to an*
* algebraic
variety X, using spaces of algebraic cycles. The case of Grassmannians is the o*
*ne treated
here. A theory related to holomorphic K - theory was also studied by Karoubi in*
* [11] and
the construction we use here coincides with the definition of "semi-topological*
* K - theory"
studied by Friedlander and Walker in [9]. Indeed their terminology reflects the*
* fact that for
a smooth projective variety X holomorphic K - theory sits between algebraic K -*
* theory of
the associated scheme and the topological K - theory of its underlying topologi*
*cal space.
More precisely, using Morel and Voevodsky's description algebraic K theory of a*
* smooth
variety X (via their work on A1 - homotopy theory [19]), Friedlander and Walker*
* showed
that there are natural transformations
Kalg(X) --ff-! Khol(X) --fi-!Ktop(X)
so that the map fi : Khol(X) ! Ktop(X) is the map induced by including the holo*
*morphic
mapping space Hol(X; Z x BU) in the topological mapping space Map(X; Z x BU), a*
*nd
where the composition fiOff : Kalg(X) ! Ktop(X) is the usual transformation fro*
*m algebraic
HOLOMORPHIC K -THEORY 3
K -theory to topological K -theory induced by forgetting the algebraic stucture*
* of a vector
bundle.
In this paper we calculate the holomorphic K -theory of a large class of vari*
*eties, including
"flag - like varieties", a class that includes Grassmannians, flag manifolds an*
*d more general
homogeneous spaces. We also give a complete calculation of the holomorphic K -*
*theory
of arbitrary symmetric products of projective spaces. Since the algebraic K -t*
*heory of
such symmetric product spaces is not in general known, these calculations shoul*
*d be of
interest in their own right. We then study the Chern character for holomorphic *
*K - theory,
using the algebraic geometric description of the Chern character constructed by*
* the authors
in [6]. The target of the Chern character transformation is the "morphic cohom*
*ology"
L*H*(X) Q, defined in terms of algebraic co-cycles in X [7]. We then prove the*
* following.
Theorem 1. For any projective variety (or appropriate colimit of project vari*
*eties) X, the
Chern character is a natural transformation
M1
ch : K-qhol(X) Q -! LkH2k-q(X) Q
k=0
which is an isomorphism for every q 0. Furthermore it preserves a natural mult*
*iplicative
structure, so that it is an isomorphism of graded rings.
In the proof of this theorem we develop techniques which will yield the follo*
*wing in-
teresting descriptions of morphic cohomology that don't involve the use of high*
*er Chow
varieties.
Consider the following quotient spaces by appropriate actions of the symmetri*
*c groups:
U= = lim-!nU(n)=n;
BU= = lim-!n;mGrm (Cnm )=m
Y
SP 1(CP1 ) = lim-!n( CP1 )=n
n
Theorem 2. If X is any projective variety, then the Quillen - Segal group com*
*pletion of
the following spaces of algebraic maps
Mor(X; U=)+ Mor(X; BU=)+ ; and Mor(X; SP 1(CP1 ))+
4 R.L. COHEN AND P. LIMA-FILHO
are all rationally homotopy equivalent. Moreover their kth- rational homotopy g*
*roups (which
we call ssk) are isomorphic to the rational morphic cohomology groups
ssk ~=1p=1LpH2p-k(X) Q:
Among other things, the relation between morphic cohomology and the morphism *
*space
into the "symmetrized" loop group allows, using loop group machinery, a geometr*
*ic descrip-
tion of these cohomology groups in terms of a certain moduli space of algebraic*
* bundles
with symmetric group action.
We then use Theorem 1 to prove the following result about "Bott periodic holo*
*morphic K
- theory". K*hol(X) is a module over K*hol(point) in the usual way, and since K*
**hol(point) =
K*top(point), we have a "Bott class" b 2 K-2hol(point). The module structure th*
*en defines a
transformation
b* : K-qhol(X) ! K-q-2hol(X):
If K*hol(X)[1_b] denotes the localization of K*hol(X) obtained by inverting thi*
*s operator, we
will then prove the following rational version of a conjecture of Friedlander a*
*nd Walker [9].
Theorem 3. The map fi : K*hol(X)[1_b] Q ! K*top(X) Q is an isomorphism.
Finally we describe a necessary conditions for the holomorphic K - theory of *
*a smooth
variety to be Bott periodic (i.e Khol(X) ~=Khol(X)[1_b]) in terms of the Hodge *
*filtration of
its cohomology. We will show that generalized flag varieties satisfy this condi*
*tion and their
holomorphic K - theory is Bott periodic. We also give examples of varieties for*
* which these
conditions fail and hence whose holomorphic K - theory is not Bott periodic.
This paper is organized as follows. In section 1 we give the definition of ho*
*lomorphic K
- theory in terms of loop groups and Grassmannians, and prove that the holomorp*
*hic K -
theory space, Khol(X) is an infinite loop space. In section 2 we give a proof o*
*f a result of
Friedlander and Walker that K0hol(X) is the Grothendieck group of the monoid of*
* algebraic
bundles over X modulo a notion of "algebraic equivalence". We prove this theore*
*m here for
the sake of completeness, and also because our proof allows us to compute the h*
*olomorphic
K - theory of flag - like varieties, which we also do in section 2. In section *
*3 we identify
Q
the equivariant homotopy type of the holomorphic K - theory space Khol( n P1),*
* where
the group action is induced by the permutation action of the symmetric group n.*
* This
will allow us to compute the holomorphic K - theory of symmetric products of pr*
*ojective
spaces, Khol(SP m(Pn)). In section 4 we recall the Chern character defined in [*
*6] we prove
that is an isomorphism of rational graded rings(Theorem 1). In section 5 we pro*
*ve Theorem
HOLOMORPHIC K -THEORY 5
2 giving alternative descriptions of morphic cohomology. Finally in section 6 r*
*ational maps
in the holomorphic K - theory spaces Khol(X) are studied, and they are used, to*
*gether with
the Chern character isomorphism, to prove Theorem 3 regarding Bott periodic hol*
*omorphic
K - theory.
The authors would like to thank many of their colleagues for helpful conversa*
*tions re-
garding this work. They include G. Carlsson, D. Dugger, E. Friedlander, M. Karo*
*ubi, B.
Lawson, E. Lupercio, J. Rognes, and G. Segal.
1. The Holomorphic K-theory space
In this section we define the holomorphic K-theory space Khol(X) for a projec*
*tive variety
X and show that it is an infinite loop space.
For the purposes of this paper we let U(n) denote the group of based algebrai*
*c loops in
the unitary group U(n). That is, an element of U(n) is a map fl : S1 ! U(n) suc*
*h that
fl(1) = 1 and fl has finite Fourier series expansion. Namely, fl can be written*
* in the form
k=NX
fl(z) = Akzk
k=-N
for some N, where the Ak's are n x n matrices. It is well known that the inclus*
*ion of the
group of algebraic loops into the space of all smooth (or continuous) loops is *
*a homotopy
equivalence of infinite dimensional complex manifolds [21].
Let X be a projective variety. It was shown by Valli in [23] that the holomor*
*phic mapping
space Hol(X; U(n)) has a C2 - operad structure in the sense of May [17]. Here C*
*2 is the
little 2-dimensional cube operad. This in particular implies that the Quillen -*
* Segal group
completion, which we denote with the superscript + (after Quillen's + - constru*
*ction),
Hol(X; U(n))+ has the structure of a two - fold loop space. (Recall that up to *
*homotopy,
the Quillen - Segal group completion of a topological monoid A is the loop spac*
*e of the
classifying space, BA.) By taking the limit over n, we define the holomorphic K*
*-theory
space to be the group completion of the holomorphic mapping space.
Definition 1.
Khol(X) = Hol(X; U)+ :
If A X is a subvariety, we then define the relative holomorphic K-theory
Khol(X; A)
to be the homotopy fiber of the natural restriction map, Khol(X) ! Khol(A):
6 R.L. COHEN AND P. LIMA-FILHO
Before we go on we point out certain basic properties of Khol(X).
1. By the geometry of loop groups studied in [21] (more specifically the "Grass*
*mannian
model of a loop group") one knows that every element of the algebraic loop grou*
*p U(n)
lies in a finite dimensional Grassmannian. When one takes the limit over n, it *
*was observed
in [4] that one has the holomorphic diffeomorphism Z x BU ~= U, where here BU is
given the complex structure as a limit of Grassmannians, and U denotes the limi*
*t of the
algebraic loop groups U(n). Thus we could have replaced U by Z x BU in the defi*
*nition
of Khol(X). That is, we have an equivalent definition:
Definition 2.
Khol(X) = Hol(X; Z x BU)+ :
This definition has the conceptual advantage that
ss0(Hol(X; BU(n))) = limm!1 ss0(Hol(X; Grn(Cm )))
where Grn(Cm ) is the Grassmannian of dimension n linear subspaces of Cm . Mor*
*eover
this set corresponds to equivalence classes of rank n holomorphic bundles over *
*X that are
embedded (holomorphically) in an m - dimensional trivial bundle.
2. It is necessary to take the group completion in our definition of Khol(X). F*
*or example,
the results of [4] imply that
a1
Hol*(P1; U) ~= BU(k)
k=0
where Hol* denotes basepoint preserving holomorphic maps. Thus this holomorphic*
* map-
ping space is not an infinite loop space without group completing. In fact afte*
*r we group
complete we obtain
Khol(P1; *) ~=Z x BU
and so we have the "periodicity" result
Khol(P1; *) ~=Khol(*):
A more general form of "holomorphic Bott periodicity" is contained in D. Rowlan*
*d's Ph.D
thesis [22] where it is shown that
Khol(X x P1; X) ~=Khol(X)
HOLOMORPHIC K -THEORY 7
for any smooth projective variety X. A more general projective bundle theorem w*
*as proved
in [9].
We now observe that the two fold loop space mentioned above for holomorphic K*
*-theory
can actually be extended to an infinite loop structure.
Proposition 4. The space Khol(X) = Hol(X; Z x BU)+ is an infinite loop space.
Proof.Let L* be the linear isometries operad. That is, Lm is the space of linea*
*r (complex)
isometric embeddings of m C1 into C1 . These spaces are contractible, and the*
* usual
operad action
Lm xm (Grn(C1 ))m - ! Grnm (C1 )
give holomorphic maps for each ff 2 Lm . It is then simple to verify that this *
*endows the
holomorphic mapping space
qnHol(X; Grn(C1 ))
with the structure of a L* - operad space. Since this is an E1 operad in the s*
*ense of May
[17], this implies that the group completion, Khol(X) = (qnHol(X; Grn(C1 ))+ ha*
*s the
*
* __
structure of an infinite loop space. *
* |__|
As is usual, we define the (negative) holomorphic K - groups to be the homoto*
*py groups
of this infinite loop space:
Definition 3. For q 0,
K-qhol(X) = ssq(Khol(X)) = ssq(Hol(X; U)+ ):
Remarks.
a. Notice that as usual, the holomorphic K-theory is a ring. Namely, the spect*
*rum (in
the sense of stable homotopy theory) corresponding to the infinite loop space K*
*hol(X), is
in fact a ring spectrum. The ring structure is induced by tensor product opera*
*tion on
Grassmannians,
Grn(Cm )k ! Grnk(Cmk ):
We leave it to the reader to check the details that this structure does indeed *
*induce a ring
structure on the holomorphic K-theory. Indeed, this parallels the well-known f*
*act that
whenever E is a ring spectrum and X is an arbitrary space, then Map (X; E) has *
*a natural
8 R.L. COHEN AND P. LIMA-FILHO
structure of ring spectrum, where Map (-; -) denotes the space of continuous ma*
*ps, with
the appropriate compact-open, compactly generated topology; cf. [18].
b. A variant of this construction was first incidentally introduced in [12], an*
*d subsequently
developed in [16] where one obtains various connective spectra associated to an*
* algebraic
variety X, using spaces of algebraic cycles. The case of Grassmannians is, up *
*to ss0 con-
siderations, the one treated here. A theory related to holomorphic K - theory *
*was also
studied by Karoubi in [11], and the definition given here coincides with the no*
*tion of "semi
- topological K- theory" introduced and studied by Friedlander and Walker in [9*
*].
2.The holomorphic K - theory of flag varieties and a general description
of K0hol(X).
The main goal of this section is to prove the following theorem which yields *
*an effective
calculation of K0hol(X), when X is a flag variety.
Theorem 5. Let X be a generalized flag variety. That is, X is a homogeneous s*
*pace of the
form X = G=P where G is a complex algebraic group and P < G is a parabolic subg*
*roup.
Then the natural map from holomorphic K - theory to topological K - theory,
fi : K0hol(X) -! K0top(X)
is an isomorphism
The proof of this theorem involves a comparison of holomorphic K - theory wit*
*h alge-
braic K - theory. As a consequence of this comparison we will recover Friedlan*
*der and
Walker's description of K0hol(X) for any smooth projective variety X in terms o*
*f "algebraic
equivalence classes" of algebraic bundles [9]. We begin by defining this notion*
* of algebraic
equivalence.
Definition 4. Let X be a projective variety (not necessarily smooth), and E0 ! *
*X and
E1 ! X algebraic bundles. We say that E0 and E1 are algebraically equivalent i*
*f there
exists a constructible, connected algebraic curve T and an algebraic bundle E o*
*ver X x T ,
so that the restrictions of E to X x {to} and X x {t1} are E0 and E1 respective*
*ly, for some
t0; t1 2 T . Here a constructible curve means a finite union of irreducible al*
*gebraic (not
necessarily complete) curves in some projective space.
HOLOMORPHIC K -THEORY 9
Notice that two algebraically equivalent bundles are topologically isomorphic*
*, but not
necessarily isomorphic as algebraic bundles.
Theorem 6. For any smooth projective algebraic variety X, the group K0hol(X) *
*is isomor-
phic to the Grothendieck group completion of the monoid of algebraic equivalenc*
*e classes of
algebraic bundles over X.
The description of Mor(X; BU(n)) given in [4] provides our first step in unde*
*rstanding
K0hol(X).
As in [4], if X is a projective variety then we call an algebraic bundle E ! *
*X embeddable,
if there exists an algebraic embedding of E into a trivial bundle: E ,! X x CN *
* for some
large N. Let OE : E ,! X x CN be such an embedding. We identify an embedding OE*
* with
the composition OE : E ,! X x CN ,! X x CN+M , where CN is included in CN+M a*
*s the
first N coordinates. We think of such an equivalence class of embeddings as an *
*embedding
E ,! X x C1 . We refer to the pair (E; OE) as an embedded algebraic bundle.
Let X be any projective variety, and let E be a rank k holomorphic bundle ove*
*r X that
is holomorphically embeddable in a trivial bundle, define HolE (X; BU(k)) to be*
* the space
of holomorphic maps fl : X ! BU(k) such that fl*(k) ~= E, where k ! BU(k) is the
universal holomorphic bundle. This is topologized as a subspace of the continuo*
*us mapping
space, which is endowed with the compact - open topology.
Let Aut(E) be the gauge group of holomorphic bundle automorphisms of E. The f*
*ollow-
ing lemma identifies the homotopy type of HolE (X; BU(k)) in terms of Aut(E).
Lemma 7. HolE (X; BU(k)) is naturally homotopy equivalent to the classifying *
*space
HolE (X; BU(k)) ' B(Aut(E)):
Proof.As was described in [4], elements in HolE (X; BU(k)) are in bijective cor*
*respondence
to isomorphism classes of rank k embedded holomorphic bundles, (i; OE). By mod*
*ifying
the embedding OE via an isomorphism between i and E, we see that HolE (X; BU(k)*
* is
homeomorphic to the space of holomorphic embeddings : E ,! XxC1 , modulo the *
*action
of the holomorphic automorphism group, Aut(E). The space of holomorphic embeddi*
*ngs
of E in an infinite dimensional trivial bundle is easily seen to be contractibl*
*e [4], and the
action of Aut(E) is clearly free, with local sections. Again, see [4] for detai*
*ls. The lemma
*
* __
follows. *
* |__|
10 R.L. COHEN AND P. LIMA-FILHO
Corollary 8. The space HolE (X; BU(k)) is connected.
Now as above, we say that two embedded algebraic bundles (E0; OE0) and (E1; O*
*E1), are
path equivalent if there is topologically embedded bundle (E; OE), over X x I, *
*which gives a
path equivalence between E0 and E1, and over each slice X x {t} is an embedded *
*algebraic
bundle. Finally, notice that the set of (algebraic) isomorphism classes of embe*
*dded algebraic
bundles forms an abelian monoid.
Lemma 9. For any projective algebraic variety X, the group K0hol(X) is isomor*
*phic to
the Grothendieck group completion of the monoid of path equivalence classes of *
*embedded
algebraic bundles over X.
Proof.Recall that !
a +
K0hol(X) = ss0 Mor(X; BU(n)) :
n
But the set of path components of the Quillen - Segal group completion of a top*
*ological
E1 space is the Grothendieck group completion of the discrete monoid of path co*
*mponents
of the original E1 - space. Now as observed above the morphism space Hol(X; BU*
*(n)) is
given by configurations of isomorphism classes of embedded algebraic bundles, (*
*E; OE). Thus
ss0(Hol(X; BU(n)) is the set of path equivalence classes of such pairs; i.e the*
* set of path
equivalence classes of embedded algebraic bundles of rank n. We may therefore c*
*onclude
that K0hol(X) is the Grothendieck group completion of the monoid of path equiva*
*lence
*
* __
classes of embeddable algebraic bundles. *
* |__|
We now strengthen this result as follows.
Lemma 10. Two embedded bundles (E0; OE0) and (E1; OE1) are path equivalent if*
* and only
if they are algebraically equivalent.
Proof.Let f : X x I ! BU(n) be the (continuous) classifying map for the topolog*
*ical
bundle E over X x I, which gives the path equivalence between E0 and E1, and de*
*note f0
and f1 the restrictions of f to X x {0} and X x {1}. Since X x I is compact, th*
*e image of
f is contained in some Grassmannian Grn(Cm ) BU(n). It follows that f0 and f1 *
*lie in
the same path component of Hol(X; Grn(Cm )). Since Hol(X; Grn(Cm )) is a disjoi*
*nt union
of constructible subsets of the Chow monoid CdimX(X x Grn(Cm )), then f0 and f1*
* lie in
HOLOMORPHIC K -THEORY 11
the same connected component of a constructible subset in some projective space*
*. Using
the fact that any two points in an irreducible algebraic variety Y lie in some*
* irreducible
algebraic curve C Y (see [20, p. 56]), one concludes that any two points in a *
*connected
constructible subset of projective space lie in a connected constructible curve*
*. Let T be
a connected constructible curve contained in Hol(X; Grn(Cm )) and containing f0*
* and f1.
Under the canonical identification Hol(T; Hol(X; Grn(Cm ))) ~=Hol(X x T; Grn(Cm*
* )), one
identifies the inclusion
i : T ,! Hol(X; Grn(Cm ))
with an algebraic morphism i: X xT ! Grn(Cm ): This map classifies the desired *
*embedded
bundle E over X x T .
*
* __
The converse is clear. *
* |__|
The above two lemmas imply the following.
Proposition 11. For any projective algebraic variety X, (not necessarily smoot*
*h), the
group K0hol(X) is isomorphic to the Grothendieck group completion of the monoid*
* of al-
gebraic equivalence classes of embedded algebraic bundles over X.
Notice that Theorem 6 implies that we can remove the "embedded" condition in *
*the
statement of this proposition. We will show how that can be done later in this *
*section.
Recall from the last section that the forgetful map from the category of coli*
*mits of
projective varieties to the category of topological spaces, induces a map of mo*
*rphism spaces,
Hol(X; Z x BU) ! Map(X; Z x BU)
which induces a natural transformation
fi : Khol(X) ! Ktop(X):
Corollary 12. Let X be a colimit of projective varieties. Then the induced ma*
*p fi :
K0hol(X) ! K0top(X) is induced by sending the class of an embedded bundle to it*
*s underlying
topological isomorphism type:
fi : K0hol(X)! K0top(X)
[E; OE]! [E]
12 R.L. COHEN AND P. LIMA-FILHO
In order to approach Theorem 1 we need to understand the relationship between*
* algebraic
K - theory, K0alg(X), and holomorphic K - theory, K0hol(X) for X a smooth proje*
*ctive vari-
ety. For such a variety K0alg(X) is the Grothendieck group of the exact categor*
*y of algebraic
bundles over X. Roughly speaking the relationship between algebraic and holomor*
*phic K
-theories for a smooth variety is the passage from isomorphism classes of holom*
*orphic bun-
dles to algebraic equivalence classes of holomorphic bundles. This relationshi*
*p was made
precise in [9] using the Morel - Voevodsky description of algebraic K - theory *
*of a smooth
scheme in terms of an appropriate morphism space. In particular, recall that
K0alg(X) = MorH((Sm=C)Nis(X; RB(tn0 BGLn(C))):
where H((Sm=C)Nis) is the homotopy category of smooth schemes over C, using the*
* Nis-
nevich topology. See [19] for details. In particular a morphism of projective*
* varieties,
f : X ! Grn(CM ) induces an element in the above morphism space and hence a cla*
*ss
2 K0alg(X): It also induces a class [f] 2 ss0(Hol(X; Z x BU)+ = K0hol(X). A*
*s seen in [9]
this correspondence extends to give a forgetful map from the morphisms in the h*
*omotopy
category H((Sm=C)Nis) to homotopy classes of morphisms in the category of colim*
*its of
projective varieties. This defines a natural transformation
ff : K0alg(X) ! K0hol(X)
for X a colimit of smooth projective varieties.
Lemma 13. For X a smooth projective variety the transformation
ff : K0alg(X) ! K0hol(X)
is surjective.
Proof.As observed above, the set of path components of the Quillen - Segal grou*
*p com-
pletion of a topological monoid Y is the Grothendieck - group completion of the*
* discrete
monoid of path components:
ss0(Y +) = (ss0(Y ))+ :
Therefore we have that K0hol(X) is the Grothendieck group completion of ss0(Hol*
*(X; Z x
BU). Thus every element fl 2 K0hol(X) can be written as
fl = [f] - [g]
where f and g are holomorphic maps from X to some Grassmannian. By the above ob*
*ser-
*
* __
vations fl = ff( - ): *
* |__|
HOLOMORPHIC K -THEORY 13
This lemma and Proposition 11 allow us to prove the following interesting spl*
*itting prop-
erty of K0hol(X) which is not immediate from its definition.
Theorem 14. Let
0 ! [F; OEF ] ! [E; OEE ] ! [G; OEG ] ! 0
be a short exact sequence of embedded holomorphic bundles over a smooth project*
*ive variety
X. Then in K0hol(X) we have the relation
[E; OEE ] = [F; OEF ] + [G; OEG ]:
Proof.This follows from Lemma 13 and the fact that short exact sequences split *
*in K0alg(X).
*
* __
*
*|__|
Lemma 13 will also allow us to prove Theorem 5 which we now proceed to do. We*
* begin
with a definition.
Definition 5. We say that a smooth projective variety X is flag - like if the f*
*ollowing
properties hold on its K - theory:
1. the usual forgetful map
: K0alg(X) ! K0top(X)
is an isomorphism, and
2. K0alg(X) is generated (as an abelian group) by embeddable holomorphic bund*
*les.
Remark: We call such varieties "flag - like" because generalized flag varieties*
* (homogeneous
spaces G=P as in the statement of Theorem 5) satisfy these conditions. We now *
*state a
strengthening of Theorem 5 which we prove.
Theorem 15. Suppose X is a flag - like smooth projective variety. Then the h*
*omomor-
phisms
ff : K0alg(X) ! K0hol(X)
and
fi : K0hol(X) ! K0top(X)
are isomorphisms of rings.
14 R.L. COHEN AND P. LIMA-FILHO
Proof.Let X be a flag - like smooth projective variety. Since every embeddable *
*holomorphic
bundle is represented by a holomorphic map f : X ! Grn(CM ), for some Grassmann*
*ian,
then property (2) implies that K0alg(X) is generated by classes , where f is*
* such a
holomorphic map. But then () 2 K0top(X) clearly is the class represented b*
*y f in
ss0(Map(X; Z x BU) = K0top(X). But this means that the map : K0alg(X) ! K0top*
*(X) is
given by the composition
fi O ff : K0alg(X) ! K0hol(X) ! K0top(X):
But since is an isomorphism this means ff is injective. But we already saw in*
* corollary
13 that ff is surjective. Thus ff, and therefore fi, are isomorphisms. Clearl*
*y from their
*
* __
descriptions, ff and fi preserve tensor products, and hence are ring isomorphis*
*ms. |__|
We now use this result to prove Theorem 6. Let X be a smooth, projective vari*
*ety and
let e : X ,! CPn be a projective embedding. We begin by describing a constructi*
*on which
will allow any holomorophic bundle E over X to be viewed as representing an ele*
*ment of
K0hol(X) (i.e E does not necessarily have to be embeddable).
So let E ! X be a holomorphic bundle over X. Recall that by tensoring E with *
*a line
bundle of sufficiently negative Chern class, it will become embeddable. (This i*
*s dual to the
statement that tensoring a holomorphic bundle over a smooth projective variety *
*with a line
bundle with sufficiently large Chern class produces holomorphic bundle that is *
*generated
by global sections.) So for sufficiently large k, the bundle E O(-k) is embedd*
*able. Here
O(-k) is the k - fold tensor product of the canonical line bundle O(-1) over CP*
*n, which,
by abuse of notation, we identify with its restriction to X. Now choose a holo*
*morphic
embedding
OE : E O(-k) ,! X x CN :
Then the pair (E O(-k); OE) determines an element of K0hol(X).
Now from Theorem 15 we know that K0alg(CPn) ~=K0hol(CPn) ~=K0top(CPn) as ring*
*s. But
since O(-k) O(k) = 1 2 K0top(CPn), this means that if
k = k : O(-k) = kO(-1) ,! kCn+1
is the canonical embedding, then the pair (O(-k); k) represents an invertible c*
*lass in
K0hol(CPn). We denote its inverse by O(-k)-1 2 K0hol(CPn), and, as before, we *
*use the
same notation to denote its restriction to K0hol(X).
Write A(E) = (E O(-k); OE) O(-k)-1e2 K0hol(X).
HOLOMORPHIC K -THEORY 15
Proposition 16. The assignment to the holomorphic bundle E the class
A(E) = [E O(-k); OE] O(-k)-1 2 K0hol(X)
is well defined, and only depends on the (holomorphic) isomorphism class of E.
Proof.We first verify that given any holomorphic bundle E ! X, that A(E) is a w*
*ell
defined element of K0hol(X). That is, we need to show that this class is indepe*
*ndent of the
choices made in its definition. More specifically, we need to show that
[E O(-k); OE] O(-k)-1 = [E O(-q); ] O(-q)-1
for any appropriate choices of k, q, OE, and . We do this in two steps.
Case 1: k = q. In this case it suffices to show that (E O(-k); OE) and (E O*
*(-k); )
lie in the same path component of the morphism space Hol(X; BU(n)), where n is *
*the
rank of E. Now using the notation of Lemma 7 we see that these two elements bot*
*h lie in
HolEO(-k) (X; BU(n)), which, as proved in Corollary 8 there is path connected.
Case 2: General Case: Suppose without loss of generality that q > k. Then cle*
*arly
the classes (E O(-k); OE)O(-k)-1 and (E O(-k)O(-(q -k)); OEq-k)O(-k)-1
O(-(q-k))-1 represent the same element of K0hol(X). But this latter class is (E*
*O(-q); OE
q-k) O(-q)-1 which we know by case 1 represents the same K - theory class as (E
O(-q); ) O(-q)-1.
Thus A(E) is a well defined class in K0hol(X). Clearly the above arguments al*
*so verify
*
* __
that A(E) only depends on the isomorphism type of E. *
* |__|
Notice that this argument implies that K0holencodes all holomorphic bundles (*
*not just
embeddable ones). We will use this to complete the proof of theorem 6.
Proof.Let X be a smooth projective variety and let HX denote the Grothendieck g*
*roup of
monoid of algebraic equivalence classes of holomorphic bundles over X. We show *
*that the
correspondence A described in the above theorem induces an isomorphism
~=
A : HX ---! K0hol(X):
We first show that A is well defined. That is, we need to know if E0 and E1 are*
* algebraically
equivalent, then A(E0) = A(E1). So let E ! X x T be an algebraic equivalence. S*
*ince T is
a curve in projective space, we can find a projective embedding of the product,*
* e : X xT ,!
CPn: Now for sufficiently large k, E O(-k) is embeddable, and given an embeddi*
*ng OEE ,
16 R.L. COHEN AND P. LIMA-FILHO
the pair (E O(-k); OEE ) defines an algebraic equivalence between the embedded*
* bundles
(E0 O(-k); OE0) and (E1 O(-k); OE1), where the OEiare the appropriate restricti*
*ons of the
embedding OE. Thus
[E0 O(-k); OE0] = [E1 O(-k); OE1] 2 K0hol(X):
Thus
[E0 O(-k); OE0] O(-k)-1 = [E1 O(-k); OE1] O(-k)-1 2 K0hol(X):
But these classes are A(E0) and A(E1). Thus A : HX ! K0hol(X) is well defined.
Notice also that A is surjective. This is because, as was seen in the proof *
*of the last
theorem, if (E; OE) is and embedded holomorphic bundle, then A(E) = [E; OE] 2 K*
*0hol(X).
The essential point here being that the choice of the embedding OE does not aff*
*ect the
holomorphic K - theory class, since the space of such choices is connected.
Finally notice that A is injective. This is follows from two the two facts:
1. The classes [O(-k)-1] are units in the ring structure of K0hol(X), and
2. If bundles of the form E0 O(-k) and E1 O(-k) are algebraically equivalen*
*t then
the bundles E0 and E1 are algebraically equivalent.
*
* __
*
*|__|
Q
3.The equivariant homotopy type of Khol( n P1) and the holomorphic
K-theory of symmetric products of projective spaces
The goal of this section is to completely identify the holomorphic K - theory*
* of symmetric
products of projective spaces, Khol(SP n(Pm )). Since the algebraic K -theory o*
*f these spaces
is not in general known, this will give us new information about algebraic bund*
*les over these
symmetric product spaces. These spaces are particularly important in this paper*
* since, as
we will point out below, symmetric products of projective spaces are representi*
*ng spaces
for morphic cohomology.
*
* Q
Our approach to this question is to study the equivariant homotopy type of Kh*
*ol( n P1),
Q
where the symmetric group n acts on the holomorphic K - theory space Khol( n P*
*1) =
Q Q
Hol( n P1; Z x BU)+ by permuting the coordinates of nP1. It acts on the topo*
*logical
Q Q
K -theory space Ktop( n P1) = Map( n P1; Z x BU) in the same way. The main re*
*sult of
this section is t the following.
HOLOMORPHIC K -THEORY 17
Q Q
Theorem 17. The natural map fi : Khol( n P1) ! Ktop( n P1) is a n - equivar*
*iant
homotopy equivalence.
Before we begin the proof of this theorem we observe the following consequenc*
*es:
Corollary 18. Let G < n be a subgroup. Then the induced map on the K - theorie*
*s of
the orbit spaces,
Y Y
ff : Khol( P1=G) ! Ktop( P1=G)
n n
is a homotopy equivalence.
Q Q
Proof.By Theorem 17, ff : Khol( n P1) ! Ktop( n P1) is a n - equivariant homo*
*topy
equivalence. Therefore it induces a homotopy equivalence on the fixed point set*
*s,
Q ' Q
ff : Khol( n P1)G---! Ktop( n P1)G:
Q Q *
* __
But these fixed point sets are Khol( n P1=G) and Ktop( n P1=G) respectively. *
* |__|
Corollary 19. ff : Khol(CPn) ! Ktop(CPn) is a homotopy equivalence.
Q *
* __
Proof.Let G = n in the above corollary. n(P1)=n = SP n(P1) ~=CPn. *
* |__|
The following example will be important because as seen earlier, symmetric pr*
*oducts of
projective spaces form representing spaces for morphic cohomology.
Corollary 20. Let r and k be any positive integers. Then
ff : Khol(SP r(CPk)) ! Ktop(SP r(CPk))
is a homotopy equivalence.
R
Proof.Let G be the wreath product G = r k viewed as a subgroup of the symmetric
group rk. The obtain an identification of orbit spaces
! Z
Y
P1 = r k = SP r(SP k(P1)) ~=SP r(CPk):
rk
*
* __
Finally, apply the above corollary when n = rk. *
* |__|
18 R.L. COHEN AND P. LIMA-FILHO
Observe that this corollary gives a complete calculation of the holomorphic K*
*- theory of
symmetric products of projective spaces, since their topological K - theory is *
*known.
In order to begin the proof of Theorem 17 we need to expand our notion of hol*
*omorphic
K - theory to include unions of varieties. So let A and B be subvarieties of CP*
*n, then define
Hol(A[B; ZxBU) to be the space of those continuous maps on A[B that are holomor*
*phic
when restricted to A and B. This space still has the action of the little isome*
*try operad
and so we can take a group completion and define Khol(A [ B) = Hol(A [ B; Z x B*
*U)+ .
If A [ B is connected, then we can define the reduced holomorphic K - theory as*
* before,
K"hol(A [ B) = the homotopy fiber of the restriction map Khol(A [ B) ! Khol(x0)*
*, where
x0 2 A \ B. With this we can now define the holomorphic K - theory of a smash p*
*roduct
of varieties.
Definition 6. Let X and Y be connected projective projective varieties with ba*
*sepoints
x0 and y0 respectively. We define "Khol(X ^ Y ) to be the homotopy fiber of the*
* restriction
map
ae : "Khol(X x Y ) ! "Khol(X _ Y )
where X _ Y = {(x; y0)} [ {(x0; y)} X x Y .
Recall that in topological K -theory, the Bott periodicity theorem can be vie*
*wed as saying
the Bott map fi : "Ktop(X) ! "Ktop(X ^ S2) is a homotopy equivalence for any sp*
*ace X. In
[22] Rowland studies the holomorphic analogue of this result. She studies the *
*Bott map
fi : "Khol(X) ! "Khol(X ^ P1) and, using the index of a family of @ operators, *
*defines a map
@ : "Khol(X ^ P1) ! "Khol(X). Using a refinement of Atiyah's proof of Bott peri*
*odicity [1],
she proves the following.
Theorem 21. Given any smooth projective variety X, the Bott map
fi : "Khol(X) ! "Khol(X ^ P1)
is a homotopy equivalence of infinite loop spaces. Moreover its homotopy invers*
*e is given
by the map
@ : "Khol(X ^ P1) ! "Khol(X):
The fact that the Bott map fi is an isomorphism also follows from the "projec*
*tive bundle
theorem" of Friedlander and Walker [9] which was proven independently, using di*
*fferent
HOLOMORPHIC K -THEORY 19
techniques. This result in the case when X = S0 was proved in [4]. The statemen*
*t in this
case is
"Khol(P1) ' "Khol(S0) = Z x BU = "Ktop(S0) ' "Ktop(S2):
Combining this with Theorem 21 (iterated several times) we get the following:
V
Corollary 22. For a positive integer k, let kP1 = (P1)(k)be the k -fold smas*
*h product of
P1. Then we have homotopy equivalences
"Khol((P1)(k)) ' Z x BU ' "Ktop(S2k);
We will use this result to prove Theorem 17. We actually will prove a splitti*
*ng result for
Q
Khol( n P1) which we now state.
Let Sn denote the category whose objects are (unordered) subsets of {1; . .;.*
*n}. Mor-
phisms are inclusions. Notice that the cardinality of the set of objects,
|Ob(Sn)| = 2n:
Notice also that the set of objects Ob(Sn) has an action of the symmetric group*
* n induced
by the permutation action of n on {1; . .;.n}.
Let X be a space with a basepoint x0 2 X. For 2 Ob(Sn), define
Y
X = X Xn
by X = {(x1; . .;.xn) such that if j is not an element of , then xj = x0}2.X *
*Notice
that if is a subset of {1; . .n.} of cardinality k, then X ~= Xk. The smash *
*product
V ()
X = X is defined similarly. The following is the splitting theorem that wi*
*ll allow us
to prove Theorem 17.
Theorem 23. Let X be a smooth projective variety (or a union of smooth projec*
*tive vari-
eties). Then there is a natural n - equivariant homotopy equivalence
Y
J : "Khol(Xn) -! K"hol(X()):
2Ob(Sn)
where the action of n on the right hand side is induced by the permutation acti*
*on of n
on the objects Ob(Sn).
Proof.In order to prove this theorem we begin by recalling the equivariant stab*
*le splitting
theorem of a product proved in [2]. An alternate proof of this can be found in *
*[3].
20 R.L. COHEN AND P. LIMA-FILHO
Given a space X with a basepoint x0 2 X, let 1 (X) denote the suspension spec*
*trum
of X. We refer the reader to [14] for a discussion of the appropriate category *
*of equivariant
spectra.
Theorem 24. There is a natural n equivariant homotopy equivalence of suspensi*
*on spec-
tra
W
J : 1 (Xn) - '--! 1 ( 2Ob(Sn)(X())):
As a corollary of this splitting theorem we get the following splitting of to*
*pological K -
theory spaces.
Corollary 25. There is a n -equivariant homotopy equivalence of topological K *
*- theory
spaces,
Y
J* : "Ktop(X()) ! "Ktop(Xn):
2ObSn
Proof.Given to spectra E and F , let sMap(E; F ) be the spectrum consisting of *
*spectrum
maps from E to F . We again refer the reader to [14] for a discussion of the a*
*ppropriate
category of spectra. If 1 is the zero space functor from spectra to infinite lo*
*op spaces, then
1 (sMap(E; F )) = Map1 (1 (E); 1 (F )); where Map1 refers to the space of infi*
*nite
loop maps.
Let bu denote the connective topological K - theory spectrum, whose zero spac*
*e is ZxBU.
Now Theorem 24 yields a n equivariant homotopy equivalence of the mapping spect*
*ra,
W '
J* : sMap(1 ( 2Ob(Sn)(X()); bu)- --! sMap(1 (Xn); bu);
and therefore of infinite loop mapping spaces,
W '
J* : Map1 (1 1 ( 2Ob(Sn)(X()); Z x BU) - --! Map1 (1 1 (Xn); Z x BU):
But since 1 1 (Y ) is, in an appropriate sense, the free infinite loop space ge*
*nerated by
a space Y , then given any other infinite loop space W , the space of infinite *
*loop maps,
Map1 (1 1 (Y ); W ) is equal to the space of (ordinary) maps Map(Y; W ). Thus w*
*e have
a n - equivariant homotopy equivalence of mapping spaces,
W '
J* : Map(( 2Ob(Sn)(X()); Z x BU)- --! Map(Xn; Z x BU):
*
* __
*
*|__|
HOLOMORPHIC K -THEORY 21
Notice that Theorem 23 is the holomorphic version of Corollary 25 . In order *
*to prove
this result, we need to develop a holomorphic version of the arguments used in *
*proving
Theorem 25. For this we consider the notion of "holomorphic stable homotopy equ*
*ivalence",
as follows. Suppose that X is a smooth projective variety (or union of varieti*
*es) and
E is a spectrum whose zero space is a smooth projective variety (or a union of *
*such),
define sHol(1 (X); E) to be the subspace of Map1 (1 1 (X); 1 (E)) consisting of*
* those
infinite loop maps OE : 1 1 (X) ! 1 (E) so that the composition
X ,! 1 1 (X) --OE-!1 (E)
is holomorphic. Notice, for example, that sHol(1 (X); bu) = Hol(X; Z x BU).
Now suppose X and Y are both smooth projective varieties, (or unions of such).
Definition 7. A map of suspension spectra, : 1 (X) ! 1 (Y ) is called a holom*
*orphic
stable homotopy equivalence, if the following two conditions are satisfied.
1. is a homotopy equivalence of spectra.
2. If E is any spectrum whose zero space is a smooth projective variety (or a*
* union
of such), then the induced map on mapping spectra, * : sMap(1 (Y ); E) !
sMap(1 (X); E) restricts to a map
*s : Hol(1 (Y ); E) ! sHol(1 (X); E)
which is a homotopy equivalence.
With this notion we can complete the proof of Theorem 23. This requires a pr*
*oof of
Theorem 24 that will respect holomorphic stable homotopy equivalences. The ver*
*sion of
this theorem given in [3] will do this. We now recall that proof and refer to [*
*3] for details.
Let X be a connected space with basepoint x0 2 X. Let X+ denote X with a disj*
*oint base-
point, and let X _S0 denote the wedge of X with the two point space S0. Topolog*
*ically X+
and X _S0 are the same spaces, but their basepoints are in different connected *
*components.
However their suspension spectra 1 (X+ ) and 1 (X _S0) are stably homotopy equi*
*valent
spectra with units (i.e via a stable homotopy equivalence j : 1 (X+ ) ' 1 (X _ *
*S0) that
respects the obvious unit maps 1 (S0) ! 1 (X+ ) and 1 (S0) ! 1 (X _ S0).) More-
over it is clear that if X is a smooth projective variety then 1 (X+ ) and 1 (X*
* _ S0) are
holomorphically stably homotopy equivalent in the above sense. Now by taking sm*
*ash prod-
ucts n -times of this equivalence, we get a n - equivariant holomorphic stable *
*homotopy
equivalence,
(n)
Jn : 1 ((X+ )(n)) = (1 ((X+ ))(n)j---!(1 (X _ S0))(n)= 1 ((X _ S0)(n)):
22 R.L. COHEN AND P. LIMA-FILHO
Now notice that the n - fold smash product (X+ )(n)is naturally (and n equiva*
*riantly)
homeomorphic to the cartesian product (Xn)+ . Notice also that the n fold itera*
*ted smash
product of X _ S0 is n - equivariantly homeomorphic to the wedge of the smash p*
*roducts,
0 1
_
(X _ S0)(n)= @ X()A _ S0:
2Ob(Sn)
Thus Jn gives a n - equivariant stable homotopy equivalence,
iW j
Jn : 1 (((Xn)+ )- '--! 1 ( 2Ob(Sn)X() _ S0)):
which gives a proof of Theorem 24. Moreover when X is a smooth projective varie*
*ty (or
a union of such) this equivariant stable homotopy equivalence is a holomorphic *
*one. In
particular, given any such X, this implies there is a n equivariant homotopy eq*
*uivalence
iW j
J*n: sHol*(1 ((Xn)+ ); bu)-'--!sHol*(1 ( 2Ob(Sn)X() _ S0)); bu):
where sHol* refers to those maps of spectra that preserve the units. If we remo*
*ve the units
from each of these mapping spectra we conclude that we have a n equivariant hom*
*otopy
equivalence
iW j
J*n: sHol(1 (Xn); bu)- -'-! sHol(1 2Ob(Sn)X() ; bu):
W
But these spaces are precisely Hol(Xn; Z x BU) and Hol( 2Ob(Sn)X(); Z x BU) =
Q () *
* __
2Ob(Sn)Hol(X ; Z x BU) respectively. Theorem 23 now follows. *
* |__|
We are now in a position to prove Theorem 17.
Proof.By theorems 23 and 25 we have the following homotopy commutative diagram:
* Q
"Khol((P1)n)-Jn--! K"hol((P1)())
? ' 2Ob(Sn)?
fi?y ?yfi
* Q
"Ktop((P1)n)-Jn--! "Ktop((P1)()):
' 2Ob(Sn)
Notice that all the maps in this diagram are n equivariant, and by the results *
*of theorems
23 and 25 the horizontal maps are n -equivariant homotopy equivalences. Further*
*more,
by Corollary 22 the maps K"hol((P1)()) ! K"top((P1)()) are homotopy equivalence*
*s. Now
Q Q
since the n action on 2Ob(Sn)K"hol((P1)()) and on 2Ob(Sn)K"top((P1)()) is g*
*iven by
permuting the factors according to the action of n on Ob(Sn), this implies that*
* the right
HOLOMORPHIC K -THEORY 23
Q Q
hand vertical map in this diagram, fi : 2Ob(Sn)K"hol((P1)()) ! 2Ob(Sn)K"top*
*((P1)()) is
a n -equivariant homotopy equivalence. Hence the left hand vertical map
fi : "Khol((P1)n) ! "Ktop((P1)n)
*
* __
is also a n - equivariant homotopy equivalence. This is the statement of Theore*
*m 17. |__|
4. The Chern character for holomorphic K - theory
In this section we study the Chern character for holomorphic K - theory that *
*was defined
by the authors in [6]. The values of this Chern character are in the rational F*
*riedlander -
Lawson "morphic cohomology groups", L*H*(X) Q. Our goal is to show that the Ch*
*ern
character is gives an isomorphism
M1
ch : K-qhol(X) Q ~= LkH2k-q(X) Q:
k=0
Recall the following basic results about the Chern character proved in [6].
Theorem 26. There is a natural transformation of functors from the category o*
*f colimits
of projective varieties to algebras over the rational numbers,
M1
ch : K-*hol(X) Q -! LkH2k-*(X) Q
k=0
that satisfies the following properties.
1. The Chern character is compatible with the Chern character for topological*
* K - theory.
That is, the following diagram commutes:
K-qhol(X) Q - fi*--! K-qtop(X) Q
? ?
ch?y ?ych
L 1 k 2k-q L 1 2k-q
k=0L H (X) Q - --!OE* k=0 H (X; Q)
where OE* is the natural transformation from morphic cohomology to singula*
*r cohomol-
ogy as defined in [7]
2. Let chk : K-qhol(X) Q ! LkH2k-q(X) Q be the projection of ch onto the kt*
*h factor.
Also let ck : K-qhol(X) ! LkH2k-q(X) be the kthChern class defined in [12,*
* x6] (see [16,
x4] for details). Then there is a polynomial relation between natural tran*
*sformations
ck = k! chk + p(ch1; . .;.chk-1)
where p(ch1; . .;.chk-1) is some polynomial in the first k - 1 Chern chara*
*cters.
24 R.L. COHEN AND P. LIMA-FILHO
As mentioned above the goal of this section is to prove the following theorem*
* regarding
the Chern character.
Theorem 27. For every q 0, the Chern character for holomorphic K - theory
M
ch : K-qhol(X) Q ! LkH2k-q(X) Q
k0
is an isomorphism.
Proof.Recall from [7] that the suspension theorem in morphic cohomology implies*
* that
morphic cohomology can be represented by morphisms into spaces of zero cycles i*
*n projective
spaces. Since zero cycles are given by points in symmetric products this can be*
* interpreted in
the following way. Let SP 1(P1 ) be the infinite symmetric product of the infin*
*ite projective
space. Given a projective variety X, let Mor(X; Z x SP 1(P1 )) denote the colim*
*it of the
the algebraic morphism spaces Mor(X; SP n(Pm )).
Lemma 28. Let X be a colimit of projective varieties. Then
M
ssq(Mor(X; (Z x SP 1(P1 ))+ ) ~= LkH2k-q(X):
k0
Similarly, Z x BU represents holomorphic K - theory in the sense that
(4.1) ssq(Mor(X; Z x BU)+ ) ~=K-qhol(X):
Thus to prove Theorem 27 we will describe a relationship between the representi*
*ng spaces
Z x SP 1(P1 ) and Z x BU.
Using the identification in Lemma 28, let
M1
2 LkH2k(Z x SP 1(P1 ))
k=1
correspond to the class in ss0((Mor(Z x SP 1(P1 ); Z x SP 1(P1 ))+ ) represente*
*d by the
identity map id : Z x SP 1(P1 ) ! Z x SP 1(P1 ).
Lemma 29. There exists a unique class o 2 K0hol(ZxSP 1(P1 ))Q with Chern char*
*acter
L 1
ch(o) = 2 k=0LkH2k(Z x SP 1(P1 )) Q.
HOLOMORPHIC K -THEORY 25
Proof.By Corollary 20 in section 3, we know that for every k and n, Khol(SP k(P*
*n)) !
Ktop(SP k(Pn)) is a homotopy equivalence. It follows that by taking limits we *
*have that
Khol(Z x SP 1(P1 )) ! Ktop(Z x SP 1(P1 )) is a homotopy equivalence. But we als*
*o know
from [15] that the natural map
M1 M1
OE : LkH2k(Z x SP 1(P1 )) Q ! H2k(Z x SP 1(P1 ); Q)
k=0 k=0
is an isomorphism. This is true because for the following reasons.
Q
1. Since products n(P1) have "algebraic cell decompositions" in the sense o*
*f [15], its
morphic cohomology and singular cohomology coincide,
Q ~= Q
OE : LkHp( n(P1))---! Hp( n P1):
2. Since both morphic cohomology and singular cohomology admit transfer maps *
*([7])
there is a natural identification of LkHp(SP r(Pm ) Q and Hp(SP r(Pm ); Q*
*) with the
R Q Q
r m invariants in LkHp( rm P1) Q and Hp( rm P1; Q) respectively. Since*
* the
Q Q
natural transformation OE : LkHp( rm P1) ! Hp( rm P1) is equivariant, th*
*en we get
an induced isomorphism on the invariants,
~=
OE : LkHp(SP r(Pm )) Q---! Hp(SP r(Pm ); Q):
3. By taking limits over r and m we conclude that
OE : LkHp(Z x SP 1(P1 )) Q -! Hp(Z x SP 1(P1 ); Q)
is an isomorphism.
Using this isomorphism and the compatibility of the Chern character maps in h*
*olomorphic
and topological K - theories, to prove this theorem it is sufficient to prove t*
*hat there exists
a unique class o 2 K0top((Z x SP 1(P1 )) Q with (topological ) Chern character
ch(o) = 2 [Z x SP 1(P1 ); Z x SP 1(P1 )] Q ~=1k=0H2k(Z x SP 1(P1 ); Q)
where 2 [Z x SP 1(P1 ); Z x SP 1(P1 )] is the class represented by the identit*
*y map. But
this follows because the Chern character in topological K - theory, ch : K"0top*
*(X) Q !
*
* __
1k=1H2k(X; Q) is an isomorphism. *
* |__|
We now show how the element o 2 K0hol(Z x SP 1(P1 )) Q defined in the above *
*lemma
will yield an inverse to the Chern character transformation.
26 R.L. COHEN AND P. LIMA-FILHO
Theorem 30. The element o 2 K0top(Z x SP 1(P1 )) Q defines natural transform*
*ations
M -q
o* : LkH2k-q(X) Q -! Khol(X) Q
k0
such that the composition
M -q M
ch O o* : LkH2k-q(X) Q ! Khol(X) Q ! LkH2k-q(X) Q
k0 k0
is equal to the identity.
Proof.The set of path components of the Quillen - Segal group completion of a t*
*opological
monoid is the Grothendieck group completion of the discrete monoid of path comp*
*onents.
If we use the notation M^to mean the Grothendieck group of a discrete monoid M,*
* this
says that
K0hol(Z x SP 1(P1 )) = ss0(Hol(Z x SP 1(P1 ); Z x BU)+) ~=(ss0(Hol(Z x SP 1(P1 *
*); Z x BU)))^;
and hence
K0hol(Z x SP 1(P1 )) Q ~=(ss0(Hol(Z x SP 1(P1 ); Z x BU)Q))^;
where the subscript Q denotes the holomorphic mapping space localized at the ra*
*tionals.
This means that o can be represented as a difference of classes,
o = [o1] - [o2]
where oi2 Hol(Z x SP 1(P1 ); Z x BU)Q.
Now consider the composition pairing
Hol(X; Z x SP 1(P1 )) x Hol(Z x SP 1(P1 ); Z x BU) ! Hol(X; Z x BU) ! Hol(X; Z*
* x BU)+:
which localizes to a pairing
Hol(X; ZxSP 1(P1 ))Q xHol(ZxSP 1(P1 ); ZxBU)Q ! Hol(X; ZxBU)Q ! Hol(X; ZxBU)+Q:
Using this pairing, o1 and o2 each define transformations
oi: Hol(X; Z x SP 1(P1 ))Q ! Hol(X; Z x BU)+Q:
Using the fact that Hol(X; Z x BU)+Qis an infinite loop space, then the subtrac*
*tion map
is well defined up to homotopy,
o1 - o2 : Hol(X; Z x SP 1(P1 ))Q ! Hol(X; Z x BU)+Q:
We need the following intermediate result about this construction.
HOLOMORPHIC K -THEORY 27
Lemma 31. For any projective variety (or colimit of varieties) X, the map
o1 - o2 : Hol(X; Z x SP 1(P1 ))Q ! Hol(X; Z x BU)+Q:
is a map of H - spaces.
Proof.Since the construction of these maps was done at the representing space l*
*evel, it is
sufficient to verify the claim in the case when X is a point. That is, we need *
*to verify that
the compositions
(4.2)
(o1-o2)x(o1-o2)
Z x SP 1(P1 )Q x Z x SP 1(P1 )Q ----------! (Z x BU)Q x (Z x BU)Q
---! (Z x BU)
Q
and
(o1-o2)
(4.3) Z x SP 1(P1 )Q x Z x SP 1(P1 )Q ---! Z x SP 1(P1 )Q - ---! (Z x BU)Q
represent the same elements of K0hol(Z x SP 1(P1 ) x Z x SP 1(P1 )) Q, where *
*and
are the monoid multiplications in ZxBU and ZxSP 1(P1 ) respectively. But by Cor*
*ollary
20 of the last section, this is the same as K0top((Z x SP 1(P1 )) x (Z x SP 1(P*
*1 ))) Q.
Now in the topological category, we know that the class o 2 K0hol(Z x SP 1(P1 )*
*) Q is
the inverse to the Chern character and hence induces a rational equivalence of *
*H - spaces
o : (Z x SP 1(P1 ))Q--'-! (Z x BU)Q:
This implies that the compositions 4.2 and 4.3 repesent the same elements of K0*
*top(Z x
*
* __
SP 1(P1 )) Q, and hence the same elements in K0hol(Z x SP 1(P1 )) Q. *
* |__|
Thus the map
o1 - o2 : Hol(X; Z x SP 1(P1 ))Q ! Hol(X; Z x BU)+Q
is an H - map from a C1 operad spaces (as described in x1), to an infinite loo*
*p space. But
any such rational H - map extends in a unique manner up to homotopy, to a map o*
*f H -
spaces of their group completions
o1 - o2 : Hol(X:Z x SP 1(P1 ))+Q! Hol(X; Z x BU)+Q:
This map is natural in the category of colimits of projective varieties X. Sinc*
*e any H - map
between rational infinite loop spaces is homotopic to an infinite loop map, thi*
*s then defines
a natural transformation of rational infinite loop spaces,
28 R.L. COHEN AND P. LIMA-FILHO
(4.4) o = o1 - o2 : Hol(X:Z x SP 1(P1 ))+Q! Hol(X; Z x BU)+Q:
So when we apply homotopy groups o defines natural transformations
M1
(4.5) o* : LkH2k-q(X) Q -! K-qhol(X) Q:
k0
Now notice that if we let X = Z x SP 1(P1 ) in (4.4), and 2 Hol(Z x SP 1(P1 *
*); Z x
SP 1(P1 ))+Qbe the class represented by the identity map, then by definition, o*
*ne has that
o() 2 Hol(Z x SP 1(P1 ); Z x BU)+Q
represents the class [o] 2 K0hol(ZxSP 1(P1 )) described in Lemma 29. Moreover t*
*his lemma
L 1
tells us that ch([o]) = [] 2 k0 LkH2k(Z x SP 1(P1 )). Now as in section 4, we*
* view the
Chern character as represented by an element ch 2 Hol(Z x BU; Z x SP 1(P1 ))+Qw*
*hich is
a map of rational infinite loop spaces, then this lemma tells us that the eleme*
*nts
ch O o() 2 Hol(Z x SP 1(P1 ); Z x SP 1(P1 ))+Q
and
2 Hol(Z x SP 1(P1 ); Z x SP 1(P1 ))+Q
are both maps of rational infinite loop spaces and lie in the same path compone*
*nt of Hol(Zx
SP 1(P1 ); Z x SP 1(P1 ))+Q. But this implies the ch O o and define the homoto*
*pic natural
transformations of rational infinite loop spaces,
ch O o ' : Hol(X; Z x SP 1(P1 ))+Q! Hol(X; Z x SP 1(P1 ))+Q:
When we apply homotopy groups this means that
M1 M1
ch O o = id : LkH2k-q(X) Q -! LkH2k-q(X) Q
k0 k0
*
* __
which was the claim in the statement of Theorem 30. *
* |__|
We now can complete the proof of Theorem 27. That is we need to prove that
M1
ch : K-qhol(X) Q -! LkH2k-q(X) Q
k0
is an isomorphism. By Theorem 30 we know that ch is surjective. In order to sho*
*w that it
is injective, we prove the following:
HOLOMORPHIC K -THEORY 29
Lemma 32. The composition of natural transformations
M1
o* O ch : K-qhol(X) Q ! LkH2k-q(X) Q ! K-qhol(X) Q
k0
is the identity.
Proof.These transformations are induced on the representing level by maps of ra*
*tional
infinite loop spaces,
ch : (Z x BU)Q ! (Z x SP 1(P1 ))Q
and
o : (Z x SP 1(P1 ))Q ! (Z x BU)Q:
The composition
o O ch : (Z x BU)Q ! (Z x BU)Q
represents an element of rational holomorphic K - theory,
[o O ch] 2 K0hol(Z x BU)Q:
Now the fact that o is an inverse of the Chern character in topological K - t*
*heory tells
us that
[o O ch] = j 2 K0top(Z x BU)Q;
where j 2 K0top(Z x BU) = ss0(Map(Z x BU; Z x BU)) is the class represented by *
*the
identity map. But according to the results in x2, we know
K0hol(Z x BU)Q ~=K0top(Z x BU)Q:
So by the compatibility of the Chern characters in holomorphic and topological *
*K - theories,
we conclude that
[o O ch] = j 2 K0hol(Z x BU)Q:
This implies that o O ch : (Z x BU)Q ! (Z x BU)Q and the identity map id : (Z x*
* BU)Q !
(ZxBU)Q induce the same natural transformations Hol(X; ZxBU)+Q! Hol(X; ZxBU)+Q.
Applying homotopy groups implies that
M1
o* O ch : K-qhol(X) Q ! LkH2k-q(X) Q ! K-qhol(X) Q
k0
*
* __
is the identity as claimed. *
* |__|
30 R.L. COHEN AND P. LIMA-FILHO
L 1
This lemma implies that ch : K-qhol(X) Q ! k0 LkH2k-q(X) Q is injective.*
* As
remarked above this was the last remaining fact to be verified in the proof of *
*Theorem
*
* __
27. *
*|__|
We end this section with a proof that the total Chern class also gives a rati*
*onal isomor-
phism in every dimenstion. Namely, recall the Chern classes
ck : K-qhol(X) ! LkH2k-q(X)
defined originally in [12]. Taking the direct sum of these maps gives us the t*
*otal Chern
class map,
M1
c : K-qhol(X) ! LkH2k-q(X):
k=0
We will prove the following result, which was conjectured by Friedlander and Wa*
*lker in [9].
Theorem 33. The total Chern class
M1
c : K-qhol(X) Q ! LkH2k-q(X) Q
k=0
is an isomorphism for all q 0.
We note that in the case q = 0, this theorem was proved in [9]. The proof in*
* general
will follow quickly from our Theorem 27 stating that the total Chern character *
*is a rational
isomorphism.
Proof.We first prove that the total Chern class
M1
c : K-qhol(X) Q -! LkH2k-q(X) Q
k=0
is injective. So suppose that for some ff 2 K-qhol(X) Q, we have that c(ff) = *
*0: So each
Chern class cq(ff) = 0 for q 0. Now recall from section 4 that in the algebra *
*of operations
L 1
between K-qhol(X) Q and k=0 LkH2k-q(X) Q, that the that the Chern classes a*
*nd
Chern character are related by a formula of the form
(4.6) ck = k! chk + p(ch1; . .;.chk-1)
where p(ch1; . .;.chk-1) is some polynomial in the first k - 1 Chern classes. S*
*o since each
cq(ff) = 0 then an inductive argument using (4.6) implies that each chq(ff) = 0*
*. Thus the
total Chern character ch(ff) = 0. But since the total Chern character is an is*
*omorphism
HOLOMORPHIC K -THEORY 31
(Theorem 27), this implies that ff = 0 2 K-qhol(X) Q. This proves that the tot*
*al Chern
class operation is injective.
L 1
We now prove that c : K-qhol(X) Q -! k=0 LkH2k-q(X) Q is surjective. To *
*do
this we will prove that for every k and element fl 2 LkH2k-q(X) Q there is a c*
*lass
ffk 2 K-qhol(X) with ck(ffk) = fl and cq(ffk) = 0 for q 6= k. We prove this by *
*induction on
k. So assume this statement is true for k m - 1, and we now prove it for k = m*
*. Let
flm 2 Lm H2m-q (X) Q. Since the total Chern character is an isomorphism, ther*
*e is an
element ffm 2 K-qhol(X) Q with chm (ffm ) = flm , and chq(ffm ) = 0 for q 6= m*
*. But formula
(4.6) implies that cq(ffm ) = 0 for q < m, and cm (ffm ) = _1_m!flm . Thus the *
*total Chern class
has value c(m!ffm ) = flm . This proves that the total Chern class is surjectiv*
*e, and therefore
*
* __
that it is an isomorphism. *
* |__|
5. Stability of rational maps and Bott periodic holomorphic K - theory
In this section we study the space of rational maps in the morphism spaces us*
*ed to
define holomorphic K - theory. We will show that the "stability property" for *
*rational
maps in the morphism space Hol(X; Z x BU) amounts to the question of whether Bo*
*tt
perioidicity holds in K*hol(X). We then use the Chern character isomorphism pro*
*ved in the
last section to prove a conjecture of Friedlander and Walker [9] that rationall*
*y, Bott periodic
holomorphic K - theory is isomorphic to topological K - theory. (Friedlander an*
*d Walker
actually conjectured that this statement is true integrally.) Given a projecti*
*ve variety Y
with basepoint y0 2 Y , let Holy0(P1; Y ) denote the space of holomorphic (alge*
*braic) maps
f : P1 ! Y satsfying the basepoint condition f(1) = y0. We refer to this spac*
*e as the
space of based rational maps in Y . In [5] the "group completion" of this space*
* of rational
maps Holy0(P1; Y )+ was defined. This notion of group completion had the proper*
*ty that if
Holy0(P1; Y ) has the structure of a topological monoid, then Holy0(P1; Y )+ is*
* the Quillen
- Segal group completion. In general Holy0(P1; Y )+ was defined to be a space o*
*f limits of
"chains" of rational maps, topologized using Morse theoretic considerations. We*
* refer the
reader to [5] for details. We recall also from that paper the following definit*
*ion.
Definition 8. The space of rational maps in a projective variety Y is said to s*
*tabilize, if
the group completion of the space of rational maps is homotopy equivalent to th*
*e space of
continuous maps,
Holy0(P1; Y )+ ' 2Y:
32 R.L. COHEN AND P. LIMA-FILHO
In [5] criteria for when the rational maps in a projective variety (or symple*
*ctic manifold)
stabilize were discussed and analyzed. In this paper we study the implications *
*in holomor-
phic K -theory of the stability of rational maps in the varieties Hol(X; Grn(CM*
* )), where
X is a smooth projective variety, and Grn(CM ) is the Grassmannian of n - dimen*
*sional
subspaces of CM . (The fact that the space of morphisms from one projective va*
*riety to
another is in turn algebraic is well known. See, for example [10, 9] for discu*
*ssions about
the algebraic structure of morphisms between varieties.) We actually study rati*
*onal maps
in Hol(X; Z x BU), which is a colimit of projective varieties. In fact we will *
*study rational
maps in the group completion Hol(X; Z x BU)+ by which we mean the group complet*
*ion
of the relative morphism space,
Hol*(P1; Hol(X; Z x BU)+ ) = Hol(P1 x X; 1 x X; Z x BU)+ :
Theorem 34. Let X be a smooth projective variety. Then the space of rational *
*maps in the
group completed morphism space Hol(X; Z x BU)+ stabilizes if and only if the ho*
*lomorphic
K - theory space Khol(X) satisfies Bott periodicity:
Khol(X) ' 2Khol(X):
Proof.The space of rational maps in the morphism space Hol(X; Z x BU)+ stabiliz*
*es if
and only if the group completion of its space of rational maps is the two fold *
*loop space,
(5.1) Hol*(P1; Hol(X; Z x BU))+ ' 2(Hol(X; Z x BU)+ ):
But by definition, the left hand side is equal to Hol(P1x X; 1 x X; Z x BU)+ = *
*Khol(P1x
X; 1 x X): But by Rowland's theorem [22] or by the more general "projective bun*
*dle
theorem" proved in [9] we know that the Bott map
fi : Khol(X) ! Khol(P1 x X; 1 x X)
is a homotopy equivalence. Combining this with property 5.1, we have that the *
*space of
rational maps in the morphism space Hol(X; ZxBU)+ stabilizes if and only if the*
* following
composition is a homotopy equivalence
B : Khol(X) - -fi-! Khol(P1 x X; 1 x X)
(5.2) '
= Hol*(P1; Hol(X; Z x BU)+ )- --! 2Hol(X; Z x BU)+ = 2Khol(X):
*
* __
*
*|__|
HOLOMORPHIC K -THEORY 33
By applying homotopy groups, the Bott map (5.2) B : Khol(X) ! 2Khol(X) define*
*s a
homomorphism
B* : K-qhol(X) ! K-q-2hol(X)
Let b 2 K-2hol(point) be the image under B*of the unit 1 2 K0hol(point): Clearl*
*y this class
lifts the Bott class in topological K - theory, b 2 K-2top(point). Observe fur*
*ther that B* :
K-qhol(X) ! K-q-2hol(X) is given by multiplication by the Bott class b 2 K-2hol*
*(point), using
the module structure of K*hol(X) over the ring K*hol(point). The homomorphism *
*B* :
K-qhol(X) ! K-q-2hol(X) was studied in [9] and it was conjectured there that if*
* K*hol(X)[1_b]
denotes the localization of K*hol(X) obtained by inverting the Bott class, then*
* one obtains
topological K - theory. We now prove the following rational version of this con*
*jecture.
Theorem 35. Let X be a smooth projective variety. Then the map from holomorph*
*ic K -
theory to topological K - theory fi : Khol(X) ! Ktop(X) induces an isomorphism
~=
fi* : K*hol(X)[1=b] -Q--! K*top(X) Q:
Proof.Consider the Chern character defined on the K-2hol(point) Q
ch : K-2hol(point) Q ! kLkH2k-2(point) Q:
Now the morphic cohomology of a point is equal to the usual cohomology of a poi*
*nt,
LkH2k-2(point) = H2k-2(point), so this group is non zero if and only if k = 1. *
* So the
Chern character gives an isomorphism
~=
ch : K-2hol(point) -Q--! L1H0(point) Q ~=Q:
Let s 2 L1H0(point) Q be the Chern character of the Bott class, s = ch(b). Sin*
*ce the
Chern character is an isomorphism, s 2 L1H0(point) Q ~=Q is a generator. We us*
*e this
notation for the following reason.
Recall the operation in morphic cohomology S : LkHq(X) ! Lk+1Hq(X) defined in*
* [7].
Using the fact that L*H*(X) is a module over L*H*(point) (using the "join" mult*
*iplication
in morphic cohomology), then this operation is given by multiplication by a gen*
*erator of
L1H0(point) = Z. Therefore up to a rational multiple, this operation on rationa*
*l morphic
cohomology, S : LkHq(X)Q ! Lk+1Hq(X)Q, is given by multiplication by the element
s = ch(b) 2 L1H0(point) Q.
34 R.L. COHEN AND P. LIMA-FILHO
In [7] it was shown that the natural map from morphic cohomology to singular *
*cohomol-
ogy OE : LkHq(X) ! Hq(X) makes the following diagram commute:
LkHq(X) - -S-! Lk+1Hq(X)
? ?
(5.3) OE?y ?yOE
Hq(X) - -=-! Hq(X):
It also follows from the "Poincare duality theorem" proved in [8] that if X i*
*s an n -
dimensional smooth variety, then LsHq(X) = Hq(X) for s n. Furthermore for k <*
* n
OE : LkHq(X) ! Hq(X) factors as the composition
(5.4) OE : LkHq(X) --S-! Lk+1Hq(X) --S-! . . .-S--!LnHq(X) = Hq(X)
Let L*Hq(X)[1=S] denote the localization of L*Hq(X) obtained by inverting the*
* trans-
formation S : L*Hq(X) ! L*+1Hq(X). Specifically
L*Hq(X)[1=S] = lim-!{L*Hq(X) --S-! L*+1Hq(X) - -S-! . .}.
Then (5.3) and (5.4) imply we have an isomorphism with singular cohomology,
~=
(5.5) OE : L*Hq(X)[1=S]---! Hq(X):
Again, since rationally the S operation is, up to multiplication by a nonzero r*
*ational number,
given by multiplication by s 2 L1H0(point) Q, we can all conclude that when ra*
*tional
morphic cohomology is localized by inverting s, we have an isomorphism with sin*
*gular
rational cohomology,
~=
(5.6) OE : L*Hq(X; Q)[1=s]---! Hq(X; Q):
Now since the Chern character isomorphism ch : K-qhol(X) Q ! 1k=0LkH2k-q(X) *
* Q
is an isomorphism of rings, then the following diagram commutes:
K-qhol(X) Q - .b--! K-q-2hol(X) Q
? ?
(5.7) ch?y~= ~=?ych
L 1 k 2k-q L 1 k+1 2k-q
k=0L H (X) Q - --!.s k=0L H (X) Q:
where the top horizontal map is multiplication by the Bott class b 2 K-2hol(poi*
*nt), and the
bottom horizontal map is multiplication by s = ch(b) 2 L1H0(point) Q.
HOLOMORPHIC K -THEORY 35
Moreover since the Chern character in holomorphic K - theory and and that for*
* topo-
logical K - theory are compatible, this means we get a commutative diagram:
K-qhol(X)[1=b] Q - -fi-! K-qtop(X) Q
? ?
ch?y ?ych
L 1 * 2k-q L 1 2k-q
k=0 L H (X; Q)[1=s]- --!OE k=0H (X; Q):
By (5.6) we know that the bottom horizontal map is an isomorphism. Moreover by *
*Theorem
33 the left hand vertical map is an isomorphism. Of course the right hand verti*
*cal map is
also a rational isomorphsim. Hence the top horizontal map is a rational isomorp*
*hism,
~= -q
fi* : K-qhol(X)[1=b]--Q-! Ktop(X) Q:
*
* __
*
*|__|
In most of the calculations of Khol(X) done so far we have seen examples of w*
*hen
Khol(X) ~= Ktop(X). In particular in these examples the holomorphic K - theory*
* is pe-
riodic, K*hol(X) ~=K*hol(X)[1_b]. As we have seen from Theorem 35, these two co*
*nditions are
rationally equivalent. We end by using the above results to give a necessary co*
*ndition for
the holomorphic K - theory to be Bott periodic, and use it to describe examples*
* where peri-
odicity fails, and therefore provide examples that have distinct holomorphic an*
*d topological
K - theories.
Theorem 36. Let X be a smooth projective variety. Then if K*hol(X) Q ~=K*hol*
*(X)[1_b]
(or equivalently K*hol(X)Q ~=K*top(X)Q), then in the Hodge filtration of its co*
*homology
we have
Hk;k(X; C) ~=H2k(X; C)
for every k 0.
Proof.Consider the commutative diagram involving the total Chern character
K0hol(X) C - fi*--! K0top(X) C
? ?
(5.8) ch?y ?ych
L k 2k OE L 2k
k0 L H (X) C - --! k0 H (X; C)
By Theorem 35, if K*hol(X) is Bott periodic, then the top horizontal map fi :*
* K0hol(X)
C ! K0top(X) C is an isomorphism. But by theorem 27 we know that the two verti*
*cal
36 R.L. COHEN AND P. LIMA-FILHO
maps in this diagram are isomorphisms. Thus if K*hol(X) is Bott periodic, then *
*the bottom
horizontal map in this diagram is an isomorphism. That is,
OE : LkH2k(X) C -! H2k(X; C)
is an isomorphism, for every k 0. But as is shown in [7], LkH2k(X) ~=Ak(X), w*
*here
Ak(X) is the space of algebraic k - cycles in X up to algebraic (or homological*
*) equivalence.
Moreover the image of OE : LkH2k(X) C ! H2k(X; C) is the image of the natural *
*map
induced by including algebraic cycles in all cycles, Ak C ! H2k(X; C), which l*
*ies in
Hodge filtration Hk;k(X; C) H2k(X; C). Thus OE : LkH2k(X) C ! H2k(X; C) is an
isomorphism implies that the composition
Ak(X) C ! Hk;k(X; C) H2k(X; C)
*
* __
is an isomorphism. In particular this means that Hk;k(X; C) = H2k(X; C). *
* |__|
We end by noting that for a flag manifold X, we know by Theorem 5 that K0hol(*
*X) ~=
K0top(X), and indeed Hp;p(X; C) ~=H2p(X; C). However in general this theorem t*
*ells us
that if have a variety X having nonzero Hp;q(X; C) for some p 6= q, then K*hol(*
*X) is not
Bott periodic, and in particular is distinct from topological K - theory. Certa*
*inly abelian
varieties of dimension 2 are examples of such varieties.
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Dept. of Mathematics, Stanford University, Stanford, California 94305
E-mail address, Cohen: ralph@math.stanford.edu
Department of Mathematics, Texas A&M University, College Station, Texas
E-mail address, Lima-Filho: plfilho@math.tamu.edu