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. References [1]M.F. Atiyah, Instantons in two and four dimensions, Comm. Math. Phys. 93 4,* * (1984), 437 - 451. [2]F.R Cohen, J.P. May, and L.R. Taylor, Splitting of certain spaces CX, Math.* * Proc. Cambridge Philos. Soc. 84 no. 3, (1978), 465-496. [3]R.L. Cohen, Stable proofs of stable splittings,Math. Proc. Camb. Phil. Soc.* * 88, (1980), 149-151. [4]R.L. Cohen, E. Lupercio, and G.B. Segal, Holomorphic spheres in loop groups* * and Bott periodicity, Asian Journal of Math. , to appear, (1999). [5]R.L. Cohen, J.D.S. Jones, and G.B. Segal, Stability for holomorphic spheres* * and Morse theory, Proc. of Int. Conf. on Topology and Geometry, Aarhus, Denmark, to appear. [6]R.L. Cohen, and P. 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Rowland, Stanford University thesis, to appear (2000). [23]G. Valli, Interpolation theory, loop groups and instantons, J. Reine Angew.* * Math. 446, (1994), 137-163. 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