AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER RALPH L. COHEN AND PAULO LIMA-FILHO Contents 1. Introduction * * 2 2. Symmetric Products and Symmetrized Grassmannnians 6 2.1. Grassmannians and their symmetrization * * 7 2.2. Infinite symmetric products of P(C1 ) * * 12 3. Rationalizations and the Chern Character * * 15 3.1. The case X = {pt} * *18 3.2. The case X arbitrary * * 21 4. Chow varieties and Chern Classes * * 26 4.1. Algebraic cycles on P(C1 )and Chern classes; the case X = {pt} * * 26 4.2. The case X arbitrary * * 30 5. Relations between the Chern character and Chern classes * * 32 5.1. Rationalization of cycle spaces * * 32 5.2. Exponential maps, the case X = {pt}. * * 35 5.3. Exponential maps, the case X arbitrary. * * 37 Appendix A. Group completions of morphism spaces * *39 References 42 ____________ 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 1. Introduction In this paper we provide a presentation, in the level of classifying spaces, * *of the Chern character and its relation to Chern classes. Although this is apparently a clas* *sical and well- known subject, our constructions have the important feature of preserving the a* *lgebraic geometric nature of the objects involved. The first manifestation of this lies * *in the fact is that all spaces and maps involved are colimits of directed systems in the categ* *ory VarC of projective algebraic varieties and algebraic maps. We will then explain the rel* *evance of this fact for the study of the morphic cohomology introduced by E. Friedlander and B* *. Lawson in [FL92 ] and of holomorhic K-theory performed by Cohen and Lima-Filho in [CLF* * ]. We expect that the constructions made here, as well as in [CLF ], can be extended * *to a broader context, such as [Fri97], once appropriate facts in motivic rational homotopy t* *heory are in place. The constructions of classifying spaces made, together with their rationaliza* *tions, involve three different algebraic geometric moduli spaces: symmetric products of projec* *tive spaces n SP q(P(Cn)), Grassmannians Grassq(C_ n Cq) and Chow varieties Chow qdP(C_ Cq * * of projective spaces. Here we use C_n to denote the dual of Cn. A fundamental link* * between these spaces are the quotient varieties Grassq(C_ n Cq)=Sq, consisting of Grass* *mannians modulo the natural action of the symmetric group Sq. These symmetrized Grassman* *nians also appear in the study of Conformal Field Theory. The departing point is the observation, made in Section 2, that the natural d* *irect sum map P(Cn)xq = Grass1(C_ n)xq ! Grassq(C_ n Cq) descends to an algebraic map SP q(P(Cn))! Grassq(C_ n Cq)=Sq. These maps fit into a directed system of algeb* *raic maps and varieties and induce a map f : SP1 (P(C1 ))! BU =S1 between the corresponding colimits. Section 2 is mostly devoted to setting up * *the di- rected systems, understanding various "algebraic geometric" filtrations giving * *the topology of SP1 (P(C1 ))and BU =S1 . The infinite symmetric product SP1 (P(C1 ))has a c* *lassical structure of an abelian topological monoid whose addition is an algebraic map, * *and we show in Proposition 2.8 that so is BU =S1 . We conclude Section 2 by proving i* *n Proposi- tion 2.15 that f : SP1 (P(C1 ))! BU =S1 is a monoid morphism and a rational ho* *motopy equivalence. Our guiding principle is to see colimits of directed systems of type {M }2 ,* * with M an algebraic variety, as representing functors from the category of algebraic v* *arieties to the category of topological spaces. In particular, whenever they come equipped with* * algebraic AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 3 maps making the colimit M1 = lim-!M into a topological monoid, then M1 repre* *sents a functor X 7! Mor(X; M1 ) from the category of algebraic varieties to the cat* *egory of topological monoids. In particular, all objects and maps in Section 2 must be * *seen as representing, respectively, functors and natural transformations in this contex* *t. We start Section 3 by introducing a method of constructing the Q-localization* * of an abelian topological monoid in the category of projective varieties. In particul* *ar we present natural directed systems in Var C to produce rationalization maps rP : SP 1(P(C* *1 )) ! SP 1(P(C1 ))Q and rB : BU ! {BU =S1 }Q, and think of the latter as a model for* * BUQ in view of Proposition 2.8. In these Q-local topological monoids, multiplication b* *y an integer is an invertible algebraic map. This localization scheme then can be used to di* *scuss monoids of the type Mor(X; M1 ) as above, along with their group completions Mor(X; M* *1 )+ ; cf. Appendix A. In particular, we discuss the canonical splittings of Mor(X; SP1 (* *Pn))+ and their rationalizations. We then study the functors represented by the constructions of Section 2 and * *their ratio- nalization. First we consider the case X = {pt}. We use the fact, shown in Prop* *osition 2.8, that the projection ae : BU ! BU =S1 is a rational equivalence, to identify H** *(BU ; Q) with H*(BU =S1 ; Q) via ae*. Then we denote by i2j the image in rational cohomolog* *y of the integral class represented by the composition Y1 SP1 (P(C1 ))-'! K(Z; 2j) --! K(Z; 2j); sp j=1 prj where sp is the canonical splitting equivalence presented in [FL92 ], and prjis* * the projection. The first main result is the following. Theorem 3.7. Let f : SP 1 (P(C1 )) ! BU=S1 be the homomorphism of Proposi- tion 2.15, and identify H*(BU=S1 ; Q) with H*(BU; Q) via the projection ae : * *BU ! BU=S1 . Then, for j 1 one has f*( j! chj) = i2j; where chj is the 2j-th component of the universal Chern character ch 2 H*(BU; Q* *). Therefore, it follows from Theorem 3.7, each Chern character can be realized * *by algebraic maps. Alternatively, these localizations are natural and so they give rise to a* * homomorphism 4 R.L. COHEN AND P. LIMA-FILHO fQ : SP1 (P(C1 ))Q! BUQ which makes the square in following diagram commute. _____________________________________* *_________________________________________________________________________@ ch ___________________BU____________________* *_________________________________________________________________________@ _____________________________________________* *_________________________________________________________________________@ ___________________________ae||__________________* *_________________________________________________________________________@ ________________________________fflffl|_____________* *_____________________________________________________________________f ________________________________________aeB___________* *________________________________________________SP1(P(C1/))/_BU=S1 _______________________________________________________* *________ ________________________________________________________* *__________|| __________________________________________________________* *_______________________rP|rB| __________________________fflffl|fQ ff""________________* *__________________lffl| SP1 (P(C1 ))Qooe__SP 1(P(C1 ))Q ____//_{BU =S1 }Q BUQ kk[W _ c g f-1Q We produce a homotopy equivalence e : SP 1(P(C1 ))Q ! SP 1(P(C1 ))Q and define Q 1 -1 -1 ch : BU ! SP1 (P(C1 ))Q= j=1K(Q; 2j) by ch = e O fQ O ae, where fQ is a hom* *otopy inverse for fQ. The equivalence e is chosen so that the following result holds. Theorem 3.8. Let ch : BU ! SP1 (P(C1 )) be the composition ch := e O f-1QO aeB * *. Then ch represents the Chern character. In other words, ch*(i2j) = ch2j2 H2j(BU; Q): We then proceed to study the case X arbitrary. Here we must initially unders* *tand the functors represented by our constructions. We first observe that the funct* *or X ! Mor(X; SP1 (P(C1 )))+ represents the morphic cohomology of X. More precisely, Y ssi( Mor(X; SP1 (P(C1 )))+ ) ~= LjH2j-i(X); j where LpHn(X) are the morphic cohomology groups introduced by Friedlander and L* *awson in [FL92 ]; see (36). Then we introduce the spaces "Khol(X) := Mor(X; BU)+ , c* *alled the (reduced) holomorphic K-theory space of X, and define the (reduced) holomorphic* * K-theory groups of X as K"-ihol(X) := ssi(K"hol(X)); cf. Definition 3.14. These groups are also studied by Friedlander and Walker in* * [FW99 ]. The main result states that the algebraic maps between the classifying spaces u* *nder consid- eration still induce uniquely determined homotopy class of maps between the cor* *responding represented objects. In particular, we prove the following. Theorem 3.16. Let X be a projective variety. The natural maps chX : Mor(X; BU* *)+ ! Mor(X; SP1 (P(C1 )))+Qinduce natural homomorphisms Y chiX: "K-ihol(X) ! LjH2j-i(X)Q j0 AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 5 from the holomorphic K-theory groups of X to its rational morphic cohomology. * *These homomorphism fit into a commutative diagram K"-ihol(X) ---! "K-itop(X) ? ? chiX?y ?ychiX Q j 2j-i Q 2j-i j0 L H (X)Q ---! j0 H (X; Q); where the right vertical arrow is the usual Chern character from topological K-* *theory to ordinary cohomology category, cf. Theorem 3.8, and the horizontal arrows are gi* *ven by the usual forgetful functors. n In Section 4 we discuss the Chow varieties Cqn;d= Chow qdP(C_ Cq) paramet* *rizing effective algebraic cycles in projective spaces, and their relation to the Cher* *n classes in the present context. We start considering the case X = {pt}, and this is essen* *tially a survey of fundamental results of Lawson [Law89 ], Lawson and Michelsohn [LM88 ]* *, Boyer, Lawson, Lima-Filho, Mann and Michelsohn [BLLF+ 93 ] and Lima-Filho [LF99 ]. The* *se form a directed system {Cqn;d; tqn;(d;e); ffl(q;k)d;n; sq(n;m);d}, whose colimit C h* *as a canonical splitting Q 1 homotopy equivalence C ' j=1K(Z; 2j); cf. [Law89 ]. An important fact is th* *at this equivalence comes from an equivalence between C and SP 1(P(C1 )) given by Lawso* *n's complex suspension theorem [Law89 ], hence one can use C as a classifying space* * for morphic cohomology in the category of varieties; cf. [FL92 ]. There is a natural map c * *: BU ! C which represents the total Chern class [LM88 ], and C has an infinite loop spac* *e structure so that c is an infinite loop space map. In the case X 6= {pt}, a subtle issue related to the group-completion functor* * requires the smoothness of X in order to define a Chern class map cX : Mor(X; BU)+ ! Mor(X; C11;*)+1: This map induces higher Chern class maps Y ciX : K"-ihol(X) ! LpH2p-i(X); p1 cf. Definition 4.12. In Section 5 we discuss relations between the Chern classes and the Chern cha* *racter. The starting point is the natural averaging map avq : Cqn;d! Cqn;dq!, coming from t* *he evident action of the symmetric group Sq. These maps assembles to give a natural ration* *alization 6 R.L. COHEN AND P. LIMA-FILHO map av1 : C ! CQ fitting into a commutative diagram BU --c-! C ? ? ae?y ?yav1 Q 1 SP 1(P(C1 ))Q ---! {BU =S1 }---! C Q' j=1K(Q; 2j): f fl1 In the case of X = {pt} we observe that the composition fl1 O f represents a * *cohomology Q 1 class R 2 j=1H2j(SP 1(P(C1 )); Q), whose identification is rather clear. Con* *sider the polynomial with rational coefficients Rj(Y1; : :;:Yj) which expresses the j-th * *elementary symmetric function ej as a polynomial ej = Rj(p1; . .;.pj) on the Newton power * *functions pk; k = 1; : :;:j; cf. [Mac79 ]. Then we have the paper. Q Proposition 5.9. Let Rj 2 H2j( k1 K(Z; 2k); Q) be the j-th component of R. Th* *en Rj = Rj(i2; : :;:i2j), where Rj is the universal polynomial in (60) and i2k is * *the rational fundamental class (20). When X is smooth we then apply the previous discussion to the universal case,* * obtaining corresponding relations between the Chern classes and Characters; cf. Theorem 5* *.10 and Corollary 5.11. In an Appendix we discuss generalities of group-completions of * *morphisms spaces and set-up the appropriate machinery to deal with the constructions deve* *loped along the paper. The invariants studied in this paper, and the relations between them are furt* *her studied in the forthcoming paper [CLF ]. Acknowledgements: The second author would like to thank Jon McCammond for useful references. 2. Symmetric Products and Symmetrized Grassmannnians In this section we present the basic results relating Grassmannians to symmet* *ric products of projective spaces. Since the objects we study fit into various directed syst* *ems of algebraic varieties, we first make some general considerations about such systems. Consider a directed system {Y }2 of projective algebraic varieties and morp* *hisms. Although the colimit lim-!Y does not exist in the category of algebraic varieti* *es, it still represents a functor from the category of projective varieties to spaces. More * *precisely, let X be a projective algebraic variety, and let {Y }2 be a directed system as ab* *ove. We denote by Map(X; Y ) the space of continuous maps for the analytic topology fr* *om X to AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 7 Y , endowed with the compact-open topology. The set of algebraic morphisms Mor(* *X; Y ) can be topologized as a closed subspace of Map(X; Y ). Definition 2.1. Given X and {Y }2 as above, define the morphism space from X * *to lim-!Y as Mor(X; Y1 ) := lim-!Mor(X; Y ); where the latter colimit is taken in* * the cate- gory of spaces. If {X } and {Y } are two such directed systems, define Mor(X1 * *; Y ) := lim-Mor(X ; Y1 ): Remark 2.2. Let Y1andenote Mor(pt; Y1 ), i.e., the topological colimit of the * *Y 's with their analytic topology. Suppose one has an operation * : Y1anx Y1an! Y1anwhi* *ch gives Y1anthe structure of a topological monoid and such that the restrictions *|Y xY* * are induced by compatible morphisms of varieties Y x Y ! YOE(;), for all and . Then, for* * each projective variety X the operation * naturally induces an structure of topologi* *cal monoid on Mor(X; Y1 ) via pointwise multiplication. 2.1. Grassmannians and their symmetrization. Here we study actions of the sym- metric group on the Grassmannians. Let Sq be the symmetric group on q letters a* *nd let C_ n denote the dual of Cn. The permutation representation of Sq on Cq induces * *a repre- sentation on C_n Cq, where Sq acts trivially on C_n . We write the canonical b* *asis of Cn as B = {e1; e2; : :e:n}, and for each J = {j1 < . .<.jk} {1; : :;:n} we let (1) eJ : Ck ,! Cn be the linear embedding sending ei to eji, for i = 1; : :;:k, whose image is de* *noted by CJ Cn. Definition 2.3. a. Given q; n 2 N, let Gr qn= Gr q(C_ n Cq) denote the Grassmannian of subspace* *s of codimension q in C_ n Cq equipped with the base point lnq= e?1 Cq, where e?1is * *the annihilator of e1 in C_n. The Grassmannian Gr1n= Gr1(C_ n) is the projective sp* *ace P(Cn) of 1-dimensional subspaces of Cn. b. The action of Sq on C_n Cq induces an algebraic action on Grqn. In particula* *r, the orbit space Grqn=Sq is a projective algebraic variety; cf. [Har92 , p. 127]. Let aenq* *: Gr qn! Grqn=Sq denote the corresponding quotient map, which is also an algebraic map. Note tha* *t the base point lnq2 Grqnis fixed under Sq, and give Grqn=Sq the base point `nq= aenq(lnq* *). 8 R.L. COHEN AND P. LIMA-FILHO Using the identification C_ n Cq+q0= (C_ n Cq) (C_ n Cq0), the direct sum of subspaces induces a based algebraic map 0 q+q0 (2) : Gr qnx Grqn! Grn (`; `0)7! ` `0: Let (3) q;q0: Sq x Sq0! Sq+q0 denote the usual inclusion, where Sq permutes the first q letters and Sq0permut* *es the last q0 letters. The particular inclusion q;1which keeps the last letter fixed * *is denoted by q : Sq ! Sq+1. Proposition 2.4. The direct sum operation is equivariant, in the sense that * *if oe 2 Sq and o 2 Sq0, then (oe * `) (o * `0) = q;q0(oe; o) * (` `0): In particular, it* * induces a based morphism between the respective quotient varieties 0 q+q0 (4) ? : Grqn=Sq x Grqn=Sq0! Grn =Sq+q0; making the following diagram commute: 0 q+q0 Gr qnx Grqn ---! Grn ? ? aenqxaenq0?y ?yaenq+q0 0 q+q0 Grqn=Sq x Grqn=Sq0---! Gr n =Sq+q0: ? Furthermore, the induced maps ? are commutative and associative, in the obvious* * sense. * * __ Proof.Evident. * *|__| We now describe classical stabilizations of the Grassmannians, which are equi* *variant for the symmetric group action, and whose presentation is necessary for our book ke* *eping. Given integers n and q k, denote by (5) fflq;kn: Gr qn! Grkn the inclusion which sends ` 2 Grqnto ` lnk-q; cf. (2). In order to define the second stabilization map, consider n m and J = {1; * *: :;:n} {1; : :;:m}, and let e_J: C_m ! C_n be the adjoint of the map eJ, defined in * *(1). Then, the surjection e_J 1 : C_m Cq ! C_n Cq defines a pull-back map (6) sqn;m: Gr qn! Grqm; AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 9 by sending ` C_n Cq to (e_J 1)-1(`) C_m Cq. Properties 2.5. Let q k and n m be positive integers. Then a. The maps fflq;knand sqn;mare algebraic embeddings and satisfy: skn;mO fflq;kn= fflq;kmO sqn;m: See diagram (8) below. b. The maps fflq;knand sqn;mare equivariant in the sense that, if oe 2 Sq then sqn;m(oe * `) = oe * (sqn;m(`)) and fflq;kn(oe * `) = q;k-q(oe; * *e) * fflq;kn; where e 2 Sk-q is the identity element; cf. (3). Since the stabilization maps ffl*;**and s**;*are equivariant, they descend t* *o morphisms of the respective quotient varieties, (7) fflq;kn: Gr qn=Sq ! Grkn=Sk and sqn;m: Gr qn=Sq ! Grqm=Sq: Definition 2.6. The colimit of the directed system (Gr qn; fflq;kn; sqn;m) of a* *lgebraic varieties is our model for BU, the classifying space for stable K-theory. This colimit comes* * with two filtrations by closed subspaces: . . .BU(n) BU(n+1) . . .BU and . . .BU(q) BU(q + 1) . . .BU: The space BU(n)is defined as the colimit lim-!Grqn, and each inclusion sn;m : * * BU(n),! q BU(m) is a cofibration and a homotopy equivalence. In particular, the inclusio* *ns sm : BU(m) ,! BU are homotopy equivalences. The space BU(q) is the classifying spac* *e of q-plane bundles, defined as the colimit lim-!Grqn. The induced maps fflq;k : BU* *(q) ,! BU(k) n are cofibrations which induce isomorphism in cohomology up to order 2q. Remark 2.7. After passing to quotients, one also has a directed system (Gr qn=S* *q; fflq;kn; sqn;m) whose colimit, denoted by BU=S1 , is called the symmetrized BU. The filtrations* * of BU described above descend, under the quotient map ae : BU ! BU=S1 , to filtrati* *ons of BU=S1 by cofibrations . . .BU(n)=S1 BU(n+1)=S1 . . .BU=S1 and . . .BU(q)=Sq BU(q + 1)=Sq+1 . . .BU=S1 : 10 R.L. COHEN AND P. LIMA-FILHO All this information can be condensed in the following commutative diagram, whe* *re the spaces in the bottom row and in the right column are colimits of their respecti* *ve column and row. fflq;kn fflkn (8) Grqn=Sq ______//_Grkn=Sk_____//BU(n)=S1 sqn;m|| skn;m|| |sn;m| fflffl|fflmq;k fflffl|fflkm fflffl| Grqm=Sq ______//_Grkm=Sk_____//BU(m)=S1 sqm|| skm|| |sm| fflffl|fflq;k fflffl|fflk fflffl| BU(q)=Sq ____//_BU(k)=Sk______//BU=S1 Recall that the (homotopy theoretic) group completion M+ , of a topological m* *onoid M, is the space BM of loops in its classifying space. See Appendix A for further d* *etails. Proposition 2.8. a: The operations ?, defined in Proposition 2.4, induce the structure of a grad* *ed abelian topological monoid on the coproduct Q(n)= qq0 Grqn=Sq, for all n. Furthermor* *e, the natural inclusions Q(n) Q(m), induced by the maps sqn;m, are abelian monoid mor* *phisms. b: The same structure assembles to make BU=S1 an abelian topological monoid, i* *n such a way that each BU(n)=S1 is a closed submonoid. c: The monoid BU(n)=S1 is homotopy equivalent to the connected component of th* *e group completion of Q(n). Similarly, the monoid BU=S1 is homotopy equivalent to the * *connected component of the group completion of Q = qq0 BU(q)=Sq: d: The natural projection ae : (BU; ) ! (BU=S1 ; ?) is a rational homotopy equi* *valence and a morphism of infinite loop spaces. Proof.Assertions a and b follow from routine verification. To prove assertion c, first recall that the connected component of the group * *completion BM of a topological monoid M is homotopy equivalent to the colimit lim-!Mff, wh* *ere the ff2 Mff's are connected components of M, and is a collection contained countably i* *nfinitely many copies of each element in ss0(M); cf. [Fri91]. This argument implies the* * following lemma. AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 11 Lemma 2.9. Let M be an abelian topological monoid, equipped with a continuous* * monoid augmentation OE : M ! Z+ onto the additive monoid of the non-negative integers,* * and let M+ be its group completion; cf. Appendix A. Define Md := -1(d) and chose 1 2* * M1. Then the colimit M1 := lim-!Mdof the system Md +1--!Md+1 given by translation * *by 1 is d also an abelian topological monoid. Furthermore, OE induces a continuous monoid* * morphism : M+ ! Z and M+0:= -1(0) is homotopy equivalent to the group-completion M+1. In our case ss0(Q(n)) = Z+ = ss0(Q), and assertion c follows from the above c* *onsiderations. To prove assertion d, we shall use the following result. Lemma 2.10. Let G be a finite group acting on a smooth, connected projective * *variety X, and let ae : X ! X=G be the quotient map. Then ae induces an isomorphism ae* ** : H*(X=G; Q) ! H*(X; Q)G , where G denote the invariants of the cohomology of X u* *nder the action of G. Furthermore, if X is simply connected and the fixed point set* * XG is non-empty, then X=G is simply connected. Proof.The first part of the theorem is well-known and follows from standard tra* *nsfer argu- ments. Consider a fixed point x 2 XG and denote __x= ae(x) 2 X=G. It follows fr* *om [Ver80] that one can find an equivariant triangulation of X in which x is a vertex, and* * such the quotient ae : X ! X=G becomes a simplicial map for the quotient triangulation o* *f X=G. In particular, given any n-simplex oe X=G, there is a simplex "oe X such that the* * restricion of ae to "oeis a homeomorphism onto oe. Standard arguments show that any loop in X=G based on __xis homotopic to a si* *mplicial loop fl : [0; 1] ! X=G. Therefore, there is a partition 0 = t0 < t1 < . .<.tn =* * 1 of [0; 1] such that fl|[ti-1;ti]maps [ti-1; ti] onto a 1-simplex oeiof the triangulation * *in such a way that O fl becomes a homeomorphism from (ti-1; ti) onto the open simplex oei. For each * *i = 1; : :;:n let "oeibe a lift of oei, and let ri : [ti-1; ti] ! X be a reparametrization of* * its characteristic map by the interval [ti-1; ti]. Since ae(r2(t1)) = fl(t1) = ae(r1(t1)) one can find g1 2 G such that g1* r2(t* *1) = r1(t1), and hence the path "fl: [0; t2] ! X defined as 8 < r1(t); t 2 [0; t1] "fl(t) = : g1 * r2(t);t 2 [t1; t2] is a lifting of fl|[0;t2]. One then proceeds inductively to produce a lifting * *"flof fl. Since fl(0) = fl(1) = __xand ae-1(__x) = {x} because x is a fixed point of the G-acti* *on, it follows that * * __ "flis a loop based on x. The result follows. * * |__| 12 R.L. COHEN AND P. LIMA-FILHO Since the action of Sq on Grqnis induced by the natural representation Sq GL(q* *; C) of the general linear group on C_n Cq, one concludes that Sq acts trivially on the coh* *omology of Gr qn. The previous lemma then implies that aeqn : Gr qn! Grqn=Sq induces an is* *omorphism between the rational cohomology of two simply-connected spaces. Hence, aeqnis a* * rational homotopy equivalence. It follows that ae is also a rational homotopy equivalenc* *e. Since ae is compatible with the direct sum operation on BU, one concludes that it is a map * *of infinite loop spaces, once we give BU=S1 the infinite loop space structure coming from t* *he abelian * * __ topological monoid structure. We leave the details to the reader. * * |__| Results such as Lemma 2.10 appear in the study of discrete transformation gro* *ups. See for example [Rat94 , Theorem 13.1.7]. Remark 2.11. Using Remark 2.2 one sees that, given a projective algebraic varie* *ty X, the morphism space Mor(X; BU=S1 ) has the structure of an abelian topological mono* *id. It then follows that the assignment X 7! Mor(X; BU=S1 ) defines a contravariant f* *unctor from projective varieties to abelian topological monoids. This can be seen as a* * presheaf of topological monoids on the site of projective algebraic varieties over C. 2.2. Infinite symmetric products of P(C1 ). Consider the projective space P(Cn) := Gr1(C_ n)of lines in Cn, and let SPq(P* *(Cn) ):= P(Cn) xq=Sq be the q-fold symmetric product of P(Cn), with natural projection (9) tnd : P(Cn) xq! SPq(P(Cn) ): The points in SPq(P(Cn) )are denoted by a1 + . .+.aq; ai 2 P(Cn), and its base * *point is q . ln1; cf. Definition 2.3. In a similar fashion to Grqn and Grqn=Sq, given q k and n m we consider* * two stabilizing maps (10) inq;k: SP q(P(Cn) )! SPk(P(Cn) ) and jn;mq: SP q(P(Cn) )! SPq(P(Cm )); defined by inq;k(oe) = oe +(k -q)ln1, and jn;mq(oe) = (s1n;m)*(oe). The latter * *map is the natural map of symmetric products induced by the inclusion s1n;mof P(Cn) as a linear su* *bspace in P(Cm ), see (6). The maps inq;kand jn;mqare algebraic embeddings which satisfy jn;mkO inq;k= imq;kO jn;mq: Definition 2.12. The colimit SP 1(P(C1 )) of the directed system (SP q(P(Cn);)i* *nq;k; jn;mq) is the infinite symmetric product of P(C1 ). This space has two filtrations by * *closed sub- spaces: . . .SP1(P(Cn) ) SP1 (P(Cn+1) ) . . .SP1(P(C1 )) AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 13 and . . .SPq(P(C1 )) SPq+1(P(C1 )) . . .SP1(P(C1 )): The subspace SP1 (P(Cn) )is defined as the colimit lim-!SPq(P(Cn),)and each ind* *uced map q jn;m : SP1 (P(Cn) ),! SP 1(P(Cm ))is a cofibration which induces an isomorphi* *sm on the homotopy groups up to order 2n. The subspace SPq(P(C1 )) is defined as the * *colimit lim-!SPq(P(Cn) )and the induced maps iq;k : SP q(P(C1 )),! SPk(P(C1 ))are cofib* *rations n which induce isomorphisms in homology up to dimension q. Proposition 2.13. The usual addition operation + between symmetric products as* *semble to give SP1 (P(C1 ))the structure of an abelian topological monoid, in such a w* *ay that each SP 1(P(Cn) )is a closed submonoid. Observe that the direct sum induces a map (11) Fnq : P(Cn) xq= {Gr 1n}xq ! Gr qn which is equivariant for the action of the symmetric group Sq, and hence induce* *s an alge- braic map fqnbetween the respective quotients making the following diagram comm* *ute. qn q P(Cn)xq --F-! Gr n ? ? (12) tnd?y ?yaeqn SPq(P(Cn) )---! Gr qn=Sq fqn Remark 2.14. Let OEi : P(Cn)xq ! P(Cn) denote the projection onto the i-th fact* *or, and let O(1) be the hyperplane line bundle over P(Cn) . It is a simple geometric f* *act that (Fqn)*(Qqn) = OE*1O(1). .O.E*qO(1), where Qqndenotes the universal quotient q-p* *lane bundle over Grqn. Proposition 2.15. a: The maps fqnform a morphism of directed systems of algebraic varieties f** : (SP q(P(Cn);)i**;*; j*;**) ! (Gr qn=Sq; ffl*;**; s**;*); so that the induced map of colimits f : SP 1 (P(C1 ))! BU=S1 is a morphism of* * abelian topological monoids. b: The map f preserves both filtrations of SP1 (P(C1 ))and BU=S1 . More precise* *ly, f(SP 1(P(Cn) )) f(BU(n)=S1 ) and f(SP q(P(C1 ))) f(BU(q)=Sq): 14 R.L. COHEN AND P. LIMA-FILHO Furthermore, the restriction fq : SP q(P(C1 ))! BU(q)=Sq is a rational homotop* *y equiv- alence for each q. Hence, f is a rational homotopy equivalence. Proof.Part a follows from a simple routine verification. See diagram (13) below. The filtration preserving property follows from the construction of the maps.* * It follows from Lemma 2.10 that pull-back under tnqgives an isomorphism between H*(SP d(P(* *Cn);)Q) and the invariants H*(P(Cn) xd; Q)Sq under the action of Sq. On the other hand, it is well-known that Fqn also induces an isomorphism (Fqn* *)* : ~= xd Hk(Gr qn; Q) -! Hk(P(Cn) ; Q)Sq, whenever q(n - 1) > k; cf. [MS74 ]. Using* * the iso- morphisms ~= k q S k q Hk(Gr qn=Sq; Q) ---! H (Gr n; Q) q ~=H (Gr n; Q); (aeqn)* exhibited in the proof of Proposition 2.8, together with aeqnOFnq= fqnOtnq, one* * concludes that fqninduces an isomorphism in the k-th rational cohomology groups, for q(n - 1) * *> k. To conclude the proof, first note that a simple inverse limit argument shows that * *fq induces an isomorphism in rational cohomology. Then, Lemma 2.10 implies that both SPq(P(C1* * ))and * * __ BU(q)=Sq are simply-connected, and hence that fq is a rational homotopy equival* *ence. |__| The following commutative diagram summarizes all the filtrations, colimits and * *maps in- volved in Proposition 2.15. (13) iq;kn ikn SP q(P(Cn))________________//_SPk(P(Cn))______________//SP1(P(Cn)) fqnrrrrr| fknrrrrr | fnrrrrr || xxrrrrr |eqn;m| xxrrrrr |ekn;m| rrrr || q | fflq;kn//k | fflkn// 1 xxr |en;m Gr n=Sq ____________________Grn=Sk ___________________Gr n=S1 | | | | | | | | |skn;m |sn;m | | fflffl iq;km || fflffl ikm | fflff* *l| sqn;m|| SP q(P(Cm ))_______|_______//_SPk(P(Cm ))_____||______//SP1(P(Cm * *)) | q rr | m rr | rr | | fm rrr | | fkrrr | | fmrrr | | rrr |eq | rrr |ek | rrr | fflffl|xxrr ||m fflq;k fflffl|xxrr ||m fflk fflffl|xxrr | Grqm=Sq _________________m_//Grkm=Sk________________m_//Gr1m=S1 |em| | | | | | | |sk |sm | || fflffl iq;k ||m fflffl ik || fflff* *l| sqm| SPq(P(C1 )) |_______//SPk(P(C1 )) |_______//SP1(P(C1 * *)) | q rr | k qq | rr | frrr | f qqq | frrr | rrr | qqq | rrr fflffl|xxrr fflq;k fflffl|xxqq fflk fflffl|xxrr BU(q)=Sq _________________//BU(k)=Sk _________________//_BU=S1 AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 15 Remark 2.16. Using the same arguments of Remark 2.11 one sees that given a proj* *ective variety X, the morphism space Mor(X; SP1 (P(C1 ))) is an abelian topological m* *onoid. Furthermore, the map f : SP1 (P(C1 ))! BU=S1 induces a morphism of abelian to* *po- logical monoids f* : Mor(X; SP1 (P(C1 ))) ! Mor(X; BU=S1 ). 3. Rationalizations and the Chern Character In this section we present a simple and concrete realization of the Chern cha* *racter in the level of classifying spaces, and explain the algebraic geometric nature of this* * realization. We first describe come general facts concerning the localization at 0 of certai* *n specific class of monoids, a process which we call the rationalization of the monoid. Consider an abelian topological monoid M, whose topology is given by an incre* *asing family . . .Mn Mn+1 . . .M of closed submonoids. Given n m, let in;m : Mn ,! Mm denote the inclusion homomorphism, and define (14) n;m : Mn ! Mm m! oe7! (___) in;m( oe ): n! Observe that the map rn : Mn ! Mn, given by multiplication by n!, makes the fol* *lowing diagram commute Mn --rn-! Mn ? ? (15) in;m?y ?yn;m Mm ---! Mm : rm It follows that the colimit MQ of the system {Mn; n;m}, has a natural structu* *re of abelian topological monoid, so that the map rM : M ! MQ induced by the rn's * *is a continuous monoid morphism. Proposition 3.1. Let M be an abelian topological monoid with a filtration as ab* *ove, where each in;m is a cofibration. Then MQ is a Q-local abelian topological monoid, in* * that mul- tiplication by any integer xm : MQ ! MQ is invertible. Furthermore, the morp* *hism rM : M ! MQ is a rational homotopy equivalence which represents the localiza* *tion at 0 of the space M in the homotopy category. 16 R.L. COHEN AND P. LIMA-FILHO Proof.It is clear that rM induces an isomorphism rM* : ssk(M) Q ! ssk(MQ), f* *or all k. * * __ Since an abelian topological monoid is simple, the result follows. * * |__| Remark 3.2. Consider the case where each member of the family { Mn } has the pr* *operty described in Remark 2.2. In other words, that each Mn has the form Ynan;1, wher* *e {Yn; } is a directed system of projective varieties. Then the rationalization MQ has t* *he structure of a colimit of algebraic varieties and the following properties hold. 1. If X is a projective variety, and M satisfies the property above, then Mo* *r(X; M) is a well-defined topological monoid (cf. Definition 2.1) whose topology is g* *iven by the increasing sequence of closed submonoids . . .Mor(X; Mn) Mor(X; Mn+1) . * *... 2. Let f : M ! N be a monoid morphism where both M and N satisfy the hypothe* *sis of the proposition above. If f is filtration preserving, i.e. f(Mn) Nn fo* *r all n, then it induces a monoid morphism fQ : MQ ! NQ so that fQ O rN = rM O f. If X is a projective algebraic variety, and M satisfies the property describe* *d in the remark above, then Mor(X; MQ) is well-defined, cf. Definition 2.1, and it is id* *entified with Mor(X; M)Q. Corollary 3.3. Let X be a projective algebraic variety, and let M satisfy the p* *roperty described in Remark 3.2. Then the group completions Mor(X; MQ)+ and ( Mor(X; M* *)Q)+ coincide. Let us recall the canonical splitting of SP1 (P(Cn)) introduced by Steenrod a* *nd subse- quently used by Friedlander and Lawson in [FL92 ]. The constructions rely on the classical identification Pn = SPn(P1). First, c* *hoose x0 = [1 : 0] 2 P1 as a basepoint. Then for n q, the canonical coordinate plane incl* *usion Pn Pq can be identified with the map (16) in;q: Pn = SPn(P1)-! Pq = SPq(P1) oe7-! oe + (q - n)x0: Now, given n q, define a morphism rq;n: Pq ! SP (qn)(Pn)by sending a1 + . .+.a* *q 2 P SP q(P1) Pq to |I|=n{ai1+ . .+.ain} 2 SP(qn)(SP n(P1)) SP(qn)(Pn): This morph* *ism, in turn, naturally induces a map rq;n: SP1 (Pq) ! SP1 (SP (qn)(Pn)): AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 17 One has an evident "trace map" (see [FL92 , Proposition 7.1]) tr : SP 1(SP (q* *n)(Pn)) ! SP 1(Pn) defined as the extension to the free monoid SP1 (SP (qn)(Pn)) of the n* *atural inclu- sion SP(qn)(Pn),! SP1 (Pn). This map can be used to define a monoid morphism aeq;n: SP1 (Pq) ! SP1 (Pn) rq;n n tr n as the composition SP 1(Pq) --! SP 1(SP (qn)(P))-! SP1 (P ). Note that aen;n* * is the identity map. Given a projective algebraic variety X, the maps iq;n and aeq;rinduce monoid * *mor- phisms iq;n*: Mor(X; SP1 (Pn)) ! Mor(X; SP1 (Pq)) and aeq;n*: Mor(X; SP1 (Pq* *)) ! Mor(X; SP1 (Pn)); which in turn induce maps between their respective group-comp* *letions iq;n*: Mor(X; SP1 (Pn))+ ! Mor(X; SP1 (Pq))+ and aeq;n*: Mor(X; SP1 (Pq))+ ! Mor(X; SP1 (Pn))+ : Definition 3.4. Given a projective variety X define Mor(X; SP1 (S2n))+ := Mor(X; SP1 (Pn))+ == Mor(X; SP1 (Pn-1))+ ; for n 1, where the latter denotes the homotopy quotient. Denote by n : Mor(X; SP1 (Pn))+ ! Mor(X; SP1 (S2n))+ the natural homotopy quotient map, and let qq;n: Mor(X; SP1 (Pq))+ ! Mor(X; SP1 (S2n))+ denote the composition n O aeq;n*. It is shown in [FL92 ] that the map Yq (17) q : Mor(X; SP1 (Pq))+ -! Mor(X; SP1 (S2j))+ j=1 Q q defined as q := j=1qj;q, is a homotopy equivalence. Definition 3.5. Let X be an algebraic variety. The colimit of the maps q is den* *oted by 1Y spX : Mor(X; SP1 (P(C1 )))+ -! Mor(X; SP1 (S2j))+ : j=1 This map is a homotopy equivalence and is functorial on X. 18 R.L. COHEN AND P. LIMA-FILHO 3.1. The case X = {pt}. Considering X = pt in the discussion above, one obtains* * the canonical splitting of SP1 (P(Cn)) Y (18) spn : SP 1 (P(Cn)) ! SP 1(S2k); 1kn-1 since the monoids in question are already group-complete. In this case, the map* *s spn are homotopy equivalences which are also monoid morphisms compatible with both incl* *usions Q Q SP 1(P(Cn)) SP1 (P(Cn+1)) and 1kn-1 SP 1(S2k) 1kn SP 1(S2k): There- fore, they induce a canonical filtration-preserving splitting homomorphism Y (19) sp : SP 1 (P(C1 )) ! SP1 (S2k): k1 We use SP1 (S2j) as our model for the Eilenberg-MacLane space K(Z; 2j) (cf. [* *DT56 ]), and denote by -2j2 H2j(SP 1(P(C1 )); Z) the class represented by the composition Y prj (20) SP 1(P(C1 )) sp-! SP 1 (S2k) --! SP1 (S2j); k1 where prj is the projection. Let i2j2 H2j(SP 1(P(C1 )); Q) denote the image of * *-2j under the coefficient homomorphism ffl* : H2j(SP 1(P(C1 )); Z) ! H2j(SP 1(P(C1 )); Q)* * induced by the canonical inclusion ffl : Z ,! Q. Q Remark 3.6. We use the notation H*(X; R) to denote the product j1 Hj(X; R), f* *or any coefficient ring R. Theorem 3.7. Let f : SP1 (P(C1 )) ! BU=S1 be the homomorphism of Propositi* *on 2.15, and identify H*(BU=S1 ; Q) with H*(BU; Q) via the projection ae : BU ! BU* *=S1 . Then, for j 1 one has f*( j! chj) = i2j; where chj is the 2j-th component of the total Chern character ch 2 H*(BU; Q). Proof.Let _nq: SPq(P(Cn) )! SP1 (P(C1 ))denote the natural map, defined as the * *com- position en O inq; cf. diagram (13). It suffices to show that (_nq)*(i2j) = (* *_nq)*(f*(j!chj)) for all q; n j + 1. Since tnq: P(Cn) x . .x.P(Cn) ! SPq(P(Cn) )induces an inje* *ction in rational cohomology, we will then show that (tnq)*O (_nq)*(i2j) = (tnq)*O (_nq)* **O f*(j!chj), for all q; n j + 1. Let x = c1(O(1)) 2 H2(P(Cn) ; Q) be the generator of the cohomology ring of P* *(Cn), and let hr = [P(Cr+1)] 2 H2r(P(Cn) ; Q), r = 1; : :;:n - 1, be the fundamental * *class of a coordinate r-plane, the Kronecker dual to xr. Define xi2 H2(P(Cn) xq; Q) as xi=* * OE*i(x), where OEi: P(Cn)xq ! P(Cn) denotes the i-th projection. Given a partition r1+ .* * .+.rq = j AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 19 where j n - 1, let hr1 . . .hrq be the associated generator of H2j(P(Cn) xq; Q* *), dual to xr11. .x.rqq. Given integers j n, one has a commutative diagram: Q j-1 P(Cj)xq --t-! SP q(P(Cj)) --i-! SP 1(P(Cj)) ~= s=1SP1 (S2s) ?? ? ? y T ?ye0 ?ye Q n-1 P(Cn) xq ---! SP q(P(Cn) )---! SP 1(P(Cn) )~= s=1SP1 (S2s) ---! SP 1(S2j); t0 i0 p where t = tjq; t0= tnq; i = ijq; i0= inq; e0= ej;nq; e = ej;n; following the no* *tation of diagram (13), and where T denotes the natural inclusion, and p is the projection. Consider a partition r1 + . .+.rq = j. If some ri is strictly less than j, t* *hen hr1 . . .hrq= T*('), where ' 2 H2j(P(Cj)xq ; Q). In this case, if 2j is the canonic* *al class of SP 1(S2j)= K(Z; 2j); and <.; .> denotes the Kronecker pairing, then = (21) = = = 0; where the last equality follows from the fact that p O e = *. On the other hand* *, it follows from the construction of the splittings spn that = 1; and hence (22) (tnq)* O (jnq)*(i2j) = xj1+ . .+.xjq: By definition, (23) xj1+ . .+.xjq= OE*1(x)j + . .+.OE*q(x)j = c1(OE*1O(1)))j + . .+.c1(OE*q* *(O(1)))j * * = j!chj OE1(O(1)) . . .OEq(O(1)): It follows from (2.14) that (24) OE*1(O(1)) . . .OE*q(O(1)) = (Fnq)*(Qqn); where Qqnis the universal quotient q-plane bundle over Gqn. Combining (22), (23* *) and (24), one gets (25) (tnq)*(jnq)*(i2j) = j!chj((Fnq)*(Qq)) = j!(Fnq)*(chj(Qqn)): 20 R.L. COHEN AND P. LIMA-FILHO Write chj(Qqn) = (aeqn)*(fflqn)*s*n(chj): Chasing diagram (13) one obtains (26) j!(Fnq)*(chj(Qqn))= j!(Fnq)*(aeqn)*(fflqn)*s*n(chj)) = j!(tnq)*(fqn)*(fflqn)*s*n(chj) = j!(tnq)*(inq)*e*nf*(chj) = (tqn)*(jnq)*f*(j!chj): * * __ This concludes the proof. * * |__| In order to obtain the actual Chern character, we apply Proposition 3.1 to ou* *r specific situation. First, define aeB : BU ! {BU=S1 }Q as the composition BU ae-!BU=S1 -* *rB! {BU=S1 }Q; where rB is the rationalization map described in Proposition 3.1. S* *ince ae is a rational homotopy equivalence, according to Proposition 2.8, we use aeB :* * BU ! {BU=S1 }Q as our model for the rationalization of BU. Then observe that the hom* *omor- phisms n : SP 1 (S2n) ! SP1 (S2n), which sends oe 2 SP1 (S2n) to n!oe, induce a* * filtration preserving endomorphism of the (weak) product Y Y (27) : SP1 (S2n) ! SP 1(S2n): n1 n1 It follows from Remark 3.2 that descends to an endomorphism Q of the rationa* *lization Q 2j j1 SP 1(S )Q which is easily seen to be a homotopy equivalence. Choose homotopy inverses f-1Q, -1Qand sp-1Qand define (28) e : SP1 (P(C1 ))Q ! SP1 (P(C1 ))Q as e := sp-1QO -1QO spQ. These fit into the following diagram whose solid arro* *ws form a commutative diagram, and which becomes homotopy commutative after including t* *he dashed ones. AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 21 (29) ae f 1 sp Q * *2k BU0___//________BU=S1oo___________SP 1(P(C ))____________________//k1 SP1 (S* * ) __00______________________________________|| ooo77o || ___00______________________________| || ooooo | ____00________________________________||| ooo | ____00aeB0_________________________________________rB|rP|QSP(S2k) |rS ______00_________________________________________||||k11| Q || _______00___________________________________|| -1 | | ________00___________________________________||fQ |rS | _______0________________________________fflffl|fflffl||--dca`_^][Zsp fflff* *l| _____________________oo_______ 1 _________|________//_Q Q 2k _______________________{BU=S1 }QSP1(P(C ))Q | k1 SP 1(S * * )Q _________________________fQ q | o77 ________________ e qqqq | Q oooo ch _________________________________________________q|qq oooo - __________________________________________________________________ffl* *ffl|xxqqq-1uutoooo _______..__________________________________________________________* *_________________________________________________________________________@ ll[Z] ^ _ ` a c sp-1Q The following result is a simple corollary of the previous constructions and * *is the desired presentation of the Chern character. Theorem 3.8. Let ch : BU ! SP1 (P(C1 )) be the composition ch := e O f-1QO ae* *B . Then ch represents the Chern character. In other words, ch*(i2j) = ch2j2 H2j(BU; Q): Proof.Just observe that the construction of implies that *(-2j) = j!-2j, and h* *ence * * __ e*(i2j) = 1_j!i2j. The result now follows from Theorem 3.7. * * |__| 3.2. The case X arbitrary. We now describe the algebraic-geometric nature of ou* *r con- struction of the Chern character. More precisely, we show that the homotopy inv* *erses f-1Q, -1Qand sp-1 yield uniquely determined homotopy class of maps between group-comp* *leted morphism spaces. Observe that, if M is any space in Diagram 29 then it represen* *ts a func- tor X 7! Mor(X; M)+ from the category of varieties to the category of spaces, a* *s one sees directly from Definition 2.1 and Remark 2.2, using the functoriality of the gro* *up completion functor. We will show that the dashed arrows induce natural transformations bet* *ween the corresponding functors, after passage to the homotopy category. Consider an algebraic variety X. The splitting map spX , introduced in Defin* *ition 3.5, is a filtration preserving morphism of topological monoids. Therefore, it induc* *es a natural monoid morphism Y (30) spXQ : Mor(X; SP1 (P(C1 )))+Q! Mor(X; SP1 (S2j))+Q; j1 according to Remark 3.2(2). Note that this is still a homotopy equivalence. 22 R.L. COHEN AND P. LIMA-FILHO Now, let Y Y (31) X : Mor(X; SP1 (S2j))+ ! Mor(X; SP1 (S2j))+ j1 j1 denote the map induced by the map , defined in (27). In other words, X ({fj}) =* * {j!fj}. In a similar fashion to the case X = pt, one sees that X induces a filtration * *preserving Q endomorphism of j1 Mor(X; SP1 (S2j))+ , and hence it induces an endomorphis* *m of the rationalized monoid Y Y (32) XQ : Mor(X; SP1 (S2j))+Q! Mor(X; SP1 (S2j))+Q: j1 j1 Since the monoid is 0-local, this is a homotopy equivalence which is natural on* * X. Definition 3.9. Given an algebraic variety X, define eX : Mor(X; SP1 (P(C1 )))+Q! Mor(X; SP1 (P(C1 )))+Q i j-1 i j -1 as the unique homotopy class of maps given by eX = spXQ O XQ O spXQ: Consider a (smooth) generalized flag variety F , i.e. a compact homogeneous s* *pace of the form F = G=P , where is a complex algebraic group and P < G is a parabolic subg* *roup. It follows from the duality results in [FL92 ] and the computations in [LF92 ] tha* *t the forgetful functor Mor(F; SP1 (P(C1 )))+ ! Map(F; SP1 (P(C1 )))+ is a homotopy equivalence. As a consequence one has the following. Proposition 3.10. If G is a finite group of automorphisms of a generalized fla* *g variety F , then the forgetful functor Mor(F=G; SP1 (P(C1 )))+Q! Map(F=G; SP1 (P(C1 )* *))+Qis a homotopy equivalence. Proof.The argument is standard. The projection F ! F=G gives a morphism F=G ! SP |G|(F )which in turn it induces a "transfer map" Mor(F; SP1 (P(C1 )))+Q! Mor(F=G; SP1 (P(C1 )))+Q which is easily seen to be a homotopy equivalence. The same applies one one re* *places Mor(; ) by Map(; ) in the construction. The observation preceding the propositi* *on completes * * __ the argument. * *|__| AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 23 Corollary 3.11. The forgetful maps Mor({BU=S1 }Q; SP1 (P(C1 )))+Q! Map({BU=S1 }Q; SP1 (P(C1 )))+Q and Mor({BU=S1 }Q x {BU=S1 }Q; SP1(P(C1 )))+Q! Map({BU=S1 }Q x {BU=S1 }Q; SP1(P(C* *1 )))+Q are homotopy equivalences. Proof.One just needs to observe that these maps are induced by morphisms of inv* *erse systems whose components are maps of the type Mor(F=G; SP1 (P(C1 )))+Q! Map(F=G; SP1 (P(C1 )))+Q; where F are generalized flag varieties and G is a finite group. Furthermore, al* *l spaces are * * __ 0-local. The result follows. * * |__| Throughout the rest of this section we use the notation B := {BU=S1 }Q and S * *:= SP 1(P(C1 ))Q. Let fQ : S ! B denote the rationalization of the map introduc* *ed in Proposition 2.15. This is a monoid morphism which belongs to Mor(S; B). Sinc* *e fQ is also a homotopy equivalence, there is a homotopy inverse f-1Q 2 Map(B; S). * * The corollary above show that the forgetful functor Mor(B; S)+ ! Map(B; S)+ is a ho* *motopy equivalence, and hence there is a unique element [ff] 2 ss0 Mor(B; S)+ which ma* *ps onto [f-1Q] 2 ss0 Map(B; S)+ . Let ff 2 Mor(B; S)+ be a representative for [ff]. Lemma 3.12. The element ff 2 Mor(B; S)+ described above is a homotopy homomo* *r- phism in the sense of Definition A.4. Proof.We know that fQ is a monoid morphism, hence B O (fQ x fQ) = fQ O S. There* *fore, f-1QO B ~ S O (f-1Qx f-1Q) as maps. In other words, [f-1QO B ] = [S O (fQ x fQ* *)] 2 ss0( Map(B; S)+ ). This is equivalent to say that (33) [(f-1Q)BxB*(B )] = [BxBS;+(fQ x fQ)]; in the language of Appendix A. We now use Proposition A.2(b) with Mor(; ) replaced by Map(; ), cf. Remark * *A.7), to conclude that ffBxB* is homotopic to (f-1Q)BxB*. This together with (33) and th* *e definition of ff gives the equalities [ffBxB*(B )] = [(f-1Q)BxB*(B )] = [BxBS;+(fQ x fQ)] = [BxBS;+(ff x ff* *)] * * __ of elements in ss0 Map(B x B; S)+ . The result now follows from Proposition A.5* *. |__| 24 R.L. COHEN AND P. LIMA-FILHO We denote a representative for [ff] by f-1Q2 Mor({BU=S1 }Q; SP1 (P(C1 )))+ . * *It follows from Proposition A.6 that f-1Qinduces an H-space map (34) (f-1Q)X* : Mor(X; {BU=S1 }Q) ! Mor(X; SP1 (P(C1 )))+ ; and this assignment is functorial on X. Theorem 3.13. The map (f-1Q)X* induces a unique homotopy class of maps (f-1Q)* *X+ : Mor(X; {BU=S1 }Q)+ ! Mor(X; SP1 (P(C1 )))+ ; such that (f-1Q)X+O u = (f-1Q* *)X*, where u : Mor(X; {BU=S1 }Q) ! Mor(X; {BU=S1 }Q)+ is the canonical map from the monoid into its group-completion. Proof.Denote M = Mor(X; {BU=S1 }Q), N = Mor(X; {BU=S1 }Q) and ff = (f-1Q)X*. The multiplicative system ss0(M) of the Pontrjagin ring H*(M) is sent by ff to * *the multi- plicative subgroup ss0(N+ ) of the units of H*(N+ ). Recall that H*(M+ ) is iso* *morphic to the localization H*(M)[ss0(M)]-1; cf. [Q ]. Therefore, there is a unique ring h* *omomorphism ff+ : H*(M+ ) ! H*(N+ ) satisfying ff+ O u* = ff*. Since both M+ and N+ are * *0-local abelian topological monoids, they are a product of rational Eilenberg-MacLane s* *paces and the homomorphism ff+ determines a unique homotopy class of maps satisfying the * *desired * * __ property. * * |__| Following the same steps as in the case X = pt, given an algebraic variety X,* * we define the map chX : Mor(X; BU)+ ! Mor(X; SP1 (P(C1 )))+Qas the composition (35) chX = eX O (f-1Q)X*O aeXB;*; where eX is introduced in Definition 3.9, and aeXB;*: Mor(X; BU)+ ! Mor(X; {BU=* *S1 }Q)+ is the map induced by the projection aeB . See Diagram 29. Let us explain the significance of these constructions. Given a projective al* *gebraic variety X, the homotopy groups (36) LjH2j-i(X) := ssi( Mor(X; SP1 (S2j))+ ) were introduced in [FL92 ] and are called the morphic cohomology groups of X. Definition 3.14. The (reduced) holomorphic K-theory space of X is defined as "K* *hol(X) := Mor(X; BU)+ , and the (reduced) holomorphic K-theory groups of X are defined as* * "K-ihol(X) := ssi(K"hol(X)): Remark 3.15. 1. Our holomorphic K-groups coincides with the "semi-topological* *" K- groups studied in [FW99 ]. We study holomorphic K-theory in greater gener* *ality and in much more depth in [CLF ]. AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 25 2. As shown in [CLF ], "Khol(X) is an infinite loop space that corresponds to* * the zero- th space of the spectrum introduced in [LLFM96 ] and whose basic properti* *es are discussed in [LF99 ]. As a main consequence of the constructions in this section, we obtain the fol* *lowing result. Theorem 3.16. Let X be a projective variety. The natural maps chX : Mor(X; * *BU)+ ! Mor(X; SP1 (P(C1 )))+Qinduce natural homomorphisms Y chiX: "K-ihol(X) ! LjH2j-i(X)Q j0 from the holomorphic K-theory groups of X to its rational morphic cohomology. * *These homomorphism fit into a commutative diagram K"-ihol(X) ---! "K-itop(X) ? ? chiX?y ?ychiX Q j 2j-i Q 2j-i j0 L H (X)Q ---! j0 H (X; Q); where the right vertical arrow is the usual Chern character from topological K-* *theory to ordinary cohomology category, cf. Theorem 3.8, and the horizontal arrows are gi* *ven by the usual forgetful functors. * * __ Proof.The result follows from the naturality of the constructions and the case * *X = {pt}. |__| Remark 3.17. The lower horizontal arrow in the diagram is the "cycle map" from * *the mor- phic cohomology of X to its singular cohomology; cf. [FL92 ]. Definition 3.18. We call the natural maps Y chiX: "K-ihol(X) ! LjH2j-i(X)Q j0 the Chern character from the (reduced) holomorphic K-theory of X to its morphic* * co- homology. Setting X = BU one obtains the the tautological Chern character ele* *ment ch 2 Mor(BU; SP1 (P(C1 )))+Qdefined as ch := chBU (Id), where Id 2 Mor(BU; BU* *)+ is the identity. Remark 3.19. The space Mor(X; SP1 (P(C1 )))+ can be made into a homotopy ring i* *n such a way that the forgetful functor Mor(X; SP1 (P(C1 )))+ ! Map(X; SP1 (P(C1 )))* *+ is a homotopy ring homomorphism when the latter space is given the multiplicative st* *ructure corresponding to the cup product. This will be explained in x4.2, after the dis* *cussion of the join pairing in Theorem 4.9. We will then see that the tensor product of bundle* *s induces 26 R.L. COHEN AND P. LIMA-FILHO a pairing X : "Khol(X) ^ "Khol(X) ! "Khol(X) which makes the Chern character i* *nto a homotopy ring homomorphism; cf. Proposition 4.11. 4. Chow varieties and Chern Classes In this section we consider spaces of algebraic cycles on projective spaces, * *together with their stabilizations and rationalizations. These spaces are natural recipients* * for Chern classes and give another algebraic-geometric model for products of Eilenberg-Ma* *cLane spaces. Most of the results presented here are an adaptation of results from [* *BLLF+ 93 ], [LLFM96 ] and [LF99 ] to the present context. 4.1. Algebraic cycles on P(C1 )and Chern classes; the case X = {pt}. n Definition 4.1. Given n > 0 and q 0, let Cqn;d= Chow qdP(C_ Cq) be the Chow variety consisting of the effective algebraic cycles of codimension q and* * degree d in P(C_ n Cq); cf. [Law95 ]. The formal addition of cycles + : Cqn;dx Cqn;d0! Cqn;d+d0 is an algebraic map which makes Cqn;*:= qd0 Cqn;d, into a graded abelian topolo* *gical monoid, called the Chow monoid of effective cycles of codimension q in P(C_ n Cq). This* * is the free abelian monoid generated by the irreducible subvarieties of codimension q in P(* *C_ n Cq). Remark 4.2. Given a complex vector space V , there is a 1-1 correspondence betw* *een ir- reducible subvarieties Z P(V ) and irreducible cones Cone (Z) V . This corre* *spon- n dence identifies the Chow variety Cqn;1= Chow q1P(C_ Cq) of cycles of degre* *e one in P(C_ n Cq) with the Grassmannian Grqn. Under this identification one obtains a * *natural embedding cq : Grqn,! Cqn;*as a connected component. An important feature of the Chow monoids is the fact that they come equipped * *with an "exterior" bilinear multiplication 0 q+q0 (37) ] : Cqn;dx Cqn;e! Cn;de given by the ruled join of cycles; cf. [Law95 ]. This operation is described as* * follows. Let i : C_ n Cq ! C_ n Cq+q0and j : C_ n Cq0! C_ n Cq+q0be the complementary embeddings induced by the inclusion of Cq into Cq+q0given by the first q coordi* *nates and of Cq0into Cq+q0as the last q0 ones. Consider a pair of subvarieties Z P(C_ n * *Cq) and W P(C_ n Cq0). One defines the subvariety Z]W P(C_ n Cq+q0) as the union of a* *ll projective lines in P(C_ n Cq+q0) which join points in i(Z) to points in j(W ).* * One extends ] to arbitrary cycles by linearity. AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 27 The join satisfies the following properties: Facts 4.3. a. The join is a strictly associative operation. b. Its restriction to the connected components yields an algebraic map 0 q+q0 (38) ] : Cqn;dx Cqn;e! Cn;de; cf. [Pl"u97] and [Bar91]. c. In the particular case of cycles of degree one (d = d0= 1), the join coincid* *es with the usual direct sum operation 0 q+q0 (39) : Grqnx Grqn! Grn ; cf. Remark 4.2. We now proceed to introduce three algebraic maps (40) tqn;(d;e): Cqn;d! Cqn;e; (41) ffl(q;k)n;d: Cqn;d! Ckn;d; (42) sq(n;m);d: Cqn;d! Cqm;d; which will define a directed system (43) {Cqn;d; tqn;(d;e); ffl(q;k)d;n; sq(n;m);d}: Given d e, we define tqn;(d;e): Cqn;d! Cqn;eby tqn;(d;e)(oe) = oe + (e - d)l* *nq. This uses the additive structure of the Chow monoid. The maps ffl(q;k)n;dand sq(n;m);dare e* *xtensions to higher degrees of the maps fflq;knand sqn;m introduced in (5) and (6), and ar* *e defined as follows. First, given J {1; : :;:k} with |J| = q, let Jc denote its complement. Then * *let shJ 2 Sk denote the shuffle permutation which sends the ordered k-tuple (J; Jc) to (1; :* * :;:k). Under the permutation representation, shJ induces an isomorphism shJ : Ck ! Ck and we* * define eJ : Cq ,! Ck and eJc : Ck-q ,! Ck as the compositions Cq -i!Ck -shJ-!Ck and Ck-q j-!Ck shJ--!Ck. These are the maps defined in (1). Define (44) fflJn;d: Cqn;d! Ckn;d by fflJn;d(oe) = shJ*(oe]lnk-q), where shJ* is the map on cycles induced by the* * shuffle map; see cf. (37). In the particular case where J = {1; : :;:q} {1; : :;:k} we denote f* *flJn;dby ffl(q;k)n;d. 28 R.L. COHEN AND P. LIMA-FILHO Remark 4.4. Observe that the maps eJn;dare all homotopic to e(q;k)n;dfor all ch* *oices of J {1; : :;:k} with |J| = q. In order to define the third stabilization map, we consider n m and J = {1; * *: :;:n} {1; : :;:m}, and let e_J: C_m ! C_n be the adjoint of the map eJ, defined in * *(1). Then, the surjection e_J 1 : C_m Cq ! C_n Cq induces a map (45) sq(n;m);d: Cqn;d! Cqm;d; defined as follows. Given an irreducible subvariety Z P(C_ n Cq) of degree d* *, let sq(n;m);d(Z) P(C_ m Cq) be the irreducible variety of whose cone Cone (sq(n;m* *);d(Z)) C_ m Cq is defined as (e_J 1)-1(Cone (Z)). Then, extend sq(n;m);dlinearly to * *arbitrary cycles. Remark 4.5. One could rephrase the last definition in terms of a suitable join * *operation, and vice-versa. We prefer this approach, for it is a direct generalization of the G* *rassmannians case. Lemma 4.6. Given q k, n m and d e, the following diagram commutes. ffl(q;k)n;d Cqn;d______________//_Ckn;d q - tk - | tn;(d;e)--|- n;(d;e)--|- --- |sq(n;m);d| ""---- | q ""- | //k sk|| Cn;e_______________Cn;e |(n;m);d | ffl(q;|k)n;e | | | | | | | | |sk | | fflffl|ffl|((n;m);eqfflffl|;k)m;d sq(n;m);e|| Cqm;d_______||_____//_Ckm;d | - | - | --- | --- | -- q | --- k fflffl|tm;(d;e)""-- fflffl|tm;(d;e)""- Cqm;e______________//_Ckm;e ffl(q;k)m;e * * __ Proof.This is just a careful diagram chase using the definitions. * * |__| Definition 4.7. The colimit C := lim-!Cqn;dof the directed system {Cqn;d; tqn;(* *d;e); ffl(q;k)d;n; sq(n;m);d} q;n;d is the "stabilized" Chow variety of effective cycles cycles in P(C_ 1 C1 ). Remark 4.8. We may fix some of the parameters q; n; d and let the other(s) go t* *o infinity, obtaining intermediate spaces. The corresponding notation will use the symbol 1* * whenever appropriate, to denote the colimits, and maps between them. For example, by fix* *ing one AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 29 of the parameters q, n or d, we have the spaces Cq1;1, C1n;1and C11;dwhen the c* *olimit is taken over the remaining two parameters. This gives three filtrations of C by c* *ofibrations: (46) : : :C11;d C11;d+1 . . .C; (47) : : :Cq1;1 Cq+11;1 . . .C; and (48) : : :C1n;1 C1n+1;1 . . .C: We have seen that Cqn;*:= qd0 Cqn;dis an abelian topological monoid and that * *sq(n;m);d and ffl(q;k)n;dare both monoid morphisms. Therefore, the colimit C11;*:= qd0 C1* *1;dis also an abelian topological monoid with a continuous augmentation OE : C11;*. In [LM88 * *], C11;dis denoted by Dd, and C is denoted by D1 . The following theorem summarizes various results proven in [Law89 ], [LM88 ],* * [FL92 ] and [BLLF+ 93 ]. The presentation here is chosen to provide a parallel with the ana* *logous results in the previous presentation of symmetric products and Grassmannians. Theorem 4.9. a. Addition of cycles gives C the structure of an abelian topological monoid. * *With this structure, Ciis homotopyjequivalent to the connected component of the group com* *pletion + (C11;*)+ = qd0 C11;d . b. Each monomorphism Cqn;1 Cqn+1;1is a homotopy equivalence, and there are comp* *atible Q q canonical splittings Cqn;1' j=1K(Z; 2j), fitting in a commutative diagram " q Cqn;1O______________________//Cn+1;1 LLL qq LLL qqqq LLL qqq QLL&& xxqq q j=1K(Z; 2j) Q q c. Each monomorphism Cqn;1 Cq+1n;1is homotopic to the inclusion of Cqn;1' j=1* *K(Z; 2j) Q q+1 as a factor in Cq+1n;1' j=1K(Z; 2j). Q q d. The natural inclusions Grqn Cqn;1,! Cqn;1' j=1K(Z; 2j) stabilize as n ! 1 * *to give Q q the truncated total Chern class cq : BU(q) ! Cq1;1' j=1K(Z; 2j). Q 1 e. By sending q ! 1 one gets the total Chern class c : BU ! C ' j=1K(Z; 2j). * *This is a bi-filtration preserving map, satisfying c(BU(q)) Cq1;1 C and c(BU(n)) C1n;* *1 C. f. Under the join operation ], the monoid C11;*= qd0 C11;dbecomes an E1 -ring s* *pace. + Therefore the group completion C11;* has an E1 -ring augmentation induced by * *the degree 30 R.L. COHEN AND P. LIMA-FILHO + of cycles : C11;* ! Z. Furthermore, the connected component of 1 in the gr* *oup + completion C11;*1 := -1(1) , which can be identified with C, has a "multiplic* *ative" infinite loop space structure for which the total Chern class map c : (BU; ) ! * *(C; ]) is a map of infinite loop spaces. Proof.Assertion a is proven in [Fri91], and the arguments in the proof are outl* *ined in the proof of Proposition 2.8. Assertions b and c follow from Lawson's complex susp* *ension theorem [Law89 ] and the splittings of [FL92 ]. Assertions d and e follow from* * [LM88 ] and a mere inspection of the definitions of the filtrations. The last assertion* * is proven in * * __ [BLLF+ 93 ]. * * |__| 4.2. The case X arbitrary. Let X be a projective algebraic variety. The identification BU(q) = Grq1 = Cq* *1;1gives an inclusion BU(q) ,! qd Cq1;d, and letting q go to infinity, one gets a map Mo* *r(X; BU) ! Mor(X; C11;*), where C11;*= qd Cq1;d. Denote by (49) cX : Mor(X; BU) ! Mor(X; C11;*)+ the composition of the map above with the universal map Mor(X; C11;*) ! Mor(X; * *C11;*)+ from Mor(X; C11;*) to its additive group-completion; cf. Appendix A. In [LLFM96 ] and [LF99 ] it is shown that Mor(X; C11;*)+ is an abelian topol* *ogical monoid with a multiplicative action of the linear isometries operad L induced by the j* *oin pairing on algebraic cycles. This gives Mor(X; C11;*)+ the structure of an augmented * *E1 -ring space, in the language of [LLFM96 ], with augmentation : Mor(X; C11;*)+ ! Z.* * De- fine Mor(X; C11;*)+d:= -1(d) and recall that Lemma 2.9 gives a homotopy equiva* *lence Mor(X; C11;*)+0~= Mor(X; C)+ . Hence one has natural equivalences (50) Mor(X; C11;*)+1~= Mor(X; C11;*)+0~= Mor(X; C)+ ; where the former one is given by translation by the element 1 2 Mor(X; C11;*)+1* *, represented by a constant map X ! C11;1: Remark 4.10. Using the complex suspension theorem of [FL92 ], one obtains a can* *onical homotopy equivalence = X : Mor(X; SP1 (P(C1 )))+ ! Mor(X; C)+ ; hence Mor(X; SP1 (P(C1 )))+ becomes a homotopy ring space, with mutiplication # : Mor(X; C)+ x Mor(X; C)+ ! Mor(X; C)+ AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 31 induced by the join of cycles. It follows from [LM88 ] that the forgetful funct* *or Mor(X; SP1 (P(C1 )))+ ! Map(X; SP1 (P(C1 )))+ is a map of ring spaces where the homotopy ring structure in the later is induc* *ed by the cup product. We make a brief digression here to explore this multiplicative structure furt* *her. The tensor product of vector bundles gives an element 2 Mor(BU x BU; BU)+ , which satisfies the following property. Let ch 2 Mor(BU x BU; BU) be the Chern char* *acter element, cf. Definition 3.18, and consider the elements #BUxBUS(ch) and chBUxBU* * () 2 Mor(BU x BU; SP1 (P(C1 ))+Q;). The fact that the topological Chern character i* *s a ring homomorphism implies that these elements represent the same element in ss0( Map* *(BU x BU; SP1 (P(C1 ))+Q; )). Then it follows from Corollary 3.11 that they represen* *t the same element in ss0( Mor(BU x BU; SP1 (P(C1 ))+Q; )). Another application of Proposi* *tion A.6 proves the following. Proposition 4.11. Given a projective algebraic variety X, the Chern character * *map chX : Mor(X; BU)+ ! Mor(X; SP1 (P(C1 )))+Qis a homotopy ring homorphism. In particu- Q i -i lar ch0X : "K0hol(X) ! j0 LjH2j(X)Q is a ring homomorphism and chX : "Khol(X* *) ! Q j 2j-i "0 j0 L H (X)Q is a homomorphism of modules over Khol(X). It is easy to see that cX factors as Mor(X; BU) ! Mor(X; C11;*)+1! Mor(X; C1* *1;*)+ . Furthermore, the map Mor(X; BU) ! Mor(X; C11;*)+1is a map of L-spaces, cf. [L* *F99 ], and hence it induces a map between their respective group completions 1 + (51) cX : Mor(X; BU)+ ! B Mor(X; C1;*)1 : Observe that the latter group-completion is taken with respect to the join pair* *ing structure on Mor(X; C11;*)+1. In [LF99 ] it is shown that, for X smooth, Mor(X; C11;*)+1is group-complete w* *ith respect to the join pairing, hence 1 + (52) Mor(X; C11;*)+1~=B Mor(X; C1;*)1 : Q The argument goes as follows. Consider ss0( Mor(X; C11;*)+1) ~= 1 x p1 LpH2* *p(X), where L*H*(X) denotes the morphic cohomology groups (36). It is shown in [FL92 * *] that whenever X is smooth then LpH2p(X) ~= A2p(X), where the latter denotes the Chow group of algebraic cycles of codimension p modulo algebraic equivalence. Furthe* *rmore, the Q multiplication on 1 x p1 LpH2p(X) induced by the join coincides with the int* *ersection pairing. Hence ss0( Mor(X; C11;*)+1) is a group under the join multiplication. 32 R.L. COHEN AND P. LIMA-FILHO Definition 4.12. Let X be a smooth algebraic variety. Using (51) and (52) we ca* *n con- struct the total Chern class map cX : Mor(X; BU)+ ! Mor(X; C11;*)+1: One can combine the suspension equivalence= X, with the splitting in Definition* * 3.5 and (36), and then take homotopy groups to define the higher Chern class maps Y ciX : K"-ihol(X) ! LpH2p-i(X) p1 from the (reduced) holomorphic K-theory of X to its morphic cohomology. The ind* *ividual components of this map are denoted by cp;iX: "K-ihol(X) ! LpH2p-i(X). Remark 4.13. a:The identification (50) allows one to identify cX : Mor(X; * *BU)+ ! Mor(X; C11;*)+1with a map cX : Mor(X; BU)+ ! Mor(X; C)+ . b: It is shown in [LF99 ] that cX is a map of spectra from the holomorphic * *K-theory spectrum of X to its morphic spectrum, in the terminology of [LLFM96 ]. c:Under the forgetful functor Mor(; ) ! Map(; ) one obtains a commutative d* *iagram i Q "K-ihol(X)cX---!p1 LpH2p-i(X) ?? ? y ?y "K-itop(X)c---!Qp1 H2p-i(X; Z) where "K-itop(X) is the reduced topological K-theory of X and c is the usu* *al Chern class map into singular cohomology. 5.Relations between the Chern character and Chern classes In this section we present a relation between the Chern characters and the Ch* *ern classes contructed in the previous sections. This requires an alternative description o* *f the rational Chern class map, and some new constructions with cycle spaces. 5.1. Rationalization of cycle spaces. We first describe a modification of the d* *irected system (43), aiming at rationalizing C. Adding all maps fflJ1;d(44) together, o* *ne defines a map 0 (53) fflqq0: Cq1;d! Cq+q1; q+q0 ( q )d X oe7! fflJ1;d(oe): J AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 33 Lemma 5.1. The following diagram commutes: fflq1 q+1 Cq1;d - --! C1;(q+1)d ? ? tq1;(d;d+1)?y ?ytq+11;(d(q+1);(d+1)(q+1)) Cq1;d+1- --! Cq+11;(q+1)(d+1): fflq1 Definition 5.2. The maps above generate a directed system {Cq1;d; tq1;(d;e); ff* *lqq0}, whose colimit we denote by CQ. For this system one can also define the additive monoi* *d C11;*Q:= 0 q lim-!qd Cq1;d. Define #av : Cq1;dx Cq1;d0! C q+q0 0by d 1; ( q )dd X oe#avoe0= shJ*(oe#oe0): |J|=q Proposition 5.3. 1.The directed system above has a cofinal subsystem given b* *y the spaces Cq1;q!together with the maps fflq1: Cq1;q!! Cq+11;(q+1)!. 2. The maps #av assemble to give a map of directed systems and satisfy oe#avo* *e0 = oe0#avoe. In particular, both colimits CQ and C11;*Qhave the structure of * *commutative topological rings, whose operations are given by algebraic maps. * * __ Proof.Follows directly from the definitions. * * |__| Theorem 5.4. The colimit CQ is homotopy equivalent to the rationalization of * *C. Proof.The directed system can be visualized with the aid of the following diagr* *am. (54) __ ____ ____ ____ ____ ____ ____ fflq __ _________________________Cq1;d1//_Cq+11;d(q+1)__//________________* *___________________ |tq1;(d;d+1)| |tq+11;(d(q+1);(d+1)(q+1))| fflffl| fflq fflffl| ______________________Cq1;d+11//_Cq+11;(d+1)(q+1)//_______________* *__________________ ___ ____ ____ ____ ____ ____ fflffl____ fflffl__q ________________________Cq1;1ffl//_Cq+11;1______//________________* *___________________ || || || || Q || Q q || Q ______________qj=1K(Z;f2j)fl//_q+1j=1K(Z;_2j)//____________1j=1K(Z* *; 2j) 34 R.L. COHEN AND P. LIMA-FILHO Note that the bottom horizontal arrow Yq q+1Y fflq : Cq1;1= K(Z; 2j) -! Cq+11;1= K(Z; 2j) j=1 j=1 is homotopic to (q + 1) . ffl(q;q+1)1;1; cf. Remark 4.4. A minor modification o* *f the arguments in * * __ Proposition 3.1 ends the proof. * * |__| We now exhibit an explicit "algebraic" rationalization map from C to CQ. The* * con- structions below provide an essential link to the previous constructions with B* *U(q) and symmetric products. The main ingredient here is the "averaging map" (55) avq: Cq1;d! Cq1;d.q!; P i q jSq which sends oe 2 Cq1;dto g2Sqg * oe 2 C1;d.q! . Lemma 5.5. The following diagrams commute: q q Cq1;d________av_______//_C1;d.q! | FFFtq1;(d;d+1) NNtq1;(dq!;(d+1)q!)NN | FFF q NNN | FFF fflq0 NNNN | ##q avq ''q ffl(q;q+1)1;d||C1;d+1____________________//C1;(d+1).q! | | | | | | | | | ffl(q;q+1)| | | fflffl|q+1 | 1;d+1 fflffl| | Cq+1___av___ | //Cq+1 |fflq 1;d | 1;d.(q+1)! | q0 EE | Ntq+1NN | EEE | 1;(d(q+1)!|;(d+1)(q+1)!)NNNN q+1 EE | NNN | t1;(d;d+1)""Efflffl| &&N fflffl| Cq+11;d+1_________________//Cq+11;(d+1).(q+1)! avq+1 and 0 q+q0 Cq1;dx Cq1;d0_#_____//_C1;dd0 | avqxavq0|| avq+q0| fflffl| fflffl| 0 q+q0 Cq1;dq!x Cq1;d0q!#//_C1;dd0(q+q0)! av AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 35 Proof.In the first diagram, the left vertical face commutes by Lemma 4.6, and t* *he right vertical face commutes by Lemma 5.1. The top and bottom faces commute because a* *vq can be seen as an additive endomorphism of Cq1;*which sends l1q to q!l1q. The commu* *tativity * * __ of last diagram follows from an inspection of the definitions. * * |__| It follows from this lemma that the averaging maps, when put together, induce* * an alge- braic map av1 : C ! CQ, from the colimit of the left vertical faces, to the col* *imit of the right vertical faces. Furthermore, this map is a map of E1 -ring spaces, for the mul* *tiplication given by the join of cycles. Q 1 Corollary 5.6. The map av1 gives the rationalization map C ' j=1K(Z; 2j) ! C* *Q ' Q 1 j=1K(Q; 2j) induced by the inclusion Z ! Q. 5.2. Exponential maps, the case X = {pt}. The averaging map (55) defines a morp* *hism (56) avq: BU(q) ! Cq1;q!: satisfying the following properties. Lemma 5.7. The averaging map (56) factors through the quotient BU(q)=Sq, indu* *cing a map flq : BU(q)=Sq ! Cq1;q!, which makes the diagram commute q q BU(q)=Sq -fl--! C1;q! ?? ? (57) y ?yfflq1 BU(q + 1)=Sq+1 ---! Cq+11;(q+1)!: flq+1 Corollary 5.8. The maps flq give, by passage to colimits, commutative diagrams * *of E1 - spaces BU - -c-! C BU --c-! C11;* ?? ? ? ? (58) y ?yav1and ?y ?yav1 BU =S1 - --! CQ; BU =S1 ---! C11;* ; fl1 fl1 Q where av1 is described in Corollary 5.6, and c is described in Theorem 4.9(e).* * Hence, fl1 represents the rational total Chern class and is a rational homotopy equivalenc* *e. In Proposition 2.15 we construct an algebraic map f : SP 1(P(C1 )) ! BU =S1 * *with the property that f*(j!chj) = i2j in rational cohomology. The homotopy class o* *f the 36 R.L. COHEN AND P. LIMA-FILHO composition Y f fl1 Y (59) R : SP 1 (P(C1 ))' K(Z; 2j) -! BU =S1 --! CQ ' K(Q; 2j); j1 j1 in the topological category, has the following evident interpretation as a coho* *mology class Q R 2 H*( j1 K(Z; 2j); Q). Let = 1n=0n denote the ring of symmetric functions p(x1; x2; : :):on infinit* *ely many variables, where n denotes the functions of degree n. Here we follow the notati* *on of [Ful97]. P * * P Let ek = i1<...