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APPEARED IN BULLETIN OF THE

arXiv:math/9204230v1 [math.AG] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 264-268A THEORY OF ALGEBRAIC COCYCLESEric M. Friedlander and H. Blaine Lawson, Jr.Abstract. We introduce the notion of an algebraic cocycle as the algebraic ana-logue of a map to an Eilenberg-MacLane space.

Using these cocycles we develop a“cohomology theory” for complex algebraic varieties. The theory is bigraded, func-torial, and admits Gysin maps.

It carries a natural cup product and a pairing toL-homology. Chern classes of algebraic bundles are defined in the theory.

There is anatural transformation to (singular) integral cohomology theory that preserves cupproducts. Computations in special cases are carried out.

On a smooth variety it isproved that there are algebraic cocycles in each algebraic rational (p, p)-cohomologyclass.In this announcement we present the outlines of a cohomology theory for alge-braic varieties based on a new concept of an algebraic cocycle. Details will appearin [FL].

Our cohomology is a companion to the L-homology theory recently studiedin [L, F, L-F1, L-F2, FM]. This homology is a bigraded theory based directly onthe structure of the space of algebraic cycles.

It admits a natural transformationto integral homology that generalizes the usual map taking a cycle to its homologyclass. Our new cohomology theory is similarly bigraded and based on the struc-ture of the space of algebraic cocycles.

It carries a ring structure coming from thecomplex join (an elementary construction of projective geometry), and it admits anatural transformation Φ to integral cohomology. Chern classes are defined in thetheory and transform under Φ to the usual ones.

Our definition of cohomology isvery far from a duality construction on L-homology. Nonetheless, there is a naturaland geometrically defined Kronecker pairing between our “morphic cohomology”and L-homology.The foundation stone of our theory is the notion of an effective algebraic cocycle,which is of some independent interest.

Roughly speaking, such a cocycle on a varietyX, with values in a projective variety Y , is a morphism from X to the space ofcycles on Y .When X is normal, this is equivalent (by “graphing”) to a cycleon X × Y with equidimensional fibres over X. Such cocycles abound in algebraicgeometry and arise naturally in many circumstances.

The simplest perhaps is thatof a flat morphism f : X →Y whose corresponding cocycle associates to x ∈X, thepullback cycle f −1({x}). Many more arise naturally from synthetic constructionsin projective geometry.

We show that every variety is rich in cocycles. Indeed if1991 Mathematics Subject Classification.

Primary 14F99, 14C05.Key words and phrases. Algebraic cycle, Chow variety, algebraic cocycle, cohomology.Received by the editors April 4, 1991The first author was partially supported by the IHES, NSF Grant DMS 8800657, and NSA GrantMDA904-90-H-4006.

The second author was partially supported by NSF DMS 8602645c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2E. M. FRIEDLANDER AND H. B. LAWSON, JR.X is smooth and projective, then every rational cohomology class that is Poincar´edual to an algebraic cycle is represented by (i.e., is Φ ⊗Q of) an algebraic cocycle.In what follows the word variety will denote a reduced, irreducible, locally closedsubscheme of some complex projective space.

Such a variety is called projective ifit is in fact closed (equivalently, compact) in some projective space.Definition 1. Given a projective variety Y ⊂P n with a fixed embedding, we de-note by Csd(Y ) the algebraic set (the “Chow set”) of effective cycles of codimension-sand degree d with support in Y .

For an arbitrary variety X we then define an ef-fective algebraic cocycle on X with values in Y to be a continuous algebraic mapϕ: X →Csd(Y )(i.e., a morphism ϕ: ˜X →Csd(Y ) from the weak normalization˜X of X). The space of all such cocycles provided with the compact-open topologywill be denoted by Csd(X; Y ).Note that Csd(X; Y ) is a priori a hybrid construction consisting of algebrogeomet-ric objects but carrying the compact-open topology.

However, for normal projectivevarieties X, it has a purely algebraic description.Proposition 2. If X is projective and normal, then the space Csd(X; Y ) admits thestructure of a locally closed, reduced subscheme of some complex projective space.Formal direct sum determines an abelian topological monoid structure on thespacesCs(X; Y )def=ad≥0Csd(X; Y ) .This structure is proved to be independent of the projective embedding chosenfor Y .In analogy with the construction of L-homology, we have the followingdefinition.Definition 3.

Let X and Y be varieties, with Y projective. Denote by Zs(X;Y) thehomotopy theoretic group completion of Cs(X; Y ) (i.e., the loops on the classifyingspace of Cs(X; Y )),Zs(X; Y ) ≡ΩB(Cs(X; Y )) .Then the bivariant morphic cohomology of X with coefficients in Y is defined to bethe homotopy groups of Zs(X; Y ),LsHq(X; Y ) ≡π2s−q(Zs(X; Y )),2s ≥q ≥0 .The first fundamental result concerning these spaces is the Algebraic Suspen-sion Theorem, which asserts that the algebraic suspension maps Csd(X; Y ) →Csd(X;ΣY ), introduced in [L], induce a homotopy equivalenceZs(X; Y )∼=−→Zs(X;ΣY ),and thus an isomorphismLsHq(X; Y ) ∼= LsHq(X;ΣY )for all s and q.

A THEORY OF ALGEBRAIC COCYCLES3Although this theory has been developed in the “bivariant context” of Definition3, we shall focus our attention here on the important special case in which Y issome projective space P N. We know from [L] that ΩB(Cs(P N)) has the homotopytype of a generalized Eilenberg-MacLane spaceΩB(Cs(P N)) ∼= K(Z, 0) × K(Z, 2) × · · · × K(Z, 2s),N ≥s .In particular, this homotopy type is independent of N ≥s. We conclude thatΩB(Cs(P N)) “modulo” ΩB(Cs−1(P N−1)) represents cohomology.

This motivatesthe followingDefinition 4. For any variety X and any s ≥0, let Zs(X) denote the homotopyfibre of the natural map BCs−1(X, P s−1) →BCs(X, P s),Zs(X) ≡htyfib{BCs−1(X, P s−1) →BCs(X, P s)} .For any 0 ≤q ≤2s, the morphic cohomology group LsHq(X) is defined byLsHq(X) ≡π2s−q(Zs(X)) .Definitions 3 and 4 are related as follows.Theorem 5.

For any variety X and any N ≥s ≥0, there is a natural homotopyequivalenceZs(X; P N) ∼= Z0(X) × Z1(X) × · · · × Zs(X) .The splitting asserted in Theorem 5 arises from natural mapsSP ∞(P N) →SP ∞(P k),k ≤Nobtained by viewing P N as SP N(P 1) and P k as SP k(P 1), where SP j(P n) denotesthe j-fold symmetric product of P n.We view morphic cohomology as the theory corresponding to “algebraic” asopposed to “arbitrary continuous” maps from X into Eilenberg-MacLane spaces.This perspective is formalized in the following.Theorem 6. For any variety X, the elementary complex join operation induces anatural ring structure onL·H·(X) ≡Ms≥0LsH·(X) .Furthermore, there is a natural transformation of graded ringsΦ: L·H·(X) →H·(X; Z),and, if X is projective thenΦ(LsH2s−j(X)) ⊗C ⊂Hs,s−j ⊕Hs−1,s−j+1 ⊕· · · ⊕Hs−j,j,where Hp,q denotes the (p, q)th Dolbeault component of Hp+q(X; C).The existence of a ring structure in morphic cohomology provides it with astructure not possessed by L-homology.

On the other hand, the natural operationson L-homology constructed in [FM] via the join operation naturally determineoperations on our morphic cohomology groups.The restriction on the image of Φ given in Theorem 6 is complemented by thefollowing existence result. A key ingredient in its proof is the total Chern class mapof [LM],BUs →Cs(P ∞),which can be viewed geometrically as the inclusion of degree 1 cycles into the spaceof all cycles on P N for N sufficiently large.

4E. M. FRIEDLANDER AND H. B. LAWSON, JR.Theorem 7.

Let E be a vector bundle over X generated by its global sections, overa variety X. Then there are naturally defined chern classesck(E) ∈LkH2k(X)with the property that Φ(ck(E)) ∈H2k(X; Z) is the usual kth chern class of E.Consequently, if X is a smooth projective variety, then the Poincare dual of thefundamental class of each algebraic subvariety lies in the subring of H∗(X; Z) gen-erated Φ(L·H·(X)).Not surprisingly, codimension-1 morphic cohomology is the easiest to compute.We have the following computation.Theorem 8.

Let X be a projective variety. ThenL1Hq(X) =Zif q = 0,H1(X; Z)if q = 1,H1,1(X; Z)if q = 2,0otherwise,where H1,1(X; Z) denotes H2(X; Z) ∩ρ−1H1,1(X; C) and where ρ: H2(X; Z) →H2(X; C) is the coefficient homomorphism.In [F] a result similar to Theorem 8 but only applying to smooth projectivevarieties was proved for L-homology.

This suggests that morphic cohomology andL-homology should satsify some form of duality.One possible candidate for apossible duality pairing is given in the following proposition.Proposition 9. For any variety X, there is a natural Kronecker pairing betweenL-homology and morphic cohomology,LsHq(X) ⊗LrHq(X) →Z,which is naturally compatible with the usual Kronecker pairingHq(X; Z) ⊗Hq(X; Z) →Z .The definition of this pairing is pleasingly geometric.

Namely, given a cycle Won X ×P N and a cycle Z on X, we take the image in H∗(P N; Z) of the fundamentalclass of the restriction of W to Z.References[F]E. Friedlander, Algebraic cycles, Chow varieties, and Lawson homology, Compositio Math.77 (1991), 55–93.[FL]E. Friedlander and H. B. Lawson, A theory of algebraic cocycles, Ann.

of Math. (to appear).[FM]E.

Friedlander and B. Mazur, Filtrations on the homology of algebraic varieties (to appear). [L-F1] P. Lima-Filho, On a homology theory for algebraic varieties, IAS preprint, 1990.

[L-F2], Completions and fibrations for topological monoids and excision for Lawson ho-mology, Compositio Math., (1991).[L]H. B. Lawson, Jr., Algebraic cycles and homotopy theory, Ann.

of Math. (2) 129 (1989),253–291.[LM]H.

B. Lawson, Jr. and M.-L. Michelsohn, Algebraic cycles, Bott periodicity, and the Cherncharacteristic map, The Mathematical Heritage of Herman Weyl, Amer. Math.

Soc., Prov-idence, RI, 1988, pp. 241–264.Department of Mathematics, Northwestern University, Evanston, Illinois 60208Department of Mathematics, State University of New York, Stony Brook, NewYork 11794

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