HIGHER ARITHMETIC CHOW GROUPS

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1 HIGHER RITHMETIC CHOW GROUPS J. I. BURGOS GIL ND E.FELIU bstract. We gve a new constructon of hgher arthmetc Chow groups for quasprojectve arthmetc varetes over a feld. Our defnton agrees wth the hgher arthmetc Chow groups defned by Goncharov for projectve arthmetc varetes over a feld. These groups are the anaue, n the rakelov context, of the hgher algebrac Chow groups defned by Bloch. The degree zero group agrees, for projectve varetes, wth the arthmetc Chow groups defned by Gllet and Soulé and n general, wth the arthmetc Chow groups of Burgos. Our new constructon s shown to be a contravarant functor and s endowed wth a product structure, whch s commutatve and assocatve. MS 2000 Mathematcs subject classfcaton: 14G40, 14C15, 14F43 Introducton Let X be an arthmetc varety,.e. a regular scheme whch s flat and quas-projectve over an arthmetc rng. In [14], Gllet and Soulé defned the arthmetc Chow groups of X, denoted as ĈHp (X), whose elements are classes of pars (Z, g Z ), wth Z a codmenson p subvarety of X and g Z a Green current for Z. Later, n [5], the frst author gave an alternatve defnton for the arthmetc Chow groups, nvolvng the Delgne complex of dfferental forms wth arthmc sngulartes along nfnty, D (X, p), that computes real Delgne-Belnson cohomoy, HD (X, R(p)). When X s proper, the two defntons are related by a natural somorphsm that takes nto account the dfferent normalzaton of both defntons. In ths paper, we follow the latter defnton. It s shown n [5] that the followng propertes are satsfed by ĈHp (X): The groups ĈHp (X) ft nto an exact sequence: (1) CH p 1,p (X) ρ D 2p 1 (X, p)/ m d D a ĈH p (X) ζ CH p (X) 0, where CH p 1,p (X) s the term E p 1, p 2 (X) of the Qullen spectral sequence (see [23], 7) and ρ s the Belnson regulator. There s a parng ĈH p (X) ĈHq (X) ĈHp+q (X) Q turnng p 0 ĈHp (X) Q nto a commutatve graded untary Q-algebra. If f : X Y s a morphsm, there exsts a pull-back morphsm f : ĈHp (Y ) ĈHp (X). ssume that X s proper and defned over an arthmetc feld. Then the arthmetc Chow groups have been extended to hgher degrees by Goncharov, n [16]. These groups Date: February 25, Ths work was partally supported by the projects MTM C02-01 and MTM C

2 2 J. I. BURGOS GIL ND E.FELIU are denoted by ĈHp (X, n) and are constructed n order to extend the exact sequence (1) to a long exact sequence of the form ĈHp (X, n) ζ CH p (X, n) ρ H 2p n D (X, R(p)) a ĈHp (X, n 1) CH p (X, 1) ρ D 2p 1 (X, p)/ m d D a ĈH p (X) ζ CH p (X) 0. Explctly, Goncharov defned a regulator morphsm where Z p (X, ) P D 2p D (X, p), Z p (X, ) s the chan complex gven by Bloch n [3], whose homoy groups are, by defnton, CH p (X, ). DD (X, ) s the Delgne complex of currents. Then the hgher arthmetc Chow groups of a regular complex varety X are defned as ĈH p (X, n) := H n (s(p )), the homoy groups of the smple of the nduced morphsm P : Z p (X, ) P D 2p D (X, p)/d2p (X, p). For n = 0, these groups agree wth the ones gven by Gllet and Soulé. However, ths constructon leaves the followng questons open: (1) Does the composton of the somorphsm K n (X) Q = p 0 CHp (X, n) Q wth the morphsm nduced by P agree wth the Belnson regulator? (2) Can one defne a product structure on p,n ĈHp (X, n)? (3) re there well-defned pull-back morphsms? The use of the complex of currents n the defnton of P s the man obstacle encountered when tryng to answer these questons, snce ths complex does not behave well under pull-back or products. Moreover, the usual technques for the comparson of regulators apply to morphsms defned for the class of quas-projectve varetes, whch s not the case of P. In ths paper we develop a hgher arthmetc ntersecton theory by gvng a new defnton of the hgher arthmetc Chow groups, based on a representatve of the Belnson regulator at the chan complex level. Our strategy has been to use the Delgne complex of dfferental forms nstead of the Delgne complex of currents n the constructon of the representatve of the Belnson regulator. The obtaned regulator turns out to be a mnor modfcaton of the regulator descrbed by Bloch n [4]. The present defnton of hgher arthmetc Chow groups s vald for quas-projectve arthmetc varetes over a feld, pull-back morphsms are well-defned and can be gven a commutatve and assocatve product structure. Therefore, ths constructon overcomes the open questons left by Goncharov s constructon. The authors, jontly wth Takeda, prove n [6] that ths defnton agrees wth Goncharov s defnton when the arthmetc varety s projectve. Moreover, by a drect comparson of our regulator wth P, t s also proved that the regulator defned by Goncharov nduces the Belnson regulator. In ths way, the open questons (1)-(3) are answered postvely. Moreover, the queston of the covarance of the hgher arthmetc Chow groups wth respect to proper morphsms wll also be treated elsewhere. Note that snce the theory of hgher algebrac Chow groups gven by Bloch, CH p (X, n), s only fully establshed for schemes over a feld, we have to restrct ourselves to arthmetc varetes over a feld. Therefore, the followng queston remans open:

3 HIGHER RITHMETIC CHOW GROUPS 3 (1) Can we extend the defnton to arthmetc varetes over an arthmetc rng? Let us now brefly descrbe the constructons presented n ths paper. Frst, for the constructon of the hgher Chow groups, nstead of usng the smplcal complex defned by Bloch n [3], we use ts cubcal ana, defned by Levne n [19], due to ts sutablty for descrbng the product structure on CH (X, ). Thus Z p (X, n) 0 wll denote the normalzed chan complex assocated to a cubcal abelan group. Let X be a complex algebrac manfold. For every p 0, we defne two cochan complexes, D,Z p (X, p) 0 and D (X, p) 0, constructed out of dfferental forms on X n wth arthmc sngulartes along nfnty ( = P 1 \ {1}). For every p 0, the followng somorphsms are satsfed: H 2p n (D,Z p (X, p) 0 ) = CH p (X, n) R, n 0, H r (D (X, p) 0) = H r D (X, R(p)), r 2p, where the frst somorphsm s obtaned by a explct quas-somorphsm D 2p,Z p (X, p) 0 Z p (X, ) 0 R (see 2.4 and 2.5). We show that there s a natural chan morphsm (see 3.1) D 2p,Z p (X, p) 0 ρ D 2p (X, p) 0 whch nduces, after composton wth the somorphsm K n (X) Q = CH p (X, n) Q descrbed by Bloch n [3], the Belnson regulator (Theorem 3.5): K n (X) Q = CH p ρ (X, n) Q H 2p n D (X, R(p)). p 0 p 0 In the second part of ths paper we use the morphsm ρ to defne the hgher arthmetc Chow group ĈHp (X, n), for any arthmetc varety X over a feld. The formalsm underlyng our defnton s the theory of dagrams of complexes and ther assocated smple complexes, developed by Belnson n [1]. Let X Σ denote the complex manfold assocated wth X and let σ be the nvoluton that acts as complex conjugaton on the space and on the coeffcents. s usual σ as superndex wll mean the fxed part under σ. Then one consders the dagram of chan complexes Ẑ p (X, ) 0 = Z p (X Σ, ) σ 0 R γ 1 Z p (X, ) 0 p 0 D 2p,Z p (X Σ, p) σ 0 D 2p ρ (X Σ, p) σ 0 ZD 2p (X Σ, p) σ where ZD 2p (X Σ, p) σ s the group of closed elements of D 2p (X Σ, p) σ consdered as a complex concentrated n degree 0. Then, the hgher arthmetc Chow groups of X are gven by the homoy groups of the smple of the dagram Ẑp (X, ) 0 (Defnton 4.3): The followng propertes are shown: ĈH p (X, n) := H n (s(ẑp (X, ) 0 )).

4 4 J. I. BURGOS GIL ND E.FELIU Theorem 4.8: Let ĈHp (X) denote the arthmetc Chow group defned n [5]. Then, there s a natural somorphsm ĈH p (X) = ĈH p (X, 0). It follows that f X s proper, ĈHp (X, 0) agrees wth the arthmetc Chow group defned by Gllet and Soulé n [14]. Proposton 4.4: There s a long exact sequence ĈHp (X, n) ζ CH p (X, n) ρ H 2p n D (X Σ, R(p)) σ a ĈH p (X, n 1) CH p (X, 1) ρ D 2p 1 (X Σ, p) σ / m d D a ĈH p (X) ζ CH p (X) 0, wth ρ the Belnson regulator. Proposton 4.12 (Pull-back): Let f : X Y be a morphsm between two arthmetc varetes over a feld. Then, there s a pull-back morphsm ĈH p (Y, n) f ĈHp (X, n), for every p and n, compatble wth the pull-back maps on the groups CH p (X, n) and H 2p n D (X, R(p)). Corollary 4.16 (Homotopy nvarance): Let π : X m X be the projecton on X. Then, the pull-back map π : ĈHp (X, n) ĈHp (X m, n), n 1 s an somorphsm. Theorem 5.46 (Product): There exsts a product on ĈH (X, ) := ĈH p (X, n), p 0,n 0 whch s assocatve, graded commutatve wth respect to the degree n. The paper s organzed as follows. The frst secton s a prelmnary secton. It s devoted to fx the notaton and state the man facts used n the rest of the paper. It ncludes general results on homocal algebra, dagrams of complexes, cubcal abelan groups and Delgne-Belnson cohomoy. In the second secton we recall the defnton of the hgher Chow groups of Bloch and ntroduce the complexes of dfferental forms beng the source and target of the regulator map. We proceed n the next secton to the defnton of the regulator ρ and we prove that t agrees wth Belnson s regulator. In sectons 4 and 5, we develop the theory of hgher arthmetc Chow groups. Secton 4 s devoted to the defnton and basc propertes of the hgher arthmetc Chow groups and to the comparson wth the arthmetc Chow group for n = 0. Fnally, n secton 5 we defne the product structure on ĈH (X, ) and prove that t s commutatve and assocatve. cknowledgments. Ths paper was orgnated durng a stay of the frst author at the CRM (Bellaterra). He s very grateful for the CRM hosptalty. Durng the elaboraton of the paper, the second author spent an academc year n the Unversty of Regensburg wth a pre-doc grant from the European Network rthmetc lgebrac Geometry. She wants to thank all the members of the rthmetc Geometry group, specally U. Jannsen and K. Künneman. We would lke to acknowledge M. Levne, H. Gllet an D. Roessler for many useful conversatons on the subject of ths paper. Fnally we would also lke to thanks the referee of the paper for hs or her work.

5 HIGHER RITHMETIC CHOW GROUPS 5 1. Prelmnares 1.1. Notaton on (co)chan complexes. We use the standard conventons on (co)chan complexes. By a (co)chan complex we mean a (co)chan complex over the category of abelan groups. The cochan complex assocated to a chan complex s smply denoted by and the chan complex assocated to a cochan complex s denoted by. The translaton of a cochan complex (, d ) by an nteger m s denoted by [m]. Recall that [m] n = m+n and the dfferental of [m] s ( 1) m d. If (, d ) s a chan complex, then the translaton of by an nteger m s denoted by [m]. In ths case the dfferental s also ( 1) m d but [m] n = n m. The smple complex assocated to an terated chan complex s denoted by s() and the anaous notaton s used for the smple complex assocated to an terated cochan complex (see [9] 2 for defntons). The smple of a cochan map f B s the cochan complex (s(f), d s ) wth s(f) n = n B n 1, and dfferental d s (a, b) = (d a, f(a) d B b). Note that ths complex s the cone of f shfted by 1. There s an assocated long exact sequence (1.1) H n (s(f) ) H n ( ) f H n (B ) H n+1 (s(f) ) If f s surjectve, there s a quas-somorphsm (1.2) ker f s( f) x (x, 0), and f f s njectve, there s a quas-somorphsm (1.3) s(f)[1] π B / (a, b) [b]. naously, equvalent results and quas-somorphsms can be stated for chan complexes. Followng Delgne [10], gven a cochan complex and an nteger n, we denote by τ n the canoncal truncaton of at degree n The smple of a dagram of complexes. We descrbe here Belnson s deas on the smple complexes assocated to a dagram of complexes (see [1]). dagram of chan complexes s a dagram of the form B n (1.4) D = (1.5) 1 B 1 γ 1 2 Consder the nduced chan morphsms n+1 =1 ϕ,ϕ 1,ϕ 2 n B, =1 γ 2 B n ϕ 1 (a ) = γ (a ) f a, γ n ϕ 2 (a ) = γ 1(a ) f a, γ n n+1. ϕ(a ) = (ϕ 1 ϕ 2 )(a ) = (γ γ 1)(a ) f a. (where we set γ n+1 = γ 0 = 0). The smple complex assocated to the dagram D s defned to be the smple of the morphsm ϕ: (1.6) s(d) := s(ϕ).

6 6 J. I. BURGOS GIL ND E.FELIU 1.3. Morphsms of dagrams. morphsm between two dagrams D and D conssts of a collecton of morphsms h, B h B B, commutng wth the morphsms γ and γ, for all. ny morphsm of dagrams D h D s(h) nduces a morphsm on the assocated smple complexes s(d) s(d ). Observe that f, for every, h and h B are quas-somorphsms, then s(h) s also a quas-somorphsm Product structure on the smple of a dagram. Let D and D be two dagrams as (1.4). Consder the dagram obtaned by the tensor product of complexes: (1.7) B 1 B 1 B 2 B B n B n (D D γ ) = 1 ξ 1 γ 1 ξ 1 γ n ξ n γ n ξ n γ 2 ξ n n In [1], Belnson defned, for every β Z, a morphsm s(d) s(d ) β s(d D ) as follows. For a, a, b B and b B, set: a β a = a a, b β a = b ((1 β)ϕ 1 (a ) + βϕ 2 (a )), a β b = ( 1) deg a (βϕ 1 (a) + (1 β)ϕ 2 (a)) b, b β b = 0, n+1 n+1 where the tensor product between elements n dfferent spaces s defned to be zero. If B, C are chan complexes, let σ : s(b C ) s(c B ) be the map sendng b c B n C m to ( 1) nm c b C m B n. Lemma 1.8 (Belnson). () The map β s a morphsm of complexes. () For every β, β Z, β s homotopc to β. () There s a commutatve dagram s(d) s(d β ) s(d D ) σ s(d ) s(d) 1 β s(d D). (v) The products 0 and 1 are assocatve specfc type of dagrams. In ths work we wll use dagrams of the followng form: (1.9) D = 1 B 1 γ 1 2 γ 2 σ B 2,

7 HIGHER RITHMETIC CHOW GROUPS 7 wth a quas-somorphsm. For ths type of dagrams, snce γ 1 s a quas-somorphsm, we obtan a long exact sequence equvalent to the long exact sequence related to the smple of a morphsm. Snce a dagram lke ths nduces a map 1 B 2 n the derved category, we obtan Lemma Let D be a dagram lke (1.9). Then there s a well-defned morphsm Moreover, there s a long exact sequence H n ( 1 ) ρ H n (B 2 ), [a 1 ] γ 2 (γ 1) 1 [a 1 ]. (1.11) H n (s(d) ) H n ( 1 ) ρ H n (B 2 ) H n 1 (s(d) ) Consder now a dagram of the form (1.12) D = 1 B 1 γ 1 2 γ 2 B 2 γ 2 3, wth γ 1 a quas-somorphsm and γ 2 a monomorphsm. Lemma Let D be a dagram as (1.12) and let D be the dagram B 1 B / 2 3 (1.14) D γ 1 γ 1 γ 2 =, 1 Then, there s a quas-somorphsm between the smple complexes assocated to D and to D : 2 s(d) s(d ). Proof. It follows drectly from the defnton that the smple complex assocated to D s quas-somorphc to the smple assocated to the dagram B 1 s( 3 γ 2 B 2 )[1] (1.15) D = γ 1 γ 2, 2. 1 Then, the quas-somorphsm gven n (1.3) nduces a quas-somorphsm s(d ) s(d ) as desred. Corollary For any dagram of the form (1.12), there s a long exact sequence (1.17) H n (s(d) ) H n ( 1 ) ρ H n 1 (s(γ 2)) H n 1 (s(d) ) Proof. It follows from the prevous lemma together wth Proposton 1.10.

8 8 J. I. BURGOS GIL ND E.FELIU 1.6. Cubcal abelan groups and chan complexes. Let C = {C n } n 0 be a cubcal abelan group wth face maps δ j : C n C n 1, for = 1,..., n and j = 0, 1, and degeneracy maps σ : C n C n+1, for = 1,..., n + 1. Let D n C n be the subgroup of degenerate elements of C n, and let C n = C n /D n. Let C denote the assocated chan complex, that s, the chan complex whose n- j=0,1 ( 1)+j δ j. th graded pece s C n and whose dfferental s gven by δ = n =1 Thus D s a subcomplex and C s a quotent complex. We fx the normalzed chan complex assocated to C, NC, to be the chan complex whose n-th graded group s NC n := n =1 ker δ1, and whose dfferental s δ = n =1 ( 1) δ 0. It s well-known that there s a decomposton of chan complexes C = NC D gvng an somorphsm NC = C. For certan cubcal abelan groups, the normalzed chan complex can be further smplfed, up to homotopy equvalence, by consderng the elements whch belong to the kernel of all faces but δ1 0. Defnton Let C be a cubcal abelan group. Let N 0 C be the complex defned by (1.19) N 0 C n = n ker δ 1 =1 n ker δ 0, and dfferental δ = δ1. 0 =2 The proof of the next proposton s anaous to the proof of Theorem n [2]. The result s proved there only for the cubcal abelan group defnng the hgher Chow complex (see 2.1 below). We gve here the abstract verson of the statement, vald for a certan type of cubcal abelan groups. Proposton Let C be a cubcal abelan group. ssume that t comes equpped wth a collecton of maps h j : C n C n+1, j = 1,..., n, such that, for any l = 0, 1, the followng denttes are satsfed: (1.21) δ 1 j h j = δ 1 j+1h j = s j δ 1 j, δj 0 h j = δj+1h 0 j = d, { δh l hj 1 δ j = l < j, h j δ 1 l > j + 1. Then, the ncluson of complexes : N 0 C NC s a homotopy equvalence. Proof. Let g j : NC n NC n+1 be defned as g j = ( 1) n j h n j f 0 j n 1 and g j = 0 otherwse. Then there s a well-defned morphsm of chan complexes H j = (Id +δg j + g j δ) : NC NC. Ths morphsm s homotopcally equvalent to the dentty.

9 Let x NC n and 0 j n 1. Then, Hence, δh n j (x) = = h n j 1 δ(x) = δg j (x) + g j δ(x) = ( 1) n j HIGHER RITHMETIC CHOW GROUPS 9 n+1 ( 1) δ 0 h n j (x) =1 n j 1 =1 ( 1) h n j 1 δ 0 (x) + n ( 1) h n j 1 δ 0 (x). =1 n =n j+1 n+1 =n j+2 ( 1) 1 h n j δ 0 (x) + ( 1) n j 1 We consder the decreasng fltraton G of NC, gven by ( 1) h n j δ 0 1(x), n =n j (1.22) G j NC n = {x NC n δ 0 (x) = 0, > max(n j, 1)}. ( 1) h n j 1 δ 0 (x). Then G 0 NC = NC and for j n 1, G j NC n = N 0 C n. If x G j+1 NC, then δg j (x) + g j δ(x) = 0 and thus, H j (x) = x. Moreover, f x G j NC, then H j (x) G j+1 NC. Thus, H j s the projector from G j NC to G j+1 NC. Thus, the morphsm ϕ : NC N 0 C gven, on NC n, by ϕ := H n 2 H 0 forms a chan morphsm homotopcally equvalent to the dentty. Moreover ϕ s the projector from NC to N 0 C. Hence, ϕ s the dentty of N 0 C whle ϕ s homotopcally equvalent to the dentty of NC. Remark To every cubcal abelan group C there are assocated four chan complexes: C, NC, N 0 C and C. In some stuatons t wll be necessary to consder the cochan complexes assocated to these chan complexes. In ths case we wll wrte, respectvely, C, NC, N 0 C and C Cubcal cochan complexes. Let X be a cubcal cochan complex. Then, for every m, the cochan complexes NX m, N 0 X m and X m are defned. Proposton Let X, Y be two cubcal cochan complexes and let f : X Y be a morphsm. ssume that for every m, the cochan morphsm X m f m Y m s a quas-somorphsm. Then, the nduced morphsms are quas-somorphsms. NX m f m NYm f m and X m Ỹ m Proof. The proposton follows from the decompostons and the fact that f m nduces cochan maps H r (X m) = H r (NX m) H r (DX m), H r (Y m) = H r (NY m) H r (DY m), NX m f m NY m, DX m f m DY m.

10 10 J. I. BURGOS GIL ND E.FELIU Proposton Let X be a cubcal cochan complex. Then the natural morphsm s an somorphsm for all n 0. H r (NX n) f NH r (X n) Proof. The cohomoy groups H r (X ) have a cubcal abelan group structure. Hence there s a decomposton H r (X ) = NH r (X ) DH r (X ). In addton, there s a decomposton X n = NX n DX n. Therefore H r (X ) = H r (NX ) H r (DX ). The lemma follows from the fact that the dentty morphsm n H r (X ) maps NH r (X ) to H r (NX ) and DH r (X ) to H r (DX ) Delgne-Belnson cohomoy. In ths paper we use the defntons and conventons on Delgne-Belnson cohomoy gven n [5] and [9], chapter 5. One denotes R(p) = (2π) p R C. Let X be a complex algebrac manfold and denote by E,R (X)(p) the complex of real dfferental forms wth arthmc sngulartes along nfnty, twsted by p. Let (D (X, p), d D) be the Delgne complex of dfferental forms wth arthmc sngulartes, as descrbed n [5]. It computes real Delgne-Belnson cohomoy of X, that s, H n (D (X, p)) = Hn D(X, R(p)). Ths complex s functoral on X. The product structure n Delgne-Belnson cohomoy can be descrbed by a cochan morphsm on the Delgne complex (see [5]): D n (X, p) Dm (X, q) D n+m (X, p + q) x y x y. Ths product satsfes the expected relatons: (1) Graded commutatvty: x y = ( 1) nm y x. (2) Lebnz rule: d D (x y) = d D x y + ( 1) n x d D y. Proposton The Delgne product s assocatve up to a natural homotopy,.e. there exsts such that h : D r (X, p) Ds (X, q) Dt (X, l) Dr+s+t (X, p + q + l) d D h(ω 1 ω 2 ω 3 ) + hd D (ω 1 ω 2 ω 3 ) = (ω 1 ω 2 ) ω 3 ω 1 (ω 2 ω 3 ). Moreover, f ω 1 D 2p (X, p), ω 2 D 2q (X, q) and ω 3 D 2l (X, l) satsfy d Dω = 0 for all, then (1.27) h(ω 1 ω 2 ω 3 ) = 0. Proof. Ths s [5], Theorem 3.3.

11 HIGHER RITHMETIC CHOW GROUPS Cohomoy wth supports. Let Z be a closed subvarety of a complex algebrac manfold X. Consder the complex D (X \Z, p),.e. the Delgne complex of dfferental forms n X \ Z wth arthmc sngulartes along Z and nfnty. Defnton The Delgne complex wth supports n Z s defned to be D,Z (X, p) = s(d (X, p) D (X \ Z, p)). The Delgne-Belnson cohomoy wth supports n Z s defned as the cohomoy groups of the Delgne complex wth supports n Z: HD,Z(X, n R(p)) := H n (D,Z (X, p)). Lemma Let Z, W be two closed subvaretes of a complex algebrac manfold X. Then there s a short exact sequence of Delgne complexes, 0 D (X \ Z W, p) D (X \ Z, p) D (X \ W, p) j D (X \ Z W, p) 0, where (α) = (α, α) and j(α, β) = α + β. Proof. It follows from [7], Theorem 3.6. In addton, Delgne-Belnson cohomoy wth supports satsfes a sempurty property. Namely, let Z be a codmenson p subvarety of an equdmensonal complex manfold X, and let Z 1,..., Z r be ts codmenson p rreducble components. Then { (1.30) HD,Z(X, n 0 n < 2p, R(p)) = r =1 R[Z ] n = 2p. For the next proposton, let δ Z denote the current ntegraton along an rreducble varety Z. In the sequel we wll use the conventons of [9] 5.4 wth respect to the current assocated to a locally ntegrable form and to the current δ Z. Proposton Let X be an equdmensonal complex algebrac manfold and Z a codmenson p rreducble subvarety of X. Let j : X X be a smooth compactfcaton of X (wth a normal crossng dvsor as ts complement) and Z the closure of Z n X. The somorphsm cl : R[Z] = H 2p D,Z (X, R(p)) sends [Z] to [(j w, j g)], for any [(w, g)] H 2p (X, R(p)) satsfyng the relaton of D,Z currents n X (1.32) 2 [g] = [w] δ Z. Proof. See [9], Proposton In partcular, assume that Z = dv(f) s a prncpal dvsor, where f s a ratonal functon on X. Then [Z] s represented by the couple (0, 1 2 (f f)) H 2p D,Z (X, R(p)). The defnton of the cohomoy wth support n a subvarety can be extended to the defnton of the cohomoy wth support n a set of subvaretes of X. We explan here the case used n the sequel. Let Z p be a subset of the set of codmenson p closed subvaretes of X, that s closed under fnte unons. The ncluson of subsets turns Z p nto a drected ordered set. We defne the complex (1.33) D (X \ Zp, p) := lm D (X \ Z, p), Z Z p

12 12 J. I. BURGOS GIL ND E.FELIU whch s provded wth an njectve map D (X, p) D (X \ Zp, p). s above, we defne D,Z p(x, p) := s() and the Delgne-Belnson cohomoy wth supports n Z p as HD,Z n p(x, R(p)) := Hn (D,Z p(x, p)) Real varetes. real varety X conssts of a couple (X C, F ), wth X C a complex algebrac manfold and F an antlnear nvoluton of X C. If X = (X C, F ) s a real varety, we wll denote by σ the nvoluton of D n (X C, p) gven by σ(η) = F η. Then the real Delgne-Belnson cohomoy of X s defned by H n D(X, R(p)) := H n D(X C, R(p)) σ, where the superndex σ means the fxed part under σ. The real cohomoy of X s expressed as the cohomoy of the real Delgne complex.e. there s an somorphsm D n (X, p) := Dn (X C, p) σ, H n D(X, R(p)) = H n (D n (X, p), d D) Truncated Delgne complex. In the rest of the work, we wll consder the Delgne complex (canoncally) truncated at degree 2p. For smplcty we wll denote t by τd (X, p) = τ 2pD (X, p). The truncated Delgne complex wth supports n a varety Z s denoted by τd,z (X, p) = τ 2p D,Z (X, p) and the truncated Delgne complex wth supports n Zp s denoted by τd,z (X, p) = τ p 2p D,Z (X, p). p Note that, snce the truncaton s not an exact functor, t s not true that τd,z (X, p) p s the smple complex of the map τd (X, p) τd (X \ Zp, p). 2. Dfferental forms and hgher Chow groups In ths secton we construct a complex of dfferental forms whch s quas-somorphc to the complex Z p (X, ) 0 R. Ths last complex computes the hgher algebrac Chow groups ntroduced by Bloch n [3] wth real coeffcents. The key pont of ths constructon s the set of somorphsms gven n (1.30). Ths complex s very smlar to the complex ntroduced by Bloch n [4] n order to construct the cycle map for the hgher Chow groups. In both constructons one consders a 2-terated complex of dfferental forms on a cubcal or smplcal scheme. Snce ths leads to a second quadrant spectral sequence, to avod convergence problems, one has to truncate the complexes nvolved. The man dfference between both constructons s the drecton of the truncaton. We truncate the 2-terated complex at the degree gven by the dfferental forms, whle n loc. ct. the complex s truncated at the degree gven by the smplcal scheme.

13 HIGHER RITHMETIC CHOW GROUPS The cubcal Bloch complex. We recall here the defnton and man propertes of the hgher Chow groups defned by Bloch n [3]. Intally, they were defned usng the chan complex assocated to a smplcal abelan group. However, snce we are nterested n the product structure, t s more convenent to use the cubcal presentaton, as gven by Levne n [19]. Fx a base feld k and let P 1 be the projectve lne over k. Let = P 1 \{1}( = 1 ). The cartesan product (P 1 ) has a cocubcal scheme structure. For = 1,..., n, we denote by t (k { }) \ {1} the absolute coordnate of the -th factor. Then the coface and codegeneracy maps are defned as δ 0(t 1,..., t n ) = (t 1,..., t 1, 0, t,..., t n ), δ 1(t 1,..., t n ) = (t 1,..., t 1,, t,..., t n ), σ (t 1,..., t n ) = (t 1,..., t 1, t +1,..., t n ). Then, nherts a cocubcal scheme structure from that of (P 1 ). n r-dmensonal face of n s any subscheme of the form δ 1 j1 δ r j r ( n r ). We have chosen to represent 1 as P 1 \ {1} so that the face maps are represented by the ncluson at zero and the ncluson at nfnty. In ths way the cubcal structure of s compatble wth the cubcal structure of (P 1 ) n [8]. In the lterature the usual representaton 1 = P 1 \ { } s often used. We wll translate from one defnton to the other by usng the nvoluton (2.1) x x x 1. Ths nvoluton has the fxed ponts {0, 2} and nterchanges the ponts 1 and. Let X be an equdmensonal quas-projectve algebrac scheme of dmenson d over the feld k. Let Z p (X, n) be the free abelan group generated by the codmenson p closed rreducble subvaretes of X n, whch ntersect properly all the faces of n. The pull-back by the coface and codegeneracy maps of endow Z p (X, ) wth a cubcal abelan group structure. Let (Z p (X, ), δ) be the assocated chan complex (see 1.6) and consder the normalzed chan complex assocated to Z p (X, ), n Z p (X, n) 0 := NZ p (X, n) = ker δ 1. Defnton 2.2. Let X be a quas-projectve equdmensonal algebrac scheme over a feld k. The hgher Chow groups defned by Bloch are =1 CH p (X, n) := H n (Z p (X, ) 0 ). Let N 0 be the refned normalzed complex of Defnton (1.18). Let Z p (X, ) 00 be the complex wth n n Z p (X, n) 00 := N 0 Z p (X, n) = ker δ 1 ker δ 0. (2.3) =1 Fx n 0. For every j = 1,..., n, we defne a map n+1 h j n (t 1,..., t n+1 ) (t 1,..., t j 1, 1 (t j 1)(t j+1 1), t j+2,..., t n+1 ). The refned normalzed complex of [2] 4.4 s gven by consderng the elements n the kernel of all faces but δ1 1, nstead of δ0 1 lke here. Takng ths nto account, together wth =2

14 14 J. I. BURGOS GIL ND E.FELIU the nvoluton (2.1), the map h j agrees wth the map denoted h n j n [2] 4.4. Therefore, the maps h j are smooth, hence flat, so they nduce pull-back maps (2.4) h j : Z p (X, n) Z p (X, n + 1), j = 1,..., n + 1, that satsfy the condtons of Proposton Therefore the ncluson s a homotopy equvalence (see [2] 4.4). Z p (X, n) 00 := N 0 Z p (X, n) Z p (X, n) Functoralty. It follows easly from the defnton that the complex Z p (X, ) 0 s covarant wth respect to proper maps (wth a shft n the gradng) and contravarant for flat maps. Let f : X Y be an arbtrary map between two smooth schemes X, Y. Let Z p f (Y, n) 0 Z p (Y, n) 0 be the subgroup generated by the codmenson p rreducble subvaretes Z Y n, ntersectng properly the faces of n and such that the pull-back X Z ntersects properly the graph of f, Γ f. Then, Z p f (Y, ) 0 s a chan complex and the ncluson of complexes Z p f (Y, ) 0 Z p (Y, ) 0 s a quas-somorphsm. Moreover, the pull-back by f s defned for algebrac cycles n Z p f (Y, ) 0 and hence there s a well-defned pull-back morphsm CH p (Y, n) f CH p (X, n). proof of ths fact can be found n [20], 3.5. See also [18] Product structure. Let X and Y be quas-projectve algebrac schemes over k. Then, there s a chan morphsm nducng exteror products s(z p (X, ) 0 Z q (Y, ) 0 ) Z p+q (X Y, ) 0 CH p (X, n) CH q (Y, m) CH p+q (X Y, n + m). More concretely, let Z be a codmenson p rreducble subvarety of X n, ntersectng properly the faces of n and let W be a codmenson q rreducble subvarety of Y m, ntersectng properly the faces of m. Then, the codmenson p + q subvarety Z W X n Y m = X Y n m = X Y n+m, ntersects properly the faces of n+m. By lnearty, we obtan a morphsm Z p (X, n) Z q (Y, m) Z p+q (X Y, n + m). It nduces a chan morphsm on the normalzed complexes and hence there s an external product s(z p (X, ) 0 Z q (Y, ) 0 ) Z p+q (X Y, ) 0, (2.5) : CH p (X, n) CH q (Y, m) CH p+q (X Y, n + m), for all p, q, n, m. If X s smooth, then the pull-back by the dagonal map : X X X s defned on the hgher Chow groups, CH p (X X, ) CH p (X, ). Therefore, for all p, q, n, m, we obtan an nternal product (2.6) : CH p (X, n) CH q (X, m) CH p+q (X X, n + m) CH p+q (X, n + m).

15 HIGHER RITHMETIC CHOW GROUPS 15 In the derved category of chan complexes, the nternal product s gven by the morphsm s(z p (X, ) 0 Z q (X, ) 0 ) Z p+q (X X, ) 0 Z p+q (X X, ) 0 Z p+q (X, ) 0. Proposton 2.7. Let X be a quas-projectve algebrac scheme over k. The parng (2.6) defnes an assocatve product on CH (X, ) = p,n CHp (X, n). Ths product s graded commutatve wth respect to the degree gven by n. Proof. See [19], Theorem Dfferental forms and affne lnes. For every n, p 0, let τd (X n, p) be the truncated Delgne complex of dfferental forms n X n, wth arthmc sngulartes at nfnty. The structural maps of the cocubcal scheme nduce a cubcal structure on τd r (X, p) for every r and p. Consder the 2-terated cochan complex D r, n (X, p) = τd r (X n, p), wth dfferental (d D, δ = n =1 ( 1) (δ 0 δ1 )). Let D (X, p) = s(d, (X, p)) be the assocated smple complex. Hence ts dfferental d s n D (X, p) s gven, for every α D r, n (X, p), by d s (α) = d D (α) + ( 1) r δ(α). Snce we are usng cubcal structures, ths complex does not compute the rght cohomoy and we have to normalze t. For every r, n, we wrte D r, n (X, p) 0 = τd r (X n, p) 0 := NτD r (X n, p). Hence D, (X, p) 0 s the normalzed 2-terated complex and we denote by D (X, p) 0 the assocated smple complex. Proposton 2.8. The natural morphsm of complexes s a quas-somorphsm. τd (X, p) = D,0 (X, p) 0 D (X, p) 0 Proof. Consder the second quadrant spectral sequence wth E 1 term gven by E r, n 1 = H r (D, n (X, p) 0 ). Snce D r, n (X, p) 0 = 0, for r < 0 or r > 2p, ths spectral sequence converges to the cohomoy groups H (D (X, p) 0). Ths s the man reason why we use the truncated complexes. If we see that, for all n > 0, the cohomoy of the complex D, n (X, p) 0 s zero, the spectral sequence degenerates and the proposton s proven. By the homotopy nvarance of Delgne-Belnson cohomoy, there s an somorphsm δ1 1 δ1 1 : H (τd (X n, p)) H (τd (X, p)). By defnton, the mage of H (τd (X n, p) 0 ) by ths somorphsm s zero. Snce H (τd (X n, p) 0 ) s a drect summand of H (τd (X n, p)), t vanshes for all n > 0.

16 16 J. I. BURGOS GIL ND E.FELIU We defne the complex D (X, p) 00 to be the smple complex assocated to the 2-terated complex wth D r, n (X, p) 00 = N 0 τd r (X n, p). Corollary 2.9. The natural morphsm of complexes τd (X, p) = D,0 (X, p) 00 D (X, p) 00 s a quas-somorphsm. Proof. It follows from Proposton 2.8, Proposton 1.20 (usng as maps {h j } the ones nduced by the maps h j defned n 2.3) and Proposton complex wth dfferental forms for the hgher Chow groups. Let Z p n,x be the set of all codmenson p closed subvaretes of X n ntersectng properly the faces of n. We consder t as a set ordered by the ncluson relaton. When there s no source of confuson, we smply wrte Z p n or even Z p. Consder the cubcal abelan group (2.10) H p (X, ) := H 2p D,Z p (X, R(p)), wth faces and degeneraces nduced by those of. Let H p (X, ) 0 be the assocated normalzed complex. Lemma Let X be a complex algebrac manfold. For every p 0, there s an somorphsm of chan complexes sendng z to cl(z). : Z p (X, ) 0 R = H p (X, ) 0, Proof. It follows from the somorphsm (1.30). Remark Observe that the complex H p (X, ) 0 has the same functoral propertes as Z p (X, ) 0 R. Let D,,Z p (X, p) 0 be the 2-terated cochan complex, whose component of bdegree (r, n) s τd r,z p n (X n, p) 0 = NτD r,z p n (X n, p) = Nτ 2p D r,z p n (X n, p), and whose dfferentals are (d D, δ). s usual, we denote by D,Z p (X, p) 0 the assocated smple complex and by d s ts dfferental. Let D 2p,Z (X, p) p 0 be the chan complex whose n-graded pece s D 2p n,z (X, p) p 0. Proposton For every p 0, the famly of morphsms D 2p n,z (X, p) p 0 H p (X, n) 0 ((ω n, g n ),..., (ω 0, g 0 )) [(ω n, g n )] defnes a quas-somorphsm of chan complexes between D 2p,Z p (X, p) 0 and H (X, n) 0. Proof. The map s well defned because (ω n, g n ) τd 2p,Z p n (X n, p) 0. Therefore, by defnton of the truncated complex (ω n, g n ) s closed. To see that t s a morphsm of complexes we compute γ 1d s ((ω n, g n ),..., (ω 0, g 0 )) = γ 1(( 1) 2p δ(ω n, g n ) + d D (ω n 1, g n 1 ),... ) = [δ(ω n, g n ) + d D (ω n 1, g n 1 )] = δ[(ω n, g n )].

17 HIGHER RITHMETIC CHOW GROUPS 17 Now we consder the second quadrant spectral sequence wth E 1 -term E n,r 1 = H r (τd,z p(x n, p) 0 ). By constructon, E n,r 1 = 0 for all r > 2p. Moreover, for all r < 2p and for all n, the sempurty property of Delgne-Belnson cohomoy mples that (2.14) H r (τd,z p(x n, p)) = 0. Hence, by Proposton 1.24, the same s true for the normalzed chan complex H r (τd,z p(x n, p) 0 ) = 0, r < 2p. Therefore, the E 1 -term of the spectral sequence s E n,r 1 = { 0 f r 2p, H 2p (τd,z (X n, p) p 0 ) f r = 2p. Fnally, from Proposton 1.25, t follows that the natural map H 2p (τd,z p(x n, p) 0 ) H p (X, n) 0 s an somorphsm. Usng the explct descrpton of the spectral sequence assocated to a double complex, t s clear that the morphsm nduced n cohomoy by agrees wth the morphsm nduced by the spectral sequence. Hence the proposton s proved. We denote CH p (X, n) R = CH p (X, n) R. Corollary Let z CH p (X, n) R be the class of an algebrac cycle z n X n. By the somorphsms of Lemma 2.11 and Proposton 2.13, the algebrac cycle z s represented, n H 2p n (D,Z p(x, p) 0 ), by any cycle such that ((ω n, g n ),..., (ω 0, g 0 )) D 2p n,z p (X, p) 0 cl(z) = [(ω n, g n )]. Remark Our constructon dffers from the constructon gven by Bloch, n [4], n two ponts: He consdered the 2-terated complex of dfferental forms on the smplcal scheme n, nstead of the dfferental forms on the cubcal scheme n. In order to ensure the convergence of the spectral sequence n the proof of last proposton, he truncated the 2-terated complex n the drecton gven by the affne schemes Functoralty of D,Z p (X, p) 0. In many ways, the complex D,Z p (X, p) 0 behaves lke the complex Z (X, ) 0. Lemma Let f : X Y be a flat map between two equdmensonal complex algebrac manfolds. Then there s a pull-back map f : D,Z p(y, p) 0 D,Z p(x, p) 0. Proof. We wll see that n fact there s a map of terated complexes f : D r, n,z p (Y, p) D r, n,z p (X, p). Let Z be a codmenson p subvarety of Y n ntersectng properly the faces of n. Snce f s flat, there s a well-defned cycle f (Z). It s a codmenson p cycle of X n

18 18 J. I. BURGOS GIL ND E.FELIU ntersectng properly the faces of n, and whose support s f 1 (Z). Then, by [14] 1.3.3, the pull-back of dfferental forms gves a morphsm τd (Y n \ Z, p) f τd (X n \ f 1 (Z), p). Hence, there s an nduced morphsm τd (Y n \ Z p Y, p) f lm τd (X n \ f 1 (Z), p) τd (X n \ Z p X, p), Z Z p Y and thus, there s a pull-back morphsm compatble wth the dfferental δ. f : D, n,z p (Y, p) D, n,z p (X, p) Remark The pull-back defned here agrees wth the pull-back defned by Bloch under the somorphsms of Lemma 2.11 and Proposton Indeed, let f : X Y be a flat map. Then, f Z s an rreducble subvarety of Y and (ω, g) a couple representng the class of [Z] n the Delgne-Belnson cohomoy wth support, then the couple (f ω, f g) represents the class of [f (Z)] (see [14], Theorem 3.6.1). Proposton Let f : X Y be a morphsm of equdmensonal complex algebrac manfolds. Let Z p f be the subset consstng of the subvaretes Z of Y n ntersectng properly the faces of n and such that X Z n ntersects properly the graph of f, Γ f. Then, () The complex D,Z p f (Y, p) 0 s quas-somorphc to D,Z p (Y, p) 0. () There s a well-defned pull-back f : D,Z p f (Y, p) 0 D,Z p(x, p) 0. Proof. rgung as n the proof of the prevous proposton, there s a pull-back map f : τd (Y n \ Z p f, p) f τd (X n \ Z p, p), nducng a morphsm f : D,Z p (Y, p) D,Z p(x, p), f and hence a morphsm f : D,Z p f (Y, p) 0 D,Z p(x, p) 0. ll that remans to be shown s that the ncluson D,Z p (Y, p) 0 D,Z p(y, p) 0 f s a quas-somorphsm. By the quas-somorphsm mentoned n paragraph 2.2 and the quas-somorphsm of Proposton 2.13, there s a commutatve dagram Z p f (Y, ) 0 R D,Z p f (Y, p) 0 Z p (Y, ) 0 R D,Z p (Y, p) 0. The proof that the upper horzontal arrow s a quas-somorphsm s anaous to the proof of Proposton Thus, we deduce that s a quas-somorphsm.

19 HIGHER RITHMETIC CHOW GROUPS lgebrac cycles and the Belnson regulator In ths secton we defne a chan morphsm, n the derved category of chan complexes, that nduces n homoy the Belnson regulator. The constructon s anaous to the defnton of the cycle class map gven by Bloch n [4], wth the mnor modfcatons mentoned n However, n [4] there s no proof of the fact that the composton of the somorphsm K n (X) Q = p 0 CHp (X, n) Q wth the cycle class map agrees wth the Belnson regulator Defnton of the regulator. Consder the map of terated cochan complexes defned by the projecton onto the frst factor D r, n,z p (X, p) = τ 2p s(d (X n, p) D (X n \ Z p, p)) r ρ τd r (X n, p) It nduces a cochan morphsm and hence a chan morphsm D,Z p(x, p) 0 (3.1) D 2p,Z p (X, p) 0 ρ D (X, p) 0, ρ D 2p (X, p) 0. (ω, g) ω. The morphsm nduced by ρ n homoy, together wth the somorphsms of Propostons 2.8, 2.11 and 2.13, nduce a morphsm (3.2) ρ : CH p (X, n) CH p (X, n) R H 2p n D (X, R(p)). By abuse of notaton, t wll also be denoted by ρ. By corollary 2.15, we deduce that, f z Z p (X, n) 0, then ρ(z) = (ω n,..., ω 0 ), for any cycle ((ω n, g n ),..., (ω 0, g 0 )) D 2p n,z p (X, p) 0 such that [(ω n, g n )] = cl(z). Proposton 3.3. () The morphsm ρ : D 2p,Z (X, p) p 0 D 2p (X, p) 0 s contravarant for flat maps. () The nduced morphsm ρ : CH p (X, n) H 2p n D (X, R(p)) s contravarant for arbtrary maps. Proof. Both assertons are obvous. Let z = ((ω n, g n ),..., (ω 0, g 0 )) D 2p n,z (X, p) p 0 be a cycle such that ts nverse mage by f s defned. Ths s the case when f s flat or when z belongs to D 2p,Z p (X, p) 0. In both cases f and the clam follows. f ((ω n, g n ),..., (ω 0, g 0 )) = ((f ω n, f g n ),..., (f ω 0, f g 0 )) Remark 3.4. Let X be an equdmensonal compact complex algebrac manfold. Observe that, by defnton, the morphsm ρ : CH p (X, 0) = CH p (X) H 2p D (X, R(p)) agrees wth the cycle class map cl. Now let E be a vector bundle of rank n over X. For every p = 1,..., n, there exsts a characterstc class Cp CH (E) CH p (X) (see [17]) and a characterstc class

20 20 J. I. BURGOS GIL ND E.FELIU Cp D (E) H 2p D (X, R(p)), called the p-th Chern class of the vector bundle E. By defnton, cl(cp CH (E)) = Cp D (E). Hence, for all p = 1,..., n. ρ(cp CH (E)) = Cp D (E), 3.2. Comparson wth the Belnson regulator. We prove here that the regulator defned n (3.2) agrees wth the Belnson regulator. The comparson s based on the followng facts: The morphsm ρ s compatble wth nverse mages. The morphsm ρ s defned for quas-projectve schemes. In vew of these propertes, t s enough to prove that the two regulators agree when X s a Grassmanan manfold, whch n turn follows from Remark 3.4. Theorem 3.5. Let X be an equdmensonal complex algebrac scheme. Let ρ be the composton of ρ wth the somorphsm gven by the Chern character ρ = : K n (X) Q CH p ρ (X, n) Q H 2p n D (X, R(p)). p 0 p 0 Then, the morphsm ρ agrees wth the Belnson regulator. Proof. The outlne of the proof s as follows. We frst recall the descrpton of the Belnson regulator n terms of homotopy theory of smplcal sheaves as n [15]. Then, we recall the constructon of the Chern character gven by Bloch. We proceed reducng the comparson of the two maps to the case n = 0 and for X a Grassmanan scheme. We fnally prove that at ths stage both maps agree. Our ste wll always be the small Zarsk ste over X. Consder X as a smooth quas-projectve scheme over C. Let B GL N be the smplcal verson of the classfyng space of the group GL N (C) vewed as a smplcal complex manfold. Recall that all the face morphsms are flat. Let B GL N,X be the smplcal sheaf over X gven by the sheaffcaton of the presheaf U B GL N (Γ(U, O U )) for every Zarsk open U X. Ths s the same as the smplcal sheaf gven by U Hom(U, B GL N ), where Hom means the smplcal functon complex. Consder the ncluson morphsms B GL N,X B GL N+1,X, for all N 1, and let B GL X = lm B GL N,X. Let Z B GL N,X and Z B GL X be the sheaves assocated to the respectve Bousfeld- Kan completons. Fnally, let Z be the constant smplcal sheaf on Z and consder the followng sheaves on X K X = Z Z B GL X, K N X = Z Z B GL N,X. By [15], Proposton 5, there s a natural somorphsm K m (X) = H m (X, K X ) = lm N H m (X, K N X).

21 HIGHER RITHMETIC CHOW GROUPS 21 Here H (, ) denotes the generalzed cohomoy wth coeffcents n K X and K N X, as descrbed n [15]. The Belnson regulator s the Chern character takng values n Delgne-Belnson cohomoy. The regulator can be descrbed n terms of homotopy theory of sheaves as follows. Consder the Dold-Puppe functor K ( ) (see [12]), whch assocates to every cochan complex of abelan groups concentrated n non-postve degrees, G, a smplcal abelan group K (G), ponted by zero. It satsfes the property that π (K (G), 0) = H (G ). In [13], Gllet constructs Chern classes whch nduce morphsms C D p H 2p (B GL N, R(p)), N 0, c D p,x : K N X, K (D X(, p)[2p]), N 0 n the homotopy category of smplcal sheaves. These morphsms are compatble wth the morphsms K N X, KN+1 X,. Therefore, we obtan a morphsm K m (X) = lm N H m (X, K N X) CD p,x H 2p m (X, R(p)). D Usng the standard formula for the Chern character n terms of the Chern classes, we obtan a morphsm K m (X) chd H 2p m (X, R(p)), whch s the Belnson regulator. The Chern character for hgher Chow groups. The descrpton of the somorphsm = K n (X) Q p 0 CHp (X, n) Q gven by Bloch follows the same pattern as the descrpton of the Belnson regulator. However, snce the complexes that defne the hgher Chow groups are not sheaves (n fact not even functors) on the bg Zarsk ste, a few modfcatons are necessary. We gve here a sketch of the constructon. For detals see [3]. If Y s a smplcal scheme whose face maps are flat, then there s a well-defned 2-terated cochan complex Z p (Y, ) 0, whose (n, m)-bgraded group s D Z p (Y n, m) 0, and nduced dfferentals. The hgher algebrac Chow groups of Y are then defned as CH p (Y, n) = H n (Z p (Y, ) 0 ). Snce the face maps of the smplcal scheme B GL N are flat, the group CH p (B GL N, n) s well defned for every p and n. Frst, Bloch constructs unversal Chern classes C CH p CH p (B GL N, 0), followng the deas of Gllet. These classes are represented by elements C CH, p Z p (B GL N, ) 0. Because at the level of complexes the pull-back morphsm s not defned for arbtrary maps, one cannot consder the pull-back of these classes Cp CH, to X, as was the case for the Belnson regulator. However, by [3] 7, there exsts a purely transcendental

22 22 J. I. BURGOS GIL ND E.FELIU extenson L of C, and classes C CH, p defned for every C-morphsm f : V B GL N. Then, there s a map of smplcal Zarsk sheaves on X defned over L, such that the pull-back f C CH, p B GL N,X K X (g Z p X L (, ) 0 ), where g : X L X s the natural map obtaned by extenson to L. There s a specalzaton process descrbed n [3], whch, n the homotopy category of smplcal sheaves over X, gves a well-defned map K X (g Z p X L (, ) 0 ) K X (Z p X (, ) 0). Therefore, there are maps Cp,X CH [B GL N,X, K X (ZX (, p))], where [, ] denotes the set of arrows n the homotopy category. Proceedng as above, we obtan the Chern character morphsm K m (X) p 0 CH p (X, m) Q. For m = 0, ths s the usual Chern character. End of the proof. Snce, at the level of complexes, ρ s functoral for flat maps, there s a sheaf map ρ : K X (Z X(, p)) K (D (X, p)) n the small Zarsk ste of X. It follows that the composton ρ C CH p s obtaned by the same procedure as the ) H 2p D (X, R(p)) Belnson regulator, but startng wth the characterstc classes ρ(cp CH nstead of the classes Cp D. Therefore, t remans to see that (3.6) ρ(c CH p ) = C D p. For ntegers N, k 0 let Gr(N, k) be the complex Grassmanan scheme of N-planes n C k. It s a smooth complex projectve scheme. Let E N,k be the rank N unversal bundle of Gr(N, k) and U k = (U k,α ) α ts standard trvalzaton. Let N U k denote the nerve of ths π cover. It s a hypercover of Gr(N, k), N U k Gr(N, k). Consder the classfyng map of the vector bundle E N,k, ϕ k : N U k B GL N, whch satsfes π (E N,k ) = ϕ k (EN ), for E N the unversal vector bundle over B GL N. Observe that all the faces and degeneracy maps of the smplcal scheme N U k are flat, as well as the ncluson maps N l U k Gr(N, k). Therefore, CH p (N U k, m) s defned and there s a pull-back map CH p (Gr(N, k), m) π CH p (N U k, m). Snce ρ s defned on N U k and s a functoral map, we obtan the followng commutatve dagram CH p (B GL N, 0) ρ H 2p D (B GL N, R(p)) s K 0 (Gr(N, k)) C CH p ϕ k CH p (N U k, 0) π CH p (Gr(N, k), 0) ρ ρ ϕ k H 2p D (N U k, R(p)) H 2p D π (Gr(N, k), R(p)) C D p

23 HIGHER RITHMETIC CHOW GROUPS 23 By constructon, Cp CH (E N,k ) s the standard p-th Chern class n the classcal Chow group of Gr(N, k), and Cp D (E N,k ) s the p-th Chern class n Delgne-Belnson cohomoy. It then follows from Remark 3.4 that (3.7) ρ(c CH p (E N,k )) = C D p (E N,k ). The vector bundle E N,k K 0 (Gr(N, k)) = lm [Gr(N, k), K M M homotopy category of smplcal sheaves, by the dagram Gr(N, k) π N U k ϕ k B GL N, ] s represented, n the where the map π s a weak equvalence of sheaves because N U k s a hypercover of Gr(N, k). Ths means that (3.8) ϕ k (CCH p (E N )) = π (C CH p (E N,k )). lso, snce π s an hypercover, π s an somorphsm n Delgne-Belnson cohomoy. Moreover, for each m 0, there exsts k 0 such that, f m m 0 and k k 0, ϕ k s an somorphsm on the cohomoy group HD 2m (, R(m)). To see ths, we frst use the computaton of the mxed Hodge structure of the cohomoy of the classfyng space gven n [11] and the well known mxed Hodge structure of the cohomoy of the Grassmanan manfolds to reduce t to a comparson at the level of sngular cohomoy. Then we use that the nfnte Grassmanan s homotopcally equvalent to the classfyng space. Fnally we use the cellular decomposton of the nfnte Grassmanan to compare ts cohomoy wth the cohomoy of the fnte Grassmanan (see for nstance [22]). Under these somorphsms, we obtan the equalty (3.9) C D p (E N,k ) = (π ) 1 ϕ k (CD p (E N )). Hence, ρ(c CH p (E N )) = C D p (E N ) ϕ k ρ(cch p (E N )) = ϕ k CD p (E N ) ρϕ k (CCH p (E N )) = ϕ k CD p (E N ). The last equalty follows drectly from (3.7), (3.8) and (3.9). Therefore, the theorem s proved. 4. Hgher arthmetc Chow groups Let X be an arthmetc varety over a feld. Usng the descrpton of the Belnson regulator gven n secton 3, we defne the hgher arthmetc Chow groups, ĈHn (X, p). The defnton s anaous to the defnton gven by Goncharov, n [16], but usng dfferental forms nstead of currents. We need to restrct ourselves to arthmetc varetes over a feld, because the theory of hgher algebrac Chow groups by Bloch s only well establshed for schemes over a feld. That s, we can defne the hgher arthmetc Chow groups for arbtrary arthmetc varetes, but snce the functoralty propertes and the product structure of the hgher algebrac Chow groups are descrbed only for schemes over a feld, we cannot gve a product structure or defne functoralty for the hgher arthmetc Chow groups of arthmetc varetes over a rng. Note however that, usng work by Levne [21], t should be possble to extend the constructons here to smooth varetes over a Dedeknd doman, at least after tensorng wth Q. In fact, when extendng the defnton to arthmetc varetes over a rng, t mght be better to use the pont of vew of motvc homoy à la Voevodsky or any of ts more recent varants.

24 24 J. I. BURGOS GIL ND E.FELIU 4.1. Hgher arthmetc Chow groups. Followng [14], an arthmetc feld s a trple (K, Σ, F ), where K s a feld, Σ s a nonempty set of complex mmersons K C and F s a conjugate-lnear C-algebra automorphsm of C Σ that leaves nvarant the mage of K under the dagonal mmerson. By an arthmetc varety X over the arthmetc rng K we mean a regular quas-projectve K-scheme X. To the arthmetc varety X we assocate a complex varety X C = ι Σ X ι, and a real varety X R = (X C, F ). The Delgne complex of dfferental forms on X s defned from the real varety X R as D n (X, p) := Dn (X C, p) σ=d, where σ s the nvoluton as n paragraph We defne anaously the chan complexes D 2p (X, p) 0, D 2p (X, p) 00, D 2p,Z p (X, p) 0, and D 2p,Z p (X, p) 00. Let be the composton : Z p (X, n) 0 R Z p (X, n) 0 R F R Z p (X R, n) 0 R = H p (X, n) 0. We consder the dagram of complexes of the type of (1.12) H p (X, ) 0 D 2p (X, p) 0 (4.1) Ẑ p γ (X, ) 0 = 1 γ 1 ρ Z p (X, ) 0 D 2p,Z (X, p) p 0 ZD 2p (X, p) where ZD 2p (X, p) s the chan complex whch s zero n all degrees except n degree zero, where t conssts of the vector subspace of cycles n D 2p (X, p). Note that t agrees wth ZE p,p,r (X)(p), the subspace of Ep,p,R (X)(p) consstng of dfferental forms wth arthmc sngulartes that are real up to a product by (2π) p, of type (p, p) and that vansh under and. The morphsm s the ncluson of chan complexes. Defnton 4.2. The hgher arthmetc Chow complex s the smple complex assocated to the dagram Ẑp (X, ) 0, as defned n (1.6): Ẑ p (X, ) 0 := s(ẑp (X, ) 0 ). Recall that, by defnton, Ẑp (X, n) 0 conssts of 5-tuples (Z, α 0, α 1, α 2, α 3 ) Z p (X, n) 0 D 2p n,z p (X, p) 0 ZD 2p (X, p) n H p (X, n+1) 0 D 2p n 1 (X, p) 0, and the dfferental s gven by Ẑ p (X, n) 0 d Ẑ p (X, n 1) 0 (Z, α 0, α 1, α 2, α 3 ) (δ(z), d s (α 0 ), 0, (Z) (α 0 ) δ(α 2 ), ρ(α 0 ) α 1 d s (α 3 )). Note that α 1 wll be zero unless n = 0. Its dfferental, however, s always zero. Defnton 4.3. Let X be an arthmetc varety over an arthmetc feld. The (p, n)-th hgher arthmetc Chow group of X s defned by ĈH p (X, n) := H n (Ẑp (X, ) 0 ), p, n 0.