(cid:1) (cid:1) (cid:1) (cid:1) Publ. RIMS,KyotoUniv. 41(2005),599–681 Higher Arithmetic K-Theory By Yuichiro Takeda∗ Abstract A concrete definition of higher K-theory in Arakelov geometry is given. The K-theory defined in this paper is a higher extension of the arithmetic K -group of 0 anarithmeticvarietydefinedbyGilletandSoul´e. Productsanddirectimagesinthis K-theory are discussed. §1. Introduction The aim of this paper is to provide a new definition of higher K-theory in Arakelov geometry and to show that it enjoys the same formal properties as the higher algebraic K-theory of schemes. Let X be a proper arithmetic variety, namely, a regular scheme which is flat and proper over Z, the ring of integers. In the research on the arithmetic Chern character of a hermitian vector bundle on X, Gillet and Soul´e defined (cid:1) the arithmetic K -group K (X) of X [9]. It can be viewed as an analogue in 0 0 Arakelov geometry of the K -group of vector bundles on a scheme. 0 (cid:1) AftertheadventofK (X),itshigherextensionwasdiscussed in[6,7,14]. 0 In these papers one common thing was suggested that higher arithmetic K- theory should be obtained as the homotopy group of the homotopy fiber of the Beilinson’s regulator map. To be more precise, there should exist a group KM (X) for each n≥0 fitting into the long exact sequence n ···→K (X)→ρ ⊕H2p−n−1(X,R(p))→KM (X)→K (X)→··· , n+1 D n n p CommunicatedbyS.Mochizuki. ReceivedApril25,2003. RevisedJune10,2004,August 10,2004. 2000MathematicsSubjectClassification(s): Primary14G40;Secondary19E08. ∗FacultyofMathematics,KyushuUniversity,6-10-1Hakozaki,Higashi-Ku,Fukuoka,812- 8581,Japan. e-mail: [email protected] (cid:1)c 2005ResearchInstituteforMathematicalSciences,KyotoUniversity. Allrightsreserved. (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) 600 Yuichiro Takeda whereHn(X,R(p))istherealDelignecohomologyofX andρistheBeilinson’s D regulator map. To get the homotopy fiber, a simplicial description of the regulator map is necessary. And it has already been given by Burgos and Wang in [6]. For a compact complex manifold M, they introduced an exact cube of hermitian vector bundles on M and associated with it a differential form called a higher Bott-Chern form. This gives a homomorphism of complexes ch:ZS(cid:1)∗(M)→D∗(M,p)[2p+1] from the homology complex ZS(cid:1)∗(M) of the S-construction of the category of hermitian vector bundles on M to the complex D∗(M,p) computing the real Deligne cohomology of M defined in [4]. It is the main theorem of [6] that the following map coincides with the Beilinson’s regulator map: ρ:Kn(M)(cid:4)πn+1(S(cid:1)(M))Hu−re→wiczHn+1(ZS(cid:1)∗(M))H−(→ch)HD2p−n(M,R(p)). ApplyingthistothecomplexmanifoldX(C)associatedwithX,wecanobtain a simplicial description of the regulator map for X. Inthispaper,wewillgiveanotherdefinitionofhigherarithmeticK-theory for a proper arithmetic variety. One of remarkable features of our arithmetic K-theory is that it is given as an extension of the algebraic K-theory by the cokernel of the regulator map. (cid:1) Before explaining our method, let us recall the definition of K (X) by 0 Gillet and Soul´e [9]. For a proper arithmetic variety X, let Ap,p(X) be the ∗ space ofreal(p,p)-formsω onX(C)such thatF ω =(−1)pω forthe complex ∞ conjugation F∞ on X(C) and let A(cid:2)(X) = ⊕pAp,p(X)/(Im∂ +Im∂). Then (cid:1) K (X)isdefined asafactorgroupof thefreeabeliangroupgeneratedby pairs 0 (E,ω) where E is a hermitian vector bundle on X and ω ∈ A(cid:2)(X). Relations on pairs are given by each short exact sequence E : 0 → E(cid:3) → E → E(cid:3)(cid:3) → 0 and ω(cid:3),ω(cid:3)(cid:3) ∈A(cid:2)(X) as follows: (E(cid:3),ω(cid:3))+(E(cid:3)(cid:3),ω(cid:3)(cid:3))=(E,ω(cid:3)+ω(cid:3)(cid:3)+c(cid:2)h(E)), where c(cid:2)h(E) is the Bott-Chern secondary characteristic class of E. (cid:1) The above definition of K (X) can be rephrased in terms of loops and 0 homotopies on |S(cid:1)(X)|, the topological realization of the S-construction of the category of hermitian vector bundles on X. Consider a pair (l,ω), where l is a pointed simplicial loop on |S(cid:1)(X)| and ω ∈ A(cid:2)(X). Two pairs (l,ω) and (l(cid:3),ω(cid:3)) are said to be homotopy equivalent if there is a cellular homotopy H : (S1×I)/({∗}×I) → |S(cid:1)(X)| from l to l(cid:3) such that the Bott-Chern secondary (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) Higher Arithmetic K-Theory 601 (cid:2) characteristic class ch(H) of H, which is defined in a natural way, is equal to ω(cid:3)−ω. Letπ(cid:1) (|S(cid:1)(X)|,c(cid:2)h)denotethesetofallequivalenceclassesofsuchpairs. 1 Then it carries the structure of an abelian group and the map K(cid:1) (X)→π(cid:1) (|S(cid:1)(X)|,c(cid:2)h) 0 1 given by (E,ω) (cid:7)→ (l ,−ω), where l is the simplicial loop on |S(cid:1)(X)| deter- E E mined by E, is proved to be bijective. Let us generalize this observation to higher homotopy groups in a general setting. Take a pointed CW-complex T and a homomorphism ρ:C∗(T)→W∗ fromthe reducedhomology complex ofT toa chain complex of abelian groups W∗. Let Sn denote the n-dimensional sphere and consider a pair (f,ω) of a pointed cellular map f : Sn → T and ω ∈ W(cid:3) = W /Im∂. Two pairs (f,ω) n n and (f(cid:3),ω(cid:3)) are said to be homotopy equivalent if there is a pointed cellular homotopy H : (Sn ×I)/({∗}×I) → T from f to f(cid:3) such that the image of the fundamental chain of H by ρ is equal to (−1)n+1(ω(cid:3) −ω). This actually gives an equivalence relation on the set of such pairs. The set of homotopy equivalence classes has the structure of a group and it becomes an abelian group when n ≥ 2. This group is denoted by π(cid:1) (T,ρ) and called the n-th n homotopy group of T modified by ρ. We define the n-th arithmetic K-theory of a proper arithmetic variety X as the (n+1)-th homotopy group of |S(cid:1)(X)| modified by the higher Bott-Chern form: K(cid:1) (X)=π(cid:1) (|S(cid:1)(X)|,ch). n n+1 (cid:1) WewillshowthatK (X)possessesthesamepropertiesastheusualhigher n K-theory of schemes. More precisely, we will show the following: (1) Fundamental exact sequence: K (X)→D(cid:2) (X)→K(cid:1) (X)→K (X)→0, n+1 n+1 n n where D(cid:2)n+1(X) = Dn+1(X)/ImdD. In the case of K(cid:1)0(X), this exact sequence has been obtained in [9]. (2) Chern class map: ch :K(cid:1) (X)→D (X). n n n If we denote KM (X) = Kerch , then we can obtain the long exact se- n n quence ···→K (X)→ρ ⊕H2p−n−1(X,R(p))→KM (X)→K (X)→ρ ··· . n+1 D n n p (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) 602 Yuichiro Takeda Weshow thatKM (X)iscanonicallyisomorphic tothehomotopy fiberof n the Bott-Chern form. (3) Arakelov K-theory: Fix an F∞-invariant Ka¨hler metric hX on X(C). The pair X = (X,h ) is called an Arakelov variety. We can define Arakelov X K-group of X as (cid:4) (cid:5) K (X)= x∈K(cid:1) (X);ch (x) is a harmonic form with respect to h . n n n X We have the exact sequence K (X)→ρ ⊕H2p−n−1(X,R(p))→K (X)→K (X)→0. n+1 D n n p (cid:1) (4) Products: K∗(X) has a product K(cid:1) (X)×K(cid:1) (X)→K(cid:1) (X). n m n+m It does not admit the associative law. But if we restrict this to K∗(X), it becomes associative. It is shown that the product is graded commutative up to 2-torsion. (5) Functoriality: For arbitrary morphism f :X →Y, we can define pull back map f(cid:1)∗ :K(cid:1) (Y)→K(cid:1) (X). n n It is compatible with product. Suppose that f is smooth and projective, and fix a Ka¨hler metric on the relative tangent bundle of f(C) : X(C) → Y(C). Then we can define direct image homomorphism f(cid:1)∗ :K(cid:1)n(X)→K(cid:1)n(Y). The projection formula for f(cid:1)∗ and f(cid:1)∗ holds. From the above properties, we can obtain a non-canonical decomposition (cid:1) of K (X) into three summands: n (cid:6) (cid:7) K(cid:1)n(X)(cid:4)Kn(X)⊕(Dn+1(X)/KerdD)⊕ ⊕HD2p−n−1(X,R(p))/Imρ . p TheBass’conjecturesaysthatthefirstsummandisafinitelygeneratedabelian group. ThesecondoneisaninfinitedimensionalR-vectorspace,andtheBeilin- son’s conjectures imply that the third one becomes a real torus. Let us describe the organization of the paper: In §2 we introduce some materials used in the paper, such as S-construction, exact cubes and higher (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) Higher Arithmetic K-Theory 603 Bott-Chern forms. In §3 we propose the notion of modified homotopy groups. In§4wegivethedefinitionofthehigherarithmeticK-groupK(cid:1)∗(X)anddeduce the fundamental exact sequence. We also define the Arakelov K-group. In §5 we prove a product formula for higher Bott-Chern forms. It provides an alternative proof of the fact that the regulator map respects the products. In §6, we discuss product in higher arithmetic K-theory. In §7, we define a direct image homomorphism in higher arithmetic K-theory. To do this we employ thehigheranalytictorsionformofanexacthermitiancubedefinedbyRoessler [13]. Moreover we establish the projection formula. §2. Preliminaries §2.1. Conventions on complexes Letusfirstsettlesomeconventionsoncomplexes. Bycomplexofanabelian category A, we mean a family of objects {Ak}k∈Z with differential dA : Ak → Ak+1. For a complex A∗ and n ∈ Z, the n-th translation A[n]∗ is defined as A[n]k =An+k and d =(−1)nd . A[n] A By chain complex we mean a family of objects {Ak}k≥0 with boundary ∂A : Ak → Ak−1. For a complex A∗ = (Ak,dA) such that Ak = 0 for k > 0, we can define a chain complex A∗ as Ak = A−k and dA = ∂A. The n-th translation A[n]∗ of a chain complex A∗ for n ≥ 0 is defined as A[n]k =Ak−n and ∂ =(−1)n∂ . A[n] A §2.2. S-construction In this subsection we recall the S-construction developed by Waldhausen [15]. Throughout this paper, we assume that any small exact category has a distinguished zero object denoted by 0. Let [n] be the finite ordered set {0,1,...,n}andAr[n]thecategoryofarrowsof[n]. Forasmallexactcategory A, let S A be the set of functors E : Ar[n] → A satisfying the following n conditions for E =E(i≤j): i,j (1) E =0 for any 0≤i≤n. i,i (2) For any i≤j ≤k, E →E →E is a short exact sequence of A. i,j i,k j,k For example, S A = {0}, S A is the set of objects of A and S A is the set of 0 1 2 short exact sequences of A. The functor SA:[n](cid:7)→ S A becomes a simplicial n set with the base point given by 0 ∈ S A. It is shown in [15] that SA is 0 homotopy equivalent to the Quillen’s Q-construction of A. Therefore the (n + (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) 604 Yuichiro Takeda 1)-th homotopy group π (SA,0) is isomorphic to K (A), the algebraic K- n+1 i theory of A. §2.3. Exact n-cubes Let us recall the notion of an exact n-cube. For more details, see [5, 6]. Let (cid:9)−1,0,1(cid:10) be the ordered set consisting of three elements. An n-cube of a small exact category A is a covariant functor from the n-th power of (cid:9)−1,0,1(cid:10) to A. For an n-cube F, we denote by F the image of an α1,...,αn object (α ,...,α ) of (cid:9)−1,0,1(cid:10)n. For integers i and j satisfying 1 ≤ i ≤ 1 n n and −1 ≤ j ≤ 1, an (n − 1)-cube ∂jF is given by (∂jF) = i i α1,...,αn−1 F . It is called a face of F. For an object α of (cid:9)−1,0, α1,...,αi−1,j,αi,...,αn−1 1(cid:10)n−1 and an integer i satisfying 1≤i≤n, a 1-cube ∂αF called an edge of F ic is Fα1,...,αi−1,−1,αi,...,αn−1 →Fα1,...,αi−1,0,αi,...,αn−1 →Fα1,...,αi−1,1,αi,...,αn−1. An n-cube F is said to be exact if all edges of F are short exact sequences. Let C A denote the set of all exact n-cubes of A. If F is an exact n-cube, n then any face ∂jF is also exact. Hence ∂j induces a map i i ∂ij :CnA→Cn−1A. Let F be an exact n-cube of A. For an integer i satisfying 1≤i≤n+1, let s1F be an exact (n+1)-cube such that its edge ∂α(s1F) is F →id F →0. i ic i α α Similarly, let s−1F be an exact (n+1)-cube such that ∂α(s−1F) is 0→F →id i ic i α F . An exact cube written as sjF is said to be degenerate. α i LetZC AbethefreeabeliangroupgeneratedbyC AandD ⊂ZC Athe n n n n subgroup generated by all degenerate exact n-cubes. Let Z(cid:2)C A = ZC A/D n n n and (cid:8)n (cid:8)1 ∂ = (−1)i+j+1∂ij :Z(cid:2)CnA→Z(cid:2)Cn−1A. i=1j=−1 Then Z(cid:2)C∗A=(Z(cid:2)CnA,∂) becomes a chain complex. In [6, §4.4], an exact (n−1)-cube Cub(E) for any E ∈S A is constructed n and it is shown that E (cid:7)→Cub(E) induces a homomorphism of complexes Cub:ZS∗A[1]→Z(cid:2)C∗A. (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) Higher Arithmetic K-Theory 605 §2.4. Higher Bott-Chern forms In this subsection we recall higher Bott-Chern forms developed by Burgos and Wang. For more details, see [5, 6]. First we introduce the recipient of higher Bott-Chern forms. Let M be a compact complex algebraic manifold, namely, the analyticspace consisting of allC-valuedpoints ofasmoothproper algebraic variety over C. Let Ep(M) be the space of real smooth differential R forms of degree p on M and Ep(M)=EpR(M)⊗RC. Let Ep,q(M) be the space of complex differential forms of type (p,q) on M. Set  EnR−1(M)(p−1)∩p(cid:1)+q⊕(cid:1)=n−1Ep(cid:1),q(cid:1)(M), n<2p, Dn(M,p)= p(cid:1)<p,q(cid:1)<p E2Rp(M)(p)∩Ep,p(M)∩Kerd, n=2p, 0, n>2p and define a differential dD :Dn(M,p)→Dn+1(M,p) by  −π(dω), n<2p−1, dD(ω)=−2∂∂ω, n=2p−1, 0, n>2p−1, where π :En(M)→Dn(M,p) is the canonical projection. Then it is shown in [4,Thm.2.6]thatthepair(D∗(M,p),dD)isacomplexofR-vectorspaceswith Hn(D∗(M,p),dD)(cid:4)HDn(M,R(p)) for n≤2p. By a hermitian vector bundle E = (E,h) on M we mean an algebraic vector bundle E on M with a smooth hermitian metric h. Let K denote the E curvatureformoftheuniqueconnectiononE thatiscompatiblewithboththe metric and the complex structure. Let us write ch (E)=Tr(exp(−K ))∈⊕D2p(M,p). 0 E p An exact hermitian n-cube on M is an exact n-cube made of hermitian vector bundles on M. Let F = {E } be an exact hermitian n-cube on α M. We call F an emi-n-cube if the metric on any E with α = 1 coin- α i cides with the metric induced from E for the surjection α1,...,αi−1,0,αi+1,...,αn E →E . α1,...,αi−1,0,αi+1,...,αn α For an emi-1-cube E : E−1 → E0 → E1, a canonical way of constructing a hermitian vector bundle tr1E on M ×P1 connecting E0 with E−1 ⊕E1 is (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) 606 Yuichiro Takeda given in [6]. More precisely, if (x : y) denotes the homogeneous coordinate of P1 and z = x/y, then tr E is a hermitian vector bundle on M ×P1 satisfying 1 the following conditions: tr1E|z=0 (cid:4)E0, tr1E|z=∞ (cid:4)E−1⊕E1. For an emi-n-cube F, let tr (F) be an emi-(n−1)-cube on M ×P1 given by 1 tr (F) = tr (∂α (F)) for α ∈ (cid:9)−1,0,1(cid:10)n−1, and tr (F) a hermitian vector 1 α 1 nc n bundle on M ×(P1)n given by (cid:13) nt(cid:14)im(cid:15)es (cid:16) tr (F)=tr tr ...tr (F). n 1 1 1 Let π : (P1)n →P1 be the i-th projection and z = π∗z. For an integer i i i i satisfying 1≤i≤n, (cid:8) dz dz dz¯ dz¯ Si = (−1)σlog|z |2 σ(2) ∧···∧ σ(i) ∧ σ(i+1) ∧···∧ σ(n), n σ(1) z z z¯ z¯ σ∈Sn σ(2) σ(i) σ(i+1) σ(n) which is a differential form with logarithmic poles on (P1)n. The Bott-Chern form of an emi-n-cube F is (cid:17) 1 ch (F)= √ ch (tr (F))∧T ∈⊕D2p−n(M,p), n (2π −1)n (P1)n 0 n n p where (−1)n(cid:8)n T = (−1)iSi. n 2n! n i=1 Aprocesstoproduceanemi-n-cubeλF fromanarbitraryexacthermitian n-cubeF isgivenin[6]. Byvirtueofthisprocess, wecanextendthedefinition of the Bott-Chern form to an arbitrary exact hermitian n-cube. Definition 2.1. The Bott-Chern form of an exact hermitian n-cube F is an element of ⊕ D2p−n(M,p) given as follows: p (cid:17) 1 ch (F)= √ ch (tr (λF))∧T . n (2π −1)n (P1)n 0 n n Theorem 2.2 ([6]). Let P(cid:1)(M) denote the category of hermitian vector bundles on M and let Z(cid:2)C(cid:1)∗(M) = Z(cid:2)C∗P(cid:1)(M). Then F (cid:7)→ chn(F) induces a homomorphism ch:Z(cid:2)C(cid:1)∗(M)→⊕D∗(M,p)[2p]. p (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) Higher Arithmetic K-Theory 607 Moreover, the following map Kn(M)=πn+1(S(cid:1)(M))Hu−re→wiczHn+1(ZS(cid:1)∗(M)) −C→ub Hn(Z(cid:2)C(cid:1)∗(M))→ch ⊕HD2p−n(M,R(p)) p coincides with the Beilinson’s regulator map. §3. Modified Homotopy Groups §3.1. Definition of modified homotopy groups Inthissectionwedevelopageneralframeworkusedlaterinthispaper. Let I be the closed interval [0,1] equipped with the usual CW-complex structure. Throughout this paper we identify the n-dimensional sphere Sn with In/∂In. Therefore Sn consists of two cells and any point of Sn except the base point is expressed by an n-tuple of real numbers (t ,...,t ) with 0<t <1. 1 n i Let T be a pointed CW-complex and ∗ ∈ T the base point. Let sk (T) n be the n-th skeleton of T when n ≥ 0 and sk−1(T) = {∗}. For n ≥ 0, let us write Cn(T) = Hn(skn(T),skn−1(T);Z), the n-th relative homology group of the pair (skn(T),skn−1(T)). Let ∂ : Cn(T) → Cn−1(T) be the connecting homomorphism forthetriple(skn(T),skn−1(T),skn−2(T)). Then(C∗(T),∂)is achaincomplexwhosehomologygroupisisomorphic tothereducedhomology group of T. Suppose that a chain complex of abelian groups (W∗,∂) and a homomor- phism of chain complexes ρ:C∗(T)→W∗ are given. Let W(cid:3)n =Wn/Im∂. Let us consider a pair (f,ω) of a pointed cellular map f :Sn →T and ω ∈W(cid:3) . n+1 A cellular homotopy from one pair (f,ω) to another pair (f(cid:3),ω(cid:3)) is a pointed cellular map H :(Sn×I)/({∗}×I)→T satisfying the following: (1) H(x,0)=f(x) and H(x,1)=f(cid:3)(x). (2) Let [Sn ×I] ∈ C (Sn ×I) denote the fundamental chain of Sn ×I, n+1 wheretheorientationonSn×I isinheritedfromthecanonicalorientation of the interval I. Then ω(cid:3)−ω =(−1)n+1ρH∗([Sn×I]). It can be shown that the cellular homotopy gives an equivalence relation on the set of pairs. Two pairs are said to be homotopy equivalent if there exists a cellularhomotopybetweenthem. Wedenotebyπ(cid:1) (T,ρ)thesetofallhomotopy n equivalence classes of pairs. (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) 608 Yuichiro Takeda Let us define a multiplication on the set π(cid:1) (T,ρ). Let T ∨T = {(x,y) ∈ n T ×T;x = ∗ or y = ∗}. Then we can define a natural map T ∨T → T by (x,∗) (cid:7)→ x and (∗,y) (cid:7)→ y. A comultiplication map µ : Sn → Sn∨Sn is given by (cid:18) ((t ,t ,...,2t ),∗), 0<t ≤ 1, µ(t ,...,t )= 1 2 n n 2 1 n (∗,(t ,t ,...,2t −1)), 1 ≤t <1, 1 2 n 2 n andahomotopyinversemapν :Sn →Sn byν(t1,...,tn−1,tn)=(t1,...,tn−1, 1−t ). For two pointed cellular maps f,g :Sn →T, let us write n f ·g :Sn →µ Sn∨Sn −f∨→g T ∨T →T, and f−1 :Sn →ν Sn →f T. A multiplication of two pairs (f,ω) and (g,τ) is (f,ω)·(g,τ)=(f ·g,ω+τ). It is easy to show that the multiplication · is compatible with the homotopy equivalencerelationonpairs. Henceitgivesrisetoamultiplicationonπ(cid:1) (T,ρ). n Letusnextverifytheassociativityofthemultiplication. Forthreepointed cellularmapsf,g,h:Sn →T,acellularhomotopyH :(Sn×I)/({∗}×I)→T 1 from (f ·g)·h to f ·(g·h) is given as follows:  f(t1,...,tn−1,u4+tn1), 0<tn ≤ u+41, H1(t1,...,tn−1,tn,u)=g(t1,...,tn−1,4tn−u−1), u+41 ≤tn ≤ u+42, h(t1,...,tn−1,4tn2−−2u−u), u+42 ≤tn <1. Since the image of H1 is contained in skn(T), we have (H1)∗([Sn×I]) = 0 in C (T). Hence H becomes a cellular homotopy from ((f,ω)·(g,τ))·(h,η) n+1 1 to (f,ω)·((g,τ)·(h,η)) for any ω,τ,η ∈W(cid:3) . n+1 Finally we show the existence of unit and inverse with respect to the mul- tiplication ·. Let 0:Sn →T be the map given by 0(Sn)={∗}. For a pointed cellular map f :Sn →T, a homotopy H from f ·0 to f is given as follows: 2 (cid:18) H2(t1,...,tn−1,tn,u)= f∗,(t1,...,tn−1,u2+tn1), 0u+<1t≤n ≤t u<+211., 2 n (cid:1) (cid:1) (cid:1) (cid:1)