T&T Proof JIMJBKK 30 August 2006 Journal of the Inst. of Math. Jussieu(2007)0(0),1–172 (cid:1)c CambridgeUniversityPress 1 doi:10.1017/S1474748007000016 PrintedintheUnitedKingdom COHOMOLOGICAL ARITHMETIC CHOW RINGS J. I. BURGOS GIL1, J. KRAMER2 AND U. KU¨HN3 1Facultad de Matema´ticas, Universidad de Barcelona, Gran V´ıa 318 4◦ 1a, 08007 Barcelona, Spain ([email protected]) 2Institut fu¨r Mathematik, Humboldt-Universita¨t zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany 3Department Mathematik, Universita¨t Hamburg, Bundesstraße 55, D-20146 Hamburg, Germany (Received 24 October 2003; accepted 8 January 2005) Abstract WedevelopatheoryofabstractarithmeticChowrings,wheretheroleofthefibresatinfinity isplayedbyacomplexofabeliangroupsthatcomputesasuitablecohomologytheory.Asparticularcases ofthisformalismwerecovertheoriginalarithmeticintersectiontheoryofGilletandSoul´eforprojective varieties.WeintroduceatheoryofarithmeticChowgroups,whicharecovariantwithrespecttoarbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a complex of differential formswithlog-logsingularitiesalongafixednormalcrossingdivisor.Thislasttheoryissuitableforthe studyofautomorphiclinebundles.Inparticular,wegeneralizetheclassicalFaltingsheightwithrespect to logarithmically singular hermitian line bundles to higher dimensional cycles. As an application we computetheFaltingsheightofHeckecorrespondencesonaproductofmodularcurves. Keywords: Arakelovgeometry;sheafcohomology;Deligne–Beilinsoncohomology; goodhermitianmetrics AMS2000Mathematicssubjectclassification:Primary14G40;14G35;14C17;14C30;11G18 Contents 0. Introduction 2 1. Background results on K-theory 13 1.1. K-theory 14 1.2. K-theory as generalized sheaf cohomology 16 1.3. λ-structure in K-theory and absolute cohomology 22 1.4. Chow Groups of regular schemes 27 1.5. Gillet cohomology and characteristic classes 33 2. Some topics from homological algebra 40 2.1. k-iterated complexes 40 2.2. Relative cohomology groups 46 2.3. Products in relative cohomology 49 2.4. Truncated relative cohomology groups 52 2.5. Products in truncated relative cohomology 54 2 J. I. Burgos Gil, J. Kramer and U. Ku¨hn 3. Green objects 58 3.1. G-complexes 58 3.2. Definition of Green objects 62 3.3. The ∗-product of Green objects 66 3.4. Associativity and commutativity 70 3.5. Functorial properties of Green objects 76 4. Abstract arithmetic Chow groups 83 4.1. Arithmetic varieties 83 4.2. Arithmetic Chow groups 85 4.3. Arithmetic intersection pairing 87 4.4. Inverse images 91 4.5. Proper push-forward 93 5. Deligne–Beilinson cohomology as a Gillet cohomology 95 5.1. Review of Deligne–Beilinson cohomology 95 5.2. Review of Deligne algebras 97 5.3. A Gillet complex: the Deligne algebra D 102 log 5.4. Deligne–Beilinson homology of proper smooth varieties 106 5.5. Deligne–Beilinson homology of arbitrary varieties 109 5.6. Classes of cycles and line bundles 116 5.7. Real varieties 119 6. Examples of D -arithmetic Chow groups 121 log 6.1. Contravariant D -arithmetic Chow rings 121 log 6.2. Covariant D -arithmetic Chow groups 130 log 6.3. Height of a cycle 135 7. Arithmetic Chow rings with pre-log-log forms 137 7.1. Pre-log-log forms 137 7.2. Pre-log forms 140 7.3. Mixed forms 141 7.4. A D -complex with pre-log-log forms 144 log 7.5. Properties of Green objects with values in D 145 pre 7.6. Push-forward of a ∗-product 149 7.7. Arithmetic Chow rings with pre-log-log forms 157 7.8. Application to products of modular curves 164 References 169 0. Introduction Background ArakelovgeometrywasinitiatedbyArakelovinhispaper[1],whereheshowsthatonecan compactify acurvedefinedoverthe(spectrumofthe)ringofintegersofanumberfieldby considering Green functions on the associated complex curve. Subsequently, Arakelov’s ideashavebeensuccessfullyrefinedbyDeligne[25]andFaltings[29],andgeneralizedto higher dimensional arithmetic varieties by Gillet and Soul´e who introduced arithmetic Cohomological arithmetic Chow rings 3 Chowgroupsforhighercodimensionalarithmeticcyclesin[36].Animportantapplication of this formalism was Vojta’s proof of Faltings’s theorem, formerly known as Mordell’s conjecture (see [74]). TheuseofanalyticalobjectssuchasGreenfunctionsallowsvariousmodificationsand extensions of the formalism of arithmetic Chow groups, which is adapted to different situations. Among others, we mention Zhang’s work [75] on admissible metrized line bundlesandMaillot’swork[60]takingintoaccounthermitianlinebundleswhosemetrics are no longer smooth but still continuous; Bost’s work [9] on L2-Green functions and 1 Moriwaki’swork[63]ondegenerateGreencurrents;Ku¨hn’swork[58]treatinghermitian line bundles on arithmetic surfaces having logarithmically singular metrics, e.g. the line bundle of modular forms equipped with the Petersson metric on a modular curve, and Burgos’s work [14,17] on arithmetic Chow groups and Deligne–Beilinson cohomology. Motivation The main motivation of these notes is to extend Ku¨hn’s generalized arithmetic inter- section pairing on arithmetic surfaces to higher dimensional arithmetic varieties. While Ku¨hn’s intersection pairing, which was motivated by preliminary results of Kramer (see [53]), is given in terms of an explicit formula for the arithmetic intersection number of two divisors (in the spirit of Arakelov), the development of a corresponding higher dimensional theory needs to be approached in a more abstract way. Moreover, we will extend this theory, not only to line bundles with singular metrics, but also to higher rank vector bundles such as the Hodge bundle that appear when considering fibrations of semi-abelian varieties. The study of higher rank vector bundles will be the subject of the forthcoming paper [19]. Arithmetic intersection theory An arithmetic ring (A,Σ,F∞) is a triple consisting of an excellent regular noetherian integral domain A, a finite non-empty set Σ of monomorphisms σ : A → C, and an antilinear involution F∞ : CΣ → CΣ of C-algebras. For simplicity, we will forget about the antilinear involution F∞ in this introduction. Then, an arithmetic variety is a flat, regularschemeX overS =Spec(A)togetherwithacomplexanalyticspaceX∞ obtained fromX bymeansofΣ;S willbecalledthebasescheme.Intuitively,theelementsofΣ are thepointsatinfinityofSprovidinga‘compactification’ofS,andtheanalyticspaceX∞is thefibreatthesepointsatinfinityorarchimedeanfibreofX.Anarithmeticintersection theory will involve three main ingredients: first, a geometric intersection theory over the scheme X, the geometric part; second, a ‘refined’ intersection theory over X∞, the analytic part; and finally an interface relating the geometric and the analytic part. The main theme of this paper is to study the second and third of these ingredients, and we will rely on existing geometric intersection theories. Geometric intersection theory It might be useful to review, at this point, the geometric intersection theories we have at our disposal. 4 J. I. Burgos Gil, J. Kramer and U. Ku¨hn The first of these theories is based on the moving lemma to reduce the intersection product of two algebraic cycles to the case of proper intersection. In order to be able to apply the moving lemma we need the scheme X under consideration to be quasi- projectiveandthearithmeticringAtobeafield.Sinceweareinterestedinmoregeneral arithmetic rings, we cannot use this method. The second approach is the deformation to the normal cone technique due to Fulton andMacPherson(see[33]).Thismethodismuchmoregeneral,sincetheschemeX need notbequasi-projectiveand,moreover,itcanalsobeappliedtothecaseinwhichthebase scheme is the spectrum of a Dedekind domain. But in this case X needs to be not only regular, but smooth over S. Since most interesting arithmetic varieties are not smooth over the base scheme, this method is not general enough for our purposes. Note however that, in the case in which X is smooth over S, this method provides an intersection pairing which is defined without tensoring with Q. We should also note that, in contrast tothecaseofanintersectionproduct,thismethodcanbeusedtodefinetheinverseimage morphism for a morphism between regular schemes of finite type over the spectrum of a regular, noetherian ring without the assumption of smoothness. The lack of a general theorem of resolution of singularities prevents us from obtaining an intersection product from this inverse image morphism. However, as a third approach, one can use de Jong’s theorem on alterations [22] to reduce the intersection of algebraic cycles to the inverse image between regular schemes, and then apply the deformation to the normal cone technique. Nevertheless, these alter- ations are finite morphisms whose degree is larger that one, in general. Therefore, this method yields an intersection product with rational coefficients. The last general approach that we will mention (and the one introduced originally by GilletandSoul´e)istousetheisomorphismbetweenK-theoryandChowgroupstotrans- fer the ring structure of K-theory to the Chow groups. This method is valid for any reg- ular, noetherian scheme X of finite Krull dimension. The main drawback of this method isthattheisomorphismbetweenK-theoryandChowgroupsisonlytrueingeneralafter tensoring with Q. Therefore, one also obtains a rational valued intersection product. Finally,wenotethattheintersectionproductwithadivisoronaregularschemecanbe defineddirectlyusingasimpleversionofthemovinglemmafordivisors(see[36,4.2.3.2]), which holds in complete generality. TheK-theoreticalmethodandthealterationmethodarethemostgeneraloftheabove methods. Since the K-theoretical approach was the one used by Gillet and Soul´e in [36] and,conceptually,fitsverywellwiththecohomologicalapproachwewillbeusingforthe analyticalpart,wealsouseitasthegeometricpartofourarithmeticintersectiontheory. But, since the geometric and the analytic part of our arithmetic intersection theory are isolated and related by a clear interface, we hope that the reader will have no difficulty in using any other of these geometric methods when applicable. Green currents and Green forms We now discuss the refined intersection theories which have been used as the analytic part of a higher dimensional arithmetic intersection theory. Cohomological arithmetic Chow rings 5 The first one is the notion of a Green current introduced in [36]. Let y be a p- codimensional cycle on X. A Green current for y is a class of currents g ∈D(cid:1)p−1,p−1 =Dp−1,p−1/(∂Dp−2,p−1+∂¯Dp−1,p−2) y X∞ X∞ X∞ X∞ satisfying the residue equation ddcg +δ =ω , (0.1) y y y where δy is the current given by integration along the cycle y∞ and ωy is a smooth form uniquely determined by (0.0). If y and z are two cycles intersecting properly (i.e. with the appropriate codimension) in X∞, the product of two Green currents, called the star-product or ∗-product, is given by the formula g ∗g =g ∧δ +ω ∧g . y z y z y z It is a Green current for the intersection cycle y·z. This approach has many analytical difficulties. For instance, some care has to be taken to define g ∧δ . Moreover, this y z product is not obviously associative and commutative, and the proof that this is indeed the case, is not trivial. As we have already mentioned this approach has been generalized in several direc- tions. Typically, these generalizations allow the presence of certain singularities for the differential form ωy. But usually only the case when X∞ is of dimension one is treated (see [9,58]), or one does not obtain a full ring structure for the generalized arithmetic Chow rings (see [60]). There are other proposals for the definition of the product of Green currents. For instance, Harris and Wang [41] have given a definition of the star-product of two Green currents for non-properly intersecting cycles that depends on a deformation of one of the cycles, and Dan [20] has given a definition of the star-product using meromorphic continuation of certain zeta functions. In [17] Burgos introduced a new definition of Green forms along the following lines. To every complex algebraic manifold X (not necessarily compact), there is associated a graded complex D∗ (X,∗), which consists of smooth forms on X with logarithmic log singularities at infinity. For instance, if X is proper, then D2p(X,p)=Ep,p(X)∩(2πi)pE2p(X), log R D2p−1(X,p)=Ep−1,p−1(X)∩(2πi)p−1E2p−2(X), log R whereEp,p(X)isthespaceofsmoothcomplexvalueddifferentialformsoftype(p,p)and E2p(X) is the space of smooth real valued differential forms. The boundary morphism R dD :Dl2opg−1(X,p)→Dl2opg(X,p) is given by dDη =−2∂∂¯η. Observe that, up to a normalization factor, this is the same differential operator as the one that appears in the residue equation (0.0). The complex D∗ (X,∗) computes the real Deligne–Beilinson cohomology of X, which log is denoted by H∗(X,R(p)). If Y is a closed subset of X, then the real Deligne–Beilinson D 6 J. I. Burgos Gil, J. Kramer and U. Ku¨hn cohomology of X with supports on Y, denoted by H∗ (X,R(p)), is the cohomology of D,Y the simple complex associated to the morphism of complexes D∗ (X,∗)→D∗ (X \Y,∗). log log Every p-codimensional algebraic cycle y with support Y defines a cohomology class cl(y)∈H2p (X,R(p)). Moreover, if W ⊆X is a closed subvariety of codimension p−1 D,Y and f ∈K(W)∗, then f defines a class cl(f)∈H2p−1(X \div(f),R(p)). D These classes satisfy the compatibility condition δ(cl(f))=cl(div(f)), where δ :H2p−1(X \div(f),R(p))→H2p (X,R(p)) D D,div(f) is the connecting homomorphism. In this setting a Green form for a p-codimensional algebraic cycle y is a representative of the class cl(y). More explicitly, we write ZDl∗og(X,∗)=Ker(dD), D(cid:1)l∗og(X,∗)=Dl∗og(X,∗)/Im(dD). ThespaceofGreenformsassociatedtoap-codimensionalcycley withsupportY isthen given by (cid:2) (cid:3) (cid:4) GE(y)= (ωy,g(cid:1)y)∈ZDl2opg(X,p)⊕D(cid:1)l2opg−1(X \Y,p)(cid:3)(cid:3)(cid:3) cl((ω d,g(cid:1)Dg(cid:1))y)==ωcly(y) . y y The star-product of Green forms is now simply the cup product in cohomology with supports. With this approach, the proof of the associativity and commutativity of the star-product is straightforward and completely formal. In [16] and [17] it is proven that, when X is projective, the arithmetic Chow groups obtained by this method agree with the ones obtained by the method of Gillet and Soul´e. It is interesting to note that all the analytical complexities appearing in the proof of the associativity and commutativity of the star-product in [36] are needed to prove the compatibility of the two definitions. In contrast, in the quasi-projective case, the groups obtained by this new method have better Hodge theoretical properties. For instance, they possess a certain homotopy invariance with respect to vector bundles. Another advantage of this new definition is that it is very easy to make variants adapted to new problems just by changing the complex D . log We should stress here that a Green form associated to a cycle is a representative of the cohomology class of the cycle with support in the same cycle. In order for the star- product of two Green forms to be a Green form for the intersection cycle, we need the Cohomological arithmetic Chow rings 7 cycles to intersect properly in X∞. Therefore, as was the case in [36], the arithmetic intersectionproductreliesonthemovinglemmaforcomplexvarieties.Inparticular,this implies that our varieties should be at least geometrically quasi-projective. In the preprint [45], Hu uses the homotopy invariance and the flexibility of the defini- tionofGreenformstogiveanewdefinitionoftheinverseimagemorphismofarithmetic Chow groups for a regular closed immersion by means of Fulton’s deformation to the normal cone technique. The main result of this paper is the construction of a special- ization morphism for Green forms. Combining this technique with de Jong’s result on alterations mentioned above, one can define the arithmetic intersection pairing without using any moving lemma, thereby removing the hypothesis of quasi-projectivity. The interface between the geometric and the analytic part In the definition of arithmetic Chow groups by means of Green currents, this inter- face is implemented by very concrete objects. Namely, to any p-codimensional algebraic cycle y, one assigns the current ‘integration along the cycle y∞’, denoted by δy, and to every rational function f, one associates the current −log|f∞|2. Compared with to the approach of Green currents, this interface is more abstract. It is implemented by the theory of characteristic classes for cycles and rational functions. The two approaches are compatible because in the appropriate complexes, the currents δy and −log|f∞|2 represent the cohomology class of y∞ and f∞, respectively. Abstract arithmetic Chow groups RecallthatourmainmotivationistoextendKu¨hn’sgeneralizedarithmeticintersection pairing on arithmetic surfaces to higher dimensional arithmetic varieties. In order to accomplish this goal we will use the flexibility of the Green form approach, changing the complex D by a complex of differential forms with certain mild singularities along a log fixed subvariety. Nevertheless, the arithmetic Chow groups that we will define in this way, and their properties, depend strongly on the actual complex used. And there is no onebestchoiceforthiscomplexofsingulardifferentialforms.Forinstance,inthispaper we introduce the complex of pre-log-log differential forms which, although it does not haveallthecohomologicalpropertiesonewouldexpect,isenoughtodefineanarithmetic intersection pairing and, in particular, the height with respect to log singular hermitian line bundles. On the other hand, in the paper [19], we introduce the complex of log- log singular differential forms. This complex has the expected cohomological properties, but is slightly more difficult to handle because one has to bound all the derivatives of the functions involved. Moreover, as we mentioned previously, in the literature there are several other variants of arithmetic Chow groups with singular differential forms. In addition, if one uses completely different kinds of complexes, one can obtain arith- metic Chow groups with new properties. For instance, in his PhD thesis [14], Burgos constructed a fully covariant version of the arithmetic Chow groups by choosing a com- plex of currents instead of the complex of smooth differential forms. Similar arithmetic Chow groups were introduced independently by Moriwaki in [63]. 8 J. I. Burgos Gil, J. Kramer and U. Ku¨hn In another direction, one may consider the following example. Let p , i= 1,...,4, be i four different points in P1. Then, the archimedean component of the height pairing of Q p −p and p −p is essentially given by the logarithm of the norm of the cross ratio of 1 2 3 4 the four points. Thus, this height pairing has a canonical lifting from R to C∗, namely the cross ratio of the four points. This suggests that one can define a finer version of arithmetic Chow groups, where the role of real Deligne–Beilinson cohomology is played by integral Deligne–Beilinson cohomology. Even more, one can think of an adelic version of the arithmetic Chow groups, similar to the one introduced in [7], but where each geometric fibre is substituted by a suitable complex that computes a certain cohomology theory, or a theory, where only certain geometric fibres are substituted by a cohomological complex. All these considerations lead us to the conclusion that it is worthwhile to develop a formalismofarithmeticChowgroupswithrespecttoanabstractcomplexandtoexplore howthepropertiesofthecomplexarereflectedbythepropertiesofthearithmeticChow groups. In this way, different variants of arithmetic Chow groups can be obtained as particular cases. Results Themainachievementofthispaperisthesystematicdevelopmentoftheformalismof abstract arithmetic Chow groups in arbitrary dimensions depending on a suitable coho- mological complex. Every choice of such a complex gives rise to new types of arithmetic Chow groups with different properties reflected by the complexes under consideration. As special cases we recover Burgos’s version of the arithmetic intersection theory devel- opedbyGilletandSoul´ein[36],weintroduceatheoryofarithmeticChowgroupswhich are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmeticChowringsusingacomplexofdifferentialformswithpre-log-logsingularities along a fixed normal crossings divisor. This latter theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to a logarithmically singular hermitian line bundle to higher dimensional cycles. AsanapplicationwecomputetheFaltingsheightofHeckecorrespondencesonaproduct of modular curves. ThisformalismofarithmeticChowgroupsisanabstractionof[17].Notehoweverthat the passage from the concrete example of [17] to the abstract version presented here is not completely straightforward. Although some constructions such as the definition of truncated cohomology classes and their product are already (at least implicitly) present in [17], others, like the notions of G-complex and of covariant f-morphism of complexes, are new. The basic idea of this paper is that the role of the complex D can be played by log any graded complex of sheaves in the Zariski topology C. We only require two proper- ties for this complex. The first is that the hypercohomology of this complex of sheaves always agrees with the cohomology of the complex of global sections. In this way we can represent cohomology classes by concrete elements of this complex. A convenient way to ensure this is to ask the complex to satisfy a Mayer–Vietoris condition. The second Cohomological arithmetic Chow rings 9 property we require is that the complex receives characteristic classes from cycles and rational functions (i.e. from K -chains). Typically, in order to ensure the existence of 1 characteristic classes, one imposes a series of axioms to the cohomology. Nevertheless, in many applications it is not convenient to impose too many conditions to the cohom- ology. Therefore, we use an auxiliary cohomology, given by a graded complex of sheaves G satisfying the axioms of [34]; we call it a Gillet cohomology. The characteristic classes will be induced by a morphism cC :G →C in the derived category. In this abstract setting, instead of Green forms, we will define Green objects that live in a space that we call ‘truncated cohomology classes’. These truncated cohomology classesaresomethingbetweenrelativecohomologyclassesandrepresentativesofrelative cohomology classes. Of course, very little can be done with this minimal set of properties. Therefore, we undertake a complete study of how the properties of the complex C are reflected by the propertiesoftheGreenobjectsandthearithmeticChowgroups.Forinstance,inorderto have an intersection product in the arithmetic Chow groups, we only need the existence of a cup product in the complex C, which is compatible with the cup product of the complex G. This implies that the cup product in C is compatible with the intersection product of cycles. We emphasize here that this abstract approach also simplifies many difficulties that appear when working with Green currents. We have already mentioned the proof of the associativity and commutativity of the star-product, but our approach also provides a new and logically independent proof of the well-definedness of the arithmetic intersec- tion product due to Gillet and Soul´e. This proof does not rely on the K -chain moving 1 lemma.WepointoutthatinBurgos’sprecedingworktheargumentsforprovingthewell- definednessofthearithmeticintersectionproductreliedonthecorrespondingarguments in [36]. We will give a more detailed discussion on how we avoid the K -chain moving 1 lemmainRemark4.22.WealsoemphasizethatsomeproblemswiththeK -chainmoving 1 lemma have been discussed and successfully solved using completely different techniques by Gubler in [40]. Once the abstract theory is developed, in the subsequent sections, we study particular cases of this construction. For instance, we recover the original arithmetic intersection theoryofGilletandSoul´eforprojectivevarieties.Wepointoutthatwewillshowin[19] howtorecoverthesearithmeticChowgroupsforquasi-projectivevarietiesasaparticular case of our construction. This example agrees with the theory developed in [17]. AsasecondexampleweintroduceatheoryofarithmeticChowgroupswhicharecovari- ant with respect to arbitrary proper morphisms. This construction was first introduced in [14], and a similar construction can be found in [63]. Furthermore,bychoosingforC acomplexofformssatisfyingcertaingrowthconditions of log- and log-log-type, we obtain a theory which is compatible with the theories devel- oped by Bost and Ku¨hn in the one-dimensional setting. This latter theory is specifically suited for the study of automorphic line bundles and allows to generalize the classical Faltingsheightwithrespecttoalogarithmicallysingularhermitianlinebundletohigher dimensional cycles. As an application, we compute the Faltings height of certain Hecke 10 J. I. Burgos Gil, J. Kramer and U. Ku¨hn correspondences. Note that the same result has been obtained recently by Autissier [3] using the one-dimensional theory. We point out that Bost’s theory can also be seen as a particular case of our abstract setting. Nevertheless, the definition of the corresponding complex C involves a mixture of L2-forms, L2-forms, L2 -currents, forms with logarithmic singularities, and currents; 1 −1 we will not write it explicitly. It would be interesting to extend the abstract setting of this paper to cover also Hu’s deformation to the normal cone technique. This would involve incorporating the special- ization functor to the axiomatic system and asking for the existence of a specialization morphism at the level of complexes. Applications Thetheorydevelopedinthispaperisextensivelyusedintheforthcomingpaper[13]by Bruinier,BurgosandKu¨hn,whereexplicitcalculationsforthearithmeticself-intersection number of the line bundle of modular forms and the Faltings height of Hirzebruch– Zagier cycles on Hilbert modular surfaces are carried out. Further calculations in this direction for other naturally metrized automorphic line bundles have been made in [12]. In his forthcoming thesis Freixas-Montplet will prove finiteness results for the height with respect to such naturally metrized line bundles, which generalize Faltings’s original result for points to cycles of higher dimensions. In the sequel [19] of this paper, we will showthatourabstractarithmeticChowgroupsattachedtothecomplexofformshaving suitablegrowthconditionsoflog-andlog-log-typecombinedwiththework[18]allowus toconstructarithmeticcharacteristicclassesforvectorbundlesequippedwithhermitian metrics, which are logarithmically singular along a divisor with normal crossings. These arithmetic characteristic classes give rise to operations on the arithmetic Chow groups, even for non-regular arithmetic varieties. In addition, we show in [19] that automorphic vectorbundlesequippedwiththenaturalinvariantmetric(Peterssonmetric)onShimura varietiesofnon-compacttypearehermitianvectorbundlesofthetypeconsideredabove. The framework of our arithmetic Chow groups attached to forms having certain log- and log-log-type singularities is one of the key ingredients in order to formulate various conjectures:inthiscontext,wementionaconjectureofKo¨hleronarithmeticintersection numbersonthemodulispaceofprincipallypolarizedabelianvarieties(see[52]);secondly, wementionaconjectureofMaillotandRoessleronarithmeticChernnumbersassociated to fibrations of motives with complex multiplication (see [61]); finally, we emphasize the conjectures of Kudla on Faltings heights and generating series for special cycles on Shimura varieties of orthogonal type (see [55,56]). Outline of the paper Let us now give a more detailed outline of the contents of each section. Section 1 In the first section we review various results relating K-theory, Chow groups and cohomology theories satisfying Gillet’s axioms [34]; these facts will be needed in the