Documenta Math. 619 Arithmetic Characteristic Classes of Automorphic Vector Bundles J. I. Burgos Gil1, J. Kramer, U. Ku¨hn Received: June23,2005 CommunicatedbyPeterSchneider Abstract. We develop a theory of arithmetic characteristic classes of (fully decomposed) automorphic vector bundles equipped with an invariant hermitian metric. These characteristic classes have values inanarithmeticChowringconstructedbymeansofdifferentialforms with certain log-log type singularities. We first study the cohomolog- ical properties of log-log differential forms, prove a Poincar´e lemma for them and construct the corresponding arithmetic Chow groups. Then, we introduce the notion of log-singular hermitian vector bun- dles, which is a variant of the good hermitian vector bundles intro- duced by Mumford, and we develop the theory of arithmetic charac- teristic classes. Finally we prove that the hermitian metrics of auto- morphicvectorbundlesconsideredbyMumfordarenotonlygoodbut also log-singular. The theory presented here provides the theoretical background which is required in the formulation of the conjectures of Maillot-Roessler in the semi-abelian case and which is needed to extend Kudla’s program about arithmetic intersections on Shimura varieties to the non-compact case. Contents 1 Introduction 620 2 Log and log-log differential forms 627 2.1 Log forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 2.2 Log-log forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 2.3 Log and log-log mixed forms. . . . . . . . . . . . . . . . . . . . 639 2.4 Analytic lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 642 2.5 Good forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 1PartiallysupportedbyGrantDGIBFM2000-0799-C02-01andDGIBFM2003-02914 Documenta Mathematica 10 (2005) 619–716 620 J. I. Burgos Gil, J. Kramer, U. Ku¨hn 3 Arithmetic Chow rings with log-log growth conditions 654 3.1 Dolbeault algebras and Deligne algebras . . . . . . . . . . . . . 655 3.2 The -complex of log-log forms . . . . . . . . . . . . . . . . 660 log D 3.3 Properties of Green objects with values in . . . . . . . . . . 662 l,ll D 3.4 Arithmetic Chow rings with log-log forms . . . . . . . . . . . . 663 3.5 The -complex of log-log forms... . . . . . . . . . . . . 665 log D 3.6 Arithmetic Chow rings with arbitrary singularities at infinity . 669 4 Bott-Chern forms for log-singular herm. vect. bundles 670 4.1 Chern forms for hermitian metrics . . . . . . . . . . . . . . . . 670 4.2 Bott-Chern forms for hermitian metrics . . . . . . . . . . . . . 672 4.3 Iterated Bott-Chern forms for hermitian metrics . . . . . . . . 676 4.4 Chern forms for singular hermitian metrics . . . . . . . . . . . 679 4.5 Bott-Chern forms for singular hermitian metrics . . . . . . . . 682 5 Arithmetic K-theory of log-singular herm. vect. bundles 696 5.1 Arithmetic Chern classes of log-singular herm. vector bundles . 697 5.2 Arithmetic K-theory of log-singular hermitian vector bundles . 699 5.3 Variant for non regular arithmetic varieties . . . . . . . . . . . 701 ∗ 5.4 Some remarks on the properties of CH (X, ) . . . . . . . . 703 l,ll,a D 6 Automorphic vector bundles d 704 6.1 Automorphic bundles and log-singular hermitian metrics . . . . 704 6.2 Shimura varieties and automorphic vector bundles . . . . . . . 711 1 Introduction The main goal. The main purpose of this article is to extend the arith- metic intersection theory and the theory of arithmetic characteristic classes `a la Gillet, Soul´e to the category of (fully decomposed) automorphic vector bundles equipped with the natural equivariant hermitian metric on Shimura varieties of non-compact type. In order to achieve our main goal, an extension of the formalism by Gillet, Soul´e taking into account vector bundles equipped with hermitian metrics allowing a certain type of singularities has to be pro- vided. The main prerequisite for the present work is the article [10], where the foundations of cohomological arithmetic Chow groups are given. Before continuing to explain our main results and the outline of the paper below, let us fix some basic notations for the sequel. Let B denote a bounded, hermitian, symmetric domain. By definition, B =G/K, where G is a semi-simple adjoint group and K a maximal compact subgroup of G with non-discrete center. Let Γ be a neat arithmetic subgroup of G; it acts properly discontinuously and fixed-point free on B. The quotient space X = Γ B has the structure of a smooth, quasi-projective, complex va- riety. The com\plexification G of G yields the compact dual Bˇ of B given by C Documenta Mathematica 10 (2005) 619–716 Arithmetic Characteristic Classes ... 621 Bˇ =G /P K ,whereP K isasuitableparabolicsubgroupofGequipped C + C + C · · with the Cartan decomposition of Lie(G) and P is the unipotent radical of + this parabolic subgroup. Every G -equivariant holomorphic vector bundle Eˇ C on Bˇ defines a holomorphic vector bundle E on X; E is called an automorphic vector bundle. An automorphic vector bundle E is called fully decomposed, if E = E is associated to a representation σ : P K GL (C), which is σ + C n · −→ trivialonP . SinceK iscompact, everyfullydecomposedautomorphicvector + bundle E admits a G-equivariant hermitian metric h. Let us recall the following basic example. Let π : (N) (N) denote the g g B −→ A universal abelian variety over themoduli spaceofprincipally polarized abelian varietiesofdimensiongwithalevel-N structure(N 3);lete: (N) (N) g g ≥ A −→B be the zero section, and Ω = Ω1 the relative cotangent bundle. The Bg(N)/A(gN) Hodgebundlee∗Ωisanautomorphicvectorbundleon (N),whichisequipped g A with a natural hermitian metric h. Another example of an automorphic vector bundleon (N)isthedeterminantlinebundleω =det(e∗Ω);thecorresponding g A hermitian automorphic line bundle (det(e∗Ω),det(h)) is denoted by ω. Background results. Let (E,h) be an automorphic hermitian vector bun- dle on X = Γ B, and X a smooth toroidal compactification of X. In [34], \ D.MumfordhasshownthattheautomorphicvectorbundleE admitsacanon- ical extension E to X characterized by a suitable extension of the hermitian 1 metric h to E . However, the extension of h to E is no longer a smooth her- 1 1 mitian metric, but inherits singularities of a certain type. On the other hand, it is remarkable that this extended hermitian metric behaves in many aspects like a smooth hermitian metric. In this respect, we will now discuss various definitions which weremadein thepastinorderto extract basicpropertiesfor these extended hermitian metrics. In [34], D. Mumford introduced the concept of good forms and good hermi- tian metrics. The good forms are differential forms, which are smooth on the complement of a normal crossing divisor and have certain singularities along this normal crossing divisor; the singularities are modeled by the singularities of the Poincar´e metric. The good forms have the property of being locally integrable with zero residue. Therefore, they define currents, and the map from the complex of good forms to the complex of currents is a morphism of complexes. The good hermitian metrics are again smooth hermitian metrics on the complement of a normal crossing divisor and have logarithmic singu- larities along the divisor in question. Moreover, the entries of the associated connectionmatrixaregoodforms. TheChernformsforgoodhermitianvector bundles,i.e.,ofvectorbundlesequippedwithgoodhermitianmetrics,aregood forms, and the associated currents represent the Chern classes in cohomology. Thus, in this sense, the good hermitian metrics behave like smooth hermitian metrics. In the same paper, D. Mumford proves that automorphic hermitian vector bundles are good hermitian vector bundles. In[14],G.Faltingsintroducedtheconceptofahermitianmetriconlinebundles Documenta Mathematica 10 (2005) 619–716 622 J. I. Burgos Gil, J. Kramer, U. Ku¨hn with logarithmic singularities along a closed subvariety. He showed that the heights associated to line bundles equipped with singular hermitian metrics of this type have the same finiteness properties as the heights associated to line bundles equipped with smooth hermitian metrics. The Hodge bundle ω on (N) equipped with the Petersson metric provides a prominent example of g A such a hermitian line bundle; it plays a crucial role in Faltings’s proof of the Mordellconjecture. RecallthattheheightofanabelianvarietyAwithrespect to ω is referred to as the Faltings height of A. It is a remarkable fact that, if A has complex multiplication of abelian type, its Faltings height is essentially given by a special value of the logarithmic derivative of a Dirichlet L-series. It is conjectured by P. Colmez that in the general case the Faltings height is essentially given by a special value of the logarithmic derivative of an Artin L-series. In [30], the third author introduced the concept of logarithmically singular hermitian line bundles on arithmetic surfaces. He provided an extension of arithmetic intersection theory (on arithmetic surfaces) adapted to such loga- rithmicallysingularhermitianlinebundles. Theprototypeofsuchalinebundle is the automorphic hermitian line bundle ω on the modular curve (N). J.- A1 B. Bost and, independently, U. Ku¨hn calculated its arithmetic self-intersection number ω2 to ζ′( 1) 1 ω2 =d ζ ( 1) Q − + ; N Q · − µζQ(−1) 2¶ here ζ (s) denotes the Riemann zeta function and d equals the degree of the Q N classifying morphism of (N) to the coarse moduli space (1). A1 A1 In [10], an abstract formalism was developed, which allows to associate to ∗ an arithmetic variety arithmetic Chow groups CH ( , ) with respect to X X C a cohomological complex of a certain type. This formalism is an abstract C versionofthearithmeticChowgroupsintroducedind[8]. In[10],thearithmetic ∗ Chow ring CH ( , ) was introduced, where the cohomological complex pre Q X D in question is built from pre-log and pre-log-log differential forms. This pre D ring allowsdus to define arithmetic self-intersection numbers of automorphic hermitian line bundles on arithmetic varieties associated to X = Γ B. It is \ expectedthatthesearithmeticself-intersectionnumbersplayanimportantrole for possible extensions of the Gross-Zagier theorem to higher dimensions (cf. conjectures of S. Kudla). In [6], J. Bruinier, J. Burgos, and U. Ku¨hn use the theory developed in [10] to obtain an arithmetic generalization of the Hirzebruch-Zagier theorem on the generatingseriesforcyclesonHilbertmodularvarieties. RecallingthatHilbert modularvarietiesparameterizeabeliansurfaceswithmultiplicationbythering of integers of a real quadratic field K, a major result in [6] is the following K O formula for the arithmetic self-intersection number of the automorphic hermi- tian line bundle ω on the moduli space of abelian surfaces with multiplication Documenta Mathematica 10 (2005) 619–716 Arithmetic Characteristic Classes ... 623 by with a fixed level-N structure K O ζ′ ( 1) ζ′( 1) 3 1 ω3 = d ζ ( 1) K − + Q − + + log(D ) ; N K K − · − µζK(−1) ζQ(−1) 2 2 ¶ here D is the discriminant of , ζ (s) is the Dedekind zeta function of K K K O K, and, as above, d is the degree of the classifying morphism obtained by N forgetting the level-N structure. As another application of the formalism developed in [10], we derived a height pairing with respect to singular hermitian line bundles for cycles in any codi- mension. Recently, G. Freixas in [15] has proved finiteness results for our height pairing, thus generalizing both Faltings’s results mentioned above and the finiteness results of J.-B. Bost, H. Gillet and C. Soul´e in [4] in the smooth case. Themainachievementofthepresentpaperistogiveconstructionsofarithmetic intersection theories, which are suited to deal with all of the above vector bundles equipped with hermitian metrics having singularities of a certain type suchastheautomorphichermitianvectorbundlesonShimuravarietiesofnon- compact type. For a perspective view of applications of the theory developed here, we refer to the conjectures of V. Maillot and D. Roessler [31], K. Ko¨hler [26], and the program due to S. Kudla [28], [29], [27]. Arithmetic characteristic classes. We recall from [36] that the arith- metic K-group K ( ) of an arithmetic variety `a la Gillet, Soul´e is defined 0 X X as the free group of pairs (E,η) of a hermitian vector bundle E and a smooth differential formbη modulo the relation (S,η′)+(Q,η′′)=(E,η′+η′′+ch( )), E for every short exact sequence of vector bundles (eequipped with arbitrary smooth hermitian metrics) : 0 S E Q 0, E −→ −→ −→ −→ andforanysmoothdifferentialformsη′,η′′;herech( )denotesthe(secondary) E Bott-Chern form of . E In [36], H. Gillet and C. Soul´e attached to the eleements of K ( ), represented 0 X byhermitianvectorbundlesE =(E,h),arithmeticcharacteristicclassesφ(E), ∗ b whichlieinthe“classical”arithmeticChowringCH ( ) . Aparticularexam- Q X ple of such an arithmetic characteristic class is the arithmetic Chern charbacter ch(E), whose definition also involves the Bott-Chdern form ch( ). E Inordertobeabletocarryovertheconceptofarithmeticcharacteristicclasses tbothecategoryofvectorbundlesE overanarithmeticvarietey equippedwith X a hermitian metric h having singularities of the type considered in this paper, Documenta Mathematica 10 (2005) 619–716 624 J. I. Burgos Gil, J. Kramer, U. Ku¨hn weproceedasfollows: Lettingh denoteanarbitrarysmoothhermitianmetric 0 on E, we have the obvious short exact sequence of vector bundles : 0 0 (E,h) (E,h ) 0, 0 E −→ −→ −→ −→ to which is attached the Bott-Chern form φ( ) being no longer smooth, but E having certain singularities. Formally, we then set e φ(E,h):=φ(E,h )+a φ( ) , 0 E ³ ´ where a is the morphibsm mappinbg differentialeforms into arithmetic Chow groups. Inordertogivemeaningtothisdefinition, weneedtoknowthesingu- laritiesofφ( );moreover,wehavetoshowtheindependenceofthe(arbitrarily E chosen) smooth hermitian metric h . 0 Oncewecaencontrolthesingularitiesofφ( ),theabstractformalismdeveloped E in [10] reduces our task to find a cohomological complex , which contains the C elements φ( ), and has all the propertiees we desire for a reasonable arithmetic E intersectiontheory. Oncethecomplex isconstructed,weobtainanarithmetic C K-theoryewith properties depending on the complex , of course. C The most naive way to construct an arithmetic intersection theory for auto- morphic hermitian vector bundles would be to only work with good forms and goodmetrics. Thisprocedureisdoomedtofailureforthefollowingtworeasons: First, the complex of good forms is not a Dolbeault complex. However, this first problem can be easily solved by imposing that it is also closed under the differential operators ∂, ∂¯, and ∂∂¯. The second problem is that the complex of good forms is not big enough to contain the singular Bott-Chern forms which occur. For example, if is a line bundle, h a smooth hermitian metric and 0 L h a singular metric, which is good along a divisor D (locally, in some open coordinate neighborhood, given by the equation z = 0), the Bott-Chern form (associatedtothefirstChernclass)c ( ;h,h )encodingthechangeofmetrics 1 0 L grows like loglog(1/z ), whereas the good functions are bounded. | | Thesolutionoftheseproblemsledusetoconsiderthe log-complexes pre made D D by pre-log and pre-log-log forms and its subcomplex consisting of log and l,ll D log-log forms. We emphasize that neither the complex of good forms nor the complex of pre-log-log forms are contained in each other. We also note that if one is interested in arithmetic intersection numbers, the results obtained by both theories agree. Discussion of results. The -complex made out of pre-log and log pre D D pre-log-log forms could be seen as the complex that satisfies the minimal re- quirements needed to allow log-log singularities along a fixed divisor as well as tohaveanarithmetic intersectiontheorywitharithmetic intersectionnumbers in the proper case (see [10]). As we will show in theorem 4.55, the Bott-Chern forms associated to the change of metrics between a smooth hermitian met- ric and a good metric belong to the complex of pre-log-log forms. Therefore, Documenta Mathematica 10 (2005) 619–716 Arithmetic Characteristic Classes ... 625 we can define arithmetic characteristic classes of good hermitian vector bun- dles in the arithmetic Chow groups with pre-log-log forms. If our arithmetic variety is proper, we can use this theory to calculate arithmetic Chern num- bers of automorphic hermitian vector bundles of arbitrary rank. However, the main disadvantage of is that we do not know the size of the associated pre D cohomology groups. The -complex made out of log and log-log forms is a subcomplex of log l,ll D D . The main difference is that all the derivatives of the component func- pre D tions of the log and log-log forms have to be bounded, which allows us to use aninductiveargumenttoproveaPoincar´elemma,whichimpliesthattheasso- ciated Deligne complex computes the usual Deligne-Beilinson cohomology (see theorem 2.42). For this reason we have better understanding of the arithmetic Chow groups with log-log forms (see theorem 3.17). Sinceagoodformisingeneralnotalog-logform, itisnottruethattheChern formsforagoodhermitianvectorbundlearelog-logforms. Hence,weintroduce thenotionoflog-singularhermitianmetrics,whichhave,roughlyspeaking,the same relation to log-log forms as the good hermitian metrics to good forms. We then show that the Bott-Chern forms associated to the change of metrics between smooth hermitian metrics and log-singular hermitian metrics are log- log forms. As a consequence, we can define the Bott-Chern forms for short exactsequencesofvectorbundlesequippedwithlog-singularhermitianmetrics. TheseBott-ChernformshaveanaxiomaticcharacterizationsimilartotheBott- Chernformsforshortexactsequencesofvectorbundlesequippedwithsmooth hermitian metrics. The Bott-Chern forms are the main ingredients in order to extend the theory of arithmetic characteristic classes to log-singular hermitian vector bundles. Thepricewehavetopayinordertouselog-logformsisthatitismoredifficult toprovethataparticularformislog-log: wehavetoboundallderivatives. Note however that most pre-log-log forms which appear are also log-log forms (see forinstancesection6). Ontheotherhand,wepointoutthatthetheoryoflog- singularhermitianvectorbundlesisnotoptimalforseveralotherreasons. The most important one is that it is not closed under taking sub-objects, quotients and extensions. For example, let 0 (E′,h′) (E,h) (E′′,h′′) 0 −→ −→ −→ −→ be a short exact sequence of hermitian vector bundles such that the metrics h′ and h′′ are induced by h. Then, the assumption that h is a log-singular hermitian metric does not imply that the hermitian metrics h′ and h′′ are log- singular, and vice versa. In particular, automorphic hermitian vector bundles that are not fully decomposed can always be written as successive extensions of fully decomposed automorphic hermitian vector bundles, whose metrics are ingeneralnotlog-singular. Arelatedquestionisthatthehermitianmetricofa unipotent variation of polarized Hodge structures induced by the polarization is in general not log-singular. These considerations suggest that one should further enlarge the notion of log-singular hermitian metrics. Documenta Mathematica 10 (2005) 619–716 626 J. I. Burgos Gil, J. Kramer, U. Ku¨hn Since the hermitian vector bundles defined on a quasi-projective variety may have arbitrary singularities at infinity, we also consider differential forms with arbitrary singularities along a normal crossing divisor. Using these kinds of differentialformsweareabletorecoverthearithmeticChowgroups`alaGillet, Soul´e for quasi-projective varieties. Finally,anothertechnicaldifferencebetweenthispaperand[10]isthefactthat inthepreviouspaperthecomplex (X,p)isdefinedbyapplyingtheDeligne log D complexconstructiontotheZariskisheafE ,which,inturn,isdefinedasthe log Zariski sheaf associated to the pre-sheaf E◦ . In theorem 3.6, we prove that log the pre-sheaf E◦ is already a sheaf, which makes it superfluous to take the log associated sheaf. Moreover, the proof is purely geometric and can be applied to other similar complexes like or . pre l,ll D D Outline of paper. The set-up of the paper is as follows. In section 2, we introduce several complexes of singular differential forms and discuss their re- lationship. Ofparticularimportancearethecomplexesoflogandlog-logforms forwhichweproveaPoincar´elemmaallowingustocharacterizetheircohomol- ogy by means of their Hodge filtration. In section 3, we introduce and study arithmetic Chow groups with differential forms which are log-log along a fixed normalcrossingdivisorD. Wealsoconsiderdifferentialformshavingarbitrary singularitiesatinfinity; inparticular, weprovethatforD beingtheemptyset, thearithmeticChowgroupsdefinedbyGillet,Soul´earerecovered. Insection5, wediscussseveralclassesofsingularhermitianmetrics;weprovethattheBott- Chern forms associated to the change of metrics between a smooth hermitian metricandalog-singularhermitianmetricarelog-logforms. Wealsoshowthat the Bott-Chern forms associated to the change of metrics between a smooth hermitian metric and a good hermitian metric are pre-log-log. This allows us to define arithmetic characteristic classes of log-singular hermitian vector bun- dles. Finally, in section 6, after having given a brief recollection of the basics of Shimura varieties, we prove that the fully decomposed automorphic vector bundles equipped with an equivariant hermitian metric are log-singular hermi- tian vector bundles. In this respect many examples are provided to which the theory developed in this paper can be applied. Acknowledgements: Incourseofpreparingthismanuscript,wehadmanystim- ulating discussions with many colleagues. We would like to thank them all. In particular, we would like to express our gratitude to J.-B. Bost, J. Bruinier, P. Guillen, W. Gubler, M. Harris, S. Kudla, V. Maillot, D. Roessler, C. Soul´e, J. Wildeshaus. Furthermore, we would like to thank EAGER, the Arithmetic Geometry Network, the Newton Institute (Cambridge), and the Institut Henri Poincar´e (Paris) for partial support of our work. Documenta Mathematica 10 (2005) 619–716 Arithmetic Characteristic Classes ... 627 2 Log and log-log differential forms In this section, we will introduce several complexes of differential forms with singularities along a normal crossing divisor D, and we will discuss their basic properties. The first one E∗ D is a complex with logarithmic growth conditions in the Xh i spirit of [22]. Unlike in [22], where the authors consider only differential forms oftype(0,q),weconsiderherethewholeDolbeaultcomplexandweshowthatit is an acyclic resolution of the complex of holomorphic forms with logarithmic poles along the normal crossing divisor D, i.e., this complex computes the cohomology of the complement of D. Another difference with [22] is that, in order to be able to prove the Poincar´e lemma for such forms, we need to imposegrowthconditionstoallderivativesofthefunctions. Notethatasimilar condition has been already considered in [24]. The second complex E∗ D contains differential forms with singularities of Xhh ii log-logtypealonganormalcrossingdivisorD,andisrelatedwiththecomplex ofgoodformsinthesenseof[34]. Asthecomplexofgoodforms,itcontainsthe Chernformsforfullydecomposedautomorphichermitianvectorbundlesandis functorial with respect to certain inverse images. Moreover all the differential forms belonging to this complex are locally integrable with zero residue. The new property of this complex is that it satisfies a Poincar´e lemma that implies that this complex is quasi-isomorphic to the complex of smooth differential forms, i.e., this complex computes the cohomology of the whole variety. The main interest of this complex, as we shall see in subsequent sections, is that it containsalsotheBott-Chernformsassociatedtofullydecomposedautomorphic vector bundles. Note that neither the complex of good forms in the sense of [34] nor the complex of log-log forms are contained in each other. The third complex E∗ D D that we will introduce is a mixture of the Xh 1h 2ii previous complexes. It is formed by differential forms which are log along a normal crossing divisor D and log-log along another normal crossing divisor 1 D . This complex computes the cohomology of the complement of D . 2 1 By technical reasons we will introduce several other complexes. 2.1 Log forms General notations. Let X be a complex manifold of dimension d. We will denote by E∗ the sheaf of complex smooth differential forms over X. X Let D be a normal crossing divisor on X. Let V be an open coordinate subset of X with coordinates z ,...,z ; we put r = z . We will say that V is 1 d i i | | adapted to D, if the divisor D V is given by the equation z z = 0, and 1 k ∩ ··· thecoordinateneighborhoodV issmallenough;moreprecisely,wewillassume that all the coordinates satisfy r 1/ee, which implies that log(1/r )>e and i i ≤ log(log(1/r ))>1. i We will denote by ∆ C the open disk of radius r centered at 0, by ∆ the r r ⊆ Documenta Mathematica 10 (2005) 619–716 628 J. I. Burgos Gil, J. Kramer, U. Ku¨hn closed disk, and we will write ∆∗ =∆ 0 and ∆∗ =∆ 0 . r r\{ } r r\{ } If f and g are two functions with non-negative real values, we write f g, if ≺ there exists a real constant C > 0 such that f(x) C g(x) for all x in the ≤ · domain of definition under consideration. multi-indices. We collect here all the conventions we will use about multi- indices. Notation 2.1. For any multi-index α=(α ,...,α ) Zd , we write 1 d ∈ ≥0 d d d α = α , zα = zαi, z¯α = z¯αi, | | i i i i=1 i=1 i=1 X Y Y d d rα = rαi, (log(1/r))α = (log(1/r ))αi, i i i=1 i=1 Y Y ∂|α| ∂|α| ∂|α| ∂|α| f = f, f = f. ∂zα d ∂zαi ∂z¯α d ∂z¯αi i=1 i i=1 i If α and β are multi-indQices, we write β α, if, foQr all i = 1,...,d, β α . i i ≥ ≥ We denote by α+β the multi-index with components α +β . If 1 i d, i i ≤ ≤ we will denote by γi the multi-index with all the entries zero except the i-th entry that takes the value 1. More generally, if I is a subset of 1,...,d , we { } will denote by γI the multi-index 1,&i I, γI = ∈ i (0,&i I. 6∈ We will denote by n the constant multi-index n =n. i In particular, 0 is the multi-index 0=(0,...,0). If α is a multi-index and k 1 is an integer, we will denote by α≤k the multi- ≥ index α , i k, α≤k = i ≤ i (0,&i>k. For a multi-index α, the order function associated to α, Φ : 1,..., α 1,...,d α { | |}−→{ } is given by k−1 k Φ (i)=k, if α <i α . α j j ≤ j=1 j=1 X X Documenta Mathematica 10 (2005) 619–716