ON THE ARITHMETIC CHERN CHARACTER H.GILLETAND C.SOULE´ 2 1 0 2 Let X be a proper and flat scheme over Z, with smooth generic fiber X . In [4] Q n weattachedtoeveryhermitianvectorbundle E =(E,kk)onX aCherncharacter a class lying in the arithmetic Chow groups of X: J p 4 ch(E)∈ ⊕ CH (X)⊗Q. 2 p≥0 b d Unlike the usual Chern character with values in the ordinary Chow groups, ch is ] G not additive on exact sequences; indeed suppose that E , i = 0,1,2 is a triple of i b A hermitian vector bundles on X, and that we are give an exact sequence h. 0→E0 →E1 →E2 →0 t a ofthe underlyingvectorbundles onX,(i.e. inwhichweignorethe hermitianmet- m rics). Then the difference ch(E )+ch(E )−ch(E ), is representedby a secondary 0 2 1 [ characteristicclasschfirstintroducedbyBottandChern[1]anddefinedingeneral b b b 1 in[2]. TheseBott-Cehernformsmeasurethe defectinadditivityofthe Chernforms v associated by Chern-Weil theory to the hermitian bundles in the exact sequence. 8 Assume now that the sequence 6 9 0→E →E →E →0 (∗) 0 1 2 4 . is exact on the generic fiber X but not on the whole of X. We shall prove here 1 Q 0 (Theorem 1) that ch(E0)+ch(E2)−ch(E1) is the sum of the class of ch and the 2 localizedCherncharacterof(∗)(see[3],18.1). Thisresultfitswellwiththeideathat b b b e 1 characteristicclasseswithsupportonthefinitefibersofX arethenon-archimedean : v analogs of Bott-Chern classes (see [6]). i X In Theorem 2 we compute more explicitly these secondary characteristic classes in a situation encountered when proving a “Kodaira vanishing theorem” on arith- r a metic surfaces ([7], 3.3). Notation. If A is an abelian group we let A =A⊗Q. Q Z 1. A general formula 1.1. Let S = Spec(Z) and f : X → S a flat scheme of finite type over S. We assume that the generic fiber X is smooth and equidimensional of dimension d. Q For every integer p ≥ 0 we denote by App(X ) the real vector space of smooth R real differential forms α of type (p,p) on the complex manifold X(C) such that Date:January25,2012. ThismaterialisbaseduponworksupportedinpartbytheNationalScienceFoundationunder GrantNo. DMS-0901373. 1 2 H.GILLETANDC.SOULE´ F∗(α) = (−1)pα, where F is the anti-holomorphic involution of X(C) induced ∞ ∞ by complex conjugation. Let A(X)= ⊕ App(X ) R p≥0 and A(X)= ⊕ Ap−1,p−1(X )/(Im(∂)+Im(∂)). R p≥1 e For every p ≥ 0 we let CH (X) be the p-th arithmetic Chow homology group of p X ([5], §2.1, Definition 2). Elements of CH (X) are represented by pairs (Z,g) d p consisting of a p-dimensional cycle Z on X and a Green current g for Z(C) on d X(C). Recall that here a Green current for Z(C) is a current (i.e. a form with distributioncoefficients)oftype(p−1,p−1)suchthatddc(g)+δ isC∞,δ being Z Z the current of integration on Z(C) ). There are canonical morphisms ([5], 2.2.1): z :CH (X) → CH (X) p p d(Z,g) 7→ Z and ω :CH (X) → App(X ) p R d(Z,g) 7→ ddc(g)+δZ. Let CHfin(X)be the ChowhomologygroupofcyclesonX the supportof which p does not meet X . There is a canonical morphism Q b:CHfin(X)→CH (X) p p mapping the class of Z to the class of (Z,0).dThe composite morphism z◦b:CHfin(X)→CH (X) p p is the obvious map. Let a:Ad−p−1,d−p−1(X )→CH (X) R p be the map sending η to the class of (0,η). We hadve ω◦a(η)=ddc(η). 1.2. We assume given a sequence 0→E →E →E →0 0 1 2 of hermitian vector bundles on X, the restriction of which to X is exact. Let Q chfin(E )∩[X]∈CH (X) = ⊕ CHfin(X) • fin Q p Q p≥0 be the localized Chern character of E ([3] 18.1), and • ch(E )∈A(X) • Q the Bott-Chern secondary characteeristic claess [2], such that 2 ddcch(E )= (−1)ich(E ), • i,C X i=0 e ON THE ARITHMETIC CHERN CHARACTER 3 where ch(E )∈A(X) is the differential form representing the Chern character of i,C the restriction E of E to X(C). Finally, if i=0,1,2, we let i,C i ch(E )∩[X]∈CH(X) = ⊕ CH (X) i Q p Q p≥0 b d d be the arithmetic Chern character of E ([4] 4.1, [5] Theorem 4). i Theorem 1. The following equality holds in CH(X) : Q 2 d (−1)ich(E )∩[X]=b(chfin(E )∩[X])+a(ch(E )). i • • X i=0 b e 1.3. This theorem is a special case of Lemma 21 in [5], though this may not be immediately apparent. Therefore, for the sake of completeness, we give a proof here. 1.4. To prove Theorem 1 we consider the Grassmannian graph construction ap- plied to E ([3] 18.1, [5] 1.1). It consists of a proper surjective map • π :W →X ×P1 such that, if φ⊂X is the support of the homology of E (hence φ is empty), the • Q restriction of π onto (X −φ)×P1 and X ×A1 is an isomorphism. The effective Cartier divisor W =π−1(X ×{∞}) ∞ is the union of the Zariski closure X of (X −φ)×{∞} with Y = π−1(φ×{∞}). The sequence E extends to a complex • e 0→E →E →E →0, 0 1 2 whichisisomorphictothe pull-baeckofEe overeX×A1. The restrictionofE toX • • is canonicallysplit exact. On W =X ×P1 the sequence E is exact;it coincides Q Q Q • e e with E (resp. 0→E →E ⊕E →E →0) when restricted to X ×{0} (resp. • 0 0 2 2 e Q X ×{∞}). WechooseametriconE forwhichtheseisomorphismsareisometries. Q • 1.5. Let e 2 x= (−1)ich(E ), i X i=0 b e and denote by t the standard parameter of A1. In the arithmetic Chow homology of W we have 0=x∩(W −W ,−log|t|2). 0 ∞ If x is the class of (Z,g), with Z meeting properly W and W , we get 0 ∞ x∩(W −W ,−log|t|2)=(Z ∩(W −W ),g∗(−log|t|2)), 0 ∞ 0 ∞ where the ∗-product is equal to g∗(−log|t|2)=g(δ −δ )−ch(E )log|t|2. W0 W∞ • Since W =X ∪Y, with Y =∅, we get e ∞ Q (1) 0 = xe∩(W −W ,−log|t|2) 0 ∞ = (Z∩W0,gδW0)−(Z ∩X,gδXe)−(Z∩Y,0)−(0,ch(E•)log|t|2). e e 4 H.GILLETANDC.SOULE´ The restriction of E to X is split exact, therefore • e e (Z ∩X,gδe)=0. X Applying π to (1) we get e ∗ (2) 0=ch(E )−π (Z∩Y),0)−(0,π (ch(E )log|t|2). • ∗ ∗ • By definition of theblocalized Chern character ([3], 18.1e, (14)) (3) π (Z∩Y)=chfin(E )∩[X] ∗ • in CHfin(X) . On the other hand we deduce from [4], (1.2.3.1), (1.2.3.2) that Q (4) − π (ch(E )log|t|2)=ch(E ). ∗ 1 • and upon replacing t by 1/t, as in tehe proof of (1.e3.2) in [4], we see that (5) π (ch(E )log|t|2)=−π (ch(E )log|t|2). ∗ • ∗ 1 Theorem 1 follows from (2)e, (3), (4), (5). e 2. A special case 2.1. We keep the hypotheses of the previous section, and we assume that X is normal,d=1, E andE haverankoneandthe metricsonE andE areinduced 0 2 0 2 by the metric on E . Finally, we assume that there exists a closed subscheme φ in 1 X whichis 0-dimensionalandsuchthatthere isanexactsequenceofsheavesonX (6) 0→E →E →E ⊗I →0, 0 1 2 φ where I is the ideal of definition of φ. φ Let c ∈A1,1(X )/(Im(∂)+Im(∂)) 2 R be the second Bott-Chern class of (6), Γ(φ,O ) the finite ring of functions on φ e φ and #Γ(φ,O ) its order. Let φ f :CH (X) →CH (S)=R ∗ 0 Q 0 be the direct image morphism.d d Theorem 2. We have an equality of real numbers f (c (E )∩[X])=f (c (E )c (E )∩[X])− c +log#Γ(φ,O ). ∗ 2 1 ∗ 1 0 1 2 Z 2 φ X(C) b b b e 2.2. To prove Theorem 2 we remark first that c (E )=c (E )+c (E ), 1 1 1 0 1 2 because the metrics on E0 band E2 arbe inducedbfrom E1. Therefore, since ch2 = −c + c21, we get 2 2 (c (E )+c (E ))2 1 0 1 2 ch (E ) = −c (E )+ 2 1 2 1 2 b b b = −cb (E )+c (E )c (E )+ch (E )+ch (E ). 2 1 1 0 1 2 2 0 2 2 By Theorem 1, this impliesbthat b c c (7) c (E )∩[X]=c (E )c (E )∩[X]+b(chfin(E )∩[X])+a(ch(E )). 2 1 1 0 1 2 • • b b b e ON THE ARITHMETIC CHERN CHARACTER 5 Since ch (E ) and ch (E ) vanish we have 0 • 1 • e e ch(E )=−c . • 2 Therefore, if we apply f∗ to (7)e, we get e f (c (E )∩[X])=f (c (E )c (E )∩[X])− c +f (b(chfin(E )∩[X])), ∗ 2 1 ∗ 1 0 1 2 Z 2 ∗ • X(C) b b b e and we are left with showing that (8) f ◦b(chfin(E )∩[X])=log#Γ(φ,O ). ∗ • φ Let|φ|={P ,··· ,P }⊂X bethesupportofφandψ =f(|φ|)⊂S. Thefollowing 1 n diagram is commutative: b // CH (φ) CH (X) 0 0 d f∗ f∗ (cid:15)(cid:15) (cid:15)(cid:15) CH (ψ) b //CH (S)=R, 0 0 where d b:CH (ψ)=Zψ →R 0 maps (n ) to n log(p). p p∈ψ p Pp For any prime p ∈ ψ we let Z be the local ring of S at p and we let ℓ = (p) p ℓ (φ)≥0 be the length of the finite Z -module Γ(φ,O )⊗Z . Clearly p (p) φ (p) log#Γ(φ,O )= ℓ log(p), φ p X p∈ψ hence it is enough to prove that (9) f (chfin(E )∩[X])=(ℓ )∈CH (ψ) =Qψ. ∗ • p 0 Q The complex E defines an element • n n [E ]= [O ]∈Kφ(X)= KPi(X). • φ,Pi 0 0 X M i=1 i=1 Toprove(9),byreplacingX byanaffineneighbourhoodofP,onecanassumethat |φ|={P}, and it is enough to show that, if p=f(P), f (chfin(O )∩[X])=ℓ (O )[p]. ∗ φ,P p φ,P Now recall that, if F is a coherent sheaf on a scheme X of finite type over S, supported on a finite set of closed points, the associated 0-cycle [F]= ℓ (F )[P]∈Z (X) p P 0 X P∈|F| is such that, if f :X →Y is a proper morphism of schemes of finite type over S, f [F]=[f (F)] ∗ ∗ ([3] , 15.1.5). Hence it is enough to show that (10) chP(O )=ℓ (O )[P]∈CH (P) ≃Q. φ,P p φ,P 0 Q 6 H.GILLETANDC.SOULE´ Replacing X by an affine neighbourhood of P, we may assume that we have an exact sequence (11) 0−→O −α→O2 −β→O −→O −→0. X X X φ HencetheidealI ⊂O (X)isgeneratedbytwoelementsβ andβ . SinceX is φ X 1 2 normal, its local rings satisfy Serre property S and, as dim(X)= 2, X is Cohen- 2 Macaulay. Sinceβ andβ spananidealofheighttwo,(β ,β )isaregularsequence 1 2 1 2 andthe sequence (11) is isomorphic to the Koszulresolutionof O =O /(β ,β ). φ X 1 2 Now (10) can be deduced from the following general fact: Lemma 1. Let X =Spec(A) be an affine scheme and Z ⊂X a closed subset such that the ideal I =(x ,...,x ) is generated by a regular sequence (x ,...,x ). Let Z 1 n 1 n K (x ,...,x ) be the Koszul complex associated to (x ,...,x ). Then • 1 n 1 n chZ(K (x ,...,x ))=[O ]∈CH (Z) . n • 1 n Z 0 Q Proof. The Grassmannian-graph construction on K (x ,...,x ) coincides with • 1 n the deformation to the normal bundle of Z in X. If W is defined as in 1.4, W =X ∪P(N ), ∞ Z/X e b where X is the blow up of X along Z, and P(N ) is the projective comple- Z/X tion of the normal bundle of Z in X. The pull back of the Koszul complex e b K (x ,...,x ) to W\W extends to a complex K (x ,...,x ) on W. The re- • 1 n ∞ • 1 n striction of K (x ,...,x ) to X is acyclic while the restriction of K (x ,...,x ) • 1 n e • 1 n to P(N ) is a resolution of the structure sheaf of the zero section Z ⊂ N ⊂ Z/X e e e Z/X P(NbZ/X). b Now observe that Z ⊂ P(N ) is an intersection of Cartier divisors D ,..., Z/X 1 D , hence n b ch(K•(x1,...,xn)|bP(NZ/X)) n e = Ych(O(−Di)→ObP(NZ/X)) i=1 n = ch(O(D )). i Y i=1 Since ch(O )=ch (O )+x =[D ]+x Di 1 Di i i i where x has degree ≥2, we get i ch(K•(x1,...,xn)|bP(NZ/X))=[D1]...[Dn]=[Z]. e This ends the proof of Lemma 1 and Theorem 2. ON THE ARITHMETIC CHERN CHARACTER 7 References [1] Bott, R., Chern, S.S.: Hermitian vector bundles and the equidistribution of the zeroes of theirholomorphicsections.Acta Math.114(1965), 71–112. [2] Bismut,J.-M.,Gillet, H.,Soul´e, C.: Analytictorsionand holomorphicdeterminant bundles I,II,III.Comm. Math. Physics115(1988), 49–78, 79–126, 301–351. [3] Fulton,W.: IntersectionTheory.Springer1984. [4] Gillet,H.,Soul´e,C.: Characteristicclassesforalgebraicvectorbundleswithhermitianmetrics I,II.Annals of Math.131(1990), 163–203, 205–238. [5] Gillet,H.,Soul´e,C.: AnarithmeticRiemann-Rochtheorem.Invent.Math.110(1992),473– 543. [6] Gillet,H.,Soul´e,C.: Directimagesinnon-archimedeanArakelovtheory.Annalesdel’institut Fourier,50,2000,363-399. [7] Soul´e, C.: A vanishing theorem on arithmetic surfaces. Invent. Math. 116 (1994), no. 1-3, 577-599. E-mail address: [email protected] DepartmentofMathematics,Statistics,andComputerScience,UniversityofIllinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago,IL 60607-7045,USA E-mail address: [email protected] IHE´S,35 route de Chartres, 91440Bures-sur-Yvette, France