On a Canonical Class of Green Currents for the Unit Sections of Abelian Schemes

On a Canonical Class of Green Currents for the Unit Sections of Abelian Schemes

For Christophe Soulé, on the occasion of his sixtieth birthday

V. Maillot and D. Rössler

Received: December 12, 2014 Communicated by Takeshi Saito

Abstract. We show that on any abelian scheme over a complex quasi-projective smooth variety, there is a Green current for the zero- section, which is axiomatically determined up to∂and∂-exact differ-¯ ential forms. On an elliptic curve, this current specialises to a Siegel function. We prove generalisations of classical properties of Siegel functions, like distribution relations and reciprocity laws. Further- more, as an application of a refined version of the arithmetic Riemann- Roch theorem, we show that the above current, when restricted to a torsion section, is the realisation in analytic Deligne cohomology of an element of the (Quillen) K1 group of the base, the corresponding denominator being given by the denominator of a Bernoulli number.

This generalises the second Kronecker limit formula and the denomi- nator12computed by Kubert, Lang and Robert in the case of Siegel units. Finally, we prove an analog in Arakelov theory of a Chern class formula of Bloch and Beauville, where the canonical current plays a key role.

2010 Mathematics Subject Classification: 14G40, 14K15, 11G16, 11G55,

1 Introduction

In this article, we show that on any abelian scheme over a complex quasi- projective smooth variety, there is a Green currentgfor the the cycle given by the zero-section of the abelian scheme, which is uniquely determined, up to ∂ and∂-exact forms, by three axioms — see Theorem 1.1.¯

We proceed to show that this Green current is naturally compatible with iso- genies (ie it satisfies distribution relations), up to ∂ and ∂-exact forms (see¯ Theorem 1.2.1), and that it intervenes in an Arakelov-theoretic generalization of a formula of Bloch and Beauville (see [2, p. 249] for the latter), which is proven here without resorting to the Fourier-Mukai transform. See Theorem 1.2.2. Furthermore, we show that if the basis of the abelian scheme is a point (ie if the abelian scheme is an abelian variety), then g is a harmonic Green current (see Theorem 1.2.3). The current g is also shown to be compatible with products (see Theorem 1.2.5).

Finally, we show that the restriction ofgto the complement of the zero-section has a spectral interpretation. Up to a sign, it is given there by the degree (g−1)part of the analytic torsion form of the Poincaré bundle of the abelian scheme. See Theorem 1.3.1 for this. In point 2 of the same theorem, we show that the restriction of the higher analytic torsion form to torsion sections, which never meet the zero-section, lies in the rational image of the Beilinson regulator from K1 to analytic Deligne cohomology and we give a multiplicative upper bound for the denominators involved. To prove Theorem 1.3.1, we make heavy use of the arithmetic Riemann-Roch theorem in higher degrees proven in [22]

and to compute the denominators described in Theorem 1.3.2, we apply the Adams-Riemann-Roch theorem in Arakelov geometry proven in [42].

If one specializes to elliptic schemes (i.e. abelian schemes of relative dimen- sion 1) the results proven in Theorems 1.1, 1.2 and 1.3 one recovers many results contained in the classical theory of elliptic units. In particular, on el- liptic schemes the currentgis described by a Siegel function and the spectral interpretation ofgspecializes to the second Kronecker limit formula. The reci- procity law for elliptic units (ie the analytic description of the action of the Galois group on the elliptic units) is also easily obtained and (variants) of the results of Kubert-Lang and Robert on the fields of definition of elliptic units are recovered as a special case of the above denominator computations. Details about elliptic schemes are given in section 5 where references to the classical literature are also given. The reader will notice that even in the case of elliptic schemes, our methods of proof are quite different from the classical ones.

The current gcan also be used to describe the realisation in analytic Deligne cohomology of the degree 0 part of the polylogarithm on abelian schemes in- troduced by J. Wildeshaus in [45] (see also [25]). The fact that this should be the case was a conjecture of G. Kings. His conjecture is proven in [26].

Here is a detailed description of the results.

Let (R,Σ)be an arithmetic ring. By definition, this means that R is an ex- cellent regular ring, which comes with a finite conjugation-invariant set Σof

embeddings intoC(see [19, 3.1.2]). For exampleRmight beZwith its unique embedding intoC, orCwith the identity and complex conjugation as embed- dings.

Recall that an arithmetic variety overRis a scheme, which is flat and of finite type over R. In this text, all arithmetic varieties overR will also be assumed to be regular, as well as quasi-projective overR. For any arithmetic varietyX overR, we write as usual

X(C) := a

σ∈Σ

(X×R,σC)(C) = a

σ∈Σ

X(C)σ.

LetD^p,p(XR)(resp. A^p,p(XR)) be theR-vector space of currents (resp. differ- ential forms)γ onX(C)such that

• γis a real current (resp. differential form) of type(p, p);

• F_∞^∗γ= (−1)^pγ,

where F∞ : X(C) → X(C) is the real analytic involution given by complex conjugation. We then define

De^p,p(XR) :=D^p,p(XR)/(im∂+ im ¯∂) (resp.

Ae^p,p(XR) :=A^p,p(XR)/(im∂+ im ¯∂) ).

All these notations are standard in Arakelov geometry. See [44] for a com- pendium. It is shown in [19, Th. 1.2.2 (ii)] that the natural mapAe^p,p(XR)→ De^p,p(XR)is an injection.

If Z a closed complex submanifold of X(C), we shall write more generally D^p,p_Z (XR)for theR-vector space of currentsγ onX(C)such that

• γis a real current of type(p, p);

• F_∞^∗γ= (−1)^pγ;

• the wave-front set of γ is included in the real conormal bundle of Z in X(C).

Similarly, we then define the R-vector spaces

De^p,p_Z (XR) :=D_Z^p,p(XR)/D^p,p_Z,0(XR)

where D_Z,0^p,p(XR) is the set of currentsγ∈D_Z^p,p(XR)such that: there exists a complex currentαof type(p−1, p)and a complex currentβ of type(p, p−1) such that γ := ∂α+ ¯∂β and such that the wave-front sets of α and β are included in the real conormal bundle ofZ in X(C).

See [24] for the definition (and theory) of the wave-front set.

It is a consequence of [11, Cor. 4.7] that the natural morphism De^p,p_Z (XR) → De^p,p(XR)is an injection.¹ Thus the real vector spaceDe^p,p_Z (XR)can be identified with a subspace of the real vector spaceDe^p,p(XR).

Furthermore, it is a consequence of [11, Th. 4.3] that for any R-morphism f : Y → X of arithmetic varieties, there is a natural morphism of R-vector spaces

f^∗:De_Z^p,p(XR)→De_f(C)^p,p ∗(Z)(YR),

provided f(C) is transverse to Z. This morphism extends the morphism Ae^p,p(XR)→Ae^p,p(YR), which is obtained by pulling back differential forms.

Fix nowS an arithmetic variety overR. Letπ:A →S be an abelian scheme over S of relative dimension g. We shall write as usual A^∨ →S for the dual abelian scheme. Writeǫ(resp. ǫ^∨) for the zero-section ofA →S (resp. A^∨→ S) and alsoS0(resp. S₀^∨) for the image ofǫ(resp.ǫ^∨). We denote by the symbol P the Poincaré bundle on A ×SA^∨. We equip the Poincaré bundle P with the unique metrichP such that the canonical rigidification ofP along the zero- sectionA^∨→ A×SA^∨is an isometry and such that the curvature form ofhPis translation invariant along the fibres of the mapA(C)×S(C)A^∨(C)→ A^∨(C).

We write P := (P, hP)for the resulting hermitian line bundle. Write P⁰ be the restriction ofP toA ×S(A^∨\S₀^∨).

The aim of this text is now to prove the following three theorems.

Theorem 1.1. There is a unique class of currents gA ∈ De^g−1,g−1(A^∨_R) with the following three properties:

(a) Any element ofgAis a Green current forS₀^∨(C).

(b) The identity(S₀^∨,gA) = (−1)^gp2,∗(ch(Pb ))^(g) holds in dCH^g(A^∨)Q. (c) The identityg_A= [n]∗g_A holds for alln>2.

Here the morphism p2 is the second projection A ×S A^∨ → A^∨ and [n] : A^∨ → A^∨ is the multiplication-by-n morphism. The symbol ch(·)b refers to the arithmetic Chern character anddCH^•(·)is the arithmetic Chow group. See [19, 1.2] for the notion of Green current.

Supplement. The proof of Theorem 1.1 given below shows that ifSis assumed proper over SpecR, then the condition (b) can be replaced by the following weaker condition :

(b)’ The identity of currentsdd^cg+δS^∨₀(C)= (−1)^gp2,∗(ch(P))^(g) holds.

Heredd^c:=_2πⁱ ∂∂¯andδS₀^∨(C)is the Dirac current associated toS₀^∨(C)inA^∨(C).

Furthermore,ch(P)is the Chern character form of the hermitian bundleP. See [20, Intro.] for this.

Remarks. (1)The condition (b) apparently makes the currentg_Adependent on the arithmetic structure ofA. We shall show in 1.2.4 below that this is not

1many thanks to J.-I. Burgos for bringing this to our attention

the case. In particular, in defining g_A, we could have assumed that R = C. The settting of arithmetic varieties is used in Theorem 1.1 because it is the natural one for property (b). Formula (15) gives a purely analytic expression forg_A. (2)It is tempting to try to refine Theorem 1.1 by using in property (b) the arithmetic Chow groups defined by J.-I. Burgos (in [10]) rather than the arithmetic Chow groups of Gillet-Soulé. One would then obtain a class of forms with certain logarithmic singularities, rather than a class of currents. Such a refinement does not seem to be easily attainable though, because of the lack of covariant functoriality of the spaces of forms mentioned in the last sentence.

The next theorem gives some properties of the class of currentsgA.

Let L be a rigidified line bundle on A. Endow L with the unique hermitian metric hL, which is compatible with the rigidification and whose curvature form is translation-invariant on the fibres ofA(C)→S(C). Let L:= (L, hL) be the resulting hermitian line bundle. Let φL :A → A^∨ be the polarisation morphism induced byL.

Theorem 1.2. 1. Let ι :A → B be an isogeny of abelian schemes over S.

Then the identityι^∨_∗(gB) =gAholds.

2. Suppose thatLis ample relatively toS and symmetric. Then the equali- ties

(S₀^∨,gA) = (−1)^gp2,∗(ch(P)) =b 1 g!p

deg(φL)φL,∗(bc1(L)^g) are verified inCHd^∗(A^∨)Q.

3. IfS→SpecRis the identity onSpecR then any element ofgAis a har- monic Green current forS₀^∨(C), whereA^∨(C)is endowed with a conjuga- tion invariant Kähler metric, whose Kähler form is translation invariant.

4. The classg_Ais invariant under any change of arithmetic rings(R,Σ)→ (R^′,Σ^′).

5. Let B → S be another abelian scheme and let πA^∨ : A^∨×SB^∨ → A^∨ (resp. πB^∨ :A^∨×SB^∨→ B^∨) be the natural projections. Then

g_A×SB=π_A^∗∨(gA)∗π_B^∗∨(gB) (1) 6. The class of currentsgAlies inDe^g−1,g−1_S∨

0(C) (A^∨_R).

7. LetT be a an arithmetic variety overRand letT →S be a morphism of schemes over R. Let AT be the abelian scheme obtained by base-change and let BC : AT → A be the corresponding morphism. Then BC(C) is tranverse toS₀^∨(C)andBC^∗g_A=g_AT.

Hereι^∨:B^∨→ A^∨is the isogeny, which is dual toι. For the notion of harmonic Green current, see [8] and [29]. The pairing∗ appearing in the equation (1) is the∗-product of Green currents. See [19, par. 2.2.11, p. 122] for the definition.

Recall that an S-isogeny between the abelian schemes Aand B is a flat and finite S-morphism A → B, which is compatible with the group-scheme struc- tures. The symbolbc1(·)refers the first arithmetic Chern class; see [20, Intro.]

for this notion.

Theorem 1.2.1 generalizes to higher degrees the distribution relations of Siegel units. See section 5 below for details. If the morphism S → SpecR is the identity on SpecR and R is the ring of integers of a number field, then it is shown in [29, Prop. 11.1 (ii)] that Theorem 1.2.3 implies Theorem 1.2.2. Still in the situation whereS→SpecRis the identity onSpecR, another construction of a Green current forS₀^∨(C)is described in A. Berthomieu’s thesis [3]. The current constructed by Berthomieu is likely to be harmonic (it is not proven in [3], but according to the author [private communication] it can easily be shown). The current constructed in [3] satisfies the identity in Theorem 1.3.1 by construction.

The last theorem relates the currentg_A to the Bismut-Köhler analytic torsion form of the Poincaré bundle (see [6, Def. 3.8, p. 668] for the definition).

Let λ be a (1,1)-form on A(C) defining a Kähler fibration structure on the fibrationA(C)→S(C)(see [6, par. 1] for this notion). With the formλ, one can canonically associate a hermitian metric on the relative cotangent bundle ΩA/S and we shall writeΩA/S for the resulting hermitian vector bundle. We suppose thatλis translation invariant on the fibres of the mapA(C)→S(C) as well as conjugation invariant. We shall write

T(λ,P⁰)∈A((Ae ^∨\S₀^∨)R) :=M

p>0

Ae^p,p((A^∨\S₀^∨)R)

for the Bismut-Köhler higher analytic torsion form ofP⁰along the fibration A(C)×_S(C)(A^∨(C)\S₀^∨(C))−→ A^∨(C)\S₀^∨(C).

For any regular arithmetic variety X over R, the (Beilinson) regulator map gives rise to a morphism of groups

reg_an:K1(X)−→M

p>0

H_D,an^2p−1(XR,R(p)).

To define the space H_D,an^2p−1(XR,R(p)) and the map reg_an, let us first write H_D,an^∗ (X,R(·))for the analytic real Deligne cohomology ofX(C). By definition,

H_D,an^q (X,R(p)) :=H^q(X(C),R(p)D,an) whereR(p)D,anis the complex of sheaves ofR-vector spaces

0→R(p)→ OX(C)

→d Ω¹_X(C)→ · · · →Ω^p−1_X(C)→0

on X(C) (for the ordinary topology). Here R(p) := (2iπ)^pR ⊆ C. We now define

H_D,an^2p−1(XR,R(p)) :={γ∈H_D,an^2p−1(X,R(p))|F_∞^∗γ= (−1)^pγ}.

By construction, the regulator map K1(X) → ⊕p>0H_D,an^2p−1(X,R(p)) (see [12] for a direct construction of the regulator and further references) fac- tors through ⊕p>0H_D,an^2p−1(XR,R(p)) and thus gives rise to a map K1(X) →

⊕p>0H_D,an^2p−1(XR,R(p)). This is the definition of the mapreg_an.

It is shown in [12, par 6.1] that there is a natural inclusion H_D,an^2p−1(XR,R(p))֒→Ae^p−1,p−1(XR).

For the next theorem, define

N2g:= 2·denominator [(−1)^g+1B2g/(2g)],

where B2g is the2g-th Bernoulli number. Recall that the Bernoulli numbers are defined by the identity of power series:

X

t>1

Btu^t

t! = u exp(u)−1.

Theorem 1.3. 1. The class of differential formsTd(ǫ^∗ΩA/S)·T(λ,P⁰)lies inAe^g−1,g−1((A^∨\S₀^∨)R)and the equality

gA|_A^∨_(C)\S^∨

0(C)= (−1)^g+1Td(ǫ^∗ΩA/S)·T(λ,P⁰) holds. In particularT(λ,P⁰)^(g−1) does not depend onλ.

2. Suppose thatλis the first Chern form of a relatively ample rigidified line bundle, endowed with its canonical metric. Letσ∈ A^∨(S)be an element of finite ordern, such thatσ^∗S^∨₀ =∅. Then

g·n·N2g·σ^∗T(λ,P⁰)∈image(reg_an(K1(S))).

A the end of section 4.2 (see the end of the proof of Lemma 4.5), we give a statement, which is slightly stronger than Theorem 1.3.2 (but more difficult to formulate).

Theorem 1.3.1 can be viewed as a generalization to higher degrees of the second Kronecker limit formula (see [31, chap. 20, par. 5, p. 276] for the latter).

Theorem 1.3.2 generalizes to higher degrees part of a classical statement on elliptic units and their fields of definition. See section 5 below.

Remark. It would be interesting to have an analogue of Theorem 1.3.2, where reg_an is replaced by the analytic cycle class cyc_an (see (3) below). If S ≃ SpecR and R is the ring of integers in a number field, thenreg_an and cyc_an can be identified but this is not true in general. In particular, this suggests

that the Bernoulli number 12 =N2/2, which appears in the denominators of elliptic units (see the last section) should be understood as coming from the natural integral structure of the groupK1(·)Qand not from the natural integral structure of the corresponding motivic cohomology groupL

pH_M^2p−1(·,Z(p))Q. Some of the results of this article were announced in [37].

We shall provide many bibliographical references to ease the reading but the reader of this article is nevertheless assumed to have some familiarity with the language of Arakelov theory, as expounded for instance in [44].

Acknowledgments. We thank J.-I. Burgos for patiently listening to our explanations on the contents of this article and for his (very) useful comments over a number of years. We also thank C. Soulé, G. Kings, as well as S. Bloch and A. Beilinson for their interest. J. Kramer made several interesting remarks on the contents of this article and his input was very useful. Many thanks also to K. Köhler for his explanations on the higher analytic torsion forms of abelian schemes. Finally, we are grateful to J. Wildeshaus for his feedback and for answering many questions on abelian polylogarithms.

Notations. Here are the main notational conventions. Some of them have already been introduced above. Recall that we wrote π : A → S for the structure morphism of the abelian schemeAoverS. We also writeπ^∨:A^∨→S for the structure morphism of the abelian scheme A^∨ overS. Write µ=µA: A×SA → Afor the addition morphism andp1:A×A^∨ → A,p2:A×A^∨→ A^∨ for the obvious projections. We shall also also write p₁,p₂ :A × A → Aand p^∨₁,p^∨₂ :A^∨× A^∨ → A^∨ for more obvious projections. Recall that we wrote ǫ (resp. ǫ^∨) for the zero-section of A →S (resp. A^∨ →S) and also that we wroteS0 (resp. S₀^∨) for the image ofǫ (resp. ǫ^∨). WriteωA:= det(ΩA/S)for the determinant of the sheaf of differentials ofAoverS. We letdd^c:= _2πⁱ ∂∂.¯ 2 Proof of Theorem 1.1

IfM is a smooth complex quasi-projective variety, we shall writeH_D^∗(M,R(·)) for the Deligne-Beilinson cohomology ofM. We recall its definition. LetM¯ be a smooth complex projective variety, which containsM as an open subscheme.

We call M¯ a compactfication of M. Suppose furthermore that M¯\M is the underlying set of a reduced divisor with normal crossings D. From now on, we view M andM¯ as complex analytic spaces and we work in the category of complex analytic spaces. Letj:M ֒→M¯ be the given open embedding. There is a natural subcomplexΩ^•M¯(logD)ofj∗Ω^•_M, called thecomplex of holomorphic differential forms on M with logarithmic singularities alongD. The objects of Ω^•M¯(logD)are locally free sheaves. We redirect the reader to [9, chap. 10] for the definition and further bibliographical references. Write F^pΩ^•M¯(logD)for the subcomplex

Ω^p_M¯(logD)−→Ω^p+1_M¯ (logD)−→ · · ·

ofΩ^•_M¯(log D). Writef_p⁰ :F^pΩ^•_M¯(logD)→j∗Ω^•_M for the inclusion morphism.

Abusing notation, we shall identify Rj∗R(p) with the complex, which is the

image by j∗ of the canonical flasque resolution of R(p). Similarly, we shall write Rj∗Ω^•_M for the simple complex associated to the image by j∗ of the canonical flasque resolution ofΩ^•_M ( the latter being a double complex). Write fp : F^pΩ^•M¯(log D) → Rj∗Ω^•_M for the morphism obtained by composing f_p⁰ with the canonical morphism j∗Ω^•_M →Rj∗Ω^•_M. There is a natural morphism of complexesR(p)→Ω^•_M (whereR(p)is viewed as a complex with one object sitting in degree0) and by the functoriality of the flasque resolution, we obtain a morphismrp: Rj∗R(p)→Rj∗Ω^•_M. We now define the complex

R(p)D:= simple Rj∗R(p)⊕F^pΩ^•M¯(log D)−→^up Rj∗Ω^•_M

whereup:=rp−fp. By definition, Deligne-Beilinson cohomology is now defined by the formula

H_D^q(M,R(p)) :=H^q(M,R(p)D).

Notice that by construction,H_D^q(M,R(p)) =H_D,an^q (M,R(p))ifM is compact (so thatD is empty). More generally, there is a natural "forgetful" morphism of R-vector spaces H_D^q(M,R(p))→ H_D,an^q (M,R(p))(what is forgotten is the logarithmic structure) ; see [10, before Prop. 1.3] for this. It can be proven that Deligne-Beilinson cohomology does not depend on the choice of the compacti- ficationM¯. By its very definition, we have a canonical long exact sequence of R-vector spaces

· · · →H^q−1(M,C)→H_D^q(M,R(p))→H^q(M,R(p))⊕F^pH^q(M,C)→ · · · (2) whereF^pH^q(M,C)is thep-th term of the Hodge filtration of the mixed Hodge structure on H^q(M,C). Furthermore the R-vector spaceH_D^q(M,R(p))has a natural structure of contravariant functor from the category of smooth quasi- projective varieties over C to the category of R-vector spaces (see [10, Prop.

1.3] for this). If we equip the singular cohomology spaces H^q(M,C) and H_D^q(M,R(p))with their natural contravariant structure, then the sequence (2) becomes an exact sequence of functors.

IfX is an arithmetic variety, then we define

H_D^q(XR,R(p)) :={γ∈H_D^q(X(C),R(p))|F_∞^∗γ= (−1)^pγ}.

Before beginning with the proof of Theorem 1.1, we shall prove the following key lemma.

Lemma 2.1. Let n> 2. The eigenvalues of the R-endomorphism [n]^∗ of the Deligne-Beilinson cohomologyR-vector spaceH_D^2p−1(A^∨(C),R(p))lie in the set {1, n, n², . . . , n^2p−1}.

Proof. The existence of the exact sequence of functors (2) shows that we have the following exact sequence ofR-vector spaces

H^2p−2(A^∨(C),C)→H_D^2p−1(A^∨(C),R(p))→

→H^2p−1(A^∨(C),R(p))⊕F^pH^2p−1(A^∨(C),C) a and that the differentials in this sequence are compatible with the natural contravariant action of [n]. Hence it is sufficient to prove the conclusion of the lemma for theR-vector spacesH^2p−2(A^∨(C),C),H^2p−1(A^∨(C),R(p))and H^2p−1(A^∨(C),C). These spaces are more easily tractable and can be approxi- mated by the Leray spectral sequence

E₂^rs=H^r(S(C),R^sπ^∨(C)∗(K))⇒H^r+s(A^∨(C), K)

where K is R or C. Now notice that since [n] is an S-morphism, this spec- tral sequence carries a natural contravariant action of[n], which is compatible with the aforementionned action of [n] on its abutment. Consider the index 2p−1. We know that [n]^∗ acts on R^sπ^∨(C)∗(K) by multiplication by n^s. This may be deduced from known results on abelian varieties using the proper base-change theorem. Hence[n]^∗ acts onH^r(S(C),R^sπ^∨(C)∗(K))by multipli- cation by n^s as well. Now the existence of the spectral sequence shows that H^2p−1(A^∨(C), K)has a natural filtration, which consists of subquotients of the spacesH^r(S(C),R^sπ^∨(C)∗(K)), wherer+s= 2p−1. Sinces62p−1, this proves the assertion for the index 2p−1. The index 2p−2 can be treated in an analogous fashion.

Proof of uniqueness. LetgAandg⁰_Abe elements ofDe^g−1,g−1(A^∨_R)satisfying (a), (b) and (c). LetκA:=g⁰_A−gA∈De^g−1,g−1(A^∨)be the error term.

Recall the fundamental exact sequence

CH^g,g−1(A^∨)−−−→^cycan Ae^g−1,g−1(A^∨_R)→^a CHd^g(A^∨)→CH^g(A^∨)→0 (3) (see [19, th. 3.3.5] for this). HereCH^g,g−1(·)is Gillet-Soulé’s version of one of Bloch’s higher Chow groups. The groupdCH^g(A^∨)is theg-th arithmetic Chow group andCH^g(A^∨)is theg-th ordinary Chow group. By construction, there are maps

CH^g,g−1(A^∨)−−→^cyc H_D^2g−1(A^∨_R,R(g))−−−−−−→H^forgetful _D,an^2g−1(A^∨_R,R(g))→Ae^g−1,g−1(A^∨_R) whose composition is cyc_an. Here cyc is the cycle class map into Deligne- Beilinson cohomology; the second map from the left is the forgetful map and the third one is the natural inclusion mentioned before Theorem 1.3.

Now letn>2. Let

V := image H_D^2g−1(A^∨_R,R(g))−−−−−−→^forgetful H_D,an^2g−1(A^∨_R,R(g))

(4) be the image of the forgetful map fromH_D^2g−1(A^∨_R,R(g))toH_D,an^2g−1(A^∨_R,R(g)).

By (b), we know thatκA∈V. FurthermoreV is invariant under[n]^∗. In fact, by Lemma 2.1,[n]^∗ restrict to an injective morphismV →V, which is thus an isomorphism, sinceV is finite dimensional. Now the projection formula shows that the equation[n]∗[n]^∗=n^2g is valid inH_D,an^2g−1(A^∨_R,R(g))and we conclude

that thatV is also invariant under[n]∗. The same equation[n]∗[n]^∗=n^2gnow shows that the eigenvalues of [n]∗ on V lie in the set {n^2g, n^2g−1, . . . , n}. In particular, [n]∗ has no non-vanishing fixed point inV. Since [n]∗κA =κA by (c), this proves thatκA= 0.

Proof of existence. As very often, the proof of existence is inspired by the proof of uniqueness. Letg∈De^g−1,g−1(A^∨_R)be a class of Green currents forS₀^∨ satisfying (b). To see that there is such ag, pick any Green currentg^′forS₀^∨(C), such thatF_∞^∗g^′ = (−1)^g−1g^′. This exists by [19, th.1.3.5]. Now a basic property of the Fourier-Mukai transformation for abelian schemes (see [32, Lemme 1.2.5]) implies that (−1)^gp2,∗(ch(P))^(g) =S₀^∨ in CH^g(A^∨)Q. Hence, looking at the sequence (3), we see that there existsα∈Ae^g−1,g−1(A^∨_R)such that(a⊗Q)(α) = (S₀^∨,g^′)−(−1)^gp2,∗(ch(Pb ))^(g). If we defineg:=g^′−α, we obtain the required class of Green currents. Now fixn>2 and letc:=g−[n]∗g. We shall prove below (see (6)) that

[n]∗p2∗(ch(Pb ))^(g)=p2∗(ch(P))b ^(g).

This implies that c lies in the space V defined in (4) above. Now recall that we proved that[n]∗ sendsV onV and that1is not a root of the characteristic polynomial of [n]∗as an endomorphism ofV. Hence the linear equation inx

x−[n]∗x=c

has a unique solution in V. Call this solutionc0. By construction the current g0:=g+c0 satisfies the equationg0−[n]∗g0= 0. Now letm>2 be another natural number. We have seen above thatg0−[m]∗g0 lies inV. On the other hand

[n]∗(g0−[m]∗g0) = [n]∗g0−[m]∗[n]∗g0=g0−[m]∗g0

henceg₀−[m]∗g₀is a fixed point of[n]∗inV. This implies thatg₀−[m]∗g₀= 0.

This proves thatg₀ satisfies (a), (b) and (c).

3 Proof of Theorem 1.2 3.1 Proof of 1.2.1

By the definition of the dual isogeny, there is a diagram A ×SB^∨ ι×Id

>B ×SB^∨ >B^∨

A ×SA^∨ Id×ι^∨

∨

>A^∨ ι^∨

∨ such that

(Id×ι^∨)^∗PA≃(ι×Id)^∗PB

and such that the outer square is cartesian. Here PA := P and PB is the Poincaré bundle ofB overS.

First notice that the arithmetic Riemann-Roch theorem [22] implies that ch((ιb ×Id)∗(OA×SB^∨)) = deg ι

in dCH^•(B ×SB^∨)Q. Now we compute

ι^∨,∗p^A_2∗(ch(Pb A)) = p^B_2∗(ch(Pb B)ch((ιb ×Id)∗(OA×SB^∨)))

= (deg ι)·p^B_2∗(ch(Pb B)). (5) Here we used the projection formula for arithmetic Chow theory (see [19, Th.

4.4.3, 7.]) and the fact that the push-forward map in arithmetic Chow theory commutes with base-change. We may now compute

ι^∨_∗ι^∨,∗p^A_2∗(ch(Pb A)) = (deg ι)·p^A_2∗(ch(Pb A)) = (deg ι)·ι^∨_∗p^B_2∗(ch(Pb B))).

In other words, we have

p^A_2∗(ch(Pb A)) =ι^∨_∗p^B_2∗(ch(Pb B))). (6) Furthermore, since ι^∨ restricts to an isomorphism between the zero-sections, the class of currentsι^∨_∗(gB)consists of Green currents forS_0,A^∨ . All this shows that gA−ι^∨_∗(gB) lies inside the space V defined in 4. Recall that V is the image of the forgetful map H_D^2g−1(A^∨_R,R(g)) −−−−−−→^forgetful H_D,an^2g−1(A^∨_R,R(g)). To conclude, notice that for anyn>2, we have

[n]∗(gA−ι^∨_∗(gB)) =gA−ι^∨_∗(gB)

since[n]commutes withι^∨. It was shown just before the proof of existence in the proof of Theorem 1.1 that[n]∗leavesV invariant and has no non-vanishing fixed points in V. ThusgA−ι^∨_∗(gB) = 0.

3.2 Proof of 1.2.2

We shall prove the equivalent identities (−1)^g

g!p

deg(φL)φL,∗(bc1(L)^g) =p2,∗(ch(Pb ))^(g) (7) and

p2,∗(ch(P))b ^(k)= 0 (8)

ifk6=g.

For the equality (8), notice that in view of (5) and the fact that[n]^∨= [n], we have

[n]^∗(p2,∗(ch(Pb ))) =n^2g·p2,∗(ch(P))b (9)

for anyn>2. On the other hand, since(Id×[n])^∗P¯ = ¯P^⊗n, we have also [n]^∗(p2,∗(ch(Pb ))) =X

k>0

n^k+g·p2,∗(ch(Pb ))^(k) (10) and comparing equations (9) and (10) as polynomials innproves equation (8).

We now proceed to the proof of equation (7). Notice that the line bundle µ^∗L⊗p^∗1L^∨⊗p^∗2L^∨onA×SAcarries a natural rigidification on the zero section A−−−→ A ×^(Id,ǫ) SA and that the same line bundle is algebraically equivalent to 0 on each geometric fibre of the morphism p2:A ×SA → A. Hence there is a unique morphismφL:A → A^∨, the polarisation morphism induced byL, such that there is an isomorphism of rigidified line bundles

(Id×φL)^∗P ≃µ^∗L ⊗p^∗₁L^∨⊗p^∗₂L^∨. (11) Furthermore, if we endow the line bundles on both sides of (11) with their nat- ural metrics, this isomorphism becomes an isometry, because both line bundles carry metrics that are compatible with the rigidification and the curvature forms of both sides are translation invariant (in fact0) on the fibres of the map p₂(C).

We shall now give a more concrete expression for p_2∗(ch(µb ^∗L)ch(pb ^∗₁L^∨)ch(pb ^∗₂L^∨)).

We first make the calculation

p_2∗(ch(µb ^∗L)ch(pb ^∗₁L^∨)ch(pb ^∗₁L)) =p_2∗(ch(µb ^∗L))

=p_2∗(α^∗p^∗₁ch(L)) =b p_2∗(p^∗₁ch(L))b

whereα:A ×SA → A ×SAis the p₂-automorphismα:= (µ,p₂). Now notice that for anyn>2,

p2∗(([n]×Id)∗([n]×Id)^∗p^∗1ch(L)) =b n^2gp2∗(p^∗1ch(L))b

= p_2∗(([n]×Id)^∗p^∗₁ch(L)) =b p_2∗(X

l>1

n^2l(p₁ch(L))b ^(l)).

Here we used the isometric isomorphism [n]^∗L ≃ L^⊗n

2 (recall that L is sym- metric). We deduce that

p2∗(p^∗1ch(L)) =b p2∗(p^∗1ch(L)b ^(g)) =p

deg(φL) Thus, using the projection formula, we see that

p_2∗(ch(pb ^∗₁L)ch(µb ^∗L)ch(pb ^∗₁L^∨)p^∗₂ch(Lb ^∨)) =p

deg(φL)ch(Lb ^∨) which implies that

p2∗(ch(pb ^∗₁L)ch(Pb )) = 1

pdeg(φL)φL,∗ch(Lb ^∨).

Now notice that in the proof of equation (7), we may assume without restriction of generality thatLis relatively generated by its sections, which is to say that the natural morphismπ^∗π∗L → Lis surjective. Indeed, for anyn>2, we have

1 g!p

deg(φL^⊗n)φ_L^⊗n_,∗(bc1(L^⊗n)^g) = 1 g!p

deg(φL)φL,∗([n]∗bc1(L)^g)

and for anyk>0,

[n]∗bc1(L)^k=n^−2k[n]∗[n]^∗bc1(L)^k =n^2g−2kbc1(L)^k. (12) Hence

1 g!p

deg(φL^⊗n)φ_L^⊗n_,∗(bc1(L^⊗n)^g) = 1 g!p

deg(φL)φL,∗(bc1(L)^g).

We may thus harmlessly replace L by L^⊗n, where n is some large positive integer. In particular, we may assume (and we do) that the morphismπ^∗π∗L → Lis surjective, sinceL is relatively ample. Now letE:=π^∗π∗L ⊗ L^∨and let

P₀^•:· · · →Λ^r(E)→Λ^r−1(E)→ · · · → E → O →0

be the associated Koszul resolution. Let

P₁^•: 0→ P →p^∗₁E^∨⊗ P → · · · →p^∗₁Λ^r−1(E)^∨⊗ P →p^∗₁Λ^r(E)^∨⊗ P → · · ·

be the complex P ⊗p^∗₁(P₀^•)^∨. All the bundles appearing in the complex P₁^• have natural hermitian metrics and we letηP¯1be the corresponding Bott-Chern class. Notice the equalities

ηP¯1 =ch(Λb −1(E^∨))ch(Pb ) =bc^top(E)Tdc⁻¹(E)ch(Pb )

indCH^•(A×SA^∨)(see [5, last paragraph]). HereΛ−1(E^∨)is the formalZ-linear combinationP

r>0(−1)^rΛ^r(E^∨). Sincerk(E)may be assumed arbitrarily large (since we may replace L by some of its tensor powers), we see that we may

assume thatηP¯1 = 0in dCH^•(A ×SA^∨)Q. Thus we may compute p2∗(ch(b P)) =p2∗(ch[b −p^∗₁Λ−1(E^∨) +O]ch(b P))

=−

rk(E)

X

r=1

(−1)^rp2∗[ch(Λb ^r(π^∗π∗(L)^∨))ch(pb ^∗₁L^⊗r)ch(b P)]

=−

rk(E)X

r=1

(−1)^rp2∗[ch(pb ^∗₁L^⊗r)ch(b P)]ch(Λb ^r(π^∗π∗(L))^∨)

=−

rk(E)X

r=1

(−1)^r 1

pdeg(φL^⊗r)φ_L⊗r,∗(ch(b L^∨,⊗r))ch(Λb ^r(π^∗π∗(L))^∨)

=− 1

pdeg(φL)φL,∗

h^rk(E)X

r=1

(−1)^rr^−g [r]∗(ch(b L^∨,⊗r))ch(Λb ^r(π^∗π∗(L))^∨)i

=− 1

pdeg(φL)φL,∗

h^rk(E)X

r=1

(−1)^rr^−g [X

s>0

r^2g−2sch(b L^∨,⊗r)^(s)]ch(Λb ^r(π^∗π∗(L))^∨)i

=− 1

pdeg(φL)φL,∗

h^rk(E)X

r=1

X

s>0

(−1)^rr^g−sch(b L^∨)^(s)ch(Λb ^r(π^∗π∗(L))^∨)i .

Now notice that the expression

[n]∗p2∗(ch(P)) =b p2∗((Id×[n])∗ch(Pb ))

=p2∗((Id×[n])∗(Id×[n])^∗X

k>0

n^−kch(Pb )^(k)) =p2∗(X

k>0

n^2g−kch(P)b ^(k))

is a Laurent polynomial inn>2. The equation (12) shows that the expression [n]∗

− 1

pdeg(φL)φL,∗

h^rk(E)X

r=1

X

s>0

(−1)^rr^g−sch(Lb ^∨)^(s)ch(Λb ^r(π^∗π∗(L))^∨)i

is also a Laurent polynomial inn>2. We may thus identify the coefficients of these polynomials. We obtain the following : ifg+kis even, then

p2∗(ch(P))b ^(k)=

− 1

pdeg(φL)φL,∗

hch(Lb ^∨)^((g+k)/2)[

rk(E)

X

r=1

(−1)^rr^g−(g+k)/2ch(Λb ^r(π^∗π∗(L))^∨)]i and

p2∗(ch(Pb ))^(k)= 0

if g +k is odd. Note that we have already proven the stronger fact that p2∗(ch(P))b ^(k)= 0ifk6=g. Thus

p2∗(ch(P))b ^(g)=− 1

pdeg(φL)φL,∗

hch(Lb ^∨)^g[

rk(E)

X

r=1

(−1)^rch(Λb ^r(π^∗π∗(L))^∨)]i .

Using furthermore that the left-hand side expression in the last equality is of pure degreegin dCH^•(A^∨)Q, we deduce that

p2∗(ch(P))b ^(g)=− 1

pdeg(φL)φL,∗

hch(Lb ^∨)^g[

rk(E)

X

r=1

(−1)^r rk(E)

r

]i .

Now notice that by the binomial formulaPrk(E)

r=1 (−1)^{r rk(E)}_r

= (1−1)^rk(E)−1 =

−1. This proves equation (7).

3.3 Proof of 1.2.3

Let Z be an analytic cycle of pure codimension c on A(C). In view of the assumption onS,A(C)is a finite disjoint union of abelian varieties and so we may (and do) choose a translation invariant Kähler form onA(C). A currentg onA(C)of type(c−1, c−1)is said to be a harmonic Green current forZ (with respect to the Kähler form), if it satisfies the following properties : gis a Green current forZ, the differential formdd^cg+δZis harmonic andR

A(C)g∧κ= 0for any harmonic formκonA(C). Notice now that a differential form onA(C)is harmonic if and only if it is translation invariant (see for instance [15, p. 648]).

Hence the concept of harmonic Green current is independent of the choice of the translation invariant Kähler form.

The property (a) in Theorem 1.1 shows thatgAis a Green current forS₀^∨(C) and the property 2 in Theorem 1.2 shows thatdd^cg+δS₀^∨(C)is harmonic. Let nowκbe a harmonic form of type(1,1)onA(C). We know thatκisd-closed and that[n]^∗κ=n²·κfor anyn>2. We may thus compute

Z

A(C)

gA∧κ= Z

A(C)

[n]∗(gA∧κ) =n⁻² Z

A(C)

[n]∗(gA∧[n]^∗κ) =n⁻² Z

A(C)

gA∧κ and henceR

A(C)gA∧κ= 0.ThusgAis harmonic.

3.4 Proof of 1.2.4

In the next section, we shall give an expression forg_A, which depends only on AC(see the formula (15)). This implies the assertion.

3.5 Proof of 1.2.5

The proof of 1.2.5 is postponed to the end of the proof of Theorem 1.3.1. See the paragraph before subsection 4.2.

3.6 Proof of 1.2.6

This is a direct consequence of Theorem 1.1.1 and [11, Cor. 4.7 (i)] (thanks to J.-I. Burgos for providing this proof).

3.7 Proof of 1.2.7

We leave the proof that BC(C) is transverse to S₀^∨(C) to the reader. The equationBC^∗gA=gAT follows from formula (15) forgA, which will be proved in the next section, together with [11, Th. 9.11 (ii)] and the fact that the higher analytic torsion forms of Bismut-Köhler are compatible with base-change.

4 Proof of Theorem 1.3 4.1 Proof of 1.3.1

This is the most difficult point to prove. We shall construct a class of currents g⁰_Awhich naturally restricts to the degree(g−1, g−1)part of the analytic tor- sion and we shall prove thatg⁰_Asatisfies the axioms definingg_A. The arithmetic Riemann-Roch theorem in higher degrees plays a crucial role here.

4.1.1 Definition of g⁰_A Let

V : 0→ OA→V0→ · · · →Vr→Vr+1 → · · ·

be a resolution ofOAbyπ∗-acyclic vector bundles. Dualising, we get a resolu- tion

V^∨:· · · →V_r+1^∨ →V_r^∨→ · · · →V₀^∨→ OA→0

ofOAon the left. The first hypercohomology spectral sequence of the complex V^∨⊗ P for the functorp2,∗provides an exact sequence

H:· · · →R^gp2∗(V_r^∨⊗ P)→R^gp2∗(V_r−1^∨ ⊗ P)→. . .

→R^gp2∗(V₀^∨⊗ P)→ǫ^∨∗(ω_A/S^∨ )→0.

Now endow the vector bundlesVrwith conjugation-invariant hermitian metrics.

The line bundle ωA/S is endowed with its L²-metric. This metric does not depend on the choice of λ. This follows from the explicit formula for the L²-metric on Hodge cohomology given in [38, Lemma 2.7]. The arithmetic Riemann-Roch [22] in higher degrees applied toP andp2 is the identity

(−1)^gch(b X

r>0

(−1)^rR^gp2∗(V^∨_r ⊗ P))−X

r>0

(−1)^rT(λ, V^∨_r ⊗ P)+

+ Z

p2

Td(Tp2) ch(P)ηV¯^∨

=p2,∗(Td(Tpc 2)ch(Pb ))− Z

p2

ch(P)R(Tp2) Td(Tp2)

in CHd^•(A^∨)Q. Here ηV¯^∨ is the Bott-Chern secondary class of V^∨, whereOA

has index0. We have identifiedλwithp^∗₁λ.

Notice first that [

Z

p2

ch(P)R(Tp2) Td(Tp2)]^(g−1)

= [ Z

p2

ch(P)R(Tπ) Td(Tπ)]^(g−1)

= [π^∨,∗(ǫ^∗(R(Tπ) Td(Tπ))) Z

p2

ch(P)]^(g−1)= 0

where we used (7).

WriteT(H)for the homogenous secondary class in the sense of Bismut-Burgos- Litcanu (see [11, sec. 6]) of the resolutionH. By its very definition,−T(H)^(g−1) is a class of Green currents forS₀^∨(C)and it is shown in [11, Th. 10.28] that

[ch(b X

r>0

(−1)^rR^gp2∗(V^∨_r ⊗ P))]^(g)= (S₀^∨,−T(H)^(g−1)) (13)

in dCH^g(A^∨)Q and equation (7) shows that

[p2,∗(Td(Tpc 2)ch(Pb ))]^(g)= [p2,∗(Td(Tπ)c ch(Pb ))]^(g)

= [p2,∗(ch(Pb ))π^∨,∗(ǫ^∗(Td(Tπ)))]c ^(g)=p2,∗(ch(Pb ))^(g) hence we are led to the equality

p2,∗(ch(Pb ))^(g)= (−1)^g(S₀^∨,−T(H)^(g−1))−X

r>0

(−1)^rT(λ, V^∨_r ⊗ P)^(g−1)

+ [ Z

p2

Td(Tp2) ch(P)ηV¯^∨]^(g−1). (14) This motivates the definition:

g⁰_A := −T(H)^(g−1)+ (−1)^g+1X

r>0

(−1)^rT(λ, V^∨_r ⊗ P)^(g−1)

+(−1)^g[ Z

p2

Td(Tp2) ch(P)ηV¯^∨]^(g−1) (15)

Lemma4.1. The class of currentsg⁰_Adoes not depend on the resolutionV, nor on the metrics on the bundlesVr, nor on the translation invariant Kähler form λ.

Proof. We first prove that the class of currentsg⁰_Adoes not depend onV and that it does not depend on the hermitian metrics or on the bundlesVr.

Suppose that there is a second resolutionV^′ dominatingV :

0 0 0 0

V^′: 0 >OA

∨

> V₀^′

∨

>· · · > V_r^′

∨

> V_r+1^′

∨

>· · ·

V : 0 >OA

Id∨

> V0

∨

>· · · > Vr

∨

> Vr+1

∨

>· · ·

Q: 0

∨

> Q0

∨

>· · · > Qr

∨

> Qr+1

∨

>· · ·

0

∨

0

∨

0

∨

By assumption the complexQis exact and we assume that its objects areπ∗- acyclic. We endow everything with hermitian metrics. We shall write H^′ for the exact sequence

H^′:· · · →R^gp2∗(V_r^′,∨⊗ P)→R^gp2∗(V_r−1^′,∨ ⊗ P)→. . . hf ill→R^gp2∗(V₀^′,∨⊗ P)→ǫ^∨∗(ω_A/S^∨ )→0.

In order to emphasize the dependence ofg⁰_Aon the resolutionV together with the collection of hermitian metrics on theVr, we shall writeg⁰

V :=g⁰

A,V instead ofg⁰_A. Recall thatη_V^∨ is the Bott-Chern secondary class of the sequenceV^∨, withOAsitting at the index0. We shall accordingly writeη

V^′,∨ for the Bott- Chern secondary class of the sequenceV

′,∨

, withOAsitting at the index0.

By definition, we have (−1)^g(g⁰_V −g⁰

V^′) = (−1)^g+1T(H)^(g−1)

−X

r>0

(−1)^rT(λ, V^∨_r ⊗ P)^(g−1)+ [ Z

p2

Td(Tp2) ch(P)ηV¯^∨]^(g−1)

−(−1)^g+1T(H^′)^(g−1)+X

r>0

(−1)^rT(λ, V

′,∨

r ⊗ P)^(g−1)

−[ Z

p2

Td(Tp2) ch(P)ηV¯^′,∨]^(g−1). Let now

Cr: 0→Q^∨_r ⊗ P →V_r^∨⊗ P →V_r^′,∨⊗ P →0

be the natural exact sequence. All the bundles appearing on Cr are endowed with natural hermitian metrics. By the symmetry formula [4, Th. 2.7, p. 271],