A fixed point formula of Lefschetz type in Arakelov geometry II: a residue formula / Une formule du point fixe de type Lefschetz en g´eom´etrie d’Arakelov II: une formule des r´esidus

A fixed point formula of Lefschetz type in Arakelov geometry II: a residue formula / Une

formule du point fixe de type Lefschetz en g´eom´etrie d’Arakelov II: une formule des r´esidus

Kai K¨ ohler

Damian Roessler

December 11, 2001

Abstract

This is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result [KR1, Th. 4.4] of the first paper to prove a residue for- mula ”`a la Bott” for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of Bismut- Goette on the equivariant (Ray-Singer) analytic torsion play a key role in the proof. / Cet article est le second d’une s´erie d’articles dont l’objet est un analogue en g´eom´etrie d’Arakelov de la formule du point fixe de Lefschetz holomorphe. Nous utilisons le r´esultat principal [KR1, Th. 4.4]

du premier article pour prouver une formule des r´esidus ”`a la Bott” pour des classes caract´eristiques vivant sur des vari´et´es arithm´etiques munis d’une action de tore; de r´ecents r´esultats de Bismut-Goette sur la torsion analytique ´equivariante (de Ray-Singer) joue un rˆole cl´e dans la preuve.

2000 Mathematics Subject Classification: 14G40, 58J52, 14C40, 14L30, 58J20, 14K15

∗Centre de Math´ematiques de Jussieu, Universit´e Paris 7 Denis Diderot, Case Postale 7012, 2, place Jussieu, F-75251 Paris Cedex 05, France, E-mail : koehler@math.jussieu.fr, URL: http://www.math.jussieu.fr/˜koehler

†Centre de Math´ematiques de Jussieu, Universit´e Paris 7 Denis Diderot, Case Postale 7012, 2, place Jussieu, F-75251 Paris Cedex 05, France, E-mail : roessler@math.jussieu.fr, URL: http://www.math.jussieu.fr/˜roessler

Contents

1 Introduction 2

2 An ”arithmetic” residue formula 3

2.1 Determination of the residual term . . . 8 2.2 The limit of the equivariant torsion . . . 13 2.3 The residue formula . . . 16 3 Appendix: a conjectural relative fixed point formula in Arakelov

theory 18

1 Introduction

This is the second of a series of four papers on equivariant Arakelov theory and a fixed point formula therein. We give here an application of the main result [KR1, Th. 4.4] of the first paper.

We prove a residue formula ”`a la Bott” (Theorem 2.11) for the arithmetic Chern numbers of arithmetic varieties endowed with the action of a diagonal- isable torus. More precisely, this formula computes arithmetic Chern numbers of equivariant Hermitian vector bundles (in particular, the height relatively to some equivariant projective embedding) as a contribution of arithmetic Chern numbers of bundles living of the fixed scheme and an anomaly term, which depends on the complex points of the variety only. Our determination of the anomaly term relies heavily on recent results by Bismut-Goette ([BGo]). The formula in 2.11 is formally similar to Bott’s residue formula [AS, III, Prop. 8.13, p. 598] for the characteristic numbers of vector bundles, up to the anomaly term.

Our method of proof is similar to Atiyah-Singer’s and is described in more de- tail in the introduction to section 2. The effective computability of the anomaly term is also discussed there.

Apart from the residue formula itself, this article has the following two side re- sults, which are of independent interest and which we choose to highlight here, lest they remain unnoticed in the body of the proof of Th. 2.11. The first one is a corollary of the residue formula, which shows that the height relatively to equivariant line bundles on torus-equivariant arithmetic varieties depends on less data than on general varieties (see corollary 2.9):

Proposition. Let Y be an arithmetic variety endowed with a torus action.

WriteYT for the fixed point scheme ofY. Suppose thatL, L⁰ are torus-equivariant hermitian line bundles. If there is an equivariant isometryLYT 'L⁰_YT overYT

and an equivariant (holomorphic) isometry LC 'L⁰_C overYC then the height ofY relatively toLis equal to the height ofY relatively toL⁰.

The second one is a conjecture which naturally arises in the course of the proof of the residue formula (see lemma 2.3):

Conjecture. Let M be aS¹-equivariant projective complex manifold, equipped with anS¹-invariant K¨ahler metric. Let E be a S¹-equivariant complex vector bundle onM, equipped with a S¹-invariant hermitian metric. LetTgt(·)(resp.

Rgt(·) of E) be the equivariant analytic torsion of E (resp. the equivariantR- genus), with respect to the automorphisme^2iπt. There is a rational function Q with complex coefficients and a pointed neighborhoodU^◦ of 0 inRsuch that

1

2Tgt(M, E)−1 2

Z

M_gt

Tdgt(T M)chgt(E)Rgt(T M) =Q(e^2πit)

ift∈U^◦

(hereMgt is the fixed point set of the automorphisme^2iπt, chgt is the equivari- ant Chern character and Tdgt is the equivariant Todd genus - see section 4 of [KR1] for more details).

The lemma 2.3 shows that this conjecture is verified, when the geometric ob- jects appearing in it have certain models over the integers but it seems unlikely that the truth of the conjecture should be dependent on the existence of such models.

The appendix is logically independent of the rest of the article. We formulate a conjectural generalisation of the main result of [KR1].

The notations and conventions of the section 4 of [KR1] (describing the main result) and 6.2 (containing a translation of the fixed point formula into arith- metic Chow theory) will be used without comment. This article is a part of the habilitation thesis of the first author.

Acknowledgments. It is a pleasure to thank Jean-Michel Bismut, Sebastian Goette, Christophe Soul´e and Harry Tamvakis for stimulating discussions and interesting hints. We are grateful to the referees for valuable comments. We thank the SFB 256, ”Nonlinear Partial Differential Equations”, at the Uni- versity of Bonn for its support. The second author is grateful to the IHES (Bures-sur-Yvette) and its very able staff for its support.

2 An ”arithmetic” residue formula

In this subsection, we consider arithmetic varieties endowed with an action of a diagonalisable torus. We shall use the fixed point formula [KR1, Th. 4.4] to obtain a formula computing arithmetic characteristic numbers (like the height relatively to a Hermitian line bundle) in terms of arithmetic characteristic num- bers of the fixed point scheme (a ”residual” term) and an anomaly term derived from the equivariant and non-equivariant analytic torsion. One can express this term using characteristic currents only, without involving the analytic torsion (see subsection 2). See equation (12) for a first version of the residue formula (where the anomaly term is expressed via the analytic torsion) and 2.11 for the final formula (where the anomaly term is expressed using a characteristic current). One can use the residue formula to compute the height of some flag varieties; there the anomaly term can be computed using the explicit values for the torsion given in [K2]. We shall nevertheless not carry out the details of this application, as the next paper [KK] gives a general formula for the height of flag varieties.

The strategy of proof we follow here is parallel to Atiyah-Singer’s in [AS, Sec- tion 8]. Notice however that our proof, which involves theγ-operations, works in the algebraic case as well. The fundamental step of the proof is a passage to the limit on both sides of the arithmetic fixed point formula, where the limit is taken on finite group schemes of increasing order inside a given torus. Both sides of the fixed point formula can be seen as rational functions of a circle ele- ment near 1 and one can thus identify their constant coefficients. The constant coefficient of the arithmetic Lefschetz trace is the arithmetic Euler characteris- tic, which can in turn be related with arithmetic characteristic numbers via the (arithmetic) Riemann-Roch formula.

Furthermore, following a remark of J.-M. Bismut, we would like to point out that a direct proof of the formula 2.11 seems tractable. One could proceed as in the proof of the fixed point formula [KR1, Th. 4.4] (by deformation to the normal cone) and replace at each step the anomaly formulae for the equivariant analytic torsion by the anomaly formulae for the integral appearing in 2.11, the latter formulae having much easier proofs (as they do not involve the spectrum of Laplace operators). One would thus avoid mentioning the analytic torsion altogether. If [KR1, Th. 4.4] and the work of Bismut-Goette was not available, this would probably be the most natural way to approach the residue formula.

LetT := Spec Z[X, X⁻¹] be the one-dimensional torus over Z. Let f : Y → SpecZbe a regular scheme, flat overZ, endowed with aT-projective action and such that the fixed schemeYT is flat overZ(this requirement is only necessary because we choose to work with arithmetic Chow theory). Let d+ 1 be the absolute dimension of Y. This action induces a holomorphic group action of the multiplicative groupC^∗ on the manifold Y(C) =: M and thus an action of the circle S¹ ⊆ C^∗. We equip Y(C) once and for all with an S¹-invariant K¨ahler metricω^{T Y(C)}=ω^{T M} (such a metric can be obtained explicitly via an embedding into some projective space). Now let m >0 be a strictly positive integer coprime to n. Consider the homomorphism sm,n : Z → Z/n, given by the formula a7→ m.(amodn). This homomorphism induces an immersion im,n:µn → T of group schemes. Let now E be aT-equivariant bundle on Y. Recall that the equivariant structure ofEinduces aZ-grading on the restriction E|YT of E to the fixed point scheme of the action of T on Y; the k-th term (k∈Z) of this grading is then denoted byEk.

Lemma 2.1 Write E^m,n for E viewed as a µn-equivariant bundle via im,n. There exists an² >0such that for allk∈Zthe natural injectionEk →E_s^n,m_m,n(k) is an isomorphism if1/n < ².

Proof: This natural injection is an isomorphism iff the equality sm,n(k) = sm,n(k⁰) (k, k⁰ ∈ Z) implies that k = k⁰. Now notice that the kernel of sm,n

is generated by n. Thus the implication is realized if we choose ² such that 1/² >2.max{|k| |k∈Z, Ek6= 0}and we are done. Q.E.D.

Corollary 2.2 Let P be a projective space overZendowed with a global action of the torusT. Write P^m,nforP viewed as a µn-equivariant scheme viaim,n. Then there exists² >0, such that if 1/n < ², then the closed immersionPT → P_T^m,n is an isomorphism.

Proof: LetM be a free Z-module endowed with a T-action, such that there is an equivariant isomorphism P ' P(M). Let us write M^m,n for M viewed as a µn-comodule via im,n. By the description of the fixed scheme given in [KR1, Prop. 2.9], we havePT =`

k∈ZP(Mk) and P_T^m,n=`

k∈Z/nP(M_k^m,n).

Furthermore, by construction the immersion P(Mk)→P factors through the immersion P(Mk) → P(M_s^m,n_m,n(k)) induced by the injection Mk → M_s^n,m_m,n(k).

By the last lemma, there exists an ² > 0 such that for all k ∈ Z the natural injection Mk → M_s^n,m_m,n(k) is an isomorphism if 1/n < ². From this, we can conclude. Q.E.D.

Let again E be a T-equivariant bundle onY, such that the cohomology of E vanishes in positive degrees. We equipECwith anS¹-invariant metric (such a metric can be obtained from an arbitrary metric by integration). Consider E andY as µn-equivariant via im,n. We shall apply [KR1, Th. 7.14] to E. For this application, we fix the primitive root of unity e^2iπm/nofµn(C). Ifα∈C^∗, we shall write g(α) for the corresponding automorphism of Y(C) and we let gm,n:=g(e^2iπm/n). SetM :=Y(C). By [KR1, Th. 7.14], we get

degd_µn(R⁰f∗(E)) = 1

2Tgm,n(M, E)−1 2

Z

M_gm,n

Tdgm,n(T M)chgm,n(E)Rgm,n(T M)

+ddeg¡

f_∗^µn¡^rkN

∨ Y /Yµn

X

i=0

(−1)ⁱchbµn(Λⁱ(N^∨_{Y /Yµn}))¢−1

.Td(T fc ^µn).chbµn(E)¢

(1) Furthermore, using the last lemma and its corollary, we see that there is an

² >0 and a formal Laurent power seriesQ(·) with coefficients in dCH(YT)C(of the formP1(z)/P2(z), whereP1(z) is a polynomial with coefficients indCH(YT)C

and P2(z) is a polynomial with rational coefficients), such that for all n, m coprime with 1/n < ², the term

¡^rkN

∨ Y /Yµn

X

i=0

(−1)ⁱchbµn(Λⁱ(N^∨_{Y /Yµn}))¢−1

.Td(T fc ^µn).chbµn(E) (2) equals Q(e^2iπm/n). Similarly , there is an ² >0 and a rational function Q(·) with complex coefficients, such that for alln, mcoprime with 1/n < ², the term

degd¡

f_∗^µn¡^rkN

∨ Y /Yµn

X

i=0

(−1)ⁱchbµn(Λⁱ(N^∨_{Y /Yµn}))¢−1

.Td(T fc ^µn).chbµn(E)¢ (3) equals Q(e^2iπm/n). Since the elements of the type e^2iπt, where t ∈ Q, form a dense subset ofS¹, we see that the functionQ(z) is uniquely determined. Let us callAT(E) the constant term in the Laurent development of Q(z) around 1. By construction, there is a polynomialP(z) with complex coefficients, such that degd_µn(R⁰f∗(E)) equals P(e^2iπm/n). By density again, this polynomial is uniquely determined. The constant term of its Laurent development around 1 (i.e. its value at 1) is the quantity deg (Rd ⁰f∗E). Using (1), we thus see that there is a uniquely determined rational functionQ⁰(z) with complex coefficients and an² >0, such that the quantity

1

2Tgm,n(M, E)−1 2

Z

M_gm,n

Tdgm,n(T M)chgm,n(E)Rgm,n(T M)

equalsQ⁰(e^2iπm/n) if 1/n < ².

Now notice the following. LetI⊂Rbe an interval such that the fixed point set Mgt does not vary for t∈I. Let gt:=g(e^2πit). Then Rgt varies continuously onI (e.g. using [K2, Remark p. 108]).

Lemma 2.3 There is a pointed neighborhoodU^◦ of0 inRsuch that 1

2Tgt(M, E)−1 2

Z

M_gt

Tdgt(T M)chgt(E)Rgt(T M) =Q⁰(e^2πit)

if t ∈U^◦. Furthermore this equality holds for all up to finitely many values of e^2πit∈S¹.

Proof: It remains to prove that the analytic torsionTgt(M, E) is continuous in tonU^◦ (see also [BGo]). LetI:=]0, ²[ be an interval on which the fixed point set Mgt does not vary. Let Mgt =S

µMµ be the decomposition of the fixed point set into connected components of dimension dimMµ=:dµ.

Let P^⊥ denote the projection of Γ^∞(Λ^qT^∗0,1M ⊗E) on the orthogonal com- plement of the kernel of the Kodaira-Laplace operator ¤q for 0 ≤q ≤d. As shown in Donnelly [Do, Th. 5.1], Donnelly and Patodi [DoP, Th. 3.1] (see also [BeGeV, Th. 6.11]) the trace of the equivariant heat kernel of¤foru→0 has an asymptotic expansion providing the formula

X

q

(−1)^q+1qTrg^∗_te^−u¤qP^⊥ ∼X

µ

X∞ k=−dµ

u^k Z

Mµ

b^µ_k(t, x)dvolx

where thebk(t, x) are rational functions intwhich are non-singular on I. Thus the analytic torsion is given by

Tgt(M, E) = Z ∞

1

X

q

(−1)^q+1qTrg_t^∗e^−u¤qP^⊥du u +

Z 1 0



X

q

(−1)^q+1qTrg^∗_te^−u¤qP^⊥−X

µ

X0 k=−dµ

u^k Z

Mµ

b^µ_k(t, x)dvolx



du u

+X

µ

X−1 k=−dµ

1 k

Z

Mµ

b^µ_k(t, x)dvolx−Γ⁰(1)X

µ

Z

Mµ

b^µ₀(t, x)dvolx .

The integrand of the first term is uniformly bounded (int) by the non-equivariant heat kernel. Hence we see in particular thatTgt(M, E) is continuous int∈I.

As the equation in the lemma holds on a dense subset ofI, it holds inIand by symmetry for a pointed neighborhood of 0. Q.E.D.

Recall that d+ 1 is the (absolute) dimension of Y. Consider the vector field K ∈Γ(T M) such that e^tK =gt onM. In [K1] the function R^rot onR\2πZ

has been defined as

R^rot(φ) := lim

s→0⁺

∂

∂s X∞ k=1

sinkφ k^s

(according to Abel’s Lemma the series in this definition converges for Res >0).

Corollary 2.4 Let Dµ ⊂ Z denote the set of all non-zero eigenvalues of the action ofK/2πonT M|Mµ at the fixed point componentMµ. There are rational functions Q⁰⁰, Qµ,k,lfork∈Dµ,0≤l≤dµ such that for all but finiteley many values ofe^2πit

Tgt(M, E) =Q⁰⁰(e^2πit) +X

µ

X

k∈Dµ

dµ

X

l=0

Qµ,k,l(e^2πit)·(∂

∂t)^lR^rot(2πikt). The functions Qµ,k,l depend only on the holomorphic structure of E and the complex structure onM.

Proof: Forζ=e^iφ∈S¹,ζ6= 1, let L(ζ, s) denote the zeta function defined in [KR1, section 3.3] withL(ζ, s) = P∞

k=1k^−sζ^k for Re s > 0. In [K2, equation (77)] it is shown thatL(ζ,−l) is a rational function inζ forl ∈N0. Also by [K2, equation (80)],

∂

∂s|s=−l(L(e^iφ, s)−(−1)^lL(e^−iφ, s)) = µ−i∂

∂φ

¶^l

2iR^rot(φ).

The corollary follows by the definition of the BismutRg-class (see [KR1, Def.

3.6]) and lemma 2.3. Q.E.D.

Remark. One might reasonably conjecture that the Lemma 2.3 is valid on any compact K¨ahler manifold endowed with a holomorphic action ofS¹.

Let us callLT(E) the constant term in the Laurent development ofQ⁰(z) around z= 1. By lemma 2.3 we obtain

deg (Rd ⁰f∗(E)) =LT(E) +AT(E).

Since for any T-equivariant bundle, one can find a resolution by acyclic (i.e.

whose cohomology vanishes in positive degrees)T-equivariant bundles, one can drop the acyclicity statement in the last equation. More explicitly, one obtains

X

q≥0

(−1)^q¡ ddeg ((R^qf∗(E))free) + log(#(R^qf∗(E))Tors)¢

=LT(E) +AT(E). (4)

Notice thatQ⁰(z) and thus LT(E) depends on the K¨ahler formω^{T M} and EC

only and can thus be computed without reference to the finite part ofY. In the next subsection, we shall apply the last equation to a specific virtual vector bundle, which has the property that its Chern character has only a top

degree term and computeAT(·) in this case. We then obtain a first version of the residue formula, which arises from the fact that the left hand-side of the last equation is also computed by the (non-equivariant) arithmetic Riemann- Roch. The following subsection then shows howLT(·) can be computed using the results of Bismut-Goette [BGo]; combining the results of that subsection with the first version of the residue formula gives our final version 2.11.

2.1 Determination of the residual term

LetF be anT-equivariant Hermitian vector bundle onY.

Definition 2.5 The polynomial equivariant arithmetic total Chern classbct(F)∈ dCHC(YT)[t] is defined by the formula

b

ct(F) := Y

n∈Z rkXFn

p=0

Xp j=0

µrkFn−j p−j

¶ b

cj(Fn)(2πint)^p−j whereiis the imaginary constant.

We can accordingly define thek-th polynomial (equivariant, arithmetic) Chern classbck,t(F) ofFas the part ofbct(F) lying in (dCH(YT)C[t])^(k), where (CH(Yd T)C[t])^(k) are the homogeneous polynomials of weighted degreek(with respect to the grad- ing ofdCH(YT)C). Define now Λt(F) as the formal power seriesP

i≥0Λⁱ(F).tⁱ. Letγ^q(F) be theq-th coefficient in the formal power series Λt/(1−t)(F); this is aZ-linear combination of equivariant Hermitian bundles. We denote bychbt(F) the polynomial equivariant Chern character and by chb^q_t(F) the component of chbt(F) lying in (dCH(YT)C[t])^(q). We recall its definition. LetNj(x1, . . . , xr) be thej-th Newton polynomial _j!¹(x^j₁+. . .+x^j_r) in the variables x1, . . . , xr. For l≥0, letσl(x1, . . . , xr) be thel-th symmetric function in the variablesx1, . . . xr. By the fundamental theorem on symmetric functions, there is a polynomial in r variablesN_j⁰, such thatN_j⁰(σ1(x1, . . . xr), . . . , σr(x1, . . . xr)) =Nj(x1, . . . xr).

We letchbt(F) :=P

j≥0N_j⁰(bc1,t(F),bc2,t(F), . . .bcrkF,t(F)).

Lemma 2.6 The element chb^p_t(γ^q(F −rkF)) is equal to bcq,t(F) if p = q and vanishes ifp < q.

Proof: It is proved in [GS3, II, Th. 7.3.4] thatch, as a map from the arithmeticb Grothendieck groupKb0(Y) to the arithmetic Chow theoryCH(Yd ) is a map of λ-rings, where the second ring is endowed with the λ-ring structure arising from its grading. Thuschb^p_t(γ^q(F−rkF)) is a polynomial in the Chern classes bc1(F),bc2(F), . . .and the variablet. By construction, its coefficients only depend on the equivariant structure of F restricted to YT. We can thus suppose for the time of this proof that the action of T on Y is trivial. To identify these

coefficients, we consider the analogous expression ch^p_t(γ^q(F−rkF)) with values in the polynomial ring CH(Y)[t], where CH(Y) is the algebraic Chow ring. By the same token this is a polynomial in the classical Chern classesc1(F), c2(F), . . . and the variable t. As the forgetful map dCH(Y) → CH(Y) is a map of λ- rings, the coefficients of these polynomials are the same. Thus we can apply the algebraic equivariant splitting principle [Thom, Th. 3.1] and suppose that F=⊕^j_i=1Lj, where theLiare equivariant line bundles. We compute ch^p_t(γ^q(F− rkF)) = ch^p_t(σq(L1−1, . . . , Lj−1)) = (σq(cht(L1)−1, . . . ,cht(Lj)−1))^(p). As the term of lowest degree in cht(Li −1) is c1,t(Li), which is of (total!) degree 1, the term of lowest degree in the expression after the last equality is σq(c1,t(L1), . . . , c1,t(Lj)) which is of degreeqand is equal to cq,t(F) and so we are done. Q.E.D.

Remark. An equivariant holomorphic vector bundleE splits at every compo- nentMµ of the fixed point set into a sum of vector bundlesL

Eθsuch thatK acts on Eθ as iθ ∈ iR. The Eθ are those En,C which do not vanish onMµ. EquipE with an invariant Hermitian metric. Then the polynomial equivariant total Chern formctK(·) is given by the formula

ctK(E)|Mµ = det(−Ω^E

2πi +itΘ^E+ Id) = Y

θ∈R rkXEθ

q=0

cq(Eθ)(1 +itθ)^rkEθ−q (5)

where Θ^E denotes the action ofKonErestricted toMµ. LetN be the normal bundle to the fixed point set. Set

(c^top_tK(N)⁻¹)⁰:= ∂

∂b|b=0crkN(−Ω^N

2πi +itΘ +bId)⁻¹

where Θ is the action of K on N. Furthermore, let r denote the additive characteristic class which is given by

rK(L)|Mµ := − 1 c1(L) +iφµ

µ

−2Γ⁰(1) + 2 log|φµ|+ log(1 + c1(L) iφµ

)

¶

= −X

j≥0

(−c1(L))^j (iφµ)^j+1

Ã

−2Γ⁰(1) + 2 log|φµ| − Xj k=1

1 k

!

forLa line bundle acted upon byK with an angleφµ∈RatMµ (i.e. the Lie derivative byK acts as multiplication byφµ).

In the next proposition, ifEis a Hermitian equivariant bundle, we writeTdcgt(T f)chbgt(E) for the formal Laurent power series development in t of the functionQ(e^2πit),

where Q(·) is the function defined in (2). Set bc^top_t (E) := bcrkE,t(E) for any equivariant Hermitian vector bundleE. Note that this class is invertible in the ring of Laurent polynomialsCH(Yd T)C[t,1/t] ifEYT has no invariant subbundle (for an explicit expression see equation (8)).

Proposition 2.7 Letq1, . . . , qkbe natural numbers such thatP

jqj=d+1. Let E¹, . . . , E^k be T-equivariant Hermitian bundles. Set x:=Q

jγ^qj(E^j−rkE^j).

The expression

Tdcgt(T f)chbgt(x) (6) has a formal Taylor series expansion int. Its constant term is given by the term of maximal degree in

b

c^top_t (N)⁻¹Y

j

b

cqj,t(E^j) (7)

which is independent oft. Also fort→0 Tdgt(T M)Rgt(T M)chgt(x)

= log(t²)·(c^top_K (N)⁻¹)⁰Y

j

cqj,K(E^j)

+rK(N) c^top_K (N)

Y

j

cqj,K(E^j) +O(t).

Note that the first statement implies that degdf_∗^T(bc^top_t (N)⁻¹Q

jbcqj,t(E^j)) = AT(x).

Proof: To prove that the first statement holds, we consider that by construc- tion, both the expression (7) and the constant term of (6) (as a formal Laurent power series) are universal polynomials in the Chern classes of the terms of the grading ofT f and the terms of the grading ofx. By using Grassmannians (more precisely, products of Grassmannians) as in the proof of 2.6, we can reduce the problem of the determination of these coefficients to the algebraic case and then suppose that all the relevant bundles split. Thus, without loss of generality, we consider a vector bundleE:=L

νLν which splits into a direct sum of line bundlesLν on whichT acts with multiplicity mν. Assume now mν 6= 0 for all ν. Setxν := ˆc1(Lν). Then

Tdcgt(E) =Y ¡

1−e^{−2πitmν−xν}¢−1

. Now

¡1−e^{−2πitmν−xν}¢−1

= 1

2πitmν+xν

+O(1)

=

d−rkXN+1 j=0

(−xν)^j

(2πitmν)^j+1 +O(1) ast→0. By definition,

b

c^top_t (E)⁻¹=Y

ν

1

2πitmν+xν =Y

ν

d−rkXN+1 j=0

(−xν)^j

(2πitmν)^j+1 . (8)

Thus,Tdcgt(E) has a Laurent expansion of the form Tdcgt(E) =bc^top_t (E)⁻¹+

d−rkXE+1 j=0

t^−rkE−jqˆj(t)

with classes ˆqjof degreej which have a Taylor expansion intand ˆqj(0) = 0. As Tdcgt(T f^T) = 1+ (terms of higher degree), we get in particular for the relative tangent bundle (assumed w.l.o.g. to be not only a virtual bundle, but a vector bundle)

Tdcgt(T f) =bc^top_t (N)⁻¹+

d−rkXN+1 j=0

t^−rkN−jpˆj(t) (9) with classes ˆpj of degree j which have a Taylor expansion int and ˆpj(0) = 0.

Let degYα denote the degree of a Chow class α and define degtt^k := k for k∈Z. Then any componentαtof the power seriesTdcgt(T f) satisfies

(degY + degt)αt≥ −rkN

and equality is achieved precisely for the summandbc^top_t (N)⁻¹. Furthermore, by Lemma 2.6

chbgt(x) = Yk j=1

³bcqj,t(E^j) +bsj(t)´

(10) with classes bsj(t) of degree larger than qj which have a Taylor expansion in t andbsj(0) = 0. Hence, any componentαtof the power series chbgt(x) satisfies

(degY + degt)αt≥d+ 1 and equality holds iffαtis in theQk

j=1bcqj,t(E^j)-part. Hence any componentαt

ofTdcgt(T f)chbgt(x) satisfies

(degY + degt)αt≥d−rkN+ 1 . (11) In particular the product has no singular terms int, as degYβ ≤d−rkN+ 1 for any Chow classβ on the fixed point scheme. In other words, by multiplying formulae (9) and (10) one obtains

Tdcgt(T f)chbgt(x) =bc^top_t (N)⁻¹ Yk j=1

b

cqj,t(E^j) +O(t),

and the first summand on the right hand side has degY ≤d−rkN+ 1, thus its maximal degree term is constant int. Hence we get formula (7). Now choose

² >0 such that the fixed point set ofgtdoes not vary ont∈]0, ²[. To prove the second formula, we proceed similarly and we formally splitT M as a topological vector bundle into line bundles with first Chern classxν, acted upon byKwith

an angleθν. The formulae for the Lerch zeta function in [K2, p. 108] or in [B2, Th. 7.10] show that theR-class is given by

Rgt(T M) =−X

θ6=0

1 xν+itθν

µ

−2Γ⁰(1) + 2 log|tθν|+ log(1 + xν

itθν

)

¶ +O(1)

fort→0. Note that the singular term is of the formα1(t) log|t|+α2(t) with (degY + degt)αµ(t)≥ −1

forµ= 1,2 (in fact, equality holds). As (c^top_tK(N)⁻¹)⁰=c^top_tK(N)⁻¹P

θ6=0

−1 xν+itθν

by definition, one obtains

Tdgt(T M)Rgt(T M)chgt(x)

= rtK(N) c^top_tK(N)

Y

j

cqj,tK(E^j) +O(t)

= Ã

(c^top_K (N)⁻¹)⁰log(t²) + rK(N) c^top_K (N)

!Y

j

cqj,K(E^j) +O(t)

because the first term on the right hand side is again independent of t ∈ R (except the log(t²)). Q.E.D.

Note that the arithmetic Euler characteristic has a Taylor expansion int. Thus we get using Proposition 2.7 and Lemma 2.3

Corollary 2.8 The equivariant analytic torsion ofxon thed-dimensional K¨ahler manifoldM has an asymptotic expansion for t→0

Tgt(M,Y

j

γ^qj(E^j−rkE^j)) = log(t²) Z

MK

Y

j

cqj,K(E^j)(c^top_K (N)⁻¹)⁰+C0+O(t)

withC0∈C.

A more general version of this corollary is a consequence of [BGo] (see the next section). We now combine our results with the (non-equivariant) arithmetic Riemann-Roch theorem. We compute

deg (fd ∗(Y

j

b

cqj(E^j)))

= deg (fd ∗(Td(T f)c Y

j

b

cqj(E^j))) =deg (fd ∗(Td(T f)c ch(b Y

j

γ^qj(E^j−rkE^j))))

= deg (fd ∗(Y

j

γ^qj(E^j−rkE^j)))−1 2T(Y

j

γ^qj(E^j−rkE^j))

= LT(Y

j

γ^qj(E^j−rkE^j))−1 2T(Y

j

γ^qj(E^j−rkE^j))

+AT(Y

j

γ^qj(E^j−rkE^j))

= LT(Y

j

γ^qj(E^j−rkE^j))−1 2T(Y

j

γ^qj(E^j−rkE^j))

+ddeg (f_∗^T( Q

jbcqj,t(E^j) b

c^top_t (N) ))

The first equality is justified by the fact that the 0-degree part of the Todd class is 1; the second one is 2.6; the third one is justified by the arithmetic Riemann- Roch theorem ([GS8, eq. (1)]); the fourth one is justified by (4) and the last one by the last proposition. Finally, we get the following residue formula:

degdf∗(Y

j

bcqj(E^j)) =LT(Y

j

γ^qj(E^j−rkE^j))

−1

2T(Y(C),Y

j

γ^qj(E^j−rkE^j)) +degdf_∗^T( Q

jbcqj,t(E^j) b

c^top_t (N) ). (12) In particular, ifLis aT-equivariant line bundle onY, one obtains the following formula for the heighthY(L) ofY relatively toL:

h_L(Y) :=deg (d bc1(L)^d+1) = LT((L−1)^d+1)

−1

2T(Y(C),(L−1)^d+1) +degdf_∗^T(bc1,t(L)^d+1 b

c^top_t (N) ).

In our final residue formula, we shall use results of Bismut-Goette to give a formula for the termLT(·)−¹₂T(Y(C),·). Notice however that the last identity already implies the following corollary:

Corollary 2.9 Let Y be an arithmetic variety endowed with aT-action. Sup- pose thatL, L⁰areT-equivariant hermitian line bundles. If there is an equivari- ant isometry LYT 'L⁰_YT over YT and an equivariant (holomorphic) isometry LC'L⁰_C overYC thenh_L(Y) =h_L⁰(Y).

2.2 The limit of the equivariant torsion

LetK⁰ denote any nonzero multiple of K. The vector fieldK⁰ is Hamiltonian with respect to the K¨ahler form as the action onM factors through a projective space. LetMK⁰ =MK denote the fixed point set with respect to the action of K⁰. For any equivariant holomorphic Hermitian vector bundleF we denote by µ^F(K⁰)∈Γ(M,End(F)) the section given by the action of the difference of the Lie derivative and the covariant derivativeL^F_K0− ∇^F_K0 onF. Set as in [BeGeV, ch. 7]

TdK⁰(T M) := Td(−Ω^{T M}

2πi +µ^{T M}(K⁰))∈A(M)

and

chK⁰(F) := Tr exp(−Ω^F

2πi+µ^F(K⁰))∈A(M).

The Chern classcq,K⁰(F) for 0≤q≤rkF is defined as the part of total degree degY + degt=qof

det(−Ω^F

2πi +tµ^F(K⁰) + Id)

att= 1, thuscq,K⁰(F) =cq(−Ω^F/2πi+µ^F(K⁰)). LetK^0∗∈T_R^∗M denote the 1-form dual toK⁰ via the metric on TRM, henceιK⁰K^0∗=kK⁰k² is the norm square inTRM. SetdK⁰K^0∗:= (d−2πiιK⁰)K^0∗and define

sK⁰(u) := −ω^{T M}

2πu exp(dK⁰K^0∗

4πiu ) =

d−1X

ν=0

−ω^{T M}(dK^0∗)^ν

2πu(4πiu)^νν!e^−kK0k2/2u. For a smooth differential formηit is shown in [BGo] (see also [B1, section C,D]) that the following integrals are well-defined:

AK⁰(η)(s) := 1 Γ(s)

Z 1 0

Z

M

ηsK⁰(u)u^s−1du for Res >1 and

BK⁰(η)(s) := 1 Γ(s)

Z ∞ 1

Z

M

ηsK⁰(u)u^s−1du

for Re s < 1. Also it is shown in [BGo] (compare [B1, Proof of theorem 7]) thats7→AK⁰(η)(s) has a meromorphic extension toCwhich is holomorphic at s= 0 and that

AK⁰(η)⁰(0) +BK⁰(η)⁰(0) = Z ∞

1

Z

M

ηsK⁰(u)du u +

Z 1 0

Z

M

η µ

sK⁰(u) + µω^{T M}

2πuc^top_K0(N)⁻¹−(c^top_K0(N)⁻¹)⁰

¶ δMK0

¶du u +

Z

M_K0

η µω^{T M}

2π c^top_K0(N)⁻¹−Γ⁰(1)(c^top_K0(N)⁻¹)⁰

¶

for the derivativesAK⁰(η)⁰,BK⁰(η)⁰ ofAK⁰(η),BK⁰(η) with respect tos; also AK⁰(η)(0) +BK⁰(η)(0) =

Z

M_K0

η(c^top_K0(N)⁻¹)⁰ . Define theBismut S-currentSK⁰(M, ω^{T M}) by the relation

Z

M

ηSK⁰(M, ω^{T M}) :=AK⁰(η)⁰(0) +BK⁰(η)⁰(0).

In particular, one notices Z

M

ηSK⁰(M, ω^{T M}) = lim

a→0⁺

" Z ∞ a

Z

M

ηsK⁰(u)du u +

Z 1 a

Z

M

η µω^{T M}

2πuc^top_K0(N)⁻¹−(c^top_K0(N)⁻¹)⁰

¶ δM_K0

du u +

Z

M_K0

η µω^{T M}

2π c^top_K0(N)⁻¹−Γ⁰(1)(c^top_K0(N)⁻¹)⁰

¶ #

= lim

a→0⁺

" Z

M

η·2iω^{T M}1−exp(^dK_4πia^0K0∗) dK⁰K^0∗

+ Z

M_K0

η µω^{T M}

2πac^top_K0(N)⁻¹−(Γ⁰(1) + loga)(c^top_K0(N)⁻¹)⁰

¶ # .

By Lemma 2.3 and Proposition 2.7, we already know that

t→0lim



Tgt(M, x)−log(t²) Z

MK

Y

j

cqj,K(E^j)·(c^top_K (N)⁻¹)⁰





exists and

2LT(x) = lim

t→0



Tgt(M, x)−log(t²) Z

MK

Y

j

cqj,K(E^j)·(c^top_K (N)⁻¹)⁰





−1 2

Z

YT(C)

Y

j

cqj,K(E^j)· rK(N)

c^top_K (N) . (13)

Now we shall compute this limit.

Theorem 2.10 The limit of the equivariant analytic torsion ofx=Q

jγ^qj(E^j− rkE^j)associated to the action of gt fort→0 is given by

limt→0



Tgt(M, x)−log(t²) Z

MK

Y

j

cqj,K(E^j)·(c^top_K (N)⁻¹)⁰





= T(M, x) + Z

M

Y

j

cqj,K(E^j)·SK(M, ω^{T M})

Proof: Let IK⁰ denote the additive equivariant characteristic class which is given for a line bundleLas follows: IfK⁰ acts at the fixed pointpby an angle θ∈RonL, then

IK⁰(L)|p:=X

k6=0

log(1 + _2πk^θ ) c1(L) +iθ+ 2kπi .

The main result of [BGo] implies that fort∈R\ {0},tsufficiently small, there is a power seriesTtint withT0=T(M, x) such that

Tgt(M, x)−Tt = Z

M

TdtK(T M)chtK(x)StK(M, ω^{T M})

− Z

Mg

Tdgt(T M)chgt(x)ItK(NMg/M).

For t → 0, both ItK(NMg/M)→ 0 and Tdgt(T M)chgt(x) → 0 (by eq. (11)), thus the last summand vanishes.

As in equation (10) chtK(x) = Q

jcqj,tK(E^j) + ˜η(t) with a form ˜η such that (degY + degt)˜η(t) > d+ 1. Thus TdtK(T M)chtK(x) = Q

jcqj,tK(E^j) +η(t) with (degY + degt)η(t)> d+ 1. Also (degY + degt)stK(t²u) =−1, hence we observe that

Z

M

TdtK(T M)chtK(x)stK(t²u) = Z

M



Y

j

cqj,K(E^j)sK(u) +η(t)stK(t²u)



 . Let ˜η(t) denote the form obtained fromη(t) by multiplying the degree degY =j part witht^−j−1for 0≤j≤d. By making the change of variable fromutot²u we get

(AtK+BtK)(TdtK(T M)chtK(x))(s) =t^2s(AK+BK)(Y

j

cqj,K(E^j) + ˜η(t))(s). Thus we find

Z

M

TdtK(T M)chtK(x)StK(M, ω^{T M})

= log(t²)·(AK+BK)(Y

j

cqj,K(E^j))(0) +(AK+BK)(Y

j

cqj,K(E^j))⁰(0) +O(tlog(t²)) which implies the statement of the theorem. Q.E.D.

2.3 The residue formula

By combining equation (12) and Theorem 2.10, we obtain the following formula.

Recall that T is the one-dimensional diagonalisable torus over SpecZ, that f : Y → SpecZ is a flat, T-projective morphism and that the fixed scheme f^T : YT → SpecZ is assumed to be flat over SpecZ. We let d+ 1 be the absolute dimension of Y. We choose T-equivariant Hermitian bundles E^j on Y and positive integersqj such thatP

jqj =d+ 1. We deduce by combining equations (12), (13) and Theorem 2.10

Theorem 2.11

ddeg µ

f∗¡ Y

j

b

cqj(E^j)¢¶

=degd µ

f_∗^T¡Q

jbcqj,t(E^j) b

c^top_t (N)

¢¶

+1 2

Z

Y(C)

Y

j

cqj,K(E^j)·SK(Y(C), ω^{T Y(C)})−1 2 Z

YT(C)

Y

j

cqj,K(E^j)· rK(N) c^top_K (N) .

Example. Assume that the fixed point scheme is flat of Krull dimension 1.

The normal bundle toYT splits as N=L

n∈ZNn. Thus ˆ

c^top_t (N)⁻¹= 1 Q

n(2πitn)^rkNn Ã

1−X

n∈Z

ˆ c1(Nn)

2πitn

!

by equation (8). Also, at a given pointp∈MK the tangent space decomposes asT M|p=LT Mθν, whereK acts with angleθν onT Mθν. Then

rK(N)

c^top_K (N)_|p= 1 Q

θiθ X

θ

2Γ⁰(1)−2 log|θ|

iθ

where theθare counted with their multiplicity. Furthermore, in this case Z

M

ηSK(M, ω^{T M}) = lim

a→0⁺

" Z

M

η·2iω^{T M}1−exp(^d_4πia^KK∗) dKK^∗

−(Γ⁰(1) + loga) Z

MK

η·(c^top_K (N)⁻¹)⁰

#

= Z

M

η·2iω^{T M} µ 1

dKK^∗ + (dK^∗)^d−1 (2πikKk)^d

¶

− lim

a→0⁺

" Z

M

η· 2iω^{T M}(dK^∗)^d−1

(2πikKk)^d (1−e^−2a1kKk2) +(Γ⁰(1) + loga)

Z

MK

η·(c^top_K (N)⁻¹)⁰

# . Now consider a line bundle L, splitting as L

kLk on the fixed point scheme (where theLk are locally free of rank≤1). We find

b

c1,t(L)^d+1 = X

k

(bc1(Lk) + 2πitkrkLk)^d+1

= X

k

¡(2πitk)^d+1rkLk+ (d+ 1)(2πitk)^dˆc1(Lk)¢ ,

thus degdf_∗^T

µbc1,t(L)^d+1 b c^top_t (N)

¶

= degdf_∗^TX

k

k^d Q

nn^rkNn Ã

(d+ 1)bc1(Lk)−krkLk·X

n∈Z

b c1(Nn)

n

! . Now notice that at a given fixed pointpoverCall but oneLk,Cvanish and set φp:= 2πk. We compute

−1 2

X

p∈MK

c1,K(L)^d+1

c^top_K (N) rK(N) = X

p∈MK

φ^d+1_p Q

θθ X

θ

−Γ⁰(1) + log|θ|

θ

and

−1

2(Γ⁰(1)+loga) X

p∈MK

c1,K(L)^d+1(c^top_K (N)⁻¹)⁰= X

p∈MK

φ^d+1_p Q

θθ X

θ

Γ⁰(1) + loga

2θ .

Hence we finally get

degdf∗bc1(L)^d+1 = degdf_∗^TX

k

k^d Q

nn^rkNn Ã

(d+ 1)bc1(Lk)−krkLk·X

n∈Z

b c1(Nn)

n

!

+ X

p∈MK

φ^d+1_p Q

θθ X

θ

−Γ⁰(1) + log(θ²) 2θ +

Z

M

c1,K(L)^d+1·iω^{T M} µ 1

dKK^∗ + (dK^∗)^d−1 (2πikKk)^d

¶

− lim

a→0⁺

" Z

M

µ^L(K)^d+1·iω^{T M}(dK^∗)^d−1

(2πikKk)^d (1−e^−2a1kKk2)

−loga X

p∈MK

φ^d+1_p Q

θθ X

θ

1 2θ

# .

3 Appendix: a conjectural relative fixed point formula in Arakelov theory

Since the first part of this series of articles was written, Xiaonan Ma defined in [Ma] higher analogs of the equivariant analytic torsion and proved curvature and anomaly formulae for it (in the case of fibrations by tori, this had already been done in [K4]). Once such formulae for torsion forms are available, one can formulate a conjectural fixed point formula, which fully generalizes [KR1, Th.

4.4] to the relative setting. LetG be a compact Lie group and letM andM⁰