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

formule du point fixe de type Lefschetz en géométrie d’Arakelov II: une formule des résidus

Kai Köhler ^{∗} 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 formula ”à 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 série d’articles dont l’objet est un analogue en géométrie d’Arakelov de la formule du point fixe de Lefschetz holomorphe. Nous utilisons le résultat principal [KR1, Th. 4.4] du premier article pour prouver une formule des résidus ”à la Bott” pour des classes caractéristiques vivant sur des variétés arithmétiques munis d’une action de tore; de récents résultats de Bismut-Goette sur la torsion analytique équivariante (de Ray-Singer) joue un rôle clé dans la preuve.

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

∗Centre de Mathématiques de Jussieu, Université 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 Mathématiques de Jussieu, Université 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 ”à la Bott” (Theorem 2.11) for the arithmetic Chern numbers of arithmetic varieties endowed with the action of a diagonalisable 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 detail 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 results, 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. Write Y_{T} for the fixed point scheme of Y. Suppose that \overline{L}, \overline{L}′ are torus-equivariant hermitian line bundles. If there is an equivariant isometry \overline{L}_{Y_{T}} ≃ \overline{L}′_{Y_{T}} over Y_{T} and an equivariant (holomorphic) isometry \overline{L}_{C} ≃ \overline{L}′_{C} over Y_{C} then the height of Y relatively to \overline{L} is equal to the height of Y relatively to \overline{L}′. 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 a S^{1}-equivariant projective complex manifold, equipped with an S^{1}-invariant Kähler metric. Let \overline{E} be a S^{1}-equivariant complex vector bundle on M, equipped with a S^{1}-invariant hermitian metric. Let T_{g_{t}}(·) (resp. R_{g_{t}}(·) of E) be the equivariant analytic torsion of \overline{E} (resp. the equivariant Rgenus), with respect to the automorphism e^{2iπt}. There is a rational function Q

with complex coefficients and a pointed neighborhood U^{∘} of 0 in R such that

\frac{1}{2}T_{g_{t}}(M, \overline{E}) − \frac{1}{2} ∫_{M_{g_{t}}} Td_{g_{t}}(T M)ch_{g_{t}}(E)R_{g_{t}}(T M) = Q(e^{2πit})

if t ∈U^{∘}

(here M_{g_{t}} is the fixed point set of the automorphism e^{2iπt}, ch_{g_{t}} is the equivariant Chern character and Td_{g_{t}} 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 objects 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 arithmetic 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 Soulé 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 University 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 numbers 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, Section 8]. Notice however that our proof, which involves the γ-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 element near 1 and one can thus identify their constant coefficients. The constant coefficient of the arithmetic Lefschetz trace is the arithmetic Euler characteristic, 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. Let T := Spec Z[X, X^{−1}] be the one-dimensional torus over Z. Let f : Y → Spec Z be a regular scheme, flat over Z, endowed with a T -projective action and such that the fixed scheme Y_{T} is flat over Z (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 group C^{∗} on the manifold Y (C) =: M and thus an action of the circle S^{1} ⊆ C^{∗}. We equip Y (C) once and for all with an S^{1}-invariant Kähler metric ω^{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 s_{m,n} : Z → Z/n, given by the formula a ↦ m.(a mod n). This homomorphism induces an immersion i_{m,n} : μ_{n} → T of group schemes. Let now E be a T -equivariant bundle on Y. Recall that the equivariant structure of E induces a Z-grading on the restriction E|_{Y_{T}} 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 by E_{k}.

Lemma 2.1 Write E^{m,n} for E viewed as a μ_{n}-equivariant bundle via i_{m,n}. There exists an ϵ > 0 such that for all k ∈ Z the natural injection E_{k} → E_{s_{m,n}(k)}^{n,m} is an isomorphism if 1/n < ϵ.

Proof: This natural injection is an isomorphism iff the equality s_{m,n}(k) = s_{m,n}(k′) (k, k′ ∈ Z) implies that k = k′. Now notice that the kernel of s_{m,n} is generated by n. Thus the implication is realized if we choose ϵ such that 1/ϵ > 2. max{|k| |k ∈ Z, E_{k} ≠ 0} and we are done. Q.E.D.

Corollary 2.2 Let P be a projective space over Z endowed with a global action of the torus T . Write P^{m,n} for P viewed as a μ_{n}-equivariant scheme via i_{m,n}. Then there exists ϵ > 0, such that if 1/n < ϵ, then the closed immersion P_{T} → P_{T}^{m,n} is an isomorphism.

Proof: Let M 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 μ_{n}-comodule via i_{m,n}. By the description of the fixed scheme given in [KR1, Prop. 2.9], we have P_{T} = ∐_{k∈Z} P(M_{k}) and P_{T}^{m,n} = ∐_{k∈Z/n} P(M_{k}^{m,n}). Furthermore, by construction the immersion P(M_{k}) → P factors through the immersion P(M_{k}) → P(M_{s_{m,n}(k)}^{m,n}) induced by the injection M_{k} → M_{s_{m,n}(k)}^{n,m}.

By the last lemma, there exists an ϵ > 0 such that for all k ∈ Z the natural injection M_{k} → M_{s_{m,n}(k)}^{n,m} is an isomorphism if 1/n < ϵ. From this, we can conclude. Q.E.D.

Let again E be a T -equivariant bundle on Y, such that the cohomology of E vanishes in positive degrees. We equip E_{C} with an S^{1}-invariant metric (such a metric can be obtained from an arbitrary metric by integration). Consider E and Y as μ_{n}-equivariant via i_{m,n}. We shall apply [KR1, Th. 7.14] to E. For this application, we fix the primitive root of unity e^{2iπm/n} of μ_{n}(C). If α ∈ C^{∗}, we shall write g(α) for the corresponding automorphism of Y (C) and we let g_{m,n} := g(e^{2iπm/n}). Set M := Y (C). By [KR1, Th. 7.14], we get

d̂eg_{μ_{n}}(R^{0}f_{∗}(\overline{E})) = \frac{1}{2}T_{g_{m,n}}(M, \overline{E}) − \frac{1}{2} ∫_{M_{gm,n}} Td_{g_{m,n}}(T M)ch_{g_{m,n}}(E)R_{g_{m,n}}(T M)

+d̂eg(f_{∗}^{μ_{n}}( ∑_{i=0}^{rk N_{Y /Yμ_{n}}^{∨}} (−1)^{i}ĉh_{μ_{n}}(Λ^{i}(\overline{N}_{Y/Y_{μn}}^{∨})))^{−1}

.T̂d(\overline{T f^{μ_{n}}}).ĉh_{μ_{n}}(\overline{E}))  (1)

Furthermore, using the last lemma and its corollary, we see that there is an ϵ > 0 and a formal Laurent power series Q(·) with coefficients in ĈH(Y_{T})_{C} (of the form P_{1}(z)/P_{2}(z), where P_{1}(z) is a polynomial with coefficients in ĈH(Y_{T})_{C} and P_{2}(z) is a polynomial with rational coefficients), such that for all n, m coprime with 1/n < ϵ, the term

( ∑_{i=0}^{rk N_{Y /Yμ_{n}}^{∨}} (−1)^{i}ĉh_{μ_{n}}(Λ^{i}(\overline{N}_{Y/Y_{μn}}^{∨})))^{−1}.T̂d(\overline{T f^{μ_{n}}}).ĉh_{μ_{n}}(\overline{E})  (2)

equals Q(e^{2iπm/n}). Similarly , there is an ϵ > 0 and a rational function Q(·) with complex coefficients, such that for all n, m coprime with 1/n < ϵ, the term

d̂eg(f_{∗}^{μ_{n}}( ∑_{i=0}^{rk N_{Y /Yμ_{n}}^{∨}} (−1)^{i}ĉh_{μ_{n}}(Λ^{i}(\overline{N}_{Y/Y_{μn}}^{∨})))^{−1}.T̂d(\overline{T f^{μ_{n}}}).ĉh_{μ_{n}}(\overline{E})) (3)

equals Q(e^{2iπm/n}). Since the elements of the type e^{2iπt}, where t ∈ Q, form a dense subset of S^{1}, we see that the function Q(z) is uniquely determined. Let us call A_{T} (\overline{E}) the constant term in the Laurent development of Q(z) around

1. By construction, there is a polynomial P (z) with complex coefficients, such that d̂eg_{μ_{n}}(R^{0}f_{∗}(\overline{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 d̂eg(R^{0}f_{∗}\overline{E}). Using (1), we thus see that there is a uniquely determined rational function Q′(z) with complex coefficients and an ϵ > 0, such that the quantity

\frac{1}{2}T_{g_{m,n}}(M, \overline{E}) − \frac{1}{2} ∫_{M_{gm,n}} Td_{g_{m,n}}(T M)ch_{g_{m,n}}(E)R_{g_{m,n}}(T M)

equals Q′(e^{2iπm/n}) if 1/n < ϵ.

Now notice the following. Let I ⊂ R be an interval such that the fixed point set M_{g_{t}} does not vary for t ∈ I. Let g_{t} := g(e^{2πit}). Then R_{g_{t}} varies continuously on I (e.g. using [K2, Remark p. 108]).

Lemma 2.3 There is a pointed neighborhood U^{∘} of 0 in R such that

\frac{1}{2}T_{g_{t}}(M, \overline{E}) − \frac{1}{2} ∫_{M_{g_{t}}} Td_{g_{t}}(T M)ch_{g_{t}}(E)R_{g_{t}}(T M) = Q′(e^{2πit})

if t ∈U^{∘}. Furthermore this equality holds for all up to finitely many values of e^{2πit} ∈ S^{1}.

Proof: It remains to prove that the analytic torsion T_{g_{t}}(M, \overline{E}) is continuous in

t on U^{∘} (see also [BGo]). Let I :=]0, ϵ[ be an interval on which the fixed point set M_{g_{t}} does not vary. Let M_{g_{t}} = ⋃_{μ} M_{μ} be the decomposition of the fixed point set into connected components of dimension dim M_{μ} =: d_{μ}.

Let P^{⊥} denote the projection of Γ^{∞}(Λ^{q}T^{∗0,1}M ⊗ E) on the orthogonal complement 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 □ for u → 0 has an asymptotic expansion providing the formula

∑_{q}(−1)^{q+1}qTr g_{t}^{∗}e^{−u□_{q}}P^{⊥} ∼ ∑_{μ} ∑_{k=−d_{μ}}^{∞} u^{k} ∫_{M_{μ}} b_{k}^{μ}(t, x) dvol_{x}

where the b_{k}(t, x) are rational functions in t which are non-singular on I. Thus the analytic torsion is given by

T_{g_{t}}(M, \overline{E}) = ∫_{1}^{∞} ∑_{q}(−1)^{q+1}qTr g_{t}^{∗}e^{−u□_{q}}P^{⊥} \frac{du}{u}

+ ∫_{0}^{1} (∑_{q}(−1)^{q+1}qTr g_{t}^{∗}e^{−u□_{q}}P^{⊥} − ∑_{μ} ∑_{k=−d_{μ}}^{0} u^{k} ∫_{M_{μ}} b_{k}^{μ}(t, x) dvol_{x}) \frac{du}{u}

+ ∑_{μ} ∑_{k=−d_{μ}}^{−1} \frac{1}{k} ∫_{M_{μ}} b_{k}^{μ}(t, x) dvol_{x} − Γ′(1) ∑_{μ} ∫_{M_{μ}} b_{0}^{μ}(t, x) dvol_{x} .

The integrand of the first term is uniformly bounded (in t) by the non-equivariant heat kernel. Hence we see in particular that T_{g_{t}}(M, \overline{E}) is continuous in t ∈ I. As the equation in the lemma holds on a dense subset of I, it holds in I and 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} = g_{t} on M. In [K1] the function R^{rot} on R \ 2πZ has been defined as

R^{rot}(ϕ) := lim_{s→0^{+}} \frac{∂}{∂s} ∑_{k=1}^{∞} \frac{sin kϕ}{k^{s}}

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

Corollary 2.4 Let D_{μ} ⊂ Z denote the set of all non-zero eigenvalues of the action of K/2π on T M_{|M_{μ}} at the fixed point component M_{μ}. There are rational functions Q′′, Q_{μ,k,l} for k ∈ D_{μ}, 0 ≤ l ≤ d_{μ} such that for all but finiteley many values of e^{2πit}

T_{g_{t}}(M, \overline{E}) = Q′′(e^{2πit}) + ∑_{μ} ∑_{k∈D_{μ}} ∑_{l=0}^{d_{μ}} Q_{μ,k,l}(e^{2πit}) · (\frac{∂}{∂t})^{l}R^{rot}(2πikt) .

The functions Q_{μ,k,l} depend only on the holomorphic structure of E and the complex structure on M.

Proof: For ζ = e^{iϕ} ∈ S^{1}, ζ ≠ 1, let L(ζ, s) denote the zeta function defined in [KR1, section 3.3] with L(ζ, s) = ∑_{k=1}^{∞} k^{−s}ζ^{k} for Re s > 0. In [K2, equation (77)] it is shown that L(ζ, −l) is a rational function in ζ for l ∈ N_{0}. Also by [K2, equation (80)],

\frac{∂}{∂s}_{|s=−l}(L(e^{iϕ}, s) − (−1)^{l}L(e^{−iϕ}, s)) = (\frac{−i∂}{∂ϕ} )^{l} 2iR^{rot}(ϕ) .

The corollary follows by the definition of the Bismut R_{g}-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 Kähler manifold endowed with a holomorphic action of S^{1}.

Let us call L_{T} (\overline{E}) the constant term in the Laurent development of Q′(z) around z = 1. By lemma 2.3 we obtain

d̂eg(R^{0}f_{∗}(\overline{E})) = L_{T} (\overline{E}) + A_{T} (\overline{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

∑_{q≥0}(−1)^{q}(d̂eg((R^{q}f_{∗}(\overline{E}))_{free}) + log(#(R^{q}f_{∗}(\overline{E}))_{Tors})) = L_{T} (\overline{E}) + A_{T} (\overline{E}). (4)

Notice that Q′(z) and thus L_{T} (\overline{E}) depends on the Kähler form ω^{T M} and \overline{E_{C}} only and can thus be computed without reference to the finite part of Y. 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 compute A_{T} (·) 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 how L_{T} (·) 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

Let \overline{F} be an T -equivariant Hermitian vector bundle on Y.

Definition 2.5 The polynomial equivariant arithmetic total Chern class ĉ_{t}(\overline{F}) ∈ ĈH_{C}(Y_{T})[t] is defined by the formula

ĉ_{t}(\overline{F}) := ∏_{n∈Z} ∑_{p=0}^{rk F_{n}} ∑_{j=0}^{p} \binom{rk F_{n} − j}{p − j}ĉ_{j}(\overline{F}_{n})(2πint)^{p−j}

where i is the imaginary constant.

We can accordingly define the k-th polynomial (equivariant, arithmetic) Chern class ĉ_{k,t}(\overline{F}) of \overline{F} as the part of ĉ_{t}(\overline{F}) lying in (ĈH(Y_{T})_{C}[t])^{(k)}, where (ĈH(Y_{T})_{C}[t])^{(k)} are the homogeneous polynomials of weighted degree k (with respect to the grading of ĈH(Y_{T})_{C}). Define now Λ_{t}(\overline{F}) as the formal power series ∑_{i≥0} Λ^{i}(\overline{F}).t^{i}. Let γ^{q}(\overline{F}) be the q-th coefficient in the formal power series Λ_{t/(1−t)}(\overline{F}); this is a Z-linear combination of equivariant Hermitian bundles. We denote by ĉh_{t}(\overline{F}) the polynomial equivariant Chern character and by ĉh_{t}^{q}(\overline{F}) the component of ĉh_{t}(\overline{F}) lying in (ĈH(Y_{T})_{C}[t])^{(q)}. We recall its definition. Let N_{j}(x_{1}, …, x_{r}) be the j-th Newton polynomial \frac{1}{j!}(x_{1}^{j} + … + x_{r}^{j}) in the variables x_{1}, …, x_{r}. For l ≥ 0, let σ_{l}(x_{1}, …, x_{r}) be the l-th symmetric function in the variables x_{1}, … x_{r}. By the fundamental theorem on symmetric functions, there is a polynomial in r variables N′_{j}, such that N′_{j}(σ_{1}(x_{1}, … x_{r}), …, σ_{r}(x_{1}, … x_{r})) = N_{j}(x_{1}, … x_{r}). We let ĉh_{t}(\overline{F}) := ∑_{j≥0} N′_{j}(ĉ_{1,t}(\overline{F}), ĉ_{2,t}(\overline{F}), … ĉ_{rk F,t}(\overline{F})).

Lemma 2.6 The element ĉh_{t}^{p}(γ^{q}(\overline{F} − rk F)) is equal to ĉ_{q,t}(\overline{F}) if p = q and vanishes if p < q.

Proof: It is proved in [GS3, II, Th. 7.3.4] that ĉh, as a map from the arithmetic Grothendieck group K̂_{0}(Y) to the arithmetic Chow theory ĈH(Y) is a map of λ-rings, where the second ring is endowed with the λ-ring structure arising from its grading. Thus ĉh_{t}^{p}(γ^{q}(\overline{F} − rk F)) is a polynomial in the Chern classes ĉ_{1}(\overline{F}), ĉ_{2}(\overline{F}), … and the variable t. By construction, its coefficients only depend on the equivariant structure of F restricted to Y_{T}. 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_{t}^{p}(γ^{q}(F − rk F)) 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 classes c_{1}(F), c_{2}(F), … and the variable t. As the forgetful map ĈH(Y) → CH(Y) is a map of λrings, 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 = ⊕_{i=1}^{j}L_{j}, where the L_{i} are equivariant line bundles. We compute ch_{t}^{p}(γ^{q}(F − rk F)) = ch_{t}^{p}(σ_{q}(L_{1} − 1, …, L_{j} − 1)) = (σ_{q}(ch_{t}(L_{1}) − 1, …, ch_{t}(L_{j}) − 1))^{(p)}. As the term of lowest degree in ch_{t}(L_{i} − 1) is c_{1,t}(L_{i}), which is of (total!) degree 1, the term of lowest degree in the expression after the last equality is σ_{q}(c_{1,t}(L_{1}), …, c_{1,t}(L_{j})) which is of degree q and is equal to c_{q,t}(F) and so we are done. Q.E.D.

Remark. An equivariant holomorphic vector bundle E splits at every component M_{μ} of the fixed point set into a sum of vector bundles ⨁ E_{θ} such that K acts on E_{θ} as iθ ∈ iR. The E_{θ} are those E_{n,C} which do not vanish on M_{μ}. Equip E with an invariant Hermitian metric. Then the polynomial equivariant total Chern form c_{tK}(·) is given by the formula

c_{tK}(\overline{E})_{|M_{μ}} = det(\frac{−Ω^{E}}{2πi} + itΘ^{E} + Id) = ∏_{θ∈R} ∑_{q=0}^{rk E_{θ}} c_{q}(\overline{E}_{θ})(1 + itθ)^{rk E_{θ}−q} (5)

where Θ^{E} denotes the action of K on E restricted to M_{μ}. Let N be the normal bundle to the fixed point set. Set

(c_{tK}^{top}(\overline{N})^{−1})′ := \frac{∂}{∂b}_{|b=0}c_{rk N}(\frac{−Ω^{N}}{2πi} + itΘ + b Id)^{−1}

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

r_{K}(L)_{|M_{μ}} := −\frac{1}{c_{1}(L) + iϕ_{μ}} (−2Γ′(1) + 2 log |ϕ_{μ}| + log(1 + \frac{c_{1}(L)}{iϕ_{μ}}))

= − ∑_{j≥0} \frac{(−c_{1}(L))^{j}}{(iϕ_{μ})^{j+1}} (−2Γ′(1) + 2 log |ϕ_{μ}| − ∑_{k=1}^{j} \frac{1}{k} )

for L a line bundle acted upon by K with an angle ϕ_{μ} ∈ R at M_{μ} (i.e. the Lie derivative by K acts as multiplication by ϕ_{μ}). In the next proposition, if \overline{E} is a Hermitian equivariant bundle, we write T̂d_{g_{t}}(\overline{T f})ĉh_{g_{t}}(\overline{E}) for the formal Laurent power series development in t of the function Q(e^{2πit}), where Q(·) is the function defined in (2). Set c_{t}^{top}(\overline{E}) := ĉ_{rk E,t}(\overline{E}) for any equivariant Hermitian vector bundle \overline{E}. Note that this class is invertible in the ring of Laurent polynomials ĈH(Y_{T})_{C}[t, 1/t] if E_{Y_{T}} has no invariant subbundle (for an explicit expression see equation (8)).

Proposition 2.7 Let q_{1}, …, q_{k} be natural numbers such that ∑_{j} q_{j} = d+1. Let

\overline{E}^{1}, …, \overline{E}^{k} be T -equivariant Hermitian bundles. Set x := ∏_{j} γ^{q_{j}}(\overline{E}^{j} − rk E^{j}). The expression

T̂d_{g_{t}}(\overline{T f})ĉh_{g_{t}}(x)  (6)

has a formal Taylor series expansion in t. Its constant term is given by the term of maximal degree in

c_{t}^{top}(\overline{N})^{−1} ∏_{j} ĉ_{q_{j},t}(\overline{E}^{j})  (7)

which is independent of t. Also for t → 0

Td_{g_{t}}(T M)R_{g_{t}}(T M)ch_{g_{t}}(x)
= log(t^{2}) · (c_{K}^{top}(N)^{−1})′ ∏_{j} c_{q_{j},K}(E^{j})

+\frac{r_{K}(N)}{c_{K}^{top}(N)} ∏_{j} c_{q_{j},K}(E^{j}) + O(t) .

Note that the first statement implies that d̂egf_{∗}^{T} (c_{t}^{top}(\overline{N})^{−1} ∏_{j} ĉ_{q_{j},t}(\overline{E}^{j})) = A_{T} (x).

Proof: To prove that the first statement holds, we consider that by construction, 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 of T f and the terms of the grading of x. 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 bundle \overline{E} := ⨁_{ν} \overline{L}_{ν} which splits into a direct sum of line bundles \overline{L}_{ν} on which T acts with multiplicity m_{ν}. Assume now m_{ν} ≠ 0 for all ν. Set x_{ν} := ĉ_{1}(\overline{L}_{ν}). Then

T̂d_{g_{t}}(\overline{E}) = ∏ (1 − e^{−2πitm_{ν}−x_{ν}} )^{−1} .

Now

(1 − e^{−2πitm_{ν}−x_{ν}} )^{−1} = \frac{1}{2πitm_{ν} + x_{ν}} + O(1)

= ∑_{j=0}^{d−rk N+1} \frac{(−x_{ν})^{j}}{(2πitm_{ν})^{j+1}} + O(1)

as t → 0. By definition,

c_{t}^{top}(\overline{E})^{−1} = ∏_{ν} \frac{1}{2πitm_{ν} + x_{ν}} = ∏_{ν} ∑_{j=0}^{d−rk N+1} \frac{(−x_{ν})^{j}}{(2πitm_{ν})^{j+1}} .  (8)

Thus, T̂d_{g_{t}}(\overline{E}) has a Laurent expansion of the form

T̂d_{g_{t}}(\overline{E}) = c_{t}^{top}(\overline{E})^{−1} + ∑_{j=0}^{d−rk E+1} t^{−rk E−j}q̂_{j}(t)

with classes q̂_{j} of degree j which have a Taylor expansion in t and q̂_{j}(0) = 0. As T̂d_{g_{t}}(\overline{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)

T̂d_{g_{t}}(\overline{T f}) = c_{t}^{top}(\overline{N})^{−1} + ∑_{j=0}^{d−rk N+1} t^{−rk N−j}p̂_{j}(t)  (9)

with classes p̂_{j} of degree j which have a Taylor expansion in t and p̂_{j}(0) = 0. Let deg_{Y} α denote the degree of a Chow class α and define deg_{t}t^{k} := k for k ∈ Z. Then any component α_{t} of the power series T̂d_{g_{t}}(\overline{T f}) satisfies

(deg_{Y} + deg_{t})α_{t} ≥ −rk N

and equality is achieved precisely for the summand c_{t}^{top}(\overline{N})^{−1}. Furthermore, by Lemma 2.6

ĉh_{g_{t}}(x) = ∏_{j=1}^{k} (ĉ_{q_{j},t}(\overline{E}^{j}) + ŝ_{j}(t))  (10)

with classes ŝ_{j}(t) of degree larger than q_{j} which have a Taylor expansion in t and ŝ_{j}(0) = 0. Hence, any component α_{t} of the power series ĉh_{g_{t}}(x) satisfies

(deg_{Y} + deg_{t})α_{t} ≥ d + 1

and equality holds iff α_{t} is in the ∏_{j=1}^{k} ĉ_{q_{j},t}(\overline{E}^{j})-part. Hence any component α_{t} of T̂d_{g_{t}}(\overline{T f})ĉh_{g_{t}}(x) satisfies

(deg_{Y} + deg_{t})α_{t} ≥ d − rk N + 1 .  (11)

In particular the product has no singular terms in t, as deg_{Y} β ≤ d − rk N + 1 for any Chow class β on the fixed point scheme. In other words, by multiplying formulae (9) and (10) one obtains

T̂d_{g_{t}}(\overline{T f})ĉh_{g_{t}}(x) = c_{t}^{top}(\overline{N})^{−1} ∏_{j=1}^{k} ĉ_{q_{j},t}(\overline{E}^{j}) + O(t) ,

and the first summand on the right hand side has deg_{Y} ≤ d − rk N + 1, thus its maximal degree term is constant in t. Hence we get formula (7). Now choose ϵ > 0 such that the fixed point set of g_{t} does not vary on t ∈]0, ϵ[. To prove the second formula, we proceed similarly and we formally split T M as a topological vector bundle into line bundles with first Chern class x_{ν}, acted upon by K with an angle θ_{ν}. The formulae for the Lerch zeta function in [K2, p. 108] or in [B2, Th. 7.10] show that the R-class is given by

R_{g_{t}}(T M) = − ∑_{θ≠0} \frac{1}{x_{ν} + itθ_{ν}} (−2Γ′(1) + 2 log |tθ_{ν}| + log(1 + \frac{x_{ν}}{itθ_{ν}})) + O(1)

for t → 0. Note that the singular term is of the form α_{1}(t) log |t| + α_{2}(t) with

(deg_{Y} + deg_{t})α_{μ}(t) ≥ −1

for μ = 1, 2 (in fact, equality holds). As (c_{tK}^{top}(N)^{−1})′ = c_{tK}^{top}(N)^{−1} ∑_{θ≠0} \frac{−1}{x_{ν}+itθ_{ν}} by definition, one obtains

Td_{g_{t}}(T M)R_{g_{t}}(T M)ch_{g_{t}}(x)
= \frac{r_{tK}(N)}{c_{tK}^{top}(N)} ∏_{j} c_{q_{j},tK}(E^{j}) + O(t)

= ((c_{K}^{top}(N)^{−1})′ log(t^{2}) + \frac{r_{K}(N)}{c_{K}^{top}(N)}) ∏_{j} c_{q_{j},K}(E^{j}) + O(t)

because the first term on the right hand side is again independent of t ∈ R (except the log(t^{2})). Q.E.D.

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

Corollary 2.8 The equivariant analytic torsion of x on the d-dimensional Kähler manifold M has an asymptotic expansion for t → 0

T_{g_{t}}(M, ∏_{j} γ^{q_{j}}(\overline{E}^{j} −rk E^{j})) = log(t^{2}) ∫_{M_{K}} ∏_{j} c_{q_{j},K}(\overline{E}^{j})(c_{K}^{top}(\overline{N})^{−1})′ +C_{0} +O(t)

with C_{0} ∈ 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

d̂eg(f_{∗}(∏_{j} ĉ_{q_{j}}(\overline{E}^{j})))

= d̂eg(f_{∗}(T̂d(\overline{T f}) ∏_{j} ĉ_{q_{j}}(\overline{E}^{j}))) = d̂eg(f_{∗}(T̂d(\overline{T f})ĉh(∏_{j} γ^{q_{j}}(\overline{E}^{j} − rk E^{j}))))

= d̂eg(f_{∗}(∏_{j} γ^{q_{j}}(\overline{E}^{j} − rk E^{j}))) − \frac{1}{2}T (∏_{j} γ^{q_{j}}(\overline{E}^{j} − rk E^{j}))

= L_{T} (∏_{j} γ^{q_{j}}(\overline{E}^{j} − rk E^{j})) − \frac{1}{2}T (∏_{j} γ^{q_{j}}(\overline{E}^{j} − rk E^{j}))

+A_{T} (∏_{j} γ^{q_{j}}(\overline{E}^{j} − rk E^{j}))

= L_{T} (∏_{j} γ^{q_{j}}(\overline{E}^{j} − rk E^{j})) − \frac{1}{2}T (∏_{j} γ^{q_{j}}(\overline{E}^{j} − rk E^{j}))

+d̂eg(f_{∗}^{T} (\frac{∏_{j} ĉ_{q_{j},t}(\overline{E}^{j})}{c_{t}^{top}(\overline{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:

d̂egf_{∗}(∏_{j} ĉ_{q_{j}}(\overline{E}^{j})) = L_{T} (∏_{j} γ^{q_{j}}(\overline{E}^{j} − rk E^{j}))

−\frac{1}{2}T (Y (C), ∏_{j} γ^{q_{j}}(\overline{E}^{j} − rk E^{j})) + d̂egf_{∗}^{T} (\frac{∏_{j} ĉ_{q_{j},t}(\overline{E}^{j})}{c_{t}^{top}(\overline{N})}). (12)

In particular, if \overline{L} is a T -equivariant line bundle on Y, one obtains the following formula for the height h_{Y} (\overline{L}) of Y relatively to \overline{L}:

h\overline{L}(Y) := d̂eg(ĉ_{1}(\overline{L})^{d+1}) = L_{T} ((\overline{L} − 1)^{d+1})

−\frac{1}{2}T (Y (C), (\overline{L} − 1)^{d+1}) + d̂egf_{∗}^{T} (\frac{ĉ_{1,t}(\overline{L})^{d+1}}{c_{t}^{top}(\overline{N})} ).

In our final residue formula, we shall use results of Bismut-Goette to give a formula for the term L_{T} (·) − \frac{1}{2}T (Y (C), ·). Notice however that the last identity already implies the following corollary:

Corollary 2.9 Let Y be an arithmetic variety endowed with a T -action. Suppose that \overline{L}, \overline{L}′ are T -equivariant hermitian line bundles. If there is an equivariant isometry \overline{L}_{Y_{T}} ≃ \overline{L}′_{Y_{T}} over Y_{T} and an equivariant (holomorphic) isometry \overline{L}_{C} ≃ \overline{L}′_{C} over Y_{C} then h\overline{L}(Y) = h\overline{L}′(Y).

2.2 The limit of the equivariant torsion

Let K′ denote any nonzero multiple of K. The vector field K′ is Hamiltonian with respect to the Kähler form as the action on M factors through a projective space. Let M_{K′} = M_{K} denote the fixed point set with respect to the action of K′. For any equivariant holomorphic Hermitian vector bundle \overline{F} we denote by μ^{F} (K′) ∈ Γ(M, End(F)) the section given by the action of the difference of the Lie derivative and the covariant derivative L_{K′}^{F} − ∇_{K′}^{F} on F. Set as in [BeGeV, ch. 7]

Td_{K′}(\overline{T M}) := Td(−\frac{Ω^{T M}}{2πi} + μ^{T M}(K′)) ∈ 𝔄(M)

and

ch_{K′}(\overline{F}) := Tr exp(−\frac{Ω^{F}}{2πi} + μ^{F} (K′)) ∈ 𝔄(M) .

The Chern class c_{q,K′}(\overline{F}) for 0 ≤ q ≤ rk F is defined as the part of total degree deg_{Y} + deg_{t} = q of

det(\frac{−Ω^{F}}{2πi} + tμ^{F} (K′) + Id)

at t = 1, thus c_{q,K′}(\overline{F}) = c_{q}(−Ω^{F} /2πi + μ^{F} (K′)). Let K^{′∗} ∈ T_{R}^{∗}M denote the 1-form dual to K′ via the metric on T_{R}M, hence ι_{K′}K^{′∗} = ∥K′∥^{2} is the norm square in T_{R}M. Set d_{K′}K^{′∗} := (d − 2πiι_{K′})K^{′∗} and define

s_{K′}(u) := \frac{−ω^{T M}}{2πu} exp(\frac{d_{K′}K^{′∗}}{4πiu}) = ∑_{ν=0}^{d−1} \frac{−ω^{T M}(dK^{′∗})^{ν}}{2πu(4πiu)^{ν}ν!} e^{−∥K′∥^{2}/2u} .

For a smooth differential form η it is shown in [BGo] (see also [B1, section C,D]) that the following integrals are well-defined:

A_{K′}(η)(s) := \frac{1}{Γ(s)} ∫_{0}^{1} ∫_{M} ηs_{K′}(u)u^{s−1}du

for Re s > 1 and

B_{K′}(η)(s) := \frac{1}{Γ(s)} ∫_{1}^{∞} ∫_{M} ηs_{K′}(u)u^{s−1}du

for Re s < 1. Also it is shown in [BGo] (compare [B1, Proof of theorem 7]) that s ↦ A_{K′}(η)(s) has a meromorphic extension to C which is holomorphic at s = 0 and that

A_{K′}(η)′(0) + B_{K′}(η)′(0) = ∫_{1}^{∞} ∫_{M} ηs_{K′}(u)\frac{du}{u}

+ ∫_{0}^{1} ∫_{M} η (s_{K′}(u) + (\frac{ω^{T M}}{2πu} c_{K′}^{top}(\overline{N})^{−1} − (c_{K′}^{top}(\overline{N})^{−1})′) δ_{M_{K′}} ) \frac{du}{u}

+ ∫_{M_{K′}} η (\frac{ω^{T M}}{2π} c_{K′}^{top}(\overline{N})^{−1} − Γ′(1)(c_{K′}^{top}(\overline{N})^{−1})′)

for the derivatives A_{K′}(η)′, B_{K′}(η)′ of A_{K′}(η), B_{K′}(η) with respect to s; also

A_{K′}(η)(0) + B_{K′}(η)(0) = ∫_{M_{K′}} η(c_{K′}^{top}(\overline{N})^{−1})′ .

Define the Bismut S-current S_{K′}(M, ω^{T M}) by the relation

∫_{M} ηS_{K′}(M, ω^{T M}) := A_{K′}(η)′(0) + B_{K′}(η)′(0) .

In particular, one notices

∫_{M} ηS_{K′}(M, ω^{T M}) = lim_{a→0^{+}} [ ∫_{a}^{∞} ∫_{M} ηs_{K′}(u)\frac{du}{u}

+ ∫_{a}^{1} ∫_{M} η (\frac{ω^{T M}}{2πu} c_{K′}^{top}(\overline{N})^{−1} − (c_{K′}^{top}(\overline{N})^{−1})′) δ_{M_{K′}} \frac{du}{u}

+ ∫_{M_{K′}} η (\frac{ω^{T M}}{2π} c_{K′}^{top}(\overline{N})^{−1} − Γ′(1)(c_{K′}^{top}(\overline{N})^{−1})′) ]

= lim_{a→0^{+}} [ ∫_{M} η · 2iω^{T M} \frac{1 − exp(\frac{d_{K′} K^{′∗}}{4πia})}{d_{K′}K^{′∗}}

+ ∫_{M_{K′}} η (\frac{ω^{T M}}{2πa} c_{K′}^{top}(\overline{N})^{−1} − (Γ′(1) + log a)(c_{K′}^{top}(\overline{N})^{−1})′) ] .

By Lemma 2.3 and Proposition 2.7, we already know that

lim_{t→0} (T_{g_{t}}(M, x) − log(t^{2}) ∫_{M_{K}} ∏_{j} c_{q_{j},K}(\overline{E}^{j}) · (c_{K}^{top}(\overline{N})^{−1})′)

exists and

2L_{T} (x) = lim_{t→0} (T_{g_{t}}(M, x) − log(t^{2}) ∫_{M_{K}} ∏_{j} c_{q_{j},K}(\overline{E}^{j}) · (c_{K}^{top}(\overline{N})^{−1})′)

−\frac{1}{2} ∫_{Y_{T} (C)} ∏_{j} c_{q_{j},K}(E^{j}) · \frac{r_{K}(N)}{c_{K}^{top}(N)} .  (13)

Now we shall compute this limit.

Theorem 2.10 The limit of the equivariant analytic torsion of x = ∏_{j} γ^{q_{j}}(\overline{E}^{j}− rk E^{j}) associated to the action of g_{t} for t → 0 is given by

lim_{t→0} (T_{g_{t}}(M, x) − log(t^{2}) ∫_{M_{K}} ∏_{j} c_{q_{j},K}(\overline{E}^{j}) · (c_{K}^{top}(\overline{N})^{−1})′)

= T (M, x) + ∫_{M} ∏_{j} c_{q_{j},K}(\overline{E}^{j}) · S_{K}(M, ω^{T M})

Proof: Let I_{K′} denote the additive equivariant characteristic class which is given for a line bundle L as follows: If K′ acts at the fixed point p by an angle θ ∈ R on L, then

I_{K′}(L)_{|p} := ∑_{k≠0} \frac{log(1 + \frac{θ}{2πk})}{c_{1}(L) + iθ + 2kπi} .

The main result of [BGo] implies that for t ∈ R \ {0}, t sufficiently small, there is a power series T_{t} in t with T_{0} = T (M, x) such that

T_{g_{t}}(M, x) − T_{t} = ∫_{M} Td_{tK}(\overline{T M})ch_{tK}(x)S_{tK}(M, ω^{T M})
− ∫_{M_{g}} Td_{g_{t}}(T M)ch_{g_{t}}(x)I_{tK}(N_{M_{g}/M}) .

For t → 0, both I_{tK}(N_{M_{g}/M}) → 0 and Td_{g_{t}}(\overline{T M})ch_{g_{t}}(x) → 0 (by eq. (11)), thus the last summand vanishes.

As in equation (10) ch_{tK}(x) = ∏_{j} c_{q_{j},tK}(\overline{E}^{j}) + η̃(t) with a form η̃ such that

(deg_{Y} + deg_{t})η̃(t) > d + 1. Thus Td_{tK}(\overline{T M})ch_{tK}(x) = ∏_{j} c_{q_{j},tK}(\overline{E}^{j}) + η(t) with (deg_{Y} + deg_{t})η(t) > d + 1. Also (deg_{Y} + deg_{t})s_{tK}(t^{2}u) = −1, hence we observe that

∫_{M} Td_{tK}(\overline{T M})ch_{tK}(x)s_{tK}(t^{2}u) = ∫_{M} (∏_{j} c_{q_{j},K}(\overline{E}^{j})s_{K}(u) + η(t)s_{tK}(t^{2}u)) .

Let η̃(t) denote the form obtained from η(t) by multiplying the degree deg_{Y} = j part with t^{−j−1} for 0 ≤ j ≤ d. By making the change of variable from u to t^{2}u we get

(A_{tK} + B_{tK})(Td_{tK}(\overline{T M})ch_{tK}(x))(s) = t^{2s}(A_{K} + B_{K})(∏_{j} c_{q_{j},K}(\overline{E}^{j}) + η̃(t))(s) .

Thus we find

∫_{M} Td_{tK}(\overline{T M})ch_{tK}(x)S_{tK}(M, ω^{T M})
= log(t^{2}) · (A_{K} + B_{K})(∏_{j} c_{q_{j},K}(\overline{E}^{j}))(0)

+(A_{K} + B_{K})(∏_{j} c_{q_{j},K}(\overline{E}^{j}))′(0) + O(t log(t^{2}))

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 Spec Z, that f : Y → Spec Z is a flat, T -projective morphism and that the fixed scheme f^{T} : Y_{T} → Spec Z is assumed to be flat over Spec Z. We let d + 1 be the absolute dimension of Y. We choose T -equivariant Hermitian bundles \overline{E}^{j} on Y and positive integers q_{j} such that ∑_{j} q_{j} = d + 1. We deduce by combining equations (12), (13) and Theorem 2.10 Theorem 2.11

d̂eg(f_{∗}( ∏_{j} ĉ_{q_{j}}(\overline{E}^{j}))) = d̂eg(f_{∗}^{T} (\frac{∏_{j} ĉ_{q_{j},t}(\overline{E}^{j})}{c_{t}^{top}(\overline{N})}))

+\frac{1}{2} ∫_{Y (C)} ∏_{j} c_{q_{j},K}(\overline{E}^{j}) · S_{K}(Y (C), ω^{T Y (C)}) − \frac{1}{2} ∫_{Y_{T} (C)} ∏_{j} c_{q_{j},K}(E^{j}) · \frac{r_{K}(N)}{c_{K}^{top}(N)} .

Example. Assume that the fixed point scheme is flat of Krull dimension 1. The normal bundle to Y_{T} splits as N = ⨁_{n∈Z} N_{n}. Thus

ĉ_{t}^{top}(\overline{N})^{−1} = \frac{1}{∏_{n}(2πitn)^{rk N_{n}}} (1 − ∑_{n∈Z} \frac{ĉ_{1}(\overline{N}_{n})}{2πitn})

by equation (8). Also, at a given point p ∈ M_{K} the tangent space decomposes as T M_{|p} = ⨁ T M_{θ_{ν}}, where K acts with angle θ_{ν} on T M_{θ_{ν}}. Then

\frac{r_{K}(N)}{c_{K}^{top}(N)}_{|p} = \frac{1}{∏_{θ} iθ} ∑_{θ} \frac{2Γ′(1) − 2 log |θ|}{iθ}

where the θ are counted with their multiplicity. Furthermore, in this case

∫_{M} ηS_{K}(M, ω^{T M}) = lim_{a→0^{+}} [ ∫_{M} η · 2iω^{T M} \frac{1 − exp(\frac{d_{K}K^{∗}}{4πia})}{d_{K}K^{∗}}

−(Γ′(1) + log a) ∫_{M_{K}} η · (c_{K}^{top}(\overline{N})^{−1})′]

= ∫_{M} η · 2iω^{T M} (\frac{1}{d_{K}K^{∗}} + \frac{(dK^{∗})^{d−1}}{(2πi∥K∥)^{d}} )

− lim_{a→0^{+}} [ ∫_{M} η · \frac{2iω^{T M}(dK^{∗})^{d−1}}{(2πi∥K∥)^{d}} (1 − e^{− \frac{1}{2a} ∥K∥^{2}})

+(Γ′(1) + log a) ∫_{M_{K}} η · (c_{K}^{top}(\overline{N})^{−1})′] .

Now consider a line bundle \overline{L}, splitting as ⨁_{k} \overline{L}_{k} on the fixed point scheme (where the L_{k} are locally free of rank ≤ 1). We find

ĉ_{1,t}(\overline{L})^{d+1} = ∑_{k}(ĉ_{1}(\overline{L}_{k}) + 2πitk rk L_{k})^{d+1}
= ∑_{k} ((2πitk)^{d+1}rk L_{k} + (d + 1)(2πitk)^{d}ĉ_{1}(\overline{L}_{k})) ,

thus

d̂eg f_{∗}^{T} (\frac{ĉ_{1,t}(\overline{L})^{d+1}}{c_{t}^{top}(\overline{N})} )

= d̂egf_{∗}^{T} ∑_{k} \frac{k^{d}}{∏_{n} n^{rk N_{n}}} ((d + 1)ĉ_{1}(\overline{L}_{k}) − k rk L_{k} · ∑_{n∈Z} \frac{ĉ_{1}(\overline{N}_{n})}{n}) .

Now notice that at a given fixed point p over C all but one L_{k,C} vanish and set ϕ_{p} := 2πk. We compute

−\frac{1}{2} ∑_{p∈M_{K}} \frac{c_{1,K}(L)^{d+1}}{c_{K}^{top}(N)} r_{K}(N) = ∑_{p∈M_{K}} \frac{ϕ_{p}^{d+1}}{∏_{θ} θ} ∑_{θ} \frac{−Γ′(1) + log |θ|}{θ}

and

−\frac{1}{2}(Γ′(1)+log a) ∑_{p∈M_{K}} c_{1,K}(L)^{d+1}(c_{K}^{top}(\overline{N})^{−1})′ = ∑_{p∈M_{K}} \frac{ϕ_{p}^{d+1}}{∏_{θ} θ} ∑_{θ} \frac{Γ′(1) + log a}{2θ} .

Hence we finally get

d̂egf_{∗}ĉ_{1}(\overline{L})^{d+1} = d̂egf_{∗}^{T} ∑_{k} \frac{k^{d}}{∏_{n} n^{rk N_{n}}} ((d + 1)ĉ_{1}(\overline{L}_{k}) − k rk L_{k} · ∑_{n∈Z} \frac{ĉ_{1}(\overline{N}_{n})}{n})

+ ∑_{p∈M_{K}} \frac{ϕ_{p}^{d+1}}{∏_{θ} θ} ∑_{θ} \frac{−Γ′(1) + log(θ^{2})}{2θ}

+ ∫_{M} c_{1,K}(L)^{d+1} · iω^{T M} (\frac{1}{d_{K}K^{∗}} + \frac{(dK^{∗})^{d−1}}{(2πi∥K∥)^{d}} )

− lim_{a→0^{+}} [ ∫_{M} μ^{L}(K)^{d+1} · \frac{iω^{T M}(dK^{∗})^{d−1}}{(2πi∥K∥)^{d}} (1 − e^{− \frac{1}{2a} ∥K∥^{2}})

− log a ∑_{p∈M_{K}} \frac{ϕ_{p}^{d+1}}{∏_{θ} θ} ∑_{θ} \frac{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. Let G be a compact Lie group and let M and M′ be complex manifolds on which G acts by holomorphic automorphisms. Let f : M → M′ be a smooth G-equivariant morphism of complex manifolds. Let ω^{T M} be a G-invariant Kähler metric on M (a Kähler fibration structure would in fact be sufficient). Let \overline{E} be a G-equivariant Hermitian holomorphic vector bundle on M and suppose for simplicity that R^{k}f_{∗}E = 0 for k > 0. Now let g be the automorphism corresponding to some element of G. The equivariant higher analytic torsion T_{g}(f, \overline{E}) is a certain element of 𝔄̃(M′_{g}), which satisfies the curvature formula

dd^{c}T_{g}(f, \overline{E}) = ch_{g}(f_{∗}\overline{E}) − ∫_{M_{g}/M′_{g}} Td_{g}(\overline{T f})ch_{g}(\overline{E}).

where \overline{T f} is endowed with the metric induced by ω^{T M}. The term in degree zero of T_{g}(f, \overline{E}) is the equivariant analytic torsion T_{g}(f^{−1}(x), \overline{E}|_{f^{−1}(x)}) of the restriction of \overline{E} to the fiber of f over x ∈ M′_{g}. Now let Y, B be μ_{n}-equivariant arithmetic varieties over some fixed arithmetic ring D and let f : Y → B be a map over D, which is flat, μ_{n}-projective and smooth over the complex numbers. Fix an μ_{n}(C)-invariant Kähler metric on Y (C). If \overline{E} is an f-acyclic (meaning that R^{k}f_{∗}E = 0 if k > 0) μ_{n}-equivariant Hermitian bundle on Y, let f_{∗}\overline{E} be the direct image sheaf (which is locally free), endowed with its natural equivariant structure and L_{2}-metric. Consider the rule which associates the element f_{∗}\overline{E} −T_{g}(f, \overline{E}) of K̂_{0}^{μ_{n}}(B) to every f-acyclic equivariant Hermitian bundle \overline{E} and the element ∫_{Y (C)_{g}/B(C)_{g}} Td_{g}(\overline{T f})η ∈ 𝔄̃(B_{μ_{n}})

to every η ∈ 𝔄̃(Y_{μ_{n}}). The proof of the following proposition is then similar to the proof of [KR1, Prop. 4.3].

Proposition 3.1 The above rule induces a group homomorphism f_{∗} : K̂_{0}^{μ_{n}}(Y) → K̂_{0}^{μ_{n}}(B).

Let R be the ring appearing in the statement of [KR1, Th. 4.4]. We are ready to formulate the following conjecture.

Conjecture 3.2 Let

M(f) = (λ_{−1}(\overline{N}_{Y/Y_{μn}}^{∨}))^{−1}λ_{−1}(f^{∗}\overline{N}_{B/B_{μn}}^{∨})(1 − R_{g}(N_{Y/Y_{μn}}) + R_{g}(f^{∗}N_{B/B_{μn}})).

The diagram

K̂_{0}^{μ_{n}}(Y) ⟶^{M(f).ρ(·)} K̂_{0}^{μ_{n}}(Y_{μ_{n}}) ⊗_{R(μ_{n})} R
↓ f_{∗}  ↓ f_{∗}^{μ_{n}}
K̂_{0}^{μ_{n}}(B) ⟶^{ρ(·)}  K̂_{0}^{μ_{n}}(B_{μ_{n}}) ⊗_{R(μ_{n})} R

commutes.

About this conjecture, we make the following comments:

(a) One can carry over the principle of the proof of [KR1, Th. 4.4] to prove this conjecture, provided a generalization of the immersion formula [B3, Th. 0.1] is available (which is not the case at the moment). We shall however not go into the details of this argument.

(b) Without formal proof again, we notice that the conjecture holds, if B_{μ_{n}}(C) has dimension 0. In that case the torsion forms are not necessary to define the direct image and the proof of [KR1, Th. 4.4] pulls through altogether.

The conjecture described in this appendix can be used to obtain explicit formulae for arithmetic (Arakelov-) characteristic classes of bundles of modular forms on Shimura varieties; see [K5] and [MR].

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