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AN ARITHMETIC RIEMANN-ROCH THEOREM FOR POINTED STABLE CURVES ʙʏ Gʀʀ FREIXAS ɪ MONTPLET Aʙʀ.-Let (O, Σ, F∞) be an arithmetic ring of Krull dimension at most 1, S = SpecO and (π : X → S; σ1,. .. , σn) an n-pointed stable curve of genus g. Write U = X \ ∪jσj(S). The invertible sheaf ω X /S (σ1 + • • • + σn) inherits a hermitian structure • hyp from the dual of the hyperbolic metric on the Riemann surface U∞. In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ω X /S (σ1 + • • • + σn) hyp. The theorem is applied to modular curves X(Γ), Γ = Γ0(p) or Γ1(p), p ≥ 11 prime, with sections given by the cusps. We show Z (Y (Γ), 1) ∼ e a π b Γ2(1/2) c L(0, MΓ), with p ≡ 11 mod 12 when Γ = Γ0(p). Here Z(Y (Γ), s) is the Selberg zeta function of the open modular curve Y (Γ), a, b, c are rational numbers, MΓ is a suitable Chow motive and ∼ means equality up to algebraic unit. R.-Soient (O, Σ, F∞) un anneau arithmétique de dimension de Krull au plus 1, S = SpecO et (π : X → S; σ1,. .. , σn) une courbe stable n-pointée de genre g. Posons U = X \ ∪jσj(S). Le faisceau inversible ω X /S (σ1 + • • • + σn) hérite une structure hermitienne • hyp du dual de la métrique hyperbolique sur la surface de Riemann U∞. Dans cet article nous prouvons un théorème de Riemann-Roch arithmétique qui calcule l'auto-intersection arithmétique de ω X /S (σ1 + • • • + σn) hyp. Le théorème est appliqué aux courbes modulaires X(Γ), Γ = Γ0(p) ou Γ1(p), p ≥ 11 premier, prenant les cusps comme sections. Nous montrons Z (Y (Γ), 1) ∼ e a π b Γ2(1/2) c L(0, MΓ), avec p ≡ 11 mod 12 lorsque Γ = Γ0(p). Ici Z(Y (Γ), s) est la fonction zêta de Selberg de la courbe modulaire ouverte Y (Γ), a, b, c sont des nombres rationnels, MΓ est un motif de Chow approprié et ∼ signifie égalité à unité près.

We generalize work of Deligne and Gillet–Soulé on a functorial Riemann–Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces of finite type \Gamma\backslash\mathbb H , for \Gamma\subset\mathrm {PSL}_{2}(\mathbb R) a Fuchsian group of the first kind, equipped with the Poincaré metric. This metric is singular at cusps and elliptic fixed points, and the original results of Deligne and Gillet–Soulé do not apply to this setting. Our theorem relates the determinant of cohomology of the trivial sheaf, with an explicit Quillen type metric defined in terms of the Selberg zeta function of \Gamma , to a metrized version of the tautological line bundle in the theory of moduli spaces of pointed orbicurves, and the self-intersection bundle of a suitable twist of the canonical sheaf \omega . We make use of surgery techniques through Mayer–Vietoris formulas for determinants of Laplacians, in order to reduce to explicit evaluations of such for model hyperbolic cusps and cones. We carry out these computations, which are of independent interest for theoretical physics, and can be adapted to other geometries. We go on to derive an arithmetic Riemann–Roch formula in the realm of Arakelov geometry, which applies in particular to integral models of modular curves with elliptic fixed points. This vastly extends previous work of the first author, whose deformation-theoretic methods were limited to torsion free Fuchsian groups. As an application, we treat in detail the case of the modular curve X(1) , and we show the interesting arithmetic content of the metrized \Psi line bundles. From this, we answer the longstanding question of evaluating the Selberg zeta special value Z'(1, \mathrm {PSL}_2 (\mathbb Z)) . The result is expressed in terms of logarithmic derivatives of Dirichlet L functions. In the analogy between Selberg zeta functions and Dedekind zeta functions of number fields, this formula can be seen as the analytic class number formula for Z(s, \mathrm {PSL}_2 (\mathbb Z)) . The methods developed in this article were conceived so that they afford several variants, such as the determinant of cohomology of a flat unitary vector bundle with finite monodromies at cusps. Our work finds its place in the program initiated by Burgos–Kramer–Kühn of extending arithmetic intersection theory to singular Hermitian vector bundles.

The main objective of the present paper is to set up the theoretical basis and the language needed to deal with the problem of direct images of hermitian vector bundles for projective non-necessarily smooth morphisms. To this end, we first define hermitian structures on the objects of the bounded derived category of coherent sheaves on a smooth complex variety. Secondly we extend the theory of Bott-Chern classes to these hermitian structures. Finally we introduce the category Sm * /C whose morphisms are projective morphisms with a hermitian structure on the relative tangent complex.

In this paper we extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for projective morphisms. The extension of the holomorphic analytic torsion classes of Bismut and Köhler is obtained as the theory of generalized analytic torsion classes associated to −R/2, R being the R-genus. As application of the axiomatic characterization, we give new simpler proofs of known properties of holomorpic analytic torsion classes, we give a characterization of the R genus, and we construct a direct image of hermitian structures for projective morphisms.