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Article Dans Une Revue Journal of the European Mathematical Society Année : 2020

RIEMANN-ROCH ISOMETRIES IN THE NON-COMPACT ORBIFOLD SETTING

Résumé

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 Γ\H, for Γ ⊂ PSL2(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 in terms of the Selberg zeta function of Γ, to a metrized version of the ψ line bundle of the theory of moduli spaces of pointed orbicurves, and the self-intersection bundle of a suitable twist of the canonical sheaf ωX. We make use of surgery techniques through Mayer-Vietoris formulae for determinants of laplacians, in order to reduce to explicit evaluations of such for model hyperbolic cusps and cones. We carry out these computations, that are of independent interest: we provide a rigorous method that fixes incomplete computations in theoretical physics, and that can be adapted to other geometries. We go on to derive an arithmetic Riemann-Roch formula in the realm of Arakelov geometry, that 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 the presence only of cusps. As an application, we treat in detail the case of the modular curve X(1), that already reveals the interesting arithmetic content of the metrized ψ line bundles. From this, we solve the longstanding question of evaluating the Selberg zeta special value Z (1, PSL2(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, PSL2(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.
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Dates et versions

hal-02365371 , version 1 (15-11-2019)

Identifiants

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Gerard Freixas Montplet, Anna-Maria von Pippich. RIEMANN-ROCH ISOMETRIES IN THE NON-COMPACT ORBIFOLD SETTING. Journal of the European Mathematical Society, 2020, 22 (11), pp.3491--3564. ⟨10.4171/jems/992⟩. ⟨hal-02365371⟩
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