Analytic Torsion for Surfaces with Cusps I: Compact Perturbation Theorem and Anomaly Formula
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Communications in
Mathematical Physics
Analytic Torsion for Surfaces with Cusps I: Compact Perturbation Theorem and Anomaly Formula Siarhei Finski UFR de Mathématiques, Case 7012, Université Paris Diderot-Paris 7, Paris, France. E-mail: [email protected] Received: 7 March 2019 / Accepted: 29 May 2020 Published online: 1 September 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: We define the analytic torsion associated with a Riemann surface endowed with a metric having Poincaré-type singularities in the neighborhood of a finite number of points and a Hermitian vector bundle with at most logarithmic singularities at those points, coming from the metric on the negative power of the canonical line bundle twisted by the divisor of the points. Then we provide a relation between this analytic torsion and the Ray–Singer analytic torsion of the compactified surface. From this relation we then establish the anomaly formula, which describes how the analytic torsion changes under the change of the metric on the surface and on the vector bundle.
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral Theory of Surfaces with Cusps . . . . . . . . . . . . . . . . . 2.1 The setting of the problem and the spectral gap theorem . . . . . . 2.2 Relative spectral theory for surfaces with cusps . . . . . . . . . . 2.3 Heat kernel on the punctured hyperbolic disc and elliptic estimates 2.4 Proofs of Theorem 2.1 and Propositions 2.4, 2.6, 2.8 . . . . . . . . 3. Compact Perturbation of the Cusp: A Proof of Theorem A . . . . . . . 3.1 General strategy of a proof of Theorem A . . . . . . . . . . . . . 3.2 Flattening the Hermitian metric: a proof of (3.1) . . . . . . . . . . 3.3 Flattening the Riemannian metric: a proof of (3.2) . . . . . . . . . 3.4 Proofs of Propositions 3.12, 3.15 . . . . . . . . . . . . . . . . . . 3.5 Existence of tight families of flattenings . . . . . . . . . . . . . . 4. The Anomaly Formula: A Proof of Theorem B . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The goal of this article is to define and study the analytic torsion associated with a Riemann surface with hyperbolic cusps and a holomorphic Hermitian vector bundle with at most logarithmic singularities around the cusps. To define the analytic torsion, we use the regularization of the heat trace, obtained by subtracting a universal contribution coming from the model case of CP1 \ {0, 1, ∞}. We provide a relation between this analytic torsion and the Ray–Singer analytic torsion of compactified surface. Then we prove the anomaly formula, which describes how this analytic torsion changes with the change of metric and Hermitian structure on the vector bundle. In our setting we do not require the metric to be of constant scalar curvature everywhere and we do not
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