In this note, we define the analytic torsion for complete non-compact Riemann surfaces with hyperbolic cusps singularities. We then show that the Quillen metric associated to it satisfies the curvature formula and study the behaviour of it for families of Riemann surfaces when the additional cusps are created by degeneration.
@article{TSG_2019-2021__36__31_0, author = {Siarhei Finski}, title = {Quillen metric theory for surfaces with cusps}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {31--50}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {36}, year = {2019-2021}, doi = {10.5802/tsg.372}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.372/} }
TY - JOUR AU - Siarhei Finski TI - Quillen metric theory for surfaces with cusps JO - Séminaire de théorie spectrale et géométrie PY - 2019-2021 SP - 31 EP - 50 VL - 36 PB - Institut Fourier PP - Grenoble UR - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.372/ DO - 10.5802/tsg.372 LA - en ID - TSG_2019-2021__36__31_0 ER -
%0 Journal Article %A Siarhei Finski %T Quillen metric theory for surfaces with cusps %J Séminaire de théorie spectrale et géométrie %D 2019-2021 %P 31-50 %V 36 %I Institut Fourier %C Grenoble %U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.372/ %R 10.5802/tsg.372 %G en %F TSG_2019-2021__36__31_0
Siarhei Finski. Quillen metric theory for surfaces with cusps. Séminaire de théorie spectrale et géométrie, Tome 36 (2019-2021), pp. 31-50. doi : 10.5802/tsg.372. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.372/
[1] Enrico Arbarello; Maurizio Cornalba; Phillip A. Griffiths Geometry of Algebraic Curves, Grundlehren der Mathematischen Wissenschaften, 2, Springer, 2011 no. 268, 488 pages | DOI | Zbl
[2] Hugues Auvray; Xiaonan Ma; George Marinescu Bergman kernels on punctured Riemann surfaces (2016) (arXiv:1604.06337, to appear in Mathematische Annalen)
[3] Nicole Berline; Ezra Getzler; Michèle Vergne Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften, 298, Springer, 1992 | DOI | MR | Zbl
[4] Jean-Michel Bismut Quillen metrics and singular fibres in arbitrary relative dimension, J. Algebr. Geom., Volume 6 (1997) no. 1, pp. 19-149 | MR | Zbl
[5] Jean-Michel Bismut; Jean-Benoît Bost Fibrés déterminants, métriques de Quillen et dégénérescence des courbes, Acta Math., Volume 165 (1990), pp. 1-103 | DOI
[6] Jean-Michel Bismut; Henri A. Gillet; Christophe Soulé Analytic torsion and holomorphic determinant bundles I. Bott–Chern forms and analytic torsion, Commun. Math. Phys., Volume 115 (1988) no. 1, pp. 49-78 | DOI | MR
[7] Jean-Michel Bismut; Henri A. Gillet; Christophe Soulé Analytic torsion and holomorphic determinant bundles II. Direct images and Bott–Chern forms, Commun. Math. Phys., Volume 115 (1988) no. 1, pp. 79-126 | DOI | MR | Zbl
[8] Jean-Michel Bismut; Henri A. Gillet; Christophe Soulé Analytic torsion and holomorphic determinant bundles III. Quillen metrics on holomorphic determinants, Commun. Math. Phys., Volume 115 (1988) no. 2, pp. 301-351 | DOI | MR
[9] Jean-Michel Bismut; Gilles Lebeau Complex immersions and Quillen metrics, Publ. Math., Inst. Hautes Étud. Sci., Volume 74 (1991) no. 1, pp. 1-291 | DOI | Zbl
[10] Jens Bolte; Frank Steiner Determinants of Laplace-like operators on Riemann surfaces, Commun. Math. Phys., Volume 130 (1990) no. 3, pp. 581-597 | DOI | MR | Zbl
[11] José I. Burgos Gil; Jürg Kramer; Ulf Kühn Arithmetic characteristic classes of automorphic vector bundles, Doc. Math., Volume 10 (2005), pp. 619-716 | DOI | MR | Zbl
[12] Eric D’Hoker; Duong H. Phong Multiloop amplitudes for the bosonic Polyakov string, Nucl. Phys., B, Volume 269 (1986) no. 1, pp. 205-234 | DOI | MR
[13] Eric D’Hoker; Duong H. Phong On determinants of Laplacians on Riemann surfaces, Commun. Math. Phys., Volume 104 (1986) no. 4, pp. 537-545 | DOI | MR | Zbl
[14] Hershel M. Farkas; Irwin Kra Riemann surfaces, Graduate Texts in Mathematics, 71, Springer, 1992 | DOI | MR | Zbl
[15] Siarhei Finski Quillen metric for singular families of Riemann surfaces with cusps and compact perturbation theorem (2019) (arXiv:1911.09087, to appear in Mathematical Research Letters)
[16] Siarhei Finski Analytic torsion for surfaces with cusps I. Compact perturbation theorem and anomaly formula, Commun. Math. Phys., Volume 378 (2020) no. 12, pp. 1713-1774 | DOI | MR | Zbl
[17] Siarhei Finski Analytic torsion for surfaces with cusps II. Regularity, asymptotics and curvature theorem, Adv. Math., Volume 375 (2020), 107409 | DOI | MR | Zbl
[18] Gerard Freixas I. Montplet Généralisations de la théorie de l’intersection arithmétique, Ph. D. Thesis, Université de Paris 11, Paris, France (2007)
[19] Gerard Freixas I. Montplet An arithmetic Riemann–Roch theorem for pointed stable curves, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009) no. 2, pp. 335-369 | DOI | MR | Zbl
[20] Gerard Freixas I. Montplet An arithmetic Hilbert–Samuel theorem for pointed stable curves, J. Eur. Math. Soc., Volume 14 (2012) no. 2, pp. 321-351 | DOI | MR | Zbl
[21] Philip A. Griffiths; Joseph Harris Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, 1994 (reprint of the 1978 original) | DOI | MR | Zbl
[22] Finn Knudsen; David B. Mumford The projectivity of the moduli space of stable curves. I: Preliminaries on ‘det’ and ‘Div’, Math. Scand., Volume 39 (1976), pp. 19-55 | DOI | MR | Zbl
[23] Werner Müller Spectral Theory for Riemannian Manifolds with Cusps and a Related Trace Formula, Math. Nachr., Volume 111 (1983) no. 1, pp. 197-288 | DOI | MR | Zbl
[24] David B. Mumford Hirzebruch’s proportionality theorem in the noncompact case, Invent. Math., Volume 42 (1977), pp. 239-272 | DOI | MR | Zbl
[25] Kazuto Oshima Notes on determinants of Laplace-type operators on Riemann surfaces, Phys. Rev. D, Volume 41 (1990) no. 2, pp. 702-703 | DOI | MR
[26] Daniel Quillen Determinants of Cauchy–Riemann operators over a Riemann surface, Funct. Anal. Appl., Volume 19 (1985) no. 1, pp. 31-34 | DOI | Zbl
[27] Daniel B. Ray; Isadore M. Singer Analytic Torsion for Complex Manifolds, Ann. Math., Volume 98 (1973) no. 1, pp. 154-177 | MR | Zbl
[28] Peter Sarnak Determinants of Laplacians, Commun. Math. Phys., Volume 110 (1987) no. 1, pp. 113-120 | DOI | MR | Zbl
[29] L. A. Takhtajan; Peter G. Zograf A local index theorem for families of -operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces, Commun. Math. Phys., Volume 137 (1991) no. 2, pp. 399-426 | DOI | MR | Zbl
[30] Scott A. Wolpert Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math., Volume 85 (1986) no. 1, pp. 119-145 | DOI | MR | Zbl
[31] Scott A. Wolpert Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Commun. Math. Phys., Volume 112 (1987) no. 2, pp. 283-315 | DOI | MR | Zbl
[32] Scott A. Wolpert The hyperbolic metric and the geometry of the universal curve, J. Differ. Geom., Volume 31 (1990) no. 2, pp. 417-472 | MR | Zbl
[33] Scott A. Wolpert Cusps and the family hyperbolic metric, Duke Math. J., Volume 138 (2007) no. 3, pp. 423-443 | DOI | MR | Zbl
[34] Ken-Ichi Yoshikawa surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space, Invent. Math., Volume 156 (2004), pp. 53-117 | DOI | MR | Zbl
[35] Ken-Ichi Yoshikawa surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space, II: A structure theorem for , J. Reine Angew. Math., Volume 677 (2013), pp. 15-70 | MR | Zbl
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