The asymptotics of the holomorphic torsion forms

Martin Puchol

doi:10.1016/j.crma.2015.11.004

Analytic geometry/Differential geometry

The asymptotics of the holomorphic torsion forms
[L'asymptotique des formes de torsion holomorphe]

Présenté par : Jean-Michel Bismut

Martin Puchol ¹

¹ Université Paris Diderot - Paris 7, campus des Grands Moulins, bâtiment Sophie-Germain, case 7012, 75205 Paris cedex 13, France

Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 301-306.

Résumé
Abstract

Dans cette note, nous utilisons la théorie des opérateurs de Toeplitz pour donner une formule asymptotique des formes de torsion analytique holomorphe associées à une famille de fibrés vectoriels holomorphes donnés par l'image directe de $L^{\otimes p}$ , où L est un fibré en droites. Pour obtenir cette asymptotique, nous faisons une hypothèse de positivité le long des fibres sur L. Ce résultat est la version en famille des résultats de Bismut et Vasserot sur la torsion holomorphe.

In this note, we use the theory of Toeplitz operators to give an asymptotic formula for the holomorphic analytic torsion forms associated with a family of holomorphic vector bundles that are the direct image of $L^{\otimes p}$ , where L is a line bundle. To obtain the asymptotics, we make a positivity assumption along the fibers on L. This result is the family version of the results of Bismut and Vasserot on the holomorphic torsion.

Reçu le : 2015-05-21
Accepté le : 2015-11-19
Publié le : 2016-01-27

DOI : 10.1016/j.crma.2015.11.004

Affiliations des auteurs :

Martin Puchol ¹

¹ Université Paris Diderot - Paris 7, campus des Grands Moulins, bâtiment Sophie-Germain, case 7012, 75205 Paris cedex 13, France

@article{CRMATH_2016__354_3_301_0,
     author = {Martin Puchol},
     title = {The asymptotics of the holomorphic torsion forms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {301--306},
     publisher = {Elsevier},
     volume = {354},
     number = {3},
     year = {2016},
     doi = {10.1016/j.crma.2015.11.004},
     language = {en},
}

TY  - JOUR
AU  - Martin Puchol
TI  - The asymptotics of the holomorphic torsion forms
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 301
EP  - 306
VL  - 354
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2015.11.004
LA  - en
ID  - CRMATH_2016__354_3_301_0
ER  -

%0 Journal Article
%A Martin Puchol
%T The asymptotics of the holomorphic torsion forms
%J Comptes Rendus. Mathématique
%D 2016
%P 301-306
%V 354
%N 3
%I Elsevier
%R 10.1016/j.crma.2015.11.004
%G en
%F CRMATH_2016__354_3_301_0

Martin Puchol. The asymptotics of the holomorphic torsion forms. Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 301-306. doi : 10.1016/j.crma.2015.11.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.11.004/

Bibliographie
Cité par

[1] A. Berthomieu; J.-M. Bismut Quillen metrics and higher analytic torsion forms, J. Reine Angew. Math., Volume 457 (1994), pp. 85-184

[2] J.-M. Bismut Hypoelliptic Laplacian and Bott–Chern Cohomology: A Theorem of Riemann–Roch–Grothendieck in Complex Geometry, Progress in Mathematics, vol. 305, Birkhäuser/Springer, Cham, 2013

[3] J.-M. Bismut; H. Gillet; C. 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

[4] J.-M. Bismut; H. Gillet; C. Soulé Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants, Commun. Math. Phys., Volume 115 (1988) no. 2, pp. 301-351

[5] J.-M. Bismut; K. Köhler Higher analytic torsion forms and anomaly formulas, J. Algebraic Geom., Volume 1 (1992), pp. 647-684

[6] J.-M. Bismut; X. Ma; W. Zhang Opérateurs de Toeplitz et torsion analytique asymptotique, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 17–18, pp. 977-981

[7] J.-M. Bismut; E. Vasserot The asymptotics of the Ray–Singer analytic torsion associated with high powers of a positive line bundle, Commun. Math. Phys., Volume 125 (1889), pp. 355-367

[8] J.-M. Bismut; E. Vasserot The asymptotics of the Ray–Singer analytic torsion of the symmetric powers of a positive vector bundle, Ann. Inst. Fourier (Grenoble), Volume 40 (1990) no. 4, pp. 835-848

[9] M. Bordemann; E. Meinrenken; M. Schlichenmaier Toeplitz quantization of Kähler manifolds and $gl (N)$ , $N \to \infty$ limits, Commun. Math. Phys., Volume 165 (1994) no. 2, pp. 281-296

[10] L. Boutet de Monvel; V. Guillemin The Spectral Theory of Toeplitz Operators, Annals of Mathematics Studies, vol. 99, Princeton University Press, Princeton, NJ, USA, 1981

[11] H. Gillet; D. Rössler; C. Soulé An arithmetic Riemann–Roch theorem in higher degrees, Ann. Inst. Fourier (Grenoble), Volume 58 (2008) no. 6, pp. 2169-2189

[12] H. Gillet; C. Soulé An arithmetic Riemann–Roch theorem, Invent. Math., Volume 110 (1992) no. 3, pp. 473-543

[13] X. Ma Formes de torsion analytique et familles de submersions. I, Bull. Soc. Math. Fr., Volume 127 (1999) no. 4, pp. 541-621

[14] X. Ma; G. Marinescu Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics, vol. 254, Birkhäuser Verlag, Basel, 2007

[15] X. Ma; W. Zhang Superconnection and family Bergman kernels, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 344 (2007) no. 1, pp. 41-44 (full version forthcoming)

[16] M. Puchol The asymptotics of the holomorphic analytic torsion forms, 2015 (forthcoming) | arXiv

[17] D.B. Ray; I.M. Singer Analytic torsion for complex manifolds, Ann. Math. (2), Volume 98 (1973), pp. 154-177

[18] C. Soulé Lectures on Arakelov Geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge University Press, Cambridge, 1992 (With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer)

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Superconnection and family Bergman kernels

Xiaonan Ma; Weiping Zhang

C. R. Math (2007)

Opérateurs de Toeplitz et torsion analytique asymptotique

Jean-Michel Bismut; Xiaonan Ma; Weiping Zhang

C. R. Math (2011)