The asymptotics of the holomorphic torsion forms

Puchol, Martin

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

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⊗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⊗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.

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

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

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

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

[7] Bismut, J.-M.; Vasserot, E. 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] Bismut, J.-M.; Vasserot, E. 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] Bordemann, M.; Meinrenken, E.; Schlichenmaier, M. Toeplitz quantization of Kähler manifolds and gl(N), N→∞ limits, Commun. Math. Phys., Volume 165 (1994) no. 2, pp. 281-296

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

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

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

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

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

[15] Ma, X.; Zhang, W. 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] Puchol, M. The asymptotics of the holomorphic analytic torsion forms, 2015 (forthcoming) | arXiv

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

[18] Soulé, C. 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)