[Asymptotique au bord de la métrique du noyau de Bergman relatif pour des courbes elliptiques]
Pour une famille de surfaces de Riemann compactes, nous étudions les comportements asymptotiques de la métrique du noyau relatif de Bergman à proximité des frontières des espaces de modules. Nous montrons que la métrique du noyau relatif de Bergman sur une famille de courbes elliptiques a une croissance hyperbolique au point singulier. La preuve est principalement basée sur la théorie des fonctions elliptiques.
For a family of compact Riemann surfaces, we study the asymptotic behaviors of the relative Bergman kernel metric near the boundaries of the moduli spaces. We have shown that the relative Bergman kernel metric on a family of elliptic curves has hyperbolic growth at the node. The proof relies largely on the elliptic function theory.
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@article{CRMATH_2015__353_7_611_0,
author = {Dong, Robert Xin},
title = {Boundary asymptotics of the relative {Bergman} kernel metric for elliptic curves},
journal = {Comptes Rendus. Math\'ematique},
pages = {611--615},
publisher = {Elsevier},
volume = {353},
number = {7},
year = {2015},
doi = {10.1016/j.crma.2015.04.015},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2015.04.015/}
}
TY - JOUR AU - Dong, Robert Xin TI - Boundary asymptotics of the relative Bergman kernel metric for elliptic curves JO - Comptes Rendus. Mathématique PY - 2015 SP - 611 EP - 615 VL - 353 IS - 7 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2015.04.015/ DO - 10.1016/j.crma.2015.04.015 LA - en ID - CRMATH_2015__353_7_611_0 ER -
%0 Journal Article %A Dong, Robert Xin %T Boundary asymptotics of the relative Bergman kernel metric for elliptic curves %J Comptes Rendus. Mathématique %D 2015 %P 611-615 %V 353 %N 7 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2015.04.015/ %R 10.1016/j.crma.2015.04.015 %G en %F CRMATH_2015__353_7_611_0
Dong, Robert Xin. Boundary asymptotics of the relative Bergman kernel metric for elliptic curves. Comptes Rendus. Mathématique, Tome 353 (2015) no. 7, pp. 611-615. doi : 10.1016/j.crma.2015.04.015. http://www.numdam.org/articles/10.1016/j.crma.2015.04.015/
[1] Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, McGraw-Hill, New York, 1979
[2] Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 6, pp. 1633-1662
[3] Curvature of vector bundles associated to holomorphic fibrations, Ann. Math. (2), Volume 169 (2009) no. 2, pp. 531-560
[4] A proof of the Ohsawa–Takegoshi theorem with sharp estimates, 2014 | arXiv
[5] Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J., Volume 145 (2008) no. 2, pp. 341-378
[6] Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, Acta Math., Volume 213 (2014) no. 1, pp. 63-106
[7] R.X. Dong, Boundary asymptotics of the relative Bergman kernel metric for elliptic curves II, submitted for publication, 2015.
[8] A solution of an extension problem with optimal estimate and applications, Ann. Math. (2), Volume 181 (2015) no. 3, pp. 1139-1208
[9] Variation of Bergman metrics on Riemann surfaces, Math. Ann., Volume 330 (2004) no. 3, pp. 477-489
[10] On the extension of holomorphic functions, Math. Z., Volume 195 (1987), pp. 197-204
[11] On Calabi conjecture for complex surfaces with positive first Chern class, Invent. Math., Volume 101 (1990) no. 1, pp. 101-172
[12] Curvature semipositivity of relative pluricanonical systems, 2007 (preprint) | arXiv
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