[Extension de la forme de courbure du faisceau canonique relatif sur les familles de variétés de Calabi–Yau et applications]
Étant donnée une application holomorphe surjective entre variétés de Kähler, dont les fibres générales sont des variétés de Calabi–Yau, les formes de volume pour les métriques Ricci-plates induisent une métrique hermitienne sur le faisceau canonique relatif sur le lieu régulier de la famille. Nous montrons que la forme de courbure s’étend comme un courant positif fermé. Par conséquent, la métrique de Weil–Petersson s’étend comme un courant positif. On sait que dans le cas projectif la forme de Weil–Petersson est la courbure d’un certain faisceau de lignes déterminant, équipé d’une métrique de Quillen. Comme application, nous obtenons qu’après éclatement du lieu singulier, le faisceau de lignes déterminant s’étend et la métrique de Quillen s’étend comme métrique hermitienne singulière, dont la courbure est un courant positif.
Given a proper surjective holomorphic map of Kähler manifolds, whose general fibers are Calabi–Yau manifolds, the volume forms for the Ricci-flat metrics induce a hermitian metric on the relative canonical bundle over the regular locus of the family. We show that the curvature form extends as a closed positive current. Consequently the Weil–Petersson metric extends as a positive current. In the projective case, the Weil–Petersson form is known to be the curvature of a certain determinant line bundle, equipped with a Quillen metric. As an application we get that after blowing up the singular locus, the determinant line bundle extends, and the Quillen metric extends as singular hermitian metric, whose curvature is a positive current.
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Keywords: Calabi–Yau manifold, Ricci-flat metric, Kähler–Einstein metric, families of Calabi–Yau manifolds, extension of Weil–Petersson metrics
Mot clés : variétés de Calabi–Yau, métriques Ricci-plates, métriques de Kähler–Einstein, familles de variétés de Calabi–Yau, extension des métriques de Weil–Petersson
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@article{AIF_2021__71_1_393_0,
author = {Choi, Young-Jun and Schumacher, Georg},
title = {Extension of the curvature form of the relative canonical line bundle on families of {Calabi{\textendash}Yau} manifolds and applications},
journal = {Annales de l'Institut Fourier},
pages = {393--406},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {71},
number = {1},
year = {2021},
doi = {10.5802/aif.3365},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3365/}
}
TY - JOUR AU - Choi, Young-Jun AU - Schumacher, Georg TI - Extension of the curvature form of the relative canonical line bundle on families of Calabi–Yau manifolds and applications JO - Annales de l'Institut Fourier PY - 2021 SP - 393 EP - 406 VL - 71 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3365/ DO - 10.5802/aif.3365 LA - en ID - AIF_2021__71_1_393_0 ER -
%0 Journal Article %A Choi, Young-Jun %A Schumacher, Georg %T Extension of the curvature form of the relative canonical line bundle on families of Calabi–Yau manifolds and applications %J Annales de l'Institut Fourier %D 2021 %P 393-406 %V 71 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3365/ %R 10.5802/aif.3365 %G en %F AIF_2021__71_1_393_0
Choi, Young-Jun; Schumacher, Georg. Extension of the curvature form of the relative canonical line bundle on families of Calabi–Yau manifolds and applications. Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 393-406. doi : 10.5802/aif.3365. https://aif.centre-mersenne.org/articles/10.5802/aif.3365/
[1] Bergman kernels and subadjuction (2010) (https://arxiv.org/abs/1002.4145)
[2] 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 | Zbl
[3] 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
[4] 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 | Zbl
[5] Kähler forms for families of Calabi–Yau manifolds, Publ. Res. Inst. Math. Soc., Volume 56 (2020) no. 1, pp. 1-13 | DOI | Zbl
[6] Analytic methods in algebraic geometry, Surveys of Modern Mathematics, 1, International Press; Higher Education Press, 2012 | MR | Zbl
[7] The moduli space of extremal compact Kähler manifolds and generalized Weil–Petersson metrics, Publ. Res. Inst. Math. Sci., Volume 26 (1990) no. 1, pp. 101-183 | DOI | Zbl
[8] Relative adjoint transcendental classes and Albanese maps of compact Kähler manifolds with nef Ricci curvature, Higher dimensional algebraic geometry. In honour of Professor Yujiro Kawamata’s sixtieth birthday. Proceedings of the conference, Tokyo, Japan, January 7–11, 2013 (Advanced Studies in Pure Mathematics), Volume 74 (2017), pp. 335-356 | DOI | Zbl
[9] Positivity of relative canonical bundles and applications, Invent. Math., Volume 190 (2012) no. 1, pp. 1-56 | DOI | MR | Zbl
[10] An extension theorem for Hermitian line bundles, Analytic and Algebraic Geometry. Selected papers based on the presentations at the international conference, Hyderabad, India 2015, Hindustan Book Agency, New Delhi, 2017, pp. 225-237 | Zbl
[11] On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I, Commun. Pure Appl. Math., Volume 31 (1978) no. 3, pp. 339-411 | Zbl
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