Extension of the curvature form of the relative canonical line bundle on families of Calabi–Yau manifolds and applications
[Extension de la forme de courbure du faisceau canonique relatif sur les familles de variétés de Calabi–Yau et applications]
Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 393-406.

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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DOI : https://doi.org/10.5802/aif.3365
Classification : 32Q25,  32Q20,  32G05,  32W20
Mots clés : variétés de Calabi–Yau, métriques Ricci-plates, métriques de Kähler–Einstein, familles de variétés de Calabi–Yau, extension des métriques de Weil–Petersson
@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/}
}
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/

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