Quasi-projectivity of the moduli space  of smooth Kähler-Einstein Fano manifolds

Li, Chi; Wang, Xiaowei; Xu, Chenyang

doi:10.24033/asens.2365

Quasi-projectivity of the moduli space of smooth Kähler-Einstein Fano manifolds
[Quasi-projectivité de l'espace de modules des variétés de Kähler-Einstein Fano régulières]

Li, Chi ; Wang, Xiaowei ; Xu, Chenyang

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 3, pp. 739-772.

Résumé
Abstract

Dans cet article, nous montrons qu'il existe une métrique hermitienne continue et canonique sur le fibré en droites CM au-dessus de l'espace de modules $\bar{ℳ}$ des variétés de Kähler-Einstein régularisables. La courbure de Chern de cette métrique hermitienne est le courant de Weil-Petersson, qui existe en tant que (1,1)-courant fermé positif sur $\bar{ℳ}$ , et étend le courant canonique de Weil-Petersson défini sur l'espace de modules $ℳ$ des variétés de Kähler-Einstein Fano régulières. Nous montrons aussi, en guise d'application de notre résultat, que le fibré des lignes CM est nef et big sur $\bar{ℳ}$ , et que sa restriction à $ℳ$ est ample.

In this paper, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space $\bar{ℳ}$ of smoothable Kähler-Einstein Fano varieties. The Chern curvature of this Hermitian metric is the Weil-Petersson current, which exists as a closed positive (1,1)-current on $\bar{ℳ}$ and extends the canonical Weil-Petersson current on the moduli space $ℳ$ of smooth Kähler-Einstein Fano manifolds. As a consequence, we show that the CM line bundle is nef and big on $\bar{ℳ}$ and its restriction on $ℳ$ is ample.

Publié le : 2018-11-12

MR Zbl

DOI : 10.24033/asens.2365

Classification : 32Q20, 14D23, 32G13.
Keywords: Kähler-Einstein metric, Fano varieties, moduli space, quasi-projectivity, Weil-Petersson current.
Mot clés : Métrique de Kähler-Einstein, variétés de Fano, espace de modules, quasi-projectivité, courant de Weil-Petersson.

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     author = {Li, Chi and Wang, Xiaowei and Xu, Chenyang},
     title = {Quasi-projectivity of the moduli space  of smooth {K\"ahler-Einstein} {Fano} manifolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {739--772},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {3},
     year = {2018},
     doi = {10.24033/asens.2365},
     mrnumber = {3831036},
     zbl = {1421.32033},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2365/}
}

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Li, Chi; Wang, Xiaowei; Xu, Chenyang. Quasi-projectivity of the moduli space  of smooth Kähler-Einstein Fano manifolds. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 3, pp. 739-772. doi : 10.24033/asens.2365. http://www.numdam.org/articles/10.24033/asens.2365/

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