In this paper, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space 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 and extends the canonical Weil-Petersson current on the moduli space of smooth Kähler-Einstein Fano manifolds. As a consequence, we show that the CM line bundle is nef and big on and its restriction on is ample.
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 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 , et étend le courant canonique de Weil-Petersson défini sur l'espace de modules 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 , et que sa restriction à est ample.
DOI: 10.24033/asens.2365
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.
@article{ASENS_2018__51_3_739_0, 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://archive.numdam.org/articles/10.24033/asens.2365/} }
TY - JOUR AU - Li, Chi AU - Wang, Xiaowei AU - Xu, Chenyang TI - Quasi-projectivity of the moduli space of smooth Kähler-Einstein Fano manifolds JO - Annales scientifiques de l'École Normale Supérieure PY - 2018 SP - 739 EP - 772 VL - 51 IS - 3 PB - Société Mathématique de France. Tous droits réservés UR - http://archive.numdam.org/articles/10.24033/asens.2365/ DO - 10.24033/asens.2365 LA - en ID - ASENS_2018__51_3_739_0 ER -
%0 Journal Article %A Li, Chi %A Wang, Xiaowei %A Xu, Chenyang %T Quasi-projectivity of the moduli space of smooth Kähler-Einstein Fano manifolds %J Annales scientifiques de l'École Normale Supérieure %D 2018 %P 739-772 %V 51 %N 3 %I Société Mathématique de France. Tous droits réservés %U http://archive.numdam.org/articles/10.24033/asens.2365/ %R 10.24033/asens.2365 %G en %F ASENS_2018__51_3_739_0
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, Serie 4, Volume 51 (2018) no. 3, pp. 739-772. doi : 10.24033/asens.2365. http://archive.numdam.org/articles/10.24033/asens.2365/
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