Relative Kähler-Einstein metric on Kähler varieties of positive Kodaira dimension

Hassan Jolany (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

In this paper we introduce a new notion of canonical metric. The notion of generalized K\"ahler-Einstein metric on the K\"ahler varieties with an intermediate Kodaira dimension is not suitable and we need to replace the twisted K\"ahler-Einstein metric (KE) to new notion of Relative K\"ahler-Einstein metric (RKE) for such varieties. This affirm a crucial error of the canonical metric introduced by Song-Tian\cite{1}\cite{2}, Tsuji \cite{51},and Zeriahi-Eyssidieux-Guedj\cite{47}.We highlight that to get C∞-solution of CMA equation of relative K\"ahler Einstein metric we need Song-Tian-Tsuji measure (which has minimal singularities with respect to other relative volume forms) be C∞-smooth and special fiber has canonical singularities. Moreover, we conjecture that if we have relative K\"ahler-Einstein metric then our family is stable in the sense of Alexeev,and Kollar-Shepherd-Barron. By inspiring the work of Greene-Shapere-Vafa-Yau semi-Ricci flat metric, we introduce fiberwise Calabi-Yau foliation which relies in context of generalized notion of foliation. In final, we give Bogomolov-Miyaoka-Yau inequality for minimal varieties with intermediate Kodaira dimension

Mots clés: Kähler-Einstein metrics; Fiberwise Calabi-Yau foliation; Weil-Petersson metric; Relative Kähler-Einstein metric

Domaines: Mathématiques [math]

Hassan Jolany. Relative Kähler-Einstein metric on Kähler varieties of positive Kodaira dimension. 2017. ⟨hal-01413754v4⟩