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 Kähler-Einstein metric on the Kähler varieties with an intermediate Kodaira dimension is not suitable and we need to replace the twisted Kähler-Einstein metric (KE) to new notion of Relative Kähler-Einstein metric (RKE) for such varieties and we highlight that to get C∞-solution of CMA equation of relative Kähler-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 Kähler-Einstein metric then our family is stable in the sense of Alexeev,and Kollar-Shepherd-Barron

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

Domains: Mathematics [math]

Hassan Jolany. Relative Kähler-Einstein metric on Kähler varieties of positive Kodaira dimension. 2017. ⟨hal-01413754v3⟩