Relative Kähler-Einstein metric on Kähler varieties of positive Kodaira dimensionReport as inadecuate




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1 LPP - Laboratoire Paul Painlevé

Abstract : 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

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





Author: Hassan Jolany -

Source: https://hal.archives-ouvertes.fr/



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