Relative K\"ahler-Einstein metric on K\"ahler varieties of positive Kodaira dimension

Abstract

For projective varieties with definite first Chern class we have one type of canonical metric which is called K\"ahler-Einstein metric. But for varieties with an intermidiate Kodaira dimension we can have several different types of canonical metrics. In this paper we introduce a new notion of canonical metric for varieties with an intermidiate Kodaira dimension. We highlight that to get $C^\infty$-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 $C^\infty$-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 dimensions which admits relative K\"ahler-Einstein metric.