Canonical metric on a mildly singular Kähler varieties with an intermediate log Kodaira dimension

Existence of canonical metric on a canonical model of projective singular variety was a long standing conjecture and the major part of this conjecture is about varieties which do not have definite first Chern class (most of the varieties do not have definite first Chern class). There is a program which is known as Song-Tian program for finding canonical metric on canonical model of a projective variety by using Minimal Model Program. In this paper, we apply Song-Tian program for mildly singular pair $(X;D)$ via Log Minimal Model Program where $D$ is a simple normal crossing divisor on $X$ with conic singularities. We show that there is a unique $C^\infty$-fiberwise conical Kähler-Einstein metric on $(X;D)$ with vanishing Lelong number which is twisted by logarithmic Weil-Petersson metric and an additional term of Fujino-Mori [72] as soon as we have fiberwise KE-stability or Kawamata's condition of Theorems 2.28, or 2.30(in $C^0$ case). In final we highlight that how the complete answer of this question can be reduced to CMA equation corresponding to fiberwise Calabi-Yau foliation(due to H.Tsuji) which is still open.