Abstract: In this talk, we plan to discuss Song-Tian's (possibly singular) generalized Keahler-Einstein metric on the canonical models of projective manifolds with semi-ample canonical line bundle. When the canonical model is one dimensional (i.e. a Riemann surface), we give the metric asymptotics of the generalized Kaehler-Einstein metric near its singular points, implying a special case of a conjecture of Song and Tian. Then we present some applications of this result in studying infinite-time singularities of the Kaehler-Ricci flow.