Variation of singular Kahler-Einstein metrics: Positive Kodaira dimension

Given a Kahler fiber space p : X -> Y whose generic fiber is of general type, we prove that the fiberwise singular Kahler-Einstein metric induces a semipositively curved metric on the relative canonical bundle K-X/Y of p. We also propose a conjectural generalization of this result for relative twisted Kahler-Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song-Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).