Variation of singular Kähler-Einstein metrics: positive Kodaira dimension

Abstract

Given a Kähler fiber space $p:X\to Y$ whose generic fiber is of general type, we prove that the fiberwise singular Kähler-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 Kähler-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 fiber-wise Song-Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).

v2: Title changed as the initial text is now split into two separate articles (authored by the same persons). The main results we obtain here are less general than their respective counterparts in the previous version, due to a gap in our arguments

References in corpus (2)

• Convergence of weak Kähler-Ricci Flows on minimal models of positive Kodaira dimension
• Stability of solutions to complex Monge-Ampere flows

Cited by in corpus (1)

• Hyperbolicity of bases of log Calabi-Yau families