Variation of singular K\"ahler-Einstein metrics: positive Kodaira dimension

TL;DR

The paper proves that fiberwise singular K"ahler-Einstein metrics induce semipositively curved metrics on the relative canonical bundle in certain fiber spaces, and explores conjectural generalizations related to twisted metrics and fiber-wise Song-Tian metrics.

Contribution

It establishes a semipositivity result for the relative canonical bundle induced by fiberwise KE metrics and proposes a conjecture extending this to twisted metrics, with partial validation.

Findings

Semipositivity of the relative canonical bundle in fiber spaces with general type fibers.

Validation of the conjecture when the twisting current has zero Lelong numbers.

Relevance of the conjecture to fiber-wise Song-Tian metrics for positive Kodaira dimension.

Abstract

Given a K\"ahler fiber space $p:X\to Y$ whose generic fiber is of general type, we prove that the fiberwise singular K\"ahler-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 K\"ahler-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).