Variation of singular Kähler-Einstein metrics: positive Kodaira dimension

2017/10/04 by Junyan Cao, Cao, Junyan, Henri Guenancia +3 · 1 citation
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.1710.01825

Abstract

Given a Kähler fiber space p:X→ 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).

Citations

• Bergman kernels and the pseudoeffectivity of relative canonical bundles

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