Variation of singular Kähler-Einstein metrics: positive Kodaira dimension

Junyan Cao (1) , Mihai Păun (2, 3, 4) , Henri Guenancia (5, 6, 7)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 KIAS Séoul - School of Mathematics
3 UIC - University of Illinois [Chicago]
4 Universität Bayreuth
5 SBU - Stony Brook University [SUNY]
6 CNRS - Centre National de la Recherche Scientifique
7 IMT - Institut de Mathématiques de Toulouse UMR5219

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).

Domaines: Géométrie différentielle [math.DG]; Géométrie algébrique [math.AG]; Variables complexes [math.CV]

Junyan Cao, Mihai Păun, Henri Guenancia. Variation of singular Kähler-Einstein metrics: positive Kodaira dimension. 2020. ⟨hal-01879013v1⟩