An Iterative Construction of Complete Kähler–Einstein Metrics

Abstract

We extend Tsuji’s iterative construction of Kähler–Einstein metrics with negative scalar curvature to non-compact Kähler manifolds with bounded geometry, using Berndtsson’s method from the compact setting. Consequently, given a holomorphic surjective map $p:X\to Y$, where $X$ is a weakly pseudoconvex Kähler manifold and $Y$ is a complex manifold, and where the smooth fibers admit Kähler–Einstein metrics with negative scalar curvature and bounded geometry, we show that the fiberwise Kähler–Einstein metrics induce a semipositively curved metric on the relative canonical bundle $K_{X/Y}$. Moreover, our approach also applies to the plurisubharmonic variation of cusp Kähler–Einstein metrics.