An iterative construction of complete Kähler--Einstein metrics

We extend Tsuji’s iterative construction of complete Kähler–Einstein metrics with negative scalar curvature to noncompact 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 $K_{X\mathrm{/}Y}$. Moreover, our approach also applies to the plurisubharmonic variation of cusp Kähler–Einstein metrics.