An Iterative Construction of Complete Kähler–Einstein Metrics

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→ 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 KX/Y . Moreover, our approach also applies to the plurisubharmonic variation of cusp Kähler–Einstein metrics.