An iterative construction of complete Kähler--Einstein metrics

Abstract

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 $K_{X/Y}$. Moreover, our approach also applies to the plurisubharmonic variation of cusp Kähler--Einstein metrics.

23 pages, minor changes, to appear in Publ. RIMS