An iterative construction of complete K\"ahler--Einstein metrics

TL;DR

This paper extends an iterative method to construct complete K"ahler--Einstein metrics on noncompact manifolds with bounded geometry, and demonstrates how fiberwise metrics induce semipositively curved bundles in complex geometry.

Contribution

It generalizes Tsuji's iterative construction to noncompact settings and shows fiberwise K"ahler--Einstein metrics induce semipositively curved relative canonical bundles.

Findings

Extended iterative construction to noncompact manifolds with bounded geometry.

Proved fiberwise K"ahler--Einstein metrics induce semipositively curved bundles.

Applied approach to plurisubharmonic variation of cusp K"ahler--Einstein metrics.

Abstract

We extend Tsuji's iterative construction of complete K\"ahler--Einstein metrics with negative scalar curvature to noncompact K\"ahler 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 K\"ahler manifold and $Y$ is a complex manifold, and where the smooth fibers admit K\"ahler--Einstein metrics with negative scalar curvature and bounded geometry, we show that the fiberwise K\"ahler--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 K\"ahler--Einstein metrics.