In 2022, I published a paper with the following abstract.
It is proved that a bounded C^2-smooth pseudoconvex domain $\Omega$ in a K\"ahler manifold $M$ can be mapped onto a locally closed analytic set in $\mathbb{C}^N$ holomorphically and properly with connected fibers if the canonical bundle of $M$ is negative on a neighborhood of $\partial\Omega$. A similar result is obtained for Zariski open domains in compact manifolds.
The purpose of my talk is to give a survey on the results closely related to this paper.