Bimeromorphic geometry of LCK manifolds

TL;DR

This paper proves that under certain conditions, bimeromorphic maps between locally conformally K"ahler manifolds are actually holomorphic, establishing the uniqueness of minimal models for a broad class of such manifolds.

Contribution

It demonstrates that if the K"ahler form is exact on the minimal K"ahler cover, then bimeromorphic maps are holomorphic, ensuring a unique minimal model for these LCK manifolds.

Findings

Bimeromorphic maps are holomorphic under the given conditions

Unique minimal models exist for a wide class of LCK manifolds

Applicable to Hopf manifolds, their submanifolds, and OT manifolds

Abstract

A locally conformally K\"ahler (LCK) manifold is a complex manifold $M$ which has a K\"ahler structure on its cover, such that the deck transform group acts on it by homotheties. Assume that the K\"ahler form is exact on the minimal K\"ahler cover of $M$. We prove that any bimeromorphic map $M'\rightarrow M$ is in fact holomorphic; in other words, $M$ has a unique minimal model. This can be applied to a wide class of LCK manifolds, such as the Hopf manifolds, their complex submanifolds and to OT manifolds.