Bimeromorphic geometry of LCK manifolds

Abstract

A locally conformally Kähler (LCK) manifold is a complex manifold $M$ which has a Kähler structure on its cover, such that the deck transform group acts on it by homotheties. Assume that the Kähler form is exact on the minimal Kähler cover of $M$. We prove that any bimeromorphic map $M^{\prime }\to M$ 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.

9 pages

References in corpus (2)

• Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds
• Non-linear Hopf manifolds are locally conformally Kahler

Cited by in corpus (1)

• Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds