Locally Conformally Kähler Manifolds of Algebraic Codimension One

Liviu Ornea; Misha Verbitsky; Victor Vuletescu

Locally Conformally Kähler Manifolds of Algebraic Codimension One

Differential Geometry 2026-06-26 v1

Authors: Liviu Ornea , Misha Verbitsky , Victor Vuletescu

Abstract

A locally conformally K\"ahler (LCK) manifold is a manifold $M$ which admits a K\"ahler structure on its universal cover $\tilde M$ , in such a way that the monodromy acts conformally on $\tilde M$ . Let $M$ be an $n$ -dimensional compact LCK manifold of algebraic dimension $n-1$ . We prove that $M$ is bimeromorphic to the total space of an isotrivial elliptic fibration. Morever, there exists an alteration of $M$ which dominates bimeromorphically a manifold admitting a free action of an elliptic curve.

Keywords

locally conformally Kähler manifolds algebraic codimension isotrivial elliptic fibration bimeromorphic equivalence

Cite

@article{arxiv.2606.27754,
  title  = {Locally Conformally Kähler Manifolds of Algebraic Codimension One},
  author = {Liviu Ornea and Misha Verbitsky and Victor Vuletescu},
  journal= {arXiv preprint arXiv:2606.27754},
  year   = {2026}
}