Locally Conformally K\"ahler Manifolds of Algebraic Codimension One

A locally conformally Kähler (LCK) manifold is a manifold $M$ which admits a Kähler 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.