Related papers: Locally conformally Kaehler manifolds with potenti…

A locally conformally K\"ahler (lcK) manifold is a complex manifold $(M,J)$ together with a Hermitian metric $g$ which is conformal to a K\"ahler metric in the neighbourhood of each point. In this paper we obtain three classification…

We study two kinds of transformation groups of a compact locally conformally Kahler (l.c.K.) manifold. First we study compact l.c.K. manifolds with parallel Lee form by means of the existence of a holomorphic l.c.K. flow. Next, we introduce…

We prove that any compact homogeneous locally conformally K\"ahler manifold has parallel Lee form.

Vaisman's theorem for locally conformally K\"ahler (lcK) compact manifolds states that any lcK metric on a compact complex manifold which admits a K\"ahler metric is, in fact, globally conformally K\"ahler (gcK). In this paper, we extend…

A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with…

We give an equivalent definition of compact locally conformally hyperk\"ahler manifolds in terms of the existence of a nondegenerate complex two-form with natural properties. This is a conformal analogue of Beauville's theorem stating that…

A locally conformally K\"ahler (LCK) manifold is a manifold which is covered by a K\"ahler manifold, with the deck transform group acting by homotheties. We show that the search for LCK metrics on Oeljeklaus-Toma manifolds leads to a (yet…

We prove that a compact toric locally conformally K\"ahler manifold which is not K\"ahler admits a toric Vaisman structure, a fact which was conjectured in \cite{mmp}. This is the final step leading to the classification of compact toric…

An LCK manifold with potential is a compact quotient of a Kahler manifold $X$ equipped with a positive Kahler potential $f$, such that the monodromy group acts on $X$ by holomorphic homotheties and multiplies $f$ by a character. The LCK…

In this article we develop a new approach to the problem of the stability of locally conformally K\"ahler structures (l.c.k structures) under small deformations of complex structures and deformations of flat line bundles. We show that under…

In this paper we show as main results two structure theorems of a compact homogeneous locally conformally Kaehler (or shortly l.c.K.) manifold, a holomorphic structure theorem asserting that it has a structure of holomorphic principal fiber…

We give a complete description of all locally conformally K\"ahler structures with holomorphic Lee vector field on a compact complex manifold of Vaisman type. This provides in particular examples of such structures whose Lee vector field is…

We consider several transformation groups of a locally conformally K\"ahler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally…

We prove that a compact lcK manifold with holomorphic Lee vector field is Vaisman provided that either the Lee field has constant norm or the metric is Gauduchon (i.e., the Lee field is divergence-free). We also give examples of compact lcK…

We will discuss in this paper homogeneous locally conformally Keahler (or shortly homogeneous l.c.K.) manifolds and locally homogeneous l.c.K. manifolds from various aspects of study in the field of l.c.K. geometry. We will provide a survey…

A locally conformally Kahler manifold is a Hermitian manifold $(M,I,\omega)$ satisfying $d\omega=\theta\wedge \omega$, where $\theta$ is a closed 1-form, called the Lee form of $M$. It is called pluricanonical if $\nabla\theta$ is of Hodge…

We prove various classification results for homogeneous locally conformally symplectic manifolds. In particular, we show that a homogeneous locally conformally Kaehler manifold of a reductive group is of Vaisman type, if the normalizer of…

We present an overview of recent results in locally conformally K\"ahler geometry, with focus on the topological properties which obstruct the existence of such structures on compact manifolds.

We study compact toric strict locally conformally K\"ahler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $-\infty$ and that the only compact complex surfaces admitting toric strict locally conformally…

We define reduction of locally conformal Kaehler manifolds, considered as conformal Hermitian manifolds, and we show its equivalence with an unpublished construction given by Biquard and Gauduchon. We show the compatibility between this…