Related papers: Currents on locally conformally K\"ahler manifolds
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 propose an approach to the existence problem for locally conformally K\"ahler metrics on compact complex manifolds by introducing and studying a functional that is different according to whether the complex dimension of the manifold is…
We study complex non-K\"ahler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties. In particular, we provide examples where the existence of locally conformal holomorphic-tamed…
We investigate isometric immersions of locally conformally Kaehler metrics into Hopf manifolds. In particular, we study Hopf-induced metrics on compact complex surfaces.
K\"ahler-Einstein currents, also known as singular K\"ahler-Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact K\"ahler spaces $X$ and their two defining…
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…
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…
This paper is devoted, first of all, to give a complete unified proof of the Characterization Theorem for compact generalized $p-$K\"ahler manifolds (Theorem 3.2). The proof is based on the classical duality between "closed" positive forms…
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…
A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering M, such that its monodromy acts on this covering by homotheties. A compact LCK manifold is called LCK with potential if M admits an authomorphic…
We characterize compact locally conformally K\"ahler (l.c.K.) manifolds under the assumption of a purely conformal, holomorphic circle action. As an application, we determine the structure of the compact l.c.K. manifolds with parallel Lee…
I present a selection of results on locally conformally K\"ahler geometry published after 1997. The proofs are mainly sketched, some of them are even omitted. Several open problems are indicated in the end.
A locally conformally K\"ahler (LCK) manifold is a complex manifold covered by a K\"ahler manifold, with the covering group acting by homotheties. We show that if such a compact manifold X admits a holomorphic submersion with positive…
We prove an extension theorem for Kahler currents with analytic singularities in a Kahler class on a complex submanifold of a compact Kahler manifold.
We obtain a local classification of complex homothetic foliations on Kaehler manifolds by complex curves. This is used to construct almost Kaehler, Ricci-flat metrics subject to additional curvature properties.
Harvey and Lawson introduced the Kaehler rank and computed it in connection to the cone of positive exact currents of bidimension (1,1) for many classes of compact complex surfaces. In this paper we extend these computations to the only…
We study many properties concerning weak K\"ahlerianity on compact complex manifolds which admits a holomorphic submersion onto a K\"ahler or a balanced manifold. We get generalizations of some results of Harvey and Lawson (the K\"ahler…
We give an account of old and new results concerning many types of non-K\"ahler metrics, with focus on the problem of their coexistence on compact complex manifolds, and their behaviour at deformations and blow-up. We also describe a…
We prove that compact complex manifolds with admitting metrics with negative Chern curvature operator either admit a $dd^c$-exact positive (1,1) current, or are K\"ahler with ample canonical bundle. In the case of complex surfaces we obtain…
On a class of compact Hermitian manifolds including compact K\"{a}hler manifolds, we prove that the the relative non-pluripolar product is always well-defined. We also prove the monotonicity of the relative non-pluripolar product in terms…