arXiv research

Products of LCK manifolds do not admit LCK structures.

Hopf manifolds can be given lcK structures, shown by constructing a family.

Paper generalizes LVMB manifolds results, finding lck with potential covers.

A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisma…

New theorem on Lee classes for LCK manifolds with potential.

In this paper, we study the properties of coverings of locally conformally Kähler (LCK) spaces with singularities. We begin by proving that a space is LCK if any only if its universal cover is Kähler, thereby generalizing a result from a previous paper. We then show that a complex space which projects over an LCK space…

A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the Bott-Chern class, of an LCK-structure. These invariants together play the same …

We investigate the relation between holomorphic torus actions on complex manifolds of LCK type and the existence of special LCK metrics. We show that if the group of biholomorphisms of such a manifold $(M,J)$ contains a non-real compact torus, then there exists a Vaisman metric on the manifold. Moreover, we show that i…

Unique minimal model for LCK manifolds proved.

Proves algebraic cones for LCK manifolds with potential.

A locally conformally Kahler (LCK) manifold is a manifold which is covered by a Kahler manifold, with the deck transform group acting by homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally conformally Kahler. We …

The article proves conditions for blow-ups of lcK spaces to remain lcK.

Let $M$ be a complex manifold and $L$ an oriented real line bundle on M equipped with a flat connection. An LCK ("locally conformally Kahler") form is a closed, positive (1,1)-form taking values in L, and an LCK manifold is one which admits an LCK form. Locally, any LCK form is expressed as an L-valued pluri-Laplacian …

Study on Calabi-Yau locally conformally Kähler manifolds proving they are Vaisman.

Integrable LCK manifolds characterized as Kähler Lie algebras.

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 Kahler potential. It is known that in this case it is an algebraic cone, that is,…

New proof shows compact homogeneous LCK manifolds are Vaisman.

Conformal vector fields on lcK manifolds are shown to be Killing or holomorphic.

An LCK manifold with potential is a compact quotient M of a Kahler manifold X equipped with a positive plurisubharmonic function f, such that the monodromy group acts on $X$ by holomorphic homotheties and maps f to a function proportional to f. It is known that M admits an LCK potential if and only if it can be holomor…

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 a Kahler form taking values in a local system $L$, called the conformal weight …

Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.

A locally conformally Kähler (LCK) manifold $M$ is one which is covered by a Kähler manifold $\tilde M$ with the deck transform group acting conformally on $\tilde M$. If $M$ admits a holomorphic flow, acting on $\tilde M$ conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vais…

Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.

Extended Vaisman theorem to compact spaces with singularities.

We give a complete description of all locally conformally Kähler 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 not homothetic to the Lee vector field of a Vaisman structure. More generally, dr…

Proves stability of lcK spaces under holomorphic mappings.

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 rank is the rank of the image of this character, considered as a function from the …

A locally conformally Kähler (LCK) manifold is a manifold which is covered by a Kähler 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 another) variation on Kronecker's theorem on units. In turn, this implies that on Oel…

A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that U(1) acts on an LCK manifold M by …

Locally conformally Kahler (LCK) manifolds with potential are those which admit a Kahler covering with a proper, automorphic Kaehler potential. Existence of a potential can be characterized cohomologically as a vanishing of a certain cohomology class, called the Bott-Chern class. Compact LCK manifolds with potential ar…

It is well known that cohomology of any non-trivial 1-dimensional local system on a nilmanifold vanishes (this result is due to L. Alaniya). A complex nilmanifold is a quotient of a nilpotent Lie group equipped with a left-invariant complex structure by an action of a discrete, co-compact subgroup. We prove a Dolbeault…

The last years have seen striking improvements on Vaisman's question about existence of locally conformally Kähler (lcK) metrics on compact complex surfaces. The aim of this paper is two-fold. We review results of different authors which, for all known examples of compact complex surfaces, give a complete answer to Vai…

An locally conformally Kahler (LCK) manifold with potential is a complex manifold with a cover which admits an automorphic Kahler potential. An LCK manifold with potential can be embedded to a Hopf manifold, if its dimension is at least 3. We give a functional-analytic proof of this result based on Riesz-Schauder theor…

For elliptic principal bundles $π:X\ra B$ over Kähler manifolds it was shown by Blanchard that $X$ has a Kähler metric if and only both Chern classes (with real coefficients) of $π$ vanish. For some elliptic principal bundles, when the span of these Chern classes is 1-dimensional, it was shown by Vaisman that $X$ carry…

The abstract discusses conjectures about metrics on complex manifolds.

New classification for Vaisman manifolds with specific properties.

In this article we introduce a generalization of locally conformally Kaehler metrics from complex manifolds to complex analytic spaces with singularities and study which properties of locally conformally Kaehler manifolds still hold in this new setting. We prove that if a complex analytic space has only quotient singul…

A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold X admits a holomorphic submersion with positive dimensional fibers at least one of which is of Kähler type, then X is globally conformall…

Book introduces principles of LCK geometry for complex manifold students.

We classify and investigate locally conformally Kähler structures on four-dimensional solvable Lie algebras up to linear equivalence. As an application we can produce many examples in higher dimension, here including lcK structures on Oeljeklaus-Toma manifolds, and we also give a geometric interpretation of some of the…

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 manifolds with holomorphic Lee vector field which are not Vaisman.

Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.

Study proposes a functional for LCK metrics on complex manifolds.

A manifold $M$ is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. Let $M$ be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on its covering. We show that $M$ admits an automorphic pote…

Oeljeklaus-Toma (OT) manifolds are complex non-Kähler manifolds whose construction arises from specific number fields. In this note, we compute their de Rham cohomology in terms of invariants associated to the background number field. This is done by two distinct approaches, one using invariant cohomology and the other…

We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally Kähler metrics. We present explicit computations for the Inoue surfaces $\mathcal{S}^0$, $\mathcal{S}^+$, $\mathcal{S}^-$ and classify the locally conformally Kähler (and the tamed loc…

A locally conformally Kaehler (LCK) manifold is a complex manifold admitting a Kaehler covering M, with monodromy acting on M by Kaehler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on M. We prove a non-Kaehler analogue of Kodaira embedding theorem: an…

We show a bijective correspondence between compact toric locally conformally symplectic manifolds which admit a compatible complex structure and pairs $(C,a)$, where $C$ is a good cone in the dual Lie algebra of the torus and $a$ is a positive real number. Moreover, we prove that any toric locally conformally Kähler me…