Proves existence of global positive LCK potential on certain manifolds.
problem Existence of global positive LCK potential on LCK manifolds.
method Analyzes L-valued pluri-Laplacian of a function (LCK potential) and uses properties of flat connections.
result Proves existence of global positive LCK potential on certain manifolds.
New theorem on Lee classes for LCK manifolds with potential.
problem Determining Lee classes on LCK manifolds with potential.
method Analyzing cohomology classes of Lee forms and proving the result for Vaisman manifolds.
result The set of Lee classes on LCK manifolds with potential forms an open half-space in H1(M,R). New proof shows compact homogeneous LCK manifolds are Vaisman.
problem Proving compact homogeneous LCK manifolds are Vaisman.
method Using homogeneous LCK manifolds with potential and a new metric construction.
result Compact homogeneous LCK manifolds are Vaisman.
A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold M~ with the deck transform group acting conformally on M~. If M admits a holomorphic flow, acting on M~ conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vais…
Paper generalizes LVMB manifolds results, finding lck with potential covers.
problem Understanding lck structures on LVMB manifolds.
method Generalization of [6] results to LVMB manifolds, constructing covers.
result Examples of LVMB manifolds that are 1-lck with potential.
Embeds LCK manifolds with potential into Hopf manifolds using Riesz-Schauder theorem.
problem Embedding LCK manifolds with potential into Hopf manifolds.
method Functional-analytic proof based on Riesz-Schauder theorem and Montel theorem; alternative argument for complex surfaces.
result Embeds LCK manifolds with potential into Hopf manifolds for dimensions at least 3.
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 …
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…
Hopf manifolds can be given lcK structures, shown by constructing a family.
problem Constructing locally conformally Kaehler structures for Hopf manifolds.
method Analytic family construction and application of Ornea-Verbitsky's theorem.
result Hopf manifolds can be endowed with lcK structures.
Products of LCK manifolds do not admit LCK structures.
problem Whether products of compact complex manifolds admit LCK metrics.
method Classifying known LCK manifolds and proving non-existence of LCK structures in product cases.
result Products of LCK manifolds do not admit LCK structures.
Proves algebraic cones for LCK manifolds with potential.
problem Characterizing algebraic cones for LCK manifolds.
method Analyzes LCK manifolds as complex submanifolds of Hopf manifolds and covers, proving algebraicity of the resulting cones.
result Affine algebraic structure on cones is independent of manifold choice.
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,…
Study describes lcK structures on Vaisman-type manifolds with holomorphic Lee vector field.
problem Characterizing lcK structures with holomorphic Lee vector field on Vaisman-type manifolds.
method Complete description through potential analysis and vector field properties.
result Examples of lcK structures with non-homothetic Lee vector field.
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…
Study on LCK metrics and their existence criteria on complex manifolds.
problem Existence of LCK metrics on complex manifolds.
method Investigation of holomorphic torus actions and biholomorphisms on LCK type manifolds.
result Existence of Vaisman and LCK metrics under specific conditions on the manifold's biholomorphism group.
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 …
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 …
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 …
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…
The article proves conditions for blow-ups of lcK spaces to remain lcK.
problem Conditions for blow-ups of locally irreducible lcK spaces to remain lcK.
method Proves conditions for blow-ups of lcK spaces to remain lcK.
result Blow-ups of locally irreducible lcK spaces are lcK if and only if the original space is induced gcK.
Integrable LCK manifolds characterized as Kähler Lie algebras.
problem Characterizing LCK manifolds with integrable anti-Lee forms.
method Examining LCK manifolds with integrable anti-Lee forms and applying to Lie algebras.
result Unimodular integrable LCK Lie algebras are Kähler Lie algebras with specific derivations.
Conformal vector fields on lcK manifolds are shown to be Killing or holomorphic.
problem Characterizing conformal vector fields on lcK manifolds.
method Analyzing properties of conformal vector fields on compact lcK manifolds.
result Conformal vector fields on compact lcK manifolds are either Killing or holomorphic.
Authors compute Morse-Novikov cohomology for Inoue surfaces and prove nonexistence of certain metrics.
problem Computing and classifying locally conformally Kähler metrics on complex surfaces.
method Review and computation of Morse-Novikov cohomology for known surfaces.
result Nonexistence of LCK metrics with potential on Inoue surfaces and Oeljeklaus-Toma manifolds.
The paper explores properties of LCK spaces and their coverings.
problem Understanding the structure and properties of LCK spaces and their coverings.
method Proving that a space is LCK if and only if its universal cover is Kähler, and showing that a complex space with discrete fibers over an LCK space also has an LCK structure.
result Spaces with discrete fibers over LCK spaces also have LCK structures.
Extended Vaisman theorem to compact spaces with singularities.
problem Generalizing Vaisman's theorem to spaces with singularities.
method Extended Vaisman's theorem to compact complex spaces with singularities.
result Vaisman's theorem extended to compact spaces with singularities.
Proves stability of lcK spaces under holomorphic mappings.
problem Stability of locally conformally Kähler spaces with singularities.
method Analyzes sufficient conditions for proper open morphisms of lcK spaces.
result Extends Varouchas' result to lcK spaces with singularities.
Unique minimal model for LCK manifolds proved.
problem Characterizing unique minimal models for LCK manifolds.
method Proving bimeromorphic maps are holomorphic.
result LCK manifolds have a unique minimal model.
Survey on LCK structures on solvmanifolds, focusing on invariant and Vaisman structures.
problem Understanding LCK structures on solvmanifolds.
method Review of recent results and presentation of new findings.
result New insights into the canonical bundle of solvmanifolds with Vaisman structures.
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 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 …
LCK metrics extended to spaces with quotient singularities, preserving key properties.
problem Extending LCK metrics to complex spaces with singularities.
method Generalization to complex analytic spaces with quotient singularities, proving conditions for existence.
result Spaces with quotient singularities admit LCK metrics if their universal cover has a specific Kaehler metric.
New metrics found on toric LCS manifolds.
problem Finding compatible complex structures on toric LCS manifolds.
method Proved a bijective correspondence between toric LCS manifolds and pairs (C,a). result Compact toric LCS manifolds have a positive potential.
Study on Calabi-Yau locally conformally Kähler manifolds proving they are Vaisman.
problem Characterizing Calabi-Yau locally conformally Kähler manifolds.
method Analyzing properties of Calabi-Yau locally conformally Kähler manifolds, proving they are Vaisman.
result Calabi-Yau locally conformally Kähler manifolds are Vaisman.
Compact lcK manifolds with holomorphic Lee field are Vaisman under certain conditions.
problem Characterizing compact locally conformally Kähler manifolds with holomorphic Lee fields.
method Analyzing conditions for a compact lcK manifold to be Vaisman when it has a holomorphic Lee vector field.
result Compact lcK manifolds with holomorphic Lee field are Vaisman if the Lee field has constant norm or the metric is Gauduchon.
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.
problem The abstract tackles the conjectures about metrics on complex manifolds, specifically balanced, SKT, and LCK.
method The abstract uses complex Hermitian manifolds, closed 1-forms, and conjectures to explore these metrics.
result The abstract verifies a conjecture about the Bott--Chern homology for all known classes of LCK manifolds.
Study characterizes toric LCK manifolds, proving conjecture and showing differences from symplectic case.
problem Characterizing compact toric locally conformally Kähler manifolds.
method Proves conjecture about toric LCK manifolds, constructs examples to show differences from symplectic case.
result Proves a conjecture about toric LCK manifolds and shows differences from symplectic case.
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.
problem Understanding locally conformally Kahler manifolds.
method Explains the concept and properties of LCK manifolds.
result Provides insights into the structure and behavior of LCK manifolds.
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.
problem Understanding non-linear Hopf manifolds and their properties.
method Holomorphic embeddings and LCK metrics.
result Non-linear Hopf manifolds admit LCK metrics.
Study proposes a functional for LCK metrics on complex manifolds.
problem Existence problem for locally conformally Kähler metrics on compact complex manifolds.
method Introducing and studying a functional that varies by complex dimension.
result Functional approach to existence problem for LCK metrics.
Study classifies LC Kahler structures on 4D solv Lie alg, with applications.
problem Classifying LC Kahler structures on 4D solvable Lie algebras.
method Investigation through linear equivalence and geometric interpretation.
result Produces many examples, including lcK structures on Oeljeklaus-Toma manifolds.
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…
No locally conformally Kähler metrics found on Oeljeklaus-Toma manifolds.
problem Existence of locally conformally Kähler metrics on Oeljeklaus-Toma manifolds.
method Analyzing Oeljeklaus-Toma manifolds with specific conditions.
result No locally conformally Kähler metrics found on these manifolds.
Nilpotent Lie algebra cohomology vanishes for certain local systems.
problem Cohomology of nilpotent Lie algebras with specific local systems.
method Proved Dolbeault cohomology vanishes for nilpotent Lie algebras.
result Dolbeault cohomology of nilpotent Lie algebras with non-trivial local systems vanishes.
Researchers compute cohomology of complex non-Kähler OT manifolds.
problem Computing de Rham and Morse-Novikov cohomology of Oeljeklaus-Toma manifolds.
method Two approaches: invariant cohomology and Leray-Serre spectral sequence.
result Chern classes vanish in real cohomology for complex vector bundles.