Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.
problem Characterizing Vaisman solvmanifolds and their properties.
method Analyzing fundamental groups and quotient structures.
result Every Vaisman solvmanifold is a finite quotient of a Kodaira-Thurston manifold.
Extends Tian theorem to Vaisman manifolds for approximations.
problem Approximating Vaisman metrics by immersions/embeddings.
method Study Vaisman metrics on compact manifolds.
result Extend Tian's theorem to Vaisman manifolds.
Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.
problem Characterize Bott-Chern cohomology of Vaisman manifolds.
method Explicit description via basic cohomology, infer relationships between cohomology groups, show invariants are unbounded, cohomological characterization of formality.
result Bott-Chern and Dolbeault numbers determine each other for Vaisman manifolds, and cohomological invariants are unbounded.
Vaisman manifolds are strongly related to Kähler and Sasaki geometry. In this paper we introduce toric Vaisman structures and show that this relationship still holds in the toric context. It is known that the so-called minimal covering of a Vaisman manifold is the Riemannian cone over a Sasaki manifold. We show that if…
Holonomy group of Bismut connection on Vaisman manifolds is studied.
problem Analyzing the holonomy group of Bismut connection on Vaisman manifolds.
method Proved and computed the holonomy group for Vaisman manifolds, including solvmanifolds and Hopf manifolds.
result Holonomy group of Bismut connection on Vaisman manifolds is contained in U(n-1).
The paper classifies Sasaki and Vaisman manifolds of unimodular Lie groups.
problem Understanding the structure of Sasaki and Vaisman manifolds on unimodular Lie groups.
method Basic structure theorem and complete classification for simply connected manifolds.
result A complete classification of simply connected Sasaki and Vaisman unimodular Lie groups.
Holomorphic tensors on Vaisman manifolds are invariant under the Lee field.
problem Characterizing holomorphic tensors on Vaisman manifolds.
method Using the parallelism of the Lee form and properties of the Lee field.
result The Kodaira dimension of Vaisman manifolds is invariant under certain quotients.
Study categorizes Vaisman manifolds with vanishing first Chern class and finds canonical metrics.
problem Characterizing Vaisman manifolds with vanishing first Chern class.
method Categorization into three types based on Bott-Chern class sign, showing canonical metrics, quasi-regularity, stability, and automorphism group behavior.
result Vaisman manifolds with non-positive Bott-Chern class admit canonical metrics and are stable under deformations.
Constructs deformations of Vaisman manifolds preserving foliations.
problem Deforming Vaisman manifolds while maintaining their canonical foliations.
method Uses a basic 1-form with specific properties to construct transverse deformations.
result Basic 1-forms exist in abundance for constructing deformations.
The abstract discusses transforming Vaisman metrics into Kähler-Einstein structures.
problem Transforming Vaisman metrics into Kähler-Einstein structures.
method Using the transverse Kähler-Ricci flow on the canonical foliation of a closed Vaisman manifold.
result Direct proof of short time existence of transverse Kähler-Ricci flow on Vaisman manifolds.
Study of unimodular Sasaki and Vaisman Lie groups, determining all modifications explicitly.
problem Classifying unimodular Sasaki and Vaisman Lie groups.
method Applying the technique of modification to determine all homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups explicitly.
result Complete classification of unimodular Sasaki and Vaisman Lie groups.
Models close to formal for Vaisman and quasi-Sasakian manifolds, including nilmanifolds.
problem Formality of quasi-Sasakian and Vaisman manifolds.
method Providing models that are as close as possible to being formal for compact manifolds with transversely Kaehler structures.
result Classification of corresponding nilmanifolds.
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.
Study Hodge theory and supersymmetry on Sasakian and Vaisman manifolds.
problem Decomposing harmonic forms and cohomology on Sasakian and Vaisman manifolds.
method Construct Lie superalgebra, use Hodge decomposition, compute supersymmetry algebra.
result Explicit computation of the supersymmetry algebra of Sasakian manifolds.
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.
We prove the non-existence of Vaisman metrics on some solvmanifolds with left-invariant complex structures. By this theorem, we show that Oeljeklaus-Toma manifolds does not admit Vaisman metrics.
New classification for Vaisman manifolds with specific properties.
problem Classifying Vaisman manifolds with large first Betti number and vanishing first basic Chern class.
method Analyzing properties and using diffeomorphism and complex structure invariance.
result Every Vaisman manifold with large first Betti number and vanishing first basic Chern class is diffeomorphic to a Kodaira-Thurston manifold.
The paper studies spectral sequences on Vaisman manifolds and finds cohomological obstructions.
problem Finding cohomological obstructions for foliations on Vaisman manifolds.
method Investigates spectral sequences associated with Riemannian foliations on Vaisman manifolds, using Betti numbers to establish lower bounds.
result Establishes cohomological obstructions for two-dimensional foliations to be induced from Vaisman structures, with quasi-regular foliations achieving the lower bound.
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.
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.
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.
A locally conformally Kaehler (l.c.K.) manifold is a complex manifold admitting a Kaehler covering M~, with each deck transformation acting by Kaehler homotheties. A compact l.c.K. manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on M~. We prove a structure theorem f…
We prove the non-existence of Vaisman metrics on some solvmanifolds with a left-invariant complex structure. By this theorem, we show that every Oeljeklaus-Toma manifold with (s,1) admits no Vaisman metric.
Vaisman algebroid explains gauge symmetry in DFT.
problem Understanding gauge symmetry in double field theory.
method Geometric realization of doubled space-time, analysis of algebras and cohomology.
result Found compatibility condition for Lie bialgebroid structures.
The study counts closed elliptic curves and Reeb orbits on Vaisman and Sasakian manifolds.
problem Counting closed orbits and elliptic curves on Vaisman and Sasakian manifolds.
method Analyzes the structure of Vaisman and Sasakian manifolds, uses quasi-regular and S1-quotients, and counts closed orbits and curves. result The number of closed elliptic curves and Reeb orbits is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold.
Calabi-Yau theorem extended to Vaisman manifolds.
problem Uniqueness of Vaisman metrics and their characterization.
method Analyzing the Lee form and Lee class properties.
result Vaisman metrics uniquely determined by volume and Lee class.
Vaisman's theorem extended to locally reducible Kähler spaces.
problem Existence of locally conformally Kähler metrics on compact Kähler spaces.
method Extended Vaisman's theorem to locally reducible Kähler spaces.
result Vaisman's theorem holds for compact Kähler spaces that are locally reducible.
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…
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…
We extend to metric compact mapping tori a splitting result for coKähler manifolds. In particular, we prove that a compact Vaisman manifold is finitely covered by the product of a Sasakian manifold and a circle.
A locally conformally Kähler (LCK) manifold is a complex manifold whose universal cover is Kähler with monodromy group acting on the universal cover by holomorphic homotheties. A Vaisman manifold M is a compact non-Kähler LCK manifold admitting an action of a holomorphic conformal flow lifting to an action on a Kähle…
Study shows compact Vaisman manifolds cannot have certain special Hermitian metrics.
problem Compact Vaisman manifolds and their compatibility with special Hermitian structures.
method Proof of non-existence of specific Hermitian metrics on compact Vaisman manifolds.
result Compact Vaisman manifolds cannot admit special Hermitian metrics like special k-Gauduchon metrics or pluriclosed metrics. Study on non-Kähler Calabi-Yau manifolds and their geometric structures.
problem Holomorphic geometric structures on non-Kähler compact complex manifolds.
method Beauville-Bogomolov decomposition and weak Bochner principle.
result Rigidity of Vaisman Calabi-Yau manifolds implies they are Kodaira manifolds.
We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) mani…
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 reduction and Kaehler reduction of the universal cover. By a recent result of Kamis…
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 Kahler manifold is a Hermitian manifold (M,I,ω) satisfying dω=θ∧ω, where θ is a closed 1-form, called the Lee form of M. It is called pluricanonical if ∇θ is of Hodge type (2,0)+(0,2), where ∇ is the Levi-Civita connection, and Vaisman if ∇θ=0. We show that a c…
Study shows Dolbeault cohomology is unchanged by complex Lie group actions.
problem Understanding how complex Lie group actions affect Dolbeault cohomology.
method Analyzes Dolbeault cohomology of compact complex manifolds with group actions.
result Induced action on Dolbeault cohomology is trivial.
Study shows certain Lie groups lead to Oeljeklaus-Toma manifolds.
problem Understanding locally conformally Kähler metrics and their relation to Oeljeklaus-Toma manifolds.
method Analyzing solvable Lie groups and their metrics, and relating them to Oeljeklaus-Toma constructions.
result Found a connection between Lie groups, locally conformally Kähler metrics, and Oeljeklaus-Toma manifolds.
Study on Vaisman metrics on Kodaira-Thurston surface using pluriclosed flow.
problem Characterizing and preserving Vaisman metrics on Kodaira-Thurston surface.
method Characterization of T2-invariant Vaisman metrics, analysis of pluriclosed flow behavior. result Pluriclosed flow preserves Vaisman condition on Kodaira-Thurston surface, including non-constant scalar curvature.
Study on compact toric locally conformally Kähler manifolds, finding specific properties.
problem Characterizing properties of compact toric locally conformally Kähler manifolds.
method Analyzing Kodaira dimension, using specific examples and mappings.
result Kodaira dimension is -∞ for underlying complex manifolds, and specific properties for surfaces and Vaisman manifolds.
Characterizes Vaisman solvmanifolds and connects them to other geometric structures.
problem Characterizing Vaisman solvmanifolds and their connections to other geometric structures.
method Characterization of unimodular solvable Lie algebras with Vaisman structures in terms of Kähler flat Lie algebras.
result Establishes algebraic restrictions for the existence of Vaisman structures and connects them to other geometric notions.
We prove that any compact homogeneous locally conformally Kähler manifold has parallel Lee form.
The pluriclosed flow preserves Vaisman condition on compact complex surfaces.
problem Preserving Vaisman condition under pluriclosed flow.
method Pluriclosed flow on compact complex surfaces.
result Preserves Vaisman condition if and only if starting metric has constant scalar curvature.
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.
We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invari…
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…
The abstract investigates convexity in locally conformally symplectic geometry.
problem Characterizing and proving convexity in locally conformally symplectic manifolds.
method Geometric characterization and proof of convexity theorems for twisted and symplectic moment maps.
result Established an analog of the symplectic convexity theorem for locally conformally symplectic manifolds.