Study blow-ups of locally conformal symplectic manifolds.
problem Blow-ups of locally conformal symplectic manifolds.
method Show existence of locally conformal symplectic structure on blow-ups.
result Existence of locally conformal symplectic structure on blow-ups.
Study locally conformal symplectic manifolds on compact nilmanifolds.
problem Characterize locally conformal symplectic structures on compact nilmanifolds.
method Left-invariant structures, foliation analysis, Lie group theory.
result Structure theorem for locally conformal symplectic manifolds of the first kind.
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.
Study cohomologies on manifolds with locally conformally symplectic structures.
problem Understanding cohomologies on manifolds with locally conformally symplectic structures.
method Introduced lcs cohomologies, studied elliptic Hodge theory, dualities, and Hard Lefschetz Condition.
result Oeljeklaus-Toma manifolds with precisely one complex place and under an arithmetic condition satisfy the Mostow property.
Reduces LCS manifolds with symplectic actions, preserving conformal structure.
problem Reduction of LCS manifolds with symplectic actions.
method Locally conformally symplectic reduction procedure.
result Compatibility with locally conformally Kähler structure and contact reduction.
We give the parallelism between locally conformal symplectic manifolds and contact manifolds. We also give the generalization of exact contact manifolds.
We obtain universal models for several types of locally conformal symplectic manifolds via pullback or reduction. The relation with recent embedding results for locally conformal Kähler manifolds is discussed.
The Darboux-Weinstein theorem is extended to LCS manifolds.
problem Extending classical results to locally conformally symplectic manifolds.
method Presentation of a version of the Darboux-Weinstein theorem in the LCS setting.
result Application to Lagrangian submanifolds.
Symplectic structures simplified for compact manifolds.
problem Locally conformally symplectic structures on compact manifolds.
method Symplectic analogue of Vaisman's theorem.
result Locally conformally symplectic structures become globally symplectic.
Reduces cotangent bundles using symplectic methods.
problem Symplectic reduction of cotangent bundles.
method Locally conformally symplectic reduction at regular values.
result Proves theorem analogous to symplectic setting.
Locally symplectic structure found on Kerr space-time.
problem Understanding Kerr space-time using geodesics.
method Identifying locally conformally symplectic structure using characteristic classes and Kerr-Schild coordinates.
result Definition of cobordism category of contact 3-manifolds and locally conformally symplectic cobordisms.
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.
We discuss a correspondence between certain contact pairs on the one hand, and certain locally conformally symplectic forms on the other. In particular, we characterize these structures through suspensions of contactomorphisms. If the contact pair is endowed with a normal metric, then the corresponding lcs form is loca…
We exhibit two three-parameter families of locally conformal symplectic forms on the solvmanifold Mn,k considered in [1], and show, using the Hodge-de Rham theory for the Lichnerowicz cohomology that that they are not dω exact, i.e. their Lichnerowicz classes are non-trivial (Theorem 1). This has several import…
Study different types of G_2-structures on Lie groups.
problem Characterize and find examples of G_2-structures on Lie groups.
method Analyze invariant exact locally conformal closed G_2-structures on simply connected Lie groups.
result Complete characterization of invariant exact locally conformal closed G_2-structures on simply connected Lie groups.
The study quantizes energy for curves in symplectic manifolds.
problem Quantization of energy for pseudo-holomorphic curves.
method Extending Topping's theorem to almost everywhere immersed submanifolds and using it to prove energy quantization.
result Energy quantization for pseudo-holomorphic curves of all genus.
The study classifies and constructs symplectic structures on Lie algebras and solvmanifolds.
problem Classifying and constructing symplectic structures on Lie algebras and solvmanifolds.
method Analyzes locally conformally symplectic Lie algebras and solvmanifolds.
result Classifies and constructs symplectic structures on Lie algebras and solvmanifolds.
In this paper, we study deformations of coisotropic submanifolds in a locally conformal symplectic manifold. Firstly, we derive the equation that governs C∞ deformations of coisotropic submanifolds and define the corresponding C∞-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies.…
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 the isotropy group is compact. We also show that such a result does not hold in t…
Constructs symplectic structures on product manifolds from LCS structures.
problem Understanding symplectic structures on product manifolds from LCS structures.
method Constructs a R+-torsor of LCS structures on a covering space. result Bounds the number of fixed points of a Hamiltonian isotopy.
Extends Gromov non-squeezing to locally conformally symplectic structures.
problem Generalizing Gromov non-squeezing to new geometric structures.
method Deformation theory applied to locally conformally symplectic structures.
result Proves a new extension of the Gromov non-squeezing phenomenon.
New definition for a special type of complex manifolds.
problem Characterizing compact locally conformally hyperkähler manifolds.
method Equivalent definition using a nondegenerate complex two-form.
result Established a conformal analogue of Beauville's theorem.
Introduces locally conformally symplectic and Kähler geometry.
problem None explicitly stated, focuses on background and applications.
method Background introduction and demonstration of applications.
result Illustrates applications of these geometries in physics.
Given a contact manifold $M_#$ together with a transversal infinitesimal automorphism ξ, we show that any local leaf space M for the foliation determined by ξ naturally carries a conformally symplectic (cs-) structure. Then we show that the Rumin complex on $M_#$ descends to a complex of differential operators on…
The paper proves a generalized Lefschetz duality for a specific type of manifold.
problem Proving the hard Lefschetz duality for a new class of manifolds.
method Generalizing Kähler identities to prove the duality for locally conformally almost Kähler manifolds.
result The hard Lefschetz duality is established for locally conformally almost Kähler manifolds.
Study Godbillon-Vey class for regular Jacobi foliations.
problem Characterizing foliations in Jacobi manifolds.
method Explicitly defined and computed Godbillon-Vey class for regular foliations.
result Expressed Godbillon-Vey class in terms of Jacobi structures.
We obtain an example of a compact locally conformal symplectic nilmanifold which admits no locally conformal Kähler metrics. This gives a new positive answer to a question raised by L. Ornea and M. Verbitsky.
This work extends locally conformal analysis to multi-Hamiltonian settings, providing new geometric structures and Hamiltonian dynamics.
problem Globalization problem in multi-Hamiltonian formalisms due to incompatibilities on chart overlaps.
method Investigation of locally conformally Nambu--Poisson and locally conformally generalized Poisson manifolds, constructing Hamiltonian-type evolution equations.
result Unified framework for classical, Nambu--Poisson, and generalized Poisson manifolds within a locally conformal context.
Study LCS structures of the second kind on Lie algebras.
problem Characterize and construct LCS Lie algebras.
method Construct new examples using representations and characterize existing ones.
result Characterize all LCS Lie algebras obtained with the construction.
Study of Gromov-Witten theory on locally conformally symplectic manifolds, focusing on non-squeezing and holomorphic curves.
problem Understanding Gromov-Witten theory in the context of locally conformally symplectic manifolds.
method Analyzing holomorphic curves and applying Gromov non-squeezing theorem to $\lcs$ manifolds.
result Gromov non-squeezing theorem has C0 rigidity in dimension 4 for $\lcs$ manifolds. Study Morse-Novikov cohomology for 1-forms on rank 1 manifolds.
problem Analyzing cohomology of closed one-forms on manifolds.
method Explicit computation and discussion of locally conformally symplectic manifolds.
result Explicit computation for Inoue surface S^0.
The Weinstein conjecture is extended to a new class of manifolds.
problem Existence of Reeb 2-curves in locally conformally symplectic manifolds.
method Extended Gromov-Witten theory and elliptic curve counts.
result Partial verification of the conjecture in higher dimensions.
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 study explores conformal symplectic foliations on closed manifolds, proving their existence in dimensions 5 and above.
problem Existence of conformal symplectic foliations on closed manifolds.
method Analysis of symplectic and conformal symplectic codimension-one foliations on closed manifolds of dimension at least 5.
result Construction of conformal symplectic foliations on closed, simply-connected, almost contact manifolds in dimension 5.
Study lattices in specific Lie groups for geometric structures.
problem Existence of lattices in Lie groups with certain geometric structures.
method Analyzing left invariant locally conformal Kähler or symplectic structures.
result Existence of lattices only in dimension 4 for Kähler structures, and in any even dimension for symplectic structures.
The paper studies LCAK metrics on complex manifolds and their properties.
problem Characterizing and understanding LCAK metrics on complex manifolds.
method Analyzes the geometric structures induced by LCAK metrics and their properties.
result Pluricanonical LCAK metrics have parallel Lee form on compact manifolds.
We prove that every compact complex surface with odd first Betti number admits a locally conformally symplectic 2-form which tames the underlying almost complex structure.
Study second-Chern-Einstein metrics on 4D manifolds, finding Killing vector fields and examples.
problem Investigate second-Chern-Einstein metrics on 4D almost-Hermitian manifolds.
method Analyze compact and unimodular almost-abelian Lie algebras, use Killing vector fields and parallel non-zero Lee forms.
result Describe 4D compact second-Chern-Einstein locally conformally symplectic manifolds and classify unimodular almost-abelian Lie algebras with second-Chern-Einstein metrics.
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.
Locally conformal symplectic (l.c.s.) groupoids are introduced as a generalization of symplectic groupoids. We obtain some examples and we prove that l.c.s. groupoids are examples of Jacobi groupoids in the sense of \cite{IM}. Finally, we describe the Lie algebroid of a l.c.s. groupoid.
Convexity theorem for Hamiltonian actions on conformal symplectic manifolds.
problem Establishing convexity in conformal symplectic geometry.
method Proving a convexity theorem for moment maps under Lee type actions.
result Analog of Kirwan's convexity theorem in conformal symplectic geometry.
We investigate the Fefferman spaces of conformal type which are induced, via parabolic geometry, by the quaternionic contact (qc) manifolds introduced by O.Biquard. Equivalent characterizations of these spaces are proved: as conformal manifolds with symplectic conformal holonomy of the appropriate signature; as pseudo-…
Establishes Hard Lefschetz Theorem for Vaisman manifolds.
problem Proving Hard Lefschetz Theorem for Vaisman manifolds.
method Analyzes de Rham and basic cohomology of compact Vaisman manifolds.
result Proves Hard Lefschetz Theorem for Vaisman manifolds.
The paper studies LCS bundles and conditions for their total spaces to have LCS forms.
problem Conditions for LCS forms on the total space of LCS fiber bundles.
method Using coupling form and twisted Hamiltonian action to construct LCS forms.
result Conditions found for LCS forms on the total space of LCS fiber bundles.
Study invariants of elliptic curves in LCS manifolds, leading to new phenomena in Riemann-Finsler geometry.
problem Defining and studying invariants of elliptic curves in locally conformally symplectic manifolds.
method Using J-holomorphic curves and Gromov-Witten theory to define and study invariants. result Found new phenomena in Riemann-Finsler geometry and an analogue of the Weinstein conjecture.
Defines compatibility between Riemannian and Jacobi structures.
problem Understanding compatibility between Riemannian and Jacobi structures.
method Introducing a notion of compatibility and proving it for specific structures.
result Fundamental examples of Jacobi structures lead to known structures like Riemann-Poisson, Kenmotsu, and locally conformally Kähler.
We study the characteristic foliation of a twisted Jacobi manifold. We show that a twisted Jacobi manifold is foliated into leaves that are, according to the parity of the dimension, endowed with a twisted contact or a twisted locally conformal symplectic structure.
Defines compatibility between Riemannian and Jacobi structures.
problem Understanding the relationship between Riemannian and Jacobi structures.
method Introducing a notion of compatibility and proving results for specific examples.
result Establishes compatibility for Poisson, contact, and locally conformally symplectic structures.