arXiv research

Symplectic structures simplified for compact manifolds.

Convexity theorem for Hamiltonian actions on conformal symplectic manifolds.

The goal of this note is to give an introduction to locally conformally symplectic and Kähler geometry. In particular, Sections 1 and 3 aim to provide the reader with enough mathematical background to appreciate this kind of geometry. The reference book for locally conformally Kähler geometry is "Locally conformal Kähl…

We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $β$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the …

Extends Gromov non-squeezing to locally conformally symplectic structures.

A new geometric model for V1 hypercolumns combines symplectic and spherical models.

The paper explores scaling symmetries in symplectic geometry and their applications to central configurations.

We study the relations between the projective and the almost conformally symplectic structures on a smooth even dimensional manifold. We describe these relations by a single almost conformally symplectic connection with totally trace--free torsion sharing the geodesics (up to parametrization) with the projective class.…

The geometry that is defined by the scalars in couplings of Einstein-Maxwell theories in N=2 supergravity in 4 dimensions is denoted as special Kaehler geometry. There are several equivalent definitions, the most elegant ones involve the symplectic duality group. The original construction used conformal symmetry, which…

The paper tackles isotropy of symplectic forms using Hodge flows.

Motivated by analogous results in locally conformal symplectic geometry, we study different classes of G$_2$-structures defined by a locally conformal closed 3-form. In particular, we give a complete characterization of invariant exact locally conformal closed G$_2$-structures on simply connected Lie groups, and we pre…

Study invariants of elliptic curves in LCS manifolds, leading to new phenomena in Riemann-Finsler geometry.

Given a closed surface S of genus at least 2, we compare the symplectic structure of Taubes' moduli space of minimal hyperbolic germs with the Goldman symplectic structure on the character variety X(S, PSL(2,C)) and the affine cotangent symplectic structure on the space of complex projective structures CP(S) given by t…

In this paper, we study the blow-up of a locally conformal symplectic manifold.We show that there exists a locally conformal symplectic structure on the blow-up of a locally conformal symplectic manifold along a compact induced symplectic submanifold.

The study explores conformal symplectic foliations on closed manifolds, proving their existence in dimensions 5 and above.

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-…

We obtain structure results for locally conformally symplectic Lie algebras. We classify locally conformally symplectic structures on four-dimensional Lie algebras and construct locally conformally symplectic structures on compact quotients of all four-dimensional connected and simply connected solvable Lie groups.

New geometric structures defined in contact metric geometry.

Study fundamental groups of geometric transformation groups using loop spaces.

Reduces cotangent bundles using symplectic methods.

We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kähler metrics and all the structures come from left-invariant locally conformal symplectic structures on the correspon…

Unified Jacobi coupling construction for various geometric settings.

The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on d…

Solves symplectic and conformal symplectic group actions equivalence problem.

This work extends locally conformal analysis to multi-Hamiltonian settings, providing new geometric structures and Hamiltonian dynamics.

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.

Locally symplectic structure found on Kerr space-time.

Paper proves h-principles for symplectic structures and foliations.

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.

We use the formalism of generalized geometry to study the generic supersymmetric AdS_5 solutions of type IIB supergravity that are dual to N=1 superconformal field theories (SCFTs) in d=4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometr…

The paper introduces polarizations in symplectic and orthogonal settings.

We exhibit two three-parameter families of locally conformal symplectic forms on the solvmanifold $M_{n,k}$ considered in [1], and show, using the Hodge-de Rham theory for the Lichnerowicz cohomology that that they are not $d_ω$ exact, i.e. their Lichnerowicz classes are non-trivial (Theorem 1). This has several import…

We give the parallelism between locally conformal symplectic manifolds and contact manifolds. We also give the generalization of exact contact manifolds.

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.

New methods prove non-squeezing in locally conformal symplectic geometry.

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.

New approach constructs symplectic structure on pseudo-Riemannian geodesics.

A locally conformally symplectic (LCS) form is an almost symplectic form $ω$ such that a closed one-form $θ$ exists with $dω= θ\wedge ω$. We present a version of the well-known result of Darboux and Weinstein in the LCS setting and give an application concerning Lagrangian submanifolds.

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…

Introduces basic facts on early vision's functional architecture.

We present a reduction procedure for locally conformally symplectic (LCS) manifolds with an action of a Lie group preserving the conformal structure, with respect to any regular value of the momentum mapping. Under certain conditions, this reduction is compatible with the existence of a locally conformally Kähler struc…

We investigate, in some details, symplectic equivalence between several conformal classes of Lorentz metrics on the hyperboloid of one sheet $H^{1,1} \cong T \times T - Δ$ and affine coadjoint orbits of the group $Diff_+(Δ)$ of orientation preserving diffeomorphisms of $Δ\cong T$ with its natural projective structure. …

Method resolves 4D symplectic orbifolds using complex geometry.

New flows introduced for symplectic geometry.

We give conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map from this manifold into a (1,2)-symplectic manifold must satisfy the fourth-order condition of being biharmonic, hence generalizing the Lichnerowicz theorem on harmonic maps. These third-order non-linear conditions …

Abstract collects open problems in billiards and symplectic geometry.

We prove that a compact toric locally conformally Kähler manifold which is not Kähler admits a toric Vaisman structure, a fact which was conjectured in \cite{mmp}. This is the final step leading to the classification of compact toric locally conformally Kähler manifolds started in \cite{p} and \cite{mmp}. We also show,…

The study constructs optimal tori on Fano manifolds and confirms mirror symmetry.