arXiv research

We study topological properties of log-symplectic structures and produce examples of compact manifolds with such structures. Notably we show that several symplectic manifolds do not admit log-symplectic structures and several log-symplectic manifolds do not admit symplectic structures, for example #m CP^2 # n bar(CP^2)…

Log-symplectic structures are Poisson structures $π$ on $X^{2n}$ for which $\bigwedge^n π$ vanishes transversally. By viewing them as symplectic forms in a Lie algebroid, the $b$-tangent bundle, we use symplectic techniques to obtain existence results for log-symplectic structures on total spaces of fibration-like maps…

Symplectic structures simplified for compact manifolds.

Symplectic 4-manifolds can be divided into three parts with a special structure.

Study on symplectic structures and their deformations.

Paper proves h-principles for symplectic structures and foliations.

The paper explores left invariant k-symplectic structures on Lie groups with bi-invariant metrics.

Classifies solvable symplectic Lie algebras via extensions and proves structural theorems.

The study classifies complex symplectic structures on 4D Lie algebras and constructs hypersymplectic structures.

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.

In this paper we introduce the notion of a smooth structure on a stratified space, the notion of a Poisson smooth structure and the notion of a weakly symplectic smooth structure on a stratified symplectic space, refining the concept of a stratified symplectic Poisson algebra introduced by Sjamaar and Lerman. We show t…

Found a new connected component in symplectic structures.

Symplectic structure found on projective structures on surfaces with boundary.

Study symplectic structures on low dimensional 2-step nilmanifolds.

The abstract proves a Moser-like theorem for C-symplectic structures and applies it to complex manifolds.

Study on symplectic Dirac operators on foliations, estimating eigenvalues.

The paper classifies symplectic fillings of lens spaces and constructs cobordisms.

New method classifies symplectic structures on Lie groups.

Classifies complex symplectic structures on Lie algebras with large abelian ideals.

We prove that an integrable system over a symplectic manifold, whose symplectic form is covariantly constant w.r.t. the Gauss-Manin connection, carries a natural hyper-symplectic structure. Moreover, a special Kaehler structure is induced on the base manifold.

This paper studies symplectic structures on elliptic surfaces with positive Euler number.

Extends Gromov non-squeezing to locally conformally symplectic structures.

New method constructs symplectic structures on 4-manifolds from trisections.

New models for symplectic structures on classifying stacks.

Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.

The study provides obstructions and examples for $p$-symplectic structures on complex manifolds.

We study deformations of symplectic structures on a smooth manifold $M$ via the quasi-Poisson theory. By a fact, we can deform a given symplectic structure $ω$ to a new symplectic structure $ω_t$ parametrized by some element $t$ in $Λ^2\mathfrak{g}$, where $\mathfrak{g}$ is the Lie algebra of a Lie group $G$. Moreover,…

We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left c…

The study confirms Gromov's speculation and provides bounds for taming symplectic structures.

For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic ana…

Log-symplectic structures are Poisson structures that are determined by a symplectic form with logarithmic singularities. We construct moduli spaces of curves with values in a log-symplectic manifold. Among the applications, we classify symplectically ruled log-symplectic $4$ manifolds (both orientable and non-orientab…

We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called \emph{complex symplectic oxidation}, to construct certain complex symplectic Lie algebras of dimension $4n+4$ from those of dimension $4n$. We specialize this construction to the nilpotent case and apply complex symplec…

Let $(M,ω)$ be a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure) and a torsion-free symplectic connection $\nabla.$ Symplectic Killing spinor fields for this structure are sections of the symplectic spinor bundle satisfying a certain first order partial dif…

Symplectic forms match on circle pattern space.

New symplectic structures found on complex manifolds without Kähler structures.

A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a \textit{$t$-symplectic Lie groupoid}; the "$t$" is motivated …

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

Geometrically interprets symplectic structure in 3-manifold triangulations.

The notions of holomorphic symplectic structures and hypercomplex structures on Courant algebroids are introduced and then proved to be equivalent. These generalize hypercomplex triples and holomorphic symplectic 2-forms on manifolds respectively. Basic properties of such structures are established.

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…

Motivated by an attempt to better understand the notion of a symplectic stack, we introduce the notion of a symplectic hopfoid, which should be thought of as the analog of a groupoid in the so-called symplectic category. After reviewing some foundational material on canonical relations and this category, we show that s…

Unified framework for classifying Sasakian, K-contact, and (κ, μ)-manifolds.

In this paper we classify symplectic Lefschetz fibrations (with empty base locus) on a four-manifold which is the product of a three-manifold with a circle. This result provides further evidence in support of the following conjecture regarding symplectic structures on such a four-manifold: if the product of a three-man…

Goldman symplectic form and complex structure compatible on $\mathrm{SL}(3,\mathbb R)$ Hitchin component.

Introduces derived Lie n-groupoids with shifted symplectic structures.

A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibration…

Develops deformation theory for symplectic foliations using $L_\infty$-algebras.

Ancient solutions found for a specific flow on symplectic half-flat structures.