arXiv research

Study on symplectic structures and their deformations.

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

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

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…

Symplectic structures simplified for compact manifolds.

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

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

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

We answer the natural question: when are a regular Poisson structure along with a complex structure transverse to its symplectic leaves induced by generalized complex structure? The leafwise symplectic form and transverse complex structure determine an obstruction class in a certain cohomology, which vanishes if and on…

This article investigates the complex symplectic geometry of the deformation space of complex projective structures on a closed oriented surface of genus at least 2. The cotangent symplectic structure given by the Schwarzian parametrization is studied carefully and compared to the Goldman symplectic structure on the ch…

The paper extends symplectic techniques to generalized complex geometry.

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

Generalized complex structures on certain torus bundles are explored.

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…

Abstract classifies Lie algebras with complex or symplectic structures.

New symplectic structures found on complex manifolds without Kähler structures.

On a smooth manifold M, generalized complex (generalized paracomplex) structures provide a notion of interpolation between complex (paracomplex) and symplectic structures on M. Given a complex manifold (M,j), we define six families of distinguished generalized complex or paracomplex structures on M. Each one of them in…

A dictionary connects symplectic to contact geometry, with applications to complex and G-structures.

Study semi-Kähler structures on specific Lie groups without symplectic structures.

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,…

New symplectic forms derived from Lagrangian fibrations on symplectic manifolds.

Symplectic forms match on circle pattern space.

Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form $\partial \bar \partial$-closed, i.e. to strong Kähler with torsion (${\rm SKT}$) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed …

We introduce a surgery for generalized complex manifolds whose input is a symplectic 4-manifold containing a symplectic 2-torus with trivial normal bundle and whose output is a 4-manifold endowed with a generalized complex structure exhibiting type change along a 2-torus. Performing this surgery on a K3 surface, we obt…

Holomorphic symplectic structure on Lagrangian moduli space.

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.

The paper explores spaces of Kähler and symplectic forms on 4-manifolds.

We shall introduce the notion of $C^\infty$ logarithmic symplectic structures on a differentiable manifold which is an analog of the one of logarithmic symplectic structures in the holomorphic category. We show that the generalized complex structure induced by a $C^\infty$ logarithmic symplectic structure has unobstruc…

This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, the second part focused on the case that the underlying struc…

We produce examples of generalized complex structures on manifolds by generalizing results from symplectic and complex geometry. We produce generalized complex structures on symplectic fibrations over a generalized complex base. We study in some detail different invariant generalized complex structures on compact Lie g…

Symplectic embedding extended to stratified spaces.

Positivity of intersections in 4-manifolds leads to taming symplectic structures.

The paper introduces polarizations in symplectic and orthogonal settings.

We study the existence of strong Kähler with torsion (SKT) metrics and of symplectic forms taming invariant complex structures $J$ on solvmanifolds $G/Γ$ providing some negative results for some classes of solvmanifolds. In particular, we show that if either $J$ is invariant under the action of a nilpotent complement o…

Let M be a 4-manifold which admits a free circle action. We use twisted Alexander polynomials to study the existence of symplectic structures and the minimal complexity of surfaces in M. The results on the existence of symplectic structures summarize previous results of the authors in [FV08a,FV08,FV07]. The results on …

A generalized complex structure is called stable if its defining anticanonical section vanishes transversally, on a codimension-two submanifold. Alternatively, it is a zero elliptic residue symplectic structure in the elliptic tangent bundle associated to this submanifold. We develop Gompf-Thurston symplectic technique…

New pseudo-Kähler Einstein spaces found with special almost complex structures.

We consider invariant symplectic connections $\nabla$ on homogeneous symplectic manifolds $(M,ω)$ with curvature of Ricci type. Such connections are solutions of a variational problem studied by Bourgeois and Cahen, and provide an integrable almost complex structure on the bundle of almost complex structures compatible…

Study on types of generalized hypercomplex structures on tori and Kodaira-Thurston surface.

Constructs a Morse-Bott function on symplectic Grassmannians.

Let $K$ be a compact group. For a symplectic quotient $M_λ$ of a compact Hamiltonian Kähler $K$-manifold, we show that the induced complex structure on $M_λ$ is locally invariant when the parameter $λ$ varies in $\mathrm{Lie}(K)^*$. To prove such a result, we take two different approaches: (i) by using the complex geom…

We show that the almost complex structure underlying a non-Kahler, nearly Kahler 6-manifold (in particular, the standard almost complex structure of S^6) cannot be compatible with any symplectic form, even locally.

The paper proves spectral convergence for a specific type of geometric quantization.

Introduces a Morse complex on symplectic manifolds using gradient flows and proves its cohomology is independent of metrics and Morse functions.

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form ω. This generalises Freed's definition …

In this paper, we construct the moduli space of marked oper structures on a closed, oriented smooth surface of negative Euler characteristic as a holomorphic fiber bundle over Teichmüller space. We prove that the holonomy map from the space of marked oper structures to the moduli space of reductive flat bundles is a ho…

Symplectic embeddings of balls into specific manifolds are studied, with restrictions and obstructions identified.

Study rigidity of Hamiltonians near a minimum in symplectic and magnetic settings.