arXiv research

Symplectic structures simplified for compact manifolds.

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…

Study symplectic structures on low dimensional 2-step nilmanifolds.

Symplectic fillings of prequantization bundles are shown to be disk bundles under certain conditions.

We study symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are associated with symplectic cohomologies of differential forms and can be of fourth-order. We introduce several natural boundary conditions on differential forms and use them to establish Hodge theory by proving various fo…

New symplectic structures found on complex manifolds without Kähler structures.

Introduces symplectic flatness for connections over symplectic manifolds.

Conditions are given under which an infinitesimal automorphism of a torsion-free connection preserving a symplectic form is necessarily a symplectic vector field. An example is given of a compact symplectic manifold admitting a flat symplectic connection and an infinitesimal automorphism that is not symplectic.

Constructing compact non-Kähler manifolds with and without the Hard Lefschetz Condition

We introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the …

The abstract explores analogues of Hodge theory in Lie algebroids.

We prove Gray--Moser stability theorems for complementary pairs of forms of constant class defining symplectic pairs, contact-symplectic pairs and contact pairs. We also consider the case of contact-symplectic and contact-contact structures, in which the constant class condition on a one-form is replaced by the conditi…

New findings show some symplectic solvmanifolds fail hard-Lefschetz condition.

The paper studies symplectic forms on projective limits of Banach bundles and their Darboux Theorem.

In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area. Moreover, the symplectic surface can be chosen to be embedded if 2n is at least 6. We also analyze the additional conditions under which e…

The paper explores positivity conditions for $χ_y$-genus and their implications on Chern numbers and symplectic manifolds.

Study on symplectic structures and their deformations.

We prove a folklore result which gives a sequence of necessary and sufficient conditions for a stably symplectic manifold to define a nilpotent element in the symplectic bordism ring.

In [11], I. M. Gelfand, V. Retakh, and M. Shubin defined the symplectic sectional curvature of a torsion-free connection preserving a symplectic form. The present article defines the corresponding notion of constant symplectic sectional curvature and characterizes this notion in terms of the curvature tensor of the sym…

The study constructs symplectic solvmanifolds satisfying the hard-Lefschetz condition.

We study the rigidity and flexibility of symplectic embeddings of simple shapes. It is first proved that under the condition $r_n^2 \le 2 r_1^2$ the symplectic ellipsoid $E(r_1, ..., r_n)$ with radii $r_1 \le ... \le r_n$ does not embed in a ball of radius strictly smaller than $r_n$. We then use symplectic folding to …

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…

The study proves symplectic quandles cannot have good involutions.

Symplectic reduction by abelian subgroups coincides under specific conditions.

Study investigates Hamiltonian systems in fibered almost-symplectic manifolds.

We regard the real symplectic group $Sp(2n,\mathbb{R})$ as a constraint submanifold of the $2n\times 2n$ real matrices $\mathcal{M}_{2n}(\mathbb{R})$ endowed with the Euclidean (Frobenius) metric, respectively as a submanifold of the general linear group $Gl(2n,\mathbb{R})$ endowed with the (left) invariant metric. For…

New insights into symplectic loops and their flux groups.

We give finiteness results and some classifications up to diffeomorphism of minimal strong symplectic fillings of Seifert fibered spaces over S^2 satisfying certain conditions, with a fixed natural contact structure. In some cases we can prove that all symplectic fillings are obtained by rational blow-downs of a plumbi…

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…

A standard convexity condition on the boundary of a symplectic manifold involves an induced positive contact form (and contact structure) on the boundary; the corresponding concavity condition involves an induced negative contact form. We present two methods of symplectically attaching 2-handles to convex boundaries of…

The paper characterizes symplectic fillings of Seifert 3-manifolds using rational blowdowns.

Study realizes symplectic algebras and homotopy types on manifolds.

In this paper we determine conditions of existence of an induced Riemannian structure on the symplectic quotient of a symplectic and Riemannian manifold following the action of a Lie group acting upon it in a hamiltonian way with equivariant momentum mapping.

Unique optimal symplectic connections found for submersions.

We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex si…

We study the Morse-Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz Condition. We consider solvmanifolds and Oeljeklaus-Toma manifolds. In…

We use the techniques of integration of Poisson manifolds into symplectic Lie groupoids to build symplectic resolutions (= desingularizations) of the closure of a symplectic leaf. More generally, we show how Lie groupoids can be used to lift singularities, in particular when one imposes a compatibility condition with a…

We generalize various symplectic reduction techniques to the context of the optimal momentum map. Our approach allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. We construct an orbit red…

A novel symplectic integrator for Hamiltonian equations on $S_2^n \times T^{\ast} \RR^m$ is developed and studied. Partitioned Runge--Kutta methods for Hamiltonian systems on products of Hamiltionian manifolds are studied, specifically, algebraic conditions for their symplecticity are derived.

We consider closed symplectically aspherical manifolds, i.e. closed symplectic manifolds $(M,ω)$ satisfying the condition $[ω]|_{π_2M}=0$. Rudyak and Oprea [RO] remarked that such manifolds have nice and controllable homotopy properties. Now it is clear that these properties are mostly determined by the fact that the s…

Study cohomologies of complex manifolds with symplectic forms and their stability.

In this paper, we show that there is a close relationship between generalized subtangent manifolds and Lie groupoids. We obtain equivalent assertions among the integrability conditions of generalized almost subtangent manifolds, the condition of compatibility of source and target maps of symplectic groupoids with sympl…

In this paper, we investigate the minimal symplectic fillings of small Seifert 3-manifolds with a canonical contact structure. As a result, we classify all minimal symplectic fillings of small Seifert 3-manifolds satisfying certain conditions. Furthermore, we also demonstrate that every such a minimal symplectic fillin…

Study of Kähler-Ricci flow on toric Fano varieties with preserved symplectic condition.

In this note we give conditions which ensure the reduction of a symplectic connection in the process of a Marsden-Weinstein reduction and of the reduction of a presymplectic manifold.

We study the Calabi-Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.

We find a complete set of local invariants of singular symplectic forms with the structurally stable Martinet hypersurface on a $2n$-dimensional manifold. In the $\mathbb C$-analytic category this set consists of the Martinet hypersurface $Σ_2$, the restriction of the singular symplectic form $ω$ to $TΣ_2$ and the kern…

Symplectic Torelli groups of positive rational surfaces are trivial or sphere braid groups.