Related papers: Nowhere vanishing primitive of a symplectic form

We study the symplectic semi-characteristic of a closed 4n-dimensional symplectic manifold. First, using the even-degree part of the primitive cohomology, we define the symplectic semi-characteristic. Second, using a vector field with…

We prove a Serre type vanishing property for the twisted primitive cohomology of a symplectic manifold. It is based on Tseng and Zhou's vanishing property under the symplectic flatness. These vanishing properties emphasizes the necessity of…

Let (m,b) a pair of natural numbers. For m even (resp. m odd and b greater than or equal to 2) we show that if there is an m-dimensional non-formal compact oriented manifold whose first Betti number equals b, there is also a symplectic…

We prove non-hyperbolicity of primitive symplectic varieties with $b_2 \geq 5$ that satisfy the rational SYZ conjecture. If in addition $b_2 \geq 7$, we establish that the Kobayashi pseudometric vanishes identically. This in particular…

A differential 1-form $\alpha$ on a manifold of odd dimension $2n+1$, which satisfies the contact condition $\alpha \wedge (d\alpha)^n \neq 0$ almost everywhere, but which vanishes at a point $O$, i.e. $\alpha (O) = 0$, is called a…

It is shown that the action for Hamiltonian equations of motion can be brought into invariant symplectic form. In other words, it can be formulated directly in terms of the symplectic structure $\omega$ without any need to choose some…

The action--Maslov homomorphism $I\co\pi_1(\text{Ham}(X,\omega))\to\R$ is an important tool for understanding the topology of the Hamiltonian group of monotone symplectic manifolds. We explore conditions for the vanishing of this…

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…

Let $X$ be a real-analytic manifold and $g\colon X\to{\mathbf R}^n$ a proper triangulable subanalytic map. Given a subanalytic $r$-form $\omega$ on $X$ whose pull-back to every non singular fiber of $g$ is exact, we show tha $\omega$ has a…

We show that the presence of a non-contractible one-periodic orbit of a Hamiltonian diffeomorphism of a connected closed symplectic manifold $(M,\omega)$ implies the existence of infinitely many non-contractible simple periodic orbits,…

Examples of nonformal simply connected symplectic manifolds are constructed.

We show, in this note, that on any symplectic supermanifold, even or odd, there exist an infinite dimensional affine space of symmetric connections, compatible to the symplectic form.

We describe the local and Steenbrink vanishing problems for singular symplectic varieties with isolated singularities. We do this by constructing a morphism $$\mathbb D_X(\underline \Omega_X^{n+p}) \to \underline \Omega_X^{n+p}$$ for a…

Let $\Omega$ be a non-singular syplectic form on some vector space $V$. Denote by $S^{n}_{k}(\Omega)$ the set of all $k$-dimensional planes $s$ in $V$ such that the restriction of $\Omega$ onto $s$ is singular. For the cases when $k=2,n-2$…

A symplectic manifold $(M,\omega)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and…

The Kodaira-Nakano Vanishing Theorem has been generalized to the relative setting by A. Sommese. We prove a version of this theorem for non-compact manifolds. As an apllication, we prove that the cohomology of a fiber of a symplectic…

We study the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold. In particular, we give conditions under which a mapping torus has a non-zero Massey product. As an application we prove that there are…

We show that if a diffeomorphism of a symplectic manifold $(M^{2n},\omega)$ preserves the form $\omega^{k}$ for $0 < k < n$ and is connected to identity through such diffeomorphisms then it is indeed a symplectomorphism.

In the definition of irreducible holomorphic symplectic manifolds the condition of being simply connected can be replaced by vanishing irregularity. We discuss finite quotients X of complex tori where the space of reflexive 2-forms is…

Given a manifold $M$, some closed $\beta\in\Omega^1(M)$ and a map $f\in C^\infty(M)$, a $\beta$-critical point is some $x\in M$ such that $d_\beta f_{x}=0$ for the Lichnerowicz derivative $d_\beta$. In this paper, we will give a lower bound…