Related papers: Symplectic Classes on Elliptic Surfaces with posit…

In this paper we present another notion of a smooth manifold with corners and relate it to the commonly used concept in the literature. Afterwards we introduce complex manifolds with corners and show that if $M$ is a compact (respectively…

In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy…

A smooth counterexample to the Hamiltonian Seifert conjecture for six-dimensional symplectic manifolds is found. In particular, we construct a smooth proper function on the symplectic 2n-dimensional vector space, 2n > 4, such that one of…

In this article we construct a new family of simply connected symplectic 4-manifolds with $b_2^+ =1$ and $c_1^2 =2$ which are not diffeomorphic to rational surfaces by using rational blow-down technique. As a corollary, we conclude that a…

Let $M$ be a closed 4-manifold with a free circle action. If the orbit manifold $N^3$ satisfies an appropriate fibering condition, then we show how to represent a cone in $H^2(M;\R)$ by symplectic forms. This generalizes earlier…

For a symplectic manifold $(M,\om)$ with exact symplectic form we construct a 2-cocycle on the group of symplectomorphisms and indicate cases when this cocycle is not trivial.

We present reformulation of Mathieu's result on representing cohomology classes of symplectic manifold with symplectically harmonic forms. We apply it to the case of foliated manifolds with transversally symplectic structure and to…

This paper studies groups of symplectomorphisms of ruled surfaces for symplectic forms with varying cohomology class. This class is characterized by the ratio R of the size of the base to that of the fiber. By considering appropriate spaces…

A symplectic cut of a manifold M with a Hamiltonian circle action is a symplectic quotient of M x C. If M is Kaehler then, since C is Kaehler, the cut space is Kaehler as well. The symplectic structure on the cut is well understood. In this…

We study symplectic structures on characteristically nilpotent Lie algebras (CNLAs) by computing the cohomology space $H^2(\Lg,k)$ for certain Lie algebras $\Lg$. Among these Lie algebras are filiform CNLAs of dimension $n\le 14$. It turns…

We study almost Hermitian 4-manifolds with holonomy algebra, for the canonical Hermitian connection, of dimension at most one. We show how Riemannian 4-manifolds admitting five orthonormal symplectic forms fit therein and classify them. In…

Making use of theory of differentiable stacks, we study symplectic vortex equations over a compact orbifold Riemann surface. We discuss the category of representable morphisms from a compact orbifold Riemann surface to a quotient stack.…

We define the notion of a hierarchically cocompact classifying space for a family of subgroups of a group. Our main application is to show that the mapping class group $\mbox{Mod}(S)$ of any connected oriented compact surface $S$, possibly…

We give an overview of various recent results concerning the topology of symplectic 4-manifolds and singular plane curves, using branched covers and isotopy problems as a unifying theme. While this paper does not contain any new results, we…

For a smooth map $f:X^4\to\Sigma^2$ that is locally modeled by holomorphic maps, the domain is shown to admit a symplectic structure that is symplectic on some regular fiber, if and only if $f^*[\Sigma]\ne0$. If so, the space of symplectic…

Suppose that $C=(C_1,..., C_m)$ is a configuration of 2-dimensional symplectic submanifolds in a symplectic 4-manifold $(X,\omega)$ with connected, negative definite intersection graph $\Gamma_C$. We show that by replacing an appropriate…

We consider classes of algebraic manifolds $\mathcal{A}$, of symplectic manifolds $\mathcal{S}$, of symplectic manifolds with the hard Lefschetz property $\mathcal{HS}$ and the class of cohomologically symplectic manifolds $\mathcal{CS}$.…

We continue our previous work to prove that for any non-minimal ruled surface $(M,\omega)$, the stability under symplectic deformations of $\pi_0, \pi_1$ of $Symp(M,\omega)$ is guided by embedded $J$-holomorphic curves. Further, we prove…

Given a closed symplectic manifold, we construct invariants which count (a) closed rational pseudoholomorphic curves with prescribed cusp singularities and (b) punctured rational pseudoholomorphic curves with ellipsoidal negative ends. We…

Let X be a holomorphic symplectic fourfold such that b_2=23 and i a symplectic involution of X . The fixed locus F of i is a smooth symplectic submanifold of X; we show that F contains at least 12 isolated points and 1 smooth surface. We…