Related papers: Symplectically replacing plumbings with Euler char…

In this note we present a new definition of the 4-manifold admitting inequivalent symplectic structures constructed by McMullen-Taubes which leads to the identification of a new symplectic structure. We prove moreover that it is…

Every exotic pair in 4-dimension is obtained each other by twisting a {\it cork} or {\it plug} which are codimension 0 submanifolds embedded in the 4-manifolds. The twist was an involution on the boundary of the submanifold. We define cork…

Let $M$ be an irreducible holomorphic symplectic (hyperk\"ahler) manifold. If $b_2(M)\geq 5$, we construct a deformation $M'$ of $M$ which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its…

Using the degeneration technique, one studies the behavior of Welschinger invariants under the blow-up, and obtains some blow-up formulae of Welschinger invariants. One also analyses the variation of Welschinger invariants when replacing a…

In terms of the gauged nonlinear $\sigma$-models, we describe some results and implications of solving the following problem: Given a smooth symplectic manifold as target space with a quasi-free Hamiltonian group action, perform the…

For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsvath and Szabo, but it has even more…

A hypersymplectic structure on a 4-manifold $X$ is a triple $\underline{\omega}$ of symplectic forms which at every point span a maximal positive-definite subspace of $\Lambda^2$ for the wedge product. This article is motivated by a…

We construct a new type of geometric knot theory, plumbers' knots, and solve the problems of distinguishing and enumerating such knots at a fixed level of complexity. (v2) Minor edits, added theorem 3.18. (v3) Substantial revisions,…

We construct a new infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as…

Despite spectacular advances in defining invariants for simply connected smooth and symplectic 4-dimensional manifolds and the discovery of effective surgical techniques, we still have been unable to classify simply connected smooth…

Based on the Euler-Lagrange cohomology groups $H_{EL}^{(2k-1)}({\cal M}^{2n}) (1 \leqslant k\leqslant n)$ on symplectic manifold $({\cal M}^{2n}, \omega)$, their properties and a kind of classification of vector fields on the manifold, we…

Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding $\iota:B_{c}\into X$ of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up $\tX_{\iota}$. In this paper, we study…

The symplectization of an overtwisted contact structure in Euclidean 3--space is shown to be an exotic symplectic structure on Euclidean 4--space. The technique can be extended to produce exotic symplectic structures in higher dimensional…

We show that the natural symplectic structure on the Milnor fiber of an isolated singularity in complex three variables whose link fibers over the circle can be modified into one which is cylindrical at the end. As a consequence we see that…

In this paper we set up the family Seiberg-Witten theory. It can be applied to the counting of nodal pseudo-holomorphic curves in a symplectic 4-manifold (especially a Kahler surface). A new feature in this theory is that the chamber…

The topology of symplectic 4-manifolds is related to that of singular plane curves via the concept of branched covers. Thus, various classification problems concerning symplectic 4-manifolds can be reformulated as questions about singular…

In this note we establish the existence of a new type of rigidity of symplectic embeddings coming from obligatory intersections with symplectic planes. More precisely, we prove that if a Euclidean ball is symplectically embedded in the…

A construction is introduced for modifying hyperkaehler manifolds with tri-Hamiltonian circle action, that in favourable situations increases the second Betti number by one. This is based on the symplectic cut construction of Lerman. In 4…

A symplectic manifold is called symplectic rationally connected if there is a non-zero genus zero Gromov-Witten invariant with two point insertions. It is conjectured that every smooth projective rationally connected variety is symplectic…

We write the Euler characteristic X(G) of a four dimensional finite simple geometric graph G=(V,E) in terms of the Euler characteristic X(G(w)) of two-dimensional geometric subgraphs G(w). The Euler curvature K(x) of a four dimensional…