Related papers: Lagrangian spheres, symplectic surfaces and the sy…

We introduce a streamlined procedure for constructing small symplectic $4$-manifolds via contact gluing, based on a technique invented by David Gay around 2000. We give several applications of this procedure, which produced results…

We show that the space of anti-symplectic involutions of a monotone $S^2\times S^2$ whose fixed points set is a Lagrangian sphere is connected. This follows from a stronger result, namely that any two anti-symplectic involutions in that…

In this article we construct a minimal symplectic 4-manifold R that has small Euler characteristic (e(R)=8) and two essential Lagrangian tori with nice properties. These properties make R particularly suitable for constructing interesting…

A surface $\Sigma \subset S^5 \subset \mathbb{C}^3$ is called \emph{special Legendrian} if the cone $0 \times \Sigma \subset \mathbb{C}^3$ is special Lagrangian. The purpose of this paper is to propose a general method toward constructing…

The following interesting quantity was introduced by K. Cieliebak and K. Mohnke for a Lagrangian submanifold $L$ of a symplectic manifold: the minimal positive symplectic area of a disc with boundary on $L$. They also showed that this…

Consider the complex linear space C^n endowed with the canonical pseudo-Hermitian form of signature (2p,2(n-p)). This yields both a pseudo-Riemannian and a symplectic structure on C^n. We prove that those submanifolds which are both…

A classical way to construct a Lagrangian in a symplectic manifold $\Sigma$ is to let $\Sigma$ appear as a smooth fiber in a Lefschetz fibration. If this is possible the singularities of the fibration induce Lagrangian spheres in $\Sigma$…

This paper develops a rigorous mathematical framework for light propagation by constructing the optical phase space with its symplectic structure and the extended phase space with its contact structure. We prove that light rays in…

Moduli spaces of stably irreducible sheaves on Kodaira surfaces belong to the short list of examples of smooth and compact holomorphic symplectic manifolds, and it is not yet known how they fit into the classification of holomorphic…

Let $(M,\omega)$ be a symplectic 4-manifold of negative Kodaira dimension. Let $C$ be an $\omega$-symplectic curve, $J$-holomorphic for some $J$ tamed by $\omega$. Then $[C]^2$ is bounded below by a constant depending only on $\omega$.…

We call a symplectic rational surface $(X,\omega)$ \textit{positive} if $c_1(X)\cdot[\omega]>0$. The positivity condition of a rational surface is equivalent to the existence of a divisor $D\subset X$, such that $(X, D)$ is a log Calabi-Yau…

We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg-Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries…

Hamiltonian symplectic actions of tori on compact symplectic manifolds have been extensively studied in the past thirty years, and a number of classifications have been achieved, for instance in the case that the acting torus is…

In this paper we introduce the notion of relative Kodaira dimension for a symplectic 4-manifold M with a possibly disconnected embedded symplectic surface F. One notable feature is that the sphere components of F have to be discarded, which…

In this article we use the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds. In particular, we construct: (1) An example of a minimal symplectic 4-manifold…

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to…

We use spectral invariants in Lagrangian Floer theory in order to show that there exist \emph{isometric} embeddings of normed linear spaces (finite or infinite dimensional, depending on the case) into the space of Hamiltonian deformations…

Let $(X, \omega)$ be a compact symplectic manifold and $L$ be a Lagrangian submanifold. Suppose $(X, L)$ has a Hamiltonian $S^1$ action with moment map $\mu$. Take an invariant $\omega$-compatible almost complex structure, we consider…

We examine symplectic topological features of certain family of monotone Lagrangian submanifolds in CP^n. Firstly, we give a cohomological restriction for Lagrangian submanifolds in CP^n whose first integral homologies are 3-torsion. In…

We consider smoothings of a complex surface with singularities of class T and no nontrivial holomorphic vector field. Under an hypothesis of non degeneracy of the smoothing at each singular point, we prove that if the singular surface…