No Arabic abstract
In this paper we study symplectic embedding questions for the $ell_p$-sum of two discs in ${mathbb R}^4$, when $1 leq p leq infty$. In particular, we compute the symplectic inner and outer radii in these cases, and show how different kinds of embedding rigidity and flexibility phenomena appear as a function of the parameter $p$.
Given two $2n$--dimensional symplectic ellipsoids whose symplectic sizes satisfy certain inequalities, we show that a certain map from the $n$--torus to the space of symplectic embeddings from one ellipsoid to the other induces an injective map on singular homology with mod $2$ coefficients. The proof uses parametrized moduli spaces of $J$--holomorphic cylinders in completed symplectic cobordisms.
A Kahler-type form is a symplectic form compatible with an integrable complex structure. Let M be a either a torus or a K3-surface equipped with a Kahler-type form. We show that the homology class of any Maslov-zero Lagrangian torus in M has to be non-zero and primitive. This extends previous results of Abouzaid-Smith (for tori) and Sheridan-Smith (for K3-surfaces) who proved it for particular Kahler-type forms on M. In the K3 case our proof uses dynamical properties of the action of the diffeomorphism group of M on the space of the Kahler-type forms. These properties are obtained using Shahs arithmetic version of Ratners orbit closure theorem.
We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.
We study the existence of symplectic resolutions of quotient singularities V/G where V is a symplectic vector space and G acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the form $K rtimes S_2$ where $K < SL_2(C)$, for which the corresponding quotient singularity admits a projective symplectic resolution. As a consequence, for $dim V eq 4$, we classify all quotient singularities $V/G$ admitting a projective symplectic resolution which do not decompose as a product of smaller-dimensional quotient singularities, except for at most four explicit singularities, that occur in dimensions at most 10, for whom the question of existence remains open.
We prove a gluing theorem for a symplectic vortex on a compact complex curve and a collection of holomorphic sphere bubbles. Using the theorem we show that the moduli space of regular stable symplectic vortices on a fixed curve with varying markings has the structure of a stratified-smooth topological orbifold. In addition, we show that the moduli space has a non-canonical $C^1$-orbifold structure.