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 KoS2 where K < SL2(C), for which the corresponding quotient singularity admits a projective symplectic resolution. As a consequence, for dimV 6= 4, we...

where the Z action is generated by (x, s, θ) 7→ (f(x), s + 1, θ). In this paper we compute the Gromov invariants of the manifolds Xf and of fiber sums of the Xf with other symplectic manifolds. This is done by expressing the Gromov invariants in terms of the Lefschetz zeta function of f and, in special cases, in terms of the Alexander polynomials of knots. The result is a large set of interesti...

We study the rigidity and flexibility of symplectic embeddings of simple shapes. It is first proved that under the condition r n ≤ 2r 1 the symplectic ellipsoid E(r1, . . . , rn) with radii r1 ≤ · · · ≤ rn does not embed in a ball of radius strictly smaller than rn. We then use symplectic folding to see that this condition is sharp and to construct some nearly optimal embeddings of ellipsoids a...

Let M be a closed symplectic manifold, with symplectic form ω. A symplectic automorphism is a diffeomorphism φ : M → M such that φω = ω. We equip the group Aut(M) = Aut(M,ω) of all such maps with the C-topology. Like the whole diffeomorphism group, this is an infinite-dimensional Lie group in a very loose sense: it has a well-defined Lie algebra, which consists of closed oneforms on M , but the...

For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action i...

Let H(q,p) be a Hamiltonian on T * n . Under suitable assumptions H, we show that the sequence (H k ) k≥1 defined by H (q,p)=H(kq,p) converges in γ-topology—defined [Vit92]—to an integrable continuous ¯(p). This is extended to case of non-autonomous Hamiltonians, and more general setting which only some variables are homogenized: consider H(kx,y,q,p) prove it has γ-limit ¯(y,q,p), thus yielding...

We prove the logarithmic concavity of the Duistermaat-Heckman measure of an Hamiltonian (n− 2)-dimensional torus action for which there exists an effective commuting symplectic action of a 2-torus with symplectic orbits. Using this, we show that any symplectic (n− 2)-torus action with non-empty fixed point set which satisfies this additional 2-torus condition must be Hamiltonian.