Deformations of Symplectic Cohomology and Ade Singularities

Let X be the plumbing of copies of the cotangent bundle of a 2−sphere as prescribed by an ADE Dynkin diagram. We prove that the only exact Lagrangian submanifolds in X are spheres. Our approach involves studying X as an ALE hyperkähler manifold and observing that the symplectic cohomology of X will vanish if we deform the exact symplectic form to a generic non-exact one. We will construct the symplectic cohomology for non-exact symplectic manifolds with contact type boundary, and we will prove a general deformation theorem: if the non-exact symplectic form is sufficiently close to an exact one then the non-exact symplectic cohomology coincides with the natural Novikov symplectic cohomology for the exact form.