Abstract

ALE spaces are the simply connected hyperkähler manifolds which at infinity look like \({\mathbb{C}^{2}/G}\), for any finite subgroup \({G \subset SL_2(\mathbb{C})}\). We prove that all exact Lagrangians inside ALE spaces must be spheres. The proof relies on showing the vanishing of a twisted version of symplectic cohomology.This application is a consequence of a general deformation technique. We construct the symplectic cohomology for non-exact symplectic manifolds, and we prove that if the non-exact symplectic form is sufficiently close to an exact one then the symplectic cohomology coincides with an appropriately twisted version of the symplectic cohomology for the exact form.

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