Deformations of symplectic cohomology and exact Lagrangians in ALE spaces

TL;DR

This paper proves that in ALE spaces, the only exact Lagrangian submanifolds are spheres, using symplectic cohomology to analyze the deformation properties of these spaces.

Contribution

It establishes a classification result for exact Lagrangians in ALE spaces, showing they are exclusively spheres, and introduces methods involving symplectic cohomology.

Findings

Exact Lagrangians in ALE spaces are only spheres.

Symplectic cohomology is used to analyze deformations.

ALE spaces are constructed via plumbing according to ADE diagrams.

Abstract

We prove that the only exact Lagrangian submanifolds in an ALE space are spheres. ALE spaces are the simply connected hyperkahler manifolds which at infinity look like C^2/G for any finite subgroup G of SL(2,C). They can be realized as the plumbing of copies of the cotangent bundle of a 2-sphere according to ADE Dynkin diagrams. The proof relies on symplectic cohomology.