Deformations of symplectic cohomology and exact Lagrangians in ALE spaces

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.

35 pages, 3 figures, minor changes and corrected typos

References in corpus (3)

• Functors and Computations in Floer homology with Applications Part II
• Novikov-symplectic cohomology and exact Lagrangian embeddings
• Lagrangian isotopies in Stein manifolds

Cited by in corpus (22)

• The monotone wrapped Fukaya category and the open-closed string map
• Floer theory for negative line bundles via Gromov-Witten invariants
• Floer theory and reduced cohomology on open manifolds
• From symplectic cohomology to Lagrangian enumerative geometry
• Subflexible symplectic manifolds
• Lagrangian homology spheres in (A_m) Milnor fibres via C^*-equivariant A_infinity modules
• Continuation homomorphism in Rabinowitz Floer homology for symplectic deformations
• Local systems on the free loop space and finiteness of the Hofer-Zehnder capacity
• Exact Lagrangians in plumbings
• Distinguishing between exotic symplectic structures
• Homotopy type of the unitary group of the uniform Roe algebra on ${\mathbb{Z}}^n$
• Two closed orbits for non-degenerate Reeb flows
• Invariance of symplectic cohomology and twisted cotangent bundles over surfaces
• Quantum Steenrod squares and the equivariant pair-of-pants in symplectic cohomology
• Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles
• $A_n$-type surface singularity and nondisplaceable Lagrangian tori
• Floer cohomology and pencils of quadrics
• Disjoinable Lagrangian tori and semisimple symplectic cohomology
• Exact Lagrangians in $A_n$-surface singularities
• Reeb Dynamics of the Link of the $A_n$ Singularity
• Index bounded relative symplectic cohomology
• Nonexistence of exact Lagrangian tori in affine conic bundles over ${\mathbb{C}}^n$