Disjoinable Lagrangian spheres and dilations

TL;DR

This paper explores restrictions on disjoint Lagrangian submanifolds in open symplectic manifolds with dilations, including stabilized Milnor fibers, revealing new geometric constraints in symplectic topology.

Contribution

It introduces new restrictions on disjoint Lagrangian submanifolds in symplectic manifolds with dilations, especially in stabilized Milnor fibers.

Findings

Restrictions on disjoint Lagrangian submanifolds in dilated symplectic manifolds

Application to stabilized Milnor fibers of hypersurface singularities

New geometric constraints in symplectic topology

Abstract

We consider open symplectic manifolds which admit dilations (in the sense previously introduced by Solomon and the author). We obtain restrictions on collections of Lagrangian submanifolds which are pairwise disjoint (or pairwise disjoinable by Hamiltonian isotopies) inside such manifolds. This includes the Milnor fibres of isolated hypersurface singularities which have been stabilized (by adding quadratic terms) sufficiently often.