Lagrangian blow-ups, blow-downs, and applications to real packing

Abstract

Given a symplectic manifold (M, ω) and a Lagrangian submanifold L, we construct versions of the symplectic blow-up and blow-down which are defined relative to L. Furthermore, we show that if M admits an anti-symplectic involution ϕ and we blow-up an appropriately symmetric embedding of symplectic balls, then there exists an anti-symplectic involution on the blow-up as well. We derive a homological condition which determines when the topology of a real Lagrangian surface L = Fix(ϕ) changes after a blow down, and we use these constructions to study the real packing numbers and packing stability for real, rank-1 symplectic four manifolds which are non-Seiberg-Witten simple.

66 pages, 1 figure

References in corpus (2)

• Rigidity and uniruling for Lagrangian submanifolds
• The homotopy type of the space of symplectic balls in rational ruled 4-manifolds

Cited by in corpus (5)

• Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds
• Coisotropic Hofer-Zehnder capacities and non-squeezing for relative embeddings
• Lagrangian Circle Actions
• Coisotropic Hofer-Zehnder capacities of convex domains and related results
• Non-orientable Lagrangian surfaces in rational 4-manifolds