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

Abstract

Given a symplectic manifold $(M,\omega)$ and a Lagrangian submanifold $L$, we construct versions of the symplectic blow-up and blow-down which are defined relative to $L$. We further show that if $M$ admits an anti-symplectic involution $\phi$, i.e., a diffeomorphism such that $\phi^2 = \mathrm{Id}$ and $\phi^* \omega = - \omega$, and we blow-up an appropriately symmetric embedding of symplectic balls, then there exists an antisymplectic involution on the blow-up $\tilde{M}$ as well. We then derive a homological condition for real Lagrangian surfaces $L = \mathrm{Fix} (\phi)$ which determines when the topology of $L$ changes after a blowdown, and we use these constructions to study the relative packing numbers and packing stability for real symplectic four manifolds which are non-Seiberg-Witten simple.