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

TL;DR

This paper develops symplectic blow-up and blow-down techniques relative to Lagrangian submanifolds, explores anti-symplectic involutions, and applies these methods to analyze real packing problems in symplectic four-manifolds.

Contribution

It introduces new constructions of symplectic blow-ups and blow-downs relative to Lagrangian submanifolds and studies their effects on real structures and packing properties.

Findings

Constructed symplectic blow-up and blow-down relative to Lagrangians.

Established conditions for anti-symplectic involutions on blow-ups.

Analyzed real packing numbers and stability in specific symplectic four-manifolds.

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. Furthermore, we show that if M admits an anti-symplectic involution {\phi} 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({\phi}) 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.