| Given a symplectic manifold (M, o) and a Lagrangian submanifold L, we construct versions of the symplectic blow-up and blow-down which are defined relative to L. Furthermore, if M admits an anti-symplectic involution &phis;, i.e. a diffeomorphism such that &phis;² = Id and &phis;*o = --o, and we blow-up an appropriately symmetric configuration of symplectic balls, then we show that there exists an antisymplectic involution on the blow-up M˜ as well. We derive a homological condition for real Lagrangian surfaces L = Fix(&phis;) which determines when the topology of L changes after a blow down, and we then use these constructions to study the real packing numbers for real Lagrangian submanifolds in ( C P², R P²).;Keywords: Symplectic, four-manifold, Lagrangian submanifold, packing, relative packing, anti-symplectic involution, real manifold. |
