Symplectic Rational Blow-up

Abstract

Fintushel and Stern defined the rational blow-down construction [FS] for smooth 4-manifolds, where a linear plumbing configuration of spheres $C_n$ is replaced with a rational homology ball $B_n$, $n \geq 2$. Subsequently, Symington [Sy] defined this procedure in the symplectic category, where a symplectic $C_n$ (given by symplectic spheres) is replaced by a symplectic copy of $B_n$ to yield a new symplectic manifold. As a result, a symplectic rational blow-down can be performed on a manifold whenever such a configuration of symplectic spheres can be found. In this paper, we define the inverse procedure, the rational blow-up in the symplectic category, where we present the symplectic structure of $B_n$ as an entirely standard symplectic neighborhood of a certain Lagrangian 2-cell complex. Consequently, a symplectic rational blow-up can be performed on a manifold whenever such a Lagrangian 2-cell complex is found.