Symplectic surgeries and normal surface singularities

TL;DR

This paper proves that negative definite symplectic surface configurations in 4-manifolds can be symplectically replaced by smoothings of complex surface singularities, extending the rational blowdown technique for constructing exotic 4-manifolds.

Contribution

It introduces a method to replace certain symplectic surface configurations with smoothings of singularities, broadening the toolkit for 4-manifold topology.

Findings

Every negative definite symplectic surface configuration has a strongly convex neighborhood.

Configurations satisfying an additional negativity condition can be replaced by the smoothing of the corresponding singularity.

Generalizes the symplectic rational blowdown procedure for constructing exotic 4-manifolds.

Abstract

We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex singularity with resolution diffeomorphic to a neighborhood of the configuration has a smoothing, then the configuration can be symplectically replaced by the smoothing of the singularity. This generalizes the symplectic rational blowdown procedure used in recent constructions of small exotic 4--manifolds.