Smoothings of singularities and symplectic surgery

TL;DR

This paper demonstrates how replacing neighborhoods of certain symplectic submanifold configurations with smoothings of surface singularities yields new symplectic 4-manifolds, extending rational blow-down techniques.

Contribution

It introduces a generalized symplectic surgery method replacing neighborhoods with smoothings of surface singularities, broadening the scope of symplectic rational blow-downs.

Findings

Constructs symplectic structures on new 4-manifolds after surgery.

Extends rational blow-down operation to more configurations.

Provides a method to produce symplectic manifolds with prescribed topology.

Abstract

Suppose that $C=(C_1,..., C_m)$ is a configuration of 2-dimensional symplectic submanifolds in a symplectic 4-manifold $(X,\omega)$ with connected, negative definite intersection graph $\Gamma_C$. We show that by replacing an appropriate neighborhood of $\cup C_i$ with a smoothing $W_S$ of a normal surface singularity $(S, 0)$ with resolution graph $\Gamma_C$, the resulting 4-manifold admits a symplectic structure. This operation generalizes the rational blow-down operation of Fintushel-Stern for other configurations, and therefore extends Symington's result about symplectic rational blow-downs.