Minimality of Symplectic Fiber Sums along Spheres

TL;DR

This paper investigates the conditions under which symplectic fiber sums along spheres are minimal, extending previous work and identifying a specific case involving rational blow-downs that affects minimality.

Contribution

It completes the classification of minimality for symplectic fiber sums along spheres, adding a new case related to rational blow-downs of -4-spheres.

Findings

Minimality is determined by previously discussed cases and a new case involving rational blow-downs.

Non-minimality occurs when a certain configuration of exceptional spheres exists after the blow-down.

The paper clarifies the conditions under which symplectic sums along spheres are minimal or not.

Abstract

In this note we complete the discussion of minimality of symplectic fiber sums. We find, that for fiber sums along spheres the minimality of the sum is determined by the cases discussed by M. Usher and one additional case: If the sum is the result of the rational blow-down of a symplectic -4-sphere in X, then it is non-minimal if X contains a certain configuration of exceptional spheres in relation to this -4-sphere.