Minimality and irreducibility of symplectic four-manifolds

Abstract

We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of Kähler surfaces. This leads to a new proof of the deformation-invariance of holomorphic minimality for complex surfaces with even first Betti number which are not Hirzebruch surfaces.

final version; cosmetic changes only; to appear in International Mathematics Research Notices