  We prove that all minimal symplectic four-manifolds are essentially
irreducible. We also clarify the relationship between holomorphic and
symplectic minimality of K\"ahler 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.

Symplectic Geometry;Algebraic Geometry;Geometric Topology