2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/102247We 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 NoticesSymplectic GeometryAlgebraic GeometryGeometric Topology57R17; 57R57; 32J15text