2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/131488We prove that any compact complex surface with positive first Chern class admits an Einstein metric which is conformally related to a Kaehler metric. The key new ingredient is the existence of such a metric on the blow-up of the complex projective plane at two distinct points.45 pages, 9 figures. Simplified proof of Lemma 25. Several minor corrections to section 5Differential GeometryGeneral Relativity and Quantum CosmologyAlgebraic GeometryAnalysis of PDEs53C25; 53C55text