2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/152243We show that there are high-dimensional smooth compact manifolds which admit pairs of Einstein metrics for which the scalar curvatures have opposite signs. These are counter-examples to a conjecture considered by Besse. The proof hinges on showing that the Barlow surface has small deformations with ample canonical line bundle.LaTeX. 14 pagesDifferential GeometryAlgebraic Geometrytext