2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/132642We describe the positive cone and the pseudo-effective cone of a non-Kählerian surface. We use these results for two types of applications: - Describe the set $σ(X)$ of possible total Ricci scalars associated with Gauduchon metrics of fixed volume 1 on a fixed non-Kähhlerian surface, and decide whether the assignment $X\mapstoσ(X)$ is a deformation invariant. - Study the stability of the canonical extension $$0\to {\cal K}_X\to {\cal A}\to{\cal O}_X\to 0$$ of a class VII surface $X$ with positive $b_2$. This extension plays an important role in our strategy to prove the GSS conjecture using gauge theoretical methods.LaTeX, 25 pages; rv: minor correction in the proof of Remark 4.2Complex VariablesAlgebraic GeometryDifferential Geometry32J15; 32Q57; 32L05; 32G13text