2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62986We extend the Cone Theorem of the Log Minimal Model Program to log varieties with arbitrary singularities.LaTex, 24 pages; title and terminology has been changed; new adjunction statement (4.9). to appear in Proc. of Steklov Institute, vol 240Algebraic Geometrytext