2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66985We define generalizations of the Albanese variety for a projective variety X. The generalized Albanese morphisms X --> Alb_r(X) contract those curves C in X for which the induced morphism Hom(π_1(X),U(r)) --> Hom(π_1(C),U(r)) has a finite image. Thus, they may be interpreted as a U(r)-version of the Shafarevich morphism.15 pages, LaTeX, major revision, the paper has been completely rewritten and greatly expanded compared to the previous versionAlgebraic Geometry14D20; 14J10text