2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/153909A toric polyhedron is a reduced closed subscheme of a toric variety that are partial unions of the orbits of the torus action. We prove vanishing theorems for toric polyhedra. We also give a proof of the $E_1$-degeneration of Hodge to de Rham type spectral sequence for toric polyhedra in any characteristic. Finally, we give a very powerful extension theorem for ample line bundles.15 pages; title changed, a completely revised and expanded version, v3: very minor modificationsAlgebraic Geometry14F17; 14F25text