2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/121816We provide sufficient conditions assuring that a suitably decorated 2-polyhedron can be thickened to a compact 4-dimensional Stein domain. We also study a class of flat polyhedra in 4-manifolds and find conditions assuring that they admit Stein, compact neighborhoods. We base our calculations on Turaev's shadows suitably "smoothed"; the conditions we find are purely algebraic and combinatorial. Applying our results, we provide examples of hyperbolic 3-manifolds admitting "many" positive and negative Stein fillable contact structures, and prove a 4-dimensional analogue of Oertel's result on incompressibility of surfaces carried by branched polyhedra.23 pages, 12 figures; published in Geometria Dedicatae (2006)Complex VariablesGeometric Topology32Q28; 57M50text