Toric Manifolds over 3-Polytopes

In this note, we gather and review several facts about the existence of toric spaces over three-dimensional simple polytopes. First, over every combinatorial simple 3-polytope, there exists a quasitoric manifold. Second, there exist combinatorial 3-polytopes that do not correspond to any smooth projective toric variety. We give the proof of the second claim, which does not refer to complicated algebro-geometrical technique. It follows from these results that any fullerene supports quasitoric manifolds but does not support smooth projective toric varieties.