Some Examples in Toric Geometry

We present two examples in toric geometry concerning the relationship between smooth toric varieties and quasitoric manifolds (or more generally unitary torus manifolds), and extend the results of [8] to prove the non-existence of almost complex quasitorics over the duals of some certain cyclic 4-polytopes. We also provide the sufficient conditions on the base polytope and the characteristic map so that the resulting quasitoric manifold is almost complex, answering the question proposed by Davis & Januszkiewicz [5] .