Classification of toric manifolds over an $n$-cube with one vertex cut

We say that a complete nonsingular toric variety (called a toric manifold in this paper) is over $P$ if its quotient by the compact torus is homeomorphic to $P$ as a manifold with corners. Bott manifolds (or Bott towers) are toric manifolds over an $n$-cube $I^n$ and blowing them up at a fixed point produces toric manifolds over $\mathrm{vc}(I^n)$ an $n$-cube with one vertex cut. They are all projective. On the other hand, Oda's $3$-fold, the simplest non-projective toric manifold, is over $\mathrm{vc}(I^n)$. In this paper, we classify toric manifolds over $\mathrm{vc}(I^n)$ $(n\ge 3)$ as varieties and also as smooth manifolds. As a consequence, it turns out that (1) there are many non-projective toric manifolds over $\mathrm{vc}(I^n)$ but they are all diffeomorphic, and (2) toric manifolds over $\mathrm{vc}(I^n)$ in some class are determined by their cohomology rings as varieties among toric manifolds.