  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.
