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We say that a complete nonsingular toric variety (called a toric manifold inthis 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 toricmanifolds over an $n$-cube $I^n$ and blowing them up at a fixed point producestoric manifolds over $\mathrm{vc}(I^n)$ an $n$-cube with one vertex cut. Theyare all projective. On the other hand, Oda's $3$-fold, the simplestnon-projective toric manifold, is over $\mathrm{vc}(I^n)$. In this paper, weclassify toric manifolds over $\mathrm{vc}(I^n)$ $(n\ge 3)$ as varieties andalso as smooth manifolds. As a consequence, it turns out that (1) there aremany non-projective toric manifolds over $\mathrm{vc}(I^n)$ but they are alldiffeomorphic, and (2) toric manifolds over $\mathrm{vc}(I^n)$ in some classare determined by their cohomology rings as varieties among toric manifolds.

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