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

Classification of Toric Manifolds over an n -Cube with One Vertex Cut