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

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.