2017/05/22 by Sho Hasui, Hasui, Sho, Hideya Kuwata +5 · 1 citation
Mathematics · #14M25 #55N10 #57S15 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1705.07530
openalex publication_date 2017/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 In and blowing them up at a fixed point produces toric manifolds over vc(In) 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 vc(In). In this paper, we classify toric manifolds over vc(In) (n≥ 3) as varieties and also as smooth manifolds. As a consequence, it turns out that (1) there are many non-projective toric manifolds over vc(In) but they are all diffeomorphic, and (2) toric manifolds over vc(In) in some class are determined by their cohomology rings as varieties among toric manifolds.