Wedge operations and torus symmetries

Abstract

A fundamental result of toric geometry is that there is a bijection between toric varieties and fans. More generally, it is known that some class of manifolds having well-behaved torus actions, called topological toric manifolds $M^{2n}$, can be classified in terms of combinatorial data containing simplicial complexes with $m$ vertices. We remark that topological toric manifolds are a generalization of smooth toric varieties. The number $m-n$ is known as the Picard number when $M^{2n}$ is a {compact smooth} toric variety. In this paper, we investigate the relationship between the topological toric manifolds over a simplicial complex $K$ and those over the complex obtained by simplicial wedge operations from $K$. As applications, we do the following. 1. We classify smooth toric varieties of Picard number 3. This is a reproving of a result of Batyrev. 2. We give a new and complete proof of projectivity of smooth toric varieties of Picard number 3 originally proved by Kleinschmidt and Sturmfels. 3. We find a criterion for a toric variety over the join of boundaries of simplices to be projective. When the toric variety is smooth, it is known as a generalized Bott manifold which is always projective. 4. We classify and enumerate real topological toric manifolds when $m-n=3$. 5. When $m-n \leq 3$, any real topological toric manifold is realizable as fixed points of the conjugation of a topological toric manifold.