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 classes of manifolds having well-behaved torus actions, say toric objects, can be classified in terms of combinatorial data containing simplicial complexes.

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 $K$. Our result provides a systematic way to classify toric objects associated with the class of simplicial complexes obtained from a given $K$ by wedge operations. As applications, we completely classify smooth toric varieties with a few generators and show their projectivity. We also study smooth real toric varieties.