Bott manifolds of Bott--Samelson type and assemblies of ordered partitions

Abstract

A Bott manifold is a smooth projective toric variety having an iterated $\mathbb{C} P^1$-bundle structure. A certain family of Bott manifolds is used to understand the structure of Bott--Samelson varieties (or Bott--Samelson--Demazure--Hansen varieties), which provide desingularizations of Schubert varieties. Indeed, each Bott--Samelson variety is diffeomorphic to a Bott manifold. However, not all Bott manifolds originate from Bott--Samelson varieties. Those that do are specifically referred to as Bott manifolds of Bott--Samelson type. In this paper, we provide a characterization of Bott manifolds of Bott--Samelson type by exploring their relationship with combinatorial objects called assemblies of ordered partitions. Using this relationship, we enumerate Bott manifolds of Bott--Samelson type and describe isomorphic Bott manifolds of Bott--Samelson type in terms of assemblies of ordered partitions.