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

Junho Jeong, Jang Soo Kim, Eunjeong Lee

TL;DR

The paper addresses the problem of characterizing Bott manifolds that arise from Bott--Samelson varieties (BS type) and provides a complete combinatorial framework for their enumeration and toric isomorphism classification. It introduces assemblies of ordered partitions (AOP) and two explicit maps that connect sequences in $[n]^m$ to Bott matrices of BS type, yielding a bijection between BS-type Bott manifolds and ${\mathsf{AOP}}(n,m)/\sim$, and a parallel isomorphism classification via ${\mathsf{AOP}}(n,m)/\approx$. A key contribution is the explicit generating function for the counts $b(n,m)$ and the demonstration that Bott matrices of BS type correspond to combinatorial equivalence classes, enabling exact enumeration across dimensions. The results also illuminate the relationship to toric Schubert varieties in type $A$, via indecomposable BS-type manifolds and reductions to standard toric classifications, thereby linking geometry, topology, and combinatorics in a concrete, computable framework.

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.

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

TL;DR

The paper addresses the problem of characterizing Bott manifolds that arise from Bott--Samelson varieties (BS type) and provides a complete combinatorial framework for their enumeration and toric isomorphism classification. It introduces assemblies of ordered partitions (AOP) and two explicit maps that connect sequences in to Bott matrices of BS type, yielding a bijection between BS-type Bott manifolds and , and a parallel isomorphism classification via . A key contribution is the explicit generating function for the counts and the demonstration that Bott matrices of BS type correspond to combinatorial equivalence classes, enabling exact enumeration across dimensions. The results also illuminate the relationship to toric Schubert varieties in type , via indecomposable BS-type manifolds and reductions to standard toric classifications, thereby linking geometry, topology, and combinatorics in a concrete, computable framework.

Abstract

A Bott manifold is a smooth projective toric variety having an iterated -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.

Paper Structure

This paper contains 5 sections, 16 theorems, 49 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Bott manifolds of Bott--Samelson type
  3. Assemblies of ordered partitions
  4. Enumeration of Bott manifolds of Bott--Samelson type
  5. Isomorphism classes of Bott manifolds of Bott--Samelson type

Key Result

Theorem 1.1

Let $n$ and $m$ be positive integers. There is a bijection between the set of Bott manifolds of BS type and the set of equivalence classes of $\mathsf{AOP}(n,m)$ under the relation $\sim$:

Figures (1)

  • Figure 1: The fan of a Bott manifold $\mathscr{B}_2$ for $b_{2,1} = -2$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1: GK94Bott
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • ...and 35 more