Uniqueness of the direct decomposition of toric manifolds

TL;DR

This paper investigates the uniqueness of decomposing toric manifolds into indecomposable factors, proving uniqueness in low dimensions and for certain 4-manifolds with specific symmetries.

Contribution

It establishes the uniqueness of smooth manifold decompositions of toric manifolds in dimensions up to two and for certain 4-manifolds with circle actions.

Findings

Decomposition as algebraic varieties is unique up to order.

Unique decomposition as smooth manifolds holds in complex dimension ≤ 2.

Decomposition into  and specific 4-manifolds with ircle actions is unique.

Abstract

In this paper, we study the uniqueness of the direct decomposition of a toric manifold. We first observe that the direct decomposition of a toric manifold as \emph{algebraic varieties} is unique up to order of the factors. An algebraically indecomposable toric manifold happens to decompose as smooth manifold and no criterion is known for two toric manifolds to be diffeomorphic, so the unique decomposition problem for toric manifolds as \emph{smooth manifolds} is highly nontrivial and nothing seems known for the problem so far. We prove that this problem is affirmative if the complex dimension of each factor in the decomposition is less than or equal to two. A similar argument shows that the direct decomposition of a smooth manifold into copies of $\mathbb{C}P^1$ and simply connected closed smooth 4-manifolds with smooth actions of $(S^1)^2$ is unique up to order of the factors.