Uniqueness of the direct decomposition of toric manifolds (Q2348673)

A toric variety is a normal algebraic variety of complex dimension \(n\) with a complex torus action having an open dense orbit. In general, a toric variety does not need to be compact or nonsingular. A compact nonsingular toric variety is called a toric manifold. A toric manifold that does not decompose into the product of two toric manifolds of positive dimension as algebraic varieties (resp. as smooth manifolds) is called algebraically indecomposable (resp. differentially indecomposable). The author studies the uniqueness of the direct decomposition of a toric manifold. He proves that the direct decomposition of a toric manifold into algebraically indecomposable toric manifolds as algebraic varieties is unique up to the order of the factors. He also proves that the direct decomposition of a toric manifold into differentially indecomposable toric manifolds is unique up to the order of the factors assuming the complex dimension of each factor is less or equal to two. He also uses a similar argument to show that the direct decomposition of a smooth manifold into copies of \({\mathbb{C}}P^{1}\) and simply connected closed smooth \(4\)-manifolds with smooth torus actions is unique up to the order of factors.