## Osaka Journal of Mathematics

- Osaka J. Math.
- Volume 52, Number 2 (2015), 439-453.

### Uniqueness of the direct decomposition of toric manifolds

#### 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 *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
*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.

#### Article information

**Source**

Osaka J. Math., Volume 52, Number 2 (2015), 439-453.

**Dates**

First available in Project Euclid: 24 March 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.ojm/1427202896

**Mathematical Reviews number (MathSciNet)**

MR3326620

**Zentralblatt MATH identifier**

1323.57016

**Subjects**

Primary: 55N40: Axioms for homology theory and uniqueness theorems

Secondary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]

#### Citation

Hatanaka, Miho. Uniqueness of the direct decomposition of toric manifolds. Osaka J. Math. 52 (2015), no. 2, 439--453. https://projecteuclid.org/euclid.ojm/1427202896