## Osaka Journal of Mathematics

### Uniqueness of the direct decomposition of toric manifolds

Miho Hatanaka

#### 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

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

Mathematical Reviews number (MathSciNet)
MR3326620

Zentralblatt MATH identifier
1323.57016

#### 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

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