I was wondering whether $(\mathbb{CP}^n, g_{FS})$, where $g_{FS}$ denotes the Fubini-Study metric, is holonomy irreducible, that is, whether or not the tangent space can be split up into subspaces invariant under the holonomy action. I computed that the holonomy group is $U(n)$. My line of thinking was the following:
If $(\mathbb{CP}^n, g_{FS})$ would not be irreducible, then the de Rham Theorem would imply (because $(\mathbb{CP}^n, g_{FS})$ is simply connected and complete) that $(\mathbb{CP}^n, g_{FS})$ is globally a product, which in turn implies that its holonomy group is a product.
So there are two things that might go wrong: Either $(\mathbb{CP}^n, g_{FS})$ can not be a product manifold, that is, either $\mathbb{CP}^n$ can not even topologically be or the metric $g_{FS}$ can not split; or $U(n)$ can not be split into a product.
But $U(n)$ is reducible, so the latter might be possible. What I am left with is the first option and I have currently no idea how to show that that's not possible. So my question is:
Why can the Fubini-Study metric $g_{FS}$ not be realized as a product metric, $h_1 + h_2 = g$? Or: Why can $\mathbb{C}\mathbb{P}^n$ not be written as a product?
