# Core components of quiver varieties as fiber bundles of flag varieties

Is there an example of Nakajima quiver variety of type A which has all core components smooth, such that at least one of them is NOT an iterated fibre bundle of flag manifolds (i.e. a space obtained by a sequence of fibre bundles whose fibres and base are flag manifolds).

• NB I have slightly changed the question after Prof. Nakajima's observation. – Filip92 Feb 14 '19 at 11:15

There is an example of an irreducible component, which is a blowup of $$\mathbb P^2$$ at a point. See Example 18 in https://arxiv.org/pdf/1611.10000.pdf.
• Indeed, but blowup of $\mathbb{P}^2$ at a point is a Hirzebruch surface $\mathbb{F}_1$ hence a (non-trivial) $\mathbb{P}^1$ bundle over $\mathbb{P}^1.$ – Filip92 Feb 13 '19 at 10:37
• Oh, sorry. There are examples of blowup of $\mathbb P^2$ at three points in E6 constructed in similar way as this example, as far as I remember. I guess, you could also find them by running my computer program in arxiv.org/pdf/math/0606637.pdf. – Hiraku Nakajima Feb 13 '19 at 13:44