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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).

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

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  • $\begingroup$ 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 $\mathbb{P}^1.$ $\endgroup$ – Filip92 Feb 13 '19 at 10:37
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    $\begingroup$ 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. $\endgroup$ – Hiraku Nakajima Feb 13 '19 at 13:44
  • $\begingroup$ I see, many thanks! So, I guess I should really think of this statement as a feature of type A quiver varieties? $\endgroup$ – Filip92 Feb 13 '19 at 22:36
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    $\begingroup$ The condition that all core components are smooth sounds very restrictive. The corresponding result for Springer fibers in type A is link.springer.com/article/10.1007%2Fs00029-010-0025-z. I do not know how much other examples one get from Spaltenstein varieties = core of type A quiver varieties. I did not study other components in the above E6 example. So I do not know whether it satisfies the condition or not. $\endgroup$ – Hiraku Nakajima Feb 15 '19 at 0:48

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