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Are deformations of Nakajima quiver varieties also Nakajima quiver varieties ?

In case the answer to this is (don't k)no(w), here are some simpler things to ask for.

  1. (If you're a differential geometer) Is any hyperkahler rotation / twistor deformation of a Nakajima quiver variety also a Nakajima quiver variety ?

  2. (An example, for algebraic geometers) Consider the Hilbert scheme $H$ of $k$ points on the minimal resolution of $\mathbb C^2/(\mathbb Z/n)$. Or restrict to $n=2=k$ and consider $Hilb^2T^*\mathbb P^1$. Its exceptional divisor over the symmetric product defines a class in $H^1(\Omega_H)$ (despite the noncompactness). Using the holomorphic symplectic form, we get a deformation class in $H^1(T_H)$. (The corresponding deformation is not so far from the twistor deformation, and can be realised as a composition of a deformation of the ALE space followed by a twistor deformation.) Is there a Nakajima quiver variety in the direction of this deformation ?

For instance if I take the quiver $\bullet^{\ \rightrightarrows}_{\ \leftleftarrows}\bullet$, dimension vector (1,1), and an appropriate stability condition (or value of the real moment map) then I get $T^*\mathbb P^1$ as the moduli space over the value $0$ of the complex moment map, and the smoothing of the surface ordinary double point over nonzero values.

Now if I take dimension vector (2,2) I can presumably get $Hilb^2$ of these surfaces, for an appropriate stability condition. However, as I vary the value of the complex moment map I simply vary the surface that I take $Hilb^2$ of, rather than getting the deformation I'm after. (The hyperkahler rotation is not a $Hilb^2$, since the exceptional divisor disappears in this deformation.)

But is there another quivery way of producing this deformation ?

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I hope you don't mind that I've made your title slightly more descriptive. – Ben Webster Jul 16 '10 at 0:40
up vote 11 down vote accepted

I do not know a general statement. I just want to give a comment:

Now if I take dimension vector $(2,2)$ I can presumably get $Hilb^2$ of these surfaces, for an appropriate stability condition.

No. You only get the symmetric product of $T^*P^1$ if you work on quiver varieties with the dimension vector $(2,2)$.

To get a $Hilb^2$ of the surface, one need to put the one-dimensional vector space $W $at the vertex 0, and take a suitable stability condition. (I hope you are familiar with convention for quiver varieties.)

Then we have two dimensional family of quiver varieties from the complex moment map deformation. Thus we get one more dimension from the deformation of the underlying surface.

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Many thanks Hiraku, I think this will resolve all my problems. By the way, I hereby start a campaign to persuade you to go through the agonies of writing another book. I think an introduction to your quiver varieties, with lots of simple examples, would be extremely popular. Perhaps other readers could express their enthusiasm, to pile on unreasonable pressure. – Richard Thomas Jul 16 '10 at 15:29
Thanks Richard. I am writing a CBMS lecture series, which unfortunately contain only few examples of quiver varieties. (The book is representation theory oriented.) I will give more examples in a next book. – Hiraku Nakajima Jul 21 '10 at 5:57

There is one sense in which all deformations of Nakajima quiver varieties are at least morally again quiver varieties. Let me not try to be precise since I will screw up if I do that.

First, note that any quiver variety $X=\mu^{-1}_{\mathbb {C}}(0)//_{\alpha}G$ written as a hyperkaehler reduction of a vector space $T^*E$ by a group $G$ has a natural deformation over $\mathfrak{g}^*$ given by $T^*E//_{\alpha}G$; the fibers are the reductions at different complex moment map levels.

There is a theorem of Kaledin and Verbitsky which implies that in any smooth symplectic family $\tilde X\to S$, (one equipped with a non-degenerate element of $\Omega^2(\tilde{X}/S)$) analytically locally at any closed point $s$ where the fiber is a Nakajima quiver variety "looks like varying the moment map parameter." Formally, this means there's a coarse moduli space for formal deformations of the Nakajima quiver variety given by $H^2(X;\mathbb{C})$ (the map is just taking the cohomology class of the symplectic form). Taking the symplectic form of the different reductions makes $H^2(X,\mathbb{C})$ a quotient of $\mathfrak{g}^*$ by Kirwan surjectivity (proven by Harada and Wilkin) so all formal deformations come from this one (in a coarse, not a fine sense, I think).

I'm not really sure how "global" this can be made. Perhaps someone who understands the deformation theory of non-compact varieties can say more.

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One more example:

Take the quiver of type $A_1$, and vector spaces $V$, $W$ with $\dim V > \dim W$. Then the quiver variety is empty regardless of (generic) stability parematers nor complex moment map parameters. But we have one dimensional deformation space for the moment map equation.

Does someone know the deformation theory of the empty set ?

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I think it should be governed by a dgla, or an $L_\infty$-algebra, perhaps carrying an action of the little discs operad. But this may only describe a formal neighbourhood of the empty set in its moduli stack. – Richard Thomas Jul 16 '10 at 15:37

Thanks for this Ben (and for the title upgrade). In fact Ivan Smith pointed the Kaledin-Verbitsky paper out to me some time ago and I forgot about it. But it sounds like the Harada-Wilkin paper might be enough to resolve this puzzle.

To make things simple and concrete, I want to concentrate on the $Hilb^2T^*\mathbb P^1$ example, i.e. the quiver $\bullet^{\ \rightrightarrows}_{\ \leftleftarrows}\bullet$ with dimension vector $(2,2)$.

If I understand correctly, I think there is only one complex moment map deformation: the dual of the lie algebra of the centre $\mathbb C^\ast\times\mathbb C^\ast$ of $GL(2)\times GL(2)$ divided by the diagonal $\mathbb C^\ast$.

But I think that $Hilb^2T^*\mathbb P^1$ has 2-dimensional $H^2$.

Does this contradict Harada-Wilkin ? I looked at their paper, but one has to know their entire paper to know how to combine their results on Nakajima quiver varieties with their results on quiver varieties (the latter having no complex moment map condition).

2 potential solutions:

(a) Harada-Wilkin seems to relate cohomology to the dual of the whole lie algebra, not just the centre. So perhaps one cannot interpret their results in terms of values of the moment map ?

(b) Their condition on the dimension vector containing a 1 in Proposition 7.6 (p42) might be a problem ? If so, would my problem go away (i.e. would I get more moment map deformations) by expressing $Hilb^2T^*\mathbb P^1$ as a Nakajima quiver variety in a different way, using a construction with a 1 in the dimension vector ? Does anyone know such a construction ?

Many thanks for your time, I'm very impressed by the generosity available on this website.

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It's not an answer, it's just to say I agree with Richard Thomas' proposal. That book would be very relevant. – Alexander Cruz Jul 18 '10 at 14:47

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