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Is it possible to obtain K3 (or any other compact hyperkahler manifold) with its hyperkahler structure as a hyperkahler quotient of an infinite-dimensional affine quaternionic vector space with an affine action?

I have seen people claiming that it is true: indeed, K3 can be obtained as the moduli space of stable bundles on another K3, and (ostensibly) this hyperkahler structure can be obtained as a hyperkahler quotient by the gauge group. The reference that I have seen is Theorem 4.1.2 from Oliver Nash thesis https://arxiv.org/abs/math/0610295. However, I don't see how it gives a K3 (and if it does give it at all).

I think I have a proof showing that this is impossible, and I am trying to see where it is wrong.

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Wow, my thesis; it's been a while! Perhaps I was/am confused but I'll add a few remarks in case it helps.

I don't know of any examples of a compact hyperkahler manifold obtained as an infinite-dimensional hyperkahler quotient.

I also don't believe that I am making the claim you state in the theorem to which you refer. Is it implicit in some subtle way?

As it happens I'm discussing the hypercomplex case, but given my motivation is to compare/contrast with the hyperkahler case I admit that this theorem is relevant to your question. Nevertheless I'm discussing a quotient that gives a non-compact space: the moduli space of irreducible instantons, plus I'm only working "formally" (i.e., brushing aside all the analysis).

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  • $\begingroup$ Dear Oliver, it was not you who made this claim! However, some people said that it implies this. Of course, if we could obtain the space of Yang-Mills connection on K3 via hyperkahler reduction, this gives a compact hyperkahler K3. Indeed, there are compact components, isomorphic to K3, in the moduli of PGL(2)-instantons. $\endgroup$
    – Misha Verbitsky
    Commented Jul 26, 2018 at 12:45
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    $\begingroup$ @MishaVerbitsky thanks for elaborating :-) I confess I never really thought seriously about gauge groups other than $SU(2)$. I'm a bit surprised but I guess I can believe that there are moduli spaces of instantons on K3 with compactifications diffeomorphic to K3. I don't have any sense for this but I wonder what would happen to the metric supplied by the quotient construction as compactification takes place: my naive guess would be that if this is the situation, the metric does not extend. $\endgroup$
    – Oliver Nash
    Commented Jul 26, 2018 at 22:23
  • $\begingroup$ @MishaVerbitsky This popped into my head again this evening and it occurred to me that I didn't see a natural metric in the $PGL(2)$ case since we don't have a definite Killing form ($SO(2, 1)$ is split). As I write this it occurs to me that you might be considering complex instantons, $PGL(2, \mathbb{C})$, but then everything is different and the Killing form is complex-valued so still no metric. Anyway, just a thought. $\endgroup$
    – Oliver Nash
    Commented Jul 28, 2018 at 21:32
  • $\begingroup$ PU(2) is just as good. There are compact K3 obtained as instanton spaces: one needs to choose a stable U(2)-bundle which cannot be deformed to reducible semistable because its c_1 is not divisible by 2. The corresponding principal PU(2)-bundle has the same deformation space, which is naturally hyperkahler and compact. $\endgroup$
    – Misha Verbitsky
    Commented Jul 29, 2018 at 6:31
  • $\begingroup$ @MishaVerbitsky Right of course $PU(2)$, I see. Very interesting in fact! $\endgroup$
    – Oliver Nash
    Commented Jul 29, 2018 at 7:19

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