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i was thinking about deformations of hyperkahler manifolds, in particular hilbert schemes of points on K3 surfaces and I think I realized something. I'm here to ask you if I'm right.

Take $X^{[n]}$ the Hilbert scheme of points on the K3 surface $X$. Now take $\pi:\mathcal{Y}\rightarrow Def(X^{[n]})$ a universal deformation: we know (from Beauville) that deformations of $X^{[n]}$ of the type $S^{[n]}$ with $S$ a K3 surface form a countable union of hyperplanes in $Def(X^{[n]})$. In other words a generic deformation of $X^{[n]}$ is not a Hilbert scheme of points on a K3 surface.

On the other side, we know that $H^2(X^{[n]},\mathbb{Z})=-2E_8\oplus 3U\oplus -2(n-1)\mathbb{Z}=:\Lambda$ and the local system $R^2\pi_*\mathbb{Z}$ is trivial on $Def(X^{[n]})$.

So am I terribly wrong or the general point $b$ of $Def(X^{[n]})$ corresponds to a hyperkahler variety $\mathcal{Y}_b$ not of type $S^{[n]}$ but with $H^2(\mathcal{Y}_b,\mathbb{Z})=\Lambda$? Correct me if I'm wrong, but I don't know examples of such hyperkahlers, so is this a way to see that there are many hyperkahlers yet to discover?

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I am not completely certain what you are asking. But if you want to see deformations of $\text{Hilb}^n$s "in nature", I recommend that you look at the (ever-growing) literature on the hyper-Kähler fourfold parameterizing lines on a smooth cubic hypersurfaces in $\mathbb{P}^5$. I recommend that you start with the seminal paper of Beauville-Donagi.

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  • $\begingroup$ Yes! thank you! So for $n=2$ the generic deformation of $X^{[2]}$ is the Fano variety of lines in a cubic fourfold and they have same integral cohomology. I'm sorry to bother you again, but I don't know the literature on the subject.. so do you know if there are any discovered analogues for $n>2$? $\endgroup$ Jul 15, 2014 at 12:16
  • $\begingroup$ For $n=4$, please see the recent article of Lehn-Lehn-Sorger-van Straten and the followup article by Addington. $\endgroup$ Jul 15, 2014 at 17:12
  • $\begingroup$ One more article about this today on the arXiv today by Genki Ouchi. Also, Ouchi cites an article by Addington-Lehn. Perhaps I misattributed this above; if so, I apologize. $\endgroup$ Jul 29, 2014 at 15:10

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