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?



Generic deformation of Hilbert scheme of points on K3 surface
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.


