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Apart the two exceptional examples given by O'Grady, the only known classes of Hyperkahler manifolds are (deformation equivalent to) Hilbert scheme of point of a K3 and generalized Kummer varieties. Both these examples were discovered and studied by Beauville (Varietes kahleriennes dont la premiere classe de Chern est nulle). Some years later, Mukai (Symplectic structure on the moduli space of sheaves on an abelian or K3 surface) discovered others Hyperkahler manifolds, namely the moduli space of (stable) sheaves on a K3 with fixed Mukai vector, which turned out to be deformation equivalents to the Hilbert scheme of points (and coincide with it for particular choices of the Mukai vector). In literature, all the authors (I have consulted) give these examples, but it seems strange to me that we have two (different) descriptions for the first deformation class and only one for the latter. There is a strange non-simmetry.

So I wonder if there is a class of examples, such as the moduli space of sheaves on a K3, for the generalized Kummer varieties that are deformation equivalent to them.

By the same article of Mukai, the moduli space of stable sheaves on an abelian surface has still a symplectic structure, so I thought that such an example can be found looking at the fibre of a morphism that generilezes the Hilbert-Chow one. At this point I am stuck.

My questions are:

1) is this argument valid or I am losing my time? And in case, how to find such a morphism?

2) are there any other class of examples and where can I find it? (I mean, of course there are explicit examples of Hyperkahelr manifolds that are deformation equivalent to a generalized Kummer variety, but I'm looking for a set of examples of the same form.)

Of course references are very welcome.

Thank you for your time!

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  • $\begingroup$ This is not a soft question! :) $\endgroup$
    – user64687
    Commented Jan 14, 2015 at 14:51
  • $\begingroup$ I thought that since I am not asking a technical question, it could be consider soft.. :) but I agree with you, this is not soft at all! Thank you. $\endgroup$
    – User3773
    Commented Jan 14, 2015 at 15:01

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I do not know whether this is what you are asking for, but there is a class of Hyperkahler that are deformation equivalent to the generalized Kummer in the same way as the moduli spaces of sheaves over a K3 surface are deformation equivalent to the Hilbert scheme.

Your guess is actually pretty good: over a polarized abelian surface $(X,H)$, you can construct the moduli space $\mathfrak{M}_v$ of $H$-semistable sheaves with Chern character $v$ which, if $v$ is primitive and $H$ does not lie on a wall with respect to $v$ is smooth and projective. Moreover, it has a symplectic structure. Only problem is that it is not going to be simply connected, since intuitively it will contain something that looks like $\text{Pic}^0(X)=\hat{X}$. The analogue of the Hilbert-Chow morphism you are looking for is not quite an analogue of the Hilbert-Chow morphism but rather of the addition map you have on the Hilbert scheme, it is called Albanese map, and it is defined as follows. Call $\mathcal{P}$ the normalized Poincare' bundle on $X\times \hat{X}$. You have a standard functor from $\mathcal{D}^b(\text{Coh}(X))$ to $\mathcal{D}^b(\text{Coh}(\hat{X}))$, which is called Fourier-Mukai transform with kernel $\mathcal{P}$ and which happens to be an equivalence (if you don't know about it, read this beautiful paper by Mukai: http://projecteuclid.org/download/pdf_1/euclid.nmj/1118786312 ). I am going to denote it by $\text{R}\mathcal{S}$. Fix a sheaf $E_0$ on $X$ with Chern character $v$. Then the Albanese map is defined as:

$$ \mathcal{M}_v \to X\times \hat{X} , \ , E\mapsto (\text{det}\text{R}\mathcal{S}(E)\otimes\text{det}\text{R}\mathcal{S}(E_0)^{-1} , \text{det}(E)\otimes \text{det}(E_0)^{-1}).$$

It can be proved that the fibers of this map are all isomorphic, that they are simply connected and then, since they have a symplectic structure from the one on the moduli space, they are compact Hyperkahlers. When the Chern character is $v=1-n[pt]$, you have that $\mathcal{M}_v\cong X^{[n]}\times \hat{X}$, so that by fixing the determinant you have slices isomorphic to the Hilbert scheme, and it can be shown by a calculation that fixing the determinant of the Fourier-Mukai transform is equivalent to fixing the sum of the points in the subschemes, so you get the generalized Kummer that you know. For a reference, see http://arxiv.org/pdf/math/0009001v2.pdf.

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  • $\begingroup$ Thank you very much! This is exactly what I was looking for! I'm going to read the references you linked to me.. $\endgroup$
    – User3773
    Commented Jan 22, 2015 at 12:38
  • $\begingroup$ I have a question: does your description of the Albanese map generalize to $g=\dim X>2$? What about fixing the determinant of the FM transform? is it still equivalent to fixing the sum of the points? $\endgroup$
    – Brenin
    Commented Jun 28, 2015 at 10:40

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