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I'm looking for a few concrete examples of vector bundles on hyperkahler varieties of dimension $\ge 4$. Here are a few examples I know already:

  • For $X$= the Hilbert scheme of points $S^{[n]}$ on a K3 surface with a line bundle $L$, one has the tautological bundle $L^{[n]}$, which has rank $n$.

  • For $X$=the variety of lines of a cubic fourfold, one has the restriction of the universal bundles on the ambient Grassmannian. Similarly for other hyperkahler varieties embedded in Grassmannians.

Are there other natural examples of interesting vector bundles like this? (I'm mostly interested in hyperkahler manifolds of $K3^{[n]}$-type, but also other cases are interesting).

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    $\begingroup$ How about the tangent bundle? Is that interesting enough for you? $\endgroup$
    – Jason Starr
    Commented Apr 3, 2018 at 0:35

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Let $S$ be a K3 surface, and consider a moduli space of sheaves $M$ where you've cooked it up so that stable = semi-stable and there are only vector bundles, no torsion-free sheaves. Suppose moreover that it's a fine moduli space, so there's a universal bundle $U$ on $M \times S$. I think Mukai vector $(5,1,0)$ on a quartic surface of Picard rank 1 should do the trick, but no promises.

Then the "wrong-way slices" $U|_{M \times \text{pt}}$ are very interesting bundles on $M$. Probably they're stable, and $S$ is a component of the moduli space of stable bundles on $M$ via this construction, but no one knows how to prove it.

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    $\begingroup$ With a lot of work you might be able to get something out of Perego and Rapagnetta's new paper 1802.01182, which cleans up some results of Yoshioka. Section 2.4.2 has to do with how these universal sheaves relate to the universal ideal sheaf on $S \times \operatorname{Hilb}^n(S)$. $\endgroup$
    – Nick Addington
    Commented Apr 3, 2018 at 21:02
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On a hyperkähler manifold $M$ with a circle action which fixes just one complex structure $I$, and rotates $J$ and $K$, and whose Kähler form $\omega_I$ is integral, there exists a hyperholomorphic line bundle $$\mathcal L\to M,$$ i.e., it admits a connection such that its $(0,1)$-part with respect to any of the complex structures in the 2-sphere generated by $I,J,K$ is a holomorphic structure. It is used in the c-map construction to obtain a correspondence between those hyperkähler manifolds and certain quaternionic Kähler manifolds.

The construction has its origin in physics, was carried out by Haydys (J. Geom. Phys. 58 (2008)), and is nicely explained in Hitchin's paper "On the hyperkähler/quaternion Kähler correspondence".

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