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This came up in a discussion I had yesterday. Since my understanding is limited, I thought I ask here, because I know there are quite a few experts lurking about. Recall that a holomorphic symplectic manifold $X$ is a complex manifold which comes equipped with a nondegenerate holomorphic $2$-form $\omega$, i.e. $\omega^{\dim X}$ is nowhere zero. Here I'll be interested exclusively in the compact simply connected Kahler (see Dmitri's answer) examples. Using Yau's work, the class of these manifolds can be identified with the class hyper-Kähler manifolds subject to the same restrictions (cf. [1]). This means a Riemannian manifold which is Kähler with respect to a triple of complex structures $I,J,K$ which behave like the quaternions, $IJ=K$ etc. Needless to say, such things are exotic. In dimension two, by the classification of surfaces, the only possible examples are K3 surfaces (or more crudely, things that behave like quartics in $\mathbb{P}^3$).

What little I know in higher dimensions can be summarized in a few sentences. Beauville [1] found a bunch of beautiful simply connected examples as Hilbert schemes of points on a K3 surface and variants for abelian surfaces: the so called generalized Kummer varieties. More generally, Mukai [2] constructed additional examples as moduli space of sheaves on the above surfaces. Huybrechts [3] mentions some further examples which are deformations of these. So now my questions:

Are there examples which are essentially different, i.e. known to not be deformations of the examples discussed above? If not, then what is the expectation? Is there any sort of classification in low dimensions?

I'm aware of some work on hypertoric varieties, which are hyper-Kähler, but I haven't followed this closely. So:

Are any of these compact? If so, how do they fit into the above picture?

While I can already anticipate one possible answer "no, none, hell no...", feel free to elaborate, correct, or discuss anything related that seems relevant even if I didn't explicitly ask for it.

Refs.

[1] Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Diff. Geom 1983

[2] Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent 1984

[3] Huybrechts, Compact hyper-Kähler manifolds: basic results, Invent 1999

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# Compact holomorphic symplectic manifolds: what's the state of the art?

This came up in a discussion I had yesterday. Since my understanding is limited, I thought I ask here, because I know there are quite a few experts lurking about. Recall that a holomorphic symplectic manifold $X$ is a complex manifold which comes equipped with a nondegenerate holomorphic $2$-form $\omega$, i.e. $\omega^{\dim X}$ is nowhere zero. Here I'll be interested exclusively in the compact simply connected examples. Using Yau's work, the class of these manifolds can be identified with the class hyper-Kähler manifolds subject to the same restrictions (cf. [1]). This means a Riemannian manifold which is Kähler with respect to a triple of complex structures $I,J,K$ which behave like the quaternions, $IJ=K$ etc. Needless to say, such things are exotic. In dimension two, by the classification of surfaces, the only possible examples are K3 surfaces (or more crudely, things that behave like quartics in $\mathbb{P}^3$).

What little I know in higher dimensions can be summarized in a few sentences. Beauville [1] found a bunch of beautiful simply connected examples as Hilbert schemes of points on a K3 surface and variants for abelian surfaces: the so called generalized Kummer varieties. More generally, Mukai [2] constructed additional examples as moduli space of sheaves on the above surfaces. Huybrechts [3] mentions some further examples which are deformations of these. So now my questions:

Are there examples which are essentially different, i.e. known to not be deformations of the examples discussed above? If not, then what is the expectation? Is there any sort of classification in low dimensions?

I'm aware of some work on hypertoric varieties, which are hyper-Kähler, but I haven't followed this closely. So:

Are any of these compact? If so, how do they fit into the above picture?

While I can already anticipate one possible answer "no, none, hell no...", feel free to elaborate, correct, or discuss anything related that seems relevant even if I didn't explicitly ask for it.