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

Question

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. $\omega^{\dim X/2}$ 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.

Thanks in advance.

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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

Answer 1

Here is my understanding of the situation. As you say, there are two known infinite families of irreducible holomorphic symplectic manifolds, namely:

• Hilbert schemes of $n$ points on a $K3$ surface (and deformations of these);
• generalised Kummer varieties, i.e. the fibre over $0$ of the addition morphism $T^n \rightarrow T$, where $T$ is a complex torus and $n$ any natural number (and deformations of these).

Apart from these, as ulrich mentions there are O'Grady's "sporadic" examples in dimension 6 and 10. These are not in either of the deformation classes above.

I think I'm right in saying that these are all the known examples. Many people are trying hard to find new ones, but I'm not aware of any results in this direction. (Unfortunately I'm not familiar with the hypertoric varieties you mention.)

A reference for the facts I quote is this article by Markman. (The article is a survey of Verbitsky's Global Torelli Theorem for holomorphic symplectic varieties, which is a big recent development in the area.)

Answer 2

I would like to make a correction. It is not true that compact complex symplectic simply-connected manifolds are Kahler, and it is even less so for non-simply-connected ones. These examples of course are not HyperKahler either, Yau's theorem does not apply to them.

I am aware of one series of simply-connected non-Kahler examples, they are given in the article of Guan

Examples of compact holomorphic symplectic manifolds which are not Kahlerian. II. Invent. Math., 121(1):135–145, 1995.

The simplest type of non-simply-connected compact complex symplectic manifolds are Kodaira surfaces, they have the structure of an elliptic fibration over an elliptic curve. This example generalises to higher dimensions.

Answer 3

Hypertoric varieties (and quiver varieties) are deformation equivalent to affine algebraic varieties. Therefore they are not compact unless they are 0-dimensional or empty sets.

Answer 4

It's probably bad form to answer one's own question -- even the software has registered its disapproval -- but I thought I'd make a small update. Regarding classification in low dimensions, according to Beauville's nice survey http://arxiv.org/pdf/1002.4321.pdf (which I wasn't aware of last week), it would appear to be open even in dimension 4. Although some bounds are known in this case. For example, Guan shown that the Betti number $b_2\in [3,8]\cup \lbrace 23\rbrace$, where $7$ and $23$ correspond to the known cases mentioned above.