Abelian fibred holomorphic symplectic manifolds

From the introduction:

Irreducible holomorphic symplectic manifolds are higher dimensional generalizations of K3 surfaces. It has been roughly twenty years since Fujiki found the first example, the Hilbert scheme S^[2] of two points on a K3 surface S, and Beauville generalized this to construct two families, S^[n] and the generalized Kummer varieties K_n. Since then there have been other constructions, but only the two examples of O'Grady have given us manifolds which are not deformations of Beauville's examples. The purpose of this article is to describe a framework for understanding irreducible holomorphic symplectic manifolds, which hopefully will lead towards some kind of classification. The main results are only conjectural, but we will describe the evidence and motivation behind them, while surveying special cases which have already been proved.

In studying the moduli space of K3 surfaces, one typically looks at Kummer surfaces or quartics in P^3, as these are dense but also relatively easy to understand. However, the structure that will generalize to higher dimensions is a fibration by abelian varieties. This suggests that we first review elliptic K3s, which also happen to be dense in the moduli space. We divide this into three main steps. Firstly, we need to know which K3 surfaces are elliptic. Secondly, we describe the family of elliptic K3s which admit a section. Thirdly, we describe the relation between elliptic K3s which don't admit sections and their relative Jacobians, which do. There is nothing new here: for the first step we recall a theorem of Pjateckii-Shapiro and Shafarevich from the 70s (c.f. also related results of Kodaira), while the second and third steps have been well understood for arbitrary elliptic surfaces for a long time.

Elliptic K3s have base P^1. There is evidence to suggest that if the 2n-dimensional irreducible holomorphic symplectic manifold X admits a non-trivial fibration, then the fibres must be abelian varieties and the base must be P^n (a large part of this has been proved by Matsushita). So the `right' generalization of an elliptic fibration on a K3 surface appears to be a fibration by n-dimensional abelian varieties over P^n, which we shall call an abelian fibration. The aim of this paper is to formulate analogues for abelian fibrations of the three main steps mentioned above for elliptic K3s.

Firstly, when does an irreducible holomorphic symplectic manifold X admit an abelian fibration? Since we are largely interested in classification up to deformation equivalence, we'd like to know whether X can always be deformed to have such a structure. Secondly, can we describe the family of all abelian fibred X which admit sections? By trying to associate a holomorphic symplectic surface (ie. a K3 surface or complex tori) to such a fibration, we can relate or possibly even deform it to Beauville's examples. Thirdly, can we relate abelian fibrations X which don't admit sections to ones that do? Once again, the goal is to deform X to a more familiar manifold, which is more amenable to classification.

The programme we will describe tends to suggest that all irreducible holomorphic symplectic manifolds can be related to Beauville's examples. By "related" we don't simply mean deformed, as O'Grady's examples would contradict that. Indeed it was through trying to understand O'Grady's ten-dimensional example and its relation to the Hilbert scheme S^[5] that the author was lead to the ideas in this paper. The author still feels that a proper understanding of this relationship will reveal many of the mysteries of holomorphic symplectic manifolds.

Appears in Volume 27 (2003) No. 1 of Turkish Journal of Mathematics. Available as math.AG/0404362 from the e-Print archive.

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Wednesday, 31-Aug-2005 14:59:11 MDT
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