Beauville Manifolds

The Beauville manifold 5constructed by means of the KZ surface S is the first nontrivial example of an irreducible four-dimensional symplectic manifold. Definition 3.4.5 Let S he slK3-type surface. The Douady space paremetrizing all zero-dimensional analytic cycles Z C 5 of length r, Igth Z = r, is called the Beauville manifold constructed by means of the KZ surface 5. The Beauville manifold is denoted by 51 ) and is a 2r-dimensional symplectic manifold [231]. 

We may present the following effective and vivid description of the Beauville manifold Let S x S he the direct produce of a KZ surface S by itself, the group Z2 acting on this product by permutation of multipliers. Then the symplectic square 5 ) = 5 X 5/Z2 of the surface S has an analytic type singularity along the diagonal... 

Theorem 3.4.3. Let M = 51 1 be an algebraic Beauville manifold. For the symplectic structure on M be Liouville integrable, it is necessary and sufficient that on S there exist a bundle of elliptic curves. Furthermore, if the symplectic structure on is integrable, then there exists an integrating morphism p ... 

Remark 2. The space of moduli of Beauville manifolds is larger than that of KZ-type surfaces. Thus, the general deformation of is not represented in the form for any surface X. Obviously, in the set of all deformations of manifolds 51 1, there exists a countable number of divisors, to the points of which there correspond integrable symplectic structures, but it is not at all clear how this should be proved. 

B-D 3] A. Beauville and O. Debarre, Sur les function theta du second ordre, Arithmetic of Complex manifolds, Proceedings, Erlangen 1988, Lecture Notes in Mathematics 1399 Springer 1989. 

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