Irreducible symplectic 4-folds which look like Hilb^2(K3)
-- Kieran O'Grady 

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Simply connected compact K&auml;hler manifolds carrying a
holomorphic symplectic form spanning H^{2,0} are called
irreducible symplectic.  In dimension 2 they are K3 surfaces;
Beauville and Huybrechts have shown that higher dimensional
irreducible symplectic manifolds share many of the properties
of K3 surfaces.  In the '60's Kodaira proved that any two K3
surfaces are deformation equivalent.  Experimental evidence
suggests that the number of deformation classes of irreducible
symplectic manifolds of a given dimension might be very small.
We will present results that should lead to a classification of
irreducible symplectic 4-folds whose 2-cohomology with 4-tuple
wedge product is isomorphic to that of Hilb2 (K3) - a typical
example of such a variety.  Certain special 4-dimensional
sextic hypersurfaces constructed by Eisenbud-Popescu-Walter
play an important role in our work.