A new six-dimensional irreducible symplectic variety

We construct a six-dimensional irreducible symplectic variety with b(2) = 8. Since the known examples of irreducible symplectic varieties have b2 = 7 or b2 = 23, our variety is in a new deformation class. The example is obtained as follows. Let J be the Jacobian of a genus-two curve with its natural principal polarization: results of another paper of ours give a symplectic desingularization of the moduli space of semistable rank-two sheaves on J with c(1) = 0 and c(2) = 2. Let M-v be this symplectic desingularization: there is a natural locally trivial fibration M-v --> J x (J) over cap. Our example is the fiber over (0, (0) over cap) of this map, we denote it by (M) over tilde. The main body of the paper is devoted to the proof that (M) over tilde is irreducible symplectic and that b(2)((M) over tilde) = 8. Applying the generalized Lefschetz Hyperplane Theorem we get that low-dimensional homotopy (or homology) groups of (M) over tilde are represented by homotopy (or homology) groups of a subset of (M) over tilde which has an explicit description. The main problem is to provide the explicit description and to extract the necessary information on homotopy or homology groups.