The main achievement of this paper is the construction of a new six-dimensional irreducible symplectic manifold \(\widetilde{\mathcal M}\). The author shows that this variety has \(b_ 2=8\). This implies that \(\widetilde{\mathcal M}\) cannot be deformed into one of the known irreducible symplectic varieties, not even up to birational equivalence.
According to the Bogomolov decomposition of compact Kähler manifolds with torsion first Chern class, there are three types of “building blocks”: complex tori, Calabi-Yau varieties, and irreducible symplectic manifolds. Compared with the other two classes, examples of irreducible symplectic manifolds are quite scarce. Similar to previous constructions, the new example is obtained as an irreducible factor in the Bogomolov decomposition of a symplectic desingularisation of a moduli space of sheaves on an Abelian surface. According to results of Yoshioka, such a construction can be successful only if the moduli space contains points which correspond to strictly semi-stable sheaves.
The construction of \(\widetilde{\mathcal M}\) is the following. Let \(J\) be the Jacobian of a genus-two curve, \(\Theta\) a Theta divisor and \(\eta\) the orientation class of \(J\). Using methods from an earlier paper of the author [J. Reine Angew. Math. 512, 49–117 (1999; Zbl 0928.14029)], it can be shown that there exists a symplectic desingularisation \(\widetilde{\mathcal M}_ v \rightarrow {\mathcal M}_ v\) of the moduli space \({\mathcal M}_ v\) of \(\Theta\)-semi-stable torsion free sheaves on \(J\) with Mukai vector \(v=2-2\eta\). The variety \(\widetilde{\mathcal M} \subset \widetilde{\mathcal M}_ v\) is the fibre over \((0,\widehat{0})\) of a natural locally trivial fibration \(\widetilde{\mathcal M}_ v \rightarrow J\times \widehat{J}\). After verifying that \(\widetilde{\mathcal M}\) is symplectic and of dimension six, the author shows that \(\widetilde{\mathcal M}\) is simply connected and has \(b_ 2=8\). He defines a refinement of S-equivalence in order to obtain a moduli theoretic interpretation of a subset of \(\widetilde{\mathcal M}_ v\). This allows him to use the generalised Lefschetz hyperplane theorem to gain information about the topology of \(\widetilde{\mathcal M}\).