Complex symplectic structures and the $\partial \bar{\partial}$-lemma

In this paper we study complex symplectic manifolds, i.e., compact complex manifolds $X$ which admit a holomorphic $(2, 0)$-form $\sigma$ which is $d$-closed and non-degenerate, and in particular the Beauville-Bogomolov-Fujiki quadric $Q_\sigma$ associated to them. We will show that if X satisfies the $\partial \bar{\partial}$-lemma, then $Q_\sigma$ is smooth if and only if $h^{2,0}(X) = 1$ and is irreducible if and only if $h^{1,1}(X) > 0$.