Complex symplectic structures and the $\mathrm{\partial }\bar{\mathrm{\partial }}$-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 $\mathrm{\partial }\bar{\mathrm{\partial }}$-lemma, then $Q_{\sigma }$ is smooth if and only if $h^{2,0}(X)=1$ and is irreducible if and only if $h^{1,1}(X)>0$.