$\text{del}\text{delbar}$-Lemma and Bott-Chern cohomology of twistor spaces

In the paper we study the Bott-Chern and Aeppli cohomologies of the twistor space of a compact self-dual 4-manifold and we characterize the validity of the $\mathrm{\partial }\bar{\mathrm{\partial }}$-lemma. We also compute explicitly the Dolbeault cohomology of the twistor space $Z$ of the flat $4$-dimensional torus, which is known to not satisfy the $\mathrm{\partial }\bar{\mathrm{\partial }}$ lemma.