<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e21" altimg="si5.svg"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mover accent="false" class="mml-overline"> <mml:mrow> <mml:mi>∂</mml:mi> </mml:mrow> <mml:mo accent="true">¯</mm

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 ∂ ∂ ¯ -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 ∂ ∂ ¯ lemma.