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

TL;DR

This paper investigates the Bott-Chern and Aeppli cohomologies of twistor spaces of self-dual 4-manifolds, characterizes the $ ext{dd}^c$-lemma, and explicitly computes the Dolbeault cohomology for the twistor space of a flat 4-torus.

Contribution

It provides a characterization of the $ ext{dd}^c$-lemma validity on twistor spaces and explicit Dolbeault cohomology calculations for the flat 4-torus case.

Findings

Characterization of the $ ext{dd}^c$-lemma on twistor spaces.

Explicit Dolbeault cohomology of the twistor space of a flat 4-torus.

Identification of cases where the $ ext{dd}^c$-lemma holds or fails.

Abstract

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 $\partial \overline \partial$-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 $\partial\overline{\partial}$ lemma.