Dolbeault and Bott-Chern formalities: deformations and $\partial\overline{\partial}$-lemma

TL;DR

This paper investigates the stability of Dolbeault and Bott-Chern formality properties under complex structure deformations and constructs a compact manifold satisfying the $ ext{∂∂̄}$-lemma but with non-vanishing Massey products.

Contribution

It demonstrates that Dolbeault and Bott-Chern formality are not preserved under deformations and provides a new example of a manifold satisfying the $ ext{∂∂̄}$-lemma with non-trivial Massey products.

Findings

Dolbeault and Bott-Chern formality are not deformation-closed.

Constructed a compact manifold with $ ext{∂∂̄}$-lemma and non-vanishing Massey products.

Abstract

It is proved that the properties of being Dolbeault formal and geometrically-Bott-Chern-formal are not closed under holomorphic deformations of the complex structure. Further, we construct a compact complex manifold which satisfies the $\partial\overline{\partial}$-lemma but admits a non vanishing Aeppli-Bott-Chern-Massey product.