Dolbeault formality for complex nilmanifolds

TL;DR

This work establishes a sharp dichotomy for Dolbeault formality on complex nilmanifolds. It proves that the Dolbeault DGA $(\Lambda^{*,*}M,\overline\partial)$ is non-formal unless the manifold is a torus, and that the Dolbeault $(0,*)$-DGA $(\Lambda^{0,*}M,\overline\partial)$ is formal precisely when the Lie algebra component ${\mathfrak g}^{0,1}$ is abelian. The authors leverage Nomizu’s quasi-isomorphism between the de Rham DGA and the Chevalley-Eilenberg DGA, together with domination techniques (MSZ) and the Console-Fino theorem for rational complex structures, to transfer non-formality from the Lie algebra to the nilmanifold. This yields a complete Lie-algebraic criterion for Dolbeault formality in the $(0,*)$-case and clarifies obstructions to Kähler-like formality in non-torus nilmanifolds, linking Dolbeault formality to abelianness of a complex Lie subalgebra. Overall, the paper provides a definitive framework for understanding Dolbeault formality in complex nilmanifolds and connects it to fundamental structural properties of the underlying Lie algebra.

Abstract

A quasi-isomorphism of differential graded algebras (DGA) is a multiplicative map inducing an isomorphism on cohomology. A DGA is called formal if it can be connected by a chain of quasi-isomorphisms to its cohomology algebra. We prove that the Dolbeault DGA of a complex nilmanifold is formal only if it is a torus, and the Dolbeault algebra of (0,p)-forms is formal if and only if the complex structure is abelian.