Verbal ideals and unobstructed complex parallelisable nilmanifolds

TL;DR

This paper characterizes when compact complex parallelisable nilmanifolds have unobstructed deformations, linking it to properties of their Lie algebras, and classifies such structures up to dimension 19.

Contribution

It establishes a criterion based on verbal ideals in free Lie algebras for unobstructed deformations and provides a partial classification of these Lie algebras.

Findings

Finitely many unobstructed nilmanifolds up to dimension 19.

Infinite families of unobstructed nilmanifolds appear from dimension 20.

Connection between reality conditions and deformation unobstructedness.

Abstract

We show that a compact complex parallelisable nilmanifold has unobstructed deformations if and only if its associated Lie algebra satisfies a reality condition and is a free Lie algebra in a variety of Lie algebras, that is, defined by a verbal ideal in a free Lie algebra. We provide a partial classification of verbal ideals and show that there are finitely many such Lie algebras up to dimension 19, whereas infinite families start to appear in dimension 20. As a consequence, there are finitely many complex homotopy types of unobstructed complex parallelisable nilmanifolds up to dimension 19, and infinitely many in dimension 20.