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arXiv · 2411.17560

Verbal ideals and unobstructed complex parallelisable nilmanifolds

Matthias Paulsen·Sönke Rollenske·Konstantin Wehler
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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.

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Matthias Paulsen, Sönke Rollenske, Konstantin Wehler. 2024-11-26. Verbal ideals and unobstructed complex parallelisable nilmanifolds. https://doi.org/10.1007/s00209-026-04035-2

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