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<p>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&#xa0;19, whereas infinite families start to appear in dimension&#xa0;20. As a consequence, there are finitely many complex homotopy types of unobstructed complex parallelisable nilmanifolds up to dimension&#xa0;19, and infinitely many in dimension&#xa0;20.</p>

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Verbal ideals and unobstructed complex parallelisable nilmanifolds

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月22日

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  • Matthias Paulsen,
  • Sönke Rollenske,
  • Konstantin Wehler

摘要

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.