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<p>We classify nilpotent Lie algebras with complex structures of weakly non-nilpotent type in real dimension eight, which is the lowest dimension where they arise. Our study, together with previous results on strongly non-nilpotent structures, completes the classification of 8-dimensional nilpotent Lie algebras admitting complex structures of non-nilpotent type. As an application, we identify those that support a pseudo-Kähler metric, thus providing new counterexamples to a previous conjecture and an infinite family of (Ricci-flat) non-flat neutral Calabi–Yau structures. Moreover, we arrive at the topological restriction <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1712_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(b_1(X)\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> for every pseudo-Kähler nilmanifold&#xa0;<i>X</i> with an invariant complex structure, up to complex dimension four.</p>

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Nilmanifolds with non-nilpotent complex structures and their pseudo-Kähler geometry

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年2月22日

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  • Adela Latorre,
  • Luis Ugarte

摘要

We classify nilpotent Lie algebras with complex structures of weakly non-nilpotent type in real dimension eight, which is the lowest dimension where they arise. Our study, together with previous results on strongly non-nilpotent structures, completes the classification of 8-dimensional nilpotent Lie algebras admitting complex structures of non-nilpotent type. As an application, we identify those that support a pseudo-Kähler metric, thus providing new counterexamples to a previous conjecture and an infinite family of (Ricci-flat) non-flat neutral Calabi–Yau structures. Moreover, we arrive at the topological restriction \(b_1(X)\ge 3\) b 1 ( X ) ≥ 3 for every pseudo-Kähler nilmanifold X with an invariant complex structure, up to complex dimension four.