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<p>We provide new families of compact complex manifolds with no Kähler structure carrying symplectic structures satisfying the <i>Hard Lefschetz Condition</i>. These examples are obtained as compact quotients of the solvable Lie group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {C}^{2n} \ltimes _{\rho } \mathbb {C}^{2m}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> <msub> <mo>⋉</mo> <mi>ρ</mi> </msub> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, for which we construct explicit lattices. By cohomological computations we prove that such manifolds carry symplectic structures satisfying the <i>Hard Lefschetz Condition</i>. Furthermore, we compute the Kodaira dimension of an almost-Kähler structure and generators for the de Rham and Dolbeault cohomologies.</p>

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Hard Lefschetz Condition on symplectic non-Kähler solvmanifolds

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  • 发表日期：

    2025年9月29日

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  • Francesca Lusetti,
  • Adriano Tomassini

摘要

We provide new families of compact complex manifolds with no Kähler structure carrying symplectic structures satisfying the Hard Lefschetz Condition. These examples are obtained as compact quotients of the solvable Lie group \(\mathbb {C}^{2n} \ltimes _{\rho } \mathbb {C}^{2m}\) C 2 n ⋉ ρ C 2 m , for which we construct explicit lattices. By cohomological computations we prove that such manifolds carry symplectic structures satisfying the Hard Lefschetz Condition. Furthermore, we compute the Kodaira dimension of an almost-Kähler structure and generators for the de Rham and Dolbeault cohomologies.