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<p>We can show that the Kuranishi space of a pair (<i>M</i>,&#xa0;<i>E</i>) of a compact Kähler manifold <i>M</i> and its flat Hermitian vector bundle <i>E</i> is isomorphic to the direct product of the Kuranishi space of <i>M</i> and the Kuranishi space of <i>E</i>. We study non-Kähler case. We show that the Kuranishi space of a pair (<i>M</i>,&#xa0;<i>E</i>) of a complex parallelizable nilmanifold <i>M</i> and its trivial holomorphic vector bundle <i>E</i> is isomorphic to the direct product of the Kuranishi space of <i>M</i> and the Kuranishi space of <i>E</i>. We give examples of pairs (<i>M</i>,&#xa0;<i>E</i>) of nilmanifolds <i>M</i> with left-invariant abelian complex structures and their trivial holomorphic line bundles <i>E</i> such that the Kuranishi spaces of pairs (<i>M</i>,&#xa0;<i>E</i>) are not isomorphic to direct products of the Kuranishi spaces of <i>M</i> and the Kuranishi spaces of <i>E</i>.</p>

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On splittings of deformations of pairs of complex structures and holomorphic vector bundles

  • 期刊论文

  • 发表日期：

    2026年4月3日

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  • Hisashi Kasuya,
  • Valto Purho

摘要

We can show that the Kuranishi space of a pair (M, E) of a compact Kähler manifold M and its flat Hermitian vector bundle E is isomorphic to the direct product of the Kuranishi space of M and the Kuranishi space of E. We study non-Kähler case. We show that the Kuranishi space of a pair (M, E) of a complex parallelizable nilmanifold M and its trivial holomorphic vector bundle E is isomorphic to the direct product of the Kuranishi space of M and the Kuranishi space of E. We give examples of pairs (M, E) of nilmanifolds M with left-invariant abelian complex structures and their trivial holomorphic line bundles E such that the Kuranishi spaces of pairs (M, E) are not isomorphic to direct products of the Kuranishi spaces of M and the Kuranishi spaces of E.