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COMPATIBILITY BETWEEN NON-KÄHLER STRUCTURES ON COMPLEX (NIL)MANIFOLDS

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Abstract

We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds (locally conformally Kähler, k-Gauduchon, balanced, and locally conformally balanced) and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a left-invariant k-Gauduchon metric is necessarily a torus. Combined with the main result in [FV16], this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics—balanced, pluriclosed and locally conformally Kähler—is a torus. Moreover, we construct a family of compact nilmanifolds in any dimension carrying both balanced and locally conformally balanced metrics and finally we show a compact complex nilmanifold does not support a left-invariant locally conformally hyper-Kähler structure.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Street, 010014, Bucharest, Romania

    L. ORNEA

  2. Institute of Mathematics, “Simion Stoilow” of the Romanian Academy, 21 Calea Grivitei Street, 010702, Bucharest, Romania

    L. ORNEA & A.-I. OTIMAN

  3. Dipartimento di Matematica e Informatica “Ulisse Dini”, Universitá degli studi di Firenze, viale Morgagni 67/A, 50134, Firenze, Italy

    A.-I. OTIMAN

  4. Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Street, 010014, Bucharest, Romania

    M. STANCIU

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  1. L. ORNEA
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  2. A.-I. OTIMAN
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  3. M. STANCIU
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Correspondence to A.-I. OTIMAN.

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Dedicated to Paul Gauduchon for his 75th birthday. Bon anniversaire!

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A.-I. Otiman is supported by GNSAGA of INdAM.

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ORNEA, L., OTIMAN, AI. & STANCIU, M. COMPATIBILITY BETWEEN NON-KÄHLER STRUCTURES ON COMPLEX (NIL)MANIFOLDS. Transformation Groups 28, 1669–1686 (2023). https://doi.org/10.1007/s00031-022-09729-5

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