Non-Kähler compact complex manifolds associated to number fields

Oeljeklaus, Karl; Toma, Matei

Oeljeklaus, Karl; Toma, Matei

Annales de l'Institut Fourier, Volume 55 (2005) no. 1, p. 161-171

Abstract
Résumé

For algebraic number fields $K$ with $s > 0$ real and $2 t > 0$ complex embeddings and “admissible” subgroups $U$ of the multiplicative group of integer units of $K$ we construct and investigate certain $(s + t)$ -dimensional compact complex manifolds $X (K, U)$ . We show among other things that such manifolds are non-Kähler but admit locally conformally Kähler metrics when $t = 1$ . In particular we disprove a conjecture of I. Vaisman.

Etant donnés des corps de nombres $K$ avec $s > 0$ plongements réels et $2 t > 0$ plongements complexes, et des sous groupes “admissibles” $U$ du groupe multiplicatif des entiers inversibles de $K$ , nous construisons et étudions certaines variétés complexes compactes $X (K, U)$ . Entre autres, nous montrons que ces variétés ne sont pas kähleriennes, mais admettent des métriques localement conformément kähleriennes lorsque $t = 1$ . En particulier, nous donnons un contre-exemple à une conjecture de I. Vaisman.

MR 2141693 | Zbl 1071.32017

DOI : https://doi.org/10.5802/aif.2093
Classification: 32J18, 32M17
Keywords: Compact complex manifolds, algebraic number fields, algebraic units, locally conformally Kähler metrics

@article{AIF_2005__55_1_161_0,
     author = {Oeljeklaus, Karl and Toma, Matei},
     title = {Non-K\"ahler compact complex manifolds associated to number fields},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {1},
     year = {2005},
     pages = {161-171},
     doi = {10.5802/aif.2093},
     zbl = {1071.32017},
     mrnumber = {2141693},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2005__55_1_161_0}
}

Oeljeklaus, Karl; Toma, Matei. Non-Kähler compact complex manifolds associated to number fields. Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 161-171. doi : 10.5802/aif.2093. http://www.numdam.org/item/AIF_2005__55_1_161_0/

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