Journal of the Mathematical Society of Japan

Kähler flat manifolds


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Using a criterion of Johnson-Rees [9] we give a list of all four and six dimensional flat Kähler manifolds. We calculate their R–cohomology, including the Hodge numbers. As a corollary, we classify all flat complex manifolds of dimension 3 whose holonomy groups are subgroups of SU(3). Moreover, we define a family of flat Kähler manifolds which are generalizations of the oriented Hantzsche-Wendt Riemannian manifolds [14].

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J. Math. Soc. Japan Volume 61, Number 2 (2009), 363-377.

First available in Project Euclid: 13 May 2009

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Mathematical Reviews number (MathSciNet)

Primary: 57N16: Geometric structures on manifolds [See also 57M50]
Secondary: 20H15: Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25] 14J32: Calabi-Yau manifolds

Bieberbach group flat manifold Kähler manifold Hantzsche-Wendt manifold Hodge diamond hyperelliptic variety


DEKIMPE, Karel; HAŁENDA, Marek; SZCZEPAŃSKI, Andrzej. Kähler flat manifolds. J. Math. Soc. Japan 61 (2009), no. 2, 363--377. doi:10.2969/jmsj/06120363.

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