Kähler flat manifolds
Karel DEKIMPE, Marek HAŁENDA, and Andrzej SZCZEPAŃSKI
Source: J. Math. Soc. Japan Volume 61, Number 2
(2009), 363-377.
Abstract
Using a criterion of Johnson-Rees [9] we give a list of all four and six dimensional flat Kähler manifolds. We calculate their $\mbi{R}$–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].
First Page:
Show
Hide
Keywords: Bieberbach group; flat manifold; Kähler manifold; Hantzsche-Wendt manifold; Hodge diamond; hyperelliptic variety
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1242220714
Digital Object Identifier: doi:10.2969/jmsj/06120363
Mathematical Reviews number (MathSciNet): MR2532893
References
L. Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume I, Math. Ann., 70 (1911), 297–336.
Mathematical Reviews (MathSciNet): MR1511623
Digital Object Identifier: doi:10.1007/BF01564500
H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic groups of four-dimensional space, Wiley, New York, 1978.
Mathematical Reviews (MathSciNet): MR484179
Zentralblatt MATH: 0381.20002
C. Cid and T. Schulz, Computation of five and six dimensional Bieberbach groups, Experiment. Math., 10 (2001), 109–115.
Mathematical Reviews (MathSciNet): MR1822856
Zentralblatt MATH: 0980.20043
Project Euclid: euclid.em/999188425
K. Dekimpe, M. Sadowski and A. Szczepański, Spin structures on flat manifolds, Monatsh. Math., 148 (2006), 283–296.
Mathematical Reviews (MathSciNet): MR2234081
C. W. Curtis and I. Reiner, Methods of Representation Theory, II, Wiley, 1987.
Mathematical Reviews (MathSciNet): MR892316
CARAT (Version 2.01.04.2003), Page of low dimensional Bieberbach groups, http://wwwb.math.rwth-aachen.de/carat/index.html
H. Hiller and C. H. Sah, Holonomy of flat manifolds with $b_{1} = 0$., Q. J. Math., 37 (1986), 177–187.
Mathematical Reviews (MathSciNet): MR841425
Digital Object Identifier: doi:10.1093/qmath/37.2.177
W. Hantzsche and H. Wendt, Dreidimensionale euklidische Raumformen, Math. Ann., 110 (1935), 593–611.
Mathematical Reviews (MathSciNet): MR1512956
Digital Object Identifier: doi:10.1007/BF01448045
F. E. A. Johnson and E. G. Rees, Kähler groups and rigidity phenomena, Math. Proc. Cambridge Philos. Soc., 109 (1991), 31–44.
Mathematical Reviews (MathSciNet): MR1075119
Digital Object Identifier: doi:10.1017/S0305004100069541
F. E. A. Johnson, Flat algebraic manifolds, Geometry of low-dimensional manifolds 1, Durham 1989, London Math. Soc. Lecture Note Ser., 150, Cambridge University Press, Cambridge, 1990, pp. 73–91.
Mathematical Reviews (MathSciNet): MR1171892
Zentralblatt MATH: 0837.57021
D. D. Joyce, Compact manifolds with special holonomy, Oxford Univ. Press, 2000.
Mathematical Reviews (MathSciNet): MR1787733
Zentralblatt MATH: 1027.53052
H. Lange, Hyperelliptic varieties, Tohoku Math. J., 53 (2001), 491–510.
Mathematical Reviews (MathSciNet): MR1862215
Digital Object Identifier: doi:10.2748/tmj/1113247797
Project Euclid: euclid.tmj/1113247797
R. J. Miatello and R. A. Podesta, The spectrum of twisted Dirac operators on compact flat manifolds, Trans. Amer. Math. Soc., 358 (2006), 4569–4603.
Mathematical Reviews (MathSciNet): MR2231389
Zentralblatt MATH: 1094.58014
Digital Object Identifier: doi:10.1090/S0002-9947-06-03873-6
J. P. Rossetti and A. Szczepański, Generalized Hantzsche-Wendt flat manifolds, Rev. Mat. Iberoamericana, 21 (2005), 1053–1070.
Mathematical Reviews (MathSciNet): MR2232676
Zentralblatt MATH: 1159.53328
Project Euclid: euclid.rmi/1136999141
A. Szczepański, Five dimensional Bieberbach groups with trivial centre, Manuscripta Math., 68 (1990), 191–208.
Mathematical Reviews (MathSciNet): MR1063225
Zentralblatt MATH: 0789.20058
Digital Object Identifier: doi:10.1007/BF02568759
A. Szczepański, Kähler flat manifolds of low dimensions, Preprint - M/05/43-IHES.
K. Uchida and H. Yoshihara, Discontinuous groups of affine transformations of $\mbi{C}^{3}$, Tohoku Math. J., 28 (1976), 89–94.
Mathematical Reviews (MathSciNet): MR400271
Digital Object Identifier: doi:10.2748/tmj/1178240881
Project Euclid: euclid.tmj/1178240881
Journal of the Mathematical Society of Japan
