×

Remarks on Hodge numbers and invariant complex structures of compact nilmanifolds. (English) Zbl 1352.53045

Summary: If \(N\) is a simply connected real nilpotent Lie group with a \(\Gamma\)-rational complex structure, where \(\Gamma\) is a lattice in \(N\), then \(H^{s,t}_{\bar{\partial}}(\Gamma\setminus N)\cong H^{s,t}_{\bar{\partial}}(\mathfrak{n}^{\mathbb{C}})\) for each \(s\), \(t\). We study relations between invariant complex structures and Hodge numbers of compact nilmanifolds from a viewpoint of Lie algberas.

MSC:

53C30 Differential geometry of homogeneous manifolds
22E25 Nilpotent and solvable Lie groups
57T15 Homology and cohomology of homogeneous spaces of Lie groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] [1] D. V. Alekseevsky: Flag manifolds, 11th Yugoslav Geometrical Seminar (Divičbare, 1996), Zb. Rad. Mat. Inst. Beograd. (N. S.) 6 (14) (1997), 3-35.; · Zbl 0946.53025
[2] S. Console and A. Fino: Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001), 111-124.; · Zbl 1028.58024
[3] I. Nakamura, Complex parallelisable manifolds and their small deformations, J. Differential Geom. 10 (1975), 85-112.; · Zbl 0297.32019
[4] M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse derMathematik und ihrer Grenzgebiete, Band 68, Springer- Verlag, New York-Heidelberg, 1972. ix+227 pp.; · Zbl 0254.22005
[5] Y. Sakane, On compact complex parallelisable solvmanifolds, Osaka J. Math. 13 (1976), 187-212.; · Zbl 0361.22005
[6] S. M. Salamon, Complex structures on nilpotent Lie algebras, J. Pure. Appl. Algebra 157 (2001), 311-333.; · Zbl 1020.17006
[7] T. Yamada, Complex structures and non-degenerate closed 2-forms of compact real parallelizable nilmanifolds, to appear in Osaka J. Math.; · Zbl 1361.53044
[8] T. Yamada, Duality of Hodge numbers of compact complex nilmanifolds, Complex manifolds 2 (2015), 168-177.; · Zbl 1333.53072
[9] T. Yamada, Hodge numbers and invariant complex structures of compact complex nilmanifolds, Complex manifolds 3 (2016), 193-206.; · Zbl 1342.53075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.