# zbMATH — the first resource for mathematics

Hodge symmetry and decomposition on non-Kähler solvmanifolds. (English) Zbl 1279.22010
Author’s abstract: Let $$G = \mathbb C^n \ltimes_\phi \mathbb C^m$$ with a semi-simple action $$\phi : \mathbb C^n \to\mathrm{GL}_m(\mathbb C)$$ (not necessarily holomorphic). Suppose that $$G$$ has a lattice $$\Gamma$$. Then we show that under some conditions on $$G$$ and $$\Gamma$$, $$G/\Gamma$$ admits a Hermitian metric such that the space of harmonic forms satisfies the Hodge symmetry and decomposition. By this result we give many examples of non-Kähler Hermitian solvmanifolds satisfying the Hodge symmetry and decomposition.

##### MSC:
 22E25 Nilpotent and solvable Lie groups 53C30 Differential geometry of homogeneous manifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds
Full Text:
##### References:
 [1] Deligne, P.; Griffiths, P.; Morgan, J.; Sullivan, D., Real homotopy theory of kahler manifolds, Invent. Math., 29, 3, 245-274, (1975) · Zbl 0312.55011 [2] Hasegawa, K., Minimal models of nilmanifolds, Proc. Amer. Math. Soc., 106, 1, 65-71, (1989) · Zbl 0691.53040 [3] C. Bock, On low-dimensional solvmanifolds, 2009. arXiv:0903.2926. · Zbl 1346.53052 [4] Oprea, J.; Tralle, A., (Symplectic Manifolds With No Kähler Structure, Lecture Notes in Math., vol. 1661, (1997), Springer) [5] Kasuya, H., Formality and hard Lefschetz property of aspherical manifolds, Osaka J. Math., 50, 2, 439-455, (2013) · Zbl 1283.53068 [6] Kasuya, H., Minimal models, formality and hard Lefschetz properties of solvmanifolds with local systems, J. Differential Geom., 93, 269-298, (2013) · Zbl 1373.53069 [7] Kasuya, H., Techniques of computations of Dolbeault cohomology of solvmanifolds, Math. Z., 273, 437-447, (2013) · Zbl 1261.22009 [8] Kotschick, D., On products of harmonic forms, Duke Math. J., 107, 3, 521-531, (2001) · Zbl 1036.53030 [9] Kasuya, H., Geometrical formality of solvmanifolds and solvable Lie type geometries, (RIMS Kôkyûroku Bessatsu, Geometry of Transformation Groups and Combinatorics, (2013)), in preparation. arXiv:1207.2390v2 · Zbl 1295.53024 [10] Hasegawa, K., A note on compact solvmanifolds with Kähler structures, Osaka J. Math., 43, 1, 131-135, (2006) · Zbl 1105.32017 [11] Tsuchiya, N.; Yamakawa, A., Lattices of some solvable Lie groups and actions of products of affine groups, Tohoku Math. J. (2), 61, 3, 349-364, (2009) · Zbl 1181.22014
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.