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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
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