Hodge symmetry and decomposition on non-Kähler solvmanifolds

Details Hodge symmetry and decomposition on non-Kähler solvmanifolds Hodge symmetry and decomposition on non-Kähler solvmanifolds

2011-09-27

arxiv.org/abs/1109.5929v4

Mathematics

Differential Geometry

7 pages

Scientific paper

Let $G=\C^{n}\ltimes_{\phi} \C^{m}$ with a semi-simple action $\phi: \C^{n}\to GL_{m}(\C)$ (not necessarilly holomorphic). Suppose $G$ has a lattice $\Gamma$. Then we show that in some conditions on $G$ and $\Gamma$, $G/\Gamma$ admits a Hermittian metric such that the space of harmonic forms satisfies the Hodge symmetry and decomposition. Moreover such $G/\Gamma$ is strictly formal and the $dd^{c}$-lemma holds. We also show that if $G/\Gamma$ admits a pseudo-K\"ahler structure then $G/\Gamma$ has the hard Lefschetz complex type. We also show that in some conditions on $G$ and $\Gamma$, $G/\Gamma$ admits a versal deformation with smooth base $(H^{1}(M, T^{1,0}),0)$. By these results we give examples of non-K\"ahler complex solvmanifolds satisfying Dolbeault-cohomological properties of compact K\"ahler manifolds.

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