Almost-complex invariants of families of six-dimensional solvmanifolds

Abstract

We compute almost-complex invariants h p,0 ∂ , h p,0 Dol and almost-Hermitian invariants h p,0 δ on families of almost-Kähler and almost-Hermitian 6-dimensional solvmanifolds. Finally, as a consequence of almost-Kähler identities we provide an obstruction to the existence of a symplectic structure on a given compact almost-complex manifold. Notice that, when (X, J, g, ω) is a compact almost Hermitian manifold of real dimension greater than four, not much is known concerning the numbers h p,q ∂ .

On the dimension of Dolbeault harmonic (1,1)-forms on almost Hermitian 4-manifolds

“…From system (12) we can easily deduce that either B, C, D are constant or k = l = 0. If k = l = 0, then for every (0, 0) ≠ (m, n) ∈ Z 2 system (11) can be rewritten as…”

“…In the remaining case when the almost Hermitian metric is not locally conformally almost Kähler, in general not much is known. For other recent and related results concerning the study of the spaces of harmonic forms on compact almost Hermitian manifolds, see [2], [8] and [12] for Dolbeault harmonic forms, and [6], [9] and [10] for Bott-Chern harmonic forms.…”

Bott-Chern harmonic forms and primitive decompositions on compact almost Kähler manifolds

“…By a direct computation the structure equations become (cf. also [13]) First, we do the following observation that will allow us to work with only leftinvariant forms (cf. [1, Lemma 5.2]).…”

Primitive decompositions of Dolbeault harmonic forms on compact almost-Kähler manifolds

“…With different techniques in [15] it was shown that also the dimension of the space of ∂-harmonic (1, 1)-forms depend on the metric on 4-dimensional manifolds (for other results in this direction see [13] and [10]). We note that explicit computations of ∂-harmonic forms is a difficult task and not much is known in higher dimension (see [16], [2], [3] for some detailed computations).…”

“…By a direct computation the structure equations become (cf. also[16])4 dφ 1 = −φ 13 − iφ 23 + φ 1 3 + φ 3 1 − iφ 2 3 + iφ 3 2 + φ 13 − iφ 23 , 4 dφ 2 = −iφ 13 + φ 23 − iφ 1 3 + iφ 3 1 − φ 2 3 − φ 3 2 − iφ 13 − φ 23 , dφ 3 = 0.Endow (X, J) with the left-invariant almost-Kähler structure given by ω = 2(e 16 + e 25 + e 34 ) = i(φ 1 1 + φ 2 2 + φ 3 3).…”