Dolbeault Harmonic $(1,1)$-forms on $4$-dimensional compact quotients of Lie Groups with a left invariant almost Hermitian structure

Abstract

We study Dolbeault harmonic (1, 1)-forms on compact quotients M = Γ G of 4-dimensional Lie groups G admitting a left invariant almost Hermitian structure (J, ω). In this case, we prove that the space of Dolbeault harmonic (1, 1)-forms on (M, J, ω) has dimension b − + 1 if and only if there exists a left invariant anti self dual (1, 1)-form γ on (G, J) satisfying id c γ = dω. Otherwise, its dimension is b − . In this way, we answer to a question by Zhang., is finite. If the almost complex structure J is integrable, i.e., if (M, J) is a complex manifold, then the numbers h p,q ∂ depend only on the complex structure and not on the metric, being the dimensions of the (p, q)-Dolbeault cohomology spaces by Hodge theory.Whether the numbers h p,q ∂ depend or not on the choice of the almost Hermitian metric ω on a given compact almost Hermitian manifold (M, J, ω) is a question by Kodaira and Spencer, appeared as Problem 20 in Hirzebruch's 1954 problem list [10]. Recently, in [12] (cf. [13]), Holt and Zhang proved that the number h 0,1 ∂ depends on the choice of the metric on a given compact almost complex 4-manifold, answering to Kodaira and Spencer. They also proved that on a compact almost Kähler 4-manifold it holds h 1,1

Primitive decomposition of Bott-Chern and Dolbeault harmonic $(k,k)$-forms on compact almost Kähler manifolds

“…by (8). Now, thanks to the last two equations, we easily deduce ω n−1 ∧ ∂∂α 0,0 = 0 and ω n−2 ∧ ∂∂α 1,1 = 0.…”

“…by (8), and this is equivalent to the first claim. Now, assume that ∂ * ψ = 0 and α 0,0 ∈ C. Since dω = 0, it follows…”

“…Similarly, for Bott-Chern harmonic forms, it yields h 1,1 BC ∶= dim C H 1,1 BC = b − + 1 for all metrics, see [4,10]. See [8,11,13] for other related results and [7,15] for two surveys on the subject.…”

Almost complex Hodge theory

“…It is natural to ask whether h1,1 characterizes almost Kähler structures, i.e., is it true that h1,1 = b − +1 if and only if J is almost Kähler? Recently, Piovani [23] shows that this is not true by a nice computation on compact quotients of Lie groups.…”

An integral condition involving $\overline\partial$-harmonic $(0,1)$-forms

“…Kodaira and Spencer (see [12,Problem 20]) asked whether the dimension of such spaces could depend on the Hermitian metric g. Very recently Holt and Zhang in [13] answered to such a question, providing a family of almost complex structures on the Kodaira-Thurston manifold such that the Hodge number h 0,1 ∂ varies with different choices of Hermitian metrics. For other results on the study of the ∂-harmonic spaces see [20], [15] and [21]. The aim of this paper is the study of Hermitian metrics on a compact 2ndimensional almost complex manifold satisfying an integral conidition involving the space of ∂-harmonic (0, 1)-forms.…”