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

Abstract

We prove that the dimension h 1,1 ∂ of the space of Dolbeault harmonic (1, 1)-forms is not necessarily always equal to b − on a compact almost complex 4-manifold endowed with an almost Hermitian metric which is not locally conformally almost Kähler. Indeed, we provide examples of non integrable, non locally conformally almost Kähler, almost Hermitian structures on compact 4-manifolds with h 1,1the space of Dolbeault harmonic (p, q)-forms. If M is compact, it is well known that the dimensions∂ depend both on the almost complex structure J and on the almost Hermitian metric ω.If J is integrable, i.e., if (M, J) is a complex manifold, then, assuming the compactness of M , by Hodge theory we know that the space of Dolbeault harmonic forms is isomorphic to the Dolbeault cohomology, i.e., H p,q ∂ ≅ H p,q ∂ ∶= ker ∂ ∩ A p,q im ∂ .

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

“…However, in the non integrable case, it might happen that h 1,1 ∂ = b − + 1 when the almost Hermitian metric is not globally conformally almost Kähler. Indeed, in [18], Tomassini and the author of the present paper proved that h 1,1 ∂ = b − + 1 on a explicit example of a 4-dimensional compact almost complex manifold endowed with a non globally conformally almost Kähler metric.…”

“…Summing ( 16) and (17), subtracting ( 16) from (17), summing (18) and (19), subtracting (18) from (19), we obtain…”

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

“…Remark 3.3. If we take the wedge product of ω k−l , for 3 ≤ l ≤ k − 1, with both equations (10) and (11), we find similar sums, but this time we have three or more addends. This does not imply, in general, that every addend is equal to 0.…”

“…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

“…On the other hand, the corresponding statement for h 1,1 no longer holds. Piovani and Tomassini [22] recently provide examples of closed almost Hermitian 4manifolds (M, J, ω) with h 1,1 = b − + 1 but ω is not locally conformally almost Kähler. However, as pointed out to the author by Tom Holt, their almost complex structures J are all almost Kähler.…”