$\overline\partial$-Harmonic forms on $4$-dimensional almost-Hermitian manifolds

Abstract

Let (X, J) be a 4-dimensional compact almost-complex manifold and let g be a Hermitian metric on (X, J). Denote by ∆If g is globally conformally Kähler, respectively (strictly) locally conformally Kähler, we prove that the dimension of the space of ∂-harmonic (1, 1)-forms on X is a topological invariant given by b − + 1, respectively b − . As an application, we provide a one-parameter family of almost-Hermitian structures on the Kodaira-Thurston manifold for which such a dimension is b − . This gives a positive answer to a question raised by T. Holt and W.Zhang.

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

“…We observe that a key ingredient in the proof of the results in [15] (see also [11]) is indeed the primitive decomposition of ∂-harmonic (1, 1)-forms on 4-dimensional manifolds. In fact, in this dimension in Proposition 4.1 we prove the general equality…”

“…Indeed they construct on the Kodaira-Thurston manifold an almost-complex structure that, with respect to different almost-Hermitian metrics, has varying dim H 0,1 ∂ . 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).…”

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

“…Arguing like in [9,Theorem 4.3] or in [12,Proposition 3.4], one can show that the differential operator L :…”

“…Introducing an effective method to solve the PDE system associated to Dolbeault harmonic forms on the Kodaira-Thurston manifold, they proved that the dimension of the space of Dolbeault harmonic forms depends on the choice of the almost Hermitian metric. In [12], Tomassini and the second author answered again to the same question, with a different approach, analyzing locally conformally almost Kähler metrics on almost complex 4-manifolds. In [9], Tomassini and the first author introduced Bott-Chern and Aeppli harmonic forms on almost Hermitian manifolds and studied their relation with Dolbeault harmonic forms.…”

“…See also [6] and [10] for recent results concerning the dimension of the spaces of Dolbeault and Bott-Chern harmonic (1, 1)-forms on compact almost Hermitian 4-manifolds. In particular, in [7,Proposition 6.1], in [12,Theorem 3.6] and in [9,Corollary 4.4], the primitive decomposition of (1, 1)-forms is used to deduce, in fact, primitive decompositions of Dolbeault and Bott-Chern harmonic (1, 1)-forms on a compact almost Hermitian 4-manifold. Given that, the study of primitive decompositions of Dolbeault, Bott-Chern, Aeppli harmonic forms can be seen as a generalisation of the just mentioned results in higher dimension 2n ≥ 4 and for every bidegree (p, q).…”