We prove that the dimension h1,1 of the space of Dol-beault 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 con-formally almost Kähler, almost Hermitian structures on compact 4-manifolds with h1,1 = b−+1. This gives an answer to [6, Question ∂ 3.3] by Holt.
Piovani, R., Tomassini, A., On the dimension of Dolbeault harmonic (1, 1)-forms on almost Hermitian 4-manifolds, <<PURE AND APPLIED MATHEMATICS QUARTERLY>>, 2022; 18 (3): 1187-1201. [doi:10.4310/PAMQ.2022.v18.n3.a11] [https://hdl.handle.net/10807/334250]
On the dimension of Dolbeault harmonic (1, 1)-forms on almost Hermitian 4-manifolds
Piovani, Riccardo;
2022
Abstract
We prove that the dimension h1,1 of the space of Dol-beault 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 con-formally almost Kähler, almost Hermitian structures on compact 4-manifolds with h1,1 = b−+1. This gives an answer to [6, Question ∂ 3.3] by Holt.
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