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Mathematical Research Letters
Volume 30 (2023)
Number 5
$\overline{\partial}$-Harmonic forms on $4$-dimensional almost-Hermitian manifolds
Pages: 1617 – 1637
DOI: https://dx.doi.org/10.4310/MRL.2023.v30.n5.a14
Authors
Abstract
$\def\op{\overline{\partial}} \def\opp{\overline{\partial\partial}}$ Let $(X, J)$ be a $4$-dimensional compact almost-complex manifold and let $g$ be a Hermitian metric on $(X, J)$. Denote by $\Delta_\op := \opp^\ast + \op^\ast \overline{\partial}$ the $\op$-Laplacian. If $g$ is globally conformally Kähler, respectively (strictly) locally conformally Kähler, we prove that the dimension of the space of $\op$-harmonic $(1,1)$-forms on $X$, denoted as $h^{1,1}_\op$, is a topological invariant given by $b\underline{} + 1$, respectively $b\underline{}$. As an application, we provide a one-parameter family of almost- Hermitian structures on the Kodaira–Hurston manifold for which such a dimension is $b\underline{}$. This gives a positive answer to a question raised by T. Holt and W. Zhang. Furthermore, the previous example shows that $h^{1,1}_\op$ depends on the metric, answering to a Kodaira and Spencer’s problem. Notice that such almost-complex manifolds admit both almost-Kähler and (strictly) locally conformally Kähler metrics and this fact cannot occur on compact complex manifolds.
The first author is partially supported by GNSAGA of INdAM and has financially been supported by the Programme “FIL-Quota Incentivante” of University of Parma and co-sponsored by Fondazione Cariparma.
The second author is partially supported by the Project PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” and by GNSAGA of INdAM.
Received 24 May 2021
Received revised 11 October 2021
Accepted 26 October 2021
Published 14 May 2024