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

Nicoletta Tardini (Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Italy)

Adriano Tomassini (Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Italy)

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.

Received 24 May 2021

Received revised 11 October 2021

Accepted 26 October 2021

Published 14 May 2024