Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 3

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

Pages: 1187 – 1201

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n3.a11

Authors

Riccardo Piovani (Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Unità di Matematica e Informatica, Università degli Studi di Parma, Italy)

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

Abstract

We prove that the dimension $h^{1,1}_{\overline{\partial}}$ 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 $h^{1,1}_{\overline{\partial}} = b^{-} +1$. This gives an answer to [6, Question 3.3] by Holt.

Keywords

almost complex 4-manifold, Dolbeault Laplacian

2010 Mathematics Subject Classification

Primary 32Q60. Secondary 53C15, 58A14.

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The first author is partially supported by GNSAGA of INdAM.

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 1 February 2022

Received revised 29 April 2022

Accepted 18 May 2022

Published 24 July 2022