∂-Harmonic forms on 4-dimensional almost-Hermitian manifolds

2021/04/21 by Nicoletta Tardini, Tardini, Nicoletta, Адриано Томассини +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · doi:10.48550/arxiv.2104.10594

openalex publication_date 2021/04/21 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

Let (X,J) be a 4-dimensional compact almost-complex manifold and let g be a Hermitian metric on (X,J). Denote by Δ_∂:=∂∂^*+∂^*∂ the ∂-Laplacian. 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, denoted as h^1,1_∂, 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. Furthermore, the previous example shows that h^1,1_∂ 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.

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