Primitive decompositions of Dolbeault harmonic forms on compact almost-Kähler manifolds

Authors

  • Andrea Cattaneo Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Unità di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy
  • Nicoletta Tardini Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Unità di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy
  • Adriano Tomassini Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Unità di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy

DOI:

https://doi.org/10.33044/revuma.3557

Abstract

Let $(X,J,g,\omega)$ be a compact $2n$-dimensional almost-Kähler manifold. We prove primitive decompositions of $\partial$-, $\bar\partial$-harmonic forms on $X$ in bidegree $(1,1)$ and $(n-1,n-1)$ (such bidegrees appear to be optimal). We provide examples showing that in bidegree $(1,1)$ the $\partial$- and $\bar\partial$-decompositions differ.

Downloads

Download data is not yet available.

References

D. Angella, N. Istrati, A. Otiman, and N. Tardini, Variational problems in conformal geometry, J. Geom. Anal. 31 no. 3 (2021), 3230–3251.  DOI  MR  Zbl

P. de Bartolomeis and A. Tomassini, On formality of some symplectic manifolds, Internat. Math. Res. Notices no. 24 (2001), 1287–1314.  DOI  MR  Zbl

A. Cattaneo, A. Nannicini, and A. Tomassini, Kodaira dimension of almost Kähler manifolds and curvature of the canonical connection, Ann. Mat. Pura Appl. (4) 199 no. 5 (2020), 1815–1842.  DOI  MR  Zbl

A. Cattaneo, A. Nannicini, and A. Tomassini, On Kodaira dimension of almost complex $4$-dimensional solvmanifolds without complex structures, Internat. J. Math. 32 no. 10 (2021), Paper No. 2150075, 41 pp.  DOI  MR  Zbl

J. Cirici and S. O. Wilson, Topological and geometric aspects of almost Kähler manifolds via harmonic theory, Selecta Math. (N.S.) 26 no. 3 (2020), Paper No. 35, 27 pp.  DOI  MR  Zbl

J. Cirici and S. O. Wilson, Dolbeault cohomology for almost complex manifolds, Adv. Math. 391 (2021), Paper No. 107970, 52 pp.  DOI  MR  Zbl

R. Coelho, G. Placini, and J. Stelzig, Maximally non-integrable almost complex structures: an $h$-principle and cohomological properties, Selecta Math. (N.S.) 28 no. 5 (2022), Paper No. 83, 25 pp.  DOI  MR  Zbl

P. Gauduchon, La $1$-forme de torsion d'une variété hermitienne compacte, Math. Ann. 267 no. 4 (1984), 495–518.  DOI  MR  Zbl

F. Hirzebruch, Some problems on differentiable and complex manifolds, Ann. of Math. (2) 60 (1954), 213–236.  DOI  MR  Zbl

T. Holt, Bott-Chern and ∂̅ harmonic forms on almost Hermitian 4-manifolds, Math. Z. 302 no. 1 (2022), 47–72.  DOI  MR  Zbl

T. Holt and W. Zhang, Harmonic forms on the Kodaira-Thurston manifold, Adv. Math. 400 (2022), Paper No. 108277, 30 pp.  DOI  MR  Zbl

R. Piovani and N. Tardini, Bott-Chern harmonic forms and primitive decompositions on compact almost Kähler manifolds, Ann. Mat. Pura Appl. (4) 202 no. 6 (2023), 2749–2765.  DOI  MR  Zbl

R. Piovani and A. Tomassini, Bott-Chern Laplacian on almost Hermitian manifolds, Math. Z. 301 no. 3 (2022), 2685–2707.  DOI  MR  Zbl

N. Tardini and A. Tomassini, Differential operators on almost-Hermitian manifolds and harmonic forms, Complex Manifolds 7 no. 1 (2020), 106–128.  DOI  MR  Zbl

N. Tardini and A. Tomassini, Almost-complex invariants of families of six-dimensional solvmanifolds, Complex Manifolds 9 no. 1 (2022), 238–260.  DOI  MR  Zbl

N. Tardini and A. Tomassini, ∂̅-harmonic forms on 4-dimensional almost-Hermitian manifolds, Math. Res. Lett. 30 no. 5 (2023), 1617–1637.  DOI  MR  Zbl

L.-S. Tseng and S.-T. Yau, Cohomology and Hodge theory on symplectic manifolds: I, J. Differential Geom. 91 no. 3 (2012), 383–416.  MR  Zbl Available at http://projecteuclid.org/euclid.jdg/1349292670.

A. Weil, Introduction à l'étude des variétés kählériennes, Publications de l'Institut de Mathématique de l'Université de Nancago, VI. Actualités Sci. Ind., no. 1267, Hermann, Paris, 1958.  MR  Zbl

Downloads

Published

2024-05-22

Issue

Vol. 67 No. 1 (2024)

Section

Article

License

Copyright (c) 2024 Andrea Cattaneo, Nicoletta Tardini, Adriano Tomassini

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following terms:

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.