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Twisted basic Dolbeault cohomology on transverse Kähler foliations

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Abstract

In this paper, we study the twisted basic Dolbeault cohomology and transverse hard Lefschetz theorem on a transverse Kähler foliation. And we give some properties for \(\Delta _\kappa \)-harmonic forms and prove the Kodaira–Serre-type duality and Dolbeault isomorphism for the twisted basic Dolbeault cohomology.

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References

  1. Ait Haddou, H.: Foliations and Lichnerowicz basic cohomology. Int. Math. Forum 2(49–52), 2437–2446 (2007)

    Article  MathSciNet  Google Scholar 

  2. Álvarez-López, J.A.: The basic component of the mean curvature of Riemannian foliations. Ann. Global Anal. Geom. 10, 179–194 (1992)

    Article  MathSciNet  Google Scholar 

  3. Banyaga, A.: Some properties of locally conformal symplectic structures. Comment. Math. Helv. 77(2), 383–398 (2002)

    Article  MathSciNet  Google Scholar 

  4. Carrière, Y.: Flots riemanniens. Astérisque 116, 31–52 (1984)

    MathSciNet  MATH  Google Scholar 

  5. Cordero, L.A., Wolak, R.A.: Properties of the basic cohomology of transversally Kähler foliations. Rend. Circ. Mat. Palermo Serie II(40), 177–188 (1991)

    Article  Google Scholar 

  6. El Kacimi-Alaoui, A.: Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications. Compos. Math. 73(1), 57–106 (1990)

    MATH  Google Scholar 

  7. El Kacimi-Alaoui, A., Hector, G.: Décomposition de Hodge basique pour un feuilletage riemannien. Ann. Inst. Fourier 36(3), 207–227 (1986)

    Article  MathSciNet  Google Scholar 

  8. Habib, G., Richardson, K.: Modified differentials and basic cohomology for Riemannian foliations. J. Geom. Anal. 23(3), 1314–1342 (2013)

    Article  MathSciNet  Google Scholar 

  9. Habib, G., Vezzoni, L.: Some remarks on Calabi–Yau and hyper-Kähler foliations. Differential Geom. Appl. 41, 12–32 (2015)

    Article  MathSciNet  Google Scholar 

  10. Hebda, J.: Curvature and focal points in Riemannian foliations. Indiana Univ. Math. J. 35(2), 321–331 (1986)

    Article  MathSciNet  Google Scholar 

  11. Jung, S.D.: The first eigenvalue of the transversal Dirac operator. J. Geom. Phys. 39, 253–264 (2001)

    Article  MathSciNet  Google Scholar 

  12. Jung, S.D., Jung, M.J.: Transversally holomorphic maps between Kähler foliations. J. Math. Anal. Appl. 416, 683–697 (2014)

    Article  MathSciNet  Google Scholar 

  13. Jung, S.D., Richardson, K.: Transverse conformal Killing forms and a Gallot–Meyer theorem for foliations. Math. Z. 270, 337–350 (2012)

    Article  MathSciNet  Google Scholar 

  14. Jung, S.D., Richardson, K.: Basic Dolbeault cohomology and Weitzenböck formulas on transversaly Kähler foliations. J. Topol. Anal. (2021). https://doi.org/10.1142/S1793525320500260

    Article  MATH  Google Scholar 

  15. Jung, S.D., Richardson, K.: The mean curvature of transverse Kähler foliations. Doc. Math. 24, 995–1031 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Kamber, F.W., Tondeur, Ph.: Duality for Riemannian foliations. In: Proceedings of Symposia in Pure Mathematics, vol. 40, no. Part I, pp. 609–618. American Mathematical Society (1983)

  17. Kamber, F.W., Tondeur, Ph.: Duality theorems for foliations. Astérisque 116, 108–116 (1984)

    MathSciNet  MATH  Google Scholar 

  18. Kamber, F.W., Tondeur, Ph.: De Rham-Hodge theory for Riemannian foliations. Math. Ann. 277, 415–431 (1987)

    Article  MathSciNet  Google Scholar 

  19. March, P., Min-Oo, M., Ruh, E.A.: Mean curvature of Riemannian foliations. Canad. Math. Bull. 39, 95–105 (1996)

    Article  MathSciNet  Google Scholar 

  20. Nishikawa, S., Tondeur, Ph.: Transversal infinitesimal automorphisms for harmonic Kähler foliations. Tohoku Math. J. 40, 599–611 (1988)

    Article  MathSciNet  Google Scholar 

  21. Park, E., Richardson, K.: The basic Laplacian of a Riemannian foliation. Amer. J. Math. 118, 1249–1275 (1996)

    Article  MathSciNet  Google Scholar 

  22. Tondeur, Ph.: Foliations on Riemannian Manifolds. Springer-Verlag, New-York (1988)

    Book  Google Scholar 

  23. Tondeur, Ph.: Geometry of Foliations. Birkhäuser Verlag, Basel (1997)

    Book  Google Scholar 

  24. Vaisman, I.: Remarkable operators and commutation formulas on locally conformal Kähler manifolds. Compos. Math. 40(3), 287–299 (1980)

    MATH  Google Scholar 

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Acknowledgements

The author would like to thank the referee for his or her valuable suggestions.

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  1. Department of Mathematics, Jeju National University, Jeju, 63243, Republic of Korea

    Seoung Dal Jung

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  1. Seoung Dal Jung
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Correspondence to Seoung Dal Jung.

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The author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2022R1A2C1003278).

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Jung, S.D. Twisted basic Dolbeault cohomology on transverse Kähler foliations. Ann Glob Anal Geom 62, 285–303 (2022). https://doi.org/10.1007/s10455-022-09851-3

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  • DOI: https://doi.org/10.1007/s10455-022-09851-3

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