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Deformation stability of p-SKT and p-HS manifolds

  • Houda BellitirEmail author
Research/Review Article
  • 7 Downloads

Abstract

We study the notions of p-Hermitian-symplectic and p-pluriclosed compact complex manifolds, which are defined as generalisations for an arbitrary positive integer p not exceeding the complex dimension of the manifold of the standard notions of Hermitian-symplectic and SKT manifolds that correspond to the case \(p=1\). We then notice that these two notions are equivalent on \(\partial \overline{\partial }\)-manifolds and go on to prove that in (smooth) complex analytic families of \(\partial \overline{\partial }\)-manifolds, the properties of being p-Hermitian-symplectic and p-pluriclosed are deformation-open. Concerning closedness results, we prove that the cones Open image in new window , resp. Open image in new window , of Aeppli cohomology classes of strictly weakly positive (pp)-forms \(\Omega \) that are p-pluriclosed, resp. p-Hermitian-symplectic, must be equal on the limit fibre if they are equal on the other fibres and if some rather weak \(\partial \overline{\partial }\)-type assumptions are made on the other fibres.

Keywords

Deformations of complex structures Positivity p-SKT manifold p-HS manifold 

Mathematics Subject Classification

32G05 32Q57 53C55 

Notes

Acknowledgements

The author is very grateful to her supervisor Dan Popovici for his continuous guidance, suggestions, support, and encouragements during her thesis and for a very careful reading of the paper, and as well as to Professor Ahmed Zeriahi for many interesting discussions. Sheng Rao, Xueyuan Wan and Quanting Zhao informed the author, after the submission on arXiv, that they showed in [20, 21] that the property of p-Kählerianity is deformation-open under the mild \(\partial \overline{\partial }\)-lemma conditions and that some previous studies in [20, 21] are related to some secondary parts of this work. The author is very grateful to them for letting her know of their works. The author is grateful to the referee for his/her helpful comments and for pointing out to Alessandrini’s work [1] where the notions of p-Hermitian-symplectic and p-pluriclosed manifolds were introduced independently. We were unfortunately unaware of this when writing the first version of this paper. However, our approach, via the Frölicher spectral sequence, seems to be new and we hope that this paper will contribute to further understanding of these problems.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesIbn Tofail UniversityKenitraMorocco

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