Geometria Complessa e Geometria Differenziale
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D. Angella - A. Tomassini

On Cohomological Decomposition of Almost-Complex Manifolds and Deformations

created by daniele on 08 Nov 2016
modified on 17 Nov 2017


Published Paper

Inserted: 8 nov 2016
Last Updated: 17 nov 2017

Journal: J. Symplectic Geom.
Volume: 9
Number: 3
Pages: 403-428
Year: 2011
Doi: 10.4310/JSG.2011.v9.n3.a5

ArXiv: 0909.3604 PDF
Links: arXiv:0909.3604


While small deformations of K\"ahler manifolds are K\"ahler too, we prove that the cohomological property to be $\mathcal{C}^\infty$-pure-and-full is not a stable condition under small deformations. This property, that has been recently introduced and studied by T.-J. Li and W. Zhang in Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom. and T. Dr\v{a}ghici, T.-J. Li, and W. Zhang in Symplectic forms and cohomology decomposition of almost complex four-manifolds, Int. Math. Res. Not., On the J-anti-invariant cohomology of almost complex 4-manifolds, to appear in Q. J. Math., is weaker than the K\"ahler one and characterizes the almost-complex structures that induce a decomposition in cohomology. We also study the stability of this property along curves of almost-complex structures constructed starting from the harmonic representatives in special cohomology classes.

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