On Cohomological Decomposition of Almost-Complex Manifolds and Deformations

Abstract

While small deformations of Kähler manifolds are Kähler 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. Drǎghici, 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 Kähler 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.

20 pages

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• Cohomology of D-complex manifolds
• On cohomological decomposition of generalized-complex structures
• On non-Kähler degrees of complex manifolds
• On Bott-Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms
• A note on tame/compatible almost complex structures on four-dimensional Lie algebras
• On second non-HLC degree of closed symplectic manifold