On cohomological decomposition of almost-complex manifolds and deformations

While small deformations of compact Kähler manifolds are Kähler too, we prove that the cohomological property to be C∞-pure-and-full is not a stable condition under small deformations. This property, which has been recently introduced and studied by Li and Zhang in [24] and Draghici et al. in [13, 14], is weaker than the Kähler one and characterizes the almost-complex structures inducing 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.