Small deformations of the complex structure do not always preserve special metric properties in the Hermitian non-Kähler setting. In this paper, we find necessary conditions for the existence of smooth curves of balanced metrics {ωt}t which start with a fixed balanced metric ω for t = 0, along a differentiable family of complex manifolds {Mt}t.

Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics {ω t } t which equals a fixed SKT metric ω for t = 0, along a differentiable family of complex manifolds {M t } t .

Let (X, J) be a nilmanifold with an invariant nilpotent complex structure. We study the existence of p-Kähler structures (which include Kähler and balanced metrics) on X. More precisely, we determine an optimal p such that there are no p-Kähler structures on X. Finally, we show that, contrarily to the Kähler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Frölicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.

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