Deformations of astheno-K\"ahler metrics

The property of admitting an astheno-K\"ahler metric is not stable under the action of small deformations of the complex structure of a compact complex manifold. In this paper, we prove necessary cohomological conditions for the existence of curves of astheno-K\"ahler metrics along curves of deformations starting from an initial compact complex manifold endowed with an astheno-K\"ahler metric. Furthermore, we apply our results providing obstructions to the existence of curves of astheno-K\"ahler metrics on two different families of real $8$-dimensional nilmanifolds endowed with invariant nilpotent complex structures.