Deformations of Strong K\"ahler with torsion metrics

Existence of strong K\"ahler 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 $\{\omega_t\}_t$ which equals a fixed SKT metric $\omega$ for $t=0$, along a differentiable family of complex manifolds $\{M_t\}_t$.