Special Hermitian metrics

We study the stability at blow-up and deformations of a class of Hermitian metrics whose fundamental two-form $\omega$ satisfies the condition $\mathrm{\partial }\bar{\mathrm{\partial }}{\omega }^k=0$, for any $k$ between 1 and $n-1$ (where $n$ is the complex dimension of the manifold). We are motivated by the existence of compact complex manifold supporting such metrics.