The form-type Calabi-Yau equation on a class of complex manifolds

The Calabi-Yau theorem says that given any smooth representative $\Phi$ of the first Chern class, there exists a unique K\”ahler metric $\omega$ cohomologous to $\alpha$ such that  $Ricci(\omega)=\Phi$. It is natural to investigate whether similar results hold when the manifolds is non-K\”ahler.  In this talk, we will introduce the form type Calabi-Yau equation which can used to prove a version of the Calabi conjecture for  balanced metrics. We define the astheno-Ricci curvature and prove that there exists a solution for the form type Calabi-Yau equation if the astheno-Ricci curvature is non-positive.