Suppose $(X,\omega)$ is a compact hermitian manifold of complex dimension $n$, and $\alpha$ is a smooth $(n-1,n-1)$ positive form, i.e., $\alpha = a \omega ^{n-1} + \displaystyle \sum _{i=1} ^m f_i \Phi _i \wedge \bar{\Phi}_i$ where $a>0$ is a constant, $f_i \geq 0$ are smooth functions, and $\Phi_i$ are smooth $(n-1,0)$ forms. Then does there exist a hermitian metric $\chi$ such that $\alpha \wedge \gamma= \chi ^{n-1} \wedge \gamma $? for all hermitian metrics $\gamma$ ?

Suppose $(X,\omega)$ is a compact hermitian manifold of complex dimension $n$, and $\alpha$ is a smooth $(n-1,n-1)$ positive form, i.e., $\alpha = a \omega ^{n-1} + \displaystyle \sum _{i=1} ^m (\sqrt{-1})^n (-1)^{n(n-1)/2}f_i \Phi _i \wedge \bar{\Phi}_i$ where $a>0$ is a constant, $f_i \geq 0$ are smooth functions, and $\Phi_i$ are smooth $(n-1,0)$ forms. Then does there exist a hermitian metric $\chi$ such that $\alpha \wedge \gamma= \chi ^{n-1} \wedge \gamma $? for all hermitian metrics $\gamma$ ?

Suppose $(X,\omega)$ is a compact hermitian manifold of complex dimension $n$, and $\alpha$ is a smooth $(n-1,n-1)$ positive form, i.e., $\alpha = a \omega ^{n-1} + \displaystyle \sum _{i=1} ^m f_i \Phi _i \wedge \bar{\Phi}_i$ where $a>0$ is a constant, $f_i \geq 0$ are smooth functions, and $\Phi_i$ are smooth $(n-1,0)$ forms. Then does there exist a hermitian metric $\chi$ such that $\alpha = \chi ^{n-1}$?

Suppose $(X,\omega)$ is a compact hermitian manifold of complex dimension $n$, and $\alpha$ is a smooth $(n-1,n-1)$ positive form, i.e., $\alpha = a \omega ^{n-1} + \displaystyle \sum _{i=1} ^m f_i \Phi _i \wedge \bar{\Phi}_i$ where $a>0$ is a constant, $f_i \geq 0$ are smooth functions, and $\Phi_i$ are smooth $(n-1,0)$ forms. Then does there exist a hermitian metric $\chi$ such that $\alpha \wedge \gamma= \chi ^{n-1} \wedge \gamma $? for all hermitian metrics $\gamma$ ?