Hi! I'm getting stuck with a proposition, please can someone help me? Let $E$ be a holomorphic vector bundle with hermitian metric $h$ over a connected compact kahler manifold $X$ with kahler form $\omega$ and associated kahler metric $g$. Let $s\in H^{0}(X,E)$ a non zero section. Consider the current $T_{s}=\frac{i\partial\bar{\partial}log(h(s,s))}{2\pi}$. Let's write $F(h)$ for the curvature of the chern connection respect to the metric $h$, and another current $R_s=\frac{ih(F(h)s,s)}{2\pi h(s,s)}$, how can i prove that $0 \leq T_{s}+R_{s}$ in the following sense: (regarding those currents as forms) the hermitian form associated to $T_{s}+R_{s}$ is semipositive definite. The Hermitian form associated to a (1,1)-real form $\delta$ is the one given in local coordinates by the matrix of coefficients of $-i\delta$ and is (semi-)positive if this matrix is.

Thank you in advance.