Let $V$ a compact normal complex analytic variety.

The space $H^{1}(V,\mathcal{PH}_{V})$ is finite dimensional.

Let $L$ a holomorphic line bundle on $V$. Every representative of $c_{1}(L)$ in $H^{1}(V,\mathcal{PH}_{V})$ is the Chern-Weil form of a smooth hermitian on $L$.

If there exists a smooth hermitian metric $h$ such that $c_{1}(L,h)$ is Kähler, then $V$ is projective-algebraic and $L$ is ample.