According to p131 corollary 1.7, a (1,1) form $u = i u_{ij} dz_i \wedge d\bar z_j$ is positive if $(u_{ij})$ is a positive semi definite matrix.
However according to definition (1.1), $\frac{u^n}{n!} = det(u_{ij})dx_1\wedge dy_1 \ldots \wedge dx_n \wedge dy_n$ should be positive, which is not necessarily true because the derterminant might be zero. What is wrong here?