I am trying to calculate the first chern class $c_1(K)$ of the canonical bundle $K = \Lambda^n(T^*\mathbb{CP}^n)^{1,0}$, where my definition of the first chern class is $c_1(K)=\frac{i}{2\pi}[F(A)] \in H^2_{dR}(\mathbb{CP}^n)$ for any connection $A$ and its curvature $F(A)$.
First, we know $K = O(-n-1)$. Cover $X=\mathbb{CP}^n$ by $U_i$, the affine patch with nonzero $i$'th coordinate. We define $h_i = (1+\sum_{j \neq i} |z_j/z_i|^2)^{-n-1}$, which patch together to give a fibrewise Hermitian inner product on $O(-n-1)$. i.e., under the trivialization $U_i$, if $e_i$ is the section corresponding to $1 \in \mathbb{C}$, and $h=|e_i|^2$, then by direct calculation $|s_ie_i|^2=|s_je_j|^2$ for all sections $s_i$, where $s_j = (z_j/z_i)^{n+1} s_i$ are the transition functions of $O(-n-1)$.
Then we know that $A = \partial \log h$ patches together to give a connection, and hence $\frac{i}{2\pi}F(A) = \frac{i}{2\pi}\bar{\partial}\partial \log h \in \Lambda^{1,1}T^*X$ gives a representative of $c_1$. Note that the Fubini-study form is $\omega|_{U_i} = \frac{i}{2\pi} \partial \bar{\partial} \log (1+\sum_{j\neq i} |z_j/z_i|^2)$, so we see that $\frac{i}{2\pi}F(A) = (n+1) \omega$ (using $\partial \bar{\partial} = -\bar{\partial}\partial$). Hence, $c_1(K)$ can be represented by a positive (1,1)-form, since $\omega$ is a positive (1,1)-form.
However, I was expecting that $c_1(X) = -c_1(K)$ can be represented by a positive $(1,1)$ form (i.e., $\mathbb{CP}^n$ is a Fano manifold) instead, which contradicts the above.
(A real (1,1) form $\omega$ is a positive if $-i\omega(a,\bar{a}) >0$ for all nonzero $a \in T^{1,0}X$.)
What's the mistake in this argument?