Integrally normalized Fubini-Study form ($[{\omega }_{\mathrm{int}}]=c_1(\mathcal{O}(1))$)

The form ${\omega }_{\mathrm{int}}=\frac{i}{2\pi }\partial \bar{\partial }\log (1+\sum |z_j{|}^2)$ patches on Complex projective space. The dual tautological metric on $\mathcal{O}(1)$ identifies it with normalized Chern curvature, so $[{\omega }_{\mathrm{int}}]=c_1(\mathcal{O}(1))$. With the unnormalized convention $i\partial \bar{\partial }\log (1+\sum |z_j{|}^2)$, divide by $2\pi$ to obtain this integral class.