Taking the product of $n$ factors equal to this differential form, we get a differential form of type $(n,n)$ on the $n$-dimensional complex space $\mathrm{X}$. Such a form can be integrated on $\mathrm{X}$ and the Wirtinger formula asserts that

$$\int_{\mathrm{X}}c_{1}(\overline{L})^{n}=\deg(L)$$

is the degree of $L$ as computed by intersection theory. As an example, if $\mathrm{X}={\mathbf{P}}^{1}({\mathbf{C}})$, we have seen that

$$c_{1}(\overline{\mathscr{O}(1)}_{\mathrm{FS}})=\frac{i}{2\pi(1+\left|{z}\right|^{2})^{2}}\mathrm{d}z\wedge d\overline{z},$$

where $z=x_{1}/x_{0}$ is the affine coordinate of $X\setminus\{\infty\}$. Passing in polar coordinates $z=re^{i\theta}$, we get

$$c_{1}(\overline{\mathscr{O}(1)}_{\mathrm{FS}})=\frac{1}{2\pi(1+r^{2})^{2}}\mathrm{d}r\wedge\mathrm{d}\theta$$

whose integral over ${\mathbf{C}}$ equals

$$\int_{{\mathbf{P}}^{1}({\mathbf{C}})}c_{1}(\overline{\mathscr{O}(1)}_{\mathrm{FS}})=\int_{0}^{\infty}\frac{1}{2\pi(1+r^{2})^{2}}2r\mathrm{d}r\int_{0}^{2\pi}\mathrm{d}\theta=\int_{0}^{\infty}\frac{1}{(1+u)^{2}}\mathrm{d}u=1.$$