An important formula is the Poincaré–Lelong equation. For any line bundle $L$ with a smooth metric, and any section $s\in\Gamma(\mathrm{X},L)$ which does not vanish identically on any connected component of $\mathrm{X}$, it asserts the following equality of currents¹¹ 1 The space of currents is the dual to the space of differential forms, with the associated grading; in the orientable case, currents can also be seen as differential forms with distribution coefficients. :

$$\mathop{\mathrm{d}\mathrm{d}^{c}}\log\left\|{s}\right\|^{-1}+\delta_{\operatorname{div}(s)}=c_{1}(\overline{L}),$$

where $\mathop{\mathrm{d}\mathrm{d}^{c}}\log\left\|{s}\right\|^{-1}$ is the image of $\log\left\|{s}\right\|^{-1}$ under the differential operator $\mathop{\mathrm{d}\mathrm{d}^{c}}$, taken in the sense of distributions, and $\delta_{\operatorname{div}(s)}$ is the current of integration on the cycle $\operatorname{div}(s)$ of codimension $1$, $\operatorname{div}(s)$.