Poincaré–Lelong formula

Remarks

Proof idea: off $f^{-1}(0)$ the function $\log\abs{f}$ is pluriharmonic, so both sides vanish there. Near a regular point of $f^{-1}(0)$ one can choose coordinates with $f = u \, z_1^{m}$ for a holomorphic unit $u$. Then $\log\abs{u}$ is pluriharmonic, and the claim reduces to the one-variable identity $\Delta \log\abs{z_1} = 2\pi \delta_0$ on each line $z' = \text{const}$. The difference of the two sides is then a closed current of order zero supported in the singular set of $f^{-1}(0)$. This set has codimension at least $2$, so such a current of bidegree $(1, 1)$ vanishes by Federer's support theorem. Example: for $X = \C$ and $f(z) = z^2 (z - 1)$ the formula says $\frac{1}{2\pi} \Delta \log\abs{f} = 2 \delta_0 + \delta_1$, the classical statement that $\log\abs{f}$ has Riesz measure given by the zeros with multiplicity. Consequence: $[Z_f]$ is $dd^c$ of a plurisubharmonic function, so it is a closed positive current. Let $s$ be a holomorphic section of a holomorphic line bundle with a smooth Hermitian metric. The same argument shows that $\frac{i}{\pi} \partial\bar\partial \log\abs{s}$ equals $[Z_s]$ minus a smooth closed form that represents the first Chern class of the bundle.

Going deeper

For a student

The formula is the bridge between divisors and positive currents. It shows that every effective divisor gives a closed positive $(1,1)$-current, and it lets one compute with zero sets through the plurisubharmonic function $\log\abs{f}$. It is used in Nevanlinna theory in several variables and in the theory of Lelong numbers. For example, the Lelong number of $\log\abs{f}$ at a point is the multiplicity of $f$ there. The line bundle version links the zero divisor of a section with the curvature of the bundle, which is the starting point of many vanishing and extension theorems.