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  "body": "\n## Statement\n\nLet $X$ be a connected [[complex manifold|JSGWAN2RJV]] and let $f$ be a [[holomorphic function|507C34W0ZT]] on $X$ that is not identically $0$.\nLet $Z_1, Z_2, \\dots$ be the irreducible components of the hypersurface $f^{-1}(0)$.\nLet $m_j \\ge 1$ be the order of vanishing of $f$ at a general point of $Z_j$.\nLet $[Z_j]$ be the current of integration over the regular part of $Z_j$, and let $[Z_f] = \\sum_j m_j [Z_j]$.\nThen, in the sense of [[currents|360H7MD9RD]] on $X$,\n$$\\frac{i}{\\pi} \\partial\\bar\\partial \\log\\abs{f} = [Z_f] .$$\n\n## Remarks\n\nProof idea: off $f^{-1}(0)$ the function $\\log\\abs{f}$ is pluriharmonic, so both sides vanish there.\nNear a regular point of $f^{-1}(0)$ one can choose coordinates with $f = u \\, z_1^{m}$ for a holomorphic unit $u$.\nThen $\\log\\abs{u}$ is pluriharmonic, and the claim reduces to the one-variable identity $\\Delta \\log\\abs{z_1} = 2\\pi \\delta_0$ on each line $z' = \\text{const}$.\nThe difference of the two sides is then a closed current of order zero supported in the singular set of $f^{-1}(0)$.\nThis set has codimension at least $2$, so such a current of bidegree $(1, 1)$ vanishes by Federer's support theorem.\nExample: 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.\nConsequence: $[Z_f]$ is $dd^c$ of a [[plurisubharmonic function|FF2P7ER3QT]], so it is a closed [[positive current|BYJHP6A906]].\nLet $s$ be a holomorphic section of a holomorphic line bundle with a smooth Hermitian metric.\nThe 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.\n\n## Explanations\n\n### L1 human\n\nIn one complex variable, $\\log\\abs{f}$ for a polynomial $f$ is harmonic except at the zeros, where it behaves like $m \\log\\abs{z - a}$.\nIts Laplacian is a sum of point masses at the zeros, weighted by their multiplicities.\nThe Poincaré–Lelong formula says the same in several variables.\nThe complex second derivative of $\\log\\abs{f}$ is concentrated on the zero set of $f$, and it equals integration over it, counted with multiplicity.\n\n### L2 human\n\nThe formula is the bridge between divisors and positive currents.\nIt shows that every effective divisor gives a closed positive $(1,1)$-current, and it lets one compute with zero sets through the plurisubharmonic function $\\log\\abs{f}$.\nIt is used in Nevanlinna theory in several variables and in the theory of [[Lelong numbers|WVB60RKC3Z]].\nFor example, the Lelong number of $\\log\\abs{f}$ at a point is the multiplicity of $f$ there.\nThe 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.\n",
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