[proofplan] The proof is local because equality of currents can be tested against compactly supported smooth forms in coordinate charts. We first prove the formula for a [holomorphic function](/page/Holomorphic%20Function) by reducing, away from the singular locus of the divisor, to the one-variable model $z_1^m$ times a nowhere-vanishing holomorphic unit. The unit contributes no $\partial\bar\partial$ term, while the model contributes $m$ times the integration current over the hypersurface. Meromorphic functions are handled by writing them as quotients of holomorphic functions, and meromorphic sections are reduced to the function case by choosing a local holomorphic frame and tracking the curvature term of the Hermitian metric. [/proofplan] [step:Reduce the current identity to local holomorphic data] Let $n=\dim_{\mathbb{C}} M$. Since the assertion is an equality of currents of type $(1,1)$, it suffices to verify it on every coordinate [open set](/page/Open%20Set) $U \subset M$ against every compactly supported smooth test form \begin{align*} \eta \in C_c^\infty(U;\Lambda^{n-1,n-1}T^*U). \end{align*} Indeed, currents are sheaves: if two currents agree after restriction to every member of an open cover, then they agree globally. First suppose that $f:U\to\mathbb{C}$ is holomorphic and not identically zero. Its divisor has the form \begin{align*} \operatorname{div}(f)=\sum_{\alpha} m_{\alpha} Z_{\alpha}, \end{align*} where each $Z_{\alpha}\subset U$ is an irreducible analytic hypersurface and $m_{\alpha}\in\mathbb{N}$ is the vanishing order of $f$ along $Z_{\alpha}$. The associated current is defined by \begin{align*} [\operatorname{div}(f)](\eta) = \sum_{\alpha} m_{\alpha}\int_{Z_{\alpha}^{\mathrm{reg}}}\eta\,d\mathcal{H}^{2n-2}(x), \end{align*} where $Z_{\alpha}^{\mathrm{reg}}$ is oriented by its complex structure and $\mathcal{H}^{2n-2}$ denotes the induced [Hausdorff measure](/page/Hausdorff%20Measure) on its smooth locus. The function $\log |f|^2$ is locally integrable on $U$ because, in local holomorphic coordinates, the logarithm of the modulus of a nonzero holomorphic function has only logarithmic singularities along an analytic hypersurface. Therefore it defines a current by \begin{align*} T_{\log |f|^2}(\psi) = \int_U \log |f|^2\,\psi\,d\mathcal{L}^{2n}(x) \end{align*} for compactly supported smooth forms $\psi$ of top degree, where $\mathcal{L}^{2n}$ is Lebesgue measure in the chosen holomorphic coordinates. The current \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |f|^2 \end{align*} means $\frac{i}{2\pi}\partial\bar\partial T_{\log |f|^2}$. The local integrability and hypersurface divisor facts used here are standard consequences of the local structure theorem for analytic hypersurfaces and [Weierstrass Preparation Theorem](/page/Weierstrass%20Preparation%20Theorem), together with [Local Integrability of Logarithms of Holomorphic Functions](/page/Local%20Integrability%20of%20Logarithms%20of%20Holomorphic%20Functions). [guided] We begin by making precise why the problem can be checked locally. A current is determined by its values on compactly supported test forms. Thus, if $U\subset M$ is a coordinate open set and \begin{align*} \eta \in C_c^\infty(U;\Lambda^{n-1,n-1}T^*U) \end{align*} is a test form, then proving the desired identity on $U$ for every such $\eta$ proves the equality of currents after restriction to $U$. Since restrictions agree on overlaps, the local identities glue to a global current identity. Now assume temporarily that $f:U\to\mathbb{C}$ is holomorphic and not identically zero. The divisor of $f$ is the analytic hypersurface of zeros, counted with multiplicity: \begin{align*} \operatorname{div}(f)=\sum_{\alpha} m_{\alpha}Z_{\alpha}. \end{align*} Here each $Z_{\alpha}$ is an irreducible analytic hypersurface, and $m_{\alpha}$ is the order to which $f$ vanishes generically along $Z_{\alpha}$. The associated current acts on a compactly supported smooth test form $\eta$ of bidegree $(n-1,n-1)$ by integration over the smooth part of the hypersurface: \begin{align*} [\operatorname{div}(f)](\eta) = \sum_{\alpha} m_{\alpha}\int_{Z_{\alpha}^{\mathrm{reg}}}\eta\,d\mathcal{H}^{2n-2}(x). \end{align*} Here $\mathcal{H}^{2n-2}$ is the Hausdorff measure induced on the smooth locus by the ambient complex coordinates. The singular part of each $Z_{\alpha}$ has complex codimension at least one inside $Z_{\alpha}$, hence real codimension at least two inside the hypersurface, so it does not affect this integration current. The other side of the identity begins with the locally integrable function $\log |f|^2$. This is locally integrable because the zero set of a holomorphic function is locally modeled, after [Weierstrass Preparation Theorem](/page/Weierstrass%20Preparation%20Theorem), by a Weierstrass polynomial times a holomorphic unit; logarithmic singularities of such functions are integrable on compact subsets by [Local Integrability of Logarithms of Holomorphic Functions](/page/Local%20Integrability%20of%20Logarithms%20of%20Holomorphic%20Functions). Thus $\log |f|^2$ defines a current, and applying $\partial\bar\partial$ to that current is meaningful. [/guided] [/step] [step:Compute the model current for a coordinate power] We use the [One-Variable Poincare-Lelong Formula](/page/One-Variable%20Poincare-Lelong%20Formula): \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |z|^2 = \delta_0 \end{align*} as currents on $\mathbb{C}$, where $\delta_0$ is the Dirac current at $0$. Let $V\subset\mathbb{C}^{n-1}$ be open, let $W\subset\mathbb{C}\times V$ be open, and define \begin{align*} g:W&\to\mathbb{C}\\ (z_1,z')&\mapsto z_1^m \end{align*} for an integer $m\in\mathbb{N}$, where $z'=(z_2,\dots,z_n)$. Since \begin{align*} \log |g(z_1,z')|^2=m\log |z_1|^2, \end{align*} linearity of currents gives \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |g|^2 = m\,\frac{i}{2\pi}\partial\bar\partial\log |z_1|^2. \end{align*} By applying the one-variable model in the $z_1$ variable and leaving the remaining variables fixed, this current acts on each \begin{align*} \eta\in C_c^\infty(W;\Lambda^{n-1,n-1}T^*W) \end{align*} as \begin{align*} m\int_{\{z_1=0\}}\eta\,d\mathcal{H}^{2n-2}(z'). \end{align*} Therefore \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |z_1^m|^2 = m[\{z_1=0\}]. \end{align*} [guided] The essential computation is the one-dimensional identity \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |z|^2 = \delta_0 \end{align*} on $\mathbb{C}$. This is the distributional form of Green's formula: away from $0$, the function $\log |z|^2$ is harmonic, and its entire distributional Laplacian is concentrated at the origin with total mass $1$ under the normalization $\frac{i}{2\pi}\partial\bar\partial$. We cite this as the [One-Variable Poincare-Lelong Formula](/page/One-Variable%20Poincare-Lelong%20Formula). Now pass from one variable to the local hypersurface model in $\mathbb{C}^n$. Let $V\subset\mathbb{C}^{n-1}$ be open, let $W\subset\mathbb{C}\times V$ be open, and define the holomorphic function \begin{align*} g:W&\to\mathbb{C}\\ (z_1,z')&\mapsto z_1^m, \end{align*} where $m\in\mathbb{N}$ and $z'=(z_2,\dots,z_n)$. Then \begin{align*} \log |g(z_1,z')|^2=\log |z_1^m|^2=m\log |z_1|^2. \end{align*} Because $\partial\bar\partial$ is linear on currents, \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |g|^2 = m\,\frac{i}{2\pi}\partial\bar\partial\log |z_1|^2. \end{align*} The one-variable current acts only in the normal coordinate $z_1$. The remaining coordinates $z'$ are passive parameters. Therefore, for every compactly supported smooth test form \begin{align*} \eta\in C_c^\infty(W;\Lambda^{n-1,n-1}T^*W), \end{align*} [Fubini's theorem](/theorems/2961) for smooth compactly supported forms reduces the action to the one-variable identity on each slice, giving \begin{align*} \left(\frac{i}{2\pi}\partial\bar\partial\log |g|^2\right)(\eta) = m\int_{\{z_1=0\}}\eta\,d\mathcal{H}^{2n-2}(z'). \end{align*} Here $\mathcal{H}^{2n-2}$ is the Hausdorff measure on the coordinate hyperplane with its complex orientation. This is precisely the integration current over the hypersurface $\{z_1=0\}$ with multiplicity $m$: \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |z_1^m|^2 = m[\{z_1=0\}]. \end{align*} [/guided] [/step] [step:Remove the contribution of holomorphic units] Let $u:U\to\mathbb{C}$ be a nowhere-vanishing holomorphic function. After shrinking $U$ if necessary, choose a holomorphic function \begin{align*} a:U&\to\mathbb{C} \end{align*} such that $u=e^a$. Then \begin{align*} \log |u|^2=\log |e^a|^2=a+\overline{a}. \end{align*} Since $a$ is holomorphic, $\bar\partial a=0$. Since $\overline{a}$ is antiholomorphic, $\partial\overline{a}=0$. Hence, as smooth forms and therefore as currents, \begin{align*} \partial\bar\partial\log |u|^2 = \partial\bar\partial(a+\overline{a}) = 0. \end{align*} Thus multiplying a holomorphic function by a nowhere-vanishing holomorphic unit does not change the current \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |f|^2 \end{align*} and does not change the divisor. [guided] The local normal form for a holomorphic function contains a nonzero holomorphic factor, and we must check that this factor contributes nothing. Let \begin{align*} u:U\to\mathbb{C} \end{align*} be holomorphic and nowhere zero. After shrinking $U$ to a simply connected coordinate neighbourhood on which $u$ has no zeros, there exists a holomorphic logarithm \begin{align*} a:U\to\mathbb{C} \end{align*} with $u=e^a$. Then \begin{align*} \log |u|^2 = \log |e^a|^2 = a+\overline{a}. \end{align*} The function $a$ is holomorphic, so $\bar\partial a=0$. The function $\overline{a}$ is antiholomorphic, so $\partial\overline{a}=0$. Therefore \begin{align*} \partial\bar\partial\log |u|^2 = \partial\bar\partial(a+\overline{a}) = \partial(\bar\partial a)+\partial(\bar\partial\overline{a}) = 0. \end{align*} Thus a holomorphic unit has zero $\partial\bar\partial$ contribution. It also has no zeros or poles, so its divisor is zero. Consequently, replacing $f$ by $uf$ leaves both sides of the Poincare-Lelong identity unchanged. [/guided] [/step] [step:Prove the formula for holomorphic functions on the regular part of the divisor] Let $p\in U$ be a smooth point of an irreducible component $Z_{\alpha}$ of the divisor of $f$ and assume $p$ lies on no other divisor component. By the local structure theorem for smooth analytic hypersurfaces, after shrinking to a coordinate neighbourhood $W\subset U$ of $p$ with $W$ disjoint from every other local divisor component, there are holomorphic coordinates \begin{align*} (z_1,\dots,z_n):W\to\mathbb{C}^n \end{align*} such that \begin{align*} Z_{\alpha}\cap W=\{z_1=0\}. \end{align*} Since $f$ has vanishing order $m_{\alpha}$ along $Z_{\alpha}$, there is a nowhere-vanishing holomorphic function \begin{align*} u:W\to\mathbb{C} \end{align*} such that \begin{align*} f(z_1,\dots,z_n)=u(z_1,\dots,z_n)z_1^{m_{\alpha}}. \end{align*} By the unit computation, \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |u|^2=0. \end{align*} By the coordinate-power computation, \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |z_1^{m_{\alpha}}|^2 = m_{\alpha}[\{z_1=0\}]. \end{align*} Therefore on $W$, \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |f|^2 = m_{\alpha}[Z_{\alpha}\cap W]. \end{align*} Summing over the finitely many local components meeting $W$ gives \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |f|^2 = [\operatorname{div}(f)] \end{align*} on the complement of the singular locus of the divisor. [guided] We now use the model computation at a regular point of the zero hypersurface. Let $p\in U$ be a smooth point of an irreducible component $Z_{\alpha}$ of the divisor of $f$, and first restrict to the case where $p$ lies on no other divisor component. The local structure theorem for smooth analytic hypersurfaces gives, after shrinking, a coordinate neighbourhood $W\subset U$ of $p$ that is disjoint from every other local divisor component and holomorphic coordinates \begin{align*} (z_1,\dots,z_n):W\to\mathbb{C}^n \end{align*} such that the hypersurface is the coordinate hyperplane: \begin{align*} Z_{\alpha}\cap W=\{z_1=0\}. \end{align*} Because $m_{\alpha}$ is the order of vanishing of $f$ along $Z_{\alpha}$, the quotient $f/z_1^{m_{\alpha}}$ is holomorphic and nowhere zero after shrinking $W$ if necessary. Thus there is a nowhere-vanishing holomorphic function \begin{align*} u:W\to\mathbb{C} \end{align*} such that \begin{align*} f(z_1,\dots,z_n)=u(z_1,\dots,z_n)z_1^{m_{\alpha}}. \end{align*} Taking logarithms of absolute values gives \begin{align*} \log |f|^2 = \log |u|^2+\log |z_1^{m_{\alpha}}|^2. \end{align*} The unit term has no contribution: \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |u|^2=0. \end{align*} The coordinate-power term is exactly the model current: \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |z_1^{m_{\alpha}}|^2 = m_{\alpha}[\{z_1=0\}]. \end{align*} Combining these two identities gives \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |f|^2 = m_{\alpha}[Z_{\alpha}\cap W]. \end{align*} If several local irreducible components of the divisor meet the chosen neighbourhood, the same argument applies to each component, and the divisor current is the sum of the corresponding integration currents with multiplicities. Therefore the formula holds on the regular part of the divisor. [/guided] [/step] [step:Extend the holomorphic identity across the singular locus] Let $S\subset U$ be the union of the singular loci of the irreducible hypersurfaces appearing in $\operatorname{div}(f)$ and their pairwise intersections. Then $S$ is an analytic subset of complex codimension at least $2$ in $U$. On $U\setminus S$ we have proved \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |f|^2 = [\operatorname{div}(f)]. \end{align*} Both sides are closed currents of order zero with locally finite mass. For the divisor current this is the local finiteness of integration over analytic hypersurfaces. For $\frac{i}{2\pi}\partial\bar\partial\log |f|^2$, it follows from [Logarithmic Potentials Define Locally Finite Currents](/page/Logarithmic%20Potentials%20Define%20Locally%20Finite%20Currents), applied to the logarithmic singularities of $\log |f|^2$ along the divisor. Their difference \begin{align*} R := \frac{i}{2\pi}\partial\bar\partial\log |f|^2 - [\operatorname{div}(f)] \end{align*} is therefore a closed current of order zero supported in $S$. By the [Support Theorem for Currents on Analytic Sets](/page/Support%20Theorem%20for%20Currents%20on%20Analytic%20Sets), a closed current of bidegree $(1,1)$ and order zero supported on such a set is zero. Hence $R=0$ on $U$, and the Poincare-Lelong formula holds for holomorphic nonzero $f$. [guided] The preceding step proves the identity away from the bad set where the divisor is singular or where distinct local components meet. Define \begin{align*} S\subset U \end{align*} to be the union of all singular loci of the irreducible hypersurface components of $\operatorname{div}(f)$ together with their pairwise intersections. Analytic hypersurfaces are smooth outside analytic subsets of complex codimension at least one inside the hypersurface, and intersections of distinct hypersurface components have complex codimension at least two in the ambient manifold. Hence $S$ has complex codimension at least $2$ in $U$. On $U\setminus S$ we have already shown \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |f|^2 = [\operatorname{div}(f)]. \end{align*} Define the difference current \begin{align*} R := \frac{i}{2\pi}\partial\bar\partial\log |f|^2 - [\operatorname{div}(f)]. \end{align*} The current $R$ is closed because both terms are closed. It is supported in $S$ because the two terms agree on $U\setminus S$. It has bidegree $(1,1)$ by construction. Its order-zero and locally finite mass properties require a regularity input: the divisor current has locally finite mass by local finiteness of analytic hypersurface volume, and $\frac{i}{2\pi}\partial\bar\partial\log |f|^2$ has locally finite mass by [Logarithmic Potentials Define Locally Finite Currents](/page/Logarithmic%20Potentials%20Define%20Locally%20Finite%20Currents), since $\log |f|^2$ has only logarithmic singularities along the divisor. Now we invoke the [Support Theorem for Currents on Analytic Sets](/page/Support%20Theorem%20for%20Currents%20on%20Analytic%20Sets): a closed current of bidegree $(1,1)$ and order zero supported on an analytic subset of complex codimension at least $2$ must vanish. Applying it to $R$ gives \begin{align*} R=0. \end{align*} Therefore \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |f|^2 = [\operatorname{div}(f)] \end{align*} on all of $U$ for every nonzero holomorphic function $f$. [/guided] [/step] [step:Pass from holomorphic functions to meromorphic functions] Let $f$ be a nonzero [meromorphic function](/page/Meromorphic%20Function) on $U$. For every point of $U$, choose a neighbourhood $W\subset U$ and holomorphic functions \begin{align*} g:W&\to\mathbb{C},\\ q:W&\to\mathbb{C} \end{align*} such that $q$ is not identically zero and \begin{align*} f=\frac{g}{q} \end{align*} as a meromorphic function on $W$. After cancelling common irreducible factors, assume $g$ and $q$ have no common hypersurface component in their zero divisors. Then \begin{align*} \operatorname{div}(f)=\operatorname{div}(g)-\operatorname{div}(q). \end{align*} On $W\setminus(\{g=0\}\cup\{q=0\})$, \begin{align*} \log |f|^2=\log |g|^2-\log |q|^2. \end{align*} Since both sides are locally integrable and agree off an analytic hypersurface, they define the same current. Therefore, by linearity and the holomorphic case, \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |f|^2 &= \frac{i}{2\pi}\partial\bar\partial\log |g|^2 - \frac{i}{2\pi}\partial\bar\partial\log |q|^2\\ &= [\operatorname{div}(g)]-[\operatorname{div}(q)]\\ &= [\operatorname{div}(f)]. \end{align*} Thus the formula holds for nonzero meromorphic functions. [guided] A meromorphic function is locally a quotient of holomorphic functions. Fix a point of $U$. There is a neighbourhood $W\subset U$ and holomorphic functions \begin{align*} g:W&\to\mathbb{C},\\ q:W&\to\mathbb{C} \end{align*} with $q$ not identically zero such that \begin{align*} f=\frac{g}{q} \end{align*} as a meromorphic function. After cancelling common irreducible hypersurface factors, the zero divisors of $g$ and $q$ have no common component. This cancellation is exactly what makes the divisor of the quotient equal to the difference of divisors: \begin{align*} \operatorname{div}(f)=\operatorname{div}(g)-\operatorname{div}(q). \end{align*} Away from the union of the zero sets of $g$ and $q$, the ordinary logarithm identity gives \begin{align*} \log |f|^2 = \log \left|\frac{g}{q}\right|^2 = \log |g|^2-\log |q|^2. \end{align*} Both sides are locally integrable, and the exceptional set is an analytic hypersurface. Since locally integrable functions that agree outside a set of Lebesgue measure zero define the same current, the identity holds as an identity of currents: \begin{align*} T_{\log |f|^2}=T_{\log |g|^2}-T_{\log |q|^2}. \end{align*} Applying the linear operator $\frac{i}{2\pi}\partial\bar\partial$ and using the holomorphic case for $g$ and $q$ gives \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |f|^2 &= \frac{i}{2\pi}\partial\bar\partial\log |g|^2 - \frac{i}{2\pi}\partial\bar\partial\log |q|^2\\ &= [\operatorname{div}(g)]-[\operatorname{div}(q)]\\ &= [\operatorname{div}(f)]. \end{align*} Because the argument is local and the local representations of $f$ agree on overlaps as meromorphic functions, these local current identities glue. Hence the Poincare-Lelong formula holds for every nonzero meromorphic function. [/guided] [/step] [step:Reduce meromorphic sections to the meromorphic function case] Let $L\to M$ be a holomorphic line bundle with smooth Hermitian metric $h$, and let $s$ be a nonzero meromorphic section of $L$. Fix a coordinate neighbourhood $U\subset M$ on which $L$ admits a nowhere-vanishing holomorphic frame \begin{align*} e:U&\to L|_U. \end{align*} There is a nonzero meromorphic function \begin{align*} f:U&\dashrightarrow\mathbb{C} \end{align*} such that \begin{align*} s=f e. \end{align*} Define the smooth real-valued weight \begin{align*} \varphi:U&\to\mathbb{R} \end{align*} by \begin{align*} |e|_h^2=e^{-\varphi}. \end{align*} Then \begin{align*} |s|_h^2=|f|^2|e|_h^2=|f|^2e^{-\varphi}, \end{align*} so, as locally integrable functions and hence as currents, \begin{align*} \log |s|_h^2=\log |f|^2-\varphi. \end{align*} Applying $\frac{i}{2\pi}\partial\bar\partial$ gives \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |s|_h^2 &= \frac{i}{2\pi}\partial\bar\partial\log |f|^2 - \frac{i}{2\pi}\partial\bar\partial\varphi\\ &= [\operatorname{div}(f)]-c_1(L,h). \end{align*} Since the frame $e$ is nowhere vanishing, the divisor of $s=f e$ on $U$ is precisely the divisor of $f$: \begin{align*} \operatorname{div}(s)|_U=\operatorname{div}(f). \end{align*} Therefore on $U$, \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |s|_h^2 = [\operatorname{div}(s)]-c_1(L,h). \end{align*} These local identities are compatible on overlaps, because both sides are globally defined currents. Hence the formula holds on $M$. [guided] We now prove the line bundle version. The key point is that a section becomes an ordinary meromorphic function after choosing a local holomorphic frame, while the Hermitian metric contributes exactly the curvature term. Fix a coordinate neighbourhood $U\subset M$ on which the line bundle $L$ is trivial, and choose a nowhere-vanishing holomorphic frame \begin{align*} e:U&\to L|_U. \end{align*} Because $e$ spans each fiber of $L|_U$, there is a unique nonzero meromorphic function \begin{align*} f:U&\dashrightarrow\mathbb{C} \end{align*} such that \begin{align*} s=f e. \end{align*} The frame has no zeros or poles, so the zeros and poles of $s$ are exactly the zeros and poles of $f$ with the same multiplicities: \begin{align*} \operatorname{div}(s)|_U=\operatorname{div}(f). \end{align*} Next encode the Hermitian metric in this frame. Define the smooth real-valued function \begin{align*} \varphi:U&\to\mathbb{R} \end{align*} by \begin{align*} |e|_h^2=e^{-\varphi}. \end{align*} Then \begin{align*} |s|_h^2 = |f e|_h^2 = |f|^2|e|_h^2 = |f|^2e^{-\varphi}. \end{align*} Taking logarithms gives the current identity \begin{align*} \log |s|_h^2=\log |f|^2-\varphi. \end{align*} Here $\log |f|^2$ is locally integrable by the meromorphic function case, and $\varphi$ is smooth. Apply $\frac{i}{2\pi}\partial\bar\partial$ to both sides: \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |s|_h^2 &= \frac{i}{2\pi}\partial\bar\partial\log |f|^2 - \frac{i}{2\pi}\partial\bar\partial\varphi. \end{align*} The meromorphic function Poincare-Lelong formula gives \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |f|^2 = [\operatorname{div}(f)]. \end{align*} By the definition of the Chern form in the frame $e$, \begin{align*} c_1(L,h)=\frac{i}{2\pi}\partial\bar\partial\varphi. \end{align*} Substituting both identities yields \begin{align*} \frac{i}{2\pi}\partial\bar\partial\log |s|_h^2 = [\operatorname{div}(f)]-c_1(L,h) = [\operatorname{div}(s)]-c_1(L,h) \end{align*} on $U$. Since both sides are globally defined currents and the local computations agree on overlaps, the equality holds on the whole manifold $M$. [/guided] [/step]