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Proposition 10.6.495892535D04F

Proposition 10.6.4.95892535D04F For a nonzero holomorphic section $s$ of $L$ and a smooth metric $h_0$,

A8F577E012B3 10.6.5
\begin{equation} c_1(L,h_0)+\mathrm{dd}^{\mathrm{c}}\log |s|_{h_0}^2=[Z(s)], \end{equation}

where $[Z(s)]$ is the current of integration along the zero divisor of $s$, counted with multiplicities. In particular $[Z(s)]$ represents $c_1(L)$.

Proof. In a local frame write $s=fe$. Since $\log |s|_{h_0}^2=\log |f|^2+\log |e|_{h_0}^2$, the left side equals $\mathrm{dd}^{\mathrm{c}}\log |f|^2$, which is the current of integration along the zero divisor of $f$ by theorem 5.5.6. These local divisors agree on overlaps, because the transition functions have no zeros. $\square$

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