Proposition 5.20.4 (Lelong–Poincaré)0C02CD3B5D59

Proof. In a holomorphic frame $e$ of $L$, write $s=fe$ and $|e|_h^2=\exp(-\rho)$. Then $\log|s|_h^2=\log|f|^2-\rho$ and $c_1(L,h)=\mathrm{dd}^{\mathrm c}\rho$. The normalized Poincaré--Lelong formula gives $\mathrm{dd}^{\mathrm c}\log|f|^2=[Z(f)]$. Substitution proves \[c_1(L,h)+\mathrm{dd}^{\mathrm c}\log|s|_h^2=[Z(s)].\] Changing the frame multiplies $f$ by a nowhere-vanishing holomorphic function; its logarithmic squared modulus is pluriharmonic, so the local identities agree on overlaps. This also checks the factor $1/(2\pi)$ in the normalization.