  We prove a generalization of the classical Poincar\'e-Lelong formula. Given a
holomorphic section $f$, with zero set $Z$, of a Hermitian vector bundle $E\to
X$, let $S$ be the line bundle over $X\setminus Z$ spanned by $f$ and let
$Q=E/S$. Then the Chern form $c(D_Q)$ is locally integrable and closed in $X$
and there is a current $W$ such that $dd^cW=c(D_E)-c(D_Q)-M,$ where $M$ is a
current with support on $Z$. In particular, the top Bott-Chern class is
represented by a current with support on $Z$. We discuss positivity of these
currents, and we also reveal a close relation with principal value and residue
currents of Cauchy-Fantappi\`e-Leray type.
