Chern currents of coherent sheaves

Given a finite locally free resolution of a coherent analytic sheaf $\mathcal{F}$, equipped with Hermitian metrics and connections, we construct an explicit current, obtained as the limit of certain smooth Chern forms of $\mathcal{F}$, that represents the Chern class of $\mathcal{F}$ and has support on the support of $\mathcal{F}$. If the connections are $(1,0)$-connections and $\mathcal{F}$ has pure dimension, then the first nontrivial component of this Chern current coincides with (a constant times) the fundamental cycle of $\mathcal{F}$. The proof of this goes through a generalized Poincar'e-Lelong formula, previously obtained by the authors, and a result that relates the Chern current to the residue current associated with the locally free resolution.