Chern Currents of Coherent Sheaves

I will present some joint work with Elizabeth Wulcan. Given a locally free resolution of a coherent analytic sheaf $F$, equipped with Hermitian metrics and connections, we construct an explicit current, obtained as the limit of smooth Chern forms of $F$, that represent the Chern class of $F$ and has support on the support of $F$. If the connections are $(1, 0)$-connections and $F$ has pure dimension, then the non-trivial part of lowest degree of this Chern current coincides with (a constant times) the fundamental cycle of $F$. The proof of this result utilizes the theory of residue currents, and goes through a generalized Poincaré-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.