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 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.

• 14C17 - Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
• 14F06 - Sheaves in algebraic geometry
• 32A27 - Residues for several complex variables
• 32C30 - Integration on analytic sets and spaces, currents
• 53C05 - Connections (general theory)

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