Chern currents of coherent sheaves

Journal article, 2022

Given a finite 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 certain smooth Chern forms of F, that represents 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 first nontrivial component of this Chern current coincides with (a constant times) the fundamental cycle of 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.