## Richard Lärkäng ; Elizabeth Wulcan - Chern currents of coherent sheaves

epiga:8653 - Épijournal de Géométrie Algébrique, July 30, 2022, Volume 6 - https://doi.org/10.46298/epiga.2022.8653
Chern currents of coherent sheaves

Authors: Richard Lärkäng ; Elizabeth Wulcan

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.

Volume: Volume 6
Published on: July 30, 2022
Accepted on: July 22, 2022
Submitted on: November 3, 2021
Keywords: Mathematics - Complex Variables,Mathematics - Algebraic Geometry,32A27, 14C17, 32C30, 14F06, 53C05