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arXiv · 2105.14843

Chern currents of coherent sheaves

Richard Lärkäng·Elizabeth Wulcan
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Abstract

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.

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Richard Lärkäng, Elizabeth Wulcan. 2021-05-31. Chern currents of coherent sheaves. https://doi.org/10.46298/epiga.2022.8653

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