Chern currents of coherent sheaves
Épijournal de Géométrie Algébrique, Tome 6 (2022)
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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.
@article{10_46298_epiga_2022_8653,
author = {L\"ark\"ang, Richard and Wulcan, Elizabeth},
title = {Chern currents of coherent sheaves},
journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
publisher = {mathdoc},
volume = {6},
year = {2022},
doi = {10.46298/epiga.2022.8653},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.46298/epiga.2022.8653/}
}
TY - JOUR AU - Lärkäng, Richard AU - Wulcan, Elizabeth TI - Chern currents of coherent sheaves JO - Épijournal de Géométrie Algébrique PY - 2022 VL - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.46298/epiga.2022.8653/ DO - 10.46298/epiga.2022.8653 LA - en ID - 10_46298_epiga_2022_8653 ER -
Lärkäng, Richard; Wulcan, Elizabeth. Chern currents of coherent sheaves. Épijournal de Géométrie Algébrique, Tome 6 (2022). doi: 10.46298/epiga.2022.8653
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