Chern currents of coherent sheaves
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866909755560165376 |
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| author | Lärkäng, Richard Wulcan, Elizabeth |
| author_facet | Lärkäng, Richard Wulcan, Elizabeth |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_14843 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Chern currents of coherent sheaves Lärkäng, Richard Wulcan, Elizabeth Complex Variables Algebraic Geometry 32A27, 14C17, 32C30, 14F06, 53C05 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. |
| title | Chern currents of coherent sheaves |
| topic | Complex Variables Algebraic Geometry 32A27, 14C17, 32C30, 14F06, 53C05 |
| url | https://arxiv.org/abs/2105.14843 |