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Let $X$ be a complex manifold and $\mathcal{O}_p$ a skyscraper sheaf.

  1. How can one compute the Chern character $ch(\mathcal{O}_p)$?
  2. For vecoto bundles, we have $ch(V)\cup ch(W)=ch(V\otimes W)$, but this does not generalize to sheaves, right? Otherwise $ch(\mathcal{O}_p)\cup ch(L)=ch(\mathcal{O}_p\otimes L)=ch(\mathcal{O}_p)$ for any line bundle $L$. This is bizarre unless $ch(\mathcal{O}_p)=0$.

Thank you for your interest and help.

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    $\begingroup$ The chern classes are easy to calculate. In your case, $c_0=1$ (as usual), $c_i=0$ for $0<i<\dim X=n$ and $c_n=(-1)^{n-1}l$ where $l$ is the length of the skyscraper sheaf. Of course, higher chern classes are zero. $\endgroup$
    – Mohan
    Apr 22, 2020 at 17:30

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  1. I'd say that one needs a finite locally free resolution of a skyscraper sheaf, but I don't think it is possible to find one explicitly enough in general.

  2. It is known than such formula is not true in general for coherent sheaves, in the correct formula we need derived tensor product of coherent sheaves.

You may find this notes http://math.columbia.edu/~dejong/seminar/note_on_algebraic_de_Rham_cohomology.pdf useful, in particular there is a calculation of a Chern character of a skyscraper sheaf on projective space on p.8, and also a short discussion of the formula for tensor product on the same page.

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  • $\begingroup$ I see. Thanks, Alex. $\endgroup$
    – user64726
    Aug 25, 2013 at 20:41

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