# Questions on Chern characters.

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.

• 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. Apr 22, 2020 at 17:30