Questions on Chern characters.

Questions on Chern characters.

Let $X$ be a complex manifold and $\mathcal{O}_p$ a skyscraper sheaf.
How can one compute the Chern character $ch(\mathcal{O}_p)$?
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.