Let $C$ be a smooth curve in a complex threefold $X$. How can I see that $$ \mathrm{ch}(\mathcal{O}_C)=(0,0,[C],\chi(\mathcal{O}_C))\in H^0\oplus H^2 \oplus H^4\oplus H^6, $$ where $H^0\cong \mathbb{Z}, H^4\cong H_2, H^6\cong \mathbb{Z}$? Here $\mathrm{ch}(E)$ stands for the Chern character of a sheaf $E$.
More generally, are there any good way to compute $\mathrm{ch}(\mathcal{O}_Z)$ for any subvariety $Z \subset Y$? Isn't the notation $\mathrm{ch}(\mathcal{O}_C)$ misleading in the sense that it depends not only on $C$ but also the ambient space $X$?
Here is a simple thought. If $D$ is a divisor on any complex manifold $Y$, then $$ \mathrm{ch}(\mathcal{O}_D)=\mathrm{ch}(\mathcal{O}_Y)-\mathrm{ch}(\mathcal{O}_Y(-D)) $$ can be easily computed by $c_1(\mathcal{O}_Y(D))=[D]$.