The main theme of my question is how to compute the holomorphic Euler characteristic of a coherent sheaf via something like Hirzebruch-Riemann-Roch
Let $X$ be smooth, projective, Calabi-Yau manifold of dimension $n \geq 3$. Let $E$ be a coherent sheaf on $X$ with one-dimensional support not necessarily pure. In other words, I would like E to be supported on a curve and points, in general.
I would like to compute the holomorphic Euler characteristic $\chi(X, E)$ of $E$. My partial attempt is as follows, but I'm not positive it's legitimate:
First I assume that $E$ is pure of dimension one, with smooth, connected support. In this case, there exists a curve $\iota :C \hookrightarrow X$ with a vector bundle $F$ on $C$ such that $E=\iota_{*}F$, and
$$\text{ch}_{k}(E) =0 \,\,\,\,\,\,\,\,\text{for all}\,\,\,\,\,\,\,\,\, k<n-1$$
Naively applying Hirzebruch-Riemann-Roch, I find
$$\chi(X, E) = \int_{X} \text{ch}(E) \text{td}(X) = \int_{X} \text{ch}_{n}(E),$$
where the final equality requires the Calabi-Yau condition. My two questions:
1. Can I apply HRR in this manner even though $E$ is not a vector bundle?
and
2. What if $E$ is not pure? If $E$ is also supported on points, I'd like to still argue that
$$ \chi(X, E)=\int_{X} \text{ch}_{n}(E).$$
Is this possible?