Chern character of a coherent sheaf over a projective manifold

Question

I am looking for a reference describing the definition/construction of the Chern character of a coherent sheaf over a projective manifold (or variety) using a resolution by vector bundle.

Answer 1

The following can be found at the end of section $2.1$ of Kobayashi's Differential Geometry of Complex Vector Bundles.

Fix a complex manifold $M$.

Let $B(M)$ be the category of holomorphic vector bundles on $M$, and form the free abelian group $\mathbb{Z}[B(M)]$ generated by isomorphism classes $[E]$ of holomorphic vector bundles $E$ in $B(M)$. For each short exact sequence $0 \to E' \to E \to E'' \to 0$, we form the element $-[E'] + [E] - [E''] \in \mathbb{Z}[B(M)]$. Let $A(B(M))$ be the subgroup of $\mathbb{Z}[B(M)]$ generated by such elements. We define the Grothendieck group $K_B(M) := \mathbb{Z}(B(M))/A(B(M))$. Note that the Chern character defines a ring homomorphism $$\operatorname{ch} : K_B(M) \to H^*(M; \mathbb{Q}).$$

Now let $S(M)$ be the category of coherent sheaves on $M$, and form the free abelian group $\mathbb{Z}[S(M)]$ generated by isomorphism classes $[\mathcal{S}]$ of coherent sheaves $\mathcal{S}$ in $S(M)$. For each short exact sequence $0 \to \mathcal{S}' \to \mathcal{S} \to \mathcal{S}'' \to 0$, we form the element $-[\mathcal{S}'] + [\mathcal{S}] - [\mathcal{S}''] \in \mathbb{Z}[S(M)]$. Let $A(S(M))$ be the subgroup of $\mathbb{Z}[S(M)]$ generated by such elements. We define the Grothendieck group $K_S(M) := \mathbb{Z}(S(M))/A(S(M))$. The map $[E] \to [\mathcal{O}(E)]$ induces a $K_B(M)$-module homomorphism $$i : K_B(M) \to K_S(M).$$

Theorem: If $M$ is projective algebraic, then $i$ is an isomorphism.

By a result of Serre (see the book for a reference), given a coherent sheaf $\mathcal{S}$, there is a global resolution $0 \to \mathcal{E}_n \to \dots \to \mathcal{E}_1 \to \mathcal{E}_0 \to \mathcal{S} \to 0$ where $\mathcal{E}_i$ are locally free coherent sheaves. Note that $\mathcal{E}_i = \mathcal{O}(E_i)$ for some holomorphic vector bundles $E_i$, so the inverse of $i$ is given by $j : K_S(M) \to K_B(M)$, $[\mathcal{S}] \mapsto \sum_{i=0}^n(-1)^i[E_i]$.

With this in mind, we define $\operatorname{ch} : K_S(M) \to H^*(M; \mathbb{Q})$ to be the composition $$K_S(M) \xrightarrow{j} K_B(M) \xrightarrow{\operatorname{ch}} H^*(M; \mathbb{Q}).$$