# Chern character of a coherent sheaf over a projective manifold [closed]

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.

• I asked about the close votes in the CURED chatroom if anyone is interested. Aug 3 at 21:02
• Insufficient context. Aug 4 at 13:43

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 $$K_S(M) \xrightarrow{j} K_B(M) \xrightarrow{\operatorname{ch}} H^*(M; \mathbb{Q}).$$