Reason for studying coherent sheaves on complex manifolds.

Question

Hello everybody! I would be interested in knowing, what the reason is for investigating coherent sheaves on complex manifolds. By definition a sheaf $F$ on a complex manifold $X$ is coherent, when it is locally finitely presented (i.e. every point possesses a neighborhood $U$, such that $F \mid U$ is the image of $O_X^n\mid U$ under some surjective morphism) and locally of finite presentation (i.e. for an open súbset $U \subseteq X$ and a surjective morphism $f: O_U^n \rightarrow F \mid U$ of sheaves the kernel of $f$ is locally finitely presented). My question now is: Why are such sheaves especially interesting? Why might somebody be interested in studying them or what is the intuition behind them? Every help will be appreciated.

Answer 1

First let me note that your definition of coherent sheaf is misleading: it implies that $\mathcal O_X$ is coherent by definition, whereas in reality coherence of $\mathcal O_X$ is a very deep theorem due to Oka .

The correct definition is that a sheaf $\mathcal F$ of $\mathcal O_X$-Modules is coherent if :
1) $\mathcal F$ is locally finitely generated : $X$ can be covered by open subsets $U$ on which there exist surjections $\mathcal O_U^N\to \mathcal F\mid U$ .
and
2) For any open subset $V\subset X$ and any morphism $f:\mathcal O_V^s\to\mathcal F\mid V $ the sheaf $Ker(f)$ on $V$ is locally finitely generated .

As for the consequences of coherence, here is a typical application: if a sequence of coherent sheaves $ \mathcal F' \to \mathcal F\to\mathcal {F''} $ is exact at $x\in X$ (stalkwise) then it is exact on an open neighbourhood of $x$.

The real power of coherence however comes through Cartan's Theorems A and B for coherent sheaves on a Stein manifold.
These theorems have innumerable consequences: on a Stein manifold
(i) Every meromorphic function is the quotient of two global holomorphic functions.
(ii) Every topological line bundle has one and only one holomorphic structure,
(iii) Every closed analytic subset is the zero set of a family of globally defined holomorphic functions.
(iv) Every holomorphic function on a closed analytic subset can be extended to a holomorphic function on the whole Stein manifold.
(v) Given global holomorphic functions $f_1,...,f_r$ without a common zero, there exist global holomorphic functions $g_1,...,g_r$ with $f_1g_1+\cdots +f_rg_r=1$.

Just to illustrate how easy theorems become with those powerful tools in hand, let me prove (ii).
Start from the exponential sequence of sheaves $ 0\to \mathbb Z \to \mathcal O\to \mathcal O^\ast \to 0 $ on the Stein manifold $X$ .
The associated long exact sequence has as a fragment $$ \cdots \to H^1(X,\mathcal O) \to H^1(X,\mathcal O^\ast) \to H^2(X,\mathbb Z) \to H^2 (X,\mathcal O) \to \cdots $$ Since $H^1(X,\mathcal O) = H^2 (X,\mathcal O)=0 $ by theorem B, we get the isomorphism $$H^1(X,\mathcal O^\ast ) \cong H^2(X,\mathbb Z)$$
It factorizes as $H^1(X,\mathcal O^\ast )\to H^1(X,\mathcal C^\ast) \to H^2(X,\mathbb Z)$ and since $ H^1(X,\mathcal C^\ast) \stackrel {\cong}{\to }H^2(X,\mathbb Z)$ is well known to be the isomorphism given by the first chern class we get the result (ii) in the more precise form:

On a Stein manifold $X$ the first Chern class induces an isomorphism of abelian groups $$\text {Pic}(X)=H^1(X,\mathcal O^\ast )\stackrel {c_1}{\cong}H^2(X,\mathbb Z)$$

Answer 2

Let me complete Georges' good answer by this. A natural class of object to study on manifolds are vector bundles. There are many reasons to be interested in them, but to name one, there are the examples of the tangent and cotangent bundles, of which the sections are vector fields and differential forms, two fundamental classes of geometric object on the manifold. Now vector bundles are essentially the same thing as locally free coherent sheaves (to a bundle one attach its sheaf which to an one set $U$ attached the section of the bundle on $U$). This is a reason to study at least certain coherent sheaves, namely the locally free ones. But it turn out that the category of locally free coherent sheaves has an important short-coming: it is not abelian. Certain morphisms have trivial kernel and cockerel, but are not isomorphisms, like the multiplication by a function which vanishes at exactly one point from the structural sheaf to itself. So one has to work with a larger category, which is abelian, namely the category of coherent sheaves.