Gaga and quasicoherent sheaf

Question

Let $X$ be a complete algebraic variety over $\mathbb{C}$. Ad Serre GAGA stases, its analytification $X^{an}$ is compact and the analytification functor induces an equivalence of categories between $Mod_c(\mathcal{O}_X)$ and $Mod_c(\mathcal{O}_{X^{an}})$. What can we say about quasicoherent modules?

Answer 1

Things can get pretty weird when dealing with quasi-coherent sheaves on analytic spaces, much weirder than you would expect if you were dealing with quasi-coherent sheaves on schemes.

Here's an algebraic perspective: every quasi-coherent sheaf on a scheme is a filtered colimit of it's coherent subsheaves, and since cohomology commutes with filtered colimits, if we know a statement for coherent sheaves, we have a recipe for figuring out what should happen on all quasi-coherent sheaves.

Unfortunately, this idea breaks completely when one enters the analytic setting. Here is an instructive example, originally due to Gabber (in the realm of rigid analytic geometry, but it works here): A quasi-coherent sheaf $\mathcal{F}$ on a Stein analytic space $X$ so that $H^1(X,\mathcal{F}) \neq 0$. (Why is this bad? For coherent sheaves $\mathcal{F}$ on a Stein space $X$, we know $H^p(X,\mathcal{F}) = 0$ for all positive $p$, and $\mathcal{F}$ is uniquely determined up to isomorphism by $H^0(X,\mathcal{F})$.)

Let $X$ be the open unit disc. Let $x',x''$ be two distinct closed points in $X$. Let $U' = X\setminus \{x'\}$, $U'' = X\setminus \{x''\}$, and $U=U'\cap U''$. Let $$\mathcal{F}' = \bigoplus_{n\in \Bbb Z} \mathcal{O}_{U'} e_n', \\ \mathcal{F}'' = \bigoplus_{n\in \Bbb Z} \mathcal{O}_{U''} e_n''$$ be two free sheaves with countably infinite rank on $U',U''$ respectively. We will now glue these two sheaves to produce an $\mathcal{O}_X$ module $\mathcal{F}$ with no nonzero global sections. This $\mathcal{F} $ will be quasi-coherent because $\mathcal{F}|_{U'}$ and $\mathcal{F}|_{U''}$ are direct limits of coherent sheaves, and if $t$ is the standard coordinate on $X$, then for $X'$ a slightly smaller open disc centered at $0$, the cohomology sequence associated to $$ 0 \to \mathcal{F} \stackrel{t}{\to} \mathcal{F} \to \mathcal{F}/t\mathcal{F}\to 0$$ provides an injection $H^0(X',\mathcal{F}/t\mathcal{F})\hookrightarrow H^1(X',\mathcal{F})$. As $\mathcal{F}/t\mathcal{F}$ is a skyscraper sheaf supported at the origin, this gives $H^1(X',\mathcal{F})\neq 0$.

To construct the correct gluing, let $h\in\mathcal{O}_X(U)$ be a function with essential singularities at $x',x''$ (for instance, $e^{\frac{1}{x'}+\frac{1}{x''}}$). We define $\mathcal{F}$ by identifying $\mathcal{F}'|_U$ and $\mathcal{F}''|_U$ with the free sheaf $\bigoplus_{n\in\Bbb Z} \mathcal{O}_U e_n$ by $$e_{2m} = e'_{2m}|_U = e_{2m}''|_U + h e_{2m+1}''|_U \\ e_{2m+1} = e_{2m+1}''|_U = e_{2m+1}'|U + h e_{2m+2}'|_U$$ for $m\in \Bbb Z$.

Let $f\in\mathcal{F}(X)$ be a global section, so that on any open set $V\subset U$ we may write it as a finite linear combination of terms of the form $f_ne_n$ for $f_n\in\mathcal{O}_X(V)$. For any open (and therefore Stein) $V\subset U$, we have that the restriction map $\mathcal{F}(U)\to \mathcal{F}(V)$ is injective, and thus $f$ is a finite $\mathcal{O}_X(U)$-linear combination of $e_n$.

By the definition of $\mathcal{F}$, the image of the injective restriction map $\mathcal{F}(X)\to\mathcal{F}(U)$ consists of finite sums $f = \sum f_n e_n$ with $f_n\in\mathcal{O}_X(U)$ so that $f_n$ is analytic at $x'$ for even $n$ and $x''$ for odd $n$, while $f_n+hf_{n-1}$ is analytic at $x'$ for odd $n$ and $x''$ for even $n$.

So if $f=\sum f_ne_n$ and $n_0$ is the maximal $n$ so that $f_n\neq 0$, we then have that $f_{n_0}$ and $hf_{n_0}$ are both analytic at one of $x',x''$ depending on the parity of $n_0$. Then $hf_{n_0}/f_{n_0}$ is meromorphic at $x'$ or $x''$, which is a contradiciton, so $\mathcal{F}(X) = 0$.

This example may also be used to produce a directed system of quasi-coherent sheaves so that it's limit is not quasicoherent.