3
$\begingroup$

What are examples of coherent sheaves $\mathfrak{F}$ on a compact complex $n$-fold with $\dim \operatorname{Supp} \mathfrak{F}=p$ , where $0\leq p \leq n$ ? And how can they be described in local coordinates under the equivalence with vector bundles?

$\endgroup$
2
  • 4
    $\begingroup$ This is equivalent to the question whether for every $p$ there is some closed subvariety of dimension $p$ (because if you have one, take the corresponding ideal sheaf $I$ and $\mathcal{O}_X / I$ is a coherent sheaf with the correct suppoert). I don't know if this is true. $\endgroup$ – Martin Brandenburg May 8 '11 at 9:24
  • $\begingroup$ I edited your question slightly. For the original question, "Who can given me an example of ...?", a perfectly valid answer would be "I can". $\endgroup$ – David Loeffler May 8 '11 at 10:03
5
$\begingroup$

Coherent sheaves are much more general than vector bundles. A vector bundle corresponds to a locally free sheaf; the support of such a sheaf is the whole variety.

If $X$ is your variety, of dimension $n$, and $Y \subset X$ is some closed subvariety of $X$, and $\mathcal{F}$ is some sheaf on $Y$, then there is a coherent sheaf $\mathcal{G}$ on $X$ whose sections over an open set $U$ are the sections of $Y$ on $U \cap Y$ (this is the "extension of $\mathcal{F}$ by 0"). If $\mathcal{F}$ is locally free (e.g. if $\mathcal{F}$ is the structure sheaf of $Y$ itself) then the support of $\mathcal{G}$ is precisely $Y$. This gives a nice supply of coherent sheaves whose support is easy to understand.

$\endgroup$
1
  • $\begingroup$ think kernels and cokernels of maps of vector bundles. $\endgroup$ – roy smith May 9 '11 at 4:13
0
$\begingroup$

Take an affine patch $S\subseteq X$, then $S\hookrightarrow\mathbb{A}^N\subseteq \mathbb{P}^N$ for some $N\geq n$. Now take a (generic) hyperplane section of $S$ in $\mathbb{P}^n$, this will have dimension $n-1$. Let $Y$ be the closure of this in $X$, then $Y$ is a codimension 1 closed subvariety of $X$. Re-applying the process enough times to $Y$ itself will give you a closed $p$-dimensional subvariety $Z$ of $X$. The ideal sheaf $\mathcal{I}_Z$ is then a coherent sheaf on $X$, and the sheaf $\mathcal{O}_Z:=\mathcal{O}_X/\mathcal{I}_Z$ is coherent and supported on $Z$.

Note that this method cannot be applied to general complex analytic varieties.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.