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?
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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 BrandenburgMay 8, 2011 at 9:24
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$\begingroup$ 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 LoefflerMay 8, 2011 at 10:03
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2 Answers
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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.
answered May 8, 2011 at 9:12
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$\begingroup$ think kernels and cokernels of maps of vector bundles. $\endgroup$ May 9, 2011 at 4:13
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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.
