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?



who can give me a example of coherent sheaf
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.


