Let $\mathcal{F}$ be a coherent sheaf on a variety $X$, and assume $\mathcal{F}$ has generic rank $n$. I expect (see e.g. [here](https://www.uni-due.de/%7Ehm0002/Artikel/CohNote_v1a.pdf)) that this actually puts no conditions on its Chern classes $c\_1(\mathcal{F}),c\_2(\mathcal{F}),...$ . Constrast this with the vector bundle case, where all Chern classes above the $n$th will vanish. However, I have no idea how to construct examples with this non-vanishing property.


**Question:** Fix $n\ge 0$. For each positive integer $k$, is there a nice example of a variety $X\_k$ with a coherent sheaf $\mathcal{F}\_k$ of generic rank $n$ and $c\_k(\mathcal{F}\_k)\ne 0$?


Even the rank $n=1$ case seems unclear to me. I imagine it might be possible to answer the question using for $X\_k$ projective spaces or partial flag varieties, but I'm not sure.



Making coherent sheaves with nonvanishing higher Chern classes
Take any affine variety $X$ of dimension $k$ which has a vector bundle $E$ of rank $k$ with $c\_k(E)\neq 0$. If $n\geq k$ take $F=E\oplus O\_X^{n-k}$. If $n<k$ take $F$ to be the quotient of $E$ by $k-n$ general sections of $E$.


