# Making coherent sheaves with nonvanishing higher Chern classes

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) 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.

• Why does the title of this question differ from the title on the list of new questions? Apr 1, 2021 at 20:56
• @JasonStarr I edited it to improve the title and I think it takes a minute to update. Apr 1, 2021 at 21:05
• Surely there's a typo in the current title? Should be "making"? Apr 1, 2021 at 23:05
• For $n=1$ and any $k$ you can consider the ideal sheaf of a subvariety of codimension $k$.
– naf
Apr 2, 2021 at 2:50

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 $$n take $$F$$ to be the quotient of $$E$$ by $$k-n$$ general sections of $$E$$.