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even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting.

Let us consider a smooth projective algebraic variety $X$ over $\mathbb{C}$. Whenever I have a coherent sheaf $F$ over $X$, I can get a resolution of $F$ by a (bounded) complex $E^\bullet$ of vector bundles on $X$ and define $\mathrm{ch}(F):=\sum (-1)^i\mathrm{ch}(E^i)$. Since I (should) know how to compute Chern classes of vector bundles via splitting principle, this makes sense. I can indeed already start with a complex of coherent sheaves $F^\bullet$ on $X$, replace by vector bundles and do the exact same thing, defining a map $\mathrm{ch}\colon D^b(CohX)\to H^\ast (X,\mathbb{Q})$, or $CH^\ast(X)$, or $HH$ something, depending on your taste (I stick to the cohomology one), which actually factors through $K(X)$ because of additivity of $\mathrm{ch}$ with respect to short exact sequences.

The other good point of vector bundles is that we have an intuition of what Chern classes are, that is some kind of objects which measure wheter certain number of generic sections of our bundle are linearly dependent.

I am now trying to reconciliate the intuition with the homological nonsense, and I would like to have some example of what can we know about some sheaf when we know its Chern character.

A precious one for me would be the possibility to characterise torsion sheaves on $X$ (e.g. $\mathcal{O_x}$ skyscrapers, $\mathcal{O_Z}$ structure sheaf of a curve/subvariety) K3 surface.

Thank you for your time!

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    $\begingroup$ Chern classes of vector bundles are essentially cohomological descriptions of the zero sets of sets of sections. The category of coherent sheaves are, in a sense, an "abelian envelope" of the category of vector bundles, and Chern classes on coherent sheaves are the canonical extension of Chern classes on vector bundles to the envelope. $\endgroup$ – Mariano Suárez-Álvarez Nov 24 '15 at 21:33
  • $\begingroup$ The Chern character gives you the Hilbert polynomial for some fixed polarization (via HRR), and that for example recovers the dimension of the support of the sheaf (as its degree). $\endgroup$ – Mattia Talpo Nov 24 '15 at 23:02
  • $\begingroup$ @MattiaTalpo: thank you for your comment, this is really what I am looking for! I think I got a grasp of the idea, but I would be glad if you could be more precise $\endgroup$ – mGb Dec 5 '15 at 18:38
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I'm expanding on my comment, not really giving a complete answer.

Say that $L$ is a polarization on your smooth projective $X$. Then HRR applied to $F\otimes L^n$ gives

$$ P_L(F)(n)=\chi(F\otimes L^{\otimes n})=\int_X ch(F\otimes L^{\otimes n})\cdot td(X)=\int_X ch(F)\cdot e^{c_1(L)}\cdot td(X) $$

where $P_L(F)$ is the Hilbert polynomial of $F$ with respect to $L$. So the Chern character recovers the Hilbert polynomial.

The Hilbert polynomial contains some information about your sheaf. For example its degree is the dimension of the support of $F$, and if $F$ is torsion-free (so that the degree of $P_L(F)$ is $dim(X)$), then, up to a constant term that depends only on $X$, the leading coefficient of $P_L(F)$ is the (generic) rank of $F$, and the next term gives you the degree of $F$.

If you didn't know already, you can find this stuff (and more) in the book "The geometry of moduli spaces of sheaves" by Huybrechts and Lehn.

I don't know about characterizing structure sheaves of subschemes on a K3 surface, but you have good chances of finding something in that same book (a large part of it is about sheaves on surfaces), or maybe someone else will comment here.

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  • $\begingroup$ Small typo: in the RHS of the equation above, it should read $e^{n\cdot c_1(L)}$. $\endgroup$ – François Mar 30 '16 at 19:39

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