Information and intuition packed in the Chern character for coherent sheaves

Question

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!

Answer 1

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.