Exact formula for $\chi(X, \, S^n \Omega^1_X)$ I have a smooth, compact complex surface $X$, and I need an explicit formula for the Euler characteristic
$$\chi(X, \, S^n \Omega^1_X),$$
where $S^n$ denotes the symmetric product, in terms of $c_1(X), c_2(X)$.
I know how to do the computation, by using the splitting principle in order to calculate the Chern classes $c_i(S^n \Omega^1_X)$ and then the Hirzebruch-Riemann-Roch formula, and I did it for small values of $n$. However, extracting a formula for general $n$ seems quite tedious.

Question. Is there a reference where I can find the value of  $\chi(X, \, S^n \Omega^1_X)$? Related to this: is there a reference
for the Chern classes $c_1(S^n \Omega^1_X)$, $c_2(S^n \Omega^1_X)$?

Notice that I need an exact formula, not an asymptotic one (which I already know).
 A: Just for reference: using the Schubert2 package from Macaulay2 this can be quite effortlessly done.
i1 : needsPackage "Schubert2";

i2 : X = abstractVariety(2, QQ[n,c1,c2,Degrees=>{0,1,2}]);

i3 : T = abstractSheaf(X, Rank=>2, ChernClass=>1+c1+c2);

i4 : X.TangentBundle = T;

i5 : chi symmetricPower(n, cotangentBundle X)

              1 3  2   1 3      1    2   1 2      1  2   1        1
o5 = integral(-n c1  - -n c2 - --n*c1  - -n c2 + --c1  - -n*c2 + --c2)
              6        6       12        2       12      4       12

If one looks at the implementation, the $n$-th symmetric power of a bundle $\mathcal F$ is computed using the pushforward of the relative $\mathcal{O}(n)$ on the projectivization $\mathbf{P}(\mathcal F^\vee)$ (as already mentioned in the other answer).
A: Following the method you indicate in your question,  if we let $x,y$ be the chern roots  then determining $c_2(Sym^{n} \Omega)$ corresponds to expanding $$\sum_{0 \leq i\neq i' \leq n} (ix + (n-i)y)(i'x + (n-i')y)$$ in terms of the symmetric functions  $xy$  (corresponding to $c_2(X)$) and $(x+y)^2$  (corresponding to $c_1(X)^2$).  It seems easier to use the function $x^2 + y^2$ instead of $(x+y)^2$  corresponding to $c_1(X)^2 - 2c_2(X)$.
Then we have to determine the two sums  $$\sum_{0\leq i\neq i' \leq n} ii'$$ and $$\sum_{0 \leq i \neq i' \leq n} i(n-i')+ i'(n-i).$$
There must be  more elegant ways of doing this,  but the brute force one is to argue that these are polynomials in $n$ of degree $4$ and compute enough values to determine them.
A: Concerning the first part of the question, my initial recollection was that Miyaoka's  famous paper On the Chern numbers of surfaces of general type has the formulas you want. In fact, he doesn't quite have them, nevertheless his basic approach should give you the formula. Set $Y= \mathbb{P}(\Omega_X^1)$, with $L= \mathcal{O}_{Y}(1)$ the relative tautological bundle. Then the Leray spectral sequence, together with Hirzebruch-Riemann-Roch and lemma 5 of Miyaoka  yields
$$\chi(X,S^n\Omega_X^1)= \chi(Y, L^n)= \int ch(L^n)\cdot todd(Y)$$
$$=\frac{n^3}{6}(c_1(X)^2-c_2(X)) + \ldots$$
Sorry, I'm too lazy to compute the rest, but hopefully that should get you started.
