I would like to compute the $\chi_y$ genus of an elliptically fibered K3 surface.

For $X$ a compact algebraic manifold, Hirzebruch's $\chi_y$ genus is defined as $\chi_y (X) = \sum_{p,q} (-1)^{p+q} h^{p,q} (X) y^p$, with $h^{p,q} = \mathrm{dim} H^{p,q} (X)$. In particular for a K3 surface, by looking at the Hodge diamond, one finds $\chi_y (K3) = 2 + 20 y + 2 y^2$ (in other conventions there might be different signs). The limit $y \rightarrow +1$ reproduces the Euler characteristic $\chi (K3) = 24$. The $\chi_y$ genus also has an integral representation in terms of characteristic classes, by using Riemann-Roch.

However I am interested in doing the computation differently. Assume that the K3 is elliptically fibered. Also to be concrete, consider the generic case where the elliptic fibration $\pi : K3 \rightarrow \mathbb{P}^1$ has 24 singular fibers, all with nodal singularities. I guess I can assume that the 24 fibers are all of type $I_1$. Let me denote by $E$ the smooth fiber and by $N$ the singular ones. Now, if I want the Euler characteristic I can write (I am stratifying the surface by the topological type of the fiber) $$ \chi (K3) = \chi (\mathbb{P}^1 \setminus 24 pt) \ \chi (E) + \chi (24 pt) \ \chi (N) = 24 $$ since $\chi (pt) = 1$ and $\chi (E) = 0$. The computation separates the contribution of the regular fibers from that of the nodal fibers, located at 24 points on the base.

The same computation for the $\chi_y$ genus fails. The reason, as far as I understand, is the presence of the nodal singularity in the fibers, which induces a monodromy in the homology of the fiber as I move around the base. How can I take this into account? For example, should I blow up the nodal fibers?

So my question: how do I generalize/correct the above computation of $\chi (K3)$ in order to compute $\chi_y (K3)$?

To put things a bit more in context: this is a simplified version of an harder computation which involves $\chi_y$ genera of Hilbert schemes of points and curves on $K3$. That's why I don't want to use the Hodge numbers or Riemann-Roch directly; I would like to understand what happens in the strata which contain the nodal fibers, and this is the simplest example I could think of.

Thanks a lot!