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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!

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    $\begingroup$ I think this is easier to understand in terms of "half $K3$ surfaces", i.e., elliptic fibrations $Y\to \mathbb{P}^1$ obtained by blowing up the $9$ base points of a general pencil of plane cubics in $\mathbb{P}^2$. By straightforward excision, $\chi_y(Y)$ equals $(y^2+y+1)+9y = y^2 + 10y+1$. For a $K3$ surface $X$ that is a double cover of $Y$, the $\chi_y$-genus equals $\chi_y(X)=2\chi_y(Y) = 2y^2+20y+2$. More pertinently, if you find your $\chi_y$-analogue of Riemann-Hurwitz for $Y$ (which is easier since it is a rational surface), that also gives the analogue for $X$. $\endgroup$
    – Jason Starr
    Commented Oct 9, 2017 at 18:25
  • $\begingroup$ I'm not sure this will be relevant, but you might want to take a look at this talk videos.birs.ca/2016/16w5153/201605051631-Nicaise.mp4 and this paper arxiv.org/pdf/1603.08424.pdf . $\endgroup$
    – Mattia Talpo
    Commented Oct 10, 2017 at 18:14
  • $\begingroup$ @MattiaTalpo. The hypothesis of that paper is that the surface is toric. A $K3$ surface is not toric. The "half K3" surface is rational, but it is not toric. $\endgroup$
    – Jason Starr
    Commented Oct 11, 2017 at 9:31
  • $\begingroup$ @JasonStarr thanks for the suggestion, that's a very good point! $\endgroup$
    – michele
    Commented Oct 11, 2017 at 12:28
  • $\begingroup$ @MattiaTalpo thanks for pointing out that paper. As Jason Starr says they assume their surface to be toric; still I'll try to see if some of their techniques can be extended. $\endgroup$
    – michele
    Commented Oct 11, 2017 at 12:30

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