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Let $X_0$ be a proper variety over a finite field $k.$ For each prime number $\ell\ne p,$ we have the $\ell$-adic intersection cohomology groups $IH^i(X).$ Due to Gabber, the alternating sum of these $$ \sum_{i=0}^{2\dim X_0}(-1)^i[IH^i(X)], $$ regarded as a virtual representation of the Weil group $W(\overline{k}/k)$ of $k$ (i.e. the cyclic group in Gal generated by Frobenius), is independent of $\ell.$ Of course it is also self-dual (up to Tate twist), and (together with Gabber's purity) each summand is independent of $\ell.$

My question is: does this alternating sum have any "geometric" interpretation? Or, to what extent (and in which way) this Euler characteristic depends on the singular locus of $X?$

E.g. when $X$ is in addition smooth, this can be interpreted in terms of the number of $k$-rational points. Being independent of $\ell,$ it's not surprising to expect it to have some "motivic" meaning, e.g. in terms of the geometry. Certainly one could also expect something for the individual $IH^i(X),$ but it seems that the Euler characteristic is always easier to relate to geometry.

Answers/comments in the general setting would be great, and those addressing to Shimura varieties and their Satake compactifications are especially appreciated.

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up vote 1 down vote accepted

It's a weighted sum over the $k$-rational points, where each point is weighted by the trace of Frobenius on its local intersection cohomology. This follows instantly from the Grothendieck trace formula. I'm not sure there's really a better description of it beyond that in general. Of course, what you can say about the local intersection cohomologies depends a lot on your setting.

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