Euler characteristic and signature of real semi-stable degenerations

Abstract

We give a motivic proof of the fact that for non-singular real tropical complete intersections, the Euler characteristic of the real part is equal to the signature of the complex part. This has originally been proved by Itenberg in the case of surfaces in $\mathbb CP^3$, and has been successively generalized by Bertrand, and by Bihan and ertrand. Our proof, different from the previous approaches, is an application of the motivic nearby fiber of semi-stable degenerations. In particular it extends the original result by Itenberg-Bertrand-Bihan to real analytic families admitting a $\mathbb Q$-non-singular tropical limit.