On the signed Euler characteristic property for subvarieties of abelian varieties

Abstract

We give an elementary proof of the fact that a pure-dimensional closed subvariety of a complex abelian variety has a signed intersection homology Euler characteristic. We also show that such subvarieties which, moreover, are local complete intersections, have a signed Euler-Poincar\'e characteristic. Both results are special cases of the fact, proved by Franecki-Kapranov, that the Euler characteristic of a perverse sheaves on a semi-abelian variety is non-negative. Our arguments use in an essential way the stratified Morse theory of Goresky-MacPherson.