On the signed Euler characteristic property for subvarieties of abelian varieties

TL;DR

This paper provides an elementary proof that pure-dimensional subvarieties of complex abelian varieties have a signed intersection homology Euler characteristic, extending to local complete intersections with a signed Euler-Poincare characteristic.

Contribution

It introduces a new elementary approach using stratified Morse theory to establish signed Euler characteristic properties for subvarieties of abelian varieties.

Findings

Pure-dimensional subvarieties have a signed intersection homology Euler characteristic.

Local complete intersection subvarieties have a signed Euler-Poincare characteristic.

Method applies to subvarieties of compact complex tori with minor modifications.

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-Poincare characteristic. Our arguments rely on the construction of circle-valued Morse functions on such spaces, and use in an essential way the stratified Morse theory of Goresky-MacPherson. Our approach also applies (with only minor modifications) for proving similar statements in the analytic context, i.e., for subvarieties of compact complex tori. Alternative proofs of our results can be given by using the general theory of perverse sheaves.