On the signed Euler characteristic property for subvarieties of abelian varieties

Journal of Singularities
volume 17 (2018), 368-387

Received: 26 February 2018. Received in revised form: 27 September 2018.

DOI: 10.5427/jsing.2018.17p

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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-Poincaré 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.

Keywords:

abelian variety, stratified Morse theory, Morse function, stratification, intersection homology, signed Euler characteristic

2000 Mathematical Subject Classification:

Primary 58K05, 32S60; Secondary 14K12.

Author(s) information: