On the Picard numbers of abelian varieties

2017/03/17 by Hulek, Klaus, Laface, Roberto · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1703.05882

Abstract

We study the possible Picard numbers of abelian varieties of given dimension g. If R_g denotes the set of realizable Picard numbers, then R_g is bounded by g². We show that, for g at least 3, the set R_g always has gaps and we analyze the nature of these gaps. We further prove that the Picard numbers are asymptotically complete in [1,g²] as g goes to infinity. Finally we show that every Picard number which can be realized over the complex numbers can already be realized by an abelian variety defined over a number field.

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• Essential dimensions of polarized endomorphisms of abelian varieties

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