Description
Title:
Pure extension of the theta divisor on a family of abelian varieties
Abstract
A symmetric line bundle L on an abelian variety has the useful property that pulling it back along the `multiplication by n’ map is the same as raising it to the n-th tensor power; we say L has pure weight 2. In particular this holds for a symmetric ample line bundle representing a polarisation. Such a line bundle extends naturally to an ample line bundle on a toroidal compactification of the universal abelian variety over the moduli space, but this extension no longer satisfies the purity property. We will show how to correct this using tropical theta functions and adelic- or b-divisors, and discuss consequences for extending the theta divisor. This is joint work with Ana Maria Botero, Jose Burgos, and Robin de Jong.